H IS TO RY
A
OF
A T H E M A T IC S
FL OR IA N
P
RO F
E SS
OR
O F H I ST
U NI VE
I
by
a
an y
m
s u re
a tt e
G L A IS H E
tha t
mp t
no
C AJO R I ,
ORY
R SITY
j
PH D
.
.
O F M ATHE MATIC S I N
O F C AL I
s u b ec t
F OR NIA
”
—J
l ose s more t ha n ma t hema t i c s
t o d i sso c ia t e i t f ro m i t s hi s t o ry
.
.
W
R
S E C O N D E D ITIC N , R E V I S E D AN D E N LAR G E D
Na n finrk
THE M AC M I L L AN C O M P ANY
L O ND O N M AC M I LL A N
C0 L m
:
A ll
.
r i g hts res erved
.
L
.
C O P Y RIG HT 1 8 93
M AC M IL LAN AN D
,
,
BY
S et
1 8 95 ;
u p a nd el ec t ro typ ed
Jan uary
Oc t o be r
mber
Febru ary
,
,
1 91 1
1 89 7 ;
N
ov e
1 8 94
,
1 90 1 ;
,
.
CO
.
Rep r i nt ed M arc h
Jan uary
,
1 90 6 ;
J ul y
.
S econd
Edi t ion
,
Rev i sed
and
Enl arg ed
C OP Y RI GHT 1 9 1 9
M AC M I LLAN C O M PANY
,
BY
Set
THE
u p a n d el ec t ro type d
.
P ubl ished
9 7 5 03
J u ly
,
1 9 19
,
,
1 909
P REFA C E TO TH E S E C ON D ED I TI ON
preparing this second edition the earlier portions of the book
have been partly r e written while the chapters on recent mathematics
are greatly enlarged and al most Wholly new The desirability of
having a reliable one volume history for the use of readers who cannot
devote the mselves to an intensive study of the history of mathe matics
is generally recogniz ed On the other hand it is a diffi cult task to
give an adequate bird s eye View of the develop m ent of mathem atics
from its earliest beginnings to the present time In compiling this
history the endeavor has been to use only the most reliable sources
Nevertheless in covering such a wide territory mi stakes are su re to
have crept in References to the sources used in the revision are
given as fully as the limitations of space woul d permit These ref
er en c es will assist the reader in following into greater detail the his
tory of any special subject Frequent use without acknowledgment
has been m ade of the following publications : A n nu ar i o B i ografi co del
Ci r col o M atemati co di P al ermo 1 91 4 ; J ahrbu ck nber di e Fortschr i tte def
M a themati k B erlin ; J C P oggen dorfi s B i ogr aphi scli L i ter ari sches
H an dworter buch Leip zig ; Geden k tagebn cli f u r M a themati k er v on Felix
M ii ll e r 3 A u fl Leip z ig und B erlin 1 9 1 2 ; Revu e S emes tri ell e des P u b
l i cati on s M a thémati qu es A ms terda m
The author is indebted to M iss Falk a M G ibson of Oakland Cal
for assistan c e in the reading of th e proofs
FL ORIA N CA J ORI
University of California
M arch 1 9 1 9
IN
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IN TRO DUCTI O N
THE B A B Y L O N IANS
TH E E GY P TIANS
THE GRE E K S
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.
.
Gr eek Geometry
The
The
The
The
The
The
Ionic School ”
School of P yth agor a s
Sop hi st School
P latonic School
First Alexandrian School
Second Alexandrian School
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.
.
.
Gr eek A r i thmeti c
THE R O MA NS
THE M AY A
and
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A l gebr a
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.
C HI NESE
THE JAP A NESE
TH E H IND U S
THE
.
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.
EUR OP E
D UR I N G
TH E
M I DD L E
A G ES
I n tr odu cti on of Roman Ma thema ti cs
Tr an sl ati on of A r a bi c M an u scr i pts
The Fi rs t A wa k en i n g a n d i ts S equ el
.
.
EURO P E
DUR IN G TH E SIX TEEN TH
EE N TH CEN TU RI ES
.
S EV EN T EEN T H A ND EI GH T
“
,
The Ren a i ssan ce
Vi eta and Desca r tes
Descartes to N ew ton
N ewton to E ul er
Eul er , L agr ange an dE
apl ac e ”
.
.
TH E NIN E T EEN T H
I n tr odu cti on
Definition O
f
.
AN D
T WEN TI E TH CEN TURI ES
a tics
Ma them
C ONT EN T S
S yn theti c Geometry
Elementary G eometry of the Triangle and Circle
Link motion
P arallel L ines Non Euclidean G e ometry and G eometry of
n D i mension s
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A n al yti c Geometry
.
Analysis Situs
Intrinsic Co ordinates
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D efi n i t ion of a C u rv e
Funda mental P ostul ates
.
Geom etric M odels
.
.
.
Al gebr a
TheoryOf Equations and Theory O f Groups
Solution of Nu merical
M agic Squares and Co mbinatory A n
aly
sis
A n al ysi s
Calcul u s f
.
O V a riations
.
Convergence of Series
P robabil ity and S tatistics
Diff erential Equations D ifference Equations ”
Integral Equations Integro differential Equatio n
s Ge
n
eral
Analysis Functional
Theories of Irrationals and Theory Of Aggre
ga tes
M athe matical Logic
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.
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,
Theory
f
o
Fu n cti on s
.
Elliptic Functions
General Theory O f Functio
n
s
Unifor m ization
.
.
.
Theory
f N u“ mber s
o
.
Fer mat s Last Theorem Waring s Theore m
O ther Recent Researches Nu mber Fields
Transcendental Nu mbers The Infinite
’
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.
.
A ppli ed M a thema ti cs
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.
Celestial M echanics
P roble m O f Three B odies
General M echanics
Fluid M otion
Sound Elasticity
Light Electricity Heat P otential
Relativity
Nomog raphy
ables
M athe matical T
imet ers I n teg raph s
Calculating M achines Pl a n
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Al phabeti cal I n dex
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A
H I S TO R Y
O F M A T H E M A TI C S
I NT ROD U CTI ON
contemplation of the various steps by which mankind has
come into possession of the vast stock of m athe matical knowledge
can hardly fail to interest the mathe matician He takes pride in the
fact that his S cience more than any other is an exact science and
that hardl y anything ev er done in m athematics has proved to be use
less The che mi st smiles a t the chi ldish e fforts of alche m ists but the
mathe matician finds the geo metry O f the Greeks and the arithmetic
of the Hindus as useful and adm irable as any research of to day He
is pleased to notice that though in course of its develop ment mathe
ma ti cs has had periods of S low growth yet in the main i t has been
pre e minently a pr ogr ess i ve science
The history of m athematics may be instructiv e as well as agreeable ;
it may not only remind u S of what we have but m ay also teach us
“
how to increase our store Says A D e M organ The early history
O f t h e m ind of m en with regard to mathe m atics leads us to point out
our own errors ; and in this respect it is well to pay attention to the
hi stor y of mathe m atics
It warns us against hasty conclusions ; it
points out the importance of a good notation upon the progress O f the
science ; it discourages excessive S pecialisation on the part of investi
gators by S howing how apparently distinct branches have been foun d
to possess unexpected connecting links ; it saves the student f rom
wasting time and energy upon proble m s which were perhaps solve d
long S ince ; it discourages him fro m attacking an unsolved problem by
the sam e method which has led other m athem aticians to fail ure ; it
teaches that fortifications can be taken in other ways than by di rect
a ttack that when repulsed fro m a direct assault it is well to recon
n oi t re and occupy the surrounding ground and to discover th e secret
1
paths by which the apparently unconquerable position can be taken
The importance of this strategic rule m ay be e mphasised by citing a
case in which it has been violated An untold a moun t of intellectual
energy has been expended on the quadrature of the circle yet no con
quest has been made by di rect assault The circle squarers have
existed in crowds ever S ince the period of Archi m edes After in
numerable failures to solve the problem at a time ev en when in
TH E
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S G un th er ,
Erl an g en , 1 8 7 6
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Zi el e und Resu l tate der
n eu er en
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M athemati sch hi stori schen For schu n g
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A
H I S T ORY O F M AT HE M ATIC S
possessed that m ost powerful tool the differential calculus
persons versed in mathe matics dropped the subj ect whi le those who
stil l persisted were completely ignorant of its history and generally
“
m isunderstood the conditions of the problem
Our problem says
A D e M organ is to square the circle with the ol d al l owan ce of means :
Euclid s postul ates and nothi ng m ore We cannot remember an
instance in which a question to be solved by a defini te method was
tried by the best heads and answered a t last by that method after
thousa nds O f co mplete failures
B ut progress w as made on this
proble m by approaching it fro m a different direc tion and by newly
di scovered paths J H Lambert proved in 1 7 6 1 that the ratio of the
circum ference O f a circle to its diameter is irrational Som e years
ago F Linde mann de monstrated that thi s ra t io is also transcendental
and that the quadrature O f the circle by means O f the ruler and c o m
passes only is i mposs i bl e He t h us showed by actual proof that which
keen m inded mathem aticians had long suspected ; namely that the
gre a t a rmy of circle squ a rers have for two thousand years been
assaulting a fort i fication whi ch is as indestructible as the fir mamen t
O f heav en
Another reason for the desi rability of historical study is the valu e
of historical knowledge to th e teacher of mathematics The interest
which pupils take in their studies may be greatly increased if the
solution O f problem s and the cold logic of geom etrical demonstrations
are interspersed with historical remarks and anecdotes A class in
arithm etic will be pleased to hear about the B abylonians and Hind us
“
and their invention of the Arabic notation ; they will m arvel at
the thousands of years which elapsed before people had even though t
—the
O f introducing into the num eral notation that Columbus egg
z ero ; they will find it astounding that it S hould hav e taken so long
to i nven t a notation which they th emselves can now l earn in a month
After the pupils h ave learned how to bisect a given angle surprise
them by telling of the many futile attempts whi ch have been made
to solve by elementar y geometry the apparently very S imple proble m
of the trisection O f an angle When they know how to construct a
square whose area is double the area O f a given square tell them about
the duplication O f the cube of its m ythical origin—how the wrath of
Apollo could be appeased only by the construction O f a cubical altar
double the given altar and how m athematicians long wrestled w ith
this problem After the class have exhausted their energies on the
theore m of the right triangle tell them the legend about its di sc ov
erer—how P ythagoras j ubilant over his great accomplishm ent
sacrifi ced a hecato mb to the M uses who inspired him When the
value of m athe matical training is called in question quote the i n
scription over the entrance into the acad em y of P lato the philosopher :
Let no one who is unacquainted with geometry enter here S tudents
in analytical geometry should know so methi ng of Descartes and
v es tig a tors
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I N TROD U CTI ON
after taking up the differential and in tegral calculus they should
become familiar with the parts that Newton Leibniz and Lagrange
played in creating that science In hi s historical talk it is possible
for the teacher to make i t plain to the student that m athematics is
1
not a dead science but a living one in which steady progr ess is made
2
A S imilar point of V ie w is taken by Henry S Wh ite : The ac
c ept ed truths of to day even the co mmonplace truths of any science
were the doubtful or the novel theories of yesterday Some i n deed
of prime importance were long esteem ed of sligh t importance and
al most forgotten The first eff ect of readi ng in the hi story of science
is a nai ve astonishment at the darkness of past centuries b ut the
ul ti m ate e ffect is a fervent ad miration for the progress achieved b y
for mer generations for the triumphs of persistence and of genius
The easy credulity with whi ch a young student supposes that of
course every algebraic equation m ust have a root gives place final ly
to a delight in the slow conquest O f the realm of imaginary numbers
and in the youthful genius of a Gauss who could demonstrate t hi s
once obscure fundamental proposition
The history of mathematics is i mportant also as a valuable con
Human progress is closely
t ri bu tio n to the history of civilisation
identified with scientific thought M athematical and physical re
search es are a reliable record of intellectual progress The history
of mathematics is one of the large windows through which the p hilo
sophic eye looks into past ages and traces the line of intellectual de
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v el opmen t
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C aj o r i , F
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The Tea
c hi n g
i ng ton , 1 8 90 , p 23 6
2
B ull A m M ath S oc
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and
Hi story of M athemati cs i n the Uni ted S tates
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1 90 9 ,
p 3 25
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Wash
THE
B A B YL ON IA NS
fertile valley of the Euphrates and Tigris was one of the
prim eval seats of hu m an society Authentic history of the peoples
inhabiting this region begins only with the foundation in Chal daea
and Babylonia of a united kingdo m out of the pre v iously disunited
tribes M uch light has been thrown on their history by the discov e ry
of the art of reading the cu n eif orm or wedge shaped syste m of writing
In the study of Babylonian m athe matics we begin with the notation
of n umbers A vertical w edg e vstood for 1 while the characters
and Y ) signified 1 0 and 1 00 respectively G ro t efen d believes the
C haracter for 1 0 originally to have been the picture of two hands as
held in prayer the pal m s being pressed together the fingers close to
each other but the thu mbs thrust out In the Babylonian notation
—
two principles were e m ployed the ad ditiv e and m ultiplicative N u m
bers below 1 0 0 were expressed by symbols whose respective values
had to be added Thus VVs tood for 2
Y for 3 w for
for 2 3
for 30 Here the symbols of higher order appear
always to the left O f those of lower order In writing the hun
dreds on the other hand a s mal l er sy mbol was placed to the left of
Thus
1 0 0 and was in that case to be mu l ti pl i ed by 1 00
I
signified 1 0 times 1 0 0 or 1 0 00 B ut this symbol for 1 000 was itself
taken for a new unit which could take s maller coe fficients to its left
Thus
denoted
not
2 0 tim es 1 00 but 1 0 ti m es 1 000
So
e
m
y
of the cuneiform nu mbers found on tablets in the ancient te mple
library at Nippur exceed a m illion ; m oreov er so m e of these Nippur
tablets exhi bit the s u btr acti ve principle ( 2 0
similar to that shown
“
in the Ro man notation XIX
If as is believ ed by m ost S pecialists the early S u m er1ans were the
in v entors O f the Cuneiform writing then they were in all probability
also the inv entors O f the notation of nu mbers M ost surprising in
this connection is the fact that Sumerian inscriptions disclose the u se
not only of the above deci mal syste m but also O f a s exages i mal one
The latter was used chiefly in constructing tables for weights a n d
measures It is full of historical interest Its consequential develop
m ent both for integers and fractions reveals a high d egree O f m athe
We possess two Babylonian tablets which exhibit
mati cal insight
i ts use O n e of t he m probably w ritten between 2 300 and 1 6 0 0 B C
2
contains a table O f square nu mbers up to 6 0 The nu mbers I 4 9
2 5 3 6 4 9 are given as the squares O f the first seven integ e r s r e
16
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B A B YL ON IA NS
THE
5
2
2
2
10
11
8
etc
We have ne xt
9
This remains unintelligible unless we assu me the sexagesimal scale
2 6O + I
which makes
The second
60 + 2 1
tablet rec ord s the magnitude of the illum inated portion of the moon s
disc for e v ery day fro m new to full moon the whole disc being assumed
to consist of 2 40 par ts The illum inated parts during the first fi v e
which is a geo m etrical
days are the series 5 1 0 2 0 40
progression From here on the series beco m es an arithm etical prog r es
sion the numbers fro m the fifth to the fifteenth day being respectiv ely
This table
4
not only exhi bi fs the use of the sexagesi mal syste m but also indicates
the acquaintance O f the B abylonians with progressions Not to be
overlooked is the fact that in the sexagesi mal notation of integers
“
the principle O f position was e mployed Thus in
the
the unit of the second or der by virtue of
I is m ade to stan d for 6 0
its position with respect to the 4 The intro d uction O f this principle
at S O early a date is the m ore re markable because in the decimal n O
t a ti on it was not regularly introduced till about the ninth century
after Christ The principle of position in its general and systematic
application requires a symbol for z ero We ask D id the Babylonians
possess one ? Had they already taken the gigantic step of representing
by a symbol the a bs en ce of units ? Neither O f the abo v e tables answers
this question for they happen to contain no nu mber in which there
was occasion to use a z ero Babylonian recor d s of many centuries
later—O i about 2 00 B C —give a symbol for z ero which denoted the
absence of a figure but apparently was not used in calculation It
consisted O f two angular m arks 5 one abov e the other roughly re
sembling two dots hastily written About I 30 A D Ptolemy in
Alexan dria use d in his A l magest the B abylonian sexagesi m al fractions
and also the o micron to represent blanks in the sexagesim al numbers
This was not used as a regular z ero It app ears therefore that the
Babyloni ans had the pr i nciple of local v alue and also a symbol for
z ero
to indicate the absence of a figure but did not use this z ero in
comp utation Their sexagesimal fractions were intro duced into India
and with th es e fractions probably passed the principle O f local value
and the restric t ed use of the z ero
The sexagesimal syste m w as used also in fractions Thus in the
Babylonian inscriptions 3, an d are d esignated by 30 and 2 0 the
“
r eader being expected in his mind to supply the word sixtieths
The astrono mer Hipparchus the geo m eter H yp si c l es and the as
t r on ome r P tole my borrowed the sexagesim al notation of fractions
from the Babylonians and introd uced it into Gree c e From that tim e
sexagesimal fractions hel d al m ost full sway in astronomical and mathe
mati cal calculations until the S i xteenth century when they finally
yielded their place to the dec imal fractions It may be asked What
l e d to the invention O f the se xagesi mal syste m ? Why was it that 6 0
2
sp e c t i v e l y
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A
6
OF M ATHE M ATIC S
H I S TORY
parts
were selected ? To this we have no positive answer Ten was
chosen in the decimal system because it r epresents the number O f
fingers B ut nothi ng of the hu man body coul d have suggested 6 0
Did the system have an astronomical origin ? It was supposed that
the early Babyloni ans reckoned the year at 3 6 0 days t hat this led
to the di v ision of the circle into 3 6 0 degrees each degree representing
the daily amount O f the supposed yearly revolution O f the sun around
the ear th Now they were very probably familiar with the fact
that the radius can be applied to its circu mference as a C hord 6 times
and that ea ch of these chords subtends an ar e measuring exactly 6 0
degrees Fix ing their attention upon these degrees the division into
6 0 par t s m ay have suggested itself to them
Thus when greater pre
c i si on necessitated a subdivision of the degree i t was partitioned into
6 0 minutes
In t h is way the sexagesim al notation was at one tim e
supposed to have originated B ut it now appears that the Babylonians
very early knew that the year exceeded 3 6 0 days M oreover i t is
highly im probable that a higher unit of 3 6 0 was chosen first and a
lower unit of 60 afterward The norm al develop ment of a number
system is from lower to higher units Another guess is that the
sexagesim al system arose as a m ixture of two earlier system s of the
bases 6 and
Certain it is that the sexagesimal system becam e
closely inte rwoven with astronomical and geometrical science The
di vision of the day into 2 4 hours and O f the hour into mi nutes and
seconds on the scale of 6 0 is attributed to the Babylonians There is
strong evidence for the belief that they had also a division of the day
into 6 0 hours The e mployment O f a sexagesi mal division in nu meral
notation in fractions in angular as well as in tim e m easurement i n
di c a t e d a beautiful har m ony which was not disturbed for thousands
O f years until Hindu and Arabic astrono m ers began to use S ines and
cosines in place O f parts of chords thereby forcing the right angle to
the front as a new angular unit which for consistency should have
been sub divided sexagesimally but was not actually so divided
It appears that the people in the Tigro Euphrates basin had m ade
very cre di table a dvance in arithm etic Their knowledge of arith
m etical and geo m etrical progressions has already been alluded to
I amb l i chu s attributes to the m also a knowledge O f proportion and
even the invention of the so called mu si cal proportion Though we
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M C an tor Vorl esung en uber Geschi chte der M athematik ,
z
B d , 3 A u fl , L e ip i g ,
Thi s w o rk h a s app ear ed i n fou r l arg e v o l um es an d c arri e s t he hi s t ory
1 90 7 , p 3 7
d ow n t o 1 7 99 Th e fou r t h v o l u me ( 1 90 8 w as w r i tt en w i t h t h e co op e ra i o n o f n i n e
M ori t Cantor ( 1 8 2 9
sc hol a r s f r om G e rma n y , I t a l y , R u ssi a an d t he U n i t ed S a t es
ran k s as t h e fore mos t g e n eral w r i t e r o f t h e n i n e t een t h ce n t u ry o n t he hi s tory
o f ma t h e ma t i cs
B or n i n M a nn h e im , a s tu de n t a t H e idelb e rg , a t G ott i ng e n u n der
G au ss a n d Web e r , a t B e rl i n u n der D i r i c hl e t , h e l e c t u r ed a t H e id elb erg w he re i n
H i s fi r s t hi s tori cal ar t i c l e w a s
1 8 7 7 h e b e came or di n a ry h o n o r a ry p r o fesso r
b roug h t o u t i n 1 8 5 6 , b u t n o t u n il 1 88 0 d id t h e fi rs t v ol u me o f hi s w ell k n ow n hi s tory
1
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t
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app ear
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t
B A B YL ON IA NS
THE
possess no conclusive proof we have nevertheless reason to believe
th at in practical calculation they used the abacu s A mong the races
of middle Asia even as far as China the abacus is as old as fable
Now Babylon was once a great commercial centre —the m etropolis of
m any nations —an d it is therefore not unreasonable to suppose that
her m erchant s e mployed this most improved aid to calculation
In 1 8 89 H V Hil precht began to make excavations at N uffar ( the
ancient Nippur)
and found brick tablets containing m ultiplicatio n and
division tables tables of squares and square roots a geo metric prog res
sion and a few computations He p ublished an account of hi s fin dings
l
in 1 90 6
“
4
The divisions in one tablet contain results like these : 6 0 divided
“
4
=
by 2
each
6 0 divided by 3 = 4 3 2 0 OOO each
and
so on using the divisors 2 3 4 5 6 8 9 1 0 1 2 1 5 1 6 1 8 The very
“
4
first division on the tablet is interpreted to be 6 0 divided by I
This strange appearance O f % as a divisor l s diffi cult to
3
explain P erhaps there 1S here a use of 2 corresponding to the Egyp tian
use of 3 as found in the Ahmes papyrus a t a perhaps conte mporaneous
4
period It is noteworthy that 6 0
which H i l pr ech t found
“
in the Nippu r brick text books is nothing less than the mystic P latonic
“
number the lord of better and worse births m entioned in P lato s
Repu bli c
M ost probably P lato received the number fro m the
2
P ythagoreans and the P ythagoreans fro m the Babylonians
In geo m etry the B abylonians accompli shed little B esides the divi
sion O f the circu mference into 6 parts by i ts radius and into 3 60 de
grees they had som e knowledge of geometrical figures such as the
triangle and quadrangle which they used in their auguries Like the
Hebrews ( 1 Kin
they took 7r = 3 Of geo metrical de monstra
“
tions there is of co u rse no trace
A S a rule in the Oriental mind
the intuitive powers eclipse the severely rational and logical
H il pre ch t con cluded from his studies that the Babylonians pos
sessed the rules for fin di ng the areas of squares rectangles right tri
angles and t r ape zoids
The astronomy of the B abylonians has attracted much attention
They worshipped the heavenly bodies from the earliest hi storic times
When Alexander the Great after the battle O f Arbela (33 1 B C took
possession of Babylon C al li sthen es found there on burned brick as
P orphyr ius says
t r on omi c al records reaching back as far as 2 2 34 B C
that these were sent to Aristotle P tole my the Alexandrian a s tr on o
mer possessed a Babylonian record of eclipses going back to 74 7 B C
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7:
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,
M athemati cal , M etrol ogi cal
of N i ppu r , b y H V H i l p r ec h t
Chron ol og i cal Ta bl ets f r om the Templ e L i brary
a r t I , S eri e s A , C u n e ifo rm Te xt s , p u b
V ol
, p
l i shed by t h e B ab yl o n i a n Exp edi t io n o f t h e U n i v e r si ty o f P e n n syl va n i a , 1 90 6 C o n
s u l t al so D E S mi t h i n B u l l A m M a th S oc , V O l 1 3 , 1 90 7 , p 3 9 2
“
2
O n t h e P l a t o n i c n u mb er c on su l t P T a n n e ry i n R evu e phi l osophi q u e, V ol I ,
V , 1 88 3 , p 5 7 3 A l so G L ori a i n L e
1 8 7 6 , p 1 7 0 ; V ol X I I I , 1 88 1 , p 2 1 0 ; V o l
s c i en e es atte n el l a n ti ca grec i a , 2 E d , M il a n o , 1 9 1 4 A pp en di ce
a nd
1
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XX
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A
8
H I S T ORY
.
OF M AT H E M ATIC S
Epping and Strassmaier have thrown considerable light on Babylon
ian chronolog y and astrono my by explaining two calendars of the
years 1 2 3 B C and 1 1 1 B C taken from cuneifor m tablets coming
presumably from an old observatory These scholars hav e succeeded
in givi ng an account of the B abyloni an calc ul ation O f the new and
full moon and have identi fied by calculations the B abyloni an names
of the planets and of the twelve z odiacal signs and tw enty eight
normal stars whi ch correspond to som e ex tent with the twenty eight
n akshatr as of the Hindus
We append part of an Assyrian astronomical
report as translat ed by Opper t :
1
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To the Ki ng my lord thy fai thful servant M ar Istar
On the first day as the new moon 5 day of the month Thammu z
d ecli ned the moon was a g ai n vi s i b le ov er the planet M ercury as I had
already predi cted t o my master the King I erred not
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1
A stronomi sches
F rei burg , 1 889
Epp i ng ,
mai er
.
.
aus
B abyl on
.
.
Unter M i twi rk u ng
von
P
.
J R S t rass
.
.
H I S T ORY
A
10
M AT HE M ATIC S
OF
papyrus—the most ancient m athematical handbook known to u s
puts us at once in contact with the m athe matical thought in Egyp t of
“
three or five thousand years ago It is entitled D irections for ob
taining the K nowledge O f all Dark Things
We see from it that the
Egyptians cared but little for theoretical results Theorems are not
found in it at all It contains hardly any general rules of procedure
but chi efly mere state ments of results intended possibly to be ex
1
plained by a teacher to his pupils
In geo metry the forte of the
Egyptians lay in m aking constructions and deter mining areas The
area of an isosceles triangle of which the sides me asu re 1 0 k hets (a
unit of length equal to
m by one guess and about thrice that
2
amount by another guess )
and the base 4 khets was erroneously given
as 2 0 square k hets or half the product O f the base by one side The
area O f an isosceles trape zoid is found similarly by multiplying half
the sum of the parallel sides by one of the non p arallel S ides The
1
area of a circle is found by deducting fro m the dia m eter ,5 of its length
1 6
=
and squaring the re mainder Here 71 is taken ( g
a
The papyrus explains also such problems
v ery fair approxi m ation
as these — TO mark out i n the field a right triangle whose sides are
1 0 a n d 4 units ; or a trape z oid whose parallel sid es are 6 and 4 and
the non —
parallel sides each 2 0 units
So m e proble m s in this papyrus see m to imply a rudimentary knowl
edge of proportion
The base —
lines O f t h e pyramids run north and south and east and
west but probably only the lines running north an d south were deter
This coupled with the fact
mined by astrono mical observations
that the word har pedon aptce applied to Egyptian geom eters means
“
rope stretchers would point to the conclusion that the Egyptian
like the Indian and C hinese geom eters constructed a right triangle
upon a given line by stretching around three pegs a rope consisting
3
of three parts in the ratios 3 :4 5 and thus forming a right triangle
If thi s e xplanation is correct then the Egyptians were fa miliar 2
years B C with the well known property of the right triangle for
the special case at least when the sides are in the ratio 3 :4 :5
On the walls of the celebrated te mple of Horus at Edfu have been
found hi eroglyp hics written about 1 00 B C which enumerate the
pieces of land owned by the priesthood and giv e their areas The
area of any quadrilateral however irregular is there found by the
form ula fl g
Thus for a quadrangle whose O pposite sides
4
are 5 and 8 2 0 and 1 5 is given the area
$ The incorrect for
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1
James Go w A S hort Hi story of Gr eek M athemati cs
C amb ri dg e ,
p 16
2
A Ei se n l o h r , E i n ma themati sches H a n dbu ch der a l ten A egypter , 2 A u sg ab e , L ei p
z ig , 1 8 9 7 , p 1 0 3 ; F L
G r i ffi t h i n P r oceedi n gs of the S oci ety of B i bli cal A rchaeol ogy,
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M C an to r , op
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4
H H a nk e l , Z u r
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1 8 74 ,
1 88 4 ,
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3
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p 86
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V ol I , 3 A u fl , 1 90 7 , p 1 0 5
Geschi chte der M athema ti k i n A l ter thu m u nd M i ttel al ter , L e ip ig ,
ci t
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z
THE
E G Y P TIA NS
mu l ae
1
of Ahme s of 300 0 years B C yield generally closer approxima
tions than those of the Edfu inscriptions written 2 0 0 years after
Euclid !
The fact that the geom etry of the Egyptians consists chiefly of
constructions goes far to explain certain of its great defects The
Egyptians failed in two essential points w i thout which a s ci en ce of
geometry in the true sense of the word cannot exist In the first
place they failed to construct a rigorously logical system of geometry
resting upon a few axio m s and postulates A great many of their
rules especially those i n solid geom etry had probably not been proved
at all but were known to be true merely from ob serya tion or a s mat
ters of fact The second great defect was their inability to bring the
num erous S pecial cases under a more general view and thereby to
ar rive at broader and more fun damental theorems Som e of the
simplest geom etrical truths were divided into numberless special cases
of which each was supposed to require separate treat ment
So me particulars about Egyptian geometry can be mentioned more
advantageously in connection with the early Greek mathe maticians
who ca me to the Egyptian priests for instruction
An insight into Egyptian methods of nu meration was obtained
through the ingenious deciphering of the hierogl yphics by Champol
lion Young and their successors The symbols used were the fol
lowing : h for 1
for 1 0 G for 1 00 ii for 1 00 0 E for
k
for
for
The
sy
bol
for
for
m
Q
if
a pointing finger ; that
1 represents a vertical staff ; that for
for
a burbot ; that for
a man in astonishm ent The
significance of the re maining symbols is very doubtful The writing
of numbers with these hieroglyphics was very cumbrous The unit
symbol of each order was repeated as m any tim es as there were units
in that order The principle employed was the addi ti ve Thus 2 3
was written
[j uH
B esides the hierogl yphics Egypt possesses the hi erati c and demoti c
writings but for want of space we pass the m by
Herodotus makes an i mportant state ment concerning the mode of
“
computing among the Egyp tians He says that they cal cu l a te wi th
pebbl es by m oving the hand fro m right to left while the Hellenes
Herein we recognise again that i nstr u
m ove it fro m left to right
men tal method o i figuring so ex tensively used by peoples of antiquity
The Egyptians used the decimal scale Since in figuring they moved
their hands hori zontally it see ms probable that they used ciphering
boards with vertical columns In each colum n there must have been
not more than nine pebbles for ten pebbles would be equal to one
pebble in the col umn next to the left
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M C a n to r
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p
o
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cit
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V ol I , 3 A u fl
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1 90
7, p 8 2
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.
OF M ATHE M ATI C S
HI S T ORY
A
12
The A hmes papyr u s con tain s interesting information on the way
in whi ch the Egyp tians e mployed fractions Their methods of opera
tion were of course radi cally diff erent from ours Fractions were a
subj ect of very great diffi culty with the ancients Simultaneous
changes in both num erator and denominator were usually avoided
In m anipul ating fractions the B abylonians kept the denominators ( 6 0 )
constant The Rom ans l i kewise kept them constant but equal to 1 2
The Egyp tians and Greeks on the other hand kept the numerators
constant and dealt with variable denom inators A hmes used the
“
term fraction in a restricted sense for he applied it only to u n i t
f r acti on s or fractions having unity for the num erator It was desig
n a t ed by writing the deno minator and then placing over it a dot
Fractional values which could not be expressed by any one unit
fracti on were exp ressed as the su m of two or more of them Thus he
1
2
_
wrote 4 4 5 in place of
Whi le Ahmes knows to be equal to 4 4 he
2
cur iously allows 43 to appear often among the unit fractions and adop ts
a special symbol for it The first important problem naturally arisi ng
was how to represent any fractional value as the sum of unit fractions
Thi s was solved by aid of a table given in the papyrus in which all
f
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fractions of the form
2
2n
+1
,
(where
designates successively all the
n
are reduced to the sum of unit fractions Thus
4 9)
2
1
1
—
h
W
en
by
who
and
how
t
h
i s table was cal
m
4 4 4 44
4 4
44
P robably i t was co mpiled e mpirically at
c ul at ed we do not know
di ff erent tim es by di fferent persons It will be seen that by repeated
application of this table a fraction whose nu merator ex ceeds two can
be expressed in the desired for m provided that there is a fraction in
the table havi ng the same deno mi nator that i t has Take for ex
a mple th e proble m to di vide 5 by 2 1 In the fir st place 5 1 + 2 + 2
2 = 1—1
5
_
Fro m the table we get 4 4 4 4 Then 2 1 2
4 4
1
1
1
2
J
i
=
=
The
papyrus
contains
prob
3
4
4
f
4 4
, 4 4
T
2 1
I
lem s i n which it is required that fractions be raised by addition or multi
plication to given whole numbers or to other fractions For example
1
1
1
1 1
—
en omi n a t or
to
The
co
mm
on
d
1
i t 18 requi red to mcr ea se
4
4 4 0 4 0
l
taken appears to be 4 5 for the numbers are stated as 1 14 5 2 4 44
1
Add
to
this
s um of these IS 2 3 1 1 i forty fifths
and
The
1
1
4 44
4
Hence the q uan tity to be added
th e sum i s g Add ?4 and we have 1
1
1
to the gi ven fracti on i s 4 4 4 4
Ah mes gives the following e xample involving an ari thmeti cal
“
of
what
the
first
persons
r o r es si on :
D
m
ivide
loaves
a
ong
1
00
;
5
p g
4
three get is what the last two get Wh at is the difference ? Ahm es
“
gi v es the solution : M ake the diff erence 54; 2 3 1 74 1 2 6 4 1
How
did
m
es co me upon
M ultiply by 1 4 ; 3 84 2 94
12
A
h
2
numbers up to
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4
2
:
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4
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1
4
4
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4
4
,
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0
,
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,
0
0
,
1
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1
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TH E E G Y P TIA NS
I3
Let a and —
d be the fir st ter m and th e di ffer
ence in the required arithm etical progression then 4[a + (a—d H
—
—
—
2
a
d
d
whence
the dif
a
d
a
e
4 )
4d)i
(
)
] (
5% (
fer ence d is 5% times the last term A ssu mi ng the last term 1 he
gets his first progression The sum is 6 0 but should be 1 00 ; hence
2
2
_
1 00
m
multiply b y 1 s for
We
have
here
a
ethod
of
solution
a
which appear s ag ain l at er among the Hindus Arabs and modern
Europeans— the famous m ethod of f al se posi ti on
Ahmes speaks of a ladder consisting of the nu mbers 7 49 343
2 40 1
1 680 7
Adjacent to t hese powers of 7 are the words pi ctu r e
cat mou s e bar l ey measu re
What is the meaning of these mysterious
data ? Upon the consideration of the problem given by Leonardo of
“
P isa in hi s Li ber abaci 3000 years later : 7 old wo men go to Ro m e ;
M oritz
each woman has 7 m ules each m ule carries 7 sacks etc
Cantor o ffers the follo w ing solution to the Ahm es riddle : 7 persons
have each 7 cats each cat eats 7 mice each mouse eats 7 ears of
barley from each ear 7 measures of co rn may grow How many
persons cats m ice ears of barley and measures of corn altogether ?
Thus
Ahm es gives 1 96 0 7 as the su m of the geometric progression
the Ahmes papyrus discloses a knowledge of both arithmetical and
geom etrical progression
Ahm e s proceeds to the solution of equations of one unknown quan
tity The unknown quantity is called hau or heap Thus the
P erhap s t hus :
54?
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problem
heap its 4 its whole it
,
,
,
,
makes
%
f = 1 9;
i
1 9,
x
.
e
.
=
x
19
+
5
.
In
B
ut
44
f
in other problems the solutions are e ffected by various other m ethods
It thus appears that the beginnings of algebra are as ancient as those
of geometry
That the period of Ahmes was a flowering time for Egyptian m athe
matie s appears from the fact that there exist other papyri (more re
c en tl y di scovered )
of the same period They were found at Kahun
south of the pyr amid of Il l ahun These docu m ents bear close t e
semblance to Ahm es They contain moreover e xamples of quadratic
2
equations the earliest of whi ch we have a record One of them is :
A given surface of say 1 00 units of area shall be represented as
the sum of two squares whose S ides are to each other as 1 :4 In
2
2
modern symbols the problem is to find as and y such that x + y
1 00 and x :y = 1
m
The
solution
rests
upon
the
ethod
of
false
4
2
2
=
=
=
position Try x 1 and y 4 then x + y 44 and
B
ut
§g 4
=
The rest of the solution canno t be made
8
V 1 9 9 1 0 and 1 0
M C antor p ci t Vol I 3 Au fl 9 7 p 7 8
C an tor op ci t V l I 90 7 pp 95 96
t hi s
case the solution
,
as follows
15
;
3
x= 16
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4
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o
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2
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o
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0
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M ATHE M ATI CS
A HISTORY OF
14
out but probably was x = 8 X 1 y = 8 X 4 6 This solution leads to
the relation
The symbol r was used to designate
square root
In so m e ways similar to the A hmes papyrus is also the Akh mim
1
papyrus writ t en ov er 2 00 0 years later at Akhm i m a city on the
Nile in Upper Egypt It is in Greek and is supposed to hav e been
written at som e time between 50 0 and 8 0 0 A D I t contains besides
“
arith metical exa mples a table for finding unit fractions like that
of Ahm e s Unlike Ahmes i t tells how the table was constructe d The
:
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rule expressed in
,
modern
symbols is as follows
,
9
A hm e s
For z = 2 this for mula reproduces part of the table in
The principal defect of Egyptian arithmetic was the lack of a
simple co mprehensive symbolism—a defect which not even the Greeks
were able to remove
The Ahmes papyrus and the other pap yri of the same period r epr e
sent the m ost advanced attain m ents of the Egyp tians in arith m etic
and geo m etry It is re m arkable that they should have reached so
great proficiency in m athe m atics at so re mote a period of a ntiquity
B ut strange indeed is the fact that during the ne xt two thousand
years they should hav e made no progress whatsoe v er in it The con
that they rese mble the Chinese in the
c l u si on forces itself upon us
not only of their govern ment but also of their
s ta ti on ar y char acter
learning All the knowledge of geometry whi ch they possessed when
Greek scholars v i sited the m s ix centuries B C was doubtless known
to the m two thousand years earlier when they built those stupendous
—
and gigantic structures the pyra mids
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L e p a py ru s m a t h éma t iq u e d Ak hmi m, M émo i r es pu bl i es par l es
B a ill e t ,
membr es de l a mi ss i on ar chéol og i q u e f r a nga i s e a u Cai r e , T I X , 1 r fasc i c u l e , P a r i s ,
—
1
S ee al so C an to r , op c i t V ol I , 1 90 7 , pp 6 7 , 50 4
2
88
1 8 9 , pp
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THE GREEKS
Gr eek Geometry
About the seventh century B C an active co mmercial intercourse
sprang up between Greece and Egyp t Naturally there arose an
interchange of ideas as well as of merchandise Greeks thirsting for
knowledge sought the Egyp tian priests for instruction Thales
P ythagoras ( En op ide s P lato De mocritus Eu d oxus all visite d the
land of the pyram ids Egyptian ideas were thus transplanted across
the sea and there stimulated Greek thought directed it into new lines
and gave to it a basis to work upon Greek culture therefore is not
primitive Not only in mathematics but also in mythology and
art Hellas owes a deb t to older countries To Egypt G reece i s i n
debted among other things for its elementary geometry B ut this
does not lessen our ad miration for the Greek m ind From the mo
ment that Hellenic philosophers applied the mselves to the study of
Egyptian geo metry this science assu med a radically different aspect
“
Whatever we Greeks receive we improve and perfect says P lato
The Egyptians carried geometry no further than was absolutely n e ces
sary for their practical wants The Greeks on the other hand had
within them a strong speculative tendency They felt a craving to
discover the reasons for things They found pleasure in the con
t empl a ti on of i deal relations and loved science a s science
Our sources of inform ation on the history of Greek geo m etry before
Euclid consist merely of scattered notices in ancient writer s The
early m athe m aticians Thales and P ythagoras left behind no written
records of their discoveries A full history of Greek geom etry and
astrono my during this period written by Eu demu s a pupil of Aris
t o tl e has been lost
It was well known to P roclus who in his com
men tar i es on Euclid gives a brief account of it This abstract con
s t i tu t es our m ost reliable infor m ation
We shall quote it frequently
under the name of Eu demi an S ummary
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The I on i c S chool
To Th al e s ( 6 40—546 B
of M iletus one of the seven wise men
an d the founder of the Ionic school falls the honor of having intro
D uring middle life he
du c ed the study of geo metry into Greece
engaged in commercial pursuits which took him to Egypt He is
said to have resided there and to have studied the physical sciences
and mathe matics with the Egyptian priests P lutarch declares that
Thales soon excelled his m asters and a ma z ed King Ama si s by meas u r
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I
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A HISTORY OF M ATHE M ATICS
16
ing the heights of the pyramids fro m their shadows Accordi ng to
P lutarc h this was done by considering that the shadow cast by a
vertical stafi of known length bears the same ratio to the shadow of
the pyramid as the height of the staff bears to the heigh t of the pyra
mi d This sol ut ion presupposes a knowledge of proportion and the
Ahmes papyrus actually sho ws that the rudiments of proportion were
kn own to the Egyp tians According to D iogenes L aer ti u s the pyra
mids were m easured by Thales in a di ff erent way ; viz by finding the
length of the shadow of the pyramid at the moment when the shadow
of a staff was equal to its ow n length P robably bo th methods were
used
The E udemi a n S u mmary ascribes to Thales the inventi on of the
theore ms on the equality of vertical angles the equality of the angles
at the base of an isosceles triangle the bisection of a circle by any
diam eter and the congruence of two triangles having a side and the two
adjacent angles equal respectively The last theorem combin ed (w e
have reason to suspect )
with the theorem on similar triangles he applied
to the measurement of the distances of ships fro m the shore Thus
Thales w as t he first to apply theoretical geometry to practical uses
The theore m that all angles inscribed in a semicircle are right angles
is attributed by so me ancient writers to Thales by others to P ythag
oras Thales was doubtl e ss famili ar with other theorem s n ot r e
corded by the ancients I t has been inferred that he knew the su m
of the thr e e angles of a triangle to be e q u l to two right angles and
1
the sides of equiangular triangles to be pr o or ti on al The E gyptians
m ust have m ade use of the above theorem s on the straight line in
some of their constructions found in the Ahmes papyrus but it was
left for the Greek phi losopher to giv e these tr uths whi ch others saw
but did not formulate into words an explicit abstract expression and
to put i nto scientifi c language and subj ect to proof that which o thers
merely felt to be true Thales m ay be said to have created the geom
etr
i
of
lines
essentially
abstract
in
its
character
wh
le the E gyp tians
y
studi ed only the geo metry of surfaces and the rudi ments of solid
2
geometry e mpirical in their character
With Thales begins also the study of scientifi c astronomy He
acquired great celebrity by the prediction of a solar eclipse in 58 5 B C
Wh ether he predicted the day of the occurrence or simply the year
is not known It is told of hi m that while conte mplating the stars
dur ing an evening walk he fell into a ditch The good old wo man
“
attendi ng him exclaimed How canst thou know what is doing in
the heavens when thou seest not what is at thy feet ?
The two most pro minent pupils of Thales were An axi man der (b 6 1 1
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1
G J Allman , Gr eek Geometry
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John s to n
G eorg e
C o ll eg e ,
2
G J
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A l lm a n ( 1 8 2 4- 1 90 4
G al w a y , I r el an d
Al lm an , op ci t , p 1 5
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D ubl i n, 1 889 , p 1 0
Thal es to Eu clid
’
w as p rofessor of mathemat i c s a t Q u ee n s
fr om
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A HISTORY OF M ATH EM ATICS
18
the teaching of philosophy m athe matics and natural science but it
was a brotherhood the me mbers of whi ch were united for life This
brotherhood had O bservances approaching m asonic peculiarity They
were forbi dden to div ulge the discov eries and doctrines of their school
Hence we are obliged to speak of the P ythagoreans as a body and
fi n d it di ffi cult to deter m ine to who m each particular discovery is to
be ascribed The P ythagoreans the mselves were in the habit of r e
ferring every discovery back to the great founder of the sect
This school grew rapidly and gained considerable political ascend
ency B ut the mystic an d secret O bservances introduced in i mitation
of E g yptian usages and the aristocratic tendencies of the school
caused it to beco m e an object of suspicio n The dem ocratic party in
Lower Italy re v olted and destroyed the buildings of the P ythagorean
school P ythagoras fled to Tarentum and thence to M e t apon t u m
where he was m urdered
P ythagoras has left behind no m athematical treatises and our
sources of infor mation are rather scanty Certain it is that in the
P yt hag
P ythagorean school m athe m atics was the principal study
oras raised mathe matics to the rank of a science Arithmetic was
courted by h im as fervently as geo metry In fact arithm etic is the
foundation of his philosophic system
The E u demi a n S u mmar y says that P ythagoras changed the study
of geom etry into the form of a liberal education for he examined its
principles to the bottom and investigated its theore m s in an i mma
t er i a l and intellectual m anner
His geo metry was connected closely
with hi s arithm etic He was especially fond of those geom etrical
relations which ad mitted of arithmetical e xpression
Like Egyptian geom etry the geo metry of the P ythagoreans is much
concerne d with areas To P ythagoras is ascribed the i mportant
theore m that the square on the hypotenuse of a right triangle is
equal to the s u m of the squares on the other two sides He had
probably learned from the Egyptians the truth of the theorem in the
special case when the sides are 3 4 5 respectively The story goes
that P ythagoras was so j ubilant over this discovery that he sacrificed
a hecato mb Its authenticity is doub ted because the P ythagoreans
believed in the transmigration of the soul and opposed the shedding
of blood In the later traditions of the Neo P ythagoreans this ob
“
j ec ti on is re m oved by replacing this bloody sacrifice by that of an
ox m ade of flour ! The proof of the law of three squares given in
Eucli d s El emen ts I 4 7 is due to Euclid hi mself and not to the
P ythagoreans
What the P ythagorean m ethod of proof was has
been a favorite topic for conj ecture
The theore m on the su m of the three angles of a triangle presu m
ably known to Thales was pro v ed by the P ythagoreans after the
manner of Euclid They de monstrated also that the plane about a
point i s co mpletely filled by six equilateral triangles four squares or
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GREEK GEO M ETRY
19
three regular hexagons so that it is possible to di vide up a plane into
figures of either kind
From the equilateral triangle and the square arise the solids nam ely
the tetraedron o c t a edro n i c osa edron and the cube
These solids
were in all probability known to the Egyp tians e xcepting perhaps
the i cosaedron In P ythagorean p hi losophy they represent r esp ec
ti v el y the four elements of the physical world ; nam ely fire air water
and earth Later another regular solid was discovered namely the
dodecaedro n which in absence of a fifth ele ment was m ade to r epr e
sent the universe itself I ambli chu s states that H ippa su s a P ytha
g or ean perished in the sea because he boasted that he first div ulged
the sphere with the twelve pentagons
The sam e story of death at
sea is told of a P ythagorean who disclosed the theory of irrationals
The star shaped pentagram was used as a symbol of recognition by
the P ythagoreans and was called by them Health
P ythagoras called the sphere the most beautiful of all solids and
the circle the most beautiful of all plane figures The treatm ent of
the subjects of proportion and of irrational quantities by hi m and
his school will be t aken up under the head of arithmetic
According to Eu demu s the P ythagoreans invented the problem s
concerning the application of areas including the cases of defect and
excess as in Euclid VI 2 8 2 9
They were also familiar with the construction of a polygon equal
in area to a given polygon and sim ilar to another given polygon This
proble m depends upon several i mportant and som ewhat advance d
theore m s and testifies to the fact that the P ythagoreans made no
mean progress in geo metry
Of the theorems generally ascribed to the Italian school so m e
cannot be attributed to P ythagoras himself nor to his earliest suc
The pro gress from e mpirical to reasoned solutions m ust of
c e ssors
nec essity have been slow It is worth noticing that on the circle
no theorem of any i mportance was discovered by this school
Though pol i tics broke up the P ythagorean fraternity yet the school
continued to exist at least two centuries longer Among the later
P ythagoreans P hilolaus and Archytas are the most pro minent
P h il ol au s wrote a book on the P ythagorean doctrines
By him were
first given to the world the teachings of the Italian school which had
been kept secret for a whole century The brilliant Ar ch ytas (4 2 8
of Tarentum known as a great statesman and general and
34 7 B C )
universally admired for his virtues was the only great geometer
among the Greeks when P lato opened his school Archytas was the
first to apply geometry to mechanics and t o t reat the latter subject
methodically He also found a very ingenious m echanical solution
to the problem of the d uplication of the cube His solution in v olves
clear notions on the generation of cones and cylinders This proble m
reduces itself to finding two m ean proportionals between two giv en
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A HISTORY OF M ATHE M ATICS
20
lines These mean proportionals were ob tained by Archytas from
the secti on of a half cylinder The doctrine of proportion was a d
v an ced through him
There is every reason to believe that the later P ythagoreans exer
c i se d a strong influence on the study and develop m ent of mathe m atics
at Athens The Sophists acquired geo metry fro m P ythagorean
sources P lato bought the works of P hilolaus and had a warm friend
in Archytas
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The S ophi s t S chool
After the defeat of the P ersians under Xerxes at the battle of
Salamis 480 B C a league was formed among the Greeks to preserv e
the freedo m of the now liberated Greek cities on the islands and coast
of the E g aean Sea Of this league Athens soon becam e leader and
dictator She caused the separate treasury of the league to be merged
into that of Athens and then spent the m oney of her allies for her
own aggran dise ment Athens was also a great comm ercial centre
Thus she became the richest and most beautiful city of antiquity
All menial work was perfor med by sla v es The citiz en of Athens was
well to do and enjoyed a large a mount of leisure The govern ment
being purely de mocratic e v ery citizen was a politician To make hi s
influence felt a mong his fellow men he mu st first of all be educated
Thus there arose a dem and for teachers The supply ca me principally
from Sicily where P ythagorean doctrines had S pread These teachers
were calle d S ophi s ts or wise m en
Unlike the P ythagoreans they
accepted pay for their teaching Although rhetoric was the principal
feature of their instruction they also taught geometry astronom y
and philosophy Athens soon becam e the headquarters of Grecian
The ho me of
m en of letters and of m athe maticians in particular
mathe m atics a m ong the Greeks was first in the Ionian Islan ds then
in Lower Italy and during the tim e now under consideration at
Athens
The geo metry of the circle which had been entirely neglected by
the P ythagoreans was taken up by the Sophists Nearly all their
disco v eries were m ade in connection with their innum erable atte mpts
to sol v e the following three fa mous problems
To trisect an are or an angle
(1)
To double the cube i e to find a cube whose vol u me is
(2)
double that of a given cube
To square the circle i e to find a square or some other
( 3)
rect i linear figure exactly equal in area to a given circle
These proble m s have p robably been the subject of more discuss i on
and research than any other proble m s in m athe matics The bisec tion
of an angle was one of the easiest proble ms in geom etr y The t ris eé
tion of an angle on the other hand presented unexpected diffi cult ies
A r i ght angle had been divided into three equal parts by the P ytha
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GREEK GEO M ETRY
B ut the
g o r e an s
cannot be eff ected
21
general construction though easy in appearance
by the aid only of rul er and co mpasses Among
the first to wrestle with it was H ippi as of El i s a conte mporary of
Socrates and born about 46 0 B C Unable to reach a solution by
ruler and compasses only he and other Greek geo meters resorted to
the use of other means P roclus mentions a man Hippias pr esum
ably Hippias of Elis as the inventor of a transcendental curve which
served to div ide an angle not only into three but into any nu mber of
equal parts Th is same curve was used later by D i nos tra t u s and
others for the quadrature of the circle On this account it i s p all ed
the qu adr atri x The curve may be described thus :The side AB of the
square shown in the figure turns uniformly about A the point B
moving along the circular arc BED In the
sa me ti me the side B C moves p arallel to i t X
self and uniformly from the position of B C B
to that of AD The locus of intersection of
AB and B C when thus m oving is the
quadratrix BFG Its equation we now write
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The ancients considered only
the part of the curve that lies inside the
quadrant of the circle ; they did not know
that x = i 2 r are asymptotes nor that there
is an infinite number of branches Accor ding to P appus D in o str a t u s
e ffected the quadrature by establishing the theore m that B ED :AD
= AD :AG
The P ythagore ans had shown that the di agonal of a square is the
side of another square having double the area of the original one
Thi s probably suggested the proble m of the d uplication of the cube
i e to find the edge of a cube having d ouble the volum e of a gi v en
cube Eratosthenes ascribes to this problem a diff erent origin The
D elians were once suffering fro m a pestilence and were ordered by
the oracle to double a certain cubical altar Thoughtless workm en
simply constructed a cube with edges twice as long but brainless
work like that did not pacify the gods The error being discovere d
He and his disciples searched
P lato was consulted on the m atter
“
eagerly for a solution to this D elian P roble m
An i mportant con
t r ib u t i o n to this proble m was m ade by Hi ppo c r ate s of C h i os ( about
m
m
He
was
a
talented
athe
atician but having been de
B
0
43
It is
frau ded of his property he was pronounced slow and stupid
also said of him that he was the first to acc ept pay for the teaching of
mathe matics He showed that the Delian P roblem could be reduced
to fi n di ng two mean proportionals between a given line and another
2
=
twice as long For in the proportion a :x x :y y 2 a since x
4
3
3
3
2 2
4
2
we have x 2 a x and x 2 a
But
a y
ay and y = 2 ax and x
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A HISTORY OF M ATHE M ATICS
22
of course he failed to find the two m ean proportionals by geo metric
construction with ruler and compasses He m ade h imself celebrated
by squaring certain lunes According to Simplicius Hippocrates
believed he actually succeeded in applying o n e of his lune quadratures
to the quadrature of the circle That Hippocrates really committed
this fallacy is not generally accepted
In the first lune which he squared he took an isosceles triangle
AB C right angled at C and drew a sem i circle on AB as a diam eter
and passing through C He drew also a se mi circle on AC as a dia m
eter and lying outside the triangle AB C The lunar area thus form ed
is half the area of the triangle AB C This is the first example of a
cur v ilinear area which ad mits of exact quadrature Hippocrates
squared other lunes hoping no doubt that he might be led to the
1
qua drature of the circle In 1 840 Th Clausen found other quadrable
lunes but in 1 90 2 E Landau of G ottingen pointed out that two of
the four lunes which Clausen supposed to be new were known to
2
Hippocrates
In his study of the quadrature and duplication problem s Hip
He showed
p o cr a t e s contributed much to the geo metry of the circle
that circles are to each other as the squares of their diam eters that
si milar segments in a circle are as the squares of their chords and
contain equal angles that in a segment less than a se mi circle the
angle is ob tuse Hippocrates contr ibuted vastly to the logic of geo m
e t ry
His inv estigations are the oldest reasoned geo metrical proofs
in existence ( Gow)For the purpose of describing geometrical figures
he used letters a practice probably introduced by the P yth agoreans
The subj ect of si milar figures as developed by Hippocrates i n
volved the theory of proportion P roportion had thus far been used
by the Greeks only in numbers They never succeeded in uniting
“
the notions of numbers and m agnitudes The term number was
used by the m in a restricted sense What we call irrational nu mbers
was not inclu ded under this notion Not even rational fractions
w ere called nu mbers
They use d the word in the same sense as we
use positive integers
Hence nu mbers were conceived as di scon
ti n u ou s while m agnitudes were c on ti n u ou s
The two notions ap
The chasm between them is ex
pea r ed therefore entirely distinct
posed to full view in the state ment of Euclid that incomm ensurable
m agnitu des do not hav e the sam e ratio as nu mbers
In Eucl id s
El emen ts we find the theory of proportion of m agnitudes de v eloped
and treated independent of that of numbers The transfer of the
theory of proportion from nu mbers to m agnitudes (and to lengths in
particular)
was a diffi cult and i mportant step
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i s g i ve n b y G L or i a i n hi s L e s ci en e esa tte n ell an ti ca Gr eci a ,
M il an o , 2 edi ti on , 1 9 1 4 , pp 7 4 94 L o r i a g i ves a l so f u ll bibliog r ap hi cal r e fe r en ce s
t o t he e x t e n s i ve l i t e r a t u r e o n H ip p oc ra t es
2
E W H ob son , S q u a r i ng the C i rcl e, C a mb ri dg e , 1 9 1 3 , p 1 6
1
A fu ll
ac cou n t
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GREEK GEO M ETRY
23
Hippocrates added to his fam e by writing a geo metrical te xt boo k
called the El emen ts This publ i cat i on shows that the P ythagorean
habit of secrecy was being abando ned ; secrecy was contrary to the
spirit of Athenian life
The sophist An tiph o n a conte mporary of Hippocrates introduce d
the pr ocess of exhaustion for the purpose of solving the proble m of
the quadrature He did hi m self credit by re marking that by i n scr ib
ing in a circle a square or an equilateral triangle and on its sides
erecting isosceles triangles with their vertices in the circum ference
and on the sides of these triangles erecting new triangles etc one
could obtain a succession of regular polygons of which each approaches
nearer to the area of the circle than the previous one until the circle
is finally exhau sted Thus is obtained an inscribed polygon whose
sides coincide with the circu mference Since there can be foun d
squares equal i n area to any polygon there also can be found a square
equal to the last polygon inscribed and therefore equal to the circle
itself B rys o n of H erac l ea a contemporary of Antiphon advance d
the proble m of the quadrature consi derably by circu mscribing poly
gons at the sa me tim e that he inscribed polygons He erred howev er
in assum ing that the area of a circle was the arithm etical m ean b e
tween circu mscribed and inscribed polygons Unlike Bryson a n d
the rest of Greek geo meters Antiphon seem s to have beli eved it
possible by continually doubling the sides of an inscribed polygon
to obtain a polygon coinciding with the circle This question gave
rise to liv ely disputes in Athens If a polygon can coincide with the
circle then says Simpliciu s we m ust put aside the notion that m agni
tudes are di v isible ad i nfi n i tu m This diflficu l t philosop hi cal q p e st ion
led to paradoxies that are difli c u l t to e xplain and that deterre d Greek
m athe maticians fro m introducing ideas of infin i ty into their geo m etry ;
rigor in geometric proofs de manded the exclusion of obscure c on c ep
tions Famous are the argu ments against the possibility of m otion
that were adv anced by Z e no of Elea the great d i alectician ( early in
the 5 th century B
None of Zeno s writings have co me down to
We know of his tenets only through his critics P lato Aristotle
uS
Si mplicius Aristotle in his P hys i cs VI 9 ascribes to Zeno four
“
“
argu ments called Zeno s paradoxies ( 1 )
The Dichotom y : You
cannot trav erse an infinite number of points in a finite time ; you
m ust traverse the half of a given distance before you traverse the
whole and the half of that again before you can traverse the whole
Th is goes on ad i nfin i tum so that (if space is m ade up of points)
there
is an in finite number in any given S pace and it cannot be traversed
“
in a finite time ( 2 )
The Achi lles : Achilles cannot overtake a tor
toise For Achilles must first reach the place from which the tortoise
started By that tim e the tortoise will have m oved on a little way
Achilles m ust then trav erse that and still the tortoise will be ahead
“
The Arrow
He is always nearer yet never m akes up to it (3)
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HISTORY
A
24
O F M ATHE M ATICS
”
An arrow in any given m om ent of its flight m ust be at rest in so m e
“
particular point (4 )
The S tade : Suppose three paral l el rows of
poi nts in j ux taposi tion as in Fig 1 One of these ( B )
is imm ov able
A
A
B
B
C
C
Fig 1
Fig 2
while A and C m ove in opposite directions with equal velocity so
as to co m e into the position in Fig 2 The m ove m ent of C relatively
to A will be double its movem ent relatively to B or in other words
any given point in C has passed twice as m any points in A as i t has
in B It cannot therefore be the case that an instant of tim e corre
spon d s to the passage fro m one point to another
“
P lato says that Zeno s purpose was
to protect the arguments of
P armenides against those who m ake fun of him ; Zeno argues that
there is no many he denies plurality That Zeno s reasoning was
wrong has been the view universally held since the tim e of Aristotle
down to the m iddle of the nineteenth century M ore recently the
opinion has been advanced that Zeno was incompletely and incor
rec t l y reported that his argu ments are serious e ff orts conducted with
logical rigor This view has been advanced by Cousin Grote and P
1
Tannery
Tanne ry clai ms that Zeno did not deny motion but
wanted to S how that motion was im possible under the P ythagorean
conception of S pace as the su m of points that the four arguments must
be taken together as constituting a dialogue between Zeno and an
adversary and that the arguments are in the form of a double dilemm a
into whi ch Zeno forces his adversary Zeno s arguments involve con
cept s of continuity of the in fi nite and infinitesi m al ; they are as m uch
the subjects of debate now as they were in the time of Aristotle
Aristotle di d not successfully explain Zeno s paradoxes He gave no
reply to the query arising in the m ind of the student how is it pos
sible for a variable to reach its limit ? Aristotle s continuum was a
sensuous physical one ; he held that since a line cannot be b ui lt up
“
of points a line cannot actually be subdiv ided into points
The
continued bisection of a quantity is unlim ited so that the unlimited
exists potentially but is actually never reached
No satisfactory
e xplanation of Zeno s argu m ents was given before the creation of
Georg Cantor s continuu m and theory of aggregates
The process of e xhaustion due to Antiphon and B ryson gave rise
“
to the cum brous but perfectly rigorous method of e xhaustion
In
deter mining the ratio of the areas between two curvilinear plane
figures say two circles geometers first inscribed or circu mscribed
S i milar polygons
and then by increasing indefinitely the number of
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S ee F C aj or i , Th e H i s to ry o f Z e no
M ath M onthly , Vol 2 2 , 1 9 1 5 , p 3
1
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s
A rg u men t s
on
M o ti o n
i n t h e A mer i c
.
A HISTORY OF M ATHE M ATICS
26
groves of the A cademi a and devoted the remainder of his life to teach
ing and writing
P lato s physical p hi losophy is partly based on that of the P yth a
Like them he sought in arithm etic and geometry the key
g or e an s
to the universe When questioned about the occupation of the D eity
“
He geom etrises continually
Accordingly a
P lato answered that
knowledge o f geo metry is a necessary preparation for the study of
philosophy To show how great a value he put on m athem atics an d
how necessary it is for higher S peculation P lato placed the i n s crip
tion over his porch Let no one who is unacquainted with geo metry
enter here
Xenocrates a successor of P lato as teacher in the
Academ y followed in his master s footsteps by declining to admit a
“
pupil who had no m athe matical training with the remark D epart
for thou hast not the grip of philosophy
P lato observed that geo m
e t ry trained the m ind for correct and vigorous thinking
Hence it
was that the E udemi an S u mmary says He filled hi s writings with
mathem atical d iscoveries and exhibited on every occasion the r e
markable connection between m athe m atics and p hilosophy
With P lato as the head m aster we need not wonder that the P l a
tonic school produced so large a nu mber of mathe maticians P lato
did little real original work but he made valuable improve ments in
the logic and m ethods employed in geo metry It is true that the
Sophist geo m eters of the prev ious century were fairly rigorous in their
proofs but as a rule they did not reflect on the inward nature of their
They used the axio ms without giv ing the m e xplicit ex
methods
pression and the geo metrical concepts such as the point line surface
1
m
fi
i
i
etc without assigning to them for al de n t on s The P ythagoreans
called a point unity in position but thi s is a statem ent of a phi lo
sophical theory rather than a definition P lato objected to calli n g a
“
point a geometrical fiction
He defined a point as the beginning
“
“
of a line or as an indivisible line and a line as length wi thout
breadth
He called the point line surface the boundaries of
the line surface solid respecti v ely M any of the d efinitions in Euclid
are to be ascribed to the P latonic school The sam e is probably true
of Euclid s axiom s Aristotle refers to P lato the axiom that equals
subtracted fro m equals leav e equals
One of the greatest achieve ments of P lato and his school is the i n
To be sure this m ethod
v en t ion of a n al ys i s as a m ethod of proof
had been used unconsciously by Hippocrates and others ; but P lato
like a true philosopher turned the instinctive logic into a conscious
legitimate method
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If a n y o n e sc i en t ifi c i n v en t i on c an c l ai m p re e mi n en c e o v e r a l l o t h ers , I sho u l d
b e i n cl i n ed m ys e l f t o e r ec t a mon u m e n t to t h e u n k n o w n i n ve n t or o f t he ma t he
m a t i cal p o in t , a s t h e s u p r eme typ e o f ha p r oc ess o f a b s t ra c t i o n w hi ch h as b ee n
Horace L amb s
a n ecessary co n di t i on o f sc i e n t i fi c w o rk from t h e v ery b eg i nn i ng
’
A ddress , S ec t ion A , B r i t A ss n , 1 90 4
1 “
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GREEK G EO M ETRY
7
The term s syn thesi s and an al ylsi s are used in mathematics in a more
special s ense than in logic In ancient mathe matics they had a dif
fe r en t meaning fro m what they now ha v e
The oldest definition of
mathe matical analysis as O pposed to synthesis is that gi v en in Euclid
“
XIII 5 which in all probability was fra m ed by Eudoxus : Analysis is
the obtaining of the thing sought by assum ing it and so reasoning up
to an admitted truth ; synthesis is the obtaining of the thing sought
by reasoning up to the inference and proof of it The analytic m ethod
is not conclusive unless all operations involved in it are know n to
be reversible To remove all doub t the Greeks as a rule added to
the analytic process a synthetic one consisting of a reversion of all
O perations occ u rring in the analysis
Thus the aim of analysis was
to aid in the discovery of synthetic pro ofs or solutions
P lato is said to have sol v ed the proble m of the duplication of the
cube B ut the solution is open to the very sam e objection whi ch he
m ade to the solutions by Archytas Eudoxus and M en ae c hm u s He
called their solutions not geom etrical but mechanical for they r e
quired the use of oth er instru ments than the ruler and co mpasses
He said that thereby the good of geo m etry is set aside a n d d estroyed
for we again reduce it to the worl d of sense instead of elev ating and
i mbuing it with the eternal and inco rporeal i mages of thought even
as it is employed by God for which reason He always is God
These
objections indicate either that the solution is wr ongly attributed to
P lato or that he wished to S how how easily non geom etric solutions
of that character can be found It is now rigorously established that
the duplication problem as well as the trisection and quadrature
proble ms cannot be solved by means of the ruler and compasses
only
P lato gave a healthful stim ulus to the study of stereo m etry which
until his tim e had been entirely neglected by the Greeks The S phere
and the regul ar solids have been stu die d to som e extent but the prism
pyramid cylinder and cone were har dly known to exist All these
solids becam e the subj ects of investigation by the P latonic school
One result of these inquiries was epoch m aking M e n ae c h mu s an
associate of P lato and pupil of Eu doxus invented the conic sections
which in course of only a century raised geo metry to the loftiest height
whi ch it was destined to reach during antiquity M en ze c hmu s cut
“
—
t hree kinds of cones the right angled
acute angled and ob tuse
angled by planes at right angles to a side of the cones and thus
obtained the three sections which we now call the parabola ellipse
and hyperbola Judging fro m the two very elegant solutions of the
Delian P roblem by means of intersections of these curves M en aech
mu s m ust have succeed ed well in investigating their properties In
what manner he carried out the graphic construction of these curve s
is not known
Another great geo meter was D i n os tratu s the brother of M en wch
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A HISTOR Y OF M ATHE M ATICS
28
and pupil of P lato Celebrated is hi s mechanical solution of the
quadrature of the circle by m eans of the qu adr a tr i x of Hippias
P erhaps the most brilliant mathe matician of t hi s period was
He was born at Cnidus about 40 8 B C studied under
Eu doxu s
Archytas and later for two m onths under P lato He was i mbued
with a true spirit of scientific inquiry and has been called the father
of scientific astronomical observation F mm the fragmentary notices
of his astronom ical researches found in later writers Ideler and
Schiaparelli succeeded in reconstructing the system of Eudoxus with
“
its celebrated representation of planetary motions by concentric
spheres
Eudoxus had a school at Cyz icus went with his pupils to
Athens visiting P lato and then returned to Cyz icus where he di ed
The fa me of the academy of P lato i s to a large extent due
3 55 B C
to Eudoxus 3 pupils of the school at Cyzicus among whom are M e n
D iogenes L aer tiu s de
wchmu s D i n os t r a t u s A t hen ze u s and Helicon
scribes Eudoxus as astronomer physician legislator as well as geo m
“
eter The E u demi an S u mmary says that Eudoxus first increased the
nu mber of general theore ms added to the three proportions three
m ore and raised to a considerable quantity the learning begun by
P lato on the subj ect of the section to which he applied the analytical
“
By this section is m eant no doub t the golden section
m ethod
(secti o au r ea )which cuts a line in e x tre me and m ean ratio The fi rs t
fi v e propositions in Euclid XIII relate t o lines cut by thi s section and
are generally attr i buted to Eudoxus Eudoxus added m uch to the
knowledge of solid geometry He proved says Archi medes that a
pyram id is exactly one third of a prism and a cone one third of a
cylinder having equal base and altitude The proof that spheres are
to each other as the cubes of their radii is probably due to hi m He
made frequent and skilful use of the m ethod of exhaustion of which
he was in all probability the inv entor A scholiast on Euclid thought
to be P roclus says further that Eudoxus practically invented the
whole of Euclid s fifth book Eudoxus also found two m ean propor
t i on al s between two given lines but the m ethod of solution is not
known
P lato has been called a maker of m athe maticians
B esides the
pupils already named the E ud emi an S u mmary m entions the following :
Th eaete tu s of Athens a m a n O f great natural gifts to who m no doubt
1
Euclid was greatly indeb ted in the composition of the r o th book
treating of i n comrn en su rab l es and of the 1 3 th book ; L e odamas of
Thasos ; N e ocl e i d e s and his pupil L e on who added m uch to the work
of their predecessors for Leon wrote an El emen ts carefully designed
both in nu mber and utility of its proofs ; Th e u di u s of M ag n e s i a who
compose d a very good book of El emen ts and generalised propositions
which had been confined to particular cases ; H e rmoti mu s of Col
oph on who discovered m any propositions of the El emen ts and c om
G J Al lma op ci t p 2 1 2
mu s
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GREEK GEO M ETRY
29
posed so me on l oci ; and finally the nam es of Amycl as of H e r acl e a
C yzi c e n u s of Ath en s and P hi l i ppu s of M e n d e
A skilful mathe matician of whose life and works we have no details
is An s taeu s the elder probably a senior conte mporary of Euclid The
fact that he wrote a work on co ni c sections tends to S how that m uch
progress had been made in their study during the tim e of M en mchmus
Aristaeus w rote also on regular solids and cultivated the analytic
m ethod His works contained probably a su mma ry of the researches
1
of the P latonic school
Ari st o tl e (3 84—3 2 2 B
the syste matiser of deductive logic though
not a professed mathematician promoted the science of geom etry by
improving so me of the most difli c u l t defin itions His P hysi cs contains
passages with suggestive hints of the principle of virtual velocities
He gave the best discussion of continuity and of Zeno s argu ments
against motion found in antiquity About his tim e there appeared a
work called M echani ca of which he is regarded by some as the author
M echanics wa s totally neglected by the P latonic school
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The Fi rst Al exandr i an S chool
In the previous pages we have seen the birth of geometry in Egyp t
its transference to the Ionian Islands thence to Lower Italy and to
Athens We have witnessed its growth in Greece from feeble child
hood to vigorous m anhood and now we shall see it return to the land
of its birth and there der ive new vigor
D uring her declining years imm e di ately following the P eIOpon
n e si an War Athens produced the greatest scientists and philosophers
of antiquity It was the time of P lato and Aristotle In 33 8 B C at
the battle of Chae ronea Athens was beaten by P hilip of M acedon
and her pow er was broken forever Soon after Alexander the Great
the son of P hi lip started out to conquer the world In eleven years
he built up a great e mpire which broke to pieces in a day Egypt
fell to the lot of P tole my Soter Alexander had founded the seaport
of Alexandria whi ch soon became the noblest of all cities P tolemy
made Alexandria the capital The history of Egypt during the nex t
three centuries is mainly the histo ry of Alexandria Literature
p hi losophy and art were di ligently cultivated P tole my created the
university of Alexandria He founded the great Library and built
laboratories m useums a zoological garden and promenades Alex
andria soon beca me the great centre of learnin g
De metrius P h al er eu s was invited from Athens to take charge of the
Library and it is probable says Gow that Eu cli d was invi ted with
him to O pen the m athe matical school
According to the studies of
H Vogt 2 Euclid was born about 3 6 5 B C and wrote his El emen ts
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G J Al l man , op ci t , p 20 5
2
B i bl i otheca mathemati ca , 3 S , V ol
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1 3 , 1 91 3,
pp
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1 93
20 2 .
O F M ATHE M ATICS
HISTORY
A
30
between 3 30 and 3 2 0 B C Of the life of Euclid little is known except
what is added by P roclus to the E u demi an S u mmar y Euclid says
P roclus was younger than P lato and olde r than Eratosthenes and
Archi medes the latter of who m mentions him He was of the P latonic
sect and well read in its doctrines He collected the El emen ts put
in order m uch that Eudoxus had prepared co mpleted many things of
Theze t e t u s and was the first who reduced to unobj ectionable de mon
When P tole my
s t r a t i o n the i mperfect atte mpts of his predecessors
Once asked hi m if geo m etry could not be m astered by an easier process
than by studying the E l emen ts Euclid returned the answer There
is no royal road to geo m etry
P appus states that Euclid was distin
g u i sh ed by the fairness and kindness o t his disposition particularly
toward those who could do anything to advance the m athematical
sciences P appus is evidently m aking a contrast to Apollonius of
1
who m he m ore than insinuates the opposite character
A pretty
2 “
little story is related by S tob mu s : A youth who had begun to read
geo metry with Euclid when he had learnt the first proposition i n
quired What do I get by learning these things ? So Euclid called
his slav e and said Give hi m threepence since he m ust m ake gain
out of what he
These are about all the personal details
preserved by Greek writers Syrian and Arabian writers clai m to
know m uch more but they are unreliable At one time Euclid of
Alexandria was universally confounded with Euclid of M egara who
liv ed a century earlier
The fa me of Euclid has at all times rested mainly upon his book on
geom etry called the El emen ts This book was so far superior to the
E l emen ts written by Hippocrates Leon and Th eu di u s that the latter
works soon perished in the struggle for existence The Greeks gave
Euclid the S pecial title of the author of the E l emen ts
It is a r e
m arkable fact in the history of geo metry t hat the El emen ts of Euclid
wr itten over two thousand years ago are still regarded by som e as the
best introduction to the m athem atical sciences In England they
were used until the present century extensively as a text book in
schools Som e editors of Euclid have however been inclined to credit
They would have us believe that a
him with m ore than is his due
finished and unassailable syste m of geometry sprang at once from the
brain of Euclid an arm ed M inerva fro m the head of Jupiter They
fail to mention the earlier eminent m athe m aticians from whom Euclid
got his material Comparatively few of the propositions and proofs
in the El emen ts are his own discoveries In fact the proof of the
“
Theore m of P ythagoras is the only one directly ascribed to him
All man conj ectures that the substance of Books I II IV comes from
the P ythagoreans that the substance of Book VI is due to the P ytha
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A D e M org an , Eucl e i des
a n d M ythol ogy
2
J G ow , op ci t , p 1 95
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i n S mi th s Di cti onary of Gr eek
’
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a nd
Roma n Bi ography
GREEK GEO M ETRY
and Eudoxus the latter contributing the doctrine of propor
tion as applicable to inco mmensurables and also the M ethod of Ex
h au s ti on s ( Book XII )that The mt e t u s contributed m uch towar d
Books X and XIII that the principal part of the original work of
1
Eucli d hi mself is to be found in Book X
Euclid was the greatest
syste matiser of his time B y careful selection fro m the m aterial before
hi m and by logical arrange m ent of the propositions selected he built
up from a few definitions an d a xio m s a proud and lofty structure
It woul d be erroneous to belie v e that he incorporated into his El ements
all the elem entary theorem s known at his ti me Archimedes Apol
l o n i u s and even he him self refer to theore m s not included in his El e
men ts as being well —
known truths
The tex t of the El emen ts that was co mmonly used in schools was
Theon s edition Theon of Alexan dria the father of Hypatia brought
out an edi t ion ab ou t 700 years after Euclid with som e alterations in
the tex t A S a consequence later comm entators especially Robert
Simson who labored under the idea that Euclid m ust be absolutely
perfect m ade Theon the S capegoat for all the defects which they
thought they c ould discover in the text as they knew it B ut a mong
the manuscripts sent by Napoleon I from the Vatican to P aris was
found a copy of the El emen ts believed to be anterior to Th eon s recen
sion M any variations fro m Th eon s version were noticed therein
but they were not at all i mportant a n d S howed that Theon generally
m ade only verbal changes
The defects in the El emen ts for which
Theon was bla med m ust therefore be due to Euclid hi mself The
El emen ts used to be considered as o ff ering m o dels of scrupulously
rigorous de monstrations It is certainly true that in point of rigor
it compares fav orably with its modern ri v als ; but when examined
in the light of strict m athe matical logic i t has been pronounced by
C S P eirce to be rid dled with fallacies
The results are correct
only because the writer s experience k eeps him on his guard In
m any proofs Euclid relies partly upon intuition
At the begi nning of our editions of the E l emen ts under the hea d of
definitions are given the assumptions of such notions as the point
line etc and so me v erbal explanations Then follow three postul ates
“
or de m ands and twel v e axio m s The term axio m was used by
P roclus but not by Euclid
He S peaks instead of co mmon no
tions —
co mmon either to all m en or to all sciences There has been
m uch contro v ersy a mong ancient and m o dern critics on the postulates
and axiom s An i mmense prepon derance of m anuscripts and the
“
testimony of P roclus place the axiom s about r i ght angl es and
2
This is i ndeed their proper place
parall el s a mong the postulates
g or ea n s
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G J Al l man , op ci t , p 2 1 1
2
A D e M org an , l oc c i t ; H H a n k e l , Theor i e der C o mpl exen Z a hl ensys teme, L e ip
z i g 1 86 7 , p 5 2
I n t he v a rio u s e d i t io n s o f E u cl id s El emen ts di f e re n t n u mb e rs a re
assig ne d t o th e axi oms
Thus t he parall el a xi om i s cal l ed by R ob er S i mso n t he
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A HISTORY OF M ATHE M ATICS
32
for they are really assu mpti on s and not common n oti ons or axioms
The post u late about par al l el s plays an i mportant rOl e in the history
of non Euclidean geo metry An important postulate which Euclid
m issed was the one of superposition according to which figures can
be moved about in S pace without any alteration in form or magnitude
The El emen ts contains t hi rteen books by Euclid and two of which
it is supposed that H yp si cl es and D amasc i us are the authors The
first four books are on plane geometry The fifth book treats of the
theory of proportion as applied to magn itudes in general It has been
greatly admired because of its rigor of treatment B eginners find the
book di fficult E xpressed in modern symbols Euclid s definition of
proportion is thus :Four magnitudes a b c d are in proportion when
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for any integers
and
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and mo
we have simul taneously
<
1 “
Says T L Heath
certain it is that there is an exact corre
spo n d e n c e
al most coincidence between Euclid s definition of equal
ratios and the m odern theory of irrationals due to D edekind H G
Zeuthen finds a close resemblance between Euclid s definition and
Weierstrass definition of equal nu mbers The S ixth book develops
the geo metry of si milar figures Its 2 7th P roposition is the earliest
m axim u m theore m known to history
The seventh eighth ninth
books are on the theory of nu mbers or on arithm etic According to
P Tannery the knowledge of the ex istence of irrationals m ust have
greatly aff ected the m ode of writing the E l emen ts The old nai v e
theory of proportion being recogniz ed as untenable proportions
are not used at all in the first four books The rigorous theory of
Eudoxus was postponed as long as possible because of its difli c u l ty
The interpolation of the arithm etical books VII—
IX is explained
as a preparation for the fuller treatment of the irrational in book X
Book VII explains the G C D of two numbers by the process
“
of division ( the so called Euclidean
The theory of
proportion of ( rational )
nu mbers i s then developed on the basis of
the definition Nu mbers are pr opor tion al w hen the first is the sam e
mul tiple part or parts of the second that the third is of the fourth
2
Thi s is believed to be the older P ythagorean theory of proportion
The tenth treats of the theory of inco mm ensurables D e M organ con
We give a fuller account of i t
s i der ed this the m ost wonderful of all
The nex t three books are on
u nder the head of Greek Arithm etic
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b y C l av i us t he 1 3 t h , by F P eyrard t he 5 th I t i s call ed
t he st h pos tu l ate i n ol d m a n u sc r ip t s , a l so by H e i be rg a n d M e n g e i n t he i r a nn ot a t e d
’
e di t i o n o f E u c lid s w o rk s , i n G r eek an d L a t i n , L ei p i g , 1 88 3 , a n d b y T L H ea t h
’
’
—
H eat h s
i n hi s Thi rteen B ook s of Eu cl i d s El emen ts , Vo l s I I I I , C a mb r idg e , 1 90 8
i s t he mo s t r ece n t ran sl a t io n i n to Eng li sh a nd i s v ery f ull y an d a bl y a n n o tat e d
1
T L H ea th , op ci t V ol II , p 1 2 4
2
R a t io , P ropor io n an d M e as uremen i n t he E eme n t s o f
R ead H B F i n e ,
Euclid, A nnal s of M athemati cs, Vo l X IX , 1 91 7, pp 70 7 6
1 2 th ,
by B ol yai t he
r 1 th ,
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l
O F M ATHE M ATICS
A HISTORY
4
”
the meaning of whi ch title is not understood Heiberg believes it to
“
m ean loci which are surfaces
The im mediate successors of Euclid in the mathe matical school at
Alexandria were probably C o n o n D os i th e u s and Z e u xi ppu s but
little is k now n of the m
212 B
the greatest mathe matician of an
Ar c h im e d e s ( 2 8 7? —
P lutarch calls him a relation of King
t i q u i ty was born in Syracuse
H i eron ; but more reliable is the sta tement of Cicero who tells us he
was of low birth Di odor u s says he visited Egypt and since he was
a great friend of Conon and Eratosthenes it is hi ghl y probable that
he studied in Alexandria This belief is strengthened by the fact that
he had the m ost thorough acquaintance with all the work previously
done in m athem atics He returned however to Syracuse where he
made hi m self useful to his ad m iring friend and patron King Hieron
by applying hi s ex traordinary inventive genius to the construction of
various war engines by which he inflicted m uch loss on the Romans
during the siege of M arcellus The story that by the use of mirrors
reflecting the sun s rays he set on fire the Ro man ships when they
came within bow shot of the walls is probably a fi c ti on The city
was taken at l ength by the Ro mans a n d Archi medes perished in the
in di scrim inate sl aughter which followed Accordi ng to tradition he
was at the ti me studying the diagram to so m e proble m drawn in the
sand As a Roman soldier approached hi m he called out Don t S poil
The soldier feeling insulted rushed upon him and killed
my circles
No blam e attaches to t he Ro man general M arcel l us who ad
hi m
m ired his g enius and raised in his honor a tomb bearing the figure
of a sphere inscribed in a cylinder When Cicero was in Syracuse
he found the to mb buried under rubbish
Archim edes was admired by his f ellow citiz ens c hi efly for hi s me
c han i c al inventions ; he hi m self pri z ed far more
ighly his discoveries
h
“
in pure science He declared that every kind of art whi ch was con
n ec t ed with daily needs was ignoble and vulgar
So m e of his works
have been lost The following are the e xtant books arranged ap
proximately in chronological order : 1 Two book s on E qu i pond er a n ce
of P l an es or C en tr es of P l an e Gr a vi ti es between which is inserted his
treatise on the Qu adr a tu r e of the P ar abol a; 2 The M ethod; 3 Two books
on the S pher e and Cyl i n der ; 4 The M eas u r emen t of the C i rcl e; 5 On
S pi r al s; 6 C on oi ds and S pher oi ds ; 7 The S a nd C ou n ter ; 8 Two books
on Fl oati ng B odi es; 9 Fifteen L emmas
In the book on the M easu r emen t of the Ci r cl e Archimedes prov es
first that the area of a circle is equal to that of a right triangle having
the length of the circumference for its base and the radius for its
altitude In this he assumes that there exists a straight line equal in
length to the circu mference—an assu mption obj ected to by som e
ancient critics on the ground that it is not ev i dent that a straight
line can equal a curved one The finding of such a line was the next
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GREEK GEO M ETRY
35
problem H e first finds an upper lim it to the ratio of the circum fer
ence to the diam eter or
To do this he starts with an equilate ral
triangle of which the b ase 1s a tangent and the verte x is the centre of
the circle By successiv ely bisecting the angle at the centre by c om
paring ratios and by taking the irrational square roots always a little
too small he finally arrived at the conclusion that 7r < 34 Nex t he
finds a lower limit by inscribing in the circle regular polygons of 6 1 2
sides finding for each successive polygon its perimeter
24
which is of course always less than the circu mference Thus he
“
finally concludes that the circu mference of a circle e xceeds three
ti mes its diameter by a part which is less than 4 but more than 44
of the diameter
Thi s approximation is e xact enough for most pur
poses
The Ou adr a tu r e of the P ar abol a contains two solutions to the prob
l em—one m echanical the other geo m etrical
The m ethod of ex
hau s tion is used in both
It is noteworthy that perhaps through the influenc e o f Zeno in
fi n i t esi mal s (i n finitely s mall constants )
were not used in rigorous
de monstration In fact the great geom eters of the period now under
consideration resorted to the radical m easure of excluding them fro m
demonstrative geo metry by a postulate This was done by Eudoxus
Euclid and Archimedes In the preface to the Ou adr atu r e of the P ar ab
ol a occurs the S O called
Archimedean postulate which Archi me des
“
himself attributes to Eu doxus : When two spaces are unequal it i s
possible to add to itself the diff erence by which the lesser 1s su rpassed
by the greater so often that every finite space will be exceeded
Euclid (El emen ts V 4)
gives the postulate in the for m of a definition :
M agnit udes are said to hav e a ratio to one another when the less
can be multiplied so as to exceed the other
Nevertheless i n fi n i t esi
m als m ay have bee n used in tentative researches That such was the
case with Archimedes is evident fro m his book The M ethod formerly
thought to be irretrievably lost but fortunately discovered by Heiberg
in 1 90 6 in Constantinople The contents of this book S hows that he
considered i n fi n i t esimal s sufficiently scientific to suggest the truths O f
theorem s but not to furnish rigorous proofs In finding the areas of
parabolic segments the volumes of S pherical segments and other solids
of revolution he uses a mechanical process consisting of the weighing
of infinitesimal ele ments which he calls straight lines or plane areas
but which are really infinitely narrow strips or infinitely thin plane
1
a
l min ae The breadth o r thickness is regarded as being the sa me in
the elem ents weighed at any one tim e The Archim e dean postulate
d i d not co m mand the interest O f m athe m aticians until t he m o de r n
arithm etic continuu m was created I t was 0 S tol z that showed that
it was a consequence of Dedek i n d s postulate relating to sections
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T L H ea t h , M ethod
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A rchi medes , C a mb r i dg e ,
191 2,
p 8
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A HISTORY OF M ATHE M ATICS
36
It would seem that in his great researches Archimedes mode of
procedure was to start with mechanics ( centre of mass of surfaces and
solids)
and by his i n fi ni t esimal m echanical m ethod to discover new
resul ts for whi ch later he deduced and published the rigorous proofs
1
Archimedes knew the integral f x3dx
Archi medes studied also the ellipse and accomplished its quadrature
but to the hyperbola he seem s to have paid less attention It is b e
l i ev ed that he wrote a book on conic sections
Of all his discoveries Archim edes priz ed most hi ghl y those in his
In it are proved the new theore m s that the
S pher e a nd Cyl i nder
surface of a sphere is equal to four times a great circle ; that the surface
segment of a sphere is equal to a circle Whose radius is the straight
line drawn from the vertex of the segment to the circ umference of its
basal circle ; that the volume and the surface of a sphere are 4 of th e
volum e and surface respectiv ely of the cylinder circum scribed about
the sphere Archimedes desired that the figure to the last proposition
be inscribed on hi s to mb Th is was ordered done by M arcellus
“
The spiral now called the spiral of Archi medes and described in
the book On S pi ral s was discovered by Archimedes and not as some
2
believ e by his friend Conon His treatise thereon is perhaps the
Nowadays subj ects of this kind
most wonderful of all his works
are m ade easy by the use of the infinitesimal calculus In its stead
the ancients used the m ethod of exh austion Nowhere is the fertility
of hi s genius m ore grandly displayed than in hi s masterly use of thi s
m ethod With Euclid and his predecessors the method of exhaustion
was only the me ans of prov ing propositions whi ch m ust have been
seen and belie v ed before they were proved B ut in the hands of
Archi medes this m ethod perhaps combined with his i nfi ni te simal
mechanical m ethod becam e an instru m ent of discovery
By the word conoid in his book on C on oi ds and S pher oi ds is
meant the solid produced by the revolution of a parabola or a hyper
bola about its axis Spheroids are produced by the rev olution of an
ellip se and are long or flat according as the ellipse revol v es around
the major or minor axis The book leads up to the cubature of these
solids A few constructions of geo
metric figures were given by A rchi
medes and Appolonius whi ch were
“
eff ected by insertions
In the
followi ng trisection of an angle at
tributed by the Arabs to Archi
m edes the insertion is achieved by the aid of a gr adu a ted ruler 3
“
To trisect the angle CAB draw the arc B CD Then insert the
—
H G Z th
i B i bl i th c m th m ti
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6
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V
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M C an to r
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F E n riq u es ,
L eip i g , 1 90 7 , p
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ca ,
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Vo l I , 3 A u fl , 1 90 7 , p 3 0 6
Fr a gen der El emen targ eometri e, deu
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2 34
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H F l e i sc he r , II ,
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GREEK GEO M ETRY
7
distance F E equal to AB marked on an edge passing through C
and moved until the points E and F are located as shown in the
figure The required angle is E FD
His arithmetical treatise and problem s wil l be considered later
We shall now notice hi s works on mechani cs Archimedes is the
author of the first sound knowledge on this subject Archytas Aris
to tl e and others attempted to form the known m echanical truths into
a science but failed Aristotle knew the property of the lever but
coul d not establish its true m athematical theory The ra di cal and
fatal defect in the speculations of the Greeks in the opinion of Whewell
was that though they had in their possession facts and ideas the
ideas wer e n ot di sti n ct and appr opri ate to the f acts
For i nstance
Aristotle asserted that when a body at the end of a lever is moving
it may be considered as having two motions ; one in the direction of
the tangent and one in the direction of the ra di us ; the former m otion
is he says accor di ng to n atu r e the latter con tr ary to n atu r e These
“
“
inappropriate notions of natural and unnatural m otions to
gether with the habits of thought whi ch dictated these spec u lations
made the perception of the true grounds of mechanical properties
1
impossible
It see ms strange that even after Archi medes had en
t er ed upon the right path thi s science should have re mained ab
s ol u t el y stationary till the tim e of Galileo—
a period of nearly two
thousand years
The proof of the property of the lever given in hi s Equ i ponder an ce
2
of P l an es
holds its place in many text books to this day M ach
“
criticiz es it
From the m ere assumption of the equilibrium of equal
weights at equal distances is derived the inverse proportionality of
weight and lever arm ! How is that possible ? Archi medes estim ate
of the efli ci en cy of the lever is expressed in the saying attribute d to
him
Give me a fulcrum on whi ch to rest and I will m ove the earth
While the E qu i pond eran ce treats of solids or the equilibrium of
solids the book o n Fl oati ng B odi es tr eats of hydrostatics His atten
tion was first drawn to the subject of specific gravity when King Hieron
asked hi m to test whether a crown professed by the m aker to be pure
gold was not alloyed with silver The story goes that our phi losopher
was in a bath when the true m ethod of solution flashed on his m ind
“
He i mmediately ran hom e naked shouting I hav e found it ! TO
solve the problem he took a piece of gold and a piece of silver each
weighing the same as the crown According to one author he deter
mined the volu m e of water displaced by the gold silver and crown
respectively and calculated from that the amount of gold and silver
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Will i am Whew ell Hi s tory of the I nducti ve S ci ences 3 rd Ed N ew York
w as M as t e r of Tr i n i ty C oll eg e
1 86 6)
V ol I p 8 7 Will i am Wh ew el l ( 1 7 94—
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1 85 8,
C am
b ri dg e
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E M ach , The S ci ence of M echa n i cs , t r b y T M c c orm ack , C h i ca g o , 1 90 7 , p 1 4
Ern st M ac h ( 1 8 3 8 1 9 1 6 w as p ro fesso r o f t h e hi story an d t heo ry o f t he i n d uc ti ve
s ci en c es at t he u n i versi ty o f V i e nn a
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A HISTORY OF M ATHE M ATICS
38
in the crown According to another writer he weighed separately
the gold sil v er an d crown while immersed in water thereby deter
m ining their loss of weight in water Fro m these data he easily found
the solution It is possible that Archim edes solved the problem by
both m ethods
After exa mining the writings of Archimedes one can well under
stand how in ancient tim es an Archi medean problem cam e to
m ean a proble m too deep for ordi na r y minds to solve and how an
“
Archi medean proof came to be the synonym for unquestionable
certainty Archim edes wrote on a ver y wide range of subj ects and
displayed great profundity in each He is the Newton of antiquity
Er ato s t h e n e s eleven years younger than Archi m edes was a native
of Cyrene He was educated in Alexandria under Callimachus the
poet who m he succeeded as custo dian of the Alexandrian Library
His many—
sided acti v ity m ay be inferred fro m his works He wrote
on Good a nd E vi l M easu r emen t of the E ar th C omedy Geogr aphy
He was
C hr on ol ogy C on s tel l a ti on s and the Du pl i ca ti on of the Cu be
also a philologian and a poet He m easured the obliquity of the
ecliptic and invented a device for finding prime numbers to be de
scribed later Of hi s geo metrical writings we possess only a letter to
P tole m y Euergetes giving a history of the duplication proble m and
also the description of a very ingenious mechanical contriv ance of his
own to sol v e it In his old age he lost his eyesight and on that account
is said to ha v e co mmitted suicide by voluntary starvation
About forty years after Archimedes flourished Apo l lo ni u s of P e r g a
whose genius nearly equalled that of his great predecessor He i n c on
t es t ab l y occupies the second place in distinction a mong ancient m athe
m a t i c i an s
Apollonius was born in the reign of P tole m y Euergetes
and died under P tole my P hi l op ator who reigned 2 2 2 —2 0 5 B C H e
studied at Alexandria under the successors of Euclid and for so m e
ti m e also at P ergamum where he m a de the acquaintance of that
E u dem u s to who m he d edicated the first three books of his C on i c
S ecti on s The brilliancy of his great work brought him the title of the
Great Geo m eter
This is all that is known of hi s life
His C on i c S ecti on s were in eight books of whi ch the first four only
have come down to us in the original Greek The nex t three b ooks
were unknown in Europe till the m iddle of the seventeenth century
when an Arabic t ranslation m ade about 1 2 50 was discov ered The
eighth book has never been found In 1 7 1 0 E Halley of Oxford pub
l i sh ed the Greek text of the first four books and a Latin translation
of the re maining three together with his conj ectural restoration of
the eighth book foun d e d on the intro ductory le mm as of P appus The
first four books contain little more than the substance of what earlier
geometers had done Eu t oc i u s tells us that H erac li des in his life of
Archim edes accused Appolonius of hav ing appropriate d in his Con i c
S ecti on s the unpublished d iscoveries of that great m athe mati cian
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GREEK GEO M ETRY
39
It is diffi cult to believe that this charge rests upon good foundation
Eu to ci u s quotes Ge m inus as replying that neither Archi medes nor
Apollonius claimed to have invented the conic sections but that
Apollonius had introduced a real i mprove ment While the first three
or four books were founded on the works of M en aechmu s Aristaeus
Euclid and Archim e des the remaining ones consisted alm ost entirely
of new matter The first three books were sent to Eu dem u s at inter
vals the other books (after E u demu s s death )
to one Attalus The
preface of the second book is interesting as showing the m ode i n
which Greek books were published at this time I t reads thus :
“
I have sent m y son Apollonius to bring you (Eu d emu s)
the secon d
book of my Conics Read it carefully and comm unicate it to such
others as are worthy of it If P hil on ide s the geom eter who m I intro
du c e d to you at Ephesus co m es into the neighbourhood of P erga m um
1
giv e it to him also
The first book says Apollonius in his preface to it contains the
m o de of pro d ucing the three sections and the conj ugate h yperbolas
and their principal characteristics more ful ly and generally worked
out than in the writings of other authors
We rem e mber that
M e n aechm u s and all his successors down to Apollonius considered onl y
sections of r i ght cones by a plane perpendicular to their si des and tha t
Apol
t he three sections were ob taine d each fro m a di fferent cone
He produce d all the
l o n i u s introduced an i mportant generalisation
sections fro m one and the sam e cone whether right or scalene and
by sections which m ay or may not be perpen dicular to its sides The
Instea d
O l d nam es for the three curves were now no longer applicable
“
“
of calling the three curves sections of the acute angled
right
“
angled and obtuse angled cone he called them ell i pse par abol a
“
To be sure we find the words parabola
a n d hyper bol a respectively
“
and ellipse in the works of Arch i medes but they are probably only
“
2
interpolati ons The word ellipse was applied because y < px p
“
being the parameter ; the word parabola was introduced because
2
2=
>
the
ter
hyperbola
because
x
a
n
d
m
y
px
p
y
The treatise of Apollonius rests on a unique property of conic sec
tions whi ch is deriv ed directly fro m the nature of the cone in wh i ch
these sections are found How this property form s the key to the
2
syste m of the ancients is told in a m asterly way by M Chasles
“
“
Conceive says he an obl i que cone on a circ u lar base ; the stra ight
l i ne draw n fro m its su mmit to the centre of the circle forming its base
is called the axi s of the cone The plane pass i ng through the axis
perpen dicular to its base cuts the cone along two lines and determi nes
in the circle a diameter ; the triangle havi ng th i s diam eter for its base
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Ze u t h e n Di e
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L ehr e
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den K eg el schn i tten i m A l ter thu m, K op e n h ag en ,
p 50 2
M C h a sl e s , Geschi chte der Geometr i c
L A S o h n ek e , H a l l e , 1 8 39 , p 1 5
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A u s d em Fr a n z os i sc h e n
u b e r t r ag e n
d u rc h ,
A HISTORY OF M ATHE M ATICS
40
and the two lines for its sides is called the tr i an gl e thr ough the axi s
In the formation of his conic sections Apollonius supposed the cutting
plane to be perpendicul ar to the plane of the triangle through the
axi s The points in whi ch thi s plane meets the two sides of this tri
angle are the ver ti ces of the c u rve ; and the straight line whi ch j oins
these two points is a diameter of it Apolloni us called this di ameter
l atu s tr an sversu m At one of the two vertices of the curve erect a per
to t he plane of the triangle thr ough the
p e n di c ul ar (l atu s r ectu m)
axis of a certain length to be determined as we shall specify later
and from the ex trem ity of this perpendicular draw a straight line to
the other vertex of the curve ; now thr ough any point whatever of
the diameter of the curve draw at right angles an ordi n ate the square
of this ordinat e co mprehended between the diameter and the curve
will be equal to the rectangle constructed on the portion of the ordi nate
co mprised between the diameter and the straight li ne and the part
of the di am eter co mprised between the first vertex and the foot of the
ordinate Such is the characteristic property which Apollonius recog
mises in hi s conic sections and whi ch he uses for the purpose of i n
ferring fro m it by adroit transformations and deductions nearly all
the rest It plays as we shall see in his hands al most the same rOl e
as the equation of the second degree with two variables ( abscissa and
ordinate )
in the syste m of analytic geom etry of D escartes
Apol
1
—
l on i u s m ade use of c o ordinates as di d M enaechmu s before hi m
Chasles continues :
It will be observed fro m thi s that the di ameter of the curve and
the perpendi cular erected at one of its extrem ities suffi ce to construct
the curve These are the two element s whi ch the ancients used with
whi ch to establish their theory of conics The perpendicul ar in ques
tion was called by the m l atu s erectu m; the m ode m s changed this nam e
first to that of l a tu s r ectu m and afterwar d s to that O f par ameter
The first book of the C on i c S ecti on s O f Apollonius is al most wholly
de v o ted to the generation of the three principal conic sections
The second book treats mainly of asymptotes axes and diameters
The third book treats of the equality or proportionality of triangles
rectangles or squares of whi ch the co mponent parts are determi ned
by portions of transversals chords asymptotes or tangents which
are frequently subj ect to a great number of con di tions It also touches
the subj ect of foci of the ellipse and hyperbola
In the fourth book Apollonius discusses the harmonic di v ision of
straight lines He also examines a syste m of two conics ; and shows
t hat they cannot cut each other in m ore than four points He i nv es
t ig a t es the various possible relative positions of two co ni cs as for
instance when they have one or two points of contact with each other
The fifth book reveals better than any other the giant intellect of
its author D iffi cult questions of maxi ma and mi n i ma of whi ch few
T L He t h A p l l ni u f P e g
C amb i dg
8 96 p C X V
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A HISTORY OF M ATHE M ATICS
2
adapted to the invention of general m ethods Instead of a cli mb to
st ill loftier heights we observe therefore on the part of later Greek
geometers a descent during which they paused here and there to look
around for details which had been passed by in the hasty ascent 1
A mong the earliest successors of Apollonius was N i c ome d e s Noth
ing definite is known of him e xcept that he invented the con choid
a curv e of the fourth order He devised a little
( m ussel
m achine by w hi ch the curve could be easily described With aid of
the conchoid he duplicated the cube The curve can also be used for
trisecting angles in a manner rese mbling that in the eighth le mma of
Archi medes P roclus ascribes this mode of trisection to N i comedes
but P appus on the other hand claims it as his own The conchoid
was used by Newton in constructing curv es of the third degree
About the tim e of N i c omedes ( say 1 8 0 B
flourished also
“
D iocl e s the in v entor of the ci ss oi d ( ivy
Thi s curve he used
for finding two mean proportionals between two given straight lines
The Greeks did not consider the co mpanion curve to the cissoid ;
in fact they considered only the part of the cissoid proper whi ch
lies insi de the circle used in constructing the curve The part of the
area of the circle left over when t he two circular areas on the concave
sides of the branches of the curve are remove d looks somewhat like
an ivy leaf Hence probably the nam e of the curve That the two
branches ex ten d to infinity appears to have been noticed first by G P
de Rob er al in 1 6 40 and then by R de Sluse 2
About the life of P e r s e u s we know as little as about that of Nico
m edes and Diocles He lived so m e tim e between 2 00 and 1 00 B C
From Heron and Ge m inus we learn that he wrote a work on the spi r e
a sort of anchor ring surface described by Heron as being produced by
the revolution O f a circle around one of its chords as an axis The
sections of this surface yield peculiar curves called spi r al s ecti ons
whi ch according to Gem inus were thought out by P erseus These
cu rves appear to be the sam e as the H i ppopede of Eudoxus
P robably som ewhat later than P erseus lived Z e n od oru s
He wrote
an interesting treatise on a new subj ect ; namely i s oper i metr i cal figu r es
Fourteen propositions are preserved by P appus and Theon Here
are a few of the m :Of isoperimetrical regular polygons the one having
the largest num ber of angles has the greatest area ; the circle has a
greater area than any regular polygon of equal periphery ; of all iso
p eri me n t r i c al polygons of n S ides the regular is the greatest ; of
all solids having surfaces equal in area the sphere has th e greatest
volu me
H yps i cl e s (between 2 00 and 1 0 0 B C )
was supposed to be the
author of both the fourteenth a n d fifteenth books of Eucli d but recent
criti cs are of opinion that the fifteenth book was written by an author
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V o l I , 3 A u fl , 1 90 7 , p 3 5 0
C ur ve/i , t ra n s l by F S c h ii tte , I ,
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G REEK G EO M ETRY
3
who lived several centuries after Christ The fourteenth book con
tains seven elegant theore m s on r eg ul ar sol i ds A treatise of Hypsi cl e s
on Ri s i ngs is of interest because it gi v es the div ision of the circum
ference into 3 60 degrees after the fashion of the B abylonians
H ippar c h u s of N i c aea i n B i thyn i a was the greatest astrono mer of
antiquity He took astronom ical observations between 1 6 1 a n d 1 2 7
B C
He established in d uctiv ely the famous theory of epicycles and
eccentrics As might be expected he was interested in m athe matics
not per s e but only as an aid to astronomical inquiry No mathe
mat i c al writings O f his are extant but Theo n of Alexandria infor m s us
that Hipparchus originate d the science of tr igon ometry and that he
calculated a table of chor d s in twelve books Such calculations
m ust have required a ready knowle dge of arithm etical and algebraical
operations He possessed arithmetical and also graphical devices for
solving geo metrical proble m s in a plane and on a S phere He gives
indication of having sei zed the idea of c o ordinate representation found
earlier in Apollonius
About 1 0 0 B C flourished H e r o n th e El d e r of Alexandria He was
the pupil of C t esib i u s who was celebrated for his ingenious mechanical
inventions such as the hydraulic organ the water clock an d catapult
It is belie v ed by some that Heron was a son of C t esi b i u s He ex
hi bi t ed talent of the sam e order as did his m ast er by the invention of
“
the eolipile and a curious mechanism known as Heron s fountain
Great uncertainty exists concerning his writings M ost authorities
believe him to be the author of an i mportant Tr eati s e on the D i optr a
of which there exist three m anuscript copies quite dissimilar B ut
1
M M arie thinks that the D i optr a is the work of H eron the Y ou nger
who lived in the seventh or eighth century after Christ an d that
Geodesy another book supposed to be by Heron is only a corrupt and
defective Copy of the former work Di optr a contains the im portant
formula for finding the area O f a triangle expressed in te r m s O f its
sides ; its derivation is quite laborious and yet excee dingly ingenious
“
It see m s to me difficult to believe says Chasles that so beautiful
a theore m sho uld be found in a work so ancient as that of Heron the
Elder without that so me Greek geo meter should have thought to
M arie lays great stress on thi s silence of the ancient writers
cite it
and argues fro m it that the true author m ust be Heron the Younger
or some writer m uch more recent than Heron the Elder B ut no r e
liable evidence has been found that there actually existed a second
mathem atician by the nam e of Heron P Tannery has shown that
i n applying this for mula Heron found the irrational square roots by
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the approximation
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x Z
4
M axi mil i en M a r i e H i s toi r e des
To me I ,
,
1 88 3 ,
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1
78
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A
where
s c i e nc es
a
2
is the square nearest to
mathemati q u es
et
phys i q u es
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P ar i s ,
A
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HISTORY
A
44
more
When a
O F M ATH E M ATICS
A
accurate value was wanted Heron took
,
in the place of a in the above form ula Apparently Heron some
“
times found square and cube roots also by the method of double
false position
“
Dioptra says Venturi were instrum ents which had g reat re
semblance to our modern theodolites The book Di optr a is a treatise
on geodesy contai ning solutions with aid of these instru ments of a
large nu mber of questions in geometry such as to find the dista nce
between two points of which one only is accessible or between two
points which are visible but both inaccessible ; from a given point to
draw a perpendi cular to a line which cannot be approached ; to find
the difference of level between two points ; to measure the area of a
field without entering it
Heron was a practical surveyor This may account for the fact
that his writings bear so little resemblance to those of the Greek
authors who considered it degrading the science to apply geometry to
surveying The character of his geometry is not Grecian but de
This fact is the m ore surprising when we consider
c idedl y Egyp tian
that Heron de monstrated his familiarity with Euclid by writing a com
m entary on the El emen ts So m e of Heron s for mulas point to an old
Egyp tian origin Thus besides the above e xact formula for the area
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of a triangle in ter ms of its sides Heron gives the formula
al
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whi ch bears a striking likeness to the formula
x
2
+ az
01
+ az
(3 1+ bz
x
fO 1
2
2
finding the area of a quadrangle found in the Edfu inscriptions
There are m oreov er points of resemblance between Heron s writings
and the ancient Ahmes papyrus Thus Ahm es used unit fractions
exclusively (except the fraction
Heron uses them oftener than o ther
fractions Like Ahmes and the priests at Edfu Heron divides c om
plicated figures into simpler ones by drawing auxiliary lines ; like the m
he shows throughout a special fondness for the isosceles trapez oid
The writings of Heron satisfied a practical want and for that reason
were borrowed extensively by other peoples We find traces of them
in Ro me in the Occident during the M iddle Ages and even in In dia
The works attributed to Heron including the newly discovere d
M el ri ca published in 1 90 3 have been edited by J H Heiberg
H S chOn e and W Schmidt
G e mi n u s of Rhodes (about 7 0 B c )
published an astronomical work
still ex tant He wrote also a book now lost on the A rr angemen t of
M a themati cs which contained m any valuable notices of the early
history of Greek mathematics P roclus and Eu toc iu s quote it fre
quently Theodosius is the author of a book of little m erit on the
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GREEK GEO M ETRY
45
geometry of the sphere Investigations due to P Tannery and A A
l
BjOrnb o see m to indicate that the m athe m atician Theodosius was
not Theodosius of Tripolis as formerly supposed but was a resident
of Bithyni a and conte mporary of Hipparchus Di onys odoru s of
Ami sus in P ontus applied the intersection of a parabola and hyperbola
to the solution of a problem whi ch Arc hi medes in his S phere and
“
Cyl i nder had left inco mplete
The proble m is to cut a sphere so
that its segments shall be in a giv en rati o
We have now sketched the progress of geometry dow n to the time
of Christ Unfortunately very little is known of the histor y of geo m
e t ry between the ti m e of Apollonius and the beginning of the Christian
era The names of qui te a number of geo meters have been mentioned
but very few of their works are now extant It is certain however
that there were no mathe maticians of real genius from Apollonius to
P tole my excep ting Hipparchus and perhaps Heron
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The S econd A l exandr i an S chool
The close of the dynasty of the L ag i des which ruled Egyp t from the
tim e of P tole my Soter the builder of Ale xandria for 3 00 years ; the
absorption of E g ypt into the Ro man E mpire ; the closer co mm ercial
relations between peoples of the East and of the West ; the gradual
decline of paganism and spread of Christianity these events were
of far reaching influence on the progress of the sciences which then
had their ho me in Alexandria Alexandria became a comm ercial and
intellectual e mporium Traders of all nations m e t in her busy streets
and in her magnificent Library museu ms lecture halls scholars fro m
the East mingled wi th those of the West ; Greeks began to stu dy older
literatures and to compare them with their own In consequence of
this interchange of ideas the Greek philosophy becam e fused with
Oriental philosophy N eo P ythagoreanism and Neo P latonism were
the names of the modified system s These stood for a ti me in Op
position to Christianity The study of P latonism and P ythagorean
mysticism led to the revival of the theory of nu mbers P erhaps the
dispersion of the Jews and their introduction to Greek learning helped
in bringing about this reviv al The theory of numbers became a
favorite study This new line of mathematical inquiry ushered in
what we may call a new school There is no doub t that even now
geometry continued to be one of the most i mportant studies in the
Alexandrian course This Second Alexandrian School may be said to
begin with the Christian era It was m ade fam ous by the na mes of
Claudius P tolemaeus D iophantus P appus Theon of S myrna Theon
of Alexandria I ambli chu s P orphyrius and others
By the side of these we may place S er e n u s of An t i noei a as ha v ing
hi t ri n of m the
Axel A t ho B j r b ( 8 7 4—9 )f C op e n hag en w
—
—
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m
t
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S
V
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S
ee
B
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9
pp 3 3 7 344
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M ATHE M ATICS
A HISTORY OF
46
been connected more or less with this new school He wrote on sec
tions of the cone and cylinder in two books one of whi ch treated
only of the triangular section of the cone through the apex He solved
“
the problem given a cone ( cylinder )to find a cylinder ( cone)so
that the section of both by the sam e plane gives similar ellipses
Of
partic ul ar interest is the following theore m which is the foundation
of the m odern theory of harmonics : If from D we draw DF cutting
the triangle A B C and choose H on it so that DE DF = EH :H E
and if we draw the line A H then every transversal through D such
as DG will be div ided by A H so that DK DG = K] J G M e n e l a u s
of Alexandria ( about 98 A D )
was the author of S phce r i ca a work
extant in Hebrew and Arabic but not in Greek In it he proves the
theorem s on the congruence of
spherical triangles and describes
their properties in m uch the sam e
way as Euclid treats plane tri
In it are also found the
D angles
theore m s that the sum of the three
sides O f a spherical triangle is less
than a great circle and that the
s u m of the three angles exceeds two right angles
Celebrated are two
theorem s of his on plane and spherical triangles The one on plane tri
“
angles is that if the three sides be cut by a stra ight line the product of
the three segm ents which ha v e no comm on ex trem ity is equal to the
product of the other three L N M Carnot m akes this proposition
“
known as the lemm a of M enelaus the base of his theory of trans
versals The corresponding theorem for spherical triangles the so
called regula s ex q u an ti ta t u m is obtained fro m the above by
“
“
rea ding chords of three segments doubled in place of three seg
ments
C laudiu s P tol e my a celebrated astrono m er was a native of Egypt
Nothing is known of his personal history except that he flourished in
Alexan dria in 1 39 A D and that he m ade the earliest astronomical
O bser v ations recor ded in his works in 1 2 5 A D
the latest in 1 5 1 A D
The chief of his works are the S yn taxi s M a themati ca (or the A l mages t
as the Arabs call it)
and the Geog r aphi ca both of which are extant
The for mer work i s based partly on his own researches but m ainly
on those of Hipparchus P tolemy see m s to have been not so m uch of
an independent investigator as a corrector and i mprover O f the work
of his great predecessors The A l mages t 1 form s the foundation of
all astrono mical science down to N Copernicus The fundam ental
“
i dea of his syste m the P tole maic System is that the earth is in the
centre of the un i verse and that the sun and p l anets revolve around
the earth P tolem y did considerable for m athematics He creat ed
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On t he i mpor tan c e
Tann ery , Recher ches
s ur
of
’
t h e Al ma g e s t i n th e hi s tory
l hi s toi r e de l
’
mi e, P a ri s,
as tron o
o
f
as t ron o
1 8 93
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my,
co n s u l t
P
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GREEK GEO M ETRY
for astrono mical use a tri gon ometry remarkably perfect in form The
foundation of this science was laid by the illustrious Hipparchus
The A l mages t is in 1 3 books Chapter 9 of the first book shows how
to calculate tables of chords The circle is div ided into 3 6 0 degrees
each of which is halved The diameter is di vided into 1 2 0 divisions ;
each of these into 6 0 parts which are again subdivided into 6 0 smaller
parts In Latin these parts were called par tes mi n u tae pr i mae and
r tes mi n u tx s ecu nd a
a
Hence our nam es m inutes and seconds
p
The se xagesimal m ethod of div iding the circle is of Bab ylonian origin
and was known to Geminus and Hipparchus B ut P tolemy s m ethod
of calculating chords seem s original with hi m He first proved the
“
proposition now appended t o Euclid VI (D)that the rectangle
contained by the diagonals of a quadrilateral figure inscribed in a
circle is equal to both the rectangles contained by its opposite S ides
He then shows how to find from the chords O f two arcs the chords of
their su m and di fference and from the chord of any arc that of its
half These theore m s he applied to the calculation of his tables of
chords The proofs of these theorems are v ery pretty P tole my s
construction of sides of a regular inscribed pentagon and decagon was
giv en later by C Clavius and L M ascheroni and now is used m uch
by engineers Let the radius B D be _
L to A C
DE = EC M ake E F = EB then B F is the side of
the pentagon and DF is the side of the decagon
Another chapter of the first book in the A l ma
g es t is de v oted to tr i gon ometry and to s pher i ca l A F
D
E
C
trigonom etry in particular P tole my proved the
“
“
lemma of M enelaus and also the regula sex q u an t i tatum
Upon these propositions he built up his trigono metry In trigono
m etric co mputations the Greeks di d not use as did the Hindus half
“
the chor d of twice the arc ( the sine
the Greeks used ins tead
the whole chord of double the arc Only in grap hic constructions
referred to again later did P tolemy and hi s predecessors use half the
chord of double the arc The fundamental theorem of plane trigo
n ome t ry that two sides of a triangle are to each other as the chords
of double the arcs m easuring the angles opposite the two sides was
not stated e xplicitly by P tole m y but was contained i mplicitly in other
theorem s M ore complete are the propositions in spherical trigo
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fact that trigonometry was cultivated not for its own sake but
to aid astrono mical inquiry explains the rather startling fact that
spherical trigonometry cam e to e xist in a developed state earlier than
plane trigono metry
The re maining books of the A l mages t are on astronomy P tolemy
has written other works which have little or no bearing on mathe
m aties except one on geo m etry Extracts fro m this book m ad e by
P roclus indicate that P tole my did not regard the parallel a xiom of
The
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A HISTORY OF M ATHE M ATICS
48
Euclid as self evident and that P tole my was the first of the long line
of geom eters fro m ancient time down to our own who toiled in the vain
attempt to pro v e it The untenable part of his de monstratio n is the
assertion that in case of parallelism the su m of the interior angles on
one side of a transversal must be the sam e as their su m on the other
side of the transversal B efore P tolemy an attempt to improve the
theory of parallels was made by P os i do ni u s (firs t cent B C )
who de
fined parallel lines as lines that are coplanar and equidistant Fro m
an Arabic writer A l N i r i zi ( ninth cent )
it appears that Si mplicius
brought forward a proof of the 5th postulate based upon this def
inition and due to his friend Ag ani s
In the m aking of maps of the earth s surface and of the celestial
sphere P tolemy ( following Hipparchus)
used stereograp hic projection
The eye is im agined to be at one of the poles the projection being
thrown upon the equatorial plane He devised an instrum ent a form
of astrolabe plani sphere whi ch is a stereographi c proj ection of the
celestial sphere 2 P tole my wrote a m onograph on the an al emrn a which
was a figure invol v ing orthographic proj ections of the celestial sphere
upon three m utually perpendi cular planes ( the horiz ontal meridian
and vertical circles)The anale mm a was used in deter mining positions
of the sun the rising and setting of the stars The procedure was
probably known to Hipparchus and the older astronom ers It fur
nished a graphic m ethod for the solution of spherical triangles and was
used subsequently by the Hindus the Arabs and Europeans as late
3
as the seventeenth century
Two prominent mathematicians of this tim e were Ni comachu s and
Their favorite study was theory of n umbers
Theon _
of S m yrna
The investigations in this science cul m inated later in the algebra of
Diophantus B ut no important geometer appeared after P tole my
for 1 50 years An occupant of this long gap was S e xtu s J u li u s
Afric an u s who wrote an u nM por t an t work on geo metry appli ed
to the art of war entitled C estes Another was the sceptic S extu s
“
he endeavored to elucidate Zeno s Arrow
Empi ri cu s ( 2 00 A
by stating another argument equally p aradoxical and therefore far
fro m illu minating : M en never die for if a man die it must either
be at a t ime when he is alive or at a time when he is not aliv e ;
hence he never dies Sextus Empiri cu s ad vanced also the paradox
that when a line rotating in a plane about one of its ends describes
a circle w ith each of its points these concentric circles are of u m
equal area yet each circle must be equal to the neighbouring circle
1
which it touches
l aw C hi g o
R B o ol
N n E cli dean G om t y t an s by H S C
9 2
—
—
R
B
l
w
i R om e
b
8
t
a
8
( 75 9 )
p of
pp 3
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1
n
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a,
er
o
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o
u
-
ono
e r
e
1
1
11
r
”
Am M
,
as
r
.
.
essor
n
.
a rs
,
ca
1
,
1
.
T h e A s t r ol a b e ,
24, 1 91 7, p
162
a th M on thl y, V ol
S e e M L a t ham ,
3
S ee A v B r au n mii hl , Geschi chte der Tr i gon ometr i e, L e ip ig , I , 1 90 0 , p 1 1
A l exan d er von B rau n m ii hl ( 1 8 5 3 1 90 8 w a s pro fe ssor a t t he t e c hn i ca l h ig h sc hoo
i n M u n i ch
“
2
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— )
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l
HISTORY
A
59
O F M AT H E M ATIC S
cylinder and imagine a cone of revolution having for its axis the side
of the cylinder passing through the initial point of the spiral then
thi s cone cuts the cylinder in a curve of double curvature The per
p en di cu l ar s to the a xis drawn through every point in this curve form
the surface of a screw which P appus here calls the pl ectoi dal s u rf ace
A plane passed t hrough one of the perpendiculars at any convenient
angle cuts that surface in a curve whose orthogonal proj ection upon
the plane of the spiral is the required qu adr atr i x P appus considers
curves of double curvature still further He produces a spher i cal
s pi r al by a point m oving unifor mly along the circu mference of a
great circle of a sphere while the great circle itself revolves un i formly
around its diameter He then finds the area of that portion of the
“
surface of the S phere determ ined by the spherical spiral a co mplana
tion which claim s the m ore lively admiration if we consider that
although the entire surface of the S phere was known since Archi m edes
tim e to m easure portions thereof such as spherical triangles was
1
then and for a long time after wards an unsol v ed proble m
A
q uestion whi ch was brought into pro m inence by D escartes and Newton
“
is the proble m of P appus
Given several straight lines in a plane
to find the locus of a point such that when perpendiculars (or more
generally straight line s at given angles)
are drawn fro m it to the
giv en lines the product of certain ones of them shall be in a given
ratio to the product of the re maining ones I t is worth noticing that
it was P appus who first found the focus of the parabola and pro
pounded the theory of the involution of points He used the directrix
and was the first to put in definite form the definition of the conic
sections as loci of those points whose distances fro m a fixed point
and from a fix ed line are in a constant ratio He solved the problem
to draw through three points lying in the same straight line three
straight lines which shall form a triangle inscribed in a given circle
From the M athemati cal C oll ecti on s m any more equally d ifli c ul t the
orem s m ight be quoted which are original with P appus as far as we
know It ought to be re marked however that he has been charged
in three instances with copying theore ms without giving due cre dit
and that he m ay have done the same thi ng in other cases in whi ch
2
we have no data by which to ascertain the real discoverer
About the tim e of P appus liv ed T h e o n of Alexandria He brought
out an e dition of Eucli d s El emen ts with notes which he probably
used as a text book in his classes His comm entary on the A l magest
is valuable for the m any historical notices and especially for the
speci mens of Greek arithm etic which it contains Theon s daughter
Hypati a a wo man celebrated for her beauty and m odesty was the
last Alexandrian teacher of reputation and is said to have been an
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M C an tor
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p
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ci t
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V ol I , 3 Au fl
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1 90 7 ,
,
2
Fo r a d e fe n ce o f P a pp u s a g a i n s t t h e s e
M ath S oc , V ol 2 3 , 1 9 1 6 , pp 1 3 1 —
1 33
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p 45 1
c ha rg e s ,
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se e
J H
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Weav er
i n B ul l A m
.
.
GREEK GEO M ETRY
5:
abler philosopher and m athematician than her father Her notes on
the works of D iophantus and Apollonius hav e been lost Her tragic
death in 4 1 5 A D is vividly described in Kingsley s H ypati a
From n ow on m athematics ceased to be cultivated in Ale xandria
The lea d ing subj ect of m en s thoughts was Christian theology
P aganism disappeared and with it pagan learning The Neo P latonic
school at Athens struggled on a century longer P roclus Isidorus and
“
P rocl u s
others kept up the gold en chain of P latonic succession
the successor O f S yr ian u s at the Athenian school wrote a co mm entary
on Euclid s El emen ts We possess only that on the first book which
is valuable for the information it contains on the history of geom et ry
D ama s ci u s of D a m ascus the pupil of Isidorus is now believ ed to be
the author of the fifteenth book of Euclid Another pupil of Isidor us
was Eu toci u s of Ascalon the co mm entator of Apollonius and Archi
m edes
S i mplici u s wrote a co mmentary on Aristotle s D e Cce l o
“
Simplicius reports Zeno as saying : That which being added to
another does not m ake it greater and being taken away fro m another
does not m ake it less is nothing
According to this the denial of
the existence of the infinitesimal goes back to Zeno This m om entous
question presented itself centuries later to Leibni z who gave d iff erent
answers The report ma d e by Simplicius of the quadratures of Anti
phon and Hippocrates of Chios is one of the best sources of historical
1
information on this point In the year 5 2 9 Justinian disapproving
heathen learni ng finally closed by i mperial edict the schools at
Athens
As a rule the geometers of the last 50 0 years showed a lack of
creative power They were commentators rather than discoverers
The principal characteristics of ancient geom etry are
A wonderful clearness and defi ni t en ess of its concepts and an
(1)
a l most perfect logical rigor of its conclusions
A co mplete want of general principles and m ethods Ancient
(2 )
geometry is decidedly speci al Thus the Greeks possessed no general
The determination of t he tangents
method of drawing tangents
to the three conic sections did not furnish any rational assistance for
drawing the tangent to any other new curve such as the conchoid
the cissoid etc
In the demonstration of a theorem there were for
the ancient geom eters as many different cases requiring separate
proof as there were diff erent positions for the lines The greatest
geometers considered it necessary t o treat all possible cases inde
pendently of each other and to prove each with equal fulness To
devi se m ethods by whi ch the various cases could all be disposed of
“
by one stroke was beyond the power of the ancients
If we c om
pare a mathem atical problem with a huge rock into the interior of
wh ich we desire to penetrate then the work of the Greek mathe
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S ee F R u dio i n B i bl i otheca ma themati ca , 3 S
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V ol 3 ,
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1 90 2 ,
pp 7 6 2
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5
A HISTORY OF M ATHE M ATICS
2
ma ti ci an s
appears to us like that of a vigorous stonecutter who wi th
chisel and ha mmer begins with indefatigable perseverance from with
out to cru mble the rock slowly into fragm ents ; the m odern m athe
m ati ci an appears like an excellent m iner who first bores through the
rock som e few passages from which he then bursts it into pieces
1
wi th one powerful blast and brings to light the treasures withi n
,
,
,
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”
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,
Greek A r i thmeti c
and
Al gebra
Greek mathematicians were in the habit of discrimina ti ng between
the s ci en ce of nu mbers and the art of calcul ation The former they
called ar i thmeti ca the latter l og i sti ca The drawing of this distinction
between the two was very natural and proper The di ff erence b e
tween the m is as m arked as that between theory an d practice Among
the Sophists the art of calculation was a favorite study P lato on
the other hand gave considerable attention to phil osop hi cal arith
metic but pronounced calculation a vulgar and c hildish art
In sketching the history of Greek calculation we shall first give a
brief account of the Greek mode of counting and O f writing n um bers
L ike the E g yptians and Eastern nations the earliest Greeks counted
In case of large num bers the pebbles
o n their fingers or with pebbles
were probably arranged in parallel vertical lines P ebbles on the
first line represented units those on the second tens those on the
thi rd hundreds and so on Later frames cam e into use in which
strings or wires took the place of lines According to tradition
P ythagoras who travelled in Egyp t and perhaps in India first
introduced this valuable instrument into Greece The abacu s as it
is called e xisted a mong di ff erent p eoples and at di fferent times in
various stages of perfection An abacus is still e mployed by the
C hi nese under the nam e of S w an pan We possess no specific inform a
tion as to how the Greek abacus looked or how it was used Boethius
says that the P ythagoreans used with the abacus certain nine signs
“
called api ces whi ch resembled in form the nine Arabic n umerals
B ut the correctness of thi s assertion is subject to grave doubts
The oldest Grecian num erical sym bols were the so called H er odi ani c
s i gn s ( after H ero di an u s a Byz antine grammarian of about 2 00 A D
who describes them) These signs occur frequently in Athenian i n
For
s c r ip t i o n s and are on that account now generally called A tti c
som e unknown reason these sym bols were afterwards replaced by the
a l pha beti c n u mer al s in which the letters of the Greek alphabet were
used together with three strange and antique letters I 9 and TD
This change was d ecide dly for the worse for the
a n d the sy mbol M
old Attic numerals were less burdensome on the memory inasm uch
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H H a n k el , Di e E n twi ck el u n g der M athemati k i n den l etzten J a hr hund er ten
T ub i ng en , 1 8 84 , p 1 6
1
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.
.
.
GREEK ARITH M ETI C AND ALGEB RA
53
as they contained fewer symbols and were better adapted to show
forth analogies in nu m erical operations The following table shows
the Greek alphabetic numerals and their respective values
.
fl
a
2
I
r
3
3
4
'
0
p
1 00
s
5
6
3 00
)
1
8
7
v
42
400
500
7
2 00
Z
e
9
t
K
it
pt
v
9
10
20
30
40
50
x
6 00
T?)
p
1»
700
8 00
1
900
,
f
0
71
Q
60
70
80
90
c
“8
o.
1 000
,
y
etc
.
3 000
2 000
It will be noticed that at 1 00 0 the alphabet is begun over again
but to prevent confusion a stroke is now placed before the lett er
and generally so mewhat below it A horiz ontal line drawn over a
number served to distinguish it m ore rea dil y from words The c o
eff icient for M was sometim es placed before or behind instead of o v er
the M Thus
was written 8M yxon It is to be observed that
the Greeks had no z ero
Fractions were denoted by first writing the num era tor m arked with
an accent then the deno minator marked with two accents an d written
twice Thus y d K 9 = 44 In case of fractions having unity for
'
the nu merator the a was omitted and the denomi nator was written
1
only once Thus
"
21
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”
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,
u
’
n
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,
.
The Greeks had the nam e
e pi
m
Archytas
1
2
proved the theorem that
if
term s
Thi s theorem is found later in the
then
an
for the ratio
mori on
ep i
or i on
[
is reduced to its lowest
m usical
writings of Euclid and of the Rom an B oethius The Euclidean for m
of arithmetic without perhaps the representation of nu mbers by lines
existed as early as the tim e of Archytas 1
Greek writers seldo m refer to calculation with alphabetic nu m erals
Addition subtraction and even m ultiplic ation were probably per
form ed on the abacus E xpert mathe m aticians m ay have used the
symbols Thus Eu t oc i u s a co mm entator of the sixth century after
Christ gives a grea t many multiplications of wh i ch the following i s
2
a specimen :
.
,
,
.
.
,
,
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.
,
-
,
P Ta n n e ry i n B i bl i otheca ma thema ti ca , 3 S
2
J G o w , op c i t , p 5 0
1
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,
V ol V I ,
.
1 90 5 ,
p
.
2 28
.
A
54
HISTORY
M ATHE M ATICS
OF
The operation is e xplained suf
fi c i en t l y by the modern nu merals
appended
In case of mixed
nu mbers , the process was still
more clum sy D ivisions are found
1 0 00
1 2 000
MM 8
400 0 0
in Theon of Alexandria s com
1 2 000
M 53 9
m entary on the A l mag es t As
3 00
3 6 00
m
xpected the process is
ight
be
e
00
2
1 000
5
3
long and tedi ous
We
have
seen
in
geo
m
etry
that
70 2 25
the more advanced m athematicians
frequently had occasion to extract the square root Thus Arc hi m edes
in his M en su r ati on of the Ci r cl e gives a large nu mber of square roots
1—
3—
2 6 5
5—
1
He states for instance that
but
he
gives
no
and
1 5 3
clue t o t lie m ethod by which he obtained these approxim ations I t
is not improbable that the earlier Greek m athematicians found the
square root by trial only E u toci u s says that the m ethod of extracting
it was given by Heron P appus Theon and other co mm entators on
the A l magest Th eon s is the only one of these m ethods known to us
It is the sam e as the one used nowadays except that sexagesimal
fractions are e mployed in place of our decimals What the m ode of
procedure actually was when sexagesimal fractions were not used has
been the subj ect of conj ecture on the part of nu merous m odern writers
Of interest in connection with arithm etical symbolism is the S and
C ou n ter (Arenarius )an essay addressed by Ar chi m e d e s to G elon
ki ng of Syracuse In it Archi m edes sho w s that people are in error who
think the sand cannot be counted or that if i t can be counted the
n umber cannot be e xpressed by arithm etical symbols He shows that
the number of grains in a heap of sand not only as large as the whole
earth but as large as the entire universe can be arithm etically ex
pressed Assum ing that
grains of sand suffice to make a little
solid of the magnitude of a pop y seed and that the diameter of a
p
1
poppy seed be not smaller than 4 part of a finger s breadth assum ing
further that the diameter of the un i verse (supposed to extend to the
sun)
be less than
diam eters of the earth and that the latter
be less than
stadia Archi medes finds a num ber which would
e xceed the number of grains of sand in the sphere O f the universe
He goes on even further S upposing the universe to reach out to the
fix e d stars he finds that the S phere having the distance fro m the
earth s centre to the fixed stars for its radi us would contain a num ber
of grains of san d less than 1 00 0 myriads of the eighth octad In our
notation this number would be 1 0 6 3 or 1 with 6 3 ciphers after it It
can hardly be doubte d that one object which Archi medes had in vie w
in making this calculation was the i mprov em ent of the Greek sym
b ol i sm It is not known whether he inve nted so m e short notation by
which to represent the abo v e num ber or not
.
”
,,
a
,
7
a
K6
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a K 6
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GREEK ARITH M ETI C AND ALGEBRA
5
We j udge fro m fragments in the second book of P appus that Apol
l on i u s proposed an i mprove m ent in the Greek m etho d of writing
numbers but its nature we do not know Thus we see that the Greeks
never posses sed the boon O f a clear co mprehensive symbolism The
honor of giving such to the world was reserved by the irony of fate
for a nam eless Indian of an unknown ti m e and we know not who m to
thank for an invention of such i mportance to the general progress of
intelligence 1
P assing fro m t he subj ect of l ogi s ti ca to that of ari thmeti ca our a t
tention is first drawn to the science O f numbers of P yt h ag or as B efore
foun ding his school P ythagoras stu die d for m any years under the
Egyptian priests and fa miliarise d him self with Egyp tian mathe matics
a n d m ysticis m
If he ever was in Babylon as so me authorities claim
he may have learned the sexagesimal notation in use there ; he may
have picked up considerable knowledge on the theory of proportion
a n d m ay have found a large nu mber of interesting astrono m ical
observations Saturated with that speculativ e spirit then pervad ing
the Greek m ind he endeavored to d iscover so me principle of ho m o
B efore hi m the philosophers of the Ionic
g e n ei t y in the univ erse
school had sought it i n the m atter of thi ngs ; P ythagoras looke d for
it in the structure of things He O bserve d var i ous nu m erical relations
or analogies betwee n num bers and the pheno m ena of the universe
B eing con v inced that i t was in nu m bers and their relations that he
was to fi n d the foundation to true philosophy he procee d ed to trace
the origin of all things to nu mber s Thus he observed that m usical
strings of equal length stretched by weights having the proportion of
pro
uced
intervals
hi
c h w e r e an octave a fifth and a fourth
d
w
444
Harm ony therefore depe n ds on m usical proportion ; it is nothing but
a m yster i ous num eri cal relation Where harm ony is there are
Hence the or der and beauty of the univ erse have their
n u mbers
origin i n n umbers There are sev en intervals in the m usical scale
and also seven planets cross i ng the heavens The sam e num erical
relations wh i ch underlie the former m ust underlie the latter B ut
where nu mbers are there is harm ony Hence his spiritual ear dis
cerne d in the planetary m otions a won derful harm ony of the spheres
The P ythagoreans inv este d particular nu mbers with ex traor dinary
attributes Thus on e is the essence of things ; it is an absolute nu mber ;
hence the origin of all nu mbers and so of all things Fou r is the m ost
erf ec t nu mber and was in so m e m ystic way conceived to correspond
to the hum an soul P hi lolaus bel i e v ed that 5 is the cause of color 6 O f
2
i
of
m
ind
and
health
and
l
ght
8
of
love
and
frien
d
ship
In
cold 7
P lato s works are evi d ences of a s i m i lar bel i ef i n rel i gious relations of
numbers E v en Aristotle referre d the virtues to nu mbers
Enough has been said about these m ystic speculations to show
what l i vely interest in m athe m ati cs they m ust have created and
.
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1
J
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G ow ,
p
o
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ci t
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,
p 6
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2
J
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G ow ,
p
o
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ci t
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,
p 69
.
.
A H ISTORY O F M ATHE M ATICS
6
m aintained
them which
time
.
Avenues of m athe matical inquiry were opened up by
otherwise would probably have remained closed at that
.
The P ythagoreans classified numbers into odd and even They
observed that the sum of the series of odd nu mbers from 1 to 2 n + 1
was always a complete square and that by ad dition of the even n u m
bers arises the series 2 6 1 2 2 0 in which every number can be de
composed into two factors diff ering from each oth er by unity Thus
These latter nu mbers were considered of
6 = 2 3 1 2 = 3 4 etc
suffi cient importance to receive the separate name of heteromeci c ( not
.
,
,
,
,
,
.
.
,
,
.
”
)
Z
.
(
equilateral ) Nu mbers of the form
.
H
,
-
were call ed
tr i an gu l ar ,
because they could always be arranged thus
Num bers which
were equal to the su m of all their possible factors such as 6 2 8 4 96
were called perf ect; those exceeding that su m excessi ve; and those
whi ch were less def ecti ve A mi cabl e numbers were those o f w hi ch
each was the sum of the factors in the other M uch attention was
paid by the P ythagoreans to the subj ect of proportion The quan
tities a b c d were said to be in ar i thmeti cal proportion when a —b
c—
d; in g eometr i cal proportion when a :b = c :d; in har mon i c propor
—
—
tion when a b :b c = a zc It is probable that the P ythagoreans
,
,
,
,
,
,
'
.
,
.
.
,
,
,
,
.
,
were also familiar with the mu si cal proportion
—
a
az
+b
2a
b
m
z
:b
.
says that P ythagoras introduced it from Babylon
In connection with arithmetic P ythagoras made ex tensive i n yesti
m
i
n
s
into
geo
etry He believed that an arithmetical fact had
a
t
o
g
its analogue in geo m etry and vi ce ver sa In connection with hi s
theore m on the right triangle he dev ised a rule by which integral
numbers could be found such that the sum of the square s of two of
the m equalled the square of the third Thus take for one side an odd
I ambl i chu s
.
,
.
.
,
,
.
2 n+
(
then
number
+ 2 n + 1 )hypotenuse
1
=
,
+
2n2
2
2n
= t he
other side and
,
If 2 n + 1 9 then the other two nu mbers
are 40 and 4 1 B ut this rul e only applies to cases in which the hy
fers
fro
one
of
the
sides
by
In
the
study
of
the
right
e
s
f
m
1
o
t
n
u
e
di
p
triangle there doubtless arose q uestions of pu zzl ing subtlety Thus
given a number equal to the side of an isosceles right triangle to find
the number which the hypotenuse is equal to The side may have
been taken equal to 1 2 4 4 or any other nu mber yet in every i n
stance all e ff orts to find a number exactly equal to the hypotenuse
m ust have re mained fruitless The problem may have been attacked
again and again until finally so me rare genius to who m i t is granted
during som e happy mom ents to soar with eagle s flight above the
lev el of human thinking grasped the happy thought that this prob
(
2n2
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“
A HISTORY OF M ATHE M ATICS
58
Euclid devot es the seventh eighth and ninth books of his El emen ts
to arithm etic E xactly how m uch contained in these books is Euclid s
ow n invention and how m uch is borrowed fro m his predecessors we
have no means of knowing Without doub t m uch is original with
Euclid The s even th book begins with twenty one definitions All
except that for prim e num bers are known to have been given by
the P ythagoreans Nex t follows a process for fin di ng the G C D
The ei ghth book deals with numbers in con
O f two or m ore nu m bers
t i n u e d proportion and with the m utual relations of squares cubes
and plane numbers Thus XXII if three num bers are in cont i nued
proportion and the first I S a square so is the thi rd In the n i n th book
the sam e subject is continued I t contain s the proposition that the
nu mber of primes is greater t han any give n number
After the dea t h of Euclid the theory of num bers remained al most
stationary for 40 0 years Geo metry monopolised the attention of all
Greek mathematicians Only two are known to have done work in
—
invented
arithm etic worthy of m ention Er ato s th e n e s ( 2 7 5 1 94 B C )
“
a sieve for finding prime nu mbers All co mposite nu mbers are
“
sifted out in the following manner : Write down the odd numbers
fro m 3 up in succession By striking out every third nu mber after
the 3 we remove all m ultiples of 3 By striking out every fifth n u m
ber after the 5 we rem ove all m ultiples of 5 In this way by rejecting
m ultiples of 7 1 1 1 3 etc we have left prim e nu mbers only H yp
worked at the subj ects of polygonal
s icl e s ( between 2 0 0 and 1 00 B C )
nu mbers and arith m etical progressions which Euclid entirely neg
“
l e c t ed
In his work on risings of the stars he showed ( 1 )
that in
an arithm etical series of 2 n term s the s u m of the last n term s exceeds
the sum of the first n by a mul tiple of n 2 ; ( 2 )
that in such a series of
2 n + 1 ter ms the s u m of the series is the number of term s multiplied
that m such a series of 2 n term s the sum is
by the m i ddle term ; (3)
1
half the number of term s m ultiplied by the two m iddle term s
For two centuries after the tim e of H ypsi cl es arithm etic disappears
fro m hi story I t is brought to light again about 1 00 A D by Ni
c omach u s a N e o P ythagorean who inaugurated the final era of Greek
Fro m now on arithm etic was a favorite study while
m athem atics
geom etry was neglected N i com achu s wrote a work entitled I n
The great
tr odu cti o A r i thmeti ca which was very fa mous in its day
nu mber of co mmentators it has received vouch for its popularity
Boethius translated it into Latin Lucian could pay no higher c om
“
i
c om ac h u s of
l
i
e
t
to
a
calculator
than
t
s
You
reckon
like
N
m
h
i
n
:
p
Gerasa
The I n tr odu cti o A ri thmeti ca was the first exhaustive work
in whi ch arith met ic was treated quite independently of geom etry
Instead of drawing lines l i ke Eucl i d he illustrates things by real
numbers To be sure in h i s book the O l d ge om etrical nomenclature is
“
retained but the m etho d is in ductive instead of deductive
Its sole
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J G ow
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p
o
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ci t
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p 87
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.
GREEK ARITH M ETIC AND ALGEB RA
59
business is classification and all its classes are derived from and
exhibited by actual numbers
The work contains few results that
are really original We mention one important proposition whi ch is
probably the author s own He states that cubical num bers are al
ways equal to the su m of successive O dd numbers Thus
3
27
1 3 + 1 5 + 1 7 4 1 9 and so on
Thi s
3
theorem was used later for finding the s um of the cubical num ber
“
themselves Th e on of S myrna is the author of a treatise on t
The work is
mathe matical rules necessary for the study of P lato
ill arranged and of little m erit Of interest is the theore m that every
square number or that nu mber m inus 1 is divisible by 3 or 4 or both
A remarkable discovery is a proposition given by Iamb li ch u s in his
treatise on P ythagorean philosophy It is founded on the observation
that the P ythagoreans called 1 1 0 1 00 1 0 00 units of the first second
“
third fourth course respectively The theore m is this : If we add
any three consecutive nu mbers of whic h the highest is di v isible by3
then add the digits of that su m then again the di gits of that su m
and so on the final su m will be 6 Thus
63
1 86
15
This discove r y was the more re markable because the
1+ 5 = 6
ordinary Greek num erical symbolism was m uch less likely to suggest
“
any such property of numbers than our Arabic notation would
have been
Hippolytus who appears to hav e been bishop at P ortus Rom ae in
Italy i nthe early part of the third century m ust be m entione d for the
“
giving of proofs by casting out the 9 s and the 7 s
The works of N i comachu s Theon of S m yrna Thymaridas and
others contain at tim es in v estigations of subjects which are really
algebraic in their nature Thym ar i das in one place uses the Greek
word m eaning unknown q uantity in a way which woul d lead one
to believe that algebra was not far distant Of interest in trac i ng the
inventio n of algebra are the arithm etical epigram s in the P al ati ne
A n thol ogy which contain about fifty proble m s leading to linear equa
tions Before the intro d uction of algebra these problem s were pro
poun ded as pu zzles A riddle attributed to Eucl i d and containe d in
the A n thol ogy is to this e ffect : A m ule an d a d onkey were walking
along la den with corn The m ule says to the donkey If you gave
m e one m easure I should carry twice as m uch as you If I gave you
one we S hould both carry equal burdens Tell m e their burdens O
m ost learned m aster of geo m etry 1
I t will be allowed says G ow that this problem if authentic was
not beyond Euclid an d the appeal to geom etry s m acks of antiquity
A far more diffi cult pu zzle was the famous cattle proble m which
Archimedes propoun de d to the Alexandrian m athe mati c ia ns The
proble m is in determ inate for fro m only sev en equations eight u m
known quantities in integral numbers are to be found I t may be
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J Gow
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p
o
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ci t
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p 99
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.
A HISTORY O F M ATH E M ATICS
60
stated thus : The sun had a herd of bulls and cows of different colors
Of B ulls the white ( W)
were in number
of the blue (B )
( 1)
: the P
of
the Y and piebald (P )
and yello w ( Y )
: the B were ( 4—)
4 4
were
of the W and Y ( 2 )
Of Cows which had the sam e colors
.
,
(w
,
,
.
)
b, y, P
,
,
,
,
(B
6
4 2 1)
1
2
(P
é
2
-
Find the nu mber of bulls and cows This leads to high numbers
but to add to its complexity the con ditions are superadded that
W+ B = a square and P + Y = a triangular n umber leading to an i n
deter minate equation o i the second degree Another problem in the
“
A n thol ogy is quite fa miliar to school boys : Of four pipes one fills the
cistern in one day the next in two days the third in three days the
fou r th in four days : if all run together how soon will they fill the
cistern ?
A great m any O f these problem s pu zzling to an ari th
m eti ci an would hav e been solved easily by an algebraist They b e
cam e very popular about the tim e of D iophantus and doubtless acted
as a powerful stim ulus on his m ind
D io ph an tu s was one of the last and m ost fertile mathematicians of
the second Alexandrian school He flourished about 2 50 A D His
age was eighty four as is known from an epitaph to this e ff ect : D io
1
1
4
m
in
youth
and
of
his
life
1n childhood
ore
as
h
a
n
t
u
s
passed
p
6
,
7
a bachelor five years after his m arriage was born a son who died four
years before his father at half his father s age The place of nativity
and parentage of D iophantus are unknow n If his works were not
written in Greek no one would think for a m om ent that they were
the product of Greek m ind There is nothing in his works that
remin ds us of the classic period of Greek m athem atics His were al
In the circle of Greek
m ost entirely new ideas ou a new subject
Except for him
m athem aticians he stands alone in his specialty
we should be constrained to say that among the Greeks al gebra was
almost an unknown science
Of his works we have lost the P ori sms but possess a fragment of
P ol ygon al N u mber s and seven books of his great work on A ri thmeti ca
said to have been written in I 3 books Recent e ditions of the A ri th
meti ca were brought out by the in defatigable historians P Tannery
and T L Heath an d by G Wertheim
If we except the Ahm e s papyrus which contains the first sug g es
tions of algebraic notation and of the solution of equations then his
A r i thmeti c a i s the earliest treatise on algebra now ex tant In this work
is introduced the idea of an algebraic equation expressed in algebraic
symbols His treat m ent is purely analytical and Completely divorced
“
from geom etrical methods He states that a num ber to be sub
1
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J G ow
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p
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ci t
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p 99
.
.
GREEK ARITH M ETIC AND AL GEB RA
”
61
tracted m ultiplied by a nu mber to be subtracted gives a nu mber to
be added
This is applied to the multipli cation of differences such
as (x I )(x
I t must be remarked th at Diophantus had no
notion whatever of negative numbers standing by themselves All
he knew were diff erences such as ( 2 x
in which 2 x could not be
s mal ler than 1 0 wi thout leading to an absurdity He appears to be
2)
the first who could perform such operations as (x 1 )
without
X (x
2 = 2
reference to geom etry Such identities as ( a + b)
a + 2 ab + b2 which
with Euclid appear in the elevated ran k of geometric theorems are
wi th Diophantus the simplest consequences of the algebraic laws of
operation His S ign for subtraction was fl‘ for equality L For u n
known quantities he had only one symbol 9 He had no sign for
addition except j uxtaposition D iophantus used but few symbols
and sometimes ignored even these by d es cribing an operation in words
when the symbol would have answered just as well
In the solution of simultaneous equations D iophantus adroitly
managed with only one symb ol for the unknown quantities and ar
rived at answers m ost commonly by the method of ten tati ve assu mp
ti on whi ch consists in assigning to som e of the unknown quantities
prelimi nary values that satisfy only one or two of the conditions
These values lead to expressions palpably wrong but which generally
suggest som e stratagem by which values can be secured satisfying
all the conditions of the problem
Diophantus also solved determinate equations of the second degree
Such eq uations were solved geo metrically by Euclid and Hippocrates
Algebrai c solutions appear to have been found by Heron of Alexandria
who gives 8 4 as an approxim ate answer to the equation 1 44x ( 1 4 x)
6 7 2 0 In the Geometry doubtfully attributed to Heron the solution of
2
2
the equation 44x + 4x = 2 1 2 is practically stated in the form x
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“( 1 54 x 2 1 2 + 84 1 )2 9
D iophantus
.
nowhere goes through with the
whole process of solving quadratic equations ; he merely states the
“
result Thus 84x2 + 7x = 7 whence x is found
Fro m partial
explanations found here and there it appears that the quadratic equa
tion was so written that all terms were positive Hence from the point
of view of D iophantus there were three cases of equations with a
positive root : d x2 + bx = c ax2 = bx + c ax2 + c = bx each case requiring
a rule slightly diff erent from the other two Notice he gives only one
root His failure to observe that a quadratic equation has two roots
eve n when both roots are positive rather surprises us I t m ust be
remembered however that this sam e inability to perceive more than
one out of the several solutions to which a proble m may po i nt is c om
mon to all Greek m athe maticians
Another point to be observ ed
is that he never accepts as an answer a q uantity which is negative
or i rration al
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A HISTORY OF M ATHE M ATICS
2
D i ophantus d e v otes only the first book of his A r i thmeti ca to the
solution of d eterminate equations The re maining books extant
\ t reat m ainly of i ndeter mi n ate qu adrati c equ ati on s of the for m A x2 +
B x + C = y2 or of two s im ultaneous equations of the sa m e for m
He
considers several but not all the possible cases which may arise in
these equations The opinion of N essel man n o n the m ethod of Dio
Indeterm inate equations
phan tu s as stated by Gow is as follows : ( 1 )
of the secon d degree are treated completely only when the quadratic
or the absolute term is wanting : his solution of the equations A x2
2 —
2 is in m any respects cra mped
2
=
=
n
A
x
B
x
For
C y a d
+C y
1
( 2)
the double equation of the second degree he has a definite rule only
when the qua dratic ter m is wanting i n both expressions : ev en the n
his solution is not general M ore co mplicated expressions occur only
under specially favourable circu mstances
Thus he solves B x + C 2
.
.
\
,
.
.
,
,
-
.
’
‘
”
.
.
=
y
+C 1
l
2
2
,
=
The extraordinaryability of Diophantus l ies rather i n another di
rection namely in his won derful ingenu i ty to reduce all sorts of
equations to particular form s which h e knows how to solve Very
great is the variety of proble m s considered The I 30 prob l em s found
in the great work of D iophantus contain over 50 different classes of
proble m s which are strung together without any attempt at classi
B ut still more m ult i farious than the problem s are the solu
fi ca ti o n
tions General m ethods are al most unknown to D ip ohan tu s Each
problem has its own distinct m ethod which is O ften useless for the
I t is therefore d iffi cult for a m odern
most close l y related proble m s
after stu dy i ng 1 0 0 Diophantine solutions to sol v e the I o r s t
This
state m ent d u e to Hankel is som ewhat overdrawn as is shown by
Heath 1
That which robs his work of m uch of its scientific value is the
fact that he always feels satisfied with one solution though his equa
tion may ad m it of an indefinite nu mber of values Another great
M o dern m athe maticians
defect is the absence of gene ral m ethods
such as L Euler J Lagrange K F Gauss had to begin the study of
in determinate analysis anew and receiv ed no direct aid from D io
In
spite
of
these
defects
u s in the for m u l ation of m ethods
h
a
n
t
p
we cannot fail to a dm ire the work for the won derful ingenuity ex
h ib i t e d therein in the solut i on O f particular equations
,
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1
T L H eat h, Di ophan tu s
.
.
f
o
Al exa ndr i a ,
2
Ed
.
,
C a mb r i dg e ,
1 910
-
,
pp 54 9 7
.
.
THE RO M ANS
Nowhere is the contrast between the Greek and Ro man m inds
shown forth more distinctly than in their attitude toward the mathe
m a ti cal science
The sway of the Greek was a flowering ti me for
m athe m atics but that of the Roman a period of sterility In philos
B ut in mathe
ophy poetry and art the Ro man was a n i m itator
m aties he did not e v en rise to the desire for i m itation The m athe
In him a
ma ti cal fruits of Greek genius lay before h im untasted
science which had no direct bearing on practical life could awake no
interest As a consequence not on l y the higher geo metry of Archi
m edes and Apollonius but e v en the El emen ts of Eucl i d were neglected
What little m athe matics the Romans possesse d did not co me altogether
fro m the Greeks but cam e in part from m ore ancient sources E xactly
where and how so m e of it originated is a m atter of doubt I t seem s
“
Ro m an notation as well as the early
m ost probable that the
practical geom etry of the Rom ans ca m e fro m the old Etruscans
who at the earliest period to which our knowledge of them ex tends
inhabited the district between the Arno and Tiber
Livy tells us that the Etruscans were in the habit of representing
the number of years elapsed by dri v ing yearly a nail into the sanc
t u a ry O f M iner v a and that the Ro m ans continued this practice
A
less pri mitive m ode of designating nu mbers presum ably of Etruscan
“
origin was a notation rese mbling the prese nt Ro man notation
This system is noteworthy from the fact that a principle is involved
in it which is rarely m e t with i n others namely the principle of sub
traction I f a letter be placed before another of greater value its
value is not to be added to but subtracted from that of the greater
In the designation of large nu mbers a horiz ontal bar place d ov er a
letter was m ade to increase its value one thousand fold In fractions
the Ro mans used the duo deci mal syste m
Of arithm etical calculations the Romans e mployed three di fferent
kin d s :
W
W
pare d or t e purpose
i n er s mb
wm a e aM
i he
p l fimm k
time of King Nu ma for he ad erecte d says P liny a statue O f the
double faced Janus of which the fingers in dicated 3 6 5 (3 5
the
number of days in a year M any other passages fro m Roman authors
point out the use of the fingers as aids to calculation In fact a fi n g er
symbolism of practically the sam e form was i n use not only in Rome
but also in Greece and throughout the East certainly as early as the
beginning of the Christian era and continued to be used in Europe
.
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1
M C an tor , op
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ci t
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V ol I , 3 A u fl
.
63
.
,
1 90 7 ,
p 5 26
.
.
A HISTORY OF M ATHE M ATICS
during the M iddle Ages We possess no knowledge as to where or when
it was invented The second mode of calculation by the abacus was
a su bject of elem entary instruction in Rom e P assages m oman
writers indicate that the kind of abacus m ost commonly used was
covered with dust an d then di v ided into colu mns by drawing straight
“
lin es Each colum n was supplie d with pebbles ( calculi whence cal
c u l ar e
and calculate )
which served for calculation
The Rom ans used also another kind of abacus consisting of a
m etallic plate hav ing grooves w ith m ovable buttons By its use all
integers between 1 and
as well as so m e fractions could be
1
represented In the two adjoining figures the lines represent grooves
.
.
,
,
.
.
”
”
,
.
,
.
,
.
i
c
w
c
ii c x i c x w i
T
the circles buttons The Ro man numerals indicat e the value of
each button in the correspon ding groove below the button in the
shorter groov e above ha v ing a fi v efold value Thus If
hence each button in th e long left—
hand groove when in use ; stands
for
and the button in the short upper groo v e stands for
The sam e holds for the other groov es labelled by Roman
num erals The eighth long groove from the left (havin g 5 buttons)
1
represents d uodecimal fractions each button indicating 44 while the
6
button above the dot m eans 4 4 In the ninth column the upper
1
1
Our
button represents 4 4 the m id dle 4 4 and two lower each
first figure represents the positions of the buttons before the operation
1
begi ns ; our second figure stands for the nu m ber 8 5 2 4 4 The eye
has here to dist i nguish the buttons in use and those left idle Those
counte d are one button abov e c
and three buttons below
c
one butto n abov e x
two buttons below I = 2 )four
1
buttons indicating duodecimals
and
the
button
for
;
3
1
1
Suppose now that
The
is
to
be
added
to
8
2
5
4
3
operator could begin with the hig est un its or the lowest units as he
pleased Naturally the hard est part is the addition of the fract i ons
an d
.
,
"
.
,
.
,
,
.
-
,
-
,
7
4
.
.
2 4
h
2 4
,
,
.
.
z
G Fri e dl ei n , Di e Z a hl ei chen u nd das el ementare Rechn en der Gr i echen a nd Romer ,
“
G o t t fr i ed Fr i edl ci n ( 1 8 2 8 1 8 7 5 w as R ek tor d er K g l
Erl an g en , 1 8 6 9 , Fi g 2 1
S t u di e n an stal t zu H ot i n B av ar i a
1
.
.
— )
”
.
.
.
A HISTOR Y OF M ATHE M ATICS
66
Ro me a science of geometry with definitions axio ms theore ms and
proofs arranged in logical ord er will be disappointed The only
geometry know n was a pr acti c al geom etry whi ch like the old Eg yp
tian consisted only of e mpirical rules This practical geo metry was
e mployed in surveying Treatises thereon have co me d own to us
co mpile d by the Ro man surveyors called agr i i n ensores or gr omati ci
One would naturally expect rules to be clearly form ulated B ut no ;
they are left to be abstracted by the reader fro m a m ass of nu merical
examples
The total impression is as though the Ro m an gromatic
were thousands of years older tha n Greek geom etry and as though
a deluge were lying between the two
So me of their rules were prob
ably inherited fro m the Etruscans but others are identical w ith those
of Heron Among the latter is that for finding the area of a triangle
fro m its sides and the approxi mate formula 44a2 for the area of
equilateral triangles (a being one of the sides ) B ut the latter area
was also calculated by the form ul as
and 4a2 the first of
which was unknown to Heron P robably the expression 4a? was de
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a
rived from the Egyptian form ula
+b
2
for the determ ination of
2
the surface of a quadrilateral This Egyptian formula was used by
the Ro mans for finding the area not only of rectangles but of any
quadrilaterals whate v er Indeed the gromatici considered it e v en
suffi ciently accurate to determ ine the areas of cities laid out i rreg u
1
s
ply
by
m
easuring
their
circu
m
ferences
Whatever Egyptian
l arl y im
geom etry the Ro mans possessed was transplanted across the M edi ter
r a n ea n at the ti m e of Ju l i u s C ws ar who ordered a survey of the whole
e mpire to secure an equitable m o d e of taxation C a sar also reform ed
the calendar and for that purpose dre w from Egyptian learnin g
He secured the services of the Alexan drian astronomer S osi gen es
Two Roman ph ilosophical writers deserve o u r attention The
in his D e r er u m
philosophical poet Ti tu s L u cr ei i u s (96 ? —
55 B
n atu r a entertains conceptions of a n infinite m ultitu d e and of an i n
finite m agnitude which accord with the m odern definitions of those
terms as being not variables but constants However the Lucretian
i nfi n i t e s are not com posed of abstract things but of m aterial particles
His infinite m ultitude is of the denu merable variety ; he m ade use
of the whole part property of infinite m ultitudes 2
Cognate topics are discussed several centuries later by the c ele
—
in
t
A
u
u
s
t
i
n
e
b ra t ed father of the Latin church S
( 3 54 4 3 0 A
g
hi s references to Zeno of Elea In a d i alogue on the question whether
or not the m ind of man m oves when the body moves a n d trav els with
the bo dy he is led to a definition of m otion in wh i ch he di splays so m e
le v ity I t has been said of scholast i c i sm that it has no sense of hum or
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H H an k el , op ci t , p 2 9 7
2
C J K eyse r i n B u l l A m Al
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al /
z
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S oc
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,
V ol
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24,
1918,
p
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2 68,
3 21
.
RO M ANS
TH E
67
Hardly does this apply to S t Augustine He says : When this dis
course was concluded a boy ca m e running from the house to call us
to dinner I the n remarked that this boy co mpels us not only to
define motion but to see it before our very eyes So let us go and
pass fro m this place to another ; for that is if I am not mistaken
nothing else than m otion
S t Augustine deserves the credit of
having accepted the e xistence of the actually infinite and to have
recogniz ed it as being not a variable but a constant He recognize d
all finite positive integers as an infinity of that type
On this point
he occupied a radically diff erent position than his forerunner the
Greek father of the church Origen of Alexandria Origen s argu ments
against the actually infinite have been pronounced by Georg Cantor
the profoundest ever advanced against the actually infinite
In the fifth century the Western Ro man E mpire was fast falling
to pieces Three great branches— Spain Gaul and the province of
Africa—broke off fro m the decaying trunk In 4 7 6 the Western
E mpire passed away and the Visigothic chief Odoacer beca m e king
Soon after I taly was conquered by the Ostrogoths under Theodor ic
It is re markable that this very period of political hu m iliation should
be the one during which Greek science was studied in Italy m ost
z ealously
School—
books began to be co mpiled fro m the elem ents of
Greek authors These compilations are very deficient but are of
absorbing inte rest fro m the fact that down to the twelfth century
they were the only sources of mathe matical knowledge in the Occident
Fore most among these writers is B o e thiu s (died
At first he
was a great favorite of King Theodoric but later being charged by
envious courtiers with treason he was imprisoned and at last decapi
t a t ed
While in prison he wrote On the C on sol ati on s of P hi l osophy As
a m athematician B oethius was a B robdingnagian a mong Roman
Scholars but a Liliputian by the si de of Greek m asters He wrote
an I n sti tu ti s A r i thmeti ca which is essentially a translation of the arith
m etic of N i c omac h u s and a Geometry in several books So m e of the
m ost beauti ful results of N i comachu s are o m itted in B oethius arith
m etic The first book on geo m etry is an extract fro m Euclid s El e
men ts which contains in addition to definitions postulates and
axioms the theore ms in the first three books without proofs How
can this omission of proofs be accounted for ? I t has been argued by
some that Boethius possessed an incomplete Greek copy of the E l e
men ts ; by others that he had Th e on s edition before hi m and b e
li ev ed that only the theore ms ca m e fro m Euclid while the proofs were
supplied by Theon The second book as also other books on geometry
attributed to Boethius teaches from numerical examples the men
su rati on of plane figures after the fashion of the agri m ensores
A celebrated portion in the geom etry of Boethius is that pertaining
to an abacus wh ich he attributes to the P ythagoreans A consi der
P ebbles
able improve m ent on the ol d abacus is there introduced
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A HISTORY OF M ATHE M ATICS
68
are discarded and api ces (probably s mall cones)
are used Upon each
of these apices is drawn a numeral giving it some value below 1 0
The nam es of these numerals are pure Arabic or nearly so but are
added apparently by a later hand The 0 is not m entioned by
Boethius in the text These numerals bear striking resemblance to
the Gubar numerals of the West Arabs which are ad mittedly of
Indian origin These facts have giv en rise to an endless controversy
Som e contended that P ythagoras was i n India and from there brought
the nine nu merals to Greece where the P ythagoreans used the m
secretly This hypothesis has been generally abandoned for it is
not certain that P ythagoras or any disciple of his e v er was in India
nor is there any e v idence in any Greek author that the apices were
known to the Greeks or that nu meral signs of any sort were used by
them with the abacus I t is improbable m oreover that the Indian
signs fro m which the apices are derived are so old as the tim e of
A second theory is that the Geometry attributed to
P ythagoras
Boethius is a forgery ; that it is not older than the tenth or possibly
the ninth century and that the apices are derived from the Arabs
B ut there is an Encyclop aedia written by C assi odori us (died about
in which both the arithm etic and geom etry of B oethius are m en
58 5)
t i on e d
Som e doubt exists as to the proper interpretation of this
passage in the Encyclop aedia At present the weight of evidence is
that the geom etry of B oethius or at least the part mentioning the
nu merals is spurious 1 A third theory (Woepck e s)is that the
Alexandrians either directly or indirectly obtained the nine nu merals
from the Hindus about the seco n d century A D and gave them to
the Ro mans on the one hand and to the Western Arabs on the other
Thi s explanation is the most plausible
It is worthy of note that C assi odori u s was the first writer to use
“
the term s rational and irrational in the sense n ow current in
arith metic and algebra 2
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t
A g ood di sc u ssi o n of t hi s so c all ed B oe hi us q uest io n , w hi ch ha s b een de
b at ed for t w o c en t u ri e s , i s g i v e n b y D E S mi t h a n d L C K a rp i n sk i i n t hei r H i nd u
A ra bi c N u mer al s , 1 9 1 1 , C hap V
2
E n cycl opédi e des s ci en ces ma themati q u es , To me I , V o l
A n il
p 2
’
l um in a t i n g a r i cl e o n an c i en fi n g e r symb o l i s m i s L J R i c ha rd so n s
Di g i ta l
R e c k o n i n g A mon g t he A n ci en s
i n the A m M ath M onthl y, Vol 2 3 , 1 8 1 6 ,
1
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’
pp
—
7 3
1
t
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T H E M AYA
The M aya of Central America and Southern M exico developed
hieroglyphic writing as found in inscriptions and codices dating ap
pa t ently fro m about the beginning of the Christian era that ranks
probably as the foremost intellectual achieve m ent of pre Colu mbian
M aya nu mber syste m s and chronology
times in the New World
are rem arkable for the extent of their early develop m ent P erhaps
five or s ix centuries before the Hindus gave a system atic e xposition
of the n deci mal nu mber system with its z ero and principle of local
value the M aya in the fl a tl an ds of Central A m erica had evolved
system atically a vi gesi mal number system e mploying a z ero and the
principle of local value In the M aya number syste m found in the
co dices the ratio of increase of successive units was not 1 0 as in the
Hindu system ; it was 2 0 in all positions e xcept the third That is
2 0 units of the lowest order ( ki n s or days )
m ake one un i t of the next
higher order (u i n al s or 2 0 days)1 8 u i n al s m ake one unit of the third
order (tu n or 3 6 0 days)2 0 tuns m ake one unit of the fourth order
(k atu n or 7 2 00 days )2 0 k at u n s m ake one unit of the fifth order
days )
and finally 2 0 cycles m ake 1 gr eat cycl e of
(cycl e or
days I n M aya codices we find symbols for 1 to 1 9 ex
pressed by bars and dots Each bar stands for 5 units each dot for
For insta nce
1 unit
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2
4
7
5
symbol
11
19
The z ero is represented by a
that looks roughly like a half
closed eye In writing 2 0 the principle of local value enters It is
expressed by a dot placed over the sym bol for z ero The nu mbers
are written vertically the lowest ord er being assigne d the lowest
position Accordingly 3 7 was expressed by the symbols for 1 7 ( three
bars and two dots )
in the kin place and one dot representing 2 0
placed above 1 7 in the uinal place T 0 write 3 6 0 the M aya dre w
two z eros one above the other with one dot higher up i n third place
The highest number found in the codices
( 1 x 1 8 x 2 0 +0 +0
is in our deci mal notation
A second numeral system is found on M aya inscriptions I t em
ploys the z ero but not the principle of local value Special sym bols
are e mployed to designate the diff erent units I t is as if we were to
1
write 2 0 3 as 2 hun dreds 0 tens 3 ones
F
n a
l g y see S G M l y
o n t o f t he M aya n m b e sy st m
d h
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”
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or a
cc
u
A n I n tr oduc ti on to the S tudy
Washi ng on , 1 9 1 5
t
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f
o
the
r-
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,
u
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e
s an
c
ro n o o
,
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or e
M aya Hi erogl i phs , G ove rn men t P ri n t i n g O ffi c e ,
M ATHE M ATICS
A HISTORY OF
70
The M aya had a sacred year of 2 6 0 days an offi cial year of 3 6 0
days a n d a solar year of 3 6 5 + days The fact that
see ms to account for the break in the vigesimal system m aking 1 8
u i n al s equal to 1 tun
The lowest common multiple
(instead of 2 0 )
“
of 2 6 0 and 3 6 5 or 1 8 980 was taken by the M aya as the calendar
“
round a period of 5 2 years which is the m ost important period in
M aya chronology
We m ay add here that the nu mber systems of Indian tribes in North
Am erica while disclosing no use of the z ero nor of the principle of
local value are of interest as exhibiting not only quinary d ecimal and
viges imal syste ms but also ternary quarternary and octonary sys
1
tem s
”
S
W C E ll
N mb
S y tem
f t h e N t h A me i
I di
i Am
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ee
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e
s,
u
er
s
s o
i ca n M a th M on thl y, V o l 2 0 , 1 9 1 3 , pp
—
l
1
1
2
ca
S
o
1
1
8
222
mati , 3 , V
3 , 9 3 , pp
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263
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—2 7 2
or
,
2 93
r can
—2 99 ;
a l so
n
an s
n
er
B ibl i othec a ma the
THE CHINESE
1
The oldest extant Chinese work of mathe matical interest is an
anonymous publication called Chou pei and written before the
second century A D perhaps long before In one of the dialogues the
Chou pei is believed to reveal the state of mathematics and astronomy
in China as early as 1 1 00 B C The P ythagorean theorem of the right
triangle appears to have been known at that early date
Next to the Chou —
pei in age is the Chi u chang S u a n—
Arith
shu
metic in Nine
commonly called the Chi u chang the most
celebrated Chinese Text on arith m etic Neither it s authorship nor
the time of its co mposition is known definitely By an edict of the
“
d espotic emperor Shih Hoang ti of the Ch in D ynasty all books were
burned and all scholars were buried i n the year 2 1 3 B C
After the
death of this e mperor learning revived again We are told that a
scholar nam ed CHA N G T S A N G found so me old writings u pon which
he based this famous treatise the Chiu chang About a century later
a revision of it was m ade by Ching Ch ou ch ang ; comm entaries on
this classic text were m ade by Liu Hui in 2 6 3 A D and by Li Ch un
“
fén g in the seventh century H ow much of the Arithm etic in Nine
Sections as it exists to day is due to the old records ante dating
21
3 B C how much to Chang T sang and h ow m uch to Ching Ch ou
ch ang it has not yet been found possible to determine
“
The Arith metic in Nine Sections begins with m ensuration ; it
gives the area of a triangle as 4 b h of a trapez oid as 4 (b
of a
1
?
circle variously as %c éd 4cd 4d and 4 4 c2 where c is the circumference
and d is the diam eter Here 7r is taken equal to 3 The area of a
2
segment of a circle is give n as
where c is the chord and a
+a )
the altitude Then follow fractions commercial arithmetic including
percentage and proportion partnership and square and cube root of
numbers Certain parts exhibit a partiality for unit fractions Divi
sion by a fraction is effected by inverting the fraction and m ultiplying
The rules of operation are usually stated in obscure language There
are given rules for finding the volu mes of the prism cylinder pyramid
truncated pyra mid and cone tetrahedron and we dge Then follow
m
roble
s in alligation There are indications of the use of positive
p
and negative nu mbers Of interest is the following problem because
centuries later it is found i n a work of the Hindu Brahmagupta :
Al l our i n fo mati on o C hi e e ma t hemati cs i s draw n from Y hi o M ik mi
Th D l pme t f M th m ti i Chi
d J apa L e ip z ig 9
a d f om D v i d
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1
eve o
e
n
E u g e n e S mi t h
1 91 4
n
r
.
o
an d
a
e
n s
a
cs
os
n
Yo shi o M ik ami
na an
’
s
n,
,
1
1 2,
n
a
r
’
s
a
Hi story of J apa n ese M athemati cs , C hi cag o ,
A
2
7
HISTORY or M ATHE M ATICS
There is a bamboo 1 0 ft high the upper end of which is broken and
reaches to the ground 3 ft from the stem What is the height of the
.
,
.
break ? In the solution
.
the
3
height of the break is taken
2
2
x 10
Here is another : A square town has a gate at th e mid point of each
side Twenty paces north of the north gate there is a tree which
is vi sible from a point reached by walking fro m the south gate 1 4
paces south and then 1 7 7 5 paces west Find the side of the square
x
The problem leads to the quadratic equation x2
10 x
+ 1 4)
The derivation and solution of thi s equation are not made
1 7 75 = 0
clear in the text There is an obscure statem ent to the eff ect tha t
the answer is obtained by evol ving the root of an expression which
is not monomial but has an additional term [ the term of the first
It has been surmised that the process here r e
ferred to was evolved more fully later and led to the method closely
resembling Horner s process of approxim ating to the roots and that
the process was carried out by the use of calculating boards Another
proble m leads to a quadratic equation the rule for the solution of
which fits the solution of li teral quadratic equations
“
—
We co me next to the S u n Tsu S u an chi ng ( Arithmetical Classic
of Sun
which belongs to the first century A D The author
“
S U N TS U says : In m aking calculations we m ust first know positions
of numbers Unity is vertical and ten hori zontal ; the hundred stands
whi le the thousand lies ; and the thousand and the ten look equally
and so also the ten thousand and the hundred
This is evidently a
reference to ab acal computation practiced from time i mmemorial in
Chi na and carried on by the use of computing rods These rods
made of s m al l bamboo or of wood were in Sun Tsu s tim e m uch longer
The later rods were about 1 4 inches long red and black in color
representing respectively positive and negative nu mbers According
to Sun Tsu units are represented by vertical rods tens by horiz ontal
rods hundreds by vertical and so on ; for 5 a single rod suffices The
n umbers 1 9 are represented by rods thus : I u inn
un
n 1, n
the n umbers i n the tens column 1 0 2 0
90 are wri tten thus
l l
|
The n umber 6 2 8 is designated
-
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z
2
,
m
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,
E
,
E
,
,
,
,
,
,
,
,
7
.
The
rods
were
placed
on
a
board
ruled
in
colu
ns
m
by
W W
and were rearranged as the computation advanced The successive
steps in the mul tiplication of 3 2 1 by 46 must have been about as
fol lows
,
.
.
1 38
46
321
1 4 7 66
46
32 1
147 2
46
32 1
.
The product was placed between the
The 46 is multiplied first by 3 then by
,
m ultiplicand
2,
and m ultiplier
and last by 1 the 46 being
.
,
A
74
or M ATHE M ATICS
HISTORY
In the first half of the seventh century WAN G H s IAo T U N G brought
for th a work the Ch i —
k u S u an —
chi n g in which nu merical cubic equa
tions appear for the first time in Chinese m athe matics This took
place seven or eight centuries after the first Chinese treatment of
quadratics Wang Hs iao t ung gives several problem s leading to
“
There is a right triangle the product of whose two S ides is
cu bi cs z
9
l
1
—
and
whose
hypotenuse
is
greater
than
the
first
side
by
0
0
6
3 44
7
5 0
It 18 requi red to k n ow the lengths of the three sides
He gives the
“
7—
l
being
answer 1 4 4 0 4 9 4 5 1 4 and the rule : The P roduct (P )
m ake the result
squared and being divided by twice the S urplus (S )
Halve the surplus and make it the li en f a
s hi h or th e constant class
or the second degree class And carry out the operation of evolution
according to the extraction of cube root The result gives the first
side Adding the surplus to it one gets the hyp otenuse Divide the
product with the first side and the quot i ent is the second side
Thi s
3
=0
The mode of solu
rule leads to the cubic equation x + S /2 x
tion is similar to the process of extracting cube roots but detail s of
the process are not revealed
In 1 2 4 7 C H I N C HI U —
S H A O wrote the S u s hu Chi u cha ng
Nine
Sections of M athe matics )
which makes a decided advance on the
solution of nu merical equations At first Ch i n Chiu—
shao led a m ili
tary lif e ; he lived at the tim e of the M ongo l ian invasion For ten
years stricken with disease he recovered and then devoted himself to
study The following problem led him to an equation of the tenth
degree : There is a circular castle of unknow n diameter having 4
gates Three miles north of the north gate is a tree which is visible
from a point 9 m iles east of the south gate The unknow n diameter
is found to be 9 He passes beyond Sun Tsu in his ability to solve
indeterminate equations arising for a number which will give the
r
residues r 1 r 2
when divided by m l m2
mm respectively
4
2—
—
Ch in C hi u shao solves the equation x + 7 6 3 2 00 x 40 6 4 2 56 0 00 0
However
0 by a process al m ost identical with Horner s method
the co mputations were very probably carried out on a computing
board divided into colu mns and by the use of computing rods
Hence the arrange ment of the work m ust have been different from
that of Ho m er B ut the operations perform ed were the sam e The
fir st digit in the root being 8 (8 hundreds )a transformation 15 ef
4 —
3—
2—
8 2 6 8 800 0 0 x + 3 8 2 0 544
fec t ed which yields x
3 2 00 3c
30 7 6 8 00 9c
the same equati on that i s obtained by Horner s process
0000 = 0
Then taking 4 as the second figure i n the root the abs olute term
Thus the C hinese had
v anishes in the operation giving the root 8 40
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t ion
Vol
of
t h e fr a c
t
1 9 14,
p
.
1 10 ,
c i rc l e w i t h
a
3 +4
2
1s
g i ven an on ymou sl y I n Gru n er t s A r chi v
P H ug h es g i v e s i n N a tu r e, V ol 93 ,
’
U si n g 34 , T M
me t hod o f con s t ru c t i ng a t ri ang l e t h at g i v es the
p
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—
5 5
3
—
io n
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g r ea t
5
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acc u rac y
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ar ea o f a
g i ve n
CHINESE
TH E
75
invented Horner s method of solving num erical equations more than
five centuries before R uffi n i and Ho m er Thi s solution of higher
numerical equations is given later in the writings of Li Y ek and others
Ch in Chiu—
shao marks an advance over Sun Tsu in the use of 0 as a
symbol for z ero M ost likely this symbol is an i mportation from
India P ositive and negative nu mbers were distinguished by the use
of red and black computing rods This author gives for the first time
a proble m which later became a favorite one a mong the Chinese ; i t
involved the trisection of a trapezoidal field under certain restrictions
in the mode of selection of boundaries
We have already m entioned a contemporary of Ch in Chiu —
shao
namely L I Y EH ; he lived far apart in a rival m onarchy and worked
“
—
independently He was the author of T sé yu an H ai chi ng ( Sea
M easurem ents
M irror of the Circle—
1 2 4 8 and of the I 4 m Y en m
an
He used the symbol 0 for z ero On account of the i n c onv en
I 2 59
i en ce of writing and printing positive and negative nu mbers in dif
fe ren t colors he designated negative nu mbers by drawing a cancella
tion m ark across the symbol Thus
stood for 6 0
stood for
—
The unknown quantity was represented by unity which was
60
probably represented on the counting board by a rod easily disti n
The term s of an equation were written
g u i she d fro m the other rods
not i n a horiz ontal but in a vertical line In Li Yeh s work of 1 2 59
as also in the work of Ch in Chiu shao the absolute term is put in the
top line ; in Li Yeh s work of 1 2 4 8 the order of the term s is reversed
so that the absolute term is in the botto m line and the highest power
of the unknown in the top line In the thirteenth century C hi nese
algebra reached a m uch hi gher develop m ent than form erly This
science with its rem arkable m ethod ( our Horner s )
of solving nu mer
“
ical equations was designated by the Chi nese the celestial elem ent
m ethod
A third prom inent thirteenth century m athe m atician was Y A N G
They deal with the
H U I of who m several books are still extant
summation of arithmetical progressions of the series I +3 + 6
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(
1
+2
+ 11
—
6
)
,
,
2 =
1
1
+
I2
§ n (n
also with proportion s imultaneous linear equations quadratic and
quartic equations
Half a century later Chinese algebra reached its height in the
“
—
treatise S u c m Izsi ao Chi mén g ( Introduction to M athe matical
S tudies
1 2 99 and the S zu yu en Vii chi en ( The P recious M irror
of the Four Ele ments
which ca me from the pen of CH U
1 30 3
S HIH CHI EH The first work contains no new results but exerted a
great stimulus on Japanese m athematics in the seventeenth century
At one time the book was lost in China but i n 1 839 it was restored
by the discov ery of a copy of a Korean reprint m ade in 1 66 0 The
is a m ore original work It treats fully of the
P recious M irror
celestial element method
He gives as an ancient method a
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A HISTORY OF M ATHE M ATICS
76
triangle (known in th e West as P ascal s arithmetical tria ng le )dis
playi ng the binomial c oeffi c i en ts which were known to the Arabs in
the eleventh century and were probably imported into China Chu
shi h Chi eh s algebraic notation was altogether diff erent from our
modern notation Thus a + 6 + 0 + d was written
’
,
,
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’
-
.
,
as shown on the left excep t that in the central position we e mploy
an asterisk in place of the Chi nese character t ai ( great extrem e ab
solute term)
and that we use the modern nu m erals in place of the
The square of a + 6 + 6 +d namely a 2 + 62 + c 2 + d2 + 2 ab
s angi for m s
is represented as shown on the right
In further illustration of the Chinese notation at the t im e of Chu
Shih Chieh we give 1
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-
2
I
=
In the fourteenth century astrono my and the calendar were studied
They involved the rudi ments of geo m etry and spheric al trigonometry
In thi s field i mportations from the Arabs are disclosed
After the noteworthy achieve ments of the thirteenth century
Chinese mathematics for several centuries was in a period of decline
“
The famous celestial elem ent m ethod in the solution of higher
equations was abandoned and forgotten M ention must be m ade
however of C H IEN G TA I WE I who in I 593 issued his S u an f a T u ng
“
which is the ol dest
tsu ng ( A Syste matised Treatise on
work now extant that contains a diagram of the form of the abacus
called m an pan and the explanation of its use The instrument was
known in China in the twelfth century Rese mbl i ng the old Roman
abacus it contained balls m ovable along rods held by a wooden
“
frame The suan pan replaced the old computing rods The Sys
is famous also for containing som e
t ema ti sed Treatise on Arith m etic
m agic squares and m agic circles Little is know n of the early history
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)
111 t he symb ol for xz n o t i c e t h a t t he
I
s on e sp a c e dow n ( x
an d o n e
I n t h e symb ol
a n d i s m ad e t o s t a n d for t he p rod u c t xz
spa ce t o t h e r i g h (2 o f
’
“
“
for zyz t he t h ree o s i n di ca t e t h e a b se n c e o f t he t erm s y, x, xy; t he s mall
2
mean s t w i ce t he p ro d u c t o f t h e t w o l e t t e r s i n t he same ro w , resp ec t i vel y o n e spac e
to t he ri g h an d to t he l e f o f
i e , 2 yz Th e li mi t a ti on s o f hi s n o t a t i on a re o b
1
”
t
v 10 u s
.
t )
.
t
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t
”
CHINESE
TH E
7
squares M yth tells us that in early ti mes the sage Y u
the enlightened emperor saw on the cala mitous Yellow River a divine
tortoise whose back was decorated with the figure made up of the
numbers from I to 9 arranged in form of a magic square or l o shu
of
magic
.
,
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,
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-
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,
The l o-s hu
.
The nu merals are indicated by knots in strings : black knots repre
sent even numbers (symbolizing i mperfection )white knots r epre
sent odd nu mbers (perfection )
Christian missionaries entered China in the six teenth century
The Italian Jesuit M a tteo Ri cci ( 1 5 5 2 —
1 6 1 0)
introduced European
astronomy and mathematics With the aid of a Chinese scholar
named H sit he brought out in 1 6 0 7 a translation of the first six books
of Euclid Soon after followed a sequel to Euclid and a treatise on
surveying The missionary M u N i k o sometim e before 1 6 60 intro
In 1 7 1 3 Adrian Vl ack s logarithmic tables to 1 1
duced logarithm s
1
places were reprinted Ferdi n and Verbi es t of West Flanders a
noted J esuit missionary and astronomer was in 1 66 9 made vice
president of the Chinese astrono mical board and in 1 6 73 its president
European algebra found its way into China M ei K u —
ch én g noticed
that the European algebra was essentially of the same principles as
“
the Chinese celestial elem ent method of form er days which had
been forgotten Through him there cam e a revival of their own
algebraic method W ithout however displacing European science
Later Chinese studies touched m ainly three subjects : The determina
tion of 71 by geom etry and by infinite series the solution of n um erical
equations and the theory of logarithm s
We shall see later that Chinese m athematics stimulated the growth
of mathematics in Japan and India We have seen that in a small
way there was a taking as well as a giving B efore the influx of
recent European science China was influenced som ewhat by H i ndu
and Arabic m athematics The Chinese achievements which stand
out most conspicuously are the solution of nu merical equations and
the origination of magic squares and magic circles
Extrac t from Revu e
C n sult H B osman s Fe di nd V bi st L ouv ai n 1 9 1 2
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o
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uesti ons sci enti
des Q
r
,
fi
q ues ,
J
na
—
Ap
an u ary
er
e
,
n l , 191 2 .
,
.
JAP ANESE
TH E
1
According to tradition there existed in Japan in rem ote ti mes a
system of num eration which extended to high powers of ten and re
sembled som ewhat the sand counter of Archim edes About 5 5 2 A D
B uddhis m was introduced into Japan This n ew m ove ment was
fostered by P rince Shotoku Taishi who was deeply interested in all
learning M athematics engaged his attention to such a degree that
he came to be called the father of Japanese m athem atics A little
later the Chinese syste m of weights and measures was adopted In
7 0 1 a university syste m was established in which m athe matics figured
prom inently Chinese science was i mported special m ention being
m ade in the offi cial Japanese records of nine Chinese texts on m athe
mati cs which include the Chou pei the S u an chi ng written by Sun
Tsu and the great arithm etical work the C hi u chang B ut this eighth
century interest in mathem atics was of sho rt duration ; the Chiu—
chang
was forgotten and the dark ages returned Calendar reckoning and
the rudiments of co mputation are the only signs of m athematical
activity until about the seventeenth century of our era On account
of the crude n umeral syste ms devoid of the principal of local value
and of a sym bol for z ero m echanical aids of co mputation became a
necessity These consisted in Japan as in China of som e form s of
the abacus In China there ca m e to be developed an instrument
called the su an—
pan in Japan it was called the s or oban The i mporta
tion of the suan pan into Japan is usually supposed to have occurred
before the close of the S ix teenth century B amboo computing rods
were used in Japan in the seventh century These round pieces were
replaced later by the square prism s (s angi pieces) Numbers were
represented by these rods in the m anner practiced by the Chinese
The nu merals were placed inside the squares of a surface ruled like a
chess board The s or oban was simply a more highly de v eloped form
of ab acal instru ment
The years 1 60 0 to 1 6 7 5 mark a period of great mathematical ac
t i vi ty
It was inaugurated by M oRI K A M B E I S H I G EY O S H I who pop u
l ari z ed the use of the s or oba n
His pupil Y O S H I DA SH I CH I B E I K OY IJ
is the author of Ji n k o ki 1 6 2 7 which attained wide popularity and
is the oldest Japanese mathematical work n ow ex tant I t explains
operations on the soroban including square and cube root In one of
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mpil ed from D av i d Eu g en e S mi t h and Yoshio M ik a mi s
Hi story of J apa n es e M a themati cs C h i c ag o , 1 9 1 4 fro m Yo sh io M i k am i s D evel op
men t of M athemati es i n Chi n a a nd J apa n L ei p z i g , 1 9 1 2 an d from T Hayashi s
A B r i ef Hi stor y of the J apa n es e M a thein a ti es , O verg edruk t u i t h et N i eu w A rchi ef
1
Thi s
is
accou n t
’
co
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voor
Wi sk u nde VI
,
pp
.
2 96
—
36
1;
V I I , pp
.
1 05
—6
1
1.
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'
JAP ANESE
TH E
9
his later editions Yoshida appended a number of ad v anced proble ms
to be solved by co mpetitors This procedure started a mong the
Japanese the practice of issui ng problem s which was kep t up until
1 8 1 3 and helped to sti mulate mathe m atical activity
Another pupil of M ori was I MA M U RA CH I S H 6 who in 1 639 pub
l ished a treatise entitled Ju gai roku written in classical Chinese
He
took up the m ensuration of the circle sphere and cone Another
author I S O MU RA KI TTO K U in his K etsu gi s hb 1 66 0 ( second edition
whe n considering problem s on m ensuration m akes a crude
approach to integration He gives magic squares both odd and even
celled and also magic circles Such squares and circles became fav or
ite topics among the Japanese In the 1 6 84 edition Isom ura gives
also magic wheels TA N A KA KI S S H I N arranges the integers 1 96 in
2
si x 4 celled m agic squares such that the su m in each row and colu m n
“
are 1 94 ; placing the s ix squares upon a cube he obtains his magic
“
1
M U RA M A TS U in
cube
Tanaka formed also m agic rectangles
2
1 6 6 3 gi v es a m ag ic square containing as m any as 1 9 cells and a m agic
circle i n volving 1 2 9 numbers M uram atsu gives also the fam ous
“
Josephus P roblem in the following form : Once upon a time there
li ved a wealthy farmer who had thirty children half being of his first
The latter wished a fa v orite son to
w ife and half of his second one
inherit all the property and accordingly she asked h im one day say
ing :Would it not be well to arrange our 30 children on a circle calli ng
one of them the first and counting out every tenth one until there
should re main on l y one who should be called the heir The hus
band assenting the wife arranged the children
the counting
resulted in the elimination of 1 4 step children at once leaving only
one Thereupon the wife feeling confident of her success said
let us reverse the order
The husband agreed again and the
counting proceeded in the reverse order with the unexpected result
that all of the second wife s children were stricken out and there r e
m ained only the step —
c hild and accordingly he inherited the property
The origi n of this problem Is not known It 15 found m uch earlier I n
of the tenth century
th e Co d ex Ei n si de l en si s ( Ei n si d el n Switz erland )
while a Latin work of Roman ti m es attributes it to Flavius Josephus
I t co mmonly appears as a problem relating to Turks and Christians
half of whom m ust be sacrificed to save a sinking S hip I t was very
co mmo n in early printed European books on arithmetic and in books
on mathe matical recreations
In 1 666 S A N )S E I K O wrote hiS Kongen ki which i n co mmon with
other works of his day considers the computation of 7r (
“
He is the first Japanese to take up the Chinese celestial elem ent
m ethod i n algebra He applies it to equations of as high a degree as
the six th His successor S A WA GU CH I and a contemporary N OZA WA
give a crude calculus resembling that of Cavalieri Sawaguchi rise s
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Y
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M ik ami i n A rchi v der M athernati k
u
.
P hys i k , Vol
.
2 0,
pp
.
—86
1 83
1
.
A HISTORY OF M ATHE M ATICS
80
above the Chinese masters in recogni sing the plurality of roots bu t
he declares problem s which yield them to be erroneous in their nature
Another evidence of a continued Chinese influence is seen in the
ii
—
m
Chinese value of 71 i 4
which
was
ade
known
in
J
a
an
by
I
KE DA
p
4
—
m
A
2
We co e now to S E KI K 6 W ( 1 6 4 1 70 8)
whom the Japanese con
sider the greatest m athematician that their country has produced
The year of hi s birth was the year in which Galileo died and Newton
was born Seki was a great teacher who attracted m any gifted pupils
L ik e P ythagoras he discouraged divulgence of m athematical dis
c ov er i es m ade by hi m self and his school
For that reason it is difficult
to determ ine with certainty the exact origin and nature of some of the
discov eries attributed to him He is said to have left hundreds of
manuscripts ; the transcripts of a few of them still re main He pub
li sh ed only one book the H a tsu bi S ampo 1 6 74 in which he solved
1 5 problem s issued by a conte mporary writer
S ek i s explanations
are quite incomplete and obscure Tak eb e one of his pupils lays
stress upon S ek i s clearness The inference is that Seki gave his ex
planations orally probably using the computing rods or s angi as he
proceeded Noteworthy a mong his m athe matical achieve ments are
th e ten zon m ethod and the yendo n m ethod
B o th of these refer to
i mpro v e m ents in algebra
The ten zan m ethod is an improvem ent of
“
the Chinese celestial elem ent m ethod and has reference to nota
tion while the yendan refers to explanations or method of analysis
The exact nature and value of these two m ethods are not altogether
“
clear By the Chinese celestial ele ment m ethod the roots of equa
tions were computed one digit at a tim e Seki re moved this lim ita
tion B uilding on results of his predecessors Seki gives also rules for
writing dow n magic squares of ( 2 n
cel ls In the case of the more
troublesom e even celled squares Seki first gives a rule for the con
2
2
struction of a m agic square of 4 cells then of 4 (n
and 1 6 n cell s
He sim plified also the treatment of magic circles P erhaps th e most
original a n d i mportant work of Seki is the invention of determinants
som etim e before 1 683 Leibniz to whom the first idea of determinants
is usually assigned m ade hi S discove ry in 1 6 93 when he stated that
three l i near equations in x and y can have the sam e ratio only when
the determ inant arising fro m the coeffi cients vanishes Seki took n
equations and gave a m ore general treatm ent Seki knew that a
th
determinant of the n order when expanded has u 1 term s and that
1
rows and colu mns are interchangeable
Usuall y attributed to Seki
“
is the invention of the yenri or circle principle which i t is claimed
accomplishes som ewhat the same things as the differential and i n
Neither the e xact nature nor the origin of the yenri
t eg ral calculus
is well u nderstood D oubt e xists whether Seki was its discoverer
TA KE B E a pupil of Seki used the yenr i and may be the chief originator
”
On t he J apan ese theory of det e mi nan ts in
F r de tail s con s l t Y M ik ami
I i s V ol II 9 4 pp 9—
36
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A HISTORY OF M ATHE M ATICS
82
can be drawn upon a folding fan Here mathematics finds application
to artistic design
After the m iddle of th e nineteenth century the native mathematics
of Japan yiel d ed to a strong infl ux of Western mathematics The
m ovem ent in Japan beca m e a part of the great international advance
In 1 9 1 1 there was started the Tohoku M athemati cal J ou r n al under
the editorship of T Hayashi I t is devoted to advanced m athematics
contains articles in many of our lea ding modern languages and is quite
1
international in character
Looking back we see that Japan produced some able mathemati
c i an s
but on account of her isolation geographically and socially
her scientific output did not aff ect or contribute to the progress of
m a them atics in the West The Babylonians Hindus Arabs and to
som e extent even the Chinese through their influence on the Hindus
contributed to the onward m arch of m athematics in the West B ut
the Japanese stand out in complete isolation
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G A M ill e r , H i s tor i cal I ntroduc ti on to M athemati cal Li teratu re,
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1 91 6 ,
p
.
24
.
THE HIND U S
After the time of the ancient Greeks the first people whose re
searches wielded a wide influence in the world m arch of m athe matics
belonged like the Greeks to the Aryan race It was however not a
European but an Asiatic nation and had its seat in far o ff India
Unlike the Greek Indian society was fix ed into castes The only
castes enjoying the privilege and leisure for advanced study an d
thinkn were the B rahmi ns whose prim e busin ess was religion and
philosophy and the K shatri yas who attended to war and govern m ent
Of the dev elop ment of Hin d u mathem atics we know but little A
few man u Sc rip ts bear testimony that the Indians had c l imbed to a
lofty height but their path of ascent is no longer traceable It woul d
seem that Greek m athematics grew up under m ore favorable con di
tions than the Hindu for in Greece it attained an independent exist
ence and was studied for its own sake while Hindu m athe m atics
always remained merely a servant to astronomy Furthermore in
Greece m athem atics was a science of the people free to be cultivated
by all who had a liking for it ; in India as in E g ypt it was in the
hands chiefly of the priests Again the Indians were in the habit of
putting into verse all m athe matical results they obtained and of
clothing them in obscure and m ystic language which though well
adapted to aid the me mory of him who already understo od the subject
Although the great Hindu
w as often unintelligible to the uninitiated
mathem aticians doubtless reasoned out m ost or all of their discoveries
yet they were not in the habit of preserving the proofs so that the
naked theorems and processes of operation are all that have com e
down to our time Very di fferent in these respects were the Greeks
Obscurity of language was generally avoided and proofs belonged
to the stock of knowledge quite as m uch as the theorem s themselves
Very striking was the difference in the bent of m ind of the Hindu and
Greek ; for while the Greek mind was pre e minently geometri cal the
Indian was first of all ari thmeti cal The Hindu dealt wi th nu mber the
Greek with form Nu merical symbolism the science of numbers
and algebra attained i n India far greater perfection than they had
previously reached in Greece On the other hand Hindu geo metry
was merely m ensuration unaccompanied by dem onstration Hindu
trigonometry is m eritorious but rests on a rith metic more than on
geomet ry
An interesting but difficult task is the tracing of the relation be
tween Hindu a n d Greek m athem at i cs It is well known that more
or less trade was carried on between Greece and In di a fro m early
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83
A HISTORY OF M ATHE M ATICS
84
times Af ter Egypt had become a Roman province a more li vely
comm ercial intercourse sprang up betw een Rome and India by way
of Al exandria A pri ori it does not seem improbable that wi th the
traffi c of merchandise there should also be an interchange of ideas
That comm unications of thought from the Hindus to the Alexandrians
actually did take place _
is e v ident from the fact that certain phi lo
sophic and theologic teachings of the M anicheans N eo P latonists
Gnostics S how un mistakable li k eness to Indian tenets Scientific
facts passed also from Alexandria to India Thi s is shown plainly
by the Greek origin of so me of the technical terms used by the Hindus
Hindu astronomy was influenced by Greek astronomy A part of
the geometrical knowledge whi ch they possessed is traceable to Alex
andria and to the writings of Heron in particular In algebra there
was probably a m utual giving and receiv ing
There is e v idence also of an intimate connection between Indian
and Chinese m athe matics In the four th and succeedin g centuries of
our era Indian embassies to China and Chinese visits to India are
1
recorded by Chinese authorities
We shall see that undoubtedly
there was an influx of Chinese m athematics into India
The history of Hindu m athem atics may be resolved into two
periods : First the S u l vasutr a peri od which terminates not later than
2 00 A D
second the as tr on omi cal and mathemati cal peri od extendin g
from about 400 to 1 2 00 A D
The term S ulvasutra m eans the rules of the cord it is the name
giv en to the supplem ents of the K al pasfi tras which explain the con
2
struction of sacrificial altars The S u l vasut ras were composed som e
ti me between 80 0 B C and 2 00 A D They are known to m odern
scholars through three quite modern m anuscripts Their aim is
prim arily not m athe m atical but religious The m athem atical parts
relate to the construction of squares and rectangles S trange to say
none of these geom etrical constructions occur in later Hindu works ;
later Indian mathematics ignores the S u l vasfi t ras !
The second period of Hindu m athe matics probably originated
with an influ x fro m Alexandria of western astronomy Z
To the fif th
j
century of our era belongs an anonymous Hindu astronom ical work
“
calle d the S urya S i ddhan ta ( Kn w l edg e from the sun )
whi ch cam e
to be regarded a standard work
11 the six th century A D
Varah a
M ihi ra wrote his P ancha S i ddhan ti k a whi ch giv es a s umm ary of the
S urya S i ddhan ta and four other astronom ical works then in use ; it
contains matters of mathem atical interest
J I n 1 8 8 1 there was found at B ak hshd l i in northwest India buried
in the earth an anonymous ari thmetic supposed from the peculiar
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Siml a ,
G R K aye , I ndi a n M a themati cs , C al c u t t a
i ng heav i l y u po n t h i s b ook w h i ch e mb odi es th e res ul t s
m a t hem a ti c s
2
G R K aye , op ci t , p 3
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p 38
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o f r ec en
t
.
We a re draw
s t u di es of
H i ndu
TH E
HIND US
85
ities of its verses to date from the third or fourth century after Christ
The document that was found is of birch bark and Is an inco mplete
copy prepared probably abo ut the eighth century of an older manu
1
s cr i pt
I t conta ins ari thmetical co mputati on
SgTh e noted Hindu astrono mer Aryab h ata was born 4 7 6 A D at
P atahpu t ra on the upp er Ganges
His celebrity rests on a work
entitled Aryabhati ya of which the third chapter is devoted to mathe
matics sfi fil b ou t one hundred years later m athem atics in India reached
the highest mark At that time flourished B rah mag u pta (born
“
—
In 6 2 8 he wrote his B r ahma sphu ta si ddhan ta ( The Revised System
of
of which the twelfth and eighteenth chap ters belong
to m athematics
5 P robably to the nin th century belongs M ah avi ra a Hindu author
on elem entary m athem atics whose wr i tin s have only recently been
g
brought to the attention of historians He 18 the author of the Gani ta
S ar a S an gr aha which throws light upon Hindu geom etry and arith
meti c j The follow ing centuries produced only two names of impor
“
—
tance ; namely S ri dh ar a who wrote a Gani ta s ara ( Q uintessence
o f C al c u l a t i on 9
f an d P adman abh a the author of an algebra The
science seems to have m ade but little progress at this t im e ; _for a
“
work enti tled S i ddhan ta S i r omani ( D iadem of an Astronom ical
7
written by B h as k axa in 1 1 50 stands little higher than that
System )
of Brahm agupta written over 50 0 years earlier The two m ost im
portant m athem atical chap ters I n thi s work are the Li l avati
the
“
beautiful i e the noble science )
and Vij a gani ta
root extrac
devoted to arithm etic and algebra From n ow on the Hindus
in the Brahmin schools seemed to content them selves with studying
the masterpieces of their predecessors Scientific intelligence de
creases continually and in m ore m odern tim es a very deficient Arabic
w ork of the sixteenth century has been held in great authority
The mathematic al chapters of the B r ahma s i ddhanta and S i ddhanta
S i r oman i were translated into English by H T Colebrooke London
The S ur ya s iddhan ta was translated by E B urgess and anno
18 1 7
M ah avi ra s
ta ted by W D Whitney N ew Haven Conn
1 860
Goni ta S ar a S angr aha was published in 1 91 2 in M adras by M Rangi
carya
We begin with geometry the field in which the Hindus were least
proficient The S ul vasutras indicate that the Hindus perhaps as
early as 80 0 B C applied geom etry in the construction of altars
2
Kaye states that the m athematical rules found in the S u l vasutras
“
relate to ( 1 )
the construction of squares and rectangles ( 2 )
the rela
equivalent rectangles and squares
tion of the diagonal to the S ides (3)
equivalent circles and squares
A k nowledge of the P y tha gorean
(4)
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The B ak hs hal i M a n u scr i pt, e di t ed b y R u dolf Hoernl y
xv11, 3 3 48 an d 2 75 2 7 9 , B omb ay, 1 888
2
G R Kaye, op ci t , p 4
1
—
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in
the I ndi an A n ti qu ar y,
A HISTORY OF M ATHE M ATICS
86
theorem is disclosed in such relations as
5
2
=
There is no evidence that these expressions were
39
obtained fro m any general rule I t will be remembered that sp ecial
cases of the P y thagorean theorem were known as early as 1 000 B C
in China and as early as 2 00 0 B C in Egyp t A curious expression
for the relation of the diagonal to a square namely
2
=
= 202
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is explained by Kaye as being an expression of a direct m easure
m ent which m ay be obtained by the use of a scale of the kind named
in one of the S ul v asu tra m anuscripts and based upon the change
ratios 3 4 34 I t is noteworthy that the fractions used are all unit
fractions and that the expression yields a result correct to fi v e decimal
places The S ulvasutra rules yield by the aid of the P ythagorean
theorem constructions for finding a square equal to the su m or dif
ference of two squares they yield a rectangle equal to a giv en square
with a s/ 2 and 2 a \/ 2 as the sides of the rectangle they yield by
geometrical construction a square equal to a given rectangle and
2
—
—
=
satisfying the relation ab (b + [a
i ( a b) corresponding
to Euclid II 5 In the S u l vasfi t ras the altar building ritual explains
the construction of a square equal to a circle Let a be the S ide of
a square and d the d iam eter of an equivalent circle then the given
1
—2 d/1 5 a
rules m ay be expressed thus d =
a =d
1
—
This third expression m ay be ob
d(1
8 2
6 +
6
t ai n e d fro m the first by the aid of the appro xi mation for V 2 given
above Strange to say none of these geo metrical constructions occur
in later Hindu works the latter completely 1g nore the ma th em atical
contents of the S u l v asu t ras
D uring the six centuries from the tim e of A ryabhata to that of
Bh askara Hindu geom etry deals m ainly with m ensuration The
Hindu gave no d efinitions no postulates no axioms no logical chain
of reasoning His knowledge of m ensuration was largely borrowed
from the M editerranean and from China through i mperfect channels of
comm unication A ryabha ta gives a rule for the area of triangle which
holds only for the isosceles triangle Brahm agupta distinguishes
between approxim ate and exact areas and gives Heron of Alexan
—
—
w
/
dria s fa m ous form ula for the triangular area
s (s
a)
(s b)
(s
2
Heron s for mula is given also by M ah avi ra who advanced b e
yond his predecessors in giving the area of an equilateral triangle as
Brah magupta and M ah avi ra m ake a remarkable ex tension
—
—
—
s
of Heron s formula by giving w’ (s —a )
as the area
b
s
c
)
( d)
)
(s
(
of a quadri lateral whose sides are a b c d and whose semi perim eter
is s That this form ula is true only for quadrilaterals that can be i n
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2
D E S mi t h , i n I s i s , Vol
1
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1,
19 1 3,
pp
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1 99 , 2 00 .
HINDUS
TH E
87
a circle was recogniz e d by Brahmagupta according to Can
l
1
2
tor s and Kaye s interpretation of B rah magupta S obscure ex
position but Hindu co mmentators did not unders tand the li mi tation
and Bhaskara finally pronounced the formula unsound Remarkable
2
—
m
m
a
is Brah agupta s theore
on cyclic quadrilaterals x ( d b)
c
2=
—
—
a
where x and
and y ( b l cd)
y are the diagonals and a b c d the lengths of the sides ; also the the
2= 2
2
2
2
orem that if a + b c and A + B
then the quadrilateral
“
is cyclic and has its
( Brahmagupta s
(aC cB bC cA )
diagonals at right angles Kaye says : From the triangles (3 4 5)
and ( 5 1 2 1 3 )
a commentator ob tains the quadrilateral (3 9 60 5 2
with diagonals 6 3 and 56 etc B rahm agup ta (says Kaye )
also i n
t rodu ces a proof of P tole my s theore m and in doing this follows D io
in constructing from right triangles (a b c)
and
phan t us ( III 1 9)
new right triangles (a y 6 7 c y)
and ( a c 6 c 7 c )
and
( a B y)
uses the actual examples given by Diophantus nam ely (3 9 5 2 , 6 5)
and ( 2 5 6 0 6
P arallelisms of this sort S how un m istakably that the
Hindus drew from Greek sources
In the mensuration of solids remarkable inaccuracies occur in
A ryabhat a He gives the volume of a pyrami d as half th e product of
3
2
the base and the height ; the volu me of a sphere as 71 r A ryab
hata gives in one pl ace an extremely accurate value for an v iz
but he him self never utiliz ed it nor did any other Hindu
m athematician before the twelfth century A frequent Indian p rac
Bh askara gives two values —the
tice was to take 7r = 3 or
3 9 2 7
and
the
inaccurate
Archi
m
e
ean
above m entioned accurate
d
5 0
value
A comm entator on L i l avati says that these values were
calculated by beginning with a regular inscribed hexagon and apply
2
—
=
ing repeatedly the form ula A D
AB
wherein A B is the
side of the given polygon and A D that of one with d ouble the number
of S ides I n this way were obtained the perim eters of the inscribe d
= 1 0 0 the
polygons of 1 2
sides
Taking
the
radius
8
3 4
perim eter of the last one gives the value which A ryab h a ta use d for 7r
The empirical nature of Hin du geom etry is illustrated by Bhaskara s
proof of the P ythagorean theorem
He draws the right triangle four
tim es in the s q u ar e of the hypo te
nuse so that in the m iddle there r e
m ains a square whose side equals
the difference between the two sides
of the right triangle Arranging this square and the four triangles in
a difl eren t way they are seen together to m ake up the su m of the
“
s quare of the two sides
B ehol d ! says Bh askara wi thout a dding
—
i
t
l
I
E
C t
V
o
d
d
6
p
3
49 6 5 3
9 7 pp
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scr ib ed I n
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an or , o
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G R K aye,
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op
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ci t.,
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20
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2 2.
1
0
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A HISTORY O F M ATHE M ATICS
88
another word of e xplanation B re tsc hnei de r conjectures that the
P ythagorean proof was substantially the sa m e as this
Rec ently it
has been shown that this interesting proof is not of Hindu origin but
was giv en m uch earlier ( early in the Christian era )
by the Chinese
writer Chang Chu n —
Ch i n g in his comm entary upon the ancient treat
1
ise the Chou pei
In another place Bhaskara gives a second dem
on s t r a t i on of this theore m by drawing fro m the verte x of the right
angl e a perpendicular to the hypotenuse and comparing th e two tri
angles thus obtained with the given triangle to which they are similar
This proof was unknown in Europe till Wallis rediscovered it The
only Indian work that touches the subj ect of the conic sections is M a
h avi ra s book which gives an inaccurate treatment of the ellipse It
is readily seen that the Hindus cared little for geome try B rah ma
gupta s cyclic quadrilaterals constitute the only g em in their geo m
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e t ry
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The grandest achieve ment of the Hindus and the one which of all
mathe matical inventions has contributed m ost to the progress of
intelligence is the perfecting of the so called Arabic Notation
That thi s notation d id not originate with the Arabs is n ow admitted
by every one Until recently the preponderance of authority favored
the hypothesis that our nu meral syste m with its concept of local
value and our symbol for z ero was wholly of Hindu origin Now i t
appears that the principal of local value was used i n the sexagesim al
syste m found on B abylonian tablets dati n g from 1 6 0 0 to 2 30 0 B C
and that Babylonian records from the centuries i mm ediately preced
ing the Christian era contain a symbol for z ero which ho w ever was
not used in computation These sexagesimal fractions appear in
P tole my s A l mages t in 1 30 A D where the o micron o is m a d e to des
i g na t e blanks in the se xagesim al nu mbers but was not used as a reg
ular zero The Babylonian origin of the sexagesim al fract i ons used by
Hindu astronomers is denied by no one The earliest form of the I n
2
dian symbol for z ero was that of a dot which according to B ii hl er
“
was commonly used in inscriptions and manuscripts i n order to
This restricted early use of the symbol for z ero r e
mark a blank
sembles som ewhat the still earlier use m ade of it by the Babylon i ans
and by P tole my I t I S therefore probable that an imperfect notation
invol ving the principl e of local value and the use of the z ero was i m
ported into India that it was there transferred from the sexagesimal
to the deci mal scale and then in the course of centuries brought to
final perfection If these views are found by further research to be
“
correct then the name Babylonic Hindu notation will be m ore
“
appropriate than eithe r Arabic or Hindu Arabic
It appears
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Yo shio M i k ami , The P ythag orean Theo rem i n A r chi e d M ath u P hysi k ,
3 S , V ol 2 2 , 1 9 1 2 , pp 1 4
2
e
m
i
t
a
n
L
C
K
r
i
n
s
k
i
n
t
h
e
i
r
H
i
A r a bi c N u mer al s ,
n
d
uo
t
d
b
D
E
S
h
d
a
i
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p
Q
B o sto n an d L on don , 1 9 1 1 , p 5 3
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A HISTORY OF M ATHE M ATICS
90
follows :Vasu (a class of 8 gods)
+ two + eight+ m ountains ( the 7 m oun
—
—
—
d
i
i
tain
the
t
s
se
v
en
l mountains H unar
(
H
9
g
days (half of which equal
The use of such notations made it pos
sible to represent a nu mber in se v eral different w ays This greatly
faci li tated the fram ing of verses containing arithmetical rules or sci
en t i fi c constants which could thus be m ore easily re me mbered
At an early period the Hindus exhibited great skill in calculating
e v en with large numbers Thus they tell us of an exa mination to
which B uddha the reform er of the Indian religion had to sub m it
when a youth in order to w i n the maiden he lov ed In arith metic
after ha ving astonished his exa miners by naming all the periods of
numbers up to the 53d he was asked whether he could determine the
num ber of primary atoms which when placed one against the other
would form a line one m ile in length B uddha found the required a n
swer in this way : 7 primary atoms make a very minute grain of dust
7 of these m ake a m inute grain of dust 7 of these a grain of dust whirled
up by the wind and so on Thus he proceeded step by step until he
finally reached the length of a mile The m ultiplication of all the fac
tors gave for the m ultitude of primary atom s in a m ile a number con
sisting of I 5 digits This problem rem inds one of the Sand—
Counter
of Archim edes
After the nu merical symboli s m had been perfected figuring was
m ade m uch easier
M any of the Indian m odes of operation diff er
from ours The Hindus were generally i n clin ed to follow the motion
fro m left to right as in writing Thus they added the left hand col
u mn s first
and m ade the necessary co rrections as they proceeded
For instance they would have added 2 54 and 6 6 3 thus :
8
which changes 8 into 9 4+ 3 7 Hence the su m 9 1 7 In
I I
Thus In 8 2 1 348 they would say
su btr ac ti on they had t w o m ethods
Or they would say 8 from
8 fro m 1 1 = 3 4 fro m 1 1
7 3 fro m
In mu l ti pl i cati on of a number by
1 1 = 3 5 fro m 1 2 = 7 4 fro m
25
another of only one digit say 56 9 by 5 they generally said 5
which changes 2 5 into 2 8
hen ce the 0 m ust be i n
creased by 4 The product is 2 84 5 In the m ultiplication with each
other o f m any fi g u red nu mbers they first multiplied in the manner
just indicated with the left hand digit of the multiplier which was
written above the m ultiplicand and placed the product abo v e the
multiplier On m ultiplying with the next digit of the m ultiplier the
product was not placed in a new row as with us but the first prod uct
obtained was corrected as the process continued by erasing when
e v er necessary the old digits a n d replacing the m by n ew ones until
finally the whole product was obtained We who possess the m odern
luxuries of pencil and paper would not be l i kely to fall in love with
“
this Hindu m ethod B ut the In dians wrote with a cane pen upon
a small blackboard with a white th i nly liquid paint which m ad e marks
that could be easily erased or upon a white tablet less than a foot
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HIND US
TH E
91
”
square strewn w ith red flour on which they wrote the figures with a
1
small stick so that the figures appeared white on a red ground
Since the digits had to be quite large to be distinctly legible an d
S ince the boards were s m all it was desirable to have a m ethod which
would not require much space Such a one was the above m ethod
of m ultiplication Figures could be easily erased and replaced by
others without sacrificing neatness B ut the Hindus had al so other
ways of m ultiplying of which we m entio n the following : The tablet
was divided into squares like a chess board Diagonals were also
drawn as seen in the figure The m ultiplication of I 2 X 7 3 5 8 8 2 0 is
3
2
exhibited in the adjoining diagram According to Kaye this mode
of m ultiplying was not of Hindu origin and was
know n earlier to the Arabs The m anuscripts
extant give no i n formation of how di vi s i ons were
executed
Hindu m athematicians of the twel fth century
8
test the correctness of arithmetical computations 8
2
0
by casting out nines but this process is not of Hindu origin
i t was kno w
n to the Rom an bishop Hippolytos in the third cen
tury
In the B ak hshal i ari th metic a knowledge of the processes of c om
putation is presupposed In fractions the nu merator is written above
the denom inator Without a dividing line Integers are written as
fractions wi th the denominator 1 In mix ed expressions the integral
1
part is written above the fraction Thus 1 1 3 In place of our
they used the word phal am abbreviated into pha Addition was i n
d ic at ed by yu abbreviated from yu ta
Numbers to be combined
7
5
were often enclosed in a rectangle Thus pha 1 2
yu jmean s T f
An unknown quantity is su nya and is des i gnated thus by a
12
“
3
heavy dot The word su nya m ea ns empty and is the word for
This double use of
z ero which is here likewise represe nted by a dot
the word and dot rested upon the idea that a position is e mpty if
not filled out It IS also to be considered e mpty so long as the n um
4
ber to be placed there has not been ascertained
The Bakhshah arithm etic contains proble ms of which so me are
solved by reduction to unity or by a sort of f al s e posi ti on E xample :
B gives twice as much as A C three t imes as much as B D four tim es as
much as C ; together they give 1 3 2 ; how much did A give ? Take 1 for
the unk nown (su nya)then A = 1 B = 2 C = 6 D = 2 4 their su m
33 D ivide 1 3 2 by 3 3 and the quotient 4 is what A gave
The method of f al se posi ti on we have encountered before among
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H H an k el , op ci t , 1 8 74 , p 1 8 6
2
M C an tor , op ci t , V ol I , 3 A u fl , 1 90 7 , p 6 1 1
3
G R K aye , op ci t , p 34
4 C an to r I
Ed , 1 90 7 , p p 6 1 3 6 1 8
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H ISTOR Y
A
92
OF MATH EMATIC S
the early Egyptians With them it was an instinc t ive procedure ;
w i th the Hindus it had risen to a conscious m ethod
Bh askara uses
it but while the Bakhsh ali docu ment preferably assum es 1 as the
u nk nown Bhaskara is partial to 3 Thus if a certain nu mber is
taken fi v e fol d 3 of the product be subtracted the rem ainder di v ided
l
1
1
by 1 0 and 3 3 and z of the original nu mber added then 6 8 is ob
t ai n ed
What IS the number ? Choose 3 then you get 1 5
and
3—
3
1 7
3 —
—
—
1 1
Then ( 6 8
1 1
4 8 t he answer
3
We shall now proceed to the consideration of som e arith m etical
problems and the Indian m odes of solution A favorite m ethod was
that of i nver si on With laconic brevity A ryabha ta describes it thus :
“
M ultiplication becomes division division becom es m ultiplicatio n;
what was gain becom es loss what loss gain ; inversion Q u i_
te d i ff erent
fro m th is quotation in style is the following proble m fro m A ryabha ta
“
which illustrates the m ethod : B eautiful maiden with bea ming eyes
tell me as thou u n d er stan ds t the right method of inversion which
3
is the number which m ultiplied by 3 then increased by 4 of the prod
u c t divided by 7 dim inished b y % of the quotient multiplied by i t
self dimi nished by 5 2 the square root extracted addition of 8 and
division by 1 0 gives the nu mber
The process consists in begin
ning with 2 and working backwards Thus
—
=
3 3
1
2 8 the answer
6
1
and
v 9
4
Here is another example taken from L i l dvati a chapter in Eb as
“
the
kara 5 great work : The square root of half
number of bees in a
S
—
swarm has flown out upon a jessamine b u sh g of the whole swarm
has remained behind one fem ale bee fli es about a m ale that 1s buzz
in g withi n a lotus fl ow e r into which he was allured in the night by its
sweet odor but is now imprisoned in it Tell m e the number of bees
Answer 7 2 The pleasing poetic garb in which all arithm etical p rob
lem s are clothed is due to the Indian practice of writing all s chool books
in verse and especially to the fact that th ese problem s propoun ded
as puz zles were a fav orite social am use m ent Says B rah magupta :
“
These proble m s are proposed simp l y for pleasure ; the wise man can
invent a thousand o thers or he can sol v e the problem s of others by
th e rules given here
As the sun eclipses the stars by hi s brilliancy
so the man of knowledge will eclipse the fam e of others in assemblies
of the people if he proposes algebraic problem s and still m ore if he
solves t hem
The Hindus solv e d proble m s in interest discount partnership
alligation summation of arith m etical and geom etric series and de
vised rul es for determining the num ber s of combinations and permu
t at i o n s
I t may here be added that chess the profoundest of all
games had its origin in India The invention of magic squares is
som etimes erroneously attributed to the Hindus Am ong the Chi
nese and Arabs m agic squares appear m uch earlier The first occur
rence of a m ag ic square among the Hindus is at D udh ai I han si in
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A H ISTORY OF M ATHE M ATICS
94
exists between the continuous an d discontinuous Yet by doing so
the Indians greatly aided the general progress of m athematics
In
deed if one understands by algebra the application of arit hm etical
operations to complex magnitudes of all sorts whether rational or irra
t i on al num bers or space m agnitudes then the learned Brahm ins of
1
Hindostan are the real inventors of algebra
Let us now examine more closely the Indian algebra In extract
2
ing the square and cube roots they used t he form ulas
a +
2
2
3= 3
2
3
2 a b+ b and ( a + b
In this connection A ryab
a + 3 a b+ 3 ab + b
ha ta speaks of dividing a nu mber into periods of two and three digits
From this we infer that the principle of position and the zero in the
nu m erical notation were already known to him In figuring wi th
z eros a state m ent of Bh askara is interesting
A fraction whose de
nominator is z ero says he admits of no alteration though m uch be
added or subtracted Indeed in the same way no change takes place
in the infinite and i mm utable D eity when worlds are destroyed or
created even though nu merous orders of beings be taken up or brought
forth Though in this he apparently evinces clear mathematical no
tions yet i n other p l aces he m akes a co mplete failure in figuring with
fractions of z ero denominator
In the Hindu solutions of deter minate equations Cantor thinks
he can see traces of D iophantine m ethods Some technical term s be
tray their Greek origin Even ii it be true that the Indians borrowed
fro m the Greeks they deserve credit for improving the solutions of
linear and quadratic equations Recogniz ing the existence of neg
ati v e nu mbers B rah magupta was able to unify the treatm ent of
the three form s of quadratic equations considered by Diophantus
2
2
=
=
vi z
ax + bx
c
b
x+ c
ax
a x + c = bx
a
and
b
c being pos
(
2
i t i v e nu m bers )by bringing the three under the one g en era l ca s e
'
“
To S ri dhara is attributed the Hin d u m ethod
of co mpleting the square which begins by m ultiplying both sides
of the equation by 4p Bh askara adv ances beyond the Greeks
“
and e v en beyond B rahm agupta when he says that the square of
a positive as also of a negative number is positive ; that the square
root of a positive nu mber is twofold positive and negative There is
no square root of a negative nu mber for it is not a square
Kaye
points out howev er that the Hindus were not the first to giv e double
2
solutions of quadratic equations The Arab Al K how ari zmi of the
2
ninth century gave both solutions of x + 2 1 = 1 0 90 Of equations of
higher degrees the Indi ans succeeded in solving only some special
cases in which both sides of the equation could be made perfect powers
by the addition of certai n term s to each
Incom parably greater progress t h an i n the solution of determinate
equations was m a de by the Hindus in the treat m ent of i nd eter mi n ate
equ ati on s
In determ inate analysis was a subject to which the Hindu
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H H an k e l ,
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o
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ci t
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,
p
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1 95 .
2
G R Kaye ,
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p
o
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ci t
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,
p 34
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.
TH E
mind
HIND US
95
showed a happy adaptation We hav e seen that this very sub
j ec t was a favorite with Diophantus and that his ingenuity was al
most ine xhaustible i n devising solutions for particular cases B ut the
glory of having invented gen er al methods in this m ost subtle branch
of mathematics belongs to the Indians The Hindu indeterm inate
analysis difl ers fro m the Greek not only in m ethod but also in aim
The obj ect of the former was to find all possible integral solutions
Greek analysis on the other hand de man ded not necessarily integral
but simply rational ans w ers D iophantus was content with a_s i ng l e
solution ; the Hindus endeavored to find all solutions possible A ryab
h ata gives solutio n s in integers to linear equations of the form axi
by = c where a b c are integers The rule e mployed is called the pu t
ver i zer
For this as for most other rules the Indians gi v e no proof
Their solution is essentially the same as the one of Euler Euler s
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process of reducing
a
b
to a cont i nued fracti on a mounts to the same as
the H indu process of findi ng the greatest co mmon di v isor of a and b
by division Thi s is frequently called the Diophantine m ethod Han
kel protests against this na me on the ground that D iophantus not
only ne ver kne w the method but did not e v en ai m at solutions purely
1
integral These equations probably grew out of proble ms in astron
They were applied for instance to determ ine the time when
omy
a certain constellation of the planets would occur in the heavens
P assing by the subject of linear equations with more than two u m
known quantities we come to indeterminate quadratic equations In
the solutio n of
they applied the m ethod r e invented
l ater by Euler of deco mposing (a b+ c)
into the product of two integers
=
m n and of placing x = m—
and
b
l
y n+ a
2
Re markable is the Hindu solution of the quadratic equation cy
2—
With great keenness of intellect they recogniz ed in the special
dx i b
2=
2
case y ax + 1 a funda mental problem in indeterm inate qua d ratics
They solved it by the cycl i c method
I t consists says D e M organ
“
2=
2
in a rule for finding an indefinite nu mber of solutions of y ax + I
(a being an integer which is not a square )by m eans of one solution
giv en or found and of feeling for one so l ution by making a solution
2
2=
2
2=
2
of y ax + b give a solution of y ax + b It amounts to the fol
2
2
lowing theore m :If p and g be one set of v alues of x and y i n y ax + b
and p and q the sam e or another set then qp+ pq and app + qq
2
2
2
From this it is ob v ious that one
ax + b
ar e values of x and y in y
2
2
solution of y ax + I may be m ade to give an y number and that if
2=
2
2
taking b at pleasure y ax + b can be solved so that x and y
2
2
are divisible by b then one preliminary sol II t i o n of y ax + I can be
be found Another m o de of trying for solutions is a combination of
the preceding with the cu ttaca
These calculations were
used in astronomy
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H Han k e l , op
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ci t
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p
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1 96 .
,
A HISTORY OF M ATHE M ATICS
96
this cyclic method constitutes the greatest invention
in the theory of nu mbers before the time of Lagrange The perver
2=
2
s i ty of fate has willed it that the equation y
ax + 1 should now be
called P el l s equ ati on; the first incisive work on it is due to Brahmin
scholarship reinforced perhaps by Greek research I t is a problem
that has exercised th e highes t fac ul ties of some of our greatest modern
analysts By them the work of the Greeks and H i ndus was done over
again ; for u nfortunately only a small portion of the Hindu algebra and
the Hindu manuscripts which we now possess were known in the
“
Occident Hankel attributed the invention of the cyclic me thod
entirely to the Hindus but later hi storians P Tan nery M Cantor
T Heath G R Kaye favor the hypothesis of ultim ate Greek origin
If the m issing parts of Diophantus are ever found light will probably
be thrown upon this question
Greater taste than for geom etry was shown by the Hindus for trig
o n o me t r y
Interesting passages are found in V araha M ihira s P ancha
1
S i ddhan ti k a of the six th century A D
which in our notation for
2
unit radius gives
S in 30
si n y
3 sin
—
—
sin
2
i
0
210
This is followed by a table of
l [ I S n (9
7 ]
]
{
2 4 sines
the angles increasing by i n terv al s of
( the eighth part
of
obviously taken from P tolemy s table of chords However
instead of dividing the radius into 6 0 parts in the m anner of P tolemy
the Hindu astronomer divides it into 1 2 0 parts which dev ice enabled
hi m to convert P tole m y s table of chords into a table of sines without
changing the numerical values A ryabha ta took a sti ll different value
for the radius nam ely 3 43 8 obtained apparently from the relation
2 X 3 1 4 I 6r =
The Hindus followed the Greeks and B abyl o
n ia n s in the practice of dividi ng the circle into quadrants each quad
—
rant into 90 degrees and 5400 m inutes thus dividing the whole circle
into
equal parts Each quadrant was divided also into 2 4 equal
parts so that each part e mbraced 2 2 5 un i ts of the whole circu mference
and corresponded to 3 3 degrees Notable is the fact that the Indians
ne v er reckoned l ike the Greeks with the whole chord of double the arc
but always with the si n e ( j oa)
and vers ed s i n e Their m ode of calcula
ting tables was theoretically very S imple The sine of 90 was equal to
the radius or 3 43 8 ; the sine of 30 was evidently half that or 1 7 1 9
D oub tl ess
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°
‘
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°
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°
.
°
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,
,
=
m
r
the
for
ula
sin
cos
they
obtained
sin
+
45
fi
fl
2 43 1
Substituting for cos a its equal sin ( 90 —a)
and making a =
Apt
2
a
2
a
ng
°
2
,
,
.
they obtained sin 6 0
°
2 97 8
.
With the sines of 90
,
6 0 , 4 5,
and 30
as starting points they reckoned the S ines of hal f the angles by the
2
=
formula v er sin 2 a 2 sin a thus obtaining the sin es of 2 2
-
,
°
,
1
G R K aye ,
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.
p ci t , p
o
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10 .
98
A
.
HISTORY
O F M ATHE M ATICS
they would have exerted had they com e two or three centu
ries earli er
At the beginni ng of the twentieth century mathematical activ ity
along mode rn lin es sprang up in In dia In the year 1 90 7 there was
founded the I ndi an M athemati cal S oci ety; in 1 90 9 there was started
1
at M adras the Jou rn al of the I n di an M athemati cal S oci ety
fl u en ce
,
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,
’
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.
1
(Th ree r ec en t
o rig i n o f o u r n u
w r i t er s
me ral s
We
a dv an c e d a rg u
r efe r
to G
(I)
t e n di n g t o di sp ro v e t h e H i n du
’
K aye s ar t i c l e s i n S ci en ti a , V ol 2 4 ,
men t s
R
—
n
I
I
0
B
l
i
J
o
u
rn
a
l
A
s
i
a
t
i
c
oc
e
n
a
I
1
S
9 7 , pp 4 7 5 5 0 8 , a l so V I I ,
pp
,
g ,
’
—
—
1 9 1 1 , p p 8 0 1 8 1 6 : i n I n di a n A n ti u ar y, 1 9 1 1 , p p 5 0 5 6 ; ( 2
t o C a r ra de V au x s
q
—
a r t i c l e i n S ci en ti a , V o l
2 1 , 1 9 1 7 , p p 2 7 3 2 8 2 ; (3
t o a R u ssi an bo o k b r o ug h t o u t
b y N ik ol B u b n ow i n 1 90 8 a n d t ran sl a t e d i n t o G e rman i n 1 9 1 4 b y J os L ez i u s
K ay e c l a i ms t o S h ow t h a t t h e p ro o f s o f t h e H i n d u or i g i n o f o u r n u me ral s a re
l arg el y l eg en dar y, t h at t he q u e st i on ha s b e en c l o u ded b y a co n fu sio n b e t w een t he
w o rd s hi ndi ( I n di an an d hi n das i ( m ea s ur e g e o m e t ri c a l , t ha t t h e sym b ol s ar e
n o t m odi fi e d l e t t e r s o f t he al p ha b e t
We m u st ho ld o u r mi n ds i n su spen se on
t hi s d i fli c u l t q u es i on an d aw ai t fu r t h e e vi den ce
19 1 8,
—
5 3 55 ;
h av e
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t
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)
.
THE ARAB S
After the flight of M oha mm ed fro m M ecca to M edina in 6 2 2 A D
an obscure people of Se mitic race began to play an important part in
the dra ma of history B efore the lapse of ten years the scattered
tribes of the Arabian peninsu l a were fused by the furnace blast of
religious enthusias m into a po w erful nation With sword in hand the
united Arabs subdued Syria and M esopota m ia Distant P ersia and
the lands beyond even unto India were added to the dom inions of
the Saracens They conquered Northern Africa and nea rl y the
whol e Spanish peninsula but were finally checke d fro m fu rther prog
ress in Western Europe by the fi rm han d of Charles M artel ( 73 2 A
The M osle m dominion extended now from In dia to Spain ; but a war
of succession to the caliphate ensued and in 7 5 5 th e M ohamm e dan
—
e mpire w as di vi ded one caliph reigning at Bagdad the other at Cor
dova in Spain Astoun ding as was the grand m arch of conquest by
the Arabs still m ore so was the ease with which they put aside their
former nomadic life adopted a higher civiliz ation an d assum ed the
sovereignty ov er cultiv ated peoples Arabic was made th e written
language throughout the conquered lands With t he rule of the Abba
sides in the East began a new period in the history of learning The
capital Bagdad situated on the Euphrates lay half way between
two old centres of scientific thought — India in the East and Greece
in the West The Arabs were destined to be the custodians of the
torch of Greek science to keep it ablaz e during the period of c on fu
sion and chaos in the Occident and afterwards to pass it over to the
Europeans This remark applies in part also to Hindu science Thus
science passed from Aryan to Sem itic races and the n back again to
the Aryan Formerl y i t was held that the Arabs a dded but little to
the knowledge of mathematics ; recent studies in di cate that they must
be credited w ith novelties once thought to be of later origin
The Abba sides at Bagdad encouraged the introduction of the
sciences by inviting able special ists to their court irrespective of na
t i on ali ty or religious belief
M e dicine and ast ronom y were their fa
v ori t e sciences
Thus Harun al Rashid the m ost d istin g uished Sara
cen ruler drew Indian physicians to Bagdad In the year 77 2 there
cam e to the court of Caliph A lm an su r a Hindu astronom er wi th as
t ron omi c al tables which were ordered to be translate d into Arabic
These tables known by the Arabs as the S i ndhi n d an d probably taken
from the B r ahma sphu ta si ddhan ta of B rah magupta stoo d in great
authority They contained the i mportant Hin du table of s i nes
Doubtless at this time and along w ith these astronomical tables
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A HISTORY OF M ATHE M ATICS
1 00
the Hi n du numerals with the z ero and the principle of position were
introduced am ong the Saracens B efore the time of M ohammed the
Ar abs had no num erals Numbers were written out in words Later
the num erous co mputations connected with the financial administra
tion over the conquered lands m ade a short symboli s m indispensable
In some localities the nu merals of the more civiliz ed conquered na
tions were used for a tim e Thus in Syria the Greek notation was
retained ; in Egypt the Coptic In some cases th e num eral adj ec
t i v e s may have been abbreviated in writing
The Di w ani —
n u mer al s
found in an Arabic P ersian dictionary are supposed to be such ab
b r evi ati on s
Gradually it becam e the practice to e mploy the 2 8 Ara
bic letters of the alphabet for num erals in analogy to the Greek sys
te n i This notation was in turn superseded by the Hindu nota tion
which quite early was adop ted by m erchants and also by writers on
arith metic Its superiority was generally recogniz ed excep t in as
Here
t r on omy where the a l phabetic notation continued to be used
the alphabetic notation o ffered no great disadvantage since in the
sexagesimal arith m etic t ak en f rom the A l mages t numbers of gen
1
e ral l y only one or two places had to be written
As regards the form of the so called Arabic numerals the state
m ent of the Arabic writer A i B i r u n i ( died
who spent m any
years in India is O f interest He says that the shape of the nu mer
als as also of the letters in India d iff ered in di fferent localities and
that the Arabs selected fro m the various form s the m ost suitable An
Arabian astronom er says there was a mong people m uch di fference in
the use of sym bols especially of those for 5 6 7 and 8 The symbols
used by the Arabs can be traced back to the tenth century We find
m aterial diff erences between those used by the Saracens in the East
and those used i n the West B ut m ost surprising is the fact that the
symbols of both the East and of the West Arabs deviate so ex tra ordi
of
n ar il y fro m the Hindu D eva n ag ar i nu m erals ( = divine nu m erals )
to day and that they resemble much m ore closely the apices of the
Rom an writer Boethius This strange similarity on the one hand
and dissimilarity on the other is difficult to explain The m ost plau
sible theory is the one of Woepc k e : ( 1 )
that about the second cen
tury after Christ before the z ero had been invented the Indian nu
meral s were brought to Alexandria whence they spread to Rom e
and also to West Africa ; ( 2 )
that in the eighth century after the no
t a t i o n in In dia had been already m uch m o d ified and perfected by the
invention of the z ero the Arabs at B agdad got it from the Hindus ;
that the Arabs of the West borrowed the Columbus egg the z ero
3)
from those in the East but retained the O l d form s of the nine num er
als if for no other reason simply to be contrary to their political ene
that the ol d form s were rem embered by the West
m ies O f the East ; ( 4)
Arabs to be of Indian origin and were hence called Gu bar n u mer ats
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H Hank el , op
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p
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2 55.
,
A HISTORY OF M ATHE MATICS
2
In astronomy great activity in original research existed as early as
the ninth century The religious O bservances demanded by M oham
me dan i sm presented to astrono mers several practical proble m s The
M oslem do minions being of such enormous ex tent it rem ained in
“
some localities for the astronomer to determine which way the B e
liever must turn during prayer that he m ay be facing M ecca The
prayers and ablutions had to take place at definite hours during the
day and night This led to m ore accurate determinations O f tirn e To
fi x the e xact date for the M oha mm edan feasts it beca m e n eces
sary to observe m ore closely the motions of the m oon In addition to
all this the O l d Oriental superstition that e xtraordinary occurrences
in the heavens in som e mysterious way a ff ect the progress of hu man
1
a ffairs adde d increased interest to the prediction of eclipses
For these reasons considerable progress was m ade Astronomical
tables and instrum ents were perfected observatories erected and a
connected series of observations instituted This intense love for as
t ron om y and astrology continued during the whole Arabic scientific
period As in India so here we hardly ever find a man exclusively
de v oted to pure m athe matics M ost of the S O called mathematicians
were first of all astronomers
The first notable author of m athematical books was M oh amme d
ib n M u s a Al Kh ow ar i z mi who lived during the reign of Caliph A l
Our chief source of information about Al Khow
M amun ( 8 1 3 —8 3
entitled K i tab A i —
Fi hr i s t written by
r iz m i is the book of chronicles
and containing biographies of learned
A i N adi m about 98 7 A D
m en A l K h ow ari z mi was engage d by the caliph in m aking ex tracts
from the S i ndhi nd in revising the tablets of P tole m y in taking O b
and in measuring a degree of
serv a t i ons at Bag da d and Da m ascus
the earth s m eridian I mportant to us is his work on algebra and
arith metic The portion on arithm etic is not extant in the original
and it was not till 1 8 5 7 that a Latin translation of it was found I t
“
begins thus : Spoken has Algorit mi Let us give deserved praise to
God our leader and defender
Here the name of the author A l
K how a r i zmi has passed into A l g ori tmi fro m w hich com e our m o dern
word al gori thm signif ying the art of computing in any particular
2
way and the obsolete form au gri m used by Chaucer The arith
m etic of Kh ow ar izmi being based on the principle of position and
“
the Hindu m ethod of calcula t ion excels says an Arabic writer
all others in brevity and easiness and exhibits the Hindu intellect
and sagacity in the grandest i n ven t ions
This book was foll owed
by a large nu mber of arithm etics by later authors which diff ered
from the earl i er ones chiefly in the g reater variety of m ethods Ara
bian arithmetics generally contained the four operations with inte
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1
2
H
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H ank el , o
p
See L
1 9 1 2,
pp
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C
206
pp
K arp i n sk i ,
—2 0 9
.
ci t
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,
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2 26
“
—2 2 8
.
A u g r i ms t o n e s
i n M oder n L a n guag e N otes , V ol
.
27,
'
THE ARAB S
1 03
gers and fractions modelled after the Indian processes They ex
pla ined the operation O f casti ng ou t the o s also the regu l a f al sa an d
the regu l a du oru m f al s oru m som etimes called the rules of false po
sition and of double position or double false position by which
algebraical examples could be solved w ithout algebra The r egu l a
o was the assigning of an assu m ed value to th e
a
l
s
a
or
a
s
s
i
t
i
l
a
o
f
f
p
unknown quantity which value if wrong was corrected by some
“
process like the rule of three
I t was known to the Hindus and to
the Egyptian Ahmes D iophantus used a m ethod alm ost identical
1
with this The regu l a du oru m f al soru m was as foll ows : To solve an
equation f (x)V assume for the m om ent two values for x; namely
= A and (b)
= 3 and deter m ine the
x = a and x = b
Then form f (a)
f
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V —A = Ea
errors
and
V
—B = Eb then
th e
,
b
required
— 12°
i 2
it
d
b
a
generally a close approxim ation but is absolutely accurate whenever
x)
is
a
linear
function
of
x
(
f
We now return to Kh o warizmi and consider the other part O f his
—
work the algebr a This is the first book known to contain this word
itself as title Really the title consists of two words al j ebr w ai mu
the
nearest
English
translation
O
f
which
is
restor
a
tion
and
a
b
a
l
a
q
“
reducti on
By restoration was m eant the transposing of negative
term s t o the other side o the equation by reduction the uniting of
f
2
2=
—
=
sim ilar terms Thus x 2 x 5x+ 6 p asses by al j eb r into x 5x+
= x+ 6
The work on alge
2 x+ 6 ; and this by a hn u q ab al a into
7
bra like the arith metic by the sam e aut hor contains little that 18
original I t explains the elem entary operations and the solutions of
linear and quadratic equations From whom did the author borro w
his knowledge of algebra ? That it cam e entirely fro m Indian sources
“
is impossible for the Hindus had no rules like the restoration and
“
reduction
They were for instance never i n the habit of m aking
all term s in an equation positive as is done by the process of r es to ra
tion
Diophantus gi v es two rules which rese m ble somewhat those
of our Arabic author but the probability that the Arab got all his al
gebra fro m D iophantus is lessened by the considerations that he rec
og n i z e d both roots of a quadratic while D iophantus noticed only one ;
and that the Greek algebraist unlike the Arab habitually rejected
irratio n al solutions It would see m therefore that the algebra of
Al
Al Kh ow a ri z mi was neither purely In dian nor purely Greek
He gave the
K ho w ar izi m i s fa m e a mong the Arabs was great
2
2
2
=
=
a mples x + 1 0 x 3 9 x + 2 1 1 o n 3x+ 4 x which are use d by later
authors for instance by the poet and m athematician O mar Khayya m
“
2
The equatio n x + 1 0 x = 3 9 runs like a thread of gol d through the
algebras of several centuries ( L C Karpinski ) I t appears in the
algebra of A bu K ami l who d rew extensiv ely upon the work of A l
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1
H H an k e l ,
.
p
o
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ci t
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,
p
.
2 59
.
A HISTORY OF M ATHE M ATICS
04
Khow ariz mi Abu Ka m il , in turn , was the source largely drawn u pon
b v the I tahan , Leonardo of P isa , in his book of 1 2 0 2
The algebra of Al -Kh ow ariz mi conta i ns also a few meagre frag
ments on geometry He gives the theorem of the right triangle , but
proves it after H i ndu fashi on and on l y for the simplest case , when the
right triangle is isosceles He then calc ulates the areas of the tri
angle , parallelogram , and circle For 71 he uses the value 34, and also
the two Indi an , ar = w/E)and 7r =
S trange to say , the last value
.
.
.
.
.
was afterwards forgotte n by the Arabs and replaced by o thers less
accurate Al Kh owarizmi prepared astronomical tables whi ch about
1 00 0 A D
were revised by M asl ama ai M aj ri ti and are of importa nce
1
as containing not only the si n e function but also the tangent function
The former is evidently of Hindu origin the latter may be an addi
tion m ade by M asl ama and was formerly attributed to Abu l Wefa
Next to be noticed are the three sons of M u s a S akir , who lived in
B agdad at the court of the Caliph A l M a mun They wrote several
works of which we m ention a geometry containing the well known
formula for the area of a triangle expressed in terms of its sides We
are told that one of the sons travelled to G reece probably to collect
astronom ical and mathematical manuscripts and that on his way back
he made acquaintance with Tabit ibn Korra Recogniz ing in him a
talented and learned astronomer M ohamm ed procured for him a place
a mong the astronom ers at the court in Bagdad Tab i t i b n K orra
—
8
was born at Harran in M esopotamia He was proficient
6
( 3 90 1 )
not onl y in astronomy and m athematics but also in the Greek Arabic
and Syrian languages His translations of Apollonius Archimedes
Euclid P tolemy Theodosius rank among the best His disserta tion
on ami cabl e n u mbers (of which each is the su m of the factors of the
other )
is the first know n specimen of original work in m athematics on
Arabic soil I t shows that he was fa miliar w ith the P ythagorea n the
ory of numbers Tabit invented the foll owing rule for finding ami cable
nu mbers which is related to Euclid s rule for perfect num bers : If
—
2 —
1—
1—
= 2 "—
=
=
r
1
1 (n being a whole n um ber )
1
p 3
q 3 2
92
are three pri mes then a = 2 pq b 2 "r are a pair of a micable numbers
Thus if n = 2 then p 1 1 q = 5 r = 7 1 and a = 2 2 0 b = 2 84 Tabit
also trisected an angle
Tabit ibn Korra is the earliest writer outside of China to discuss
magic squares Other Arabic tracts on this subj ect are due to Ibn
2
A l Haitam and later writers
Foremost among the astronomers of the ninth century ranked Al
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S ee H S u t e r ,
D i e ast ron o mi sc he n Tafel n des M u hammed ib n M ii sa Al
K hwari mi i n der B e a rb ei t u n g des M asl a ma ib n A hme d Al M adj r i ti u n d de r L a t e i n
’
U eb er se t zu n g d es A t hel h ard v on B a t h , i n M emoi res de l A cad émi e R des S ci ences
et des L ettr es de Dan emar k , C op en hag u e , 7 1116 S , S ec t io n des L e t res, t I I I , no 1 ,
1
z
”
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t
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1 9 14
2
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S ee H S u t e r , Di e M a themati k er
PD 3 6 . 93 . 1 3 6 . 1 39. 1 40 . 1 46 , 2 1 8
.
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u
.
A stron omen der A raber
u
.
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i hre
Werk e
,
1 900 ,
A HISTORY OF M ATHE M ATICS
06
last Arabic translators and co mmentators of Greek authors The fact
that he estee m ed the algebra of M ohamm ed ibn M usa Al K how ariz imi
worthy of his comm entary indicates that thus far algebra had m ade
little or no progress on Arabic soil Abu l Wefa invented a method for
computing tables of sines which gives the sine of half a degree correct
to nine deci mal places He used th e tangent and calculated a table of
tangents In considering the sha dow triangle of sun dials he intro
d u c e d also the secan t and cos ecan t
Unfortunately these new trigo
n ome t r i c fu n ctions and the discovery O f the m oon s variation ex
cited appare n tly no notice a m ong his conte mporaries and follo w ers
“
A treatise by Abu l Wei a on geometric constructions indicates that
e ff orts were being m ade at that ti me to i mprove draughting I t con
tains a neat construction of the corners of the regular poly edron s on
the circum scribed sphere Here for the first tim e appears the con
dition which afterwards becam e very famous in the Occident that
the construction be e ffected with a single O pening of the compasses
K u hi the second astrono mer at the observatory of the em ir at
Al —
Bagdad was a close stu dent of Archim edes and Apollonius He solved
the proble m to construct a segm ent of a sphere equal in volu me to
a given segment and having a curved surface equal in area to that of
a nother giv en segment He Al S ag an i and Al B iru ni m ade a stu dy
of the trisection of angles Ab u l Ju d an able geometer solved the
problem b v the intersection of a parabola with an equilateral hyper
bola
The Arabs had already discovered the theore m that the su m of two
cubes can ne v er be a cube This is a special case of the last theore m
of F erma t Ab u M oh am m e d Al Kh oj an di of C ho ra ssan thought he
had prov ed this His proof now lost is said to have been defective
Several centuries later B eha Eddin declared the impossibility of
3
3= 3
x +y
z
Creditable work in theory of nu mbers and algebra was
done by Al Kar k hi of Bagdad who liv ed at the beginning of the ele v
His treatise on algebra is the greatest algebraic work
e n th century
of the Arabs In it he appears as a discip l e of D iophantus He was
the first to operate with higher roots and to sol v e equations of the
2
"
form x + ax = b For the solution of quadratic equations he gives
both arithmetical and geo m etrical proofs He was the first Arabic
author to give and prove the theorem s on the summation of the se
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r Ies :
1
2
( + l
+
1
1
3
g
a
( 1+
+
Karkhi also busied hi mself with in determ inate analysis He
showe d S kill in han dling the m ethods of D iophantus but ad ded no
thing whatever to the stock of knowledge already on hand Rather
surprising is the fact that A l K a rk h i s algebra S hows no traces what
Al
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THE
ARAB S
19
7
ever of Hindu indeterminate analysis B ut most astonishing it is
that an arithm etic by the sam e author completely excludes the Hindu
numerals I t is constructed wholly after Greek pattern Abu l Wefa
also in the second half of the tenth century wrote a n arith metic in
which Hindu nu merals find no place This practice is the very oppo
site to that of other Arabian authors The question why the Hindu
nu merals were ignored by so e minent authors is certainly a puzz le
Cantor suggests that at one tim e there may have been rival schools
O f which one followed al most exclusively Greek m athe m atics
the
other Indian
The Arabs were fam iliar with geometric solutions of quadratic equa
tions Attempts were n ow m a de to solve cubic equations geo metri
cally They were led to such solutions by the stu dy of questions like
the Archi medean problem de man ding the section of a sphere by a
plane so that the two segments shall be in a prescribe d ratio The
first to state this proble m in form of a cubic equation was Al M ah ani
of Bagdad while Ab u J a f ar Al ch azin was the first Arab to sol v e the
equatio n by co nic sections Solutions were given also by A l Kuhi
Al Hasan ibn A l Haita m and others
Another diffi cult proble m to
determine the S ide of a regular heptagon required the construction of
3— 2 —
the S ide from the equation x x 2 x+ 1 = 0 I t was attempted by
m any and at last sol v ed by Abu l Jud
The one who did m ost to elevate to a method the solution of alge
b rai c e quations by intersecting conics was the poet O mar Kh ayyam
of Chorassan (a bout 1 0 4 5
He divides cubics into two classes
the tri nomial and quadrinomial and each class into fa milies and spe
cies Each species is treated separately but according to a general
plan He believed that cubics could not be solved by calculation nor
quadratics by geom etry He rejected negative roots an d often
b i—
failed to discover all the positive ones Attempts at b i qua drat i c
4
1
equations were made by Abu l Wefa who solved geom etrically x a
4
3
and x + ax b
The solution of c u bi c e q ua t i on s by intersecting conics was the great
est achievement O f the A rabs in algebra The foundation to this work
had been laid by the Greeks for it was M en ae chm u s who first con
3
3— =
3
structed the roots of x a o or x 2 a 0 It was not his aim to
find the number corresponding to x but simply to determine the side
The Arabs on the other
a
t of a cube double another cube of S ide a
hand had another object in vie w : to find the roots of given num erical
equations In the Occident the Arabic solutions of cubics remained
u nknown until quite recently D escartes and Thom as B aker invented
these constructions ane w The works of Al Khayyam A l Karkhi
Abu l Jud show how the Arabs departed further and further fro m
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L M att hi essen , Gr u ndzu ge der A n ti ken a nd M oder n en A l gebra der L i tteral en
hi
n
1
c
L
i
i
u
w
i
M
a
tt
esse
8
0
l
e
i
h
u
n
e
n
e
1
8
8
2
L
d
G
( 3 1 90 6 w as p rofesso r
p g,
7 , p 9 3
g
g ,
o f physi c s a t Ros o c k
1
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z
t
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.
)
A H ISTORY
108
OF M ATHE M ATICS
-
the Indian methods and placed themselves more immediately under
Greek infl uences
With Al Karkh i and O mar Khayya m mathematics among the
Arabs of the East reached fl ood mark and now it begins to ebb B e
tween 1 1 0 0 and 1 30 0 A D com e the crusades with war and bloodshed
during which European Christians profited m uch by their contact with
Arabian culture then far superior to their o wn The crusaders were
not the only adversaries of the Arabs D uring the first half of the
thirteenth century they had to encounter the wild M ongol ian hordes
and in 1 2 56 were conquered by them under the leadership of H u l aga
The caliphate at Bagdad now ceased to exist At the clos e of the four
t ee n th century still another e mpire was form ed by Ti m ur or Tamer
D uring such s w eeping turm oil it is not surprising
l a n e the Tartar
that science declined Indeed it is a marv el that it existed at all
D uring the supre macy of H ul ag u lived N a sir-Edd in ( 1 2 0 1
a man of broad cul ture and an able astronom er He persuaded H u
lagu to build hi m and his associates a large O bservatory at M araga
Treatises on algebra geometry arith m etic and a translation O f Eu
c l i d s El emen ts were prepared by hi m
He for the first tim e elabo
rated trigonom etry independently of astronomy and to such great
perfection that had his work been known Europeans of the fif teenth
1
century m ight have spared their labors
He tried hi s skill at a proof
of the paral lel postulate His proof ass umes that if AB is perpen di c
ular to CD at C and if another straight line ED P makes an angle
ED C acute then the perpendiculars to AB co mprehended betwee n
AB an d EF and drawn on the side of CD toward E are shorter and
shorter the further they are fro mCD His proof in Latin translation
was published by Wallis in
Even at the court of Ta m erlane in
Samarkand the sciences were by
no means neglected A group of
astronomers was draw n to this
a
Ul e g B e g ( 1 393
B court
grandson of Tamerlane was hi m
self an astronomer M ost prominent at this tim e was Al Kas h i the
author O f an arithm etic Thus during intervals of peace science
continued to be cultivated in the East for several centuries The
last Oriental writer was B eha Eddi n ( 1 54 7
His E ssence of
A r i thmeti c stands on about the sa m e level as the work of M ohamm ed
ibn M usa Kh ow a r iz mi written nearly 8 0 0 years before
“
Wonderful is the expansive power of Oriental peoples w ith which
upon the wings of the w ind they conquer half the world but more
wonderful the energy with which in less than two generations they
raise the mselves from the lowest stages O f cultivation to scientifi c
,
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B i bli otheca ma thema ti ca
7 , 1 8 93 , p 6
2
R B o n o l a , N on-E u cli dean Geometry, t ran s l by H S C a rsl aw , C h i cag o ,
pp 1 0 1 2
1
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1 9 1 7,
A HISTORY OF M ATHE M ATICS
1 10
Fro m this he derives the formulas for spherical right triangles This
sine form ula was probably known before this to Tabit ibn Korra and
1
others To the four fundamental formulas already given by P tolemy
he added a fifth discov ered by hi mself If a b c be the sides and
A B C the angles of a spherical triangle right angled at A then
cos E = cos b sin C This is frequently called Geber s Theorem
Radical and bold as were his innovations in spherical trigonometry
in plane trigono metry he foll owed slavishly the old beaten path of
“
“
the Greeks Not even did he adopt the Indian sine and cosine
“
but still used the Greek chord of double the angle
So painful was
the departure fro m old ideas even to an independent Arab !
It is a re markable fact that a mong the early Arabs no trace what
e v er of the use of the abacus can be discov ered At the close of the
thirteenth century for the first tim e do we find an Arabic writer Ib n
Al b anna who uses processes which are a mix ture of ab acal and Hindu
computation Ibn Al ban n a lived in B ugia an African seaport and it
is plain that he ca me under European influences and thence got a
knowledge of the abacus Ibn A lbann a and Abraham ibn Esra be
fore h im solv ed equations of the first degree by the rule of double
false position
After Ibn A lban na we find it used by A i —
K al s adi
a n d B eha Eddi n ( 1 54 7
If ox+ b = o let m and n be any two
nu mbers ( double false position
let also am+ b = M a n + b = N
—
—
=
+
then x (u M mN ) (M N )
3
Of interest is the approxi mate solution of the cubic x + Q = P x
which gre w out of the computation of x = si n
The m ethod is
shown only in this one nu merical example I t is giv en in M i r am
C hel ebi in 1 4 98 in his annotations of certain Arabic astrono mical
3
tables The solution is attributed to A tabeddi n Ja mshi d Write
x=
If
then a is the first app r oxi ma
tion x being sna l l We have Q = aP + R and consequently x = a +
3
say Then a + b is the second app roxima
( R+ a )
3
3
—
=
=
P
S
a
Hence
tion We have R bP + S a and Q (a + b)
3
3
—
=
x
P say
Here a + b
a + b+ (S
a + (a l b) P =
+ c Is the third approximation a n d so on In general the a mount of
computation is considerable though for find i ng x = sin I the m ethod
answered v ery well This example is the onl y known approximate
arithm etical solution O f an a ffected equation due to Arabic writers
Nearly three centuries before this the Italian Leonardo of P isa
carried the solution of a cubic to a high degree of approximation but
without disclosing his method
The latest prominent Spanish Arabic scholar was Al K al s adi of
Grana da who died in 1 486 He wrote the Rai si ng of the Vei l of the
“
“
The word gubar m eant originally dust and
S ci en ce of Gu bar
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S ee B i bli otheca ma thema ti ca , 2 S , Vol 7 , 1 893 , p 7
2
zen u moder n en A l g ebra , L eip i g ,
L M a tthi essen , Gr u ndzug e d A n ti /
3
S e e C an to r , op ci t V o l I , 3 rd Ed , 1 90 7 , p 7 8 2
1
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1 8 78,
p
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2 75 .
THE ARAB S
111
stands here for written arithmetic with nu m erals in contrast to men
tal arithmetic In addition subtraction and mu l tiplication the
result is written a bove the other figures The square root was indi
c a t ed by the initial Arabic letter of the word
idre m eaning root
j
“
particularly square root
He had symbols for the unkn own and
His
h ad in fact a considerable a mount of algebraic symbolis m
2
2
3
approximation for the square root wl a + b nam ely (4a + 3ab)
/ ( 4a +
is believed by S G unther to disclose a m ethod of continued frac
b)
tions without our m odern notation since
Al K al sadi s work e xcels other Arabic works in
the am ount of algebraic s ymbolism used Arabic algebra before h im
contained m uch less symboli sm then Hindu al gebra With Nessel
1
mann we di v ide algebras with respect to notation into three classes :
Rhetori cal al gebr as in which no s ymbols are used everything
(I )
being written out in words ( 2 )
S yn copa ted a l gebr as in which as in
S S e v erything is written out in words excep t that abbrev ia
sed for certain frequently recurring O perations and ideas
al gebr as
in which all forms and operations are repre
fully developed al gebraic sym bo l is m as for exa mple
2
x + 1 0 x+ 7
According to this classification Arabic works ( e xcepting
those of the later western Arabs)the Greek works of I amb l i chu s an d
Thymari das and the works of the early Italian writers and of Regio
montanus are r hetor i cal in form ; the works of the later western Ara b s
of Diophantus and of the later European writers down to about the
m iddle of the seventeenth century ( excepting Vieta s and O u g h t r e d s )
are syn copated in form ; the Hin d u works and those of Vieta and
O u gh t r ed and of the Europeans since the m i ddle of the se v enteenth
century
mboti c in form It is thus seen that the western Arabs
took an
n in m atters of algebraic notation and were
inferior
predecessors or contemporaries excep t the
Hindus
I n the year
Columbus discovered Am erica the M oors
lost their last
on Spanish soil ; the productive period of
Arabic science was passed
We have witnessed a laudable intellectual activity among the
Arabs They had the good fortune to possess rulers who by their
m u n ifi cen ce furthered scientific research
At the courts of the c a
l iph s scientists were supplied with libraries and observatories : A
large number of astronomical and m athem atical works were written
by Arabic authors I t has been said that the Arab s were learned
but not original With our present knowledge of their work this
dictu m needs revision they have to their credit several substantial
accomplishments They solved cubic equations by geometric con
struction perfected trigonom etry to a m arked degree and made nu
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G H F N esselmann , Di e A l gebra der Gr i echen , B erl in ,
.
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.
1 84 2 ,
pp
.
30 1
—
3 6
0
.
A HISTORY OF M ATHE M ATICS
merou s s maller
and astronomy
advances all along the line of m athematics physics
Not least of their services to science consists in this
that they adopted the learning of Greece and India and kept what
they received with care When the love for science began to grow
in the Occident they transmitted to the Europeans the valuable treas
ures of antiquity Thus a Se mitic race was during the D ark Ages
the custodian of the Aryan i ntellectual po ssessions
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A H ISTOR Y O F M AT H E MATI CS
1 14
ance of B e de th e Ven er abl e ( 6 7 2 —73 the most learned man of hi s
tim e He was a native of Wea rmou th in England His works con
tain treatises on the C ompu tu s or the computation of Easter time
and on fi ng er reckoning It appears that a fi ng er symbolism was then
w idely used for calculation
The corre c t determ ination of the time
of Easter was a proble m which in those days greatly agitated the
Church I t became desirable to have at least one m onk at each mon
as t ery who could determ ine the day of religious festivals and could
compute the calendar Such determ inations required some knowledge
of arith metic Hence we find that the art of calculating always found
som e little corner in the curriculum for the education of m onks
The year in which B ede died is also the year in which Al c u i n ( 73 5
was born Alcui n was educated in Ireland and was called to the
80 4)
court of Charle m agne to direct the progress of education in the great
Frankish E mpire Charle magne was a great patron of learning and
of learned m en In the great sees and m onasteries he founded schools
in which were taught the psal ms writing singing computation (com
s
and
gra
mm
ar
By
was
here
m
eant
probably
not
m
u
t
u
c
t
u
s
u
o
)
p
p
m erely the deter m ination of Easter ti m e but the art of computation
in general E xactly what m odes of reckoning were then e mployed
we have no m eans of knowing I t is not likely that Alcuin was fa miliar
with the apices of Boethius or with the Roman m ethod of reckoning
on the abacus He belongs to that long list of scholars who dragged
the theory of nu mbers into theology Thus the nu mber of beings
created by God who created all things well is 6 because 6 is a perfect
nu mber ( the s u m of its divisors being 1 + 2 +
8 on the other
hand is an i mperfect nu mber ( 1 +
hence the second origin
of m ankind e m anated fro m the nu mber 8 which is the number of souls
said to have been in Noah s ark
There is a collection of P roble m s for Quickenin g the M ind (pr op
which are certainly as old as 1 0 0 0 A D
os i ti on es ad acu endos i u ven es )
and possibly older Cantor is of the opinion that they were written
much earlier and by Alcuin The following is a speci m en of these
P roble m s : A d og chasing a rabbit which has a start of 1 50 feet
ju mps 9 feet every ti me the rabbit ju mps 7 In order to determine in
how many leaps the dog overtakes the rabbit 1 50 is to be di v id ed by 2
In this collection of proble ms the areas of triangular and quadrangular
pieces of land are found by the same formulas of approxim ation as
those used by the Egyp tians and give n by B oethius in his geo metry
“
An old problem is the cistern proble m (given the ti me in which
several pipes can fill a cistern singly to fi n d the time in which they
fi ll it jointly )
which has been foun d previously in Heron in the Greek
A n thol ogy and in Hin d u works
M any of the proble ms sho w that
the col lection was compile d ch i efly fro m Ro man sources The prob
l em which on account of its un i queness gi v es the m ost pos i tiv e testi
mony regarding the Roman origin is that on the interpretation of a
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INTROD UCTION OF RO M AN M ATHE M ATICS
1 15
wil l in a case where twins are born The problem is identical with the
Roman except that diff erent ratios are chosen Of the exercises for
recreation we mention the one of the wolf goat an d cabbage to be
rowed across a river in a boat hol ding only one besides the ferry man
so that the goat shall not eat
Q uery : H ow m ust he carry the m across
1
the cabbage nor the wolf the goat ? The solutions of the proble ms
for quickening the m ind require no further knowledge than the rec ol
lection of so me few form ulas used in surveying the ability to solve
linear equations and to perform the four funda mental operations with
integers E xtraction of roots was nowhere de manded ; fractions hardly
2
ever occur
The great e mpire of Charlemagne tottered and fell a l most i mm e
War and confusion ensued Scientific pur
dia t el y after his death
suits were abandoned not to be resu m ed until the close of the tenth
century when under Saxon rule in Germ any and Capetian in France
more peacef u l tim es began The thick gloo m O f ignorance co mmenced
to disappear The zeal with which the study o f mathe matics was now
taken up by the m onks is due principally to the energy and influence
—
of one mam G e rb e rt He was born in Aurillac in Auvergne After
receiving a m onastic education he engaged in study chiefly of m athe
m a ti c s in Spain On his return he taught school at Rhei m s for ten
years and beca me distinguished for his profound scho l arship By
King Otto I and his successors G e rb e r t was held in highest esteem
He w as elected bishop of Rheim s then O f Ravenna and finally was
made P ope under the nam e of Sylvester II by his form er E mperor
Otho III He died in 1 00 3 after a life intricately involved in many
political and ecclesiastical quarrels Such was the career of the great
est m athe matician oi the tenth century in Europe By his conte m
po ra ri es his mathe matical knowledge was considered wonderful
M any even accused him O f cri minal intercourse with evil spirits
G erb er t enlarged the stock O f his knowledge by procuring copies
O f rare books
Thus in M antua he found the geometry of B oethius
Though this i s of s mall scienti fic value yet it is of great importance
in history It was at that ti m e the principal book fro m which Euro
pean scholars could learn the elements of geom etry G e rb er t studied
it w ith z eal and is ge n erally believed hi mself to be the author O f a g e
H Weissenborn denied his authorship and clai med that the
ometry
book in question consists of three parts which cannot co me from one
and the sam e author M ore recent study favors the conclusion that
3
G erb e r t is the author and that he co mpiled it fro m diff erent sources
This geo metry contains little more tha n the one of B oethius but the
fact that occasional errors in the latter are herein corrected s hows that
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S G u n the r , Ges chi chte des mathem Un ter r i chts i m deu ts chen M i ttel al ter
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1 88 7 ,
2
p 32
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M C a n t or ,
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3
S G un t her ,
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V o l I , 3 A u fl , 1 90 7 , p 8 3 9
Ges chi chte der M athemati k , 1 Tei l , L e i p i g ,
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2 49
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B e rl i n ,
A HISTORY OF M ATHE M ATICS
1 16
”
the author had mastered the subj ect
The first mathe matical paper
“
of the M iddle Ages which deserves thi s nam e says Hankel is a
letter of G e rb e r t to A dal bol d bishop of Utrecht in which is explained
“
the reason why the area of a triangle ob tai n ed geometrically by
taking the product of the base by half its altitude diff ers fro m the
area calculated arithm etically according to the formula
used by surveyors where a stands for a side of an equilateral triangle
He gives the correct explanation that in the latter form ula all the
s mall squares in which the triangle is supposed to be divided are
counted in wholly e v e n though parts of the m proj ect beyond it
1
D E S mith calls attention to a great m edi eval num ber game called
I t was played
r i thm om ac h i a clai med by so me to be of Greek origin
as late as the six teenth century I t called for considerab l e ari thme ti
cal ability and was kno w n to G erb er t O ron c e Fi n é Thomas B rad
A board rese mbling a chess board was used R e
w a rdi n e and others
1
=
2 4 3 of 7 2 4 2
l a t ion s like 8 1
3 6 + 7g of 3 6 were involved
7
G e rb e r t m ade a careful study of the arith metical works of B oethius
He himself published the first perhaps both of the foll owing two
works —A S mal l B ook on the D ivi si on of N u mbers and Rul e of Compu
They giv e an insight into the methods of cal c u
tal i on on the A bacu s
lation practised in Europe before the introduction of the Hindu nu
meral s G e rb er t used the abacus which was probably unknown to
Al cuin B e rn el i nu s a pupil of G erb e r t describes it as consisting O f
a s mooth board upo n which geom etricians were accustomed to stre w
blue sand and then to draw their diagra ms For arithmetical pur
poses the board was divided into 3 0 colu mns O f which 3 were reserved
for fractions while the re maining 2 7 were divi d ed into groups wi th
colu
ns
in
each
In
every
group
the
colu
m
ns
were
m
arked
r
s
e
c
m
e
p
3
and S (si n gu l ari s )
or
t i v el y by the letters C (cen tu m)D (decem)
M (mon as ) B e rn eli n u s gives the nine nu m erals used which are the
apices of Boethius and then re marks that the Greek letters may be
used in their place By the use of these columns any nu mber can be
written without introducing a z ero and all operations in arithmetic
can be perform ed in the sa me way as we execute ours without the col
u mu s but wi th the symbol for z ero Indeed the m ethods of adding
subtracting and m ultiplying in vogue among the abac i s ts agree sub
B ut in a di vision there is very great
s tant i all y with those O f to day
difference The early rules for division appear to have been fram ed
to satisfy the follo w ing three conditions : ( I )
The use of the multipli
cation table shall be restricted as far as possible ; at least it shall never
be required to m ultiply m entally a figure of two di gits by another of
one digit ( 2 )
Subtractions shall be avoided as m uch as possible and
The operation shall proceed in a purely
replaced by additions (3)
2
m echanical way without requiring trials
That it should be meces
sary to m ake such conditions see ms strange to us ; but it m ust be re
—
A m M ath M thl y Vol
8 9
H Hank el op ci t p 3 1 8
pp 7 3 8
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A HISTORY OF
1 18
f
Tr ansl ati on
o
M ATHE M ATICS
A r a bi c M a nu s cr i pts
By his great erudition and pheno menal acti v ity G erb e r t infused
new lif e into the study not only of m athe matics but also Of philosophy
P upils fro m France Germ any and I taly gathered at Rhei ms to enj oy
his instruction When they themselves becam e teachers they taught
of course not only the use of the abacus and geometry but also what
they had learned of the philosophy of Aristotle His philosophy was
known at first only through the writings of B oethius B ut the grow
ing enthusias m for it created a demand for his complete works Greek
texts were wanting B ut the Latins heard that the Arabs too were
great ad mirers of P e ripa t e ti sm and that they possessed translations
of Aristotle s works and comm entaries thereon This led them finally
to search for and translate Arabic m anuscripts D uring this search
m athe m atical works also came to their notice and were translated
into Latin Though som e f ew uni mportant works m ay have been
translated earlier yet the period of greatest activity began about 1 1 0 0
The z eal displayed in acquiring the M oha mmedan treasures of knowl
edge excelled even that of the Arabs the mselves when i n the eighth
century they plun dered the rich co ffers of Greek and Hindu science
A m ong the earliest scholars engaged i n translating manuscripts into
Latin was Ath el ar d of B ath The period of his activity is the first
quarter of the t w elfth century He travelled e xtensiv ely in Asia
M inor Egypt perhaps also in Spai n and braved a thousa n d perils
that he m ight acquire the language and science of the M oha mmedans
He made one of the earli est translations from the Arabic of Eucli d s
He translated the astronom ical tables of A l K ho w a rizmi
El emen ts
In 1 8 57 a m anuscript was found i n the l ibrary at Ca mbr i dge which
prov ed to be the arithm etic by A l K how a riz mi in Latin This trans
lation also is very probably due to A thel a rd
At about the sa m e tim e flourished P l ato of Ti voti or P l ato Ti bu r ti n u s
He e ff ected a translation of the astronomy of A l Battani and of the
S phceri ca of Theodosius
About the m iddle of the twelfth century there was a group of Chris
tian scholars busily at work at Tole do un der the lea d ersh ip of Ray
m ond then archbishop of Toledo A mong those who worke d un der
his d irection Joh n of S evi ll e was m ost prom inent He translated
works chiefly on Aristotelian ph i losoph y Of i mportance to u s is a
The rule for
l iber al ghoa ri s mi co mpiled by him from Arabic authors
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the
ad
bd
di v I S I on
bc
bd
ad
bc
of one
Th I S
f rac t I on
'
me
sa
by another
expl a n a t IO n I s
Is
proved as follows :
gi v en by the
a
c
d
d
t h Ir t ee n t h
cen
tury Germ an writer Jor dan u s N emorariu s On comparing works
like this with those of the a b a c i s t s we notice at once th e most striking
diff erence which shows that the two parties d rew from i n dependent
.
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T RANSLATION OF ARABI C M ANUSCRI P TS
1 19
sources It is argued by some that G erb er t got his apices and his arith
metical knowledge not fro m Boethius but fro m the A rab s in Spain
and that part or the whole of the geo metry of B oethius is a forgery
dating from the time of G erbe rt If this were the case then the writ
ings of G erber t would betray Arabic sources as do those of John of
Seville B ut no points of rese mblance are found G erb e r t could not
have learned from the Arabs the use of the abacus because all evidence
we have goes to S how that they did not employ it Nor is it probable
that he borrowed fro m the Arabs the apices because they were never
used in Europe except on the abacus In illustrating an example in
division m athe maticians of the tenth and eleventh centuries state
an example in Ro man nu merals the n draw an abacus and insert in it
the necessary nu mbers with the apices Hence it see ms probable that
the abacus and apices were borro w e d fro m the sam e source The
contrast between authors like John of Se v ille drawing fro m Arabic
works and the abac is ts consists in this that unl i ke the latter the
former mentio n the Hindus use the term al gori sm calculate with the
z ero and do not e mploy the abacus
The form er teach the extraction
of roots the ab aci s ts do not ; they teach the sexages im al fractions used
by th Arabs while the abaci s t s employ the duodecim als of the R o
q
mans
A little late r than John of Seville flourishe d G e r ard of C r e mo n a i n
Lo mbardy B eing desirous to gain possession of the A l mages t he
went to Toledo and there in 1 1 7 5 translated this great work of P t ol
emy
Inspired by the richness of M oha mm e d an literature he gave
hi mself up to its stu d y He translate d into Latin over 70 Arabic works
Of mathe matical treatises there were a mong these besides the A i
magest the 1 5 books O f Euclid the S phcer i ca of Theo dosius a work of
M enelaus the algebra of A l —
K ho w a ri z mi the astrono my of Jabir ibn
Afl ah and others less i mportant
Through Gerard of Cre m ona the
term si n u s was introduced into trigono metry Al Kh aw ar izmi s a l
gebra was translated also by Robert of Chester ; his translation prob
ably antedated Cre mona s
In the thirteenth century the z eal for the acquisition of Arabic
learning continued Fore most a m ong the patrons of science at this
ti m e ranked E mperor Frederick II of Hohenstaufen (died
Through frequent contact with M ohamm edan scholars he becam e
familiar with Arabic science He e mploye d a nu mber of scholars i n
tra n slating Arabic manuscripts and it was through him that we came
i n possess i on of a n ew translation of the A l mages t
Another royal
head d eserv ing m ention as a z ealous pro moter of Arabic science was
Alfonso X of Castile (di ed
He gathe re d aroun d hi m a nu m ber
of Jewish and Christian scholars who transl ated an d co mpiled astro
Astrono m ical tables prepare d
n omi c al works fro m Arabic sources
by two Jews spread rapi dly in the Occi dent and constitute d the bas i s
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M C an tor
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V ol I , 3 A u fl
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1 90 7 ,
p 8 79 ,
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c h ap t e r
40
.
A HISTORY OF M ATHE M ATICS
1 20
of all astronomical calculation till the sixteenth century The n u m
ber of scholars who aided in transplanting Arabic scie nce upon Chris
tian soil was large B ut we m ention only one G i ovanni C ampano of
Novara (about
who brought out a new translation of Euclid
which drove the earlier ones from the field and which formed the
1
basis of the printed editions
At the m iddle of the twelfth century the Occident was in possession
of the so called Arabic notation At the close of the century the
Hindu m ethods of calculation began to supersede the cu mbrous me th
ods inherited from Ro me Algebra with its rules for solving linear
and quadratic equations had been m ade accessible to the Latins The
geometry of Euclid the S phceri ca of Theodosius the astronomy of
P tole m y and other works were n ow accessible in the Latin tongue
Thus a great a mount of n ew scientific m aterial had com e into the
hands of the Christians The talent necessary to digest this hetero
g e n eo u s m ass of knowle d ge was not wanting The figure of Leonardo
of P isa adorns the vestibule of the thirteenth century
It is i mportant to notice that no work either on m athematics or
astronom y was translated direc tly from the Greek previous to the
fif teenth century
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The Fi r
s t A w ak en i n g
a nd
i ts S equ el
Thus far France and the B ritish Isles hav e been the headquarters
of m athe m atics in Christian Europe B ut at the beginning of the
thirteenth century the talent and activity of one man was suffi cient to
assign the m athe matical science a new ho me in I taly This man was
not a monk like B ede Alcuin or G e rb e r t but a laym an who found
time for scientific study L e o n ardo of P i s a is the man to whom we
owe the first renaissance of m athem atics on Christian soil He is also
called Fi bon acc i i e son of B on ac c i o His father was secretary at one of
the nu merous factories erected on the south and east coast of the M ed
i t err an ea n by the enterprising m erchants of P isa He m ade Leonar do
when a boy learn the use of the abacus The boy acquire d a strong
taste for m athematics and in later years during extensive travels in
Egypt Syria Greece and Sicily collected from the various peoples
all the knowle dge he could get on this subj ect Of all the m ethods of
calculation he found the Hindu to be unquestionably the best R e
turning to P isa he published in 1 2 0 2 his great work the Li ber A baci
A revised edition of this appeared in 1 2 2 8 This work contains the
knowledge the Arabs possessed in arith m etic and algebra and treats
the subject in a free and indepen dent way This together with the
other books of Leonardo shows that he was not merely a compiler
nor like other writers of the M iddle Ages a slavi sh im itator of the
form in which the subject had been previously presented The extent
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pp 33 8 , 339
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A HISTORY OF M ATHE M ATICS
122
—
B ritish M useum one English
is of about 1 2 30 50 another
is of 1 2 4 6 The earliest undoubted Arabic numerals on a grave s tone
are at P forzheirn in Baden of 1 3 7 1 and one at U l m of 1 3 88 The
earli est coins dated in the Arabic n umerals are as follows : Swiss 1 4 2 4
Austrian 1 4 84 French 1 48 5 German 1 4 8 9 Scotch 1 53 9 Engli sh I 5 5 1
The earl iest calendar with Arabic figures is that of KObel 1 5 1 8 The
form s of the nu m erals varied considerably The 5 w as the most
freakish An upright 7 was rare in the earlier centuries
In the fifteenth century the abacus with its counters ceased to be
used in Spain and I taly In France it was used later and it did not
disappear in England and Germany before the middle O f the seven
1
The m ethod of abac al computation is found in the
t een t h century
English exchequer for the last tim e i n 1 6 76 In the reign of Henry I
the exchequer was distinctly organized as a court of l aw but the fi nan
The term ex
c i al business O f the crown was also carried on there
chequer is derived fro m the chequered cloth which covered the table
at which the accounts were m ade up Suppose the sheriff was s u m
mon ed to answer for the full annual dues in m oney or in tallies
The liabilities and the actual payments of the sheri ff were balanced
by m eans of counters placed upon the squares of the chequered table
those on the one S ide of the table representing the value of the tallies
warrants and specie presented by the sheriff and those on the other
the a mount for which he was liable so that it was easy to see whether
“
the sheriff had m e t his O bligatio n s or not In Tudor times pen and
ink dots took the place O f counters These dots were used as late as
The tally upon which accounts were kept was a peeled
wooden rod split in such a way as to divide certain notches previously
cut in it One piece O f the tally was given to the payer ; the other piece
was kept by the exchequer The transaction could be verified easily
by fitting the two halves together and noticing whether the notches
”
tallied or nor Such tallies rem ained in use as late as 1 7 83
In the Wi n ter s Tal e (I V
Shakespeare lets the clown be embar
r assed by a proble m which he could not do without counters
Iago (i n
expresses his contempt for M ichael Cassio forsooth a
Othel l o i 1 )
“
3
great mathematician by calling h i m a counter caster
S O gen
eral indeed says P eacock appears to have been the practice of this
species of arith metic that its rules and principles form an essential
part of the arithm etical treatises of that day The real fact seem s to
be that the old methods were used long after the Hindu nu merals were
m anuscript
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i n t he En cycl ope di a of P u re M a themati cs ,
A ri thm e t i c
G eo rg e P eacoc k ,
L on don , 1 8 4 7 , p 40 8
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2
A r t i cl e
Exc heq u er
i n P al g rav e s D ic ti on ary of P ol i ti cal E con omy, L o n do n ,
1
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1 8 94
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i n for ma t io n , c o n s u l t F P B a r n a r d , The Cas ti n g Cou n ter and
H e g i ve s a l i st o f 1 5 9 e xt ra c t s f ro m E ng l is h
the Cou n ti n g B oar d, O xfo r d , 1 9 1 6
i n v en to ri es re ferr i n g t o co u n t i n g bo ard s a n d a l so pho t og rap h s o f r ec k o n i ng t a bl e s
a t B ase l a n d N ifrn be rg , o f rec k o n i n g c l o h s a t M u n i ch, e tc
3
For
a ddi ti o n a l
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FIRST AWAKENIN G
THE
SE Q UEL
A ND I TS
1 23
in co mmon and general use With such dogged persistency does man
c li ng to the old !
The L i ber A baci was for centuries one of the storehouses from
which authors got material for works on arith metic and algebra In
it are set forth the most perfect methods of calculation w ith integers
and fractions known at that tim e ; the square and cube root are ex
plained cube root nor having been considered in the Christian o eci
dent before ; equations of the first and second degree leading to prob
le ms either determinate or indeterm inate are solved by the methods
“
“
of s ingle or double position and also by real algebra He recog
2
n i zed that the quadratic x + c = bx may be satisfied by two values of x
He took no cognizance O f negative and im aginary roots The book
contains a large nu mber of problem s The following was proposed to
Leonardo of P isa by a m agister in Constantinople as a d iffi cult prob
l em : If A gets from B 7 denare then A s su m is fi v e f ol d B s ; if B gets
from A 5 denare then B s su m is seven fold A s H ow m uch has each ?
The Li ber A baci contains another problem which is of historical ih
t er es t because it was given with so m e variations by Ahme s 3 0 0 0 years
earlier : 7 old women go to Ro me ; each woman has 7 m ules each m ule
carries 7 sacks each sack contains 7 loaves with each loaf are 7 knives
each knif e is p ut up in 7 sheaths What is the s um total of all named ?
l
A ns
Following the practice of Arabic and of Greek and
Egyp tian writers Leonardo frequently uses unit fractions This was
d one also by other European wr i ters of the M iddle Ages
He ex
plained how to resolve a fraction into the su m O f unit fractions He
was one of the first to separate the nu merator from the denom inator
by a fractional line B efore his time when fractions were written in
Hindu Arabic nu merals the deno minator was written beneath the
numera tor wi thout any S ign of separation
In 1 2 2 0 Leonardo of P isa published his P r acti ca Geometri co which
contains all the knowledge of geo metry and trigonometry transm itted
to him The writings of Euclid and of so m e other Gr eek m asters were
known to him either from Arabic manuscripts directly or from the
translations made by his countrym en Gerard of Cremona and P lato
of Tivoli A S previously stat ed a principal source of his geom etrical
knowledge was P lato of Tivolis translation in 1 1 1 6 fro m the Hebre w
2
into Latin , of the L i ber embadoru m of Abraham S avasor da Leo
nardo s Geometry contains an elegant geom etrical demonstration of
Heron s formula for the area of a triangle as a function of its three
sides ; the proof rese mbles Heron s Leonardo treats the rich material
before hi m with S kill some originality an d Eu c l idean rigor
Of still greater interest than the preceding works are those contain
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M C an tor
S ee a p ro b l em i n t h e A hme s
V ol II , 2 A u fl , 1 90 0 p 2 6
p apy r u s b e l i e ve d t o b e o f t h e sa m e t yp e as t h i s
2
S ee M C u r t e , Ur k u nden zu r G eschi chte der M a thema ti k I Th e i l , L e i p i g , 1 90 2 ,
1
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A H ISTOR Y O F M ATH E M ATICS
1 24
ing Fibonacci s more original investigations We must here preface
that after the publication of the L i ber A baci Leonardo was presented
by th e astronom er D ominicus to E mperor Frederick II of Hohen
staufen On that occasion John of P alermo an imperial notary
proposed several proble ms which Leonardo solved promptly The
first (probably an old fa m iliar problem to him)
was to find a nu mber x
2
2
such that x + 5 and x 5 are each square nu mbers The answer is
5
7
=
=
His m asterly S O
x
3 2 ; for (3 1
5 (21
l u t i on of this I s g i ven i n his l i ber qu ad r a toru m a manuscript which was
not printed but to which reference is m ade in the second edition of
his L i ber A baci The problem was not original with J ohn of P alerm o
since the Arabs had already solved similar o nes Som e parts of Leo~
nar do s solution may have been borrowed fro m the Arabs but the
m ethod which he e m ployed of building squares by the su mm ation of
o dd nu m bers I S original with hi m
The second proble m proposed to Leonardo at the fa mous scientific
tournam ent which acco mpanied the presentation of this celebrated al
r a i s t to that great patron of learning E mperor Fre d erick II was
e
b
g
3
2
the sol v in g of the equation x + 2 x + 1 0 x = 2 0 As yet cubic equations
Instead of brooding stubbornly
h ad not been sol v ed a l gebraically
ov er this knotty proble m and after many failures still entertaining
new hopes of success he changed his m ethod of inquiry and showed
by clear and rigorous dem onstration that the roots of this equatio n
could not be represented by the Euclidean irrational quantities or in
other words that they could not be constructed with the ruler and
co mpass only He contented him self with finding a very close ap
proxi mation to the required root His work on this cubic is found in
the Fl os together with the solution of the fol lowing third probl em
given him by John O f P alerm o : Three m en possess in common an u n
know n sum of m oney t; the share of the first is E; that of the second 1;
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D esirous of depositing the
each takes at haz ard a certain a m ount ;
th e
2
,
deposi t s
z
6
'
su
3
at a safer place
m
first takes
only x the second carries y but deposits only
,
,
,
but deposits
x,
the third takes z and
,
3
Of the am ount deposited each one m ust receive exactly 3
,
in order to possess his share of the whole sum Find x y z Leonardo
shows the problem to be indeterminate Assuming 7 for the su m
drawn by each fro m the deposit he finds t = 4 7 x = 33 y = I 3 z = 1
One woul d have thought that after so brilliant a beginning the
sciences transplanted from M ohamm edan to Christian soil would
have enjoyed a stea dy a n d vigorous develop m ent B ut this was not
the case D uring the fourteenth a n d fifteenth centuries th e mathe
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A HISTORY OF M ATHE M ATICS
1 26
in abbreviations of Italian words such as p for pi n (m ore )
m for men o
2
co for cos a ( the unknown x)
c c for cen s o (x )
cece for cen s oce n s o
(less )
“
3
(x ) Our present notation has arisen by al most insensible degrees
as convenience suggested d ifferent m arks of abbreviation to different
authors ; and that perfect symbolic language which addresses itself
solely to the eye and enables us to take in at a glance the m ost c om
plicated relations of quantity is the resul t of a large series of sm all im
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We shall now m ention a few authors who lived during th e thirteenth
and fourteenth and the first half of the fifteenth centuries
We begi n with the philosophic wri tings of Tho mas Aq u in as ( 1 2 2 5
the great I talian philosopher of the M iddle Ages who gave in
the completest form the ideas of Origen on infinity A q ui n a s notion
of a continuu m particularly a linear continuu m m a de it potenl i al l y
di v isible to infinity since practically the d ivisions coul d not be carried
out to infinity There was therefore no minimu m line On the other
hand the poi nt is not a constituent part of the line since it does not
possess the property of infinite divisibility that parts of a line possess
nor can the continuum be constructed out of points However a
2
point by its m otion has the capacity of generating a line This con
ti n u um held a fi rm ascendancy over the ancient ato m istic d octrine
whi ch assu m ed m atter to be composed of very s mall in divisible par
t i c l es
No continuu m superior to this was created bef ore the nine
t een th century
Aquinas explains Zeno s argu ments against m otion
as they are given by Aristotle but hardly presents an y new point of
—
m
view The English an Ro g e r B a co n
1 2 94] l i kewise argued
against a continuum of in divisible parts different from points R e
newing argum ents presented by the Greeks and early Arabs he held
that the doctrine of indivisible parts of uniform siz e would m ake the
diagonal of a square commensurable with a side Likewise if through
the ends of an indivisible arc of a circle ra dii are drawn these ra dii
intercept an arc on a concentric circle of smaller radius ; from this it
would follow that the i nner circle is of the sam e length as the outer
circle which is i mpossible Bacon argued against infinity If tim e
were infinite the absurdity would follow that the part is equal to the
whole B acon s views were m ade known more wid ely through D u n s
S co tu s ( 1 2 6 5
the theological and philosophical opponent of
Thomas Aquinas However both argued against the existence of
in divisible parts (points ) D uns Scotus wrote on Zeno s paradoxi es
but without reaching new points of view Hi s commentaries were
annotated later by the Italian theol ogian Fr an c i s c u s d e P i ti g i am s
who expressed hi mself in favor of the a dmi ssion of the actual infinity
“
“
to explain the D ichotom y and the Ach i lles but fa i ls to ade
l
u
a
t
e
Sc holastic ideas on in fi n i tv and th e
q
y elaborate the subj ect
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C
W H ersch el M a t hema t i c s i n Edi n bur gh En cycl ope di a
R Wa ll n e r i n B i bl i otheca mathemati ca 3 F B d I V 1 90 3 pp
F
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THE FIRST AWAKENING AND ITS SE Q UEL
12
7
continuum find expression in the writings of B radw ardi n e the Eng
1
lish doctor pr of u ndi s
About the tim e of Leonardo of P isa ( 1 2 0 0 A
lived the German
monk J or dan u s N e m or an u s —1
who wrote a once fam ous work
on th e properties of nu mbers printed in 1 496 an d m odelled after the
arithm etic of B oethius The m ost trifling num eral properties are
treated with nauseating pedantry and prolixity A practical arith
metic based on the Hin d u notation was also written by hi m J ohn
H alif ax ( Sacro B osco died 1 2 5 6 )
taught in P aris and m ade an extract
from the A l mages t containing onl y the m ost elem entary parts of that
work This extract was for nearly 400 years a work of great popular
ity and standard authority as was also his arithm etical work the
Tr acl a l n s de axl e nn mer an dl
Other prom inent writers are A lb e r tu s
M agn u s ( 1 1 93 ? —1 2 80 )
a n d G e or g P e ur b ach ( 1 4 2 3 1 4 6 1 )
in Ger
m any I t appears that here and there som e of our m odern ideas were
anticipated by writers of the M id dle Ages Thus N i c ol e O r e sme
a bishop in Normandy first conceived the notion of
(about 1 3 2 3
fractional powers afterwar ds re discovered by S tevin and suggested a
3=
1%
%
8 O r esm e concluded that 4
notation Since 4 6 4 and 6 4
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In his notation
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154
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expressed
Is
or
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ma them atician s of the M iddle Ages possessed som e
O r esme even atte mpted a graphic representation
idea of a function
B ut of a nu meric
d ependance of one quantity upon another as found in D escartes
2
there is no trace a m ong the m
In an unpublished m anuscript O r esme found the su m of the infinite
5
series
an i nf
Such recurrent infinite series were
35 +
formerly suppose d to have m a de their first appearance in the eight
The use of infinite series is explained also in the L i ber
een th century
3
de tri pl i ci motn by the P ortuguese m athe m atician A l vor a s Thomas
i n 1 50 9 He gi v es the d ivision of a line segm ent into parts represent
ing the term s of a convergent geometric series ; that is a segm ent A B
P iB :P i + 1
is d iv ided into parts such that A B :P IB = P 1B :P 2B
B
Such a d ivision of a line segm ent occurs later in Napier s
kinematical d iscussion of logarithm s
Th o mas B r adw ar din e ( about 1 2 90
archbishop of Canter
bury studied star polygons The first appearance of such polygons was
with P ythagoras and his school We next m eet with such polygons
in the geometry of B oethius an d also in the translation of Euclid from
th e Arabic by A th el ard of Bath
To England falls the honor of hav
ing produced the earliest European writers on trigonom etry The
—
F C j i A m i M t/ M
thl y V l
9 5 pp 4 5 4 7
—
H W l t
B i bl i th
S
l
V
m ll m l
3
9 3 pp
3
5 45
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1e e I n e r 1n
Etu des
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2 2,
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L eon ar d da Vi nc i , V o l I I I , P ari s , 1 9 1 3 , pp 393 , 5 40 , 5 4 1 , by
P i e rr e D u h em ( 1 8 6 1 1 9 1 6 o f t h e U n i v e r si ty o f B or d eau x ; se e a l so Wi el e i t n er i n
168
B i bl i otheca ma thema ti ca , V o l 1 4 , 1 9 1 4 , pp 1 50 —
3
S ee
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A HISTORY OF M ATHE M ATICS
1 28
writings of B radw ardi n e of Richard of Wallin gford and John M aud
i th both professors at Oxford and of Si m on B redon of Win che comb e
contain trigonom etry drawn fro m Arabic sources
The works of the Greek monk M axi mu s P l an u de s (about 1 2 6 0
are of interest only as showing that the Hindu nu m erals were
then known in Greece A writer belonging like P l an u des to the B y
z an t i n e school was M an u e l M os c h opu l u s who lived in Constantino
ple in the early part of the fourteenth century To him appears to be
due the introduction into Europe of magic squares He wrote a treatise
on this subj ect M agic squares were kno w n before this to the Arabs
and Japanese ; they originated w ith the Chinese M edimv al astrol
og er s and physicians believed the m to possess m ystical properties and
to be a charm against plague when engraved on silver plate
Recently there has been printed a Hebrew arith m etical work by
the French J ew L evi ben Ger s on writte n in
and handed do w n
in several m anuscripts I t contains for mulas for the number of per
m utations and co mbinations of n things taken 13 at a ti m e It is worthy
of note that the earliest practical arith metic known to have been
brought out in print appeared anonymously in Treviso I taly in 1 4 7 8
“
and is referred to as the Tre v iso arithm etic
Four years later in
cam e out at B a mberg the first printed German arithm etic I t
1 48 2
is by Ul ri ch Wagn er a teacher of arith metic at N urnberg I t was
2
printed on parchm ent but only f ragments of one copy are now extant
According to E n e s trom P h C al an dri s De ari thmetri ca opn s cnl u rn
“
Florence 1 49 1 is the first printed treatise contain i ng the word z ero
it is found in so m e fourteenth century m anuscripts
In 1 494 was pr i n ted the S n mma de A ri thmeti ca Geometr i a P r opor
ti on e et P r opor ti on al i ta written by the Tuscan m onk Lu c a P a ci oli
who as we rem arked introduced several symbols in
( 1 44 5
algebra This contains all the knowledge of his day on arithm etic
algebra and trigono metry a n d is the first com prehensive work which
appeare d affi fi fi é f fl baci of Fibonacci I t contains li ttle of im
portance which can n o t b e found in F i b o n acci s great work published
three centuries earlier P ac i oli ca m e in personal touch with two ar
3
—
i
t s t s who were also m athe m aticians L eon a rdo da Vi n ci
1
2
( 4 5 1 5 1 9)
and P i er del l a Fr an ces ca ( 1 4 1 6
D a Vinci inscribed regular
polygons in circles but did not distinguish between accurate and ap
I t is interesting to note that da Vinci was
proxi mate constructions
familiar with the Greek tex t of Archim edes on the measurement of
the circle P ier della Francesca advanced the theory of perspectiv e
and left a manuscript on regular solids which was published by
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B i bl i otheca ma themati ca , 3 S , V ol 1 4 , 1 9 1 6 , p 2 6 1
2
S ee D E S mi t h , Rar a ar i thmeti ca , B os ton a n d L on don , 1 908 , pp 3 , 1 2 , 1 5 ;
F U n g e r , M ethodi k der P r ak ti s chen A r i thmeti k i n H i s tor i s cher E n twi c k el u n g , L e ip
z ig , 1 8 88 , p 39
’
3
C o n su l t P D u hem s Etudes s ur L eonard dc Vi nci , P a ri s, 1 90 9
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EURO P E D URING TH E SIXTEENTH SE VE NTEENTH
A N D EIGHTEENTH CENTUR IES
,
We find it conv enient t o choose the time of the capture of Constan
t i n opl e by the Turks as the date at which the M id dl e Ages ended and
M o dern Ti m es began In 1 4 53 the Turks battere d the walls of this
celebrated m etropolis with cannon and finally capt u red the city ; the
Byzantine E mpire fel l to ri se no m ore Calam itous as was this event
to the East it acted fa v orably u pon the progress of learning in the
West A great nu mber of learned Greeks fled into Italy bringing with
the m precious m anuscripts of Greek literature This contri buted
vastly to the revi v ing of classic learning Up to this ti m e Greek mas
ters were known only through the often very corrupt Arabic m a nu
scripts but n ow they began to be studied from original sources and
in their own language The first English transl ation of Euclid was
m ade in 1 5 7 0 fro m the Greek by S i r H en ry B i l l i ngsl ey assisted by
1
John D ee
About the m id dle of the fifteenth century printing was
invented ; books beca m e cheap an d plentiful ; the printing press trans
form ed Europe into an au dience —
roo m Near the close of t he fifteenth
century A m erica was discov ered and soon after the earth was cir
The pulse and pace of the world began to quicken
c u m n av i g a t ed
M en s m inds beca m e less serv i le ; they becam e clearer and stronger
The in di st inctness of thought which was the characteristic feature of
m edi aeval learni n g began to be re m e died chiefly by the steady cultiva
tion of P ure M athe m atics and Astronomy D ogmatism was attacked ;
there arose a long struggle with the authority of the Church and the
establish ed schoo l s of ph il osophy The Copernican Syste m was set
up in opposition to the tim e honored P tolemaic Syste m The long
and eager contest between the two cul minated in a crisis at the ti m e
of Ga l i leo and resulte d in the victory of the new syste m Thus by
slow degrees the m in ds of m en were cut adrift from their old scholastic
moorings and sent forth on the wide sea of scientific inquiry to dis
cover new islands and continents of truth
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The Ren ai ss an ce
With the sixteenth century began a period of increased intellectual
activity The hu man m ind m ade a vast e ffort to achieve its freedom
Atte mpts at its em ancipation from Church authority had been m ade
before but they were st ifle d an d re n dere d abort ive The first great
and successful re v olt against ecc l es i astical authority was made in
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1
G B H al s t e d i n A m f
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ou r
1 30
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f M ath
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V ol I I ,
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1 8 79.
THE RENAISSANCE
31
Germany The n ew desire for j udging freely and in depen dently I n
matters of religion was prece ded and acco mpa n ied by a grow i ng spirit
of scientific inquiry Thus it was that for a ti m e Germany led the
van in science She produced Regi omon tan u s C oper ni cu s Rhwti cu s
and K epl er at a period when France and England had as yet brought
forth hardly any great scientific thinkers This re markable scientific
producti v eness was no doubt d u e to a great extent to the co mmercial
prosperity of Germany M aterial prosperity is an essential condition
for the progress of knowle dge As long as every individual is obliged
to collect the necessaries for his subsistence there can be no leisure
for higher pursuits At this ti m e Germany had accu mulated con
s i der abl e wealth
The Hanseatic League commanded the trade of
the North Close co mmercial relations existed between Germany and
Italy Italy too excelled in comm ercial activ ity and enterprise
We need only m ention Venice whose glory began with the crusades
and Florence with her ban k ers and her manufacturers of silk and wool
These two cities becam e great intellectual centres Thus Italy too
prod uced m en in art literature and science who shone forth in fullest
splen dor In fact Italy was the fatherland of what is term ed the R e
naissance
For the first great contributions to the m athe matical sciences we
m ust therefore look to Italy and Ger many In Italy brill i ant acces
sions were m a de to al gebra in Germany progress was m a de in astron
o my a n d trigono metry
On the threshold of this new era we m eet in Germany with the figure
of John M ueller more general l y called R e g io mon tan u s ( 1 43 6
He studied as
C hiefly to him we owe the re v i v al of trigono m etry
t ro n omy and trigono m etry at Vienna u nder the ce l ebrated George
P eu rb ach
The l atter percei v ed that the existing Latin translations
of the Al magest were full of errors and that Arabic authors had not
rem ained true to the Greek original P eu rb ach therefore began to
m ake a tra nslation directly fro m the Greek B ut he di d not live to
finish it His work was continued by Regiomontanus who went be
yond his m aster Regiomontanus learned the Greek language from
Cardinal B essarion whom he followed to Italy where he remained
eight years collecting manuscripts from Greeks who had fled thither
from the Turks In addition to the transl ation of and the c omm en
tary on the A l mag es t he prepared transl ations of the C on i cs of Apol
l on i u s of Archi m e d es and of the m echanical works of Heron Regio
m ontanus and P e u rb ach adopte d the Hindu s i n e in p l ace of the Greek
c hord of dou bl e the a rc
The Greeks and afterwards the Arabs divided
the radius into 6 0 equal parts and each of these again into 6 0 sm aller
ones The Hin du expressed the length of the rad ius by parts of the
circu mference saying that of the 2 1 6 0 0 equal d iv isions of the latter
i t took 343 8 to m easure the radius Regiomontanus to secure greater
precision constructed one table of sines on a radius divided into
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HISTORY
A
13 2
or M ATHE M ATICS
parts and another on a ra di us divi ded decirn al l y into
divisions He emphasiz ed the use of the tangen t in trigo
Following out some ideas of his master he calcu l ated a
n om e t ry
table of tangents Germ an m athematicians were not the first Euro
peans to use this function In England it was known a century earlier
to B radw ar din e who speaks of tangent (u mbr a vers a)
and cotangen t
(u mbr a r ecta)and to John M au di th Even earlier in the twelf th
century the u mbra versa and u mbr a r ecta are used in a translation from
Arabic into Latin e ff ected by Gerard of Cremona of the Tol edian
Tables of Al Z ark ali who lived in Toledo about 1 0 80 Regiomontanus
was the author of an arithm etic and also of a co mplete treatise on
trigonom etry containing solutions of both plane and spherical tri
angles So m e innovations in trigonom etry formerly attributed to
Regiomontanus are now known to have been introduced by th e Arabs
before him Nevertheless m uch credit is due to hi m His complete
m astery of astronom y and m athematics and his enthusiasm for them
were of far reaching influence throughout Germ any So great was his
reputation that P ope Six tus I V call ed him to I taly to improve the
calendar Regiomontanus left his beloved city of N ii rn b erg for Rom e
where he died in the following year
montanus
trig onom etry and
A fter the tim e of P e u rbach and Regio—
—
t
h
e calculation of tables c ofit in u edl t o occupy Germ an schol
l
e sp e i
”
g al y
ars M ore refined as tron omi c ar ifl s t rm en t s were made which gave
observations of greater precision ; but these would have been useless
without trigonometrical tables of correspon ding accuracy Of the sev
eral tables calculated that by Georg J oachi m of Feldkirch in Tyrol gen
—
e ti c u s ( 1 5 1 4 1 5 6 7 )
deserves special m ention He cal
e ral l y called Rh a
and from 1 0
c u l a t e d a table of S ines with the radius =
to 1 0 and later on another with the radius =
"
and proceeding fro m 1 0 to
He began also the construction of
tables of tangents and secants to be carried to the sam e degree of
accuracy ; but he died before finishing them For twelve years he had
had in continual e mploym ent several calculators The work was c om
This was
pl e t ed in 1 5 96 by his pupil Val e n ti n e O th o ( 1 5 50 ?
—
indeed a gigantic work a monument of Germ an d iligence and in de
fatigable perseverance The tables were rep ubl i shed in 1 6 1 3 by B ar
—
m
i
s
u
of Heidelberg who spared no pains
th ol o alu s P ti c s £1 56 1 1 6 1 3 )
to free theri ifi eiiY ors P itiscus was perhaps the first to use the word
trigonometry
Astronomical tables of so great a degree of accu
racy had never been dream ed of by the Greeks Hindus or Arabs
That Rhaet ic u s was not a ready calculator only is indicated by h i s
views on trigonometrical lines Up to his ti me the trigonometric
functions had been considered always with relation to the arc ; he was
the first to construct the right triangle and to make the m depend d i
It was fro m the right triangle that Rhee ti c u s
re c tl y upon its angles
got his idea of calculating the hyp otenuse ; i e he was the first to plan
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HISTORY
A
1 34
O F M ATHE M ATICS
Each contestant proposed thirty problems The one who could
solve th e greatest num ber within fifty days should be the victor Tar
taglia solved the thirty proble ms proposed by Floridas in two hours ;
Floridas could not solve any of Tartaglia s From now on Tartaglia
studied cubic equations with a will In 1 54 1 he discovered a general
3i
2= t
solution for the cubic x px
o by transfor ming it into the for m
3i
x
The news of Tartaglia s victory spread all over Italy
Tartaglia was entreated to m ake known his m ethod but he declined
to do so saying that after his completion of the translation from the
Greek of Euclid and Archimedes he would publish a large al gebra
containing his m ethod B ut a scholar from M ilan nam ed H i e r oni mo
after many solicitations and after giving the
C ar d an o ( 1 50 1
most solemn and sacred pro mises of secrecy succeeded in obtaining
from Tartaglia a knowle dge of his rules Cardan was a singular mix
ture of genius folly self conceit and mysticism He was successively
professor of m a them atics and m edicine at M ilan P avia and Bologna
In 1 5 70 he was imprisoned for debt Later he went to Rom e was
admitted to the college of physicians and was pensioned by the pope
At this tim e Cardan was writing his A r s M agn a and he knew no
better way to crown his work than by inserting the m uch sought for
rules for solving cubics Thus Card an broke his m ost solem n vows
and publi shed i n 1 54 5 in his A r s M agn a Tartaglia s solution of cubics
However Cardan did cred i t his friend Tartaglia with the discov ery
of the rule Nevertheless Tartagl i a beca m e d esperate His most
cherishe d hope of giving to the world an imm ortal work which should
be the m onum ent of his deep learning and power for original research
was suddenly d estroyed ; for the crow n intended for hi s work had
been snatched away His first step was to write a history of his i n
to completely annih ilate his enem ies he challenged
v e n t i on ; but
Cardan and his pupil Lodovico Ferrari to a contest : each party
S hould propose thirty one questions to be solved by the other within
fifteen days Tartaglia solved most questions in seven days but the
other party did not send in their solutions before the expiration of th e
fifth m onth ; m oreover all their so l utions except one were wrong A
replication and a rejoinder followed Endless were the problem s pro
pose d and sol v ed on both sides The dispute produce d m uch chagrin
and heart burnings to the parties and to Tartaglia especially who
m e t with m any other d isappoint m ents After having reco v ere d hi m
self again Tartaglia began in 1 556 the publ i cation of the work which
he had h ad in hi s m in d for so long ; but he die d before he reached the
consideration of cubic equations Thus the fon dest wish of his life r e
m ained unfulfilled H O W mu c h c r e di t for the algebraic solution of the
general cubic I S due to Tartaglia an d how m uch to D el Ferro it is now
impossible to ascertain d efinitely D el Ferro s researches were ne v er
publ i she d and were lost We k now of the m only through the rem arks
of Car dan and his pupil L Ferrari who say that Del Ferro s an d Tar
22
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THE RENAISSAN CE
”
were alike Certain it is that the customary des ig
“
nation Cardan s soluti on of the cubic ascribes to Cardan what
belongs to one or the other of his predecessors
Remarkable is the great interest that the solution of cubics excited
throughout I taly I t is but natural that after this great conquest
mathe maticians S hould attack b i quadratic equations A S in the case
of cubics so here the first impulse was given by Colla who in 1 540
4
2
proposed for solution the equation x + 6 x + 3 6 = 6 0 x To be sure
Cardan had studied particular cases as early as 1 539 Thus he solved
2
4
3
the equation 1 3x x + 2 x + 2 x+ 1 by a process similar to that em
ployed by D iophantus and the Hindus ; nam ely by adding to both
2
S ides 3 x and thereby rendering both nu m bers co mplete squares
B ut
Cardan failed to find a general solution ; it re mained for his pupil
Lo dovi c o Ferrari ( 1 5 2 2 —1 56 5 )
of B ologna to make the brilliant di s
c o ve ry of the general solution of bi quadratic equations
Ferrari re
2— 2 =
2
d u c ed Colla s equation to the form (x l 6 ) 6 0 x+ 6 x
In order to
give also the right m ember the form of a co mplete square he added to
both members the expression
containing a new u n
2
known quantity y This gave him
2 y)
x + 6 0 x+
2
—
—
( 1 2y l y )The condition that the right m e mber be a complete square
2 =
is expressed by the cubic equation ( 2 y+ 6 )
( 1 2y+ y )90 0 Ex tract
2
—
ing the square root of the bi quadratic he got x + 6 + y= x xf 2y+ 6
t ag l i a
’
methods
135
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Solving the cubic for
y
and substituting it remained
,
only to determine x from the resulting quadratic L Ferrari pursued
1
a similar m ethod with other nu merical b i quadratic equations Car
dan ha d the pleasure of publishing this discovery in his A rs M agn a
in 1 54 5 Ferrari s solution is som etim es ascribed to R B ombel l i but
he is no m ore the discoverer of it than Cardan is of the solution called
by his name
To Cardan algebra I s m uch indebted Inhis A rs M agn a he takes
notice of negative roots of an equation calling them fi cti ti ou s while
the positive roots ar e called real He paid some attention to compu
t a t i o n s involving the square root of negative nu mbers but failed
to recogni z e i maginary roots Cardan also observed the diffi culty
in the irre d ucible case i n the cubics which l ike th e quadrature of the
“
circle has S ince so m uch torm ented the perverse ingenuity of m athe
m aticians
B ut he di d not un derstan d i ts nature It rem aine d for
Raph a e l B omb ell i of B ologna who published in 1 5 7 2 an algebra of
great m erit to point out the reality of the apparently imaginary ex
pression which a root assu mes also to assig n i ts value when rational
and thus to lay the foundation of a more intim ate knowledge of imagi
nary quantities Cardan was an inveterate gambler In 1 6 6 3 there
was published posthu mously his gambler s m anual De l u do al ece
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H H an k el ,
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op ci t
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p 36 8
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A HISTORY OF M ATHE M ATICS
1 36
which conta ins discussions relating to the chances favorable for throw
ing a particular nu mber with two dice and also with three dice Car
dan considered another problem in probabilities S tated in general
terms the problem is :What is the proper div ision of a stake between
two players if the gam e is interrup ted and one player has taken s 1
1
points the o ther s z points s points being required to win Cardan
2+
gives the ratio ( 1 + 2 +
Tar taglia
+ [s
+ [s
—
gives (s + s l —s 2)
s
s
B oth of these answers are wrong Car
/( + 2 s
dan considered also what later became know n as the f P eter sbu rg
p roblem
After the brilliant success in solving equations of th e third and
fourth degrees there was probably no one who doubted that wi th
aid of irrationals of hi gher degrees the solution of equations of any
degree whatever could be found B ut all atte mpts at the algebraic
solution of the quintic were fruitless and finally Abel demonstrated
that all hopes of finding algebraic solutions to equations of higher
than the fourth degree were purely Utopian
Since no solution by radicals of equations of higher degrees co uld
be found there rem ained nothing else to be done than the d evising of
processes by which the real roots of nu merical equations could be
found by approximation The Chinese m ethod used by them as early
as the thirteenth century w as unknown in the Occid ent We have
seen that in the early part of the thirteenth century Leonardo of P isa
solved a cubic to a high degree of approxi mation but we are ignorant
of his m ethod The earliest known process in the Occident of ap
m
to
a
root
of
an
a
fected
nu
erical equation was invented by
ro
ac
h
i
n
f
g
p
Nicolas Chu q u e t who in 1 4 84 at Lyons w rote a work of high rank
entitled L e tri party en l a sci en ce des n ombr es It was not printed until
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If
then
C hu q u e t
takes the interm ediate value
it;
as a closer approxim ation to the root x He finds a series of successive
interme di ate values We stated earlier that in 1 4 98 the Arabic writer
3
M iram Chelebi gave a method of solv ing x + Q = P x which he attrib
utes to A tab eddi n J amshid This cubic arose in the computation of
.
.
.
x = S In
The earl iest printed m ethod of approxim ation to the roots of af
f ec t ed equations is that of Cardan who gave it in the A rs M agn a
1 54 5 under the title of r egu l a au r ea
It is a skilful application of
“
the rule of false position and is applicable to equations of any de
gree This m ode of approxim ation was exceedingly rough yet this
fact hardly explains why Clavius S te v in and Vieta did not refer to it
”
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1
M C an tor , I I
Au fl , 1 900 , pp 50 1 , 5 2 0 , 5 3 7
2
P ri n t ed i n t h e B u l l eti n o B on compagn i , T xii i , 1 880 ; see pp 6 5 3 6 5 4 S ee a l so
“
F C aj o ri ,
A H i s t o ry of t he A ri t hme t i cal M e t ho d s o f A pp ro xima t io n to the
Roo t s of N u meri c al Eq u a t i o n s o f o n e U n k n ow n Q uan ti ty
i n C ol or ado Col l eg e
P u bl i cati on , G en eral S eri es N os 5 1 an d 5 2 , 1 9 1 0
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A HISTORY OF M ATHE M ATICS
1 38
also to goniom etry de v eloping such relations as sin a = sin
a
a
—w 1th
—
—sm ( 6 0 a )csc
ctn a + cs c a = t a n
‘
a
,
.
°
,
)
th e
2
aid of whi ch he could compute from the functions of angles below
essentially
th e functions of the re maining angles below
3 0 or 4
by addition and subtraction alone Vieta is the first to apply alge
braie transformation to trigonometry particularly to the m ultisection
of angles Letting 2 cos a = x he expresses cos n a as a function of x
for all integers n < 1 1 ; letting 2 sin a = x and 2 S in 2 a = y he expresses
“
2
Vieta exclaims : Thus the analysis
2x
s i n n a in ter m s of x and y
of angular sections involves geom etric and arithm etic secrets which
hitherto have been penetrated by no one
An ambassador fro m Netherlands once told Henry IV that France
did not possess a S ingle geom eter capable of sol v ing a problem pro
pounded to geo meters by a B el gian mathe m atician A dri an u s R o
m anus I t was the solution of the equation of the forty fif th degree
5
3
11
43
45 =
—
C
+ o4 sy
4 sr 3 7 95y + 95 6 34y
4 5y + r
°
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Henry IV called Vieta who having al ready pursued S i m ilar inv esti
g a t i o n s saw at once that this awe inspiring proble m was simply the
equation by which C 2 S i n cl)was expressed in terms of y = 2 si nf g d);
it was necessary only to di v ide an angle once
that S ince 4 5 = 3
—
into 5 equal parts and then twice into 3 a division which could be
efl ec t ed by corresponding equations of the fifth and third degrees
Bril liant was the discovery by Vieta of 2 3 roots to this equation i n
stead of only one The reason why he did not find 4 5 solutions is
that the remaining ones involve negative S ines which were u n in tel
ligible to him D etailed investigations on the famous old problem
Of the section of an angle into an odd nu mber of equal parts led Vieta
to the discovery of a trigonom etrical solution of Cardan s irreducible
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—
cos q5) 3
case in cubics He appl ied the equation ( 2
2 = 2
3
>
x
a
a
6
when
a
2 c os qS to the solution of x
3b by placing x
3
and determining q rom b= 2 a c os qb
2a
The m ain principle e mployed by him in the solution of equations
is that of r edu cti on He solves the quadratic by making a suitable
substitution which will rem ove the term containing x to the first de
gree Like Cardan he reduces the general expression of the cubic to
z
—
the form
then assu m ing
z )z and su b s ti
G— 3
3=
tut ing he gets z bz
P utting z y he has a quadratic
In the solut i on of b i—
qua d ratics Vieta still re m ai ns true to his principle
of reduction This giv es hi m the well known cubic resolvent He
thus a dheres throughout to his fav orite principle and thereby i n
t ro du c e s into algebra a un i fo rm ity of m etho d which clai m s our lively
a dmiration In V i eta s a l gebra we d iscov er a partial knowl e dge of
the relations existing between the c oefli c i en ts and the roots of an equa
3
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RENAISSANCE
TH E
1 39
tion He S hows that if the coeffi cient of the second term in an equa
tion of the second degree is minus the su m of two nu mbers whose
product is the third term then the two nu mbers are roots of the equa
tion Vieta rej ected all except positi v e roots ; hence it was impossible
for him to fully perceive the relations in question
The most epoch m aking innovation in algebra due to Vieta is the
denoting of general or indefinite quantities by letters of the alphabet
To be sure Regiom ontanus and S tifel in Germany and Cardan in
Italy used letters before him but Vieta extended the idea and first
m ade it an essential part of algebra The new algebra was called by
hi m l ogi s ti ca s peci os a in distinction to the old l ogi s ti ca n u merosa
Vieta s form alism diff ered consid erably from that of to day The
“
equation
was written by him a c u bus
cubo aeq u al ia a + b cubo
In
+ 6 i n a quadr 3 + a in b quadr
num erical equations the unknown quantity was denoted by N its
2
3
square by Q and its cube by C Thus the equation x 8 x + 1 6 x = 40
—
was written 1 C 8 O+ I 6 N ( equ al 40 Vieta use d the term c o
effi c i en t
but it was little used before the close of the seventeenth
“
1
century So metimes he uses also the term polynom ial
Observe
that exponents and our symbol
for equality were not yet in use ;
but that Vieta employed the M altese cross
as the short hand
symbol for a ddition and the
for subtraction These two Char
a c t e r s had not bee n in very general use before the ti m e of Vieta
It
“
is very s ingular says Hallam that discoveries of the greatest con
v e n i e n ce and apparently not above the ingenuity of a village school
m aster S hould hav e been overlooked by men of extraor dinary acute
nes s lik e Tartaglia Cardan and L Ferrari ; and har dly less so that by
d int of that acuteness they dispensed with the aid of these c on t ri v
ane c s in which we suppose that so much of the utility of algebraic ex
pressio n consists
Ev en after improvements in notat i on were once
proposed it was with extre m e slowness that they were admitted into
general use They were m ade oftener by accident than design and
their authors had little notion of the effect of the Change which they
were m aking The intro duction of the and
symbols seem s to be
du e to the Germ ans who although t h ey d i d not enrich algebra d u r
ing the Renaissance with great inventions as did the Italians still cul
The arith m etic of Joh n Wi dmann brought
ti v at ed it with great z eal
out in 1 48 9 in Leip z ig is the earliest printed book in wh i ch the and
symbols have been found The sign is not restricted by hi m to
“
ordinary addition ; it has the more general m eaning e t or and
“
as in the heading regula au g m en t i
decrem enti
The
S ign is
“
used to indicate subtraction but not regular ly so The word plus
“
d oes not occur in Wi dm an n s text ; the word m inus is use d only two
or three t imes The symbols
are use d regularly for ad d i
an d
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1
Encycl oped i c des
s c i en ces
mathémal i q u es , Tome I , V ol
.
2 , 1 90 7 ,
p
.
2.
A HISTORY OF M ATHE M ATI CS
1 40
tion and subtraction in
in the arithm etic of Gr ammateu s
a teacher at the University of Vienna
(Heinrich Schreiber died I 5 2 5 )
His pupil Ch ri s tofi Ru dol fl the writer of the first text book on algebra
in the German language (printed in
e mploys these symbols also
So did S tifel who brought out a second edition of R u dol fl s C oss in
Thus by slow degrees their adoption became universal Sev
1 553
eral independent p aleographic studies of Latin m anuscripts of the
fourteenth and fifteenth centuries m ake it alm ost certain that the
comes from the Latin et as it was cursively written in m anu
S ign
2
scripts j ust before the time of the i nvention of printing The ori
gin of the S ig n
is still uncertain There is another short hand
symbol of which we ow e the origin to the Germans In a manu
script published sometim e i n the fifteenth century a dot placed
before a num ber is m ade to signify the extractio n of a root of
that n umber This dot is the e mbryo of our present symbol for the
“
square root Christoff R u dolff in his algebra re marks that the
radix quadrata is for brevity designated in his algorithm wi th the
Here the dot has grown into a symbol m uch
character w/ as V4
like our own This sam e symbol was used by M i chael S tif el Our
—
the author of
S i g n of equality is due to Rob e rt R e c or d e ( 1 5 1 0 1 5
which is the first E nglish treatise on
The Whets ton e of Wi tte ( I 5
algebra He selected this symbol because no two things could be
m ore equal than two parallel lines
The S ign
for division was
first used by Johan n H ei nri ch Rahn a Swiss in his Teu tsche A l gebr a
Zurich 1 6 59 and was introduced in England through Thom as
B ran ck e r s translation of Rahn s book London 1 668
M i ch ae l S ti f e l ( 1 4 86 P
the greatest German algebraist of the
sixteenth century was born in Esslinge n and died i n Jena He was
educated in the monastery of his native place and afterwards be
came P rotestant m inister The study of the S ignificance of m ystic
numbers in Revelation and in Daniel drew him to m athe matics He
studied Germ an and I talian works and publishe d in 1 544 in Latin
a book entitled A ri thmeti ca i ntegr a M elanchthon wrote a preface to
it Its three parts treat respectively of rational nu mbers irrational
numbers and algebra S tifel gives a table containing the nu merical
values of the binomial coeffi cients for powers below the 1 8 th He ob
serves an advantage in letting a geom etric progression correspond to
an arithmetical progression and arriv es at the designation of integral
powers by numbers Here are the germ s of the theory of expone n ts
and of logarith ms In 1 54 5 S t ifel published an arithm etic in German
His edition of R u dol ff s C oss contains rules for solving cubic equations
d erived from the writings of Cardan
—
—
G En est m i n B i bl i th ca m th m ti
S
l
V
8
9 pp
9
9
55 5 7 ;
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ro
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V ol
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1 4,
1914 ,
p
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2 78
For re fe re n ces see M
op ci t , Vol I , 1 90 2 , pp
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a
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a
ca ,
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o
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1
0
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e
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C a n tor ,
1 3 3 , 1 34
p
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ci t
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V o l I I,
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2
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Ed
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1 9 00 ,
p
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23 1
;
J
.
Tr op fk e,
A HISTORY OF M ATHE M ATICS
42
century From the notes of P appus he attempted to restore the miss
ing fifth book of Apollonius on maxi ma and mi ni ma His Chief work is
his m asterly and original treat ment of the conic sections wherein he
discusses tangents and asymptotes m ore fully than Apollonius had
done and applies them to various physical and astronom ical problems
To M au rol yc u s has been ascribed also the discov ery of the inference
1
by mathematical induction I t occurs in his introduction to his Opu s
1 575
Later m athematical induction was
cu l a ma thema ti ca Venice
used by P ascal in his Tr ai l e du tr i angl e ari thméti qu e
P rocesses
akin to m athematical induction som e of which would yield the mod
ern m athematical induction by introducing som e S light change in the
mode of presentation or in the point of view were given before M au
of Novara
r ol yc us
Gi ovan ni C a mpa n o (latini z ed for m Cam panus )
in Italy in his edition of Euclid
proves the irrationality of the
golden section by a recurrent m ode of inferen ce resulting in a reductio
ad absurdum B ut he does not descend by a regular progression from n
to n 1 n 2 etc but leaps irregularly over perhaps sev eral integers
Campano s process was used later by F erma t A recurrent mo de of
“
inference is found in B hask ara s cyclic m ethod of solving inde
term inate equations i n Theon of S myrna (about 1 30 A D )
and in
P ro c l u s s process for finding nu m bers representing the sides and di
that
a g on al s of squares ; i t is found in Euclid s proof (E l emen ts IX 2 0 )
the nu mber of prim es is infinite
2
The foremost geom etrician of P ortugal was P e d r o N u n e s ( 1 50 2
or N on i u s He showed that a ship sailing so as to m ake equal
1 5 7 8)
angles with the m eri dians does not travel in a straight line nor usually
along the arc of a great circle but describes a path called the l oxo
“
Nunes invented the nonius and described it i n
d rom ic cur v e
his De cr epu scu l i s Lisbon 1 54 2 I t consists in the juxtapositio n of
equal arcs one a re di v i ded into m equal parts and the other into m+ 1
equal parts Nonius took m = 8 9 The instrum ent is also called
“
a vernier after the French m an P i erre Ver n i er who r e invente d it
in 1 6 3 1 The forem ost French m athem atician before Vieta was P e te r
who perished in the m assacre of S t B artholom ew
Ramu s ( 1 5 1 5
The n ew
Vi eta possessed great fam iliarity with ancient geom etry
form which he gave to algebra by representing general quantities by
letters enabled him to point out m ore easily how the construction of
the roots of cubics depended upon the celebrated ancient problem s of
the duplication of the cube and the trisection of an angle He reached
the interesting conclusion that the former problem includes the solu
tions of all cubics in which the radical in Tartaglia s form ula is real
but that the latter problem includes o nly those leading to the i rredu
cible case
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G V acca i n B ull eti n A m M ath S oci ety, 2 S , V ol
F C aj o ri i n V ol 1 5 , pp 40 7 —
40 9
2
S e e R G ui mar ae s, P ed ro N u n es , C oi mp r e, 1 9 1 5
1
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1 909 ,
p
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70
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S ee
al so
'
THE RENAISSANCE
1 43
The problem of the quadrature of the circle was revived in this age
and was z ealously studied eve n by men of e minence and m athematical
ability The ar my of circ l e squarers becam e most formidable q ing
the seventeenth century A mong the first to revive this problem was
the Germ an Cardinal N i c ol au s Cu san u s ( 1 40 1
who had the
reputation of being a great logician His fallacies were expose d to
full view by Regio montanus A S in this case so in others e v ery quad
rator of note raised up an opposing mathematician : O ron c e Fine was
—
m et by Jean B uteo ( c 1 49 2 1 57 2 )
and P Nunes ; Joseph Scaliger by
Vieta A drian u s Romanus and Cla v ius ; a Qu erc u by Adriaen A n
Two mathem aticians of Netherlands Adri anu s
thon i sz ( 1 5 2 7
Roman u s ( 1 56 1 —1 6 1 5)
and Lu d ol ph van Geu l e u ( 1 540
oce n
pie d themsel v es with appro xi mating to the ratio between the circumfer
ence and the diam eter The form er carried the value 71 to 1 5 the lat
ter to 3 5 p l aces The value of 7r is t herefore often nam ed L u dolph s
nu mber His performance was considered so e xtraordinary that the
nu mbers were cut on his to mb stone ( now lost )
in S t P eter s church
yard at Leyden These m en had used the A rch imedian m ethod of
i n and C ircu m S cribed polygons a m ethod refined in 1 6 2 1 by Wi l l e
brord S n el l i u s ( 1 5 80 —1 6 2 6 )
who showed how narrower limits m ay be
obtained for 71 without increasing the nu mber of S ides of the poly
gon s Snellius used two theore ms equivalent to 1 ( 2 sin 6 tan 0)
A 94
csc 0+ cot
The greatest refinem ents in the use of the geo
metrical m ethod of Archi m edes were reached by C Huyghens in his
De ci r cu l i magn i tudi n e i n ven ta 1 6 54 and by J am e s G r e g or y ( 1 6 3 8
professor at S t Andre w s and Edinburgh in his E xer ci tati on es
167
geometri cae 1 668 and Ver a ci r cu l i et hyper bol a e q u adr a tu r a 1 6 6 7
Gregory gave s everal form ulas for approximating to 71 and in the
second of these publications bol dly attem pted to prov e by the Ar
c hi m e dean algorith m that the quadrature of the circle is i mpossible
Huyghens S howed that Gregory s proof is not conclusive although
he himself believed that the quadrature is impossible Other attempts
to prove this impossib i lity were m ade by Thom as Fau ta t D e Lagny
—
—
of P aris in 1 7 2 7 Joseph Saurin ( 1 6 59 1 7 3 7)
in 1 7 2 0
( 1 6 6 0 1 734)
Isaac Newton in his P ri n ci pi a I 6 lemm a 2 8 E Waring L Euler
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1 77 1
That these proofs w ould lack rigor was al most to be expected as
long as no distinction was made between algebraical and t ran sc en
dental nu mbers
The earliest explicit expression for i r by an infinite nu mber of op
Considering regular polygons of 4 8
e ra ti on s was found by Vieta
sides inscribed in a circle of unit r adi us he found that the
16
area of the circle i s
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,
A HISTORY OF M ATHE M ATICS
1 44
from which we obtain
W
I
1
x/i
2
«
S in
which may be derived from Euler s formul a
’
1
5
0
cos 9/4 C05 0/8
As m entioned earlier it was Adri anu s Roman u s ( 1 56 1—
1 6 1 5)
of L ou
vain who propounded for solution that equation of the forty—
fif t h d egree
solved by Vieta On receiving Vieta s solution he at once departed for
P aris to m ake his acquaintance with so great a master Vieta proposed
to him the Apollonian problem to draw a circle touching three given
“
circles Adri an u s Rom anus solved the proble m by the intersection of
two hyperbolas but this solution did not possess the rigor of the ancient
geom etry Vieta caused him to see this and then in his turn pre
2
sented a solution which had all the rigor desirable
Rom anus
did m uch toward S implifying spherical trigonom etry by reducing by
means of certain proj ections the 2 8 cases in triangles then considered
to only six
M ention m ust here be m ade of the i mprovem ents of the J ulian
cal endar The yearly determination of the movable feasts had for
a long tim e been connected with an untold am ount of confusion The
rapid progress of astronomy led to the consideration of this subject
and many new calenda rs were proposed P ope Gregory XIII con
v ok ed a large nu mber of m athe m aticians astrono m ers and prelates
who decided upon th e adoption of the calendar proposed by the Jesuit
—
of Rom e To rectify the errors of
h
r
i
t
o
ho
ru
s
C
l
a
v
iu
s
1
s
C
( 53 7 1 6 1 2 )
p
the Julian calendar it was agreed to write in the new calendar the 1 5 th
of October imm e diately after the 4 th of October of the year 1 58 2
The Gregorian calen dar m e t with a great deal of opposition both
among scientists and a mong P rotestants Clavius who ranked high
as a geom eter m e t the obj ections of the form er m ost ably and eff ec
ti v el y ; the prejudices of the latter passed away with ti m e
The passion for the study of mystical properties of nu mbers de
scended fro m the ancients to the moderns M uch was written on
numerical m ysticism even by such e minent men as P ac i ol i and Stifel
The N u mer oru m M ysteri a of P eter B ungus covered 70 0 q uarto pages
He worked wi th great industry and satisfaction on 6 66 which is the
number of the beast in Revelation (xiii
the sym bol of Antichrist
He reduced the nam e of the i mpious M artin Luther to a form whi ch
may express this form idable number
P lacing a = 1 b = 2 etc k = 1 0
l 2 0 etc he finds after m isspelling the name that M
constitutes the number required
These attacks on the great reform er were not unprovoked for his
c 0 5 9/2
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W
H ob son , S q u ari ng the Ci rcl e, C amb r idg e , 1 9 1 3 , pp 2 6 , 2 7 , 3 1
Q ue t el e t , Hi stoi r e des S ci ences mathémati qu es cl phys i q u es chez l es B el ges
.
.
B ru xel l es ,
1 86 4 ,
p
.
1 37.
.
.
A HISTORY OF M ATHE M ATI CS
1 46
now began to blossom Cardinal Richelieu during the reign of Louis
XIII pursued the broad policy of n o t favoring th e opinions of any
sect but of pro moting the interests of the nation His age was re
m arkable for the progress of knowle dge It produced that great secu
lar literature the counterpart of which was found in England in the
S ixteenth century
The seventeenth century was m ade illustrious
also by the great French mathe maticians Roberval D escartes D es
argues F erma t and P ascal
M ore gloomy is the picture in Ger many The great changes which
revolutioni z ed the world in the S ixteenth century and which led Eng
land to national greatness led Germ any to degradation The first
e ffects of the Reform ation there were salutary At the close of the
fifteenth an d during the sixteenth century Germany had been con
spi cu ous for her scientific pursuits
She had been a leader in as
t ro n omy and trigono metry Algebra also excep t ing for the disco v eries
in cubic equations was before the ti me of Vieta in a m ore a dvanced
state there than elsewhere B ut at the beginning of the seventeenth
century when the sun of science began to rise in France it set in Ger
m any
Theologic disputes and religious strife ensued The Thirty
Years War ( 1 6 1 8—1 64 8 )
proved ruinous The German e mpire was
S hattered and beca m e a m ere lax confederation of petty despotis m s
Co mmerce was d estroye d ; national feeling died out Art disappeared
and in literature there was only a sl avish i mitation of French arti
fi c i ali ty Nor di d Ger m any recover fro m this low state for 2 0 0 years ;
for in 1 7 56 began another stru ggle the Seven Years War which
turned P russia into a wasted l and Thus i t followed that at the b e
ginn i ng of the seventeenth century the great Kepler was the only
German mathe matician of e m inence and that in the interval of 2 00
years between Kepler and Ga uss there arose no great mathematician
in Germany excepting Leibniz
Up to the sev enteenth century mathe matics was cul tivated but l ittle
in Great Britain D uring the sixteenth century she brought forth
no mathe m atician comparable with Vieta S tifel or Tartaglia B ut
with the tim e of Recorde the English becam e conspicuous for nu meri
cal S kil l The first i mportant arith metical work of English authorship
was published in Latin in 1 5 2 2 by Cu thb e rt Ton s tal l ( 1 4 74
He
h ad stu d ied at Oxfor d Ca mbri dge and P a dua and dre w freely fro m
the works of P a c i ol i and Regiomontanus Reprints of his arith metic
appeared in England and France After Recorde the higher branches
of mathematics began to be stu died Later Scotland brought forth
John Napier the inventor of logarith m s The instantaneous app re
c i a t i on of their value is d oubtless the result of superiority in calcula
tion In Italy and especially in France geo metry which for a l ong
ti me had been an al most stationary science began to be studied with
success Galileo To rri c el l i Roberval F e rma t Desargues P ascal
Descartes a nd the English Wallis are the great revoluti oners of thi s
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VIETA TO D ES CARTES
1 47
science Theoretical mechanics began to be studied The foundations
were laid by F erma t and P ascal for the theory of numbers and the
theory of probability
We S hall first consider the i mprovements made in the a r t of c al c u
lating The nations of antiquity experimented thousands of years
upon numeral notations before they happened to stri ke upon the so
“
called Arabic notation
In the si mple expedient of the cipher
which was perm anently introduced by the Hindus mathe matics re
I t would see m that after
c ei v ed one of the m ost powerful im pulses
“
the Arabic notation was once thoroughly understood decim al
fractions would occur a t once as an obvious extension of it B ut it
is curious to think how m uch science had attempted in _
physical re
search and how deeply nu mbers had been pondered before it was per
powerful simplicity of the Arabic notation was as
c ei v e d that the all —
valuable and as m anageable in an infi nitely descending as in an i n
1
finitely ascending progression
Simple as decim al fractions ap
pear to us the inv ention of them is not the result of one mind or even
of one age They came into use by al m ost imperceptib l e degrees The
first m athematicians identified with their history did not perceive
their true nature and i mportance and failed to invent a suitable no
The idea of decim al fractio ns m akes its first appearance in
t a ti on
Thus
methods for appro xi mating to the square roots of nu mbers
John of Seville presu mably in imitation of Hindu rules adds 2 n ci
m
h
r
to
the
nu
ber
t
en finds the square root and takes this as the
h
e
s
p
nu merator of a fraction whose deno minator is 1 followed by n ciphers
The same method was followed by C ardan but it failed to be generally
adopted even by his Italian contemporaries ; for otherwise it would
certainly have been at least mentioned by P i etr o C a taldi ( died 1 6 2 6 )
in a work devoted exclusively to the extraction of roots Cataldi
and before him Bombelli in 1 5 7 2 find the square root by m eans of
continued fractions— a m ethod ingenious and novel but for p ractical
purposes inferior to Cardan s Or once Fi ne ( 1 4 94—1 55 5)
in France
( called al so O ron ti u s F i n aeu s)and Wil l i am B u c k l e y ( died about
in England extracted the square root in the sa m e way as
1 550 )
Cardan and John of Seville The invention of decim als has been
frequently attributed to Regiom ontanus on the ground that i n
stead O f placing the sinus to t u S in trigonometry equal to a m ultiple
of 6 0 like the Greeks he put i t = 1 0 0 o oo B ut here the t rig o n ome t
and not in fractions Though
r i cal lines were e xpressed in i n teg er s
he adopted a decim al division of the rad ius he and his successors
d id not app l y the idea outside of tr i gono m etry an d indeed had no
notion whatever of d eci mal f r acti on s To S i mo n S tevi n ( 1 548
1 6 20)
of Bruges in Belgiu m a m an who did a great d eal of work in
most d iverse fields of science we owe the first system atic treat m ent of
he describes in very express
d ecim al fractions In his L a Di s me ( r 5 8 5)
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M ark N api er , M emoi rs of J ohn N api er of M erchi ston
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Edi n b u rg h,
1 8 34
.
A HISTORY OF M ATHE M ATICS
1 48
terms the advantages not only of decimal fractions but also of the
S tevin applied
decimal division in systems of weights and measures
“
1
the new fractions to all the operations of ordinary arithmetic
What he lacked was a suitable notation In place of our decimal point
he used a cipher ; to each place in the fraction was attached the cor
responding index Thus i n his notation the nu mber
wo uld be
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0 1 2
,
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3
or
These indices though cum brous in practice are
of interest because they e mbody the notion of powers of nu mbers
“
Stevin considered also fractional powers He says that g placed
within a circle would mea n x / but he does not actually use his nota
tion This notion had been advanced m uch earlier b y O resme but
it had re mained unnoticed Stevin found the greatest common di
2
2
3
visor of x +x and x + 7x + 6 by the process of continual division
thereby applying to polyno mials Euclid s mode of finding the greatest
comm on divisor of nu mbers as explained in B ook VII of his El emen ts
Stevin was enthusiastic not only over decimal f ractions but also over
the decimal div ision of weights and m easures He considered it the
duty of gov ern ments to establish the latter He advocated the deci
m al subdivisio n of the degree
N 0 i mprove ment was made in the
notation of deci mals till the beginning of the seventeenth century
After Stevin decimals were used by J oos t B urg i ( 1 55 2
a Swiss
by birth who prepared a manuscript on arithmetic soon after 1 592 and
by Joh ann H art mann B ey er who assu mes the invention as his own
In 1 6 0 3 he published at Frankfurt on the M ain a L ogi sti ca Deci mali s
Historians of m athe matics do not yet agree to whom the first intro
duction of the decim al point or c omma should be ascribed Among
the candidates for the honor are P ell os
B ii rg i
P itiscus
Kepler
Napier ( 1 6 1 6
This divergence
( 1 60 8
of opinion is due m ainly to different standards of j udgm ent If the
require ment m ade of candidates is not only that the decimal point or
co mma was actually used by them but that they must give evidence
that the numbers used were actually deci mal fractions that the point
or comma was with them not m erely a general symbol to indicate
a sepa ration that they must actually use the dec imal po int i n opera
tions including m ultiplication or div ision of deci mal fractions then
it would seem that the honor falls to John Napier who e xhibits such
use in his Rabdol ogi a 1 6 1 7 P erhaps Napier received the suggestion
2
for this notation from P itiscus who according to G En estrOm uses
the point in his Tr i gon ometri a of 1 60 8 and 1 6 1 2 not as a regular deci
mal point but as a more general S ign of separation N api er s decimal
point did not meet with i mm ediate adoption W Ou ghtred in 1 63 1
designates the fraction 56 thus 0 | 56 Al ber t Gi r ard a pupil of S tevin
in 1 6 2 9 uses the point on one occasion J ohn Wal lis in 1 6 57 writes
A Q ue tel e t p ci t p
58
59 1 2
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B i bl i otheca ma themati ca , 3 S
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Vo l 6 ,
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1 90 5 ,
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p
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1 09 .
,
A HISTORY OF M ATH E M ATICS
1 50
its logarithm We notice that as the m otion proceeds B C
decreases in geo metrical progression while DE increases In arith
7
=
=
m etical progression
Let A B a 1 0 let x = DF y= B C then
d ( a —y)
The velocity of the point C is
AC = a—
y; this gives
y
DE
,
.
,
.
—nat
log
.
dx
dt
=a
,
,
,
.
dt
y
= t+ c
When
.
then
t= o ,
y
be the velocity of the point
=a
and
F,
then
and c their values and reme mbering that
tion x = N ap log y we get
t
.
Nap log
.
c
.
nat log
a
Again
.
,
Substituting for
x = at
.
7
=
10
a
,
= 1 07
y
—nat log
and that by
defini
—
.
y
I t is e v ident from this formula that Napier s logarithms are not the
sam e as the natural logarith m s N api e r s logarithm s increase as the
nu mber itself decreases He took the logarith mof sin
i e
the logarith m of
The logarith m of S i n or increased from z ero
as a decreased from
Napier s genesis of logarith ms from the con
c ep ti on of two flowing points re m inds us of Newton s doctrine of
The relation between geo metric an d arith metical pr og res
fl u xion S
sions so skilfully utiliz ed by Napier had been observed by Archi
m edes Stifel and others What was the base of Napier s system of
logarith m s ? To this we reply that not only did the notion of a
“
base nev er suggest itself to him but it is inapplicable to his
system This n ot io n deman ds that z ero be the logarithm of 1 ; in
7
Napier s syste m z ero is the logarith m of 1 0
Napier s great i n
v en t i on was given to the world in 1 6 1 4 in a work entitled M i r ifi ci
In it he exp l ained the nature of
l oga ri thmor u m ca non i s descri pti o
his logarithm s and gave a logarith mic table of the natural S ines of
a quadrant fro m m inute to m inute In 1 6 1 9 appeared Napier s
M i r ifi ci l ogar i thmor u m ca n on i s con s tr u cti o as a posthu m ous work i n
which his m ethod of calculating logarith ms is explained An English
translation of the Con s tru cti o by W R M acdonald appeared in
Edinburgh in 1 88 9
H e n ry B rigg s ( 1 556
in Napier s ti me professor of geometry
at Gresham College London , and afterwards professor at Oxford
was so struck with ad m iration of N apier s book that he left his studies
in Lon don to do ho mage to the Scottish ph ilosopher Briggs was de
“
layed in his journey and Napier complained to a comm on friend Ah
John M r Briggs will not com e
At that very mom ent knocks were
heard at the gate and B riggs was brought into the lord s chamber
Alm ost one quarter of an hour was spent each beholding the other
“
w ithout speaking a word
At last Briggs began : M y lord I have
undertaken this long j ourney purposely to see your person and to
know by what engine of wit or ingenuity you cam e first to think of
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VIETA TO DES CARTES
1 51
this m ost excellent h elp in astrono my v iz the l ogarithm s ; but my
lord being by you foun d out I wonder nobody found it out before
when now kno w n it is so easy
B riggs suggested to Napier the ad
vanta ge that would result fro m retaining z ero for the logarithm of the
whole sine but choosing 1 0 0 00 00 0 0 00 for the logarith m of the r o th
Napier said that he had al
part of that same S ine i e of 5 44
ready thought o i the change and he pointed out a slight i mprovem ent
on Briggs idea ; vi z that z ero
that of the whole
i s ti c of nu mbers greater than uni
gested by Briggs Briggs ad mitted
“
invention of B r igg ian logarith ms occurred therefore to Briggs
and Napier indepen dently The great practical advantage of the n ew
system was that its fun da mental progression was acco mm odated to
the base 1 0 of our nu merical scale B riggs d e v oted all his energies
to the construction of tables upon the new plan Nap ier died in 1 6 1 7
with the satisfaction of having found in Briggs an able fri en d to bring
to completion his unfinished plans In 1 6 2 4 B r i ggs published his
A r i thmeti ca l ogari thmi ca c ontaining the logarit hm s to 1 4 places of
numbers from 1 to
and from
to
The gap fro m
to
was filled up by that illustrious successor of Napier
and Briggs A dri an Vl aca ( 1 60 0 ?
He was born at Gouda in
Holland and lived ten years i n Lon don as a bookseller and publishe r
B eing dri v en out by London bookdealers he settled I n P aris where he
m e t opposition aga i n for selling foreign books He died at The Hague
John M ilton in his Def en s i o s ecu nda publishe d an abuse of hi m
of
V l ac q published in 1 6 2 8 a table of logarith m s fro m 1 to
which
were calcul ated by him self The first publ i cation of
B r ig gi a n logarith m s of t rigono m etric functions was ma d e in 1 6 2 0 by
Ed mu n d G u nt e r ( 1 58 1 —
1 6 26)
of London a colleague of Br i ggs who
found the logarith m ic S ines and tangents for every m inute to seven
places Gunter was the inventor of the words cosi n e and cotangen t
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The word cosi n e was an abbreviation of compl emen tal si n e The
invention of the words tan gen t and s ecan t is d u e to the physician and
m athematician Thomas Fi n ch a native of Flensburg who use d them
in his Geometr i a r otu ndi Basel 1 583 Gunter is known to engineers
“
for his Gunter s chain
I t is tol d of hi m that When he was a stu
d ent at Christ College it fell to his lot to preach the P assion ser m on
which so me ol d divines that I knew di d hear but they said that i t
was said of hi m then in the Univ ersity that our Savior never su ffered
so m uch since his passion as in that sermon it was such a lamented
1
one
Briggs devoted the last years of h i s l i fe to calculating more
extensive B r igg ia n logarith m s of tri gonom etric functions but he d ie d
in 1 63 1 leaving his work unfinished It was carried on by H e nr y G el
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Au br ey
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B r i ef Li ves , Edi t i on A C l ark ,
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1 8 98 ,
V ol I , p
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2 76 .
A HISTORY OF M ATHE M ATICS
1 52
—
i
r
l b an d ( 1 5 9 7 1 6 3 7
li shed by V l ac q at
)of
Gresham College in London and then pub
his own e xpense Briggs divided a degree into
1 0 0 parts as was done also by N Roe in 1 6
33 W O u g h t r ed in 1 6 5 7
John Newton in 1 6 58 but owing to the publication by V l ac q of tr igo
n ome t r i cal tables constructed on the old sexagesim al division Briggs
innovation did not prevail B riggs and V l ac q published four funda
m ental works the results of which have not been superseded by any
subsequent calculations until very recently
The word characteristic as used in logarithm s first occurs in
“
Briggs A r i thmeti ca l ogari thmi ca 1 6 2 4 ; the word mantissa was i n
t ro du ced by J ohn Wallis in the Latin edition of his A l gebr a 1 6 93
p 4 1 and was used by L Euler in his I n tr odu cti o i n anal ysi n in 1 748
p 85
The onl y riv al of John Napier in the invention of logarith ms was the
Swiss J oo s t B ii rg i ( 1 55 2 —1 6 3
He publishe d a table of logarith ms
A r i thmeti s che u nd Geometri sche P r ogr ess tabu l en P rague 1 6 2 0 but he
conceived the idea and constructed his table in dependently of Napier
He neglected to have it published until Napier s logarithm s were
know n and adm ired throughout Europe
Among the various inventions of Napier to assist the m em ory of
“
the student or calculator is N api er s rule of circular parts for the
solution of S pherical right triangles I t is perhaps the happiest
example of artificial m e m ory that is known
Napier gives in the
Des cr i pti o a proof of his rule ; proofs were given later by Johann
Heinrich Lambert ( 1 7 6 5)
and Leslie Ellis
Of the four for
“
m ulas for oblique S pherical triangles which are so m eti m es called N a
pier s Analogies only two are due to Napier hi m self ; they are giv e n
in hi s C on str u cti o The other t wo were added by B riggs in his an
notations to the C on stru cti o
A m od ification of Napier s logarith m s was m ade by J oh n S pe i d e ll
a teacher of m athem atics in London who published the N ew L og a
r i thmes
London 1 6 1 9 containing the logarithm s of S ines tangents
and secants Speidell did not advance a new theory He simply
aimed to i mprove on N api er s tables by m aking all logarithm s posi
tive To achie v e this en d he subtracted Napier s logarithm ic nu mbers
8
from 1 0 and then discarded the last two digits Napier gave l og si n
8
Subtracting this from 1 0 leaves 5 2 586 1 48 Speidell
wrote l og s i n
I t has been said that S peid el l s logarith ms
5 25 86 1
of 1 6 1 9 are logarithm s to the natural base e This is not quite true
on account of co mplications arising from the fact that the logarith ms
in Spei del l s table appear as i n tegr al nu mbers and that the natural
trigonom etric values ( not printed in Sp ei del l s tables )
are likewise
written as integral nu mbers If the last five figures in Sp ei del l s log
then the l ogarith ms are
ari thms are taken as deci m als (m antissas )
t he natural logar i th m s (with 1 0 a dd ed to e v ery negati v e character
R M o t i z A m M th M thl y V l
9 5 p 22
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A HISTORY OF M ATHE M ATICS
1 54
Third Table
with
the
as the comm on r atio
1
69
Each column is a geom etric progression of
.
The
.
69
terms
21
first or top nu mbers in
colum ns the mselves constitute a geometric progression hav ing
the rat
the first top num ber being
10
the second 990 0 0 0 0
7
,
”
,
and so on The last nu mber in the 6 9th colu m n is 4998 60 9 40 34
Thus this Third Table giv es a series of nu mbers very nearly but
7
not exactly i n geom etrical progression and lying between 1 0 and
“
Says Hutton these tables were found in the
v ery nearly
The
m ost si mple m anner by little more than easy subtractions
nu mbers are taken as the S ines of angles between 90 and
Kine
ma t i cal considerations yield hi m an upper and a lower li m it for the
logarith m of a giv e n sine B y these lim its he obtains the logari th m
“
of each nu mber in his Third Table
To obtain _the logarithm s of
sines between 0 and 30 Nap i er in dicates two methods By one of
“
the m he co mputes log sin 6
by the aid of his Third
S in d sin ( 90
Table and the form ula sin 2
A repetition
of this process gives the logarith ms of S ines down to
and
so on
Burgi s m ethod of computation was m ore primitive than N api er s
“
In his table the logarith m s were printed in red and were called red
nu mbers the antilogarith m s were in black The expressions r = I o n
.
.
,
,
,
.
,
°
'
”
.
.
°
°
.
,
°
.
’
’
.
n
.
where
indicate the
m ode
of computation
series is obtained by
of that term
p o l a t i on
.
and
ro = o,
addi n g
Any term
.
bn
u = 1, 2,
of
th e
to the preceding term b _ the
n
P roceeding
,
3,
geom etric
1
1,
4
th part
thus Burgi arrives at r = 2 30 2 70 0 2 2 and
this last pair of nu mbers being obtained by inter
,
,
.
In the Appendix to the C on stru cti o there are described three m eth
ods of co mputing logarith m s which are probably the result of the
j oint lab ors of Napier and B riggs The first method rests on the
successive extractions of fifth roots The second calls for square
roots only
Taking log
and log
find the logarithm
of the m ean proportion between 1 and 1 0 There follows log
x 10
log
then log
and so on Substantially this m ethod was
i
used by Kepler in his book on logari th m s of 1 6 2 4 and by V l ac q The
third m ethod i n the Appen dix to the C on stru cti o lets log 1 = 0 log 1 0 =
10
and takes 2 as a factor 1 0 ti m es yielding a number composed
of 30 1 0 2 9996 figures ; hence log
.
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-
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VIETA
DESCARTES
TO
1 55
”
A famous method of co mputing logarith ms is th e so called rad ix
method
I t requires the aid of a table of radices or nu mbers of the
-
.
for m
with their logarithms
It
The logarithm of a number is
.
found by resolving the nu mber into factors of the form
”
then adding the logarith ms of the factors The earliest appearance
“
of this method is in the anonym ous Appendix (very probably due
1
to O u g h t red)
t o Ed w a rd Wright s 1 6 1 8 e d ition of N ap i er s D es cri pti o
I t is fully d eveloped by Briggs who in his A ri thmeti ca l og ari thmi ca
1 6 24
gives a table of radices The m etho d has been frequently re
2
discovered and given in various form s
A S light S im plification of
Briggs process was gi ven as one of three m ethods by Robert Fl ow er in
a tract The Radi x a n ew w ay of maki ng L ogari thms London 1 7 7 1
He di v ides a given number by a power of 1 0 and a S ingle digi t so as
to reduce the first figure to 9 and then m ultipli es by a procession of
ra dices until all the d igits beco m e nines The ra dix m etho d was re
d iscovered in 1 7 8 6 b l Geor ge A tw ood ( 1 7 4 6
the inventor of
Atwood s machine in A n ess ay on the A ri thmeti c of Factors and
again by Z ecchi n i L eon el l i in 1 80 2 by Thom as M anning ( 1 7 7 2
scholar of Caius College Cambridge in 1 8 0 6 by Thomas Weddle in
1 84 5 Hearn in 1 8 4 7 and Orchard in 1 8 4 8
Extensions and v ariations
of the radix m ethod have been published by P eter Gray ( 1 80 7 ?
a writer on life contingencies Thom an A J Ellis ( 1 8 1 4
and
others The three distinct methods of its application are due to B riggs
Flower and Wed dle
Another m ethod of co mputing common logarith m s is by the re
= I
= 10
ea
t
e
d
for
m
ation
of
geo
m
etric
m
eans
If
A
then C
B
p
V A B = 3 1 6 2 2 7 8 has the logarithm 5 D = V B C = 5 6 2 3 4 1 3 has the
logarith m 7 5 etc P erhaps suggested by Napier s rem arks in the
C on s tru cti o this m ethod was d eveloped by French writers of whom
3
—
in 1 6 7 0 was perhaps the fi r s t O zan am
Jacques Oz an am ( 1 6 40 1 7 1 7 )
is best known for his Recr ea ti on s mathéma ti qu es cl phy s i qu es 1 6 94
Still d iff erent d evices for the co mputation of logarith m s were i n
vented by Brook Taylor
John Long
William Jones
Roger Cotes
An d rew Reid
J am es D odson
Abel
4
Burja
and others
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W
L Gl ai sh e r , i n Quar terl y f ou r of M a th s , V ol 4 6 , 1 9 1 5 , p 1 2 5
2
For t h e de t ai l ed hi s to ry o f t hi s m e t hod co n su l t a l so A J E lli s i n P r oceedi n gs
—
of the Roya l S oc i ety ( L o n do n , V o l 3 1 , 1 88 1 , pp 3 98 4 1 3 ; S L u p to n , M a thema ti cal
’
—
Ga ette, Vo l 7 , 1 9 1 3 , pp 1 4 7 1 5 0 , 1 7 0 1 7 3 ; C h H u t t o n 5 I n t r o d u c t i o n t o hi s
M athemati cal Ta bl es
’
3
S e e J W L G l ai sh er I n Q u ar terl y J ou r n al of P u re and A ppl M d th s , V o l 4 7 ,
1
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1 91 6
pp
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30 1
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2 49
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Fo r d e t ail s see C h H u t t o n 3 I n t r o du c t i o n t o h i s M a thema ti ca l Ta bl es , al so t he
“
En cycl opedi c des s c i ences mathé mati q u es , 1 90 8 ; I , 2 3 , T a b l es d e l o
g ar i t h me s
4
’
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A H ISTORY O F M ATH EM ATICS
1 56
After the labor of computing logarithm s was practically over the
facile methods of computing by infinite series came to be discovered
Ja mes Gregory Lord William B rounk er ( 1 6 2 0
Nicholas M er
cator ( 1 6 2 0
John Walli s and Ed mund Halley are the pioneer
workers M ercator in 1 668 derived what amounts to the infinite
series for log
Transformations of this series yielded rapidly
—
converging results Wallis in 1 6 95 obtained 1 log
I
z)
(
5
2
=
z+ §
2 +
G Vega in his Thes au ru s of 1 7 94 lets z 1 /( 2y
The theoretic view point of the logarith mwas broadened som ewhat
during the seventeenth century by the graphic representation both
in rectangular and polar co or d inates of a variable and its variable
logarithm Thus were invented the logarith mic curve and the loga
r i th mi c spiral
I t has been thought that the earliest refere nce to the
logarith m ic curve was m ade by the Italian Evangelista Torricelli i n
a letter of the year 1 6 44 but P aul Tannery m ade it practically certain
that Descartes knew the curve in
Descartes described the log
arithmic spiral in 1 6 3 8 in a letter to P M ersenne but does not give its
equation nor connect it with logarithms He describes it as the curve
which makes equal angles with all the radii drawn through the origin
“
The name logarith mic S piral was coined by P i err e Va r i gn on in a
paper presented to the P aris acade my in 1 7 0 4 and published in
The m ost brilliant conquest in algebra during the six teenth century
had been the solution of cubic and biquadratic equations All a t
tempts at solving al gebraically equations of higher degrees rem aining
fruitless a n ew line of inquiry—the properties of equations and their
roots— was gradually opened up We hav e seen that Vieta had at
t ai n ed a partial knowle dge of the relations between roots and c o
J a cqu es P el eti er ( 1 5 1 7
a French man of letters poet
efli c i en t s
and m athem atician had observ ed as early as 1 558 that the root of
an equation is a div isor of the last term In passing he writes equa
tions with all term s on one S ide and equated to z ero This was done
also by B uteo and Harriot One who extended the theory of equa
tions somewhat further than Vieta was Al b e rt G ir ar d ( 1 590 ?
a mathe matician of Lorraine Like Vieta this ingenious author ap
plied algebra to geo metry and was the first who understood the use
of negative roots in the solution of geom etric problem s He spoke of
i m aginary quantities inferred by in duction that ev ery equation has
as many roots as there are units in the num ber expressing its degree
and first S howed how to express the su ms of their powers in term s of
the c oefli c i en t s Another algebraist of considerable power was the
English Th oma s H arr i ot ( 1 56 0
He acco mpanied the first
colony sent out by Sir Walter Raleigh to Virginia Af ter having sur
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S ee G L o r i a B i bl i otheca ma th , 3 S , V ol 1 , 1 900 , p 7 5 ; L i n ter medi a i r e des
mathémati ci ens , Vo l 7 , 1 900 , p 95
2
Fo r d e t a i l s a n d r e fe re n ce s see F C a j o r i ,
H i s to ry o f t he Exp o n e n t i al ' a n d
Log a ri thmi c C o n c ep t s , A m M a th M on thl y, V o l 20 , 1 9 1 3 , p p 1 0 , 1 1
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O F M ATHE M ATICS
HISTORY
A
1 58
by Z C L M Hence in des i gnating ratio I use not one point but two
points which I use at the sam e tim e for division ; thus for your
I write dy :x = dt :a; for dy is to x as dt is to a is indeed the
sa me as dy d iv i d ed by x is equal to dt d ivided by a From this equa
tion follow then all the r ul es of proportion
This conception of
ratio and proportion was far in advance of that in contempora ry
arith m etics Through the aid of C hr i s ti an Wolf the dot was generally
adopted in the eighteenth century as a symbol of m ultiplication
P resu m ably Leibniz had no knowle dge that Harriot in his A r ti s
an al yti cw pr a xi s
1 63 1
used a dot for multiplication as in aaa 3
bba = + 2 ccc Harriot s dot received no attention not even fro m Wallis
O u g h t r ed and so m e of his English conte mporaries Richard N o r
wood John Speidell and others were prominent in introducing abb re
v i a t i on s for the trigono m etric functions : s s i or s i n for s i n e; s co or
s i co for
sine co mple m ent or cosi n e se for s ecan t etc O u gh t red
did not use parentheses Term s to be aggregated were enclosed b e
tween double c ol ons He wrote x/ (A +E )
thus wl q l—
E : The two
—
d ots at the end were som eti m es o m itted
Thus C :A +B E m eant
3
—
(A +B E ) B efore O u gh t r e d the use of parentheses had been sug
gested by Clavius in 1 6 0 8 and Girard in 1 6 2 9 In fact as early as
_
’
F
w
‘
/
28
ID thus R v ( R 2 8 m en R 1 0 )
1 55 6 Tartaglia wrote x
where
“
R v m eans radix universalis
but he did not use parentheses in i n
1
di ca t i n g the product of two expressions
P arentheses were used by
I Er rar d de B ar l e —
D uc
Jacobo de Billy
Richard
Norwood
Sam uel Foster
nevertheless parentheses did
not beco m e popular in algebra before the time of Leibniz and the
B ernoullis
I t is noteworthy that O ugh t red denotes 3—
and 1 33 the approxi
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m ate
ratios of the circumference to the diameter by the symbol
,
occurs in the
S
’ it
edition and I n the later editions of his Cl ovi s mathe
mati cce
O u g h t re d s n otation was a dopted and used extensively by
Isaac B arrow I t was the forerunner of the notation 7r =
first used by William Jones I n 1 70 6 in his S ynopsi s pal mari oru m ma
thes eos London 1 70 6 p 2 6 3
L Euler first used 7r = 3 1 4 1 59
1 73 7
In his tim e the symbol m e t with general adoption
O u gh t red stands out pro m inently as the in v entor of the circular
and the rectilinear S lide rules
The circular slide rule was described
in print in hi s book the C i rcl es of P ropor ti on 1 6 3 2 His rectilinear
slide rule was described in 1 633 in an A ddi ti on to the above work
B ut O u gh tred was not the first to describe the circular slide rule in
print ; this was done by one of his pupils Richard D elamain in 1 6 3 0
2
in a booklet entitled Gr ammel ogi a A b i tter controv ersy arose b e
1 64 7
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G En est ro m i n B i bl i otheca mathemati ca , 3 S , V ol 7 , p 2 96
2
S ee F C aj o r i , Wi l l i a m O u ghtr ed, C h i c a g o a n d L on d on , 1 9 1 6 , p 46
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VIETA TO D ESCARTES
59
tween Dela m ain and O ug h t red Each accused the other of having
stolen the inv ention fro m him M ost probably each was an i n
d ependent inventor
TO the inventio n of the rectilinear slide rule
O ug h t red has a clear title He states that he designed his sli d e rules
as early as 1 6 2 1 The S li de rule was improv ed in England during the
1
seventeenth and eighteenth centuries and was used quite e xtensiv el y
Som e of the stories told about O ug h tr ed are doubtless apocryphal
as for instance that his econom ical wife denied him the use of a candle
for study in the evening and that he died of joy at the Restoration
after drinking a glass of sack to his M aj esty s heal th D e M organ
“
hum orously rem arks It S hould be added by way of excuse that h e
was eighty—
S i x years old
Algebra was now in a state of s ufficient perfection to enable Des
cartes and o thers to take that importa nt step which form s one of the
grand epochs in the history of m athem atics —the application of alge
brai e analysis to define the nature and investigate the properties of
algebraic curves
In geometry the determ ination of the areas of curv ilinear figures
was diligently studied at this period P au l G u l di n ( 1 5 7 7
a
Swiss m athe matician of considerable note re di scov ered the following
theorem published in hi S _C en tr obaryca which has been nam ed after
hi m though first found i n the M athema ti cal C ol l ecti on s of P appus :
The volu me of a solid of revolution is equal to the area of the g en er a t
ing figure m ultiplied by the C ircu m ference described by the centre of
grav ity We S hall see that this m ethod excels that of Kepler and
Cav alieri in following a m ore exact and natural course ; but it has the
disadvantage of necessitating the determination of the centre of grav
ity which in itself m ay be a more diffi cult problem than the original
one of finding the v olu m e Gul din made som e atte mpts to pro v e his
theorem but Cavalieri pointed out the weakness of his de monstration
was a native of Wii r temb e rg a n d i m
Joh ann e s K e pl e r ( 1 57 1 —1 6 30 )
b ib ed Copernican principles while at the University of T ubi n g en His
pursuit of science was repeate dly interrupted by war religious perse
pecuniary e m barrassm ents frequent changes of resid ence
c u t i on
and family troubles In 1 60 0 he beca me for one year assistant to the
Danish astrono mer Tycho B rahe i n the observatory near P rague
The relation between the two great astronom ers was not always of an
agreeable character Kepler s publications are v olum inous His first a t
te mpt to explain the solar system was m ade in 1 596 when he thought
he had discovered a curious relation between the five regular solids
and the number and distance of the planets The publication of this
pseudo discov ery brought him m uch fam e : At one tim e he tried to
represent the orbit of M ars by the ov al curv e which we now write i n
3
=
polar coordinates p 2 r cos 0 M aturer reflection and intercourse
with Tycho Brahe an d G al i l eo l ed hi m to investigations and results
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S ee F Caj ori , Hi s tory
.
f
o
the Logar i thmi c S lide Ru l e, N ew York ,
1 90 9 .
A HISTORY OF M ATHE M ATICS
160
wor thy of his genius
Kepler s laws
He enriched pure math e
mati cs as well as astronomy I t is not strange that he was interested
in the m athem atical science which had done him so much serv ice ; for
“
if the Greeks had not cultivated conic sections Kepler could not
1
have superseded P tolemy
The Greeks never dream ed that these
curves would ever be of practical use ; Arist ae us and Apoll onius
studied them m erely to satisfy their intellectual cravings after the
ideal ; yet the conic sections assisted Kepler in tracing the march of
the planets in their elliptic orbits Kepler m ade also extended use of
logarith m s and decim al fractions and was enthusiastic in d iffusin g
a knowledge of them At one time while purchasin g wine he was
struck by the inaccuracy of the ordinary modes of d eterm ining the
contents of kegs This led h im to the study of the vol um es of solids
of rev olution and to the publication of the S tereometri a Dol i oru m in
In it he deals first with the solids know n to Archimedes and
1615
then takes up others Kepler m ade wide application of an old but
neglected idea that of infinitely great and infinitely small quantities
Greek m athem aticians usually shunned this notion but with it modern
mathe maticians co mpletely revolutioniz ed the science In comparing
rectilinear figures the method of superposition was employed by the
ancients but in comparing rectilinear and curvilinear figures wi th
each other this m ethod failed because no addition or subtraction of
rectilinear figures could ever produce curvilinear ones To meet this
case they devised the M ethod of Exhaustion which was long and
d ifficult ; it was purely syn thetical and in general required that the
conclusion should be known at the outset The new notion of infinity
led gradually to the invention of m ethods irnm easu rabl y m ore power
ful Kepler conceived the circle to be composed of an infinite nu mber
of triangles hav ing their comm on vertices at the centre and their
bases in the C ircumf erence ; and the sphere to consist of an infinite
nu mber of pyram ids H e appl i ed conceptions of this kind to the de
term ination of the areas and volu mes of figures generated by curv es
revolving about any line as axis but succeeded in solving only a f ew
of the simplest out of the 84 problem s which he proposed for investi
g a t i on in his S ter eometri a
Other points of m athem atical interest in Kepler s works are ( 1 )
the
assertion that the circu mference of an ellipse whose axes are 2 a and
2 b is nearly 7r
a passage from which it has been i nk n ed
(2)
that Kepler knew the variation of a function near its m axim um value
the assu mption of the principle of continuity (which
to disappear ; (3)
diff erentiates m odern from ancient geo m etry)when he shows that
a parabola has a focus at infinity that lines radiating from this c ee c u s
focus are parallel and have no other point at i n finity
The S ter eometri a led Cavalieri an Italian Jesuit to the consideration
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Will i am Wh ew ell
Vol I , p 3 1 1
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H i story
f
o
the I nducti ve
,
S ci ences , 3 rd Ed
.
,
N ew York ,
1 8 58 ,
A HISTORY OF M ATHE M ATICS
62
There is an important curve not known to the ancients which now
began to be studied with great z eal Roberval gave it the na me of
“
trochoid P ascal the nam e of roulette Galileo the name of cy
c l o id
The invention of this cu rve see m s to be due to Charl es B ou
vel l es who in a geo m etry publi shed in P aris in 1 50 1 refers to this cu r ve
in connection with the problem of the squaring of the circle Galileo
value d it for the graceful form it woul d give to arches in architecture
He ascertained its area by weighing paper figures of the cycloid against
that of the generating circle and found thereby the first area to be
nearly but not exactly thrice the latter A m athematical determina
t i on was made by his pupil Evan g el i s ta Torri c elli ( 1 6 0 8
who
is more widely known as a physicist than as a m athem atician
By the M ethod of Indivisibles he dem onstrated its area to be triple
that of the revolving circle and pub l ished his solution This sam e
quadrature had been e ffe cted a few y ears earlier ( about 1 6 3 6 )
by
Roberval in France but his solution was not known to the Italians
Roberval being a m an of irritable and violent disposition unjustly
accused the m ild and am iable Torricelli of stealing the proof This
accusation of pla giarism created so m uch chagrin with Torricelli that
it is considered to have been the cause of his early death Vin c e n zo
Vivi ani ( 1 6 2 2
another prom inent pup il of Galileo determined
the tangent to the cycloid Thi s was accompli shed in France by
Descartes and F ermat
In France where geom etry began to be cultivated with greatest
success Roberval F ermat P ascal e mployed the M ethod of Indivis
i bl es and m ade new i mpro v e m ents in it G il e s P e r s o n e d e R o b e rval
for forty years professor of m athematics at the College
( 1 60 2
of France in P aris clai med for hi m self the invention of the M ethod of
Indivi sibles Since his complete works were not published until after
his death it is difli c u l t to settle questions of priority M on tu cl a and
Chasl es are of th e Opinion that he invented the m e thod independently
of and earlier than the I talian geom eter though the work of the latter
was published much earli er than Roberval s M arie finds it diffi cult
to believe that the Frenchman borrowed nothing whatever from the
Italian for both could not have hit independently upon the word
I n di vi si bl es which is applicable to infinitely sm all quantities as con
cei v ed by Cavalieri
but not as conceived by Roberval Roberval
and P ascal improved the rational basis of the M ethod of In divisibles
by consi dering an area as m a d e up of an indefinite nu mber of rectangles
instead of lines and a solid as composed of indefinitely sm all solids
instead of surfaces Roberval applied the m ethod to the finding of
areas volumes and centres of grav ity He e ffected the quadrature
1
m
m
m
of a parabola of any degree y a x and also of a parabola y
m
We have already m entioned his qua drature of the cycloid
a
x
Roberval is best known for his m ethod of drawing tangents which
however was invented at the same tim e if not earlier by T orri c el l i
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VIETA
D ES CARTES
To
163
Torricelli s appeared in 1 6 44 under the title Oper a geometri ca Rober
Som e of his special applications
v a l gives the fuller exposition of it
were published at P aris as early as 1 6 44 in M er se nn e s C ogi ta ta physi co
Roberval presented the full develop m ent of the sub
mathemati ca
m
A
c
e
f
e
i
on c es in 1 66 8 which published it
to
the
French
a
O
ec
t
d
S
y
j
in its M émoi r es This academy had grown out of scient ific m eetings
held with M ersenne at P aris I t was founded by M inister Richelieu
in 1 6 3 5 and reorganiz ed by M inister Colbert in 1 666 M ari n M er sen n e
—
rendered great services to science His polite and e n
( 1 588 1 6 4 8 )
gaging m anners procured him m any friends inclu ding D escartes and
F erma t He encouraged scientific research carried on an ex tensiv e
correspondence a n d thereby was the medium for the i n ter c ommu n ic a
tion of scientific intelligence
Roberv al s m ethod of drawing tangents is allied to Newton s prin
c ip l e of fl uxi on s : Arch im edes conceived his spiral to be generated by
a double m otion This idea Roberv al exten ded to all curves P lane
curves as for instance the conic sections m ay be generated by a point
acted upon by two forces and are the resultant of two m otions If
at any point of the curve the resultant be resolv ed into its components
then the diagonal of the parallelogram determ ined by them is the tan
gent to the curve at that point The greatest difficulty connecte d
wi th this ingenious m ethod consisted in resolv ing the resultant into
components hav ing the proper lengths and directions Roberval did
not always succee d i n doing this yet his new idea was a great step in
advance He broke off from the ancient definition of a tangent as
—
a straight line hav ing only one point in comm on with a curve a defi
n i ti on which by the m etho d s then available was not adapted to bri ng
out the properties of tangents to curves of higher degrees ; nor e v en of
curves of the second degree and the parts they may be made to play
in the generation of the curves The subj ect of tangents receive d
special attention also from F erma t D escartes and Barrow and
reached its highest develop m ent after the invention of the differential
calculus Fermat and D escartes d efined tangents as secants whose
two points of intersection with the curve coincide ; B arrow considered
a curve a polygon and called one of its sides produced a tangent
A profound schola r in all branches of learning and a m athem atician
of exceptional powers was P i e rr e d e F e r mat ( 1 6 0 1
He studied
law at Toulouse and in 1 6 3 1 was m ade councillor for the parliament
of Toulouse His leisure tim e was m ostly devoted to m athem atics
which he stu died with irresistible passion Unlike D escartes and
Fermat has left the im
P ascal he led a quiet and unaggressive l i fe
press of his genius upon all branches of mathem atics then known
A great contribution to geom etry was his D e maxi mi s et mi ni mi s
About twenty years earl i er Kepler had first observ e d that the incre
m ent of a var i able as for instance the or di nate of a curv e is evan
e sce n t for values very near a m axi m u m or a m in i m um value of the
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A HISTORY OF M ATHE M ATICS
1 64
ariable D eveloping this idea F erma t obtained his rule for maxima
an d minima
He substituted x+ e for x in the given function of x and
then equated to each other the two consecutive values of the function
and div i d ed the equation by e If e be taken 0 then the roots of this
equation are the v alues of x m aking the fun ction a maximum or a
m inim u m F e rma t was in possession of this rule in 1 6 2 9 The main
difference between it and the rule of the differential calculus is that it
introduces the indefinite quantity e instead of the in fi nitely s mall dx
F erma t m ade it the basis for his m ethod of d rawing tangents which
inv ol v ed the determ ination of the length of the subtangent for a giv en
point of a curve
Owing to a want of explicitness in statement Ferm at s method of
m axim a and m ini m a and of tangents was severely attacked by hi s
great conte mporary D escartes who cou l d never be brough t to render
due j ustice to his m erit In the ensuing dispute F erm a t found t w o
z ealous d efenders in Roberval and P ascal
the father ; whi le C M y
dorge G D esargues an d Claude Hardy Supported D escartes
Since F e rm a t intro d uce d the conception of infinitely small di ffer
e n c e s between consecutive values of a function and arrived at the
principle for fin ding the m axim a and m inim a it was m aintained by
Lagrange Laplace and Fourier that F erma t may be regarded as the
first in v entor of the diff erential calculus This point is not well taken
as will be seen from the words of P oisson him self a Frenchman who
rightly says that the differential calculus consists in a system of rules
proper for finding the di fferentials of all functions ra th er than in the
use which may be m ade of these infinitely s mall variations in the s o
l u t io n of one or two isolated proble m s )
A conte mporary m athe m atician whose genius perhaps equalled that
of the great F erma t was B l ai s e P as c al ( 1 6 2 3
He was born at
Clerm ont in Au v ergne In 1 6 2 6 his father retired to P aris where he
dev oted hi m self to teaching his son for he would not trust his educa
tion to others Blaise P ascal s genius for geometry showed itself when
he was but twel v e years old His fa ther was well skilled in m athe
m aties but di d not wish his son to study it until he was perfect l y
acquainted with Latin and Greek All m athem atical books were
hidde n out of his sight The boy once asked his father what m athe
“
m at i c s treated of and was answered i n general
that it was the
m ethod of m aking figures with exactness and of finding out what
proportions they relati v ely had to one another
He was at the same
ti me forb i dden to talk any m ore about it or ever to think of it B ut
his genius could not sub m i t to be confined within these bounds Start
ing with the bare fact that m athematics taught the m eans of making
figures infall i bly exact he e mployed his thoughts about it and with
a piece of charcoal drew figures upon the tiles of the pave ment trying
the metho d s of d raw i ng for example an exact circle or equ i lateral
triangle He gave names of his own to these figures and then formed
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M ATHE M ATICS
A HISTORY OF
1 66
Huygens Wren and F erma t solved som e of the questions The chief
disco v eries of C hri s toph e r Wr e n ( 1 6 3 2
the celebrated architect
of S t P aul s Cathe d ral in Lon don were the rectification of a cycloidal
arc and the determ ination of its centre of gravity F erma t found the
area generated by an arc of the cycloid Huygens invented the c y
c l oi dal pendulu m
The beginning of th e seventeenth century witnessed also a revival of
synthetic geo m etry One who treated conics still by ancient m e thods
but who succeeded in greatly simplifying m any proli x proofs of Apoll o
i n P aris a friend of D escartes
nius was C l au d e M yd or g e ( 1 58 5—1 6 4 7 )
B ut it re m ained for G ir ar d D e s ar g u e s ( 1 593—1 6 6 2 )
of Lyons and for
P ascal to leave the beaten track and cut o ut fresh paths They intro
d u c e d the important m ethod of P erspective All conics on a cone with
c i rcular base appear circular to an eye at the apex H en ce D e sarg u es
and P ascal concei v ed the treatment of the conic sections as proj ections
of circles Two i mportant and beautiful theorem s were given by D es
“
argues : The one is ou the involution of the s i x points in which a
transversal m eets a conic and an inscribed quadrangle ; the other is
that if the vertices of two triangles S ituated either in S pace or in
a plane lie on three lines m eeting in a point then their S ides m eet in
three points lying on a line ; and conversely This last theore m has
been employed i n recent tim es by B rian ch on C S turm G erg onn e
an d P oncelet
P oncelet m a d e it the basis of his beautiful theory of
ho mol ogical figures We owe to D esargues the theory of involution
and of transv ersals ; also the beautiful conception that the two ex
t r emi t i es of a straight line m a y be considered as m eeting at infi nity
a n d that parallels d i ff er fro m othe r pairs of lines only in having their
points of intersection at infinity lH e r e—
invented the epicycloid and
showed its applicat i on to the construction of gear teeth a subj ect
elaborated m ore fully later by La Hire P ascal greatly admired
Desargues results saying ( in his E ss ai s pou r l es C on i qu es ) I wish to
acknowle dge that I owe the little that I have discovered on this sub
P ascal s and D esargues writings contained
j ec t to his writings
some of the fundam ental ideas of mo dern synthetic geom etry In
P ascal s wonderful work on conics written at the age of S ix teen and
now lost were given the theorem on the anharm onic ratio first found
in P appus and also that celebrated proposition on the mystic hexagon
“
known as P ascal s theore m viz that the oppos i te s i des of a hexa
gon inscribed in a conic intersect in three points which are collinear
This theore m form ed the keystone to his theory He him self said
that from this alone he deduced over 40 0 corollaries e mbracing the
conics of Apollonius and m any other results Less gi fte d than D es
argues a n d P ascal was P h ili ppe d e l a H ir e ( 1 6 40
At first
activ e as a pa i nter he afterward s d e v ote d him self to astronomy and
m athe matics a n d beca m e professor of the Co l l ege de France in P aris
He wrote three works on conic secti ons published in 1 6 73 1 6 7 9 and
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VIETA TO
D ES CARTES
167
es C on i ca e was best known
The
last
of
these
the
ec
t
i
o
n
La
S
5
Hire gav e the polar properties of circles and by proj ection transferred
his polar theory from the circle to the conic sections In the c on struc
tion of maps D e la Hire used globular projection in which the eye
is not at the pole of the sphere as in the P tole maic stereographic pro
j ec ti on but on the radius pro duced through the pole at a distance
r sin 4 5
outside the sphere Globular projection has the advantage
that everywhere on the map there is approxim ately the same degree
of exaggeration of distances This m ode of projection was modified
by his countryman A P arent D e la Hire wrote on roulettes on
graphic methods epic y cloids c on choidsr md on m agic squares The
labors of D e la Hire the genius of D esargues and P ascal uncovered
several of the rich treasures of m o dern synthetic geom etry ; but owing
to the absorbing interest taken in the analytical geom etry of D escartes
and later in the di fferential calculus the subject was a l most entirely
neglected until the nine t eenth century
In the theory of numbers no new results of scientific value had been
reached for over 1 0 0 0 years extending fro m the times of D iophantus
and the H indus until the beginning of the seventeenth century B ut
the illustrious period we are now consi dering produced m en who
rescued this science from the real m of m ysticis m and superstition
in which i t had been so long i mprisoned ; the properties of nu mbers
bega n again to be stu died scientifically Not being in possession of
t h e Hin d u indeter m inate analysis m any beautiful results of the
B rahmi ns had to be r e discovered by the Europeans Thus a solution
in integers of linear indeterminate equations was r e discovered by the
Frenchm an B a c h e t d e M ézir i ac ( 1 58 1
who was the earliest
noteworthy European D i opha n t i s t In 1 6 1 2 he published P r obl emes
pl ai san ts ci dél ecta bl es qu i s e f on t par l es n ombr es and in 1 6 2 1 a Greek
edition of Di ophan tu s with notes An interest in pri m e nu mbers is
disclosed in the so called M ers en n e s nu mbers of th e form M p =
—1 with p pri m e M arin M ersenne asserted in the preface to his
29
C ogi ta ta P hysi co M a thema ti ca 1 6 44 that the only values of p not
greater than 2 5 7 which m ake M D a prim e are 1 2 3 5 7 1 3 1 7 1 9
Four mistakes have n ow been detecte d in
3 1 6 7 1 2 7 and 2 57
M erse n n e s classification v i z M 6 7 is compos i te ; M 6 1 M 8 9 and M I O7
are prim e M 1 8 1 has been found to be compos i te M ersenne gave in
2 8 4 96
8 1 2 8 2 3 5 50 33 6
1 6 44 also the first eight perfect nu m bers 6
In Eucl i d s El e
8 58 986 90 56 1 3 7 43 86 9 1 3 2 8 2 3 0 5 8 43 0 0 8 1 3995 2 1 2 8
men ts Bk 9 P rop 3 6 is g i v en the form ula for perfect num bers
1
1
P
P
P
2
The above eight perfect nu mbers
where 2
1 is pri m e
(2
are repro d uced by taking p= 2 3 5 7 1 3 1 7, 1 9 3 1 A ninth perfect
nu m ber was found i n 1 8 8 5 by P S eelh off for which p= 6 1 a tenth
in 1 9 1 2 by R E P owers for which p= 8 9 The father of the m o dern
theory of numbers is F e rm at He was S O uncommunicative in di s
position that he generally concealed his m ethods and m ade known
1 68
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A HISTORY OF M ATHE M ATICS
1 68
res ul ts only In some cases later analysts have been greatly
pu zzled in the attempt of supplying the proofs Fermat owned a copy
of B ach e t s Di ophantus in which he entered num erous marginal notes
In 1 6 7 0 these notes were incorporated in a new edition of Di ophantus
brought out by his son Other theore ms on numbers due to F ermat
were publi shed in his Oper a vari a ( edited by hi s son )
and i n Walli s s
C ommerci u m epi s tol i cu m of 1 6 58
Of the foll ow ing theorems the
1
first seven are found in the marginal notes :
is i mpossible for integral values of x y and 2 when
( 1)
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This famous theorem was appended by Fermat to the problem of
D iophantus II 8 : To divide a given square number into two squares
“
Fermat s m arginal note is as foll ows : On the other hand it is im
possible to separate a cube into two cubes or a biquadrate into two
biquadrates or generally any power e xcept a square into two powers
with the sam e exponent I have discovered a truly m arvelous proof
of this which however the m argin is not large enough to contai n
That Ferm at actually possessed a proof is doubtful N 0 general
proof has yet been published Euler proved the theorem for n = 3
and n = 4 ; D irichlet for n = 5 and u = 1 4 G Lamé for n = 7 and K um
m er for m any other values
Repeatedly was the theorem m ade the
priz e question of learned societies by the Academy of Sciences in
The
P aris in 1 8 2 3 and 1 8 50 by the Acade m y of B russels in 1 883
recent history of the theorem foll ows later
A prim e of the form 4u + 1 is only once the hypo thenuse of a
(2 )
right triangle ; its square is twice ; its cube is three times etc Ex
a mple :
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A prime of the form
(3)
can be expressed once and only
once as the su m of two squares P roved by Euler
A n um ber composed of two cubes can be resolved into two
(4)
other cubes in an infinite multiplicity of ways
Every nu mber is either a triangular number or the sum of two
5)
or three trian g ular n umbers ; either a square or the sum of two three
or four squares ; either a pentagonal number or the sum of two three
four or five pentagon al nu mbers ; sim ilarly for polygonal numbers
in general The proof of this and other theorem s is promised by
F erm at in a future work which never appeared This theorem is
also given with others in a letter of 1 63 7
addressed to P ater
4u + 1
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M er sen n e
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A S m any num bers as you please may be found such that the
(6)
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square of each re mains a square on the addition to or sub traction from
it of the sum of all the numbers
.
fu l l er hi s t o ri c al acco u n t o f F e rm a t s D i op han t in e t heo re ms an d p rob
S ee
l e m s , s e e T L H ea t h , Di opha n tu s of A l exa n dr i a , 2 E d , 1 9 1 0 , pp 2 6 7 3 2 8
a l so A n n a l s of M a themati cs , 2 S , Vol
1 6 1 1 87
1 8 , 1 9 1 7 , pp
1
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1
A HISTORY OF
70
M ATHE M ATICS
factor of the form 4u + 1 From the above supposition it would follo w
that 5 is not the sum of two squares—a conclusion contrary to fact
Hence the supposition is false and the theorem is estab l ished F erma t
applied this m ethod of descent with success in a large nu mber of
theorem s By this method L Euler A M Legendre P G L Di ri ch
let proved se v eral of his enunciations and many other nu merical
propositions
F erma t was interested in magic squares These squares to which
the Chinese and Arabs were so partial reached the Occi d ent not l ater
than the fifteenth century A m agic square of 2 5 cells was found by
M Curtz e in a Ger man m anuscrip t of that ti m e The artist Albrecht
“
D ii re r exhibits one of 1 6 cells in 1 5 1 4 in his painting calle d M ela n
Choli e
The abo v e na med B er n ha rd Fr eni cl e de B essy ( about 1 6 0 2
1 6 7 5)
brought out the fact that the nu mber of m agic squares increased
enormously with the order by writing do w n 8 8 0 magic squares of
the or der four Ferm at gav e a general rule for finding the number of
m agic squares of the order n such that for n = 8 this nu mber was
but he seem s to hav e recogniz ed the falsity of his
ru l e B achet de M ézi riac i n his P r obl emes pl ai s an ts cl dél ectabl es
Lyon 1 6 1 2 gave a rule des terrasses for writ i ng down m agic
squares of odd order Fr en icl e de B essy gave a process for those of
1
e v en order In the seventeenth century magic squares were studied
by Antoine Arnauld Jean P re s t e t J O z anam ; in the eighteenth cen
tury by P oignard D e la Hire J Sauveur L L P ajot J J Rallier
des Ourm es L Euler and B enja m in Franklin In a letter B Franklin
“
2
said of his m agic square of 1 6 cells I m ake no question but you
will read ily allo w the square of 1 6 to be the m ost m agically m agical
of any m agic square ev er m a d e by any m agician
A correspondence between B P as cal and P Fermat relating to a
certain ga me of chance was the germ of the theory of probabilities
of which som e anticipations are foun d in Cardan Tartaglia J Kepler
and Galileo Che v alier de M ere proposed to B P ascal the funda
“
2
m ental P roble m of P oints
to determ ine the probability which
each player has at any given stage of the gam e of winning the game
P ascal and F e rma t supposed that the players have equal chances of
winning a S ingle point
The former co mmunicated this proble m to F ermat who studied
it with li v ely interest and sol v ed it by the theory of combinations a
theory which was diligently stu died both by him and P ascal The
calculus of probabilities engaged the attention also of C Huygens
The m ost i mportant theore m reached by him was that if A has p
chances of wi nning a su m a and q chances of winning a su m b then
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En cycl opedi c des s ci en ces math s , T I , V ol 3 , 1 90 6 , p 6 6
2
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P
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s
1
86
6
2
2
0
23 7
v
r
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m
l
t
B
l
a
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s
e
P
a
sc
a
l
T
S ee a l so I
Oeu es c
,
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, pp
,
p e es de
To d h u n te r , H i s tor y of the M a themati cal Theory of P r oba bi li ty, C a mb ri dg e a n d
L o n do n , 1 86 5 , C h ap t er I I
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VIETA TO D ES CARTES
he
may
expect to
the
win
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0
m
6
1
15
2
1 71
Huygens ga v e his results in
.
a treatise on probabili ty ( 1 6
which was the best account of the
subj ect until the appearance of Jakob B ernoulli s A r s conj ectandi
which containe d a reprint of Huygens treatise An absurd abuse of
m athem atics in connection with th e probability of testi mony was
made by J ohn C r ai g who in 1 6 99 concluded that faith in the Gospel
so far as it depended on oral tradition expired about the year 80 0
and so far as it depended on written tradition it would expire in the
year 3 1 50
Connected with the theory of probability were the investigations
on mortal ity and insurance The use of tables of mortality does not
seem to have been altogether unknow n to the ancients but the first
na m e usually m entioned in this connection is Captain John Gr au nt
who published at London in 1 6 6 2 his N atu ral an d P ol i ti cal Obs erva
ti on s
made u pon the bi l l s of mor tal i ty basing his de d uctions upon
records of deaths which began to be kept i n London in 1 592 and were
first intended to make known the progress of the plague Graunt was
careful to publish the actual figures on which he based his conc l usions
“
comparing him self when so doing to a S illy schoolboy com ing to
say his lessons to the world ( that peevish and tetchie m aster)who
brings a bundle of rods wherewith to be whipped for every m istake
1
he has co m mitted
Nothing of marked i mportance was done after
1
Graun t until 1 6 93 when Ed m und Halley published in the P hi l o
s ophi cal Tr an s acti on s ( London )
his celebrated m e m oir on the D egr ees
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f M or tal i ty of M an k i nd
A ttempt to as cer ta i n the P r i ce of
A n n u i ti es u pon Li ves
To find the val u e of an annuity , m ultiply the
C hance that the individual concerne d will be alive after n years by
w i th
o
an
.
the present value of the annual paym en t due at the end of n years ;
the n su m the results thus obtaine d for all values of n from 1 to the
extrem e possible age for the life of that individual Halley considers
also annuities on joint lives
Am ong the ancients Archim edes was the only one who attained
clear and correct notions on theoretical statics He had acquired
fi rm possession of the idea of pressure wh i ch lies at the root of me
c h an i c al science
B ut his ideas slept nearly twenty centuries until
the ti me of S S tevi n a n d G al il e o G al il e i ( 1 56 4
Stevin deter
m i ned accura t el y the force necessary to sustai n a body on a plane
inclined at any angle to the horizon He was in possession of a c om
While Stevin inv estigated statics
pl e t e doctrine of equilibriu m
Gal i leo pursued principally dynam ics Galil eo was the first to abandon
the idea usually attributed to Aristotle that bodies descend more
q uickly in proportion as they are heav ier ; he established the first law
of m otion ; determ ined the laws of falling bo dies ; and h avi ng obtained
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f
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P roba bi l i ty, p p 3 8 , 4 2
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A H ISTORY OF M ATHE M ATICS
1 72
a clear notion of acceleration and of the indepen dence of diff erent
m otions was able to prove that proj ectiles move in parabolic curves
Up to his tim e it was believ ed that a cannon ball moved forward at
first in a straight line and then suddenly fell vertically to the ground
Galileo had an understanding of cen tri fu gal f orces and gave a correct
definition of momen tu m Though he formulated the fundam ental
principle of statics ; known as the par al l el ogr am of f orces yet he did
not fully recogniz e its scope The principle of virtual velocities was
partly conceived by G u i d o Ub al do (died
and afterwards more
fully by Galil eo
Galileo is the founder of the science of dynamics Am ong his con
t e mporaries it was chiefly the novelties he detected in the sky that
m ade h i m celebrated but J Lagrange clai m s that his astrono mical
discoveries required only a telescope and perseverance while it took
an extraor dinary genius to discover laws from phenom ena which we
see constantly and of which th e true e xplanation escaped all earlier
philosophers Galileo s dialogues on m echanics the Di scors i e demos
tr azi on i ma tema ti che 1 6 3 8 touch also the subject of infinite aggregates
The author displays a keenness of vision and an originality which
was not equalled before the tim e of D edekind and Georg Cantor
Salviati who i n general represents Galileo s own ideas in these dia
1 “
l og u e s says
infinity and indivisibility are in their very nature i n
comprehensibl e to us
Simplicio who is the spokes man of Aris
“
t o t el i a n scho l astic philosophy re marks that
the infinity of points
in the long line is greater than the infinity of points in the short line
“
Then co m e the rem arkable words of Salviati : This is one of the
diffi culties which arise when we atte mpt with our finite m in ds to
discuss the infinite assigning to it those properties which we give to
the finite and unlim ited ; but this I think is wrong for we cannot
speak of infinite quantities as being the one greater or less than or
equal to another
We can on l y infer that the totality of all
nu mbers is infinite and that the n u mber of squares is infinite and
that the nu mber of the roots is infinite ; neither is the nu mber of squares
less than the totality of all nu mbers nor the latter greater than the
former ; and fi nally the attributes equal greater and l eS S are
not app l icable to infinite but only to finite quantities
O ne
l i ne does not contain m ore or less or just as man y poi n t s as another
e ac h l i n e contains an infinite nu m ber
but
From the time of
Galileo an d D escartes to Sir W i ll i am Hamilton there was held the
doctrine of the fi n i tu de of the hu man mind and its consequent in
ability to conceive the infinite A D e M organ ridiculed this saying
“
the argument a mounts to this who drives fat oxen should himself
be fat
Infin i te series which sprang into pro m i nence at the tim e of the
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S ee G al i l eo s Di a l og u es con cer n i n g two n ew S ci en ces , t ra n sl a t ed b y H e n ry C rew
“
a n d Al fo n so d e S a l v i o , N ew Y o rk , 1 9 1 4,
F i rs D ay, pp 30 3 2
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A
1 74
O F M ATHE M ATICS
HISTORY
D escartes considered m athe matical studies absolutely pernicious as a
m eans of internal culture In a letter to M ersenne D escartes says :
“
M D esargues puts me under obligations on account of the pains
that it has pleased h i m to have in me i n that he S hows that he is
sorry that I do not wish to study m ore in geometry but I have re
sol v ed to quit only abstract geo metry that is to say the consideration
of questions which s er ve onl y to exer ci se the mi nd and this in order to
study another kind of geo metry which has for its obj ect the e xp l ana
tion of the phenom ena of nature
You know that all my ph y sics
is nothing else than geo metry
The years between 1 6 2 9 and 1 649
were passed by h im i n Holland in the study principally of physics
and m etaphysics His residence in Holla nd Was during the m ost
brilliant days of the D utch state In 1 63 7 he published his Di scou rs
de l o M éthode containing a mong others an essay of 1 0 6 pages on
geo metry His Geometri c is not easy reading An edition appeared
subsequently with notes by his fr iend De B eau n e which were inten ded
to re mov e the difficulties The Geometr i c of D escartes is of epoch
m aking i mportance ; ne v ertheless we cann ot accept M ichel Chasles
state ment that this work is pr ol es s i ne matre crea ta—a child brought
into being without a m other I n part D escartes ideas are found in
Apollonius ; the applicatio n of algebra to geo metry is found in Vieta
G he tal di O u g h t red and e v en a mong the Arabs F e rma t D escartes
contemporary advanced i deas on analytical geom etry akin to his
own in a treatise entitled A d l ocos pl an os cl s ol idos i s agoge which
howe v er was not published until 1 6 7 9 in Ferm at s Vari a oper a In
D escartes Géométri e there is no syste matic dev elop ment of the
m ethod of analytics The m ethod m ust be constructe d fro m iso l ate d
state ments occurring in di ff erent parts of the treatise In the 3 2
geometric d ra w ings illustrating the text the a xes of coord inates are
in no case explicitly set forth The treatise consists of three book s
The first deals with p roble ms which can be constructed by the a id
of the circle and straight line only
The second book is on the
“
nature of curved lines
The third book treats of the construction
of proble ms solid and more than solid
I n the first book it is ma d e
clear that if a proble m has a finite nu mber of solutions the final
equation ob tained will have o nly one unknown that if the final
“
equation has t w o or more unknowns the problem is not whol l y
“
1
If the final equation has two unkno w ns then since
determined
there is always an infinity of diff erent points which satisfy th e d e
m and it is therefore required to recogniz e and trace the line on which
To acco mplish this D escartes
all of the m m ust be located (p
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C J K eyser , M athemati cs ,
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1 90 7 ,
pp 3 6 0 3 7 2
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D esc a r t e s Géometr i e, ed
B i bl i otheca ma thema ti ca , 3 S
191 2,
.
pp
20
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—
44 ; F
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C aj o r i i n P opul ar S ci ence M on thl y,
.
We a re h ere g u ided by G En e st rom i n
p 4
—
l
1
2
0
2 43 ; V ol
1 2 , p p 2 73 , 2 74 ; Vo l
14,
V
o
1
4
,
, pp
n
n
l
1
2
1
2 43 , 3 2 9 , 33 0
a
n
b
H
i
e
l
e
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t
e
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V
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d y
W
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4 , pp
p 35 7 ,
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1 886 ,
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D ES CARTES
NEWTON
TO
1 75
selects a straight l ine which he som eti mes calls a diam eter (p 3 1 )
and associates each of its points with a point sought in such a way that
the latter can be constructed whe n the form er point is assumed as
“
known Thus on p 1 8 he says J e c hoises une ligne droite co mm e
A B pour rapporter a ses d ivers points tous ceux de cette l igne courbe
EC
Here D escartes follows Apollonius who related the points of
a conic to the points of a dia meter by distances ( ordinates )
which
m ake a constant angle with the diam eter and are determined in length
by the positio n of the point on the dia meter This constant angle is
with D escartes usually a right angle The n e w feature introduced by
Descartes was the use of an equ ati on wi th mor e than one u n k n own so
that (in case of two unknowns )
for any value of one unknown (ab
scissa )the length of the other ( ordinate )
could be co mputed He
uses the letters x and y for the abscissa and ordinate He makes it
plain that the x and y may be represen t ed by other distances t han the
ones selected by hi m (p
that for instance the angle form ed by
x and y need not be a right angle
I t is noteworthy that Descartes
and F e rma t and their successors down to the mid dle of the eighteenth
century used oblique co ordinates m ore frequently than did later
analysts I t is also note w orlthy that D escartes does not formally
introduce a second a xis our y axis Such formal introduction is found
in G Cra mer s I n tr odu cti on d l an al yse des l i gn es cou r bes al gébri qu es
1 750 ; earlier publications by de Gua L Euler W M ur d och and others
contai n o nly occasional references to a y—
axis The words abscissa
ordinate were not used by D escartes In the strictly techni cal
sense of anal ytics as one of the coordinates of a point th e word
ordinate was used by Leibniz i n 1 6 94 but i n a less restricted sense
such e xpressions as o rdi n a tim a ppl i ca tae occur m uch earl ier in
F C o mman din u s and others The technical use of abscissa is
observed i n the eighteenth century by C Wo l f and others In the
“
m ore general sense of a d istance it was used earlier by B Cavalier i
in his Indiv i sibles by S tef a no degl i A ngel i ( 1 6 2 3
a professor
of m athe matics in Ro me and by others Leibniz introduced the word
To guard aga inst certain current historical
; c oOrdi n a tae i n 1 6 92
“
errors we qu ote the following fro m P Tannery : One frequentl y
attributes wrongly to D escartes the introduction of the convention
of reckoning coordinates positiv ely and negati v ely in the sense in
which we start the m fro m the origin The truth is that in this respect
the Geometri c of 1 6 3 7 contains only certain re marks touching the
interpretation of real or false (positive or negative )
roots of equations
If then w e examine with care the rules given by D escartes in
his Geometri c as well as his application of the m we notice that he
adopts as a principle that an equation of a geometric locus is not
vali d except for the angle of the coordinates ( quadrant )
in which it
was established and all his contemporaries do likewise The extension
of an equation to other angles ( quadrants )
was freely made in par ti cu
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A
1 76
O F M ATHE M ATICS
HISTORY
lar cases for the interpretation of the negative roots of equations ;
but while it served particular conventions (for e xample for reckoning
distances as positive and negative )it was in reality quite long in
co mpletely establishing itself and one cannot attribute the honor for
it to any particular geom eter
analytical geometry partly
D escartes geo metry was called
because unlike the synthetic geometry of the ancients it is actually
a n al yti ca l in the sense that the word is used in logic ; and partly b e
cause the practice had then already arisen of designating by the term
a n al ys i s the calculus with general quantities
The first important example solved by D e scartes in hi s geometry
“
“
is the proble m of P appus ; v iz Given several straight li nes in a
plane to find the locus of a point such that the perpendiculars or more
generally straight lines at given angles drawn from the point to the
given lines S hall satisfy the condition that the product of certain of
them S hall be in a give n ratio to the product of the rest
Of this
celebrated proble m the Greeks solved only the special case when the
nu mber of giv en lines is four in which case the locus of the point
turns out to be a conic section By D escartes it was solv ed c om
pl e t el y and it a fforded an excellent exa mple of the use which can be
m ade of his analytical m ethod in the study of loci Another solution
was give n later by Newton in the P ri nci pi a D escartes illustrates
“
his analytical m ethod also by the ovals n ow nam ed after him cer
taines ovales que vous verrez etre tres utiles pour la th éorie de la
catoptrique
These curves were studied by D escartes probably as
early as 1 6 2 9 ; they were intended by him to serve in the construction
of converging lenses but yielded no results of practical value In
1
the nineteenth century they received m uch attention
The power of D escartes anal ytical m ethod in geom etry has been
vividly set forth recently by L Bolt zmann in the rem ark that the
form u l a appears at ti mes cleverer than the man who in v ented it Of
all the problem s which he solved by his geom etry none gave him as
great pleasure as his m ode of constructing tangents It was published
earlier than the m ethods of F ermat and Roberval which were noticed
on a preceding page
D escartes m ethod consisted in first finding the norm al Through
a given point x y of the curve he drew a circle which had its centre
at the intersection of the norm al and the x—
axis Then he imposed
the condition that the circle cut the curve in two coincident points
In 1 63 8 D escartes indicated in a letter that in place of the
x y
circle a straight line may be used This idea is elaborated by Fl ori
mond de B eau n e in his notes to the 1 6 49 edition of D escartes Géométr i e
In finding the point of intersection of the normal and x axis D escartes
used the method of I ndeter mi n ate Coeffi ci en ts of which he bears the
honor of invention Indeterm inate coeffi cients were e mployed by
I 19
S ee G Lori a Ebene Cu rvou ( F S c hut te)
74
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H ISTORY
A
1 78
O F M AT H EM ATIC S
of its beautiful properties and the controversies which their discovery
occasioned Its quadrature by Roberval was generally considered a
brilliant achieve ment but D escartes co mmented on it by saying that
any one m oderately well versed in geom etry might ha v e done this
He then sent a S hort demonstration of his own On Roberval s i n
t i ma t i n g that he had been assisted by a knowledge of the solution
D escartes constructed the tangent to the curve and challenged
Roberval and F erm a t to do the sam e Fermat acco mplished it but
Roberval never succeeded in solving this p roblem which had cost
the genius of D escartes but a m oderate degree of attention
The application of algebra to the doctri ne of curve d lines reacted
favorably upon algebra A S a n abstract Science D escartes i mproved
it by the introduction of the modern exponential notation In his
“
2
Géométr i e 1 6 3 7 he writes ao o n a pour m ultiplier a par s oi meme ;
3
pour l e m ultiplier encore une fois par a e t ainsi a l i nfi n i
et a
“
3
3
Thus while F Vieta represented A by A c u b u s and Ste vin x
3
by a figure 3 within a s mall circle D escartes wrote a In his Geometri c
he does not use negative and fractional exponents nor l iteral ex
His notation was the outgrowth and an i mprovem ent of
p on en t s
notations e mployed by writers before him Nicolas C hu q u e t s m anu
3
1
script work L e Tri party en l a sci ence des n ombr es 1 4 84 gives 1 2 x
8
3
5
8
5
and r ox and their product 1 2 0 x by the symbols 1 2 1 0 1 2 0
respectiv ely C hu q u e t goes even further and writes 1 2 x and 7x
2
3
1
thus
he represents the product of 8x and 7x by 56
J
B urgi R eym er and J Kepler use Ro man nu m erals for the exponen
I
I
2
Thom as Harriot S imply
tial symbol J Burgi writes 1 6x thus
16
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repeats the letters ; he writes in his A rti s an al yti coe praxi s
"
2
1 0 2 4 aa + 6 2 54 a
a
1 0 2 4 a + 6 2 54 a thus : a aaa
D escartes exponent i al notation sprea d rap i dly ; about 1 6 60 or
1 6 7 0 the positive integral exponent h a d w o n an un d i sputed place in
algebraic notation In 1 6 5 6 J Wallis S peaks of negative a n d fractional
indices in his A r i thmeti ca i nfi n i toru m but he does not actually
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write
_ 1
a
for
or
a
’
/s
,
—
3
for x/a It was I Newton who in his fam ous
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letter to H Oldenburg dated June 1 3 1 6 7 6 and containing his a n
n ou n c em en t of the bino m ial theorem first uses negative and fractional
exponents
With D escartes a letter represented always only a positiv e nu mber
I t was Johann H u dde who in 1 6 5 9 first let a letter stan d f o r negative
as well as positive values
D escartes also estab l ished so m e theore m s on the theory of equa
tions Celebrate d is his rule of signs for determining the nu mber
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L e Tr ip a r ty ,
B u l l etti n o B on compag n i , V ol
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1 3,
1 8 80 ,
p
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7 40
.
D ESCARTES TO
NEWTON
1 79
of positiv e and negative roots He gives the rule after pointing out
the roots 2 3 4
5 a n d the corresponding bino m ial factors of the
1—
2
3—
equation x 4x 1 9x + 1 0 6 x 1 2 0 = 0
His exact words are as
follo ws
On c o nn oi t aussi de ceci combien il peut y avoir de vraies racines
e t co mbien de fausses en chaque eq u a t i o n : a sa v oir i l y en peut a v oir
et
autant de v raies que les S ignes
s y trouvent de fois etre
c ha n g es
e t autant de fausses qu il s
trou
e
de
fois
eux
ignes
v
d
S
y
ou d eux S ig n es
qui s e n t re suivent Co mm e en la derniere a cause
4
qui est u n change ment d u signe e n
q u apres + x il y a
2
—
e t apr es
1 9x
i l y a + r o 6 x e t apres + r o 6 x il y a —1 2 0
sont
qui
encore d eux autres ch ange ments ou c onn oi t qu il y a tro i s vraies
3
2
racines ; e t une fausse a cause que les deux S ignes
de 4x e t 1 9x
s e n t re—
sui v ent
This state me n t lacks co mpleteness For this reason he has been
frequently critici z ed J Wallis clai med that D escartes failed to
notice that the rule breaks down in case of i maginary roots but
D escartes does not say that the equation al w ays has but that it may
have so m any roots
D id D escartes recei v e any suggestion of his
rule fro m earlier writers ? He m ight have received a hint from H
Cardan whose re marks on this subj ect have been sum mariz ed by
1
G En e s t r6 m as follows : If m an equation of the secon d third or
fourth degree ( 1 )
th e last ter m is negative then one variation of S ig n
signifies one and only one positive root ( 2 )
the last term is positive
the n t wo variations indicate either several positive roots or none
Cardan does not consider equations ha v ing more than two variations
G W Leibniz was the first to erroneously attribute the rule of signs
to T Harriot D escartes was charged by J Wallis with availing
himself without acknowledgment of Harriot s theory of equations
particularly his m ode of generating equations ; but there see ms to be
no goo d ground for the charge
In m echanics D escartes can hardly be said to have advanced b e
yond Gal ileo The l atter had overthrown the ideas of Aristotle on
“
this subject and D escartes S i mply threw himself upon the enem y
“
that had already bee n put to the rout
His state ment of the first
and second laws of motion was an improvem ent in form but his third
law is false in substa n ce The m otions of bodies i n their direct i mpact
was i mperfectly understood by Galileo erroneously giv en by D escartes
an d first correctly stated by C Wren J Wall i s and C Huygens
One of the most devoted pupil s of D escartes was the learned
P ri n cess El i za beth daughter of Frederick V
She applied the new
analytical geo m etry to the so l utio n of the Apollon ian problem
His second royal follower was Qu een C hri s ti n a the daughter of Gus
tav us Adolphus She urged upon D escartes to com e to the Swe dish
court After m uch hesitatio n he accepted the invitation in 1 6 49
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1
B i bl i otheca
mathema ti ca ,
3 rd
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1 90 6
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2 93
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A HISTORY OF M ATHE M ATICS
1 80
He died at S tockhol m one year later His life had been one long war
fare against the prej udices of men
I t is most re markable that the m a thematics and philoso
phy of
D escartes should at first have been appreciated less by his country
m en than by foreigners The indiscreet temper of D escartes alienated
the great contemporary French m athem aticians Roberv al F erma t
P ascal
They continued in in v estigations of their own and on som e
points strongly opposed D escartes The univ ersities of France were
under strict ecclesiastical control and did nothing to introduce his
m athe matics and philosophy It was in the youthful universities of
Holland that the eff ect of Cartesian teachi ngs was most immediate
and strongest
The only prominent French man who i mme diately followed in the
footsteps of the great master was F l o ri mon d d e B eau n e ( 1 6 0 1 1 6 5
He was one of the first to point out that the properties of a curve
can be deduced fro m the properties of its tangent This m ode of
inquiry has been called the i nver se method of tangen ts He contributed
to the theory of equations by consid ering for the first tim e the upper
and lo w er limits of the roots of nu merical equations
In the Netherlands a large nu mber of d istinguished m athe maticians
were at once struck with ad miration for the Cartesian geo metry
Fore most a m ong these are va n S chooten John de Witt van H eu rae t
professor
S l u z e and H u dde Fr an c i s c u s van S c hoo t e n ( died
of m athe m atics at Leyden brought out a n edition of D escartes
geom etry together with the notes thereon by D e B eaune His chief
work is his Exerci tati on es M athemati cce 1 6 57 in which he applies the
analyt i cal geom etry to the solutio n of m any interesting and d ifficult
proble ms The noble hearted Joh ann d e Wi tt ( 1 6 2 5
grand
pensioner of Holland celebrated as a states man and for his tragical
end was an ardent geometrician He conceived a new and ingenious
way of generating conics which is essentially the same as that by
projective pencils of rays in m odern syn thetic geometry He treated
the subj ect not synthetically but with aid of the Cartesian analysis
R e n é F r an coi s d e S l u s e ( 1 6 2 2 —
1 6 8 5)
and Joh ann H u dd e ( 1 633
1 7 0 4)
m ade so m e i mprove m ents on D escartes and Ferm at s m ethods
of drawing tangents and on the theory of maxim a and minima With
H u dde we find the first use of three variables i n analytical geom etry
He is the author of an ingenious rule for fin ding equal roots We
3
illustrate it by the equation x
Taking an arith
metical progression 3 2
of which the highest term is equal to
the degree of the equation and m ultiplying each term of the equation
respecti v ely by the correspon ding term of the progression we get
3—
2—
2—
=
2x
8 x 0 or 3 x
2x
This last equation is by one
3x
d egree lower than the original one
Find the G O D of the two
equations This i s x —2 ; hence 2 is on e of the two equal roots Had
there been no common divisor then the original equation woul d not
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1
A HISTORY OF M ATHE M ATI CS
1 82
and obtained a geometric series that was i nfi ni te How
ever infinite series had been obtained before him by A l varu s Thomas
a nati v e of Lisbon in a work L i ber de tri pl i ci motu
and possibly
by others B ut S t Vincent was the first to apply geom etric series to
“
the Achill es and to look upon the paradox as a question in the
s ummation of an infinite series M oreover St Vincent was the first
to state the e xact tim e and place of overtaking the tortoise He
spoke of the li mit as an obstacle against further advance similar to
a rigid wall Apparently he was not troubled by the fact that in hi s
theory the variable does not r ea ch its l im it His exposition of the
“
Achi l les was favorably received by G W Leibniz and by writers
over a century afterw ard The fullest account and discussion of
Zeno s argu ments on m otion that was published before the nineteenth
century was given by the noted French skeptical philosopher P i er r e
B ayl e i n an article
Zenon d El ée in his Di cti on n a i r e hi stori qu e et
ad
i nfi ni tu m
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cr i ti qu e ,
The prince of philosophers in Holland and one of the greatest
scientists of the seventeenth century was C h ri s tian H u yg e n s ( 1 6 2 9
1 69
a native of The Hague E m inent as a physicist and astronomer
as well as m athe matician he was a worthy predecessor of Sir Isaac
Newton He studied at Leyden under Fr a ns Va n S chooten The
perusal of som e of his earliest theore ms led R D escartes to predict
his future greatness In 1 6 5 1 Huygens wrote a treatise in which he
pointed out the fallacies of Gregory S t Vincent on the subject of
quadratures He hi mself gave a re markably close and convenient
approxi m ation to the length of a circular arc In 1 66 0 and 1 663 he
went to P aris an d to London In 1 6 6 6 he was appointe d by Louis
XIV m e mber of the French Academ y of Sciences He was induced
to rem ain in P aris from that time until 1 68 1 when he returned to hi s
native city partly for consideration of his health and partly on ac
count oi the revocation of the Edict of Nantes
The m ajority of his profound discoveries were m ade with aid of the
ancient geo metry though at ti mes he used the geo m etry of R D es
cartes or of B Cavalieri and P Fermat Thus like his illustrious
friend Sir Isaac Newton he al w ays sho w ed partiality for the Greek
geom etry Ne w ton and Huygens were kind r ed min d s and had the
greatest ad miration for each other Newton always speaks of hi m
“
as the Sum mus H ug en i u s
To the two curves ( cubical parabola and cycloid )
previously recti
fi e d he adde d a third —
the cissoid A French physician Clau dius
P errault proposed the question to d eter mine the path in a fixed plane
of a heavy point attached to one en d of a taut string whose other end
m oves along a straight line in that plane Huygens and G W Leibniz
studied this proble m in 1 6 93 generaliz ed it and thus worked out the
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1
2
H
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Wi el ei tn e r
S ee
i n B i bli otheca ma themati ca , 3 F , B d 1 9 1 4 , 1 4 , p
F C aj or i i n A m M ath M on thl y, V o l 2 2 , 1 9 1 5 , p p 1 0 9 1 1 2
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152
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D ESCARTES
TO NEWTON
”
tractrix
Huygens solved
geometry of the
the problem of th e
catenary determined the surface of the parabolic and hyperbolic
conoid and disco v ered the properties of the logarith mic curve an d
the solids generated by it Huygens De horol og i a os ci l l atori o (P aris
1 6 73)
is a work that ranks second only to the P ri n ci pi a of Newton
and constitutes h istorically a necessary introduction to it The book
opens with a description of pendulum clocks of which Huygens IS the
inventor Then follows a treatm ent of accelerated m otion of bodies
fall ing free or sl iding on inclined planes or on given curves —cul
m i n a ti n g in the brilliant discovery that the cycloid is the tautochronous
curve To the theory of curves he added the important theory of
“
evolutes
After explaining that the tangent of the evolute is
normal to the involute he applied the theory to the cycloid and
showed by si mple reasoning that the evolute of this curve is an equal
cycloid Then co mes the co mplete general discussion of the centre
of oscillation This subject had been proposed for investigation by
M M ersenne and discussed by R D escartes and G P Roberval
In Huygens assumption that the common centre of gravity of a
group of bodies oscillating about a horiz ontal axis rises to its original
height but no higher is expressed for the first time one of th e most
beautif ul principles of dynamics afterwards called the principle of
the conserv ation of vi s vi va The thirteen theore m s at the close of
the work relate to the theory of centrifugal force in circular motion
2
This theory aided Ne w ton in d isco v ering the l aw of gravitation
Huygens wrote the first formal treatise on probability He pro
posed the wave—
theory of light and with great sk i ll applie d geom etry
to its develop m ent This theory was long neglecte d but was revived
and elaborated by Thomas Young and A J Fresnel a century later
Huygens an d his brother i mproved the telescope by devising a better
way of grin ding and pol i sh i ng lenses With m ore effi cient i n s tru
m ents he d eterm ined the nature of Saturn s appendage and sol v ed
other astronom ical questions Huygens Opu s cu l a pos thu ma appeared
1
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I n 1 70 3
.
The theory of combinations the prim itive notions of which go
back to ancient Greece received the attention of Wi l li am B u ckl ey
of Ki ng s College Cambridge (d ied 1 5
and especially of Blaise
P ascal who treats of it in his A r i thmeti cal Tr i a n gl e
B efore P ascal
thi s Triangle had been constructe d by N Tartaglia an d M Stifel
Ferm at applied co mb i nations to the stu dy of probability The earliest
m athe m atical work of Leibniz was his De a r te combi n a tor i a
The
subject was treated by John Wal l is in his A l gebra
—
was one of the m ost original ma th e ma t i
Joh n Wal li s ( 1 6 1 6 1 70 3 )
c i an s of his day
He was e ducated for the Church at Cambridge an d en
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)
G L o ri a , Eben e C u r vou ( F S c h u tt e I I 1 9 1 1 , p 1 8 8
2
E D uhri n g , K r i ti sche Geschi chte der A ll geme i n en P ri nc i pi en der M echa n i k
L e ip i g , 1 88 7 , p 1 3 5
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A HISTORY OF M ATHE M ATICS
1 84
Holy Orders B ut his genius was e mployed chiefly in the study of
mathe matics In 1 6 4 9 he w as appointed S av i l ia n professor of geo metry
at Oxford He was one of the original m e mbers of the Royal Society
which was founded in 1 6 6 3 He ranks as one of the world s greatest de
1
Walli s thoroughl y grasped the mathema t
C ipherers of c ryptic writing
ical m ethods both of B Cavalieri and R D escartes His C oni c S ecti ons
is the earliest work in which these curves are no longer considered as
sections of a cone but as curves of the second degree and are treated
analytically by the Cartesian m ethod of c o ord inates In this work
Wallis S peaks of D escartes in the highest term s b ut in his Al gebr a
( 1 6 8 5 Latin edition 1 6 9 he without good reason accuses D escartes
of plag i ariz ing fro m T Harriot I t is interesti ng to observe that in
his A l gebr a Wallis discusses the possibility of a fourth dimension
“
Whereas nature says Wallis doth not admit of m ore than three
dimensions
it may justly see m very improp er to talk of
(local )
a solid
drawn into a fourth fi fth s i xth or further dim ension
Nor can our fan si e im agine h ow there should be a fourth local dimen
2
The first to busy himself with the nu mber
sion beyond these three
of dimensions of space was P tole my Wall i s felt the need of a m ethod
of representing i maginaries graphically but he failed to discover a
3
general and consistent representation He published Nasir—
Eddin s
proof of the parallel postulate and abandoning the idea of equi
distance that had been e mployed without success by F C om ma n di n o
C S C l a v io P A Cataldi and G A B ore l li gave a proof of his own
based on the axiom that to e v ery figure there exists a S i milar figure
4
of arbitrary m agnitude
The existence of si milar triangles was as
su med 1 00 0 years before Wallis by Aga mis who was probably a
teacher of Sim plicius We h av e a l ready m entioned else w here Wal l is s
solution of the priz e questions on the cycloid which were proposed by
P ascal
The A r i thmeti ca i nfi n i tor u m published in 1 6 5 5 is his greatest work
By the application of analysis to the M etho d of In d iv isibles he greatly
increased the power of this instru ment for e ffecting quadratures He
created the arith metical conception of a li mit by considering the
successive values of a fraction for med in the study of certain ratios ;
these fractional values steadi ly approach a li miting v alue so that
the d iff erence becom es less than any assignable one and v anishes
when the process is carr i e d to infinity He a dv ance d beyond J Kepler
by making more extended use of the law of continuity and placing
t er e d
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D E S mi t h i n B u ll A m M a th S oc , Vol 2 4 , 1 9 1 7 , p 8 2
2
G En es t r Orn i n B i bli otheca ma thema ti ca , 3 S , V o l 1 2 , 1 9 1 1 1 2 p 8 8
’
3
S e e Wal l i s A l gebr a , 1 6 8 5 , p p 2 6 4 2 7 3 ; see al so En es t r Om i n B i bl i otheca mathe
269
mati ca , 3 S , V ol 7 , pp 2 6 3—
4
S ee al so F En g el u P S t ack e l , Theor i e der
R B on o l a , op ci t , p p 1 2 1 7
Thi s t rea t is e
P ar al l elli ni en von E u cl id bi s a uf Ga uss , L e i p ig , 1 8 95 , pp 2 1 3 6
n o f S ac c h e r i , a l so t h e e ss ays o f L amb e r an d Ta u r inus,
n
n
G
r
a
i
m
n
l
a
t
i
o
s
t
o
e
r
a
s
v
i
es
t
g
an d l e t t ers of G auss
1
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t
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A HISTORY OF M ATHE M ATICS
1 86
responding to
and the curves represented by the equations y
2 3
2 1
2 2
—
—
—
=
=
=
(1
y ( 1 x )y ( 1 x )y ( 1 x )etc are expressed in
functions of the circum scribed rectangles havi ng x and y for their
sides by the quantities forming the series
x,
.
,
,
,
,
,
3
—
x
1x 3
_
,
1
5
§ x3+ x 5
7
1x + § x 1x etc
x
,
x
,
When x = 1 these values becom e respectively
.
etc N ow
1 15
2 i
—
=
since the ordinate of the circle is y ( 1 x )the exponent of which is
1 or th e m ean value between 0 and 1 the question of this quadrat ure
reduce d itself to this If 0 1
etc oper ated upon by a certain law
1
4 8
—
give 1
give when operated upon by the sam e
3 15 5 what will
law ? He atte mpted to solve this by i n terpol ati on a _m ethod first
brought into prominence by him and arrived by a highly complicated
and difficult analysis at the following very remarkable expression :
,
1,
3
,
.
,
,
,
.
,
.
,
,
,
,
,
,
,
,
"
71
2
He did not succeed in m aking the interpolation itself because he
did not e mploy li teral or general exponents and could not conceiv e a
series with m ore than one term and less than two which it seeme d
to him the interpolated series m ust have The consideration of this
dimc u l ty led I Newton to the discovery of the B inom ial Theore m
This is the best place to S peak of that discovery Newton virtually
assum ed that the sam e conditions which underlie the general ex
pressions for the areas given above m ust also hold for the expression
to be interpol ated In the first place he observed that i n each ex
pression the first term is x that x i ncreases in odd powers that t e
3
3
3
h
3
signs alternate and
and that the second term s 1x 1x 1x 1x
are in arithm etical progression Hence the first two terms of the
,
,
,
.
.
.
.
.
,
,
,
,
,
m ust
interpolated series
be
,
,
He nex t considered that the de
x
3
no minators 1 3 5 7 etc are in arith m etical progression and that
the coefli c i en t s in the num erators in each expre ssion a re the digits
0
of some power of the number 1 1 ; nam ely for the first expression 1 1
1
2
or 1 ; for the seco nd 1 1 or 1 1 ; for the third 1 1 or 1 2 1 ; for the
3
fourth 1 1 or 1 3 3 1 ; etc He then discovered that having give n
the second digit ( call it m)the remaining digits can be found by con
—
—
—
m o m r m 2
t I n u al m u l ti pl I c a tI on of the term s of the ser i es
,
,
,
,
.
,
,
‘
’
,
,
,
,
,
,
,
,
,
.
,
,
,
,
etc
.
Thus if
,
m= 4 ,
then
4
.
gives
6; 6
773
.
—2
gives
4;
D ESCARTES
gi v es
1
second term is
To
NEWTON
18
Applying this rule to the required series since
.
,
we hav e
3
m
7
t he
1 and then get for the succeeding
,
coeffi cients in the num erators respectiv ely
-
1
“
,
1 111
1x
hence the required area for the circular segm ent is x
etc
1
3
1x
;
7
5
x
1
5
3
.
7
etc Thus he found the interpolated expression to be an infinite series
instead of one having m ore than one term and less than two as Wallis
believ e d it m ust be This interpolation suggeste d to Newton a mode
2 i
—
of expanding ( 1 x )or m ore generally ( 1
into a series He
observed that he had only to o mit from the expression just found the
denominators 1 3 5 7 etc and to lower each power of x by unity
and he had the desired expression In a letter to H Oldenburg
Newton states the theorem as follows :The ex traction
(June 1 3
of roots is much shortened by the theorem
,
.
,
.
,
,
,
,
,
,
.
,
.
,
,
.
.
,
n
B Q+
—
2 17
111
,
where A means the first term P B the second term C th e third
term etc He verified it by actual m ultiplication but gave no regular
proof of it He gave it for any exponent whatever but m a d e no dis
tinction between th e case when the exponent is positive and integral
and the others
I t S hould here be mentioned that very rude beginnings of the bi
nom ial theorem are found very early The Hindus and Arabs use d
2
3
the expansions of ( a + b)and (a + b)for extracting roots ; Vieta knew
the expansion of
but these were the results of S im ple multi
plication without the discovery of any law The binomial coefficients
for positive whole exponents were known to som e Arabic and Euro
pean m athem aticians B P ascal derive d the coeffi cients from the
“
m ethod of what is called the
arithm etical triangle
Lucas de
Burgo M Stifel S S te v inus H B riggs and others all possessed
s omething from which one would think the binomial theorem could
“
have been gotten with a little attention if we did not know that
such S imple relations were diffi cult to discover
Though Wallis had ob ta ined an entirely n ew expression for 7r he
was not satisfied with it ; for instead of a finite number of term s yield
ing an absolute value it contained an infinite number approaching
nearer and nearer to that value He therefore induced his friend Lor d
B r ou n c k e r the first presi d ent of the Royal Society to investigate
this subject Of course Lord B rou n c k e r did not find what they were
after but he ob tained the following beaut iful equality
,
,
n
,
,
,
.
,
.
,
.
.
.
.
”
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,
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,
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,
,
,
.
”
,
.
,
,
,
,
.
,
.
,
,
A H ISTO RY O F M ATH EMATICS
1 88
49
etc
Continued fractions both ascending and descending appear to have
been known alrea dy to the Greeks a nd Hindus though not in our
present notation B rou n c k e r s e xpression gave birth to the theory of
continued fractions
Wallis m ethod of quadratures was diligently studied by his dis
c ipl es
Lord B rou n c k e r obtained the first infinite series for the area
of the equilateral hyperbola xy= 1 between one of its asymptotes and
2+
.
,
,
,
’
.
.
’
.
the ordi nates for
and
x= 1
x= 2
(London
L ogari thmotechni a
;
V IZ
1 668
,
.
1
the area
)of N i
1
1
M er cator
col au s
is often said
3
to conta i n the ser i es log
2
In
3
con
real I ty I t
tains the nu m erical values of the first few term s of that series tak
ing a = 1 also a = 2 1 He adhered to the m ode of exposition which
favored the concrete special case to the general form ula Wallis was
the first to state M ercator s logarithm ic series in general symbols
M ercator d e d uce d his results from the grand property of the hyper
bola deduced by Gregory S t Vincent in B ook VII of his Opu s geo
metri cu m Antwerp 1 6 4 7 : If parallels to one asym ptote are drawn
between the hyperbola and the other asymptote so that the successive
areas of the m ixtilinear qua drilaterals thus form ed are equal then
the lengths of the parallels form a geom etric progression Apparently
the first writer to state this theorem in the language of logarithm s
was the B elgian Jesuit A lf ons A n ton de S aras a who defended Gregory
S t Vincent against attacks m ade by M ersenne M ercator showed
how the construction of logarithm ic tables could be reduced to the
quadrature of hyperbolic spaces Following up som e suggestions of
Wallis Wi l l i am N ei l succeeded in rectifying the cubical parabola and
C Wr en in rectifying any cycloidal arc
Gregory St Vincent i n
P art X of h i s Opu s describes the construction of certain quartic cu rves
often called virtual parabolas of S t Vincent one of which has a shap e
2
2 —2 = 4
—
m uch l ike a lem niscate and in Cartesian c o ordinates is d (y
x )
y
Curves of this type are mentioned in the correspondence of C Huy
gens with R d e Sluse and with G W Leibniz
A prom inent English m athem atician and contemporary of Wallis
was I s aac B arr ow ( 1 6 30
He was professor of m athematics
in Lon don and then in Cambri dge but in 1 6 6 9 he resigned his chair
,
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Wallis in
1 68
and Edm und Halley in
5,
B arrow gav e also the integral
1 64 7
M ATHE M ATICS
A HISTORY OF
1 90
tan 0 d 9= log sec 0; B Cav al ieri in
.
”
established the i ntegral
x
dx
by E Torricelli Gregory S t V i ncent
2
B P ascal J
.
.
,
.
.
Ja m es Gregory and
,
Sim il ar results were obtained
.
P
.
Fermat G
,
.
P
.
Roberval and
,
N ewton to Eu l er
I t has been seen that in France prodigious scientific progress was
m ade during the beginning and m iddle of the seventeenth century
The toleration which m arked the reign of Henry IV and Louis XIII
was acco mpanied by intense intell ectual activity Extraordinary con
fiden c e cam e to be placed in the power of the hu m an m ind The bold
intellectual conquests of R D escartes P Fer mat an d B P ascal en
riched m ath em atics with i mperishable treasures D uring the early
part of the reign of Louis XIV we behold the sunset splendor of this
glorious period Then followed a night of mental effe m inacy This
lack of great scien t ific thinkers during the reign of Louis XIV may be
due to the S imple fact that no great m inds were born ; but according
to Buckle it was due to the paternalism to the spirit of dependenc e
a n d subordination
and to the lack of toleration which marked the
policy of Louis XIV
In the absence of great French thinkers Louis XIV surrounded
him self by em inent foreigners O ROmer from Den m ark C Huygens
from Holland D om in i c Cassini from Italy were the m athem atician s
and astronomers a dorning his court They were in possession of a
brilliant reputation before going to P aris Simply because they per
formed scientific work i n P aris that work belongs no more to France
than the disco v eries of R D escartes belong to Holland or those of
J Lagrange to Ger m any or those of L Euler and J V P oncelet to
Russia We m ust look to other countries than F ran c e f or the great
scientific m en of the latter part of the seventeenth century
About the tim e when Louis XIV assu med the direction of the
French govern m en t Charles II beca m e king of England At this
tim e England was extending her comm erce an d navigation and a d
v an c i n g considerably in m ater i al prosper i ty
A strong intellectual
m o v e m ent took place which was unwittingly supporte d by the king
The age of poetry was soon followed by an age of science and philos
o phy
In two successive centuries England produced Shakespeare
and I Newton !
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S ee F C aj or i i n B i bli otheca mathema ti ca , 3 S , Vol 1 4 , 1 9 1 5 , pp 3 1 2 - 3 1 9
2
H G Z e u t h e n , Ges chi chte der M ath ( d e u t sc h v R M eye r , L eip ig , 1 90 3 ,
pp 2 5 6 ff
1
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)
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z
.
NEWTON
To
EULER
1 91
Germany still continued in a state of national degra dation The
Thirty Years War had dism embered the e mpire and brutalized the
people Yet this darkest period of Germany s history produced G W
Leibniz one of the greatest geniuses of m odern times
There are certai n focal points in history toward which the lines of
past progress converge and fro m which rad iate the adv ances of the
future Such was the age of Newton and Leibniz in the history of
D uring fifty years preceding this era several of the
mathematics
brightest and acutest m athem aticians bent the force of their genius
in a direction which finally led to the discovery of the infinites imal
calculus by Newton and Leibniz B Cavalieri G P Roberval P
F erma t R Descartes J Wallis and others had each contributed to
So great was the adv ance m ade and so near was
the new geo m etry
their approach toward the invention of the infinitesimal analysis that
both ] Lagrange and P S Laplace pronounced their countryman
P F erm a t to be the first inventor of it
The differential calculus
therefore was not so m uch an individual discov ery as the grand result
of a succession of discover i es by different m in d s Indeed no great
discovery eve r flashed upon the mind at once and though those of
Newton will influence mankind to the end of the world yet it must be
“
adm itted tha t P ope s lines are only a poetic fancy :
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N ature an d Nature s l aw s lay h i d i n n ig ht ;
’
‘
Go d sai d , Let Newton be , an d all was l i g ht
’
I s aac N e w ton ( 1 6 4 2 —
was born at
1 727
shire , the sam e year in which Galileo died
s mal l and weak that his life was despaired
.
)
Woolsthorpe in L incoln
At his birth he was so
of His m other sent him
at an early age to a village school and in his twelf th year to the public
school at Grantham A t first he see ms to have been very inattentiv e
to his studies and very low in the school ; but when one day the little
Isaac received a severe kick upon his sto m ach fro m a b oy who was
above him he labored har d till he ran k e d higher in school than his
antagonist From that tim e he continued to rise until he was the
1
head boy At Grantham Isaac showed a decided taste for m echan
ical inventions He constructed a water clock a wind mill a carriage
m oved by the person who sat in it and other toys When he had a t
t ain e d his fifteenth year his m other took h i m ho m e to assist her in
the m anagement of the farm but h is great dislike for farmw ork and
his irresistible passion for study in d uced her to send h im back to
Grantham where he re mained till his e ighteenth year when he en
t ered Trinity College Ca mbridge
Ca mbridge was the real
b i rthplace bf Newton s genius So m e id ea of his strong intuiti v e
powers may be drawn from the fact that he regarded the theorems of
ancient geom etry as self ev ident truths and that without any pre
li minary study he m ade h i m se l f m aster of D escartes Geometry He
,
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l
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-
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’
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-
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’
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,
1
D B r ew s te r , The M emo i rs
.
f
o
N ew ton , Edin b u rg h , V ol I ,
.
1 8 55,
p 8
.
.
92
.
HISTORY OF M ATHE M ATICS
A
afterwards regarded thi s neglect of ele m entary geometry a mistake
in his m athem atical studies and he expressed to Dr H P e mberton
“
his regret that he had applied him self to the works of D escartes and
other algebraic writers before he had consid ered the El emen ts of Euclid
with that attention which S O e xcellent a writer deserves
B esides R
D escartes Geometry he studied W O ug h t red s Cl avi s J Kepler s
Opti cs the works of F Vieta v a n S c ho o t en s M i scel l an i es I B arrow s
L ectu res and the works of J Wallis
He was particularly delighted
with Wallis A ri thmeti c of I nfi n i tes a treatise fraught with rich and
varied suggestions Newton had the good fortune of ha v ing for a
teacher and fast friend the celebrate d D r Barrow who had been
elected professor of Greek in 1 660 and was made L u casian professor
of mathe m atics in 1 6 6 3 The mathe m atics of Barrow and of Wallis
were the starting points fro m which Newton with a higher power
than his masters moved onward into wider fields Wallis had ef
f ec te d the quadrature of curves whose ordinates are expressed by any
2
—
integral and positive power of ( 1 x ) We have seen how Wallis
attemp ted but failed to interpolate between the areas thus calculat ed
the areas of other curves such as that of the circle ; how Newton at
tacked the proble m e ff ected the interpolation and discovered the
B inom ial Theorem which aff orded a much easier and direct access to
the quadrature of curves than did the m ethod of interpolation ; for
even though the bino mial expression for the ordi n a te b e raised to a
fractional or negativ e power the binomial could at once be expanded
into a series and the quadrature of each separate term of that series
could be e ffected by the m ethod of Wallis Newto n introduced the
system of literal indices
Newton s study of quadratures soon led hi m to another and most
profound invention He himself says that in 1 6 6 5 and 1 6 6 6 he c on
c eiv ed the m ethod of fl u xi o n s and applied the m to t he quadrature of
curves Newton did not co mmunicate the in v ention to any of his
friends till 1 6 6 9 when he placed i n the hands of Barro w a trac t e n
titled De A n al ys i per E q u a ti on es N u mer o Termi n oru m I nfi n i tas which
was sent by Barrow to John Collins who greatly ad mired it In
this treatise the principle of fl uxion s though distinctly pointed out
is on l y partially developed a ri d explained Supposing the abscissa to
increase uniformly in proportion to the ti me he looked upon the area
of a curve as a nascent quantity increasing by continued fl u xi on in
the proportion of the length of the ordinate The expression which
was obtained for the fl uxi on he expanded into a finite or infinite series
of m ono mial term s to which Wallis rule was applicable Barrow
urged Newton to publish this treatise ; but the m odesty of the author
of which the excess if not culpable was certainly in the present i n
1
Had this tract
stance very unfortunate prevented his compliance
J h P l yf ai
P rog ess Of t he M athemati cal and P hysi ca l S cien ces i n Eu
.
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o
n
a
r,
”
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,
r
pa di a B r i ta n ni ca , 7 th Edi tion
.
,
A HISTORY OF M ATHE M ATICS
1 94
which ( time )
another space x b y I ncreasing with an uniform celerity
x measures and exhibits as described : then 2 xx will represent the
celerity by which the space y at the same mo ment of tim e proceeds
to be described ; and con traryw i se
“
B ut whereas we need not consider the time here any farther than
it is expounded and m easured by an equable local motion ; and b e
sides whereas only quantities of the sa m e kind can be compared to
gether and also their velocities of Increase and decrease therefore in
what follo w s I shall have no regard to ti me formally co n sidered but
I Shall suppose so me one of the quantities proposed being of the same
kin d to be increased by an equable fl u xi on to which the rest may be
referred as it were to time ; and therefore by way of analogy it
may not i mproperly receive the na me of tim e
In this statem ent of
Newto n there is contained his answer to the obj ection w hich has been
raised against his m ethod that it introduces into analysis the foreign
idea of m otion A quantity thus increasing by uniform fluxi on is
what we now call an i ndepen dent variable
Newton conti n ues : N ow those quantities which I consider as
gradually and i n definitely increasing I S hall hereafter call fl u en ts or
fl owi n g q u an ti ti es and S hall represent them by the final letters of the
alphabet v x y and z;
and the velocities by which every fluent
is increase d by its generating m otion (which I may call fl u xi ons or
simply velocities or celerities )I shall represent by the same letters
pointed thus v x y z That is for the celerity of the quantity v
I S hall put v and so for the celerities of the other quantities x y and
z I shall put x y a n d z respectively
I t m ust here be observ ed that
Newton does not take the fl u xi on s the mselves infinitely sm all The
“
m om ents of fl u x i on s
a term introduced further on are infinitely
s m all quantities These mom ents as defined and used in the
M ethod of Fl u xi on s are substantially the d i fferentials of Leibniz D e
M organ poi n ts out that no s m al l am ount of confusion has arisen fro m
the use of the word fl u xi on an d the notatio n x by all the English writers
previous to 1 70 4 excepting Newton and George Cheyne i n the sense
1
of an infinitely sm all increm ent
S trange to say even in the C om
mer ci u m epi s tol i cu m the words momen t and fl u xi on appear to be used
as synon ymous
Af ter show ing by examples how to solve the first problem Newto n
proceeds to the demonstration of his solution
The mom ents of flowi n g quantities ( that is their indefinitely
small parts by the accession of which in infinitely sm all portions of
time they are continually increased )
are as the velocities of their
flowing or increas i ng
“
Wherefore if the m oment of any one (a s x)
be represented by the
product of its celerity x into a n i nfinitely sm all quantity 0 (i e by
,
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D e M o rg a n ,
O n t he E a rl y H i s t o ry
M agazi n e, N o v emb e r , 1 8 5 2
1
A
.
.
of
I n fin i t e sirn al s ,
.
i n P hil osophi cal
NEWTON
)the moments of the others
EULER
To
1 95
will be represented by v0 yo
20 because v0 x0 yo and 230 are to each other as v x y and
Now S ince the m o ments as x0 and yo are the indefinitely little
accessions of the flo w ing quantities x a nd y by which those quantities
are increased through the several indefinitely little intervals of tim e
it follows that those quantities x and y after any in definitely small
interval of tim e becom e x+ xo and y+y0 and therefore the equation
which at all times indifferently expresses the relation of the flowing
quantities will as well express the relation between x + x0 and y+ y0
as between x and y; so that x+ x0 and y+ y0 m ay be substituted in
the sam e equation for those quantities instead of x and y Thus let
3— 2
3=
—
any equation x ax + axy y 0 be giv en and substitute x+ x0 for
x and y+y
0 for y and there will arise
x0
,
v,
,
y,
z,
,
,
'
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
.
,
,
,
,
x
3
3 3
o
+ 3x x0 + 3xx0 xo + x
2
x axo xo
+ axy+ ayxo + ax0 y0
dx
2
2 ax 0
+ a xyo
—y3 —sy ro —syyoyo —y 0
“
Now by supposition x ax + axy y 0 which therefore being
expunged and the rem aining term s being divid ed by 0 there will
rem ai n
2
3
,
,
3=
,
,
,
2
3x x
x+ ayx+ axy 3y y+3xxx0
3
3
—
+ x0 0 y0 0 = 0
2
2 ax
xx0 + axy0
a
3yyyo
.
B ut whereas z ero is supposed to be infinitely little that it may r epr e
sent the m o ments of quantities t he term s that are m ultiplied by it
will be nothing in respect of the rest (ter mi n i i n com du cti pr o n i hi l o
; therefore I rej ect the m and there
possu n t haberi cu m al i i s cal l al i )
remains
,
,
,
3x x
2
x+ ayx+ axy 3y y
2 ax
2
0,
as above in E xampl e I‘ Newton here uses i n fi n i t e simal s
M uch greater t han g n the first problem were the difficulties en
countered i n the sdII‘It i on of the second proble m involving as it does
in verse operations which ha v e been taxing the skill of the best ana
l ys t s since his tim e Newton gives first a S pecial solution to the second
roble
i
which
he
resorts
to
a
rule
for
which
he
has
given
no
proof
m
n
p
I n the general solution of his second proble m Newton assu m ed
homogeneity with respect to the fl u xi on s and then considered three
when the equation contains two flu xi on s of quantities and
ca ses : ( 1 )
but one of the fl uen t s ; ( 2 )
when the equation involves both the fl u en t s
when the equation contains the flu
as well as both the fl u xio n s ; (3 )
ents and the fluxion s of three or more quantities The first case is the
.
e
,
,
,
.
.
,
.
easiest since it requires s im ply the integration of
dy
d}
f (x)to which
,
A HISTORY OF M ATHE M ATICS
1 96
k
special solution is applicable The second case demanded
nothing less than the general solution of a di ff erential equation of the
first order Those who kn ow what e ff orts were afte rwards needed
for the complete exploration of this field in analysis wil l not depre
ciate Newton 5 work even though he resorted to solutions In form of
in fin ite series Newton 5 thi rd case co mes now under the solution of
partial differential equations He took the equation 2 x z+xy= 0
and succeeded In finding a particular integral of i t
The rest of the treatise is devoted to the determination Of m axim a
and m ini ma the radius of curvature of curves and o ther geom etrical
applications of his fl uxion ary calcul us All this was done previous
to the year 1 6 7 2
It must be observed that in the M ethod of Fl u xi on s (as well as in
his De A n al ysi and all earli er papers)
the m ethod e mployed by New
ton is strictly i nfi n i tesirn al and i n substance like that of Leibniz
Thus the original conception of the calculus in England as well as
on the Continent was based on infi n i tesimal s The fundam ental
principles of the fl uxion ary calculus w ere first given to the world in
the P ri n ci pi a but i t s peculiar notation did not appear until publi shed
in the second vol um e of Wall is A l gebr a in 1 6 93 The expositi on
given in the A l gebr a was a contribution of Newton ; it rests on i n
fi n i tesimal s
In the first edition of the P ri nci pi a ( 1 6 8 7 )
the descrip
tion of fluxi on s is l ikewise founded on i nfi ni t esimal s but in the second
the foun d ation is som ewhat altered In B ook II Le mm a II
( 1 7 1 3)
of the first e dition we read : Cave tam en i n tel l exeri s par ti cu l as
hi s
.
.
,
’
’
.
.
.
,
,
.
.
.
,
,
,
.
,
,
’
.
.
,
.
fi n i ta s
.
M omen ta q u am pri mu m fi n i tce
,
,
magn i tu di n i s , desi n u n t
a l i qu aten u s
perpetu o eoru m
sunt principia jamjam nas
su n t
momen ta
Fi n i ri en i m repu gn at
i n cr emen to vel decremen to
I n t elli g en da
centia fi n i taru m mag ni tud i n um
In the
es se
.
.
second edition the two
sentences whi ch we print in italics are replaced by the foll owing :
“
P artic ul ar: fi ni tm non sunt mom enta sed q uantitates ip sas ex mo
mentis g en i tw
Through th e diffi culty of the phrases in bo th ex
tracts this much distinctly appears that in the first m oments are
infin i tely sm all quantities What else they are in the second is not
1
C lear
In the Qu adr atu re of Cu rves of 1 70 4 the i nfinitely smal l
quantity is completely abandoned I t has been shown that in the
M ethod of Fl u xi on s Newton rej ected term s invol v ing the quantity 0
because they are i nfi nitely small compared with other terms This
reasoning is unsatisfactory ; for as long as 0 is a quantity though
ever so small this rej ection cannot be m ade w ithout aff ecting the
result Newton seems to have felt this for in the Qu adr atu e of C u rves
“
he remarked that in m athematics the m inutest errors are not to be
neglected (errores quam m inimi in rebus ma th emat i cis non sunt
contem nendi )
The early distinction between the syste m of Newton and Leibniz
.
”
.
,
,
,
.
.
,
.
,
.
,
,
.
r
,
”
.
1
A De
.
M org an l oc
,
.
ci t
.
,
1 85 2 .
O F M ATHE M ATICS
HISTORY
A
1 98
apart by a sm al l angle fro m the tangent CH B ut when CK c o
incides with CH and the lines CE E c cC reach their ultimate
ratios then the points C and c accurately coincide and are one
and the sam e
Newton then adds that in mathematics the
minutest errors are not to be neglected
Thi s is plainly a re
j ec ti on of the postulates of Leibniz The doctrine of infinitely
small quantities is here renounced in a manner which would lead
one to suppose that Newton had never held it himself Thus it
appears that Newton s doctrine was diff erent in difl er en t periods
Though in the above reasoning the Charyb dis of i nfi n i tesimal s is
safely avoided the dangers of a Scyll a stare us in the face We are
required to believe that a point m ay be consi dered a triangle or that
a triangle can be inscribed in a point ; nay that three dissim ilar tri
angles becom e S i milar and equal when they have reached their ulti
m ate for m In one and the sa m e point
In the introduction to the Qu adr atu re of Cu rves the fi uxi on of x
is deter mined as follows
In the sam e tim e that x by flowing becom es x+ 0 the power
"
x beco m es
i e by the m ethod of infinite series
2 —
n
n
2
—
—
x +n o x
+
0 x
+ etc
.
,
,
,
,
”
“
.
.
.
.
’
.
,
,
.
,
,
,
”
.
.
,
.
”
.
n
n
,
,
1
2
.
2
,
and the increments
0
and
no
1
n
n
+
2
are to one another as
1
"
x
‘
2-
to
”+
ux
‘
1
n
2
n
2
Let now the incre ments vanish and their last proportion will b e
1 to n x
: hence the fl u x i on of the quantity x is to the fl u xi on of the
—
"
quantity x as 1 : n ae
“
The fl u xion of lines straight or curved in all cases whatev er as
also the fl u xio n s of su pe rfi c i es angles and other quantiti es can be
obtaine d in the sa m e m anner by the m ethod of prim e and ultimate
ratios B ut to establ i sh in this way the analysis of infinite quantities
an d to i n v est i gate pri m e and ultim ate ratios of finite quant i t i es nas ~
cent or ev anescent is i n har m ony with the geo m etry of the ancients ;
a n d I ha v e en d eavored t o S how that in the m etho d of fl u x i on s it is
not necessary to intro d uce into geo m etry infinitely sm al l quantities
This mo d e of d ifferent i ating d oes not rem ove all the difficulties con
n e c t e d with the subj ect
When 0 beco m es nothing then we get the
n
‘
,
”
I
I
,
,
,
,
,
,
.
,
,
”
,
,
,
.
,
.
r a tI O
0
n
nx
‘
I
,
0
t
ch
nee ds further
e l u c I da t l o n
.
In dee d the
,
m ethod
of Newton as delivered by hi m self is encu mbered with diffi culties
,
,
NEWTON
To
EULER
9
and obj ections Later we S hall state B ishop B erkeley s obj ection to
this reasoning Even among the ablest ad mirers of Newton there
have been obstinate disputes respecting his explanation of hi s m ethod
“
of prime and ultim ate ratios
The so called me thod of limits is frequently attributed to New
ton but the pure meth od of limits was never adopted by him as hi s
method of constructing the calculus All he did was to establish in
hi s P ri n ci pi a certain principles which are applicable to that me thod
but which he used for a different purpose The first lemma of the
first book has been m ade the foundation of the m ethod of limits
“
Q uantities and the ratios of quantities which in any fini te tim e
converge continually to equality and before the end of that tim e ap
proach nearer the one to the other than by any given diff erence b e
come u l tirn atel y equal
In this as well as in the lemmas follow ing thi s there are obscurities
and diffi culties Newton appears to teach that a variable quantity
and its limit will ultimately coincide and be equal
The full title of Newton s P r i n ci pi a is P hi l os ophi w N atu ral i s P r i n
ci pi a M a themati ca
I t was printed in 1 6 8 7 under the direction and
at the expense of Edm und Halley A second edition was brought
out in 1 7 1 3 with m any alterations and improvem ents and aecom
It was sold out in a few
pan i ed by a preface fro m Roger Cotes
m onths but a pirated edition published in Am sterda m supplied the
d e mand The third and last edition which appeared in England during
Newton s lifetim e was published in 1 7 2 6 by Henry P e mberton The
P r i n ci pi a consists of three books of which the first two constituting
the great bulk of the work treat of the mathe matical principles of
natural philosophy namely the laws and conditions of motions and
forces In the third book is draw n up the constitution of the universe
as deduced from the foregoin g principles The great principle under
lying this m em orable work is that of universal gravitation The first
book was completed on April 2 8 1 686 After the remarkably short
period of three months the second book was finished The third book
is the result of the next nine or ten m onths labors I t is only a sketc h
of a m uch m ore extended elaboration of the subj ect which he had
p lanned but which was never brought to com pletion
The law of gravitation is enunciated in the first book Its discovery
env elops the name of Newton in a halo of perpetual glory The cur
rent version of the discovery is as follows : it was conj ectured by
Robert Hooke ( 1 6 3 5
C Huygens E Halley C Wren I N ew
ton and others that if J Kepler s third l aw was true ( its absolute
accuracy was doubted at that tim e)then th e attraction betw een the
earth and other m embers of the solar system varied i n vers ely as the
square of the d istance B ut the proof of the truth or falsity of the
guess was wanting In 1 6 6 6 Newton reasoned in substance that if
r
i
represent
the
acceleration
of
gra
ty
on
the
surface
of
the
earth
v
g
’
.
”
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,
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-
,
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”
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’
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,
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,
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,
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’
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,
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'
,
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,
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,
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,
’
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,
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‘
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,
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,
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’
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,
,
,
M ATHE M ATICS
A HISTORY OF
200
be the earth s radius R the distance of the m oon fro m the earth T
the tim e of lunar re v olution and a a degree at the equator then if
the law is true
’
,
,
,
,
,
,
R
2_
T
'
71
4
_
_
2
T
or
2’
1 80 a
.
The data at Newton s command gave R = 6 o 4r T=
seconds
but a only 6 0 instead of 6 9 1 English miles This w rong valu e of a
rendered the calculated value of g smaller than its true value as
known from actual m easure m ent It looked as though the l aw of
inverse squares were not the true l aw and Newton laid the calculatio n
aside In 1 6 84 he casually ascertai ned at a m eeting of the Royal
Society that Jean P icard had m easured an arc of the m eridian and
obtained a m ore accurate value for the earth s radius Taking the
corrected value for a he found a figure for g which correspon ded to
the known value Thus the law of inverse squares was verified In a
scholiu m in the P ri n ci pi a Newton acknowle dged his indebted n ess to
Huygens for the laws on centrifugal force e mployed in his calculation
The perusal by the astrono m er Adam s of a great mass of u n pub
l i shed letters and m anuscripts of Newton for m ing the P orts m outh
collection (which re mained private property until 1 8 7 2 when its
owner placed it i n the hands of the University of Cam bridge )
see m s to
indicate that the d iffi culties encountered by Ne w to n i n the above
calculation were of a diff erent nature According to Adam s Newto n s
nu m erical veri fication was fairly complete in 1 6 6 6 but Newton had
not been able to determ ine what the attraction of a spherical shell
upon an ex ternal point would be His letters to E Halley S ho w
that he did not suppose the earth to attract as though all its m ass
were concentrated into a point at the centre He could not have
asserted therefore that the assu m ed law of grav ity was verified by
the figures though for long d istances he m ight have claim ed that it
yielded close approxim ations When Halley vis i ted Ne w ton in 1 6 84
he requested Newton to deter mine what the orbit of a planet would
be if the law of attraction were that of inverse squares Newton had
solved a si m ilar problem for R Hooke i n 1 6 79 and replied at once
that it was an ellipse Af ter Halley s visit Newton with P icard s
n ew value for the earth s ra di us reviewed his early ca l culation a n d
was able to S how that if the distances between the bodies i n the solar
syste m were so great that the bo dies m ight be considered as points
then their m otions were in accordance with the assu m ed law of gravi
t a t io n
In 1 6 8 5 he co mplete d his disco v ery by showing that a sphere
whose density at any point depends only on the d istance from the
centre attracts an external point as though its whole m ass were con
c en t ra t e d at the centre
Newton s unpublished m anuscripts in the P orts m outh collect io n
show that he had worked out by m eans of fl uxi on s and fl u en t s hi s
’
.
,
,
,
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,
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,
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,
’
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,
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,
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,
’
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,
,
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.
.
,
,
,
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,
.
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,
’
’
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,
,
’
,
,
,
.
.
’
,
,
A HISTORY OF M ATHE M ATICS
20 2
by Albert Girard ta ke the explicit form as do also the general formul a
d erived later by E Waring Newto n uses his formul ae for fix ing an
upper lim it of real roots ; the su m of any even power of all the roots
m ust exceed the same even power of any one of the roots He estab
li shed also another lim it : A nu mber is an upper li m it if when sub
stituted for x it gives to f (x)
and to all its derivatives the sa me S ign
In 1 74 8 Col in M aclauri n proved that an upper limit is obtained by
adding unity to the absolute value of the large st negative c oefli ci en t
of the equation Newton showed that in equations w ith real c o
effi cients im aginary roots al w ays occur in pairs His inventive genius
is grandly displaye d in his rule for determ ining the inferior li mit of the
nu mber of im aginary roots and the superior limits for the number
of positiv e and negative roots Though less expeditious than D es
cartes Newton s rule always gives as close and generally closer
limits to the number of positive and negative roots Newton did
not prove his rul e
Som e light was thrown upon it by George Campbell and Col in
M aclaurin in the P hi l os ophi c al Tr an s acti on s of the years 1 7 2 8 and
B ut no co mplete de m onstration was found for a century and a
1 72 9
half until at last Sylvester established a re markable general theorem
which includes Newton s rule as a special case Not without interest
is Newton s suggestion that the conchoid be adm itted as a curve to
be used in geom etric constructions along with the straight line and
circle since the conchoid can be used for the duplication of a cube and
trisectio n of an angle—to one or the other of which every problem
in v olving curves of the third or fourth degree can be reduced
The treatise on M ethod of Fl u xi on s contains Newton s method of
approxim ating to the roots of nu m erical equations Substantiall y
the sam e explanation is given in his D e an al ysi per cequ ati on es nu mero
ter mi n oru m i nfi n i tas
He explains it by working one exa mple nam ely
1
3
the now fam ous cubic y 2y
The earl i est printed account
appeared in Wallis A l gebr a 1 6 8 5 chapter 94 Ne w ton assu mes that
an approxim ate value is already known which d iffers from the true
value by less than one tenth of that value He takes y= 2 and sub
3
2
=
s t i tu t es y
2 + p in the equation which beco m es p + 6 p + 1 0 p
Neglecting the higher powers of p he gets 1 op
Taking
he gets
From
he gets q = o o 54 + r and by the sa me process r = 0 0 0 0 4 8 53
F i nally y = 2 + 1 0 0 54
Newton arranges
5 147
his work in a paradigm He seem s quite aware that his m ethod m ay
fail If there is d oubt he says whether p= 1 is sufficiently close to
2
the truth find p fro m 6p + r o p 1 = o He does not show that even
th is latter m ethod will always ans w er By the sam e m ode of pro
,
.
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,
,
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‘
‘
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,
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’
’
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’
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’
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’
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’
,
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-
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,
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,
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,
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”
Fo r q u ot a t io n s f ro m N ew t o n , see F C aj o r i ,
H i s t o ri ca l N o t e o n t he N e w t on
R ap h so n M e t hod o f App roxi ma t io n , A mer M a th M on thl y, V o l 1 8 , 1 9 1 1 , pp 2 9
1
.
.
.
.
.
NEWTON TO EULER
203
Newton finds by a rapidly converging series the value of y
3
3
3=
—
in terms of a and x i n the equation y + axy+ aay x 2 a o
In 1 6 90 Joseph R aphs on ( 1 6 4 8
a fello w of the Royal Society
of London p ublished a tract A n al ysi s wqu ati onu m u n i versal i s His
m e thod closely resembles that of Newton
The only diff erence is
this that N ewton deriv es each successive step p q r of approach to
the root fro m a n ew equation while R aph son finds it each time by
substitution in the origi n al equation I n Newton s cubic R aphson
2
3
would not find the second correction by the use of x + 6 x + 1 0 x 1 = 0
but would substitute 2 1 + q in the original equation finding q
He would then substitute 2 o g46 + r in the original equation
0 0 54
fin ding r
R aph s on does not m ention
53 and so on
Newton ; he evidently considered the difference suffi cient for his
method to be classed independently To be e mphasiz ed is the fact
that the process which in modern texts goes by the nam e of New
ton s method of approxim ation is really not Newto n s m ethod but
c e du re ,
,
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,
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’
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,
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,
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,
”
’
.
’
,
,
Raph so n
’
m odification
s
of it The f or m now so fa miliar
.
,
f (a )
was
(a )
a
not used by Newton but was used by R aphso n To be sure Raphs on
out i n full as poly
does not use this notation ; he writes f (a )
and f (a )
It is d oubtful whether this m ethod S hould be nam ed after
n omi al s
Newton alone Though not identical with Vieta s process it re
se mbles Vieta s The chief diff erence lies in the d ivisor used The
divisor
is m uch simpler and easier to compute than Vieta s
divisor Raphs on s version of the process represents what J Lagrange
recogniz e d as an advance on the schem e of Ne w ton The m ethod is
“
“
1
plus S imple que celle de Newton
P erhaps the na m e Newton
R aph s o n m ethod would be a designation m ore nearly representing
the facts of history We m ay a dd that the solution of nu m erical
equations was considered geom etrically by Thom as B aker in 1 6 84
“
and E dmund Halley in 1 6 8 7 but i n 1 6 94 Halley had a very great
desire of doing the sam e in numbers
The only difference betwe en
Hall ey s and Newton s own m ethod is that Halley solves a quadratic
equation at each step Newton a linear equation Halley m odified
also certain al gebraic expressions yielding approxim ate cube and
fif th roots give n i n 1 6 9 2 by the French man Thomas Fan tet de L agny
I n 1 7 0 5 and 1 7 0 6 Lagny outlines a m ethod of di fferences ;
( 1 66 0 1 7
such a m ethod less syste matically developed had bee n previously
if a b c
explained i n England by John Collins B y this m ethod _
are in arith m etical progression then a root m ay b e foun d app roxi
m ately from the first second and higher diff erences of f ( a)
f (b)
'
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-
,
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,
,
,
,
,
f (a)
,
,
”
,
,
Newton s M ethod of Fl u xi on s contai ns also
gram wh i ch enable d hi m in an equation f (x
,
’
,
1
L ag rang e , R es o l u ti on des
,
eq u al
.
nu
m
.
,
1
,
,
Newto n s parallelo
= 0 to find a ser i es
y)
7 98 , N o t e
’
,
V
,
p
.
1 38
.
A HISTORY OF M ATHE M ATICS
4
in powers of x equal to the variable y The great utility of this rul e
lay in its determ ining the f or m of the series ; for as soon as the law was
k n own by which the e xponents in the series v ary then the e xpansion
could be effected by the m ethod of indeterm inate coefficients The
rule is stil l used in determining the infinite branches to curves or their
figure at m ultiple points Newton gave no proof for it nor any clue
as to how he discovered it The proof was supplied half a century
1
later by A G K astner and G Cram er independently
In 1 704 was published as an appendix to the O pti cks the E nu
merati a l i nearu m terti i ordi n i s which contains theore ms on the theor y
of curv es Newton divi d es cubics into seventy tw o S pecies arranged
in larger groups for which his co mmentators have supplied the nam es
“
“
genera and classes recogniz ing f o urteen of the form er and seven
of the latter He overlooked six S pecies dem anded by his
( or four)
principles of classification and after w ards added by J S tirling Wil
liam M urdoch ( 1 7 54
and G Cram er He enunciates the r e
“
m arka ble theorem that the five S pecies which he nam es divergent
parabolas give by their p rojection every cubic curve whatever AS
a rule the tract con tains no proofs I t has been the subj ect of frequent
conjecture how Newton de d uced his results Recently we have gotten
at the facts since m uch of the analysis used by Newton and a few
additional theorems hav e been discov ered am ong the P ortsm outh
papers An account of the four holograph m anuscripts on this sub
jcet has been published by W W Rouse B all in the Tr a n s acti ons of
the L ondon M a thema ti ca l S oci ety ( vol xx pp 1 0 4
It is inter
esting to observe how Newton begins his research on the classification
of cubic curves by the algebraic m ethod but finding it laborious
attacks the p roblem geometrically and after w ards returns again t o
analysis
Space d oes not perm it us to do m ore than m erely mention Newton s
prolonged researches in other departm ents of science He con ducted
a long series of experim ents in optics and is the author of the corpus
c u l ar theory of light
The last of a nu mber of papers on optics
which he contributed to the Royal Society 1 6 8 7 elaborates the theory
“
of fits
He explained the decomposition of light and the theory
of the rainbow By him were invented the reflecting telescope and
the sextant (afterwards r e invented by Thom as Godfrey of P hila
2
delphia and by John Hadley)He d educed a theoretical e xpression
for the velocity of sound in air engaged in experi ments on chem istry
elasticity m agnetism and the l aw of cooling and entered upon geo
logical speculations
D uring the two years following the close of 1 6 92 Newton su ff ered
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,
1
S G u n t h er , Vermi schte U n ters uchu ngen
.
1 8 76 ,
2
pp
1 36
-
1 87
p 42
.
Ges chi chte d math
.
.
F C aj or i , Teac hi n g
.
.
.
z ur
a nd
H i s tory of M a themati cs i n the U S
.
.
Wi s s
.
,
z
L eip i g ’
Wa s h i n g ton
,
1 8 90 ,
A HISTORY OF M ATHE M ATICS
206
went to London and re mained there fro m January till M arch He
there becam e incidentally acquainted with the m athe matician John
P ell to who m he explained a m ethod he ha d found on the su mm ation
of series of nu mbers by their diff erences P el l told h im that a similar
form ula had been published by Gabriel M outon ( 1 6 1 8—1 6 94)as
early as 1 6 70 and then called his attention to N M ercator s work
on the rectification of the parabola While in London Leibniz ex
hi bi t ed to the Royal Society his arith m etical m achine which was
S im ilar to B
P ascal s but m ore e ffi cient and perfect
After his re
turn to P aris he had the leisure to study m athe m atics m ore syste m
at i cal l y
With in d o mitable energy he set about re m o ving his ig n o
rance of higher m athe matics C Huygens was his principal master
He studied the geo metric works of R D escartes H on orari us Fabri
Gregory S t Vincent and B P ascal A careful st udy of infinite
series led hi m to the discovery of the following expression for the
ratio of the C ircu mference to the dia meter of the circle previously
discovered by J ames Gregory :
,
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,
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’
,
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,
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,
’
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,
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,
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.
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,
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,
,
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,
—+
.
77
1
= I
z
s
I
+s
l
l
r
s
etc
.
This elegant series was found in the same way as N M ercator s on
the hyperbola C Huygens was highly pleased with it and urged
him on to new invest i gations In 1 6 73 Leibniz derived the seri es
’
.
.
.
.
a r e tan x = x
—1
x
3
+ 1x
3
from which m ost of the practical m etho ds of computing 7T have been
obtained This series had been previously disco v ered by Ja mes
Gregory and was used by A br aham S ha r p ( 1 6 5 1 —1 74 2 )under i n
In 1 7 0 6
s t ru c t i o n s fro m E Halley for calculating i t to 7 2 places
—
John M achi n ( 1 6 80 1 7
professor of astronomy at Gresha m Col
lege in London obtained 1 0 0 places by using a n expression that is
obtained from the relation
.
,
.
.
,
—
4
4
are
tan —arc tan 2 11
,
by substituting Gregory s infinite series for
ar e ta n 1 and a re tan 1 1g
—
m
i
i
2
M achin s for ula was used I n 1 8 74 by W l l am S han ks ( 1 8 1 1 88 2 )
for computing
to 70 7 places
Leibniz entered into a detailed study of the quadrature of curves
and thereby became inti mately acquainted with the higher m ath
Am ong the papers of Leibniz is still found a manuscript
ema ti cs
on quadratures written before he left P aris in 1 6 7 6 but which was
never printed by him The m ore important parts of it were e mbodied
in articles published later i n the A cta er u di tor u m
In the study of Cartesian geometry the attention of Leibniz was
’
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’
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,
,
.
.
NEWTON TO EULER
20 7
drawn early to the direct a n d inverse proble ms of tangents The
direct proble m had been sol v ed by D escartes for the si mplest curves
only ; while the inverse had co mpletely transcended the power of his
analysis Leibniz investigated both proble ms for any curve ; he
—
constructed what he called the tri an gu l u m char acteri s ti cu m an
infinitely s mall triangle between the infinitely small part of the curve
coinciding with the tangent and the diff erences of the ordinates and
abscissas A curve is here considered to be a polygon The tri angu l u m
char acter i sti cu m is si m ilar to the triangle for m ed by the tangent the
ordinate of the point of contact and the sub tangent as well as to
that between the ordinate norm al and sub norm al It was e mployed
by I Barrow i n Englan d but Leibniz states that he obtained it from
P ascal From it Leibniz observed the connection e xisting between the
direct and inverse proble ms of tangents He saw also that the latter
could be carried back to the quadrature of cu rves All these results
are contained in a m anuscript of Leibniz written in 1 6 7 3 O n e mode
used by h im i n e ffecting quadratures was as follows : The rectangle
form ed by a sub normal p and an elem ent a (i e infinitely s mall part
of the abscissa)
is equal to the rectangle form ed by the ordinate y
and the elem ent l of that ordinate ; or in symbols pa = yl _B ut the
summation of these rectangles from z ero on gives a right triangle
equal to half the square of the ordinate Thus usin g Cavalieri s no
ta t i on he gets
.
.
,
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.
,
-
,
,
-
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.
,
,
,
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.
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,
-
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,
’
.
,
,
o
B ut
a = o mn
mn p
.
’
y
= om n
o
mn
.
.
l;
o
.
yl
=y
2
1
2
( omn m eaning omni a all )
.
.
,
hence
mn
l
o
mn l
.
.
This equation is especially interesting since it is here that Leibniz
first introduces a n ew notation He says : I t will be useful to write
,
“
.
for
writes
mn
o
th e
.
,
as
l
for
mn l ,
o
.
that is the
,
su
m
of the
l
’
S
”he then
;
equation thus :
Fro m this he deduced
th e
S
implest integrals such as
,
Since the sy mbol of su mm ation
raises the
d im ensions ,
cluded that the opposite calculus or that of diff erences
,
he con
d,
would
A HISTORY OF M ATHE M ATICS
20 8
lower them
Thus if
.
then
l = ya ,
,
ya
d
l
The symbol
d
was at
first placed by Leibniz In the denom inator because the lowering of
the power of a term was brought about in ordinary calculation by
division The m anuscript giving the abov e is dated October 2 9th
This then was the m e morable day on which the notatio n
—
of the n ew calculus cam e to b e a notation which contributed enor
mou sl y to the rapid gro w th and perfect develop m ent of the calculus
Leibniz proceeded to apply his n ew calculus to the solution of
certain proble ms then grouped together u n der the name of the In
v erse P roble m s o f Tangents
He found the cubical parabola to be
the solution to the following : To find the c u rv e in which the sub
norm al is reciprocally proportional to the ordinate The correctness
of his solution was tested by him by applying to the result the m ethod
of tangents of B aron R en e Fra n goi s de Sluse ( 1 6 2 2 1 6 8 5)
and reason
ing backwards to the original supposition In the solution of the
,
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,
,
,
,
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'
.
-
.
third problem he changes his notation from
x
d
to the now usual nota
tion dx I t is worthy of rem ark that i n these investigations Leibniz
nowhere explains the S ignificance of dx and dy except at one place
.
,
,
a
In
m arg i nal
note : Idem est
dx
x
et
d
,
id est differentia inter duas
,
proxim as
Nor does he use the term difi er en ti al but al w ays differ
en ce
Not till ten years later in the A cta eru di tor u m did he give
further explanations of these symbols What he aim ed at principally
was to d eterm i ne the change an expression undergoes when the sym
x
.
,
.
,
,
.
or
d
is placed before it
.
It
m ay
be a consolation to students
wrestling with the elem ents of the differential calculus to know that
2
it required Leibniz considerable thought and attentio n to determ ine
whether
dx dy
is the sa m e as d (xy)and
,
dx
dy
j
the sam e as df
After
.
y
considering these questions at the close of one of his manuscripts he
conclude d that the expressions were not the sam e though he could
not give the true value for each Ten days later i n a m anuscript
xdy
dated November 2 1 1 6 7 5 he found the equation ydx = dxy —
giving an expression for d (xy)which he observed to be true for all
curves He succeed ed also in eliminating dx fro m a diff erential
equation so that it contained only dy and thereby led to the solution
“
of the problem under consideration
B ehold a most elegant way
C J G h d t E td ck g d h h r n A l ysi s Hal l e 8 55 p
5
h
8 48
C J G h d t E td k g d r D ifl enzi l
g d ch L ei b i z H al l
,
,
,
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,
,
,
,
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,
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,
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1
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er
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,
n
e
un
er
o e e
na
.
,
1
12
.
,
.
'
2
pp
ar
2 5,
er
.
41
.
ar
,
n
ec
un
e
er
a rec
nun
ur
n
,
e,
1
,
A HISTORY OF M ATHE M ATICS
2 10
by Leibniz to Tschi rn hausen during a controversy which they had
had on this subject Fearing that Ts chi rnh au sen might claim as his
own an d publish the notation and rules of the differential calculus
Leibniz d ecide d at last to m ake public the fruits of his inventions
In 1 684 or nine years after the new calculus first dawned upon the
m ind of Leibniz and nineteen years after Ne w to n first worked at
fl uxi on s and three years before the publicatio n of Newton s P ri n ci pi a
Leibniz published in the A cta eru di toru m his first paper on the differ
He was unwilling to give to the world all his treasures
en t i al calculus
but chose those parts of his work which w ere most abstruse and least
perspicuous This epoch m aking paper of only S ix pages bears the
“
title : Nova me thodu s pro maxi mi s e t m inim is i t emq u e t an g en tib u s
qu a nec frac tas nec irrationales quantitates moratur e t S ingulare
pro illis calculi genus
The rules of calculation ar e briefly stated
without proof and the meaning of dx and dy is not m ade clear
P rinter s errors increased the diffi culty of comprehending the subj ect
It has been inferred fro m this that Leibniz him self had no definite
Are dy and dx finite or infinitesi m al
a n d settled ideas on this subj ect
quantities ? At first they appear i n deed to have been taken as finite
when he says : We now call any line selected at random dx then
we designate the line which is to dx as y is to the sub tangent by dy
which I S the d i fference of y Leibn iz then ascertains by his calculus
in what way a ray of light passing through two di ff erently refracting
m edia can trav el easiest fro m one point to another and then closes
h i s article by giv ing his solution in a few words of F de B e au n e s
proble m T w o years later ( 1 6 86 )Leibniz published in the A cta
er u di tor u m a paper containing the rudi m ents of the integral calculus
The quantities dx and dy are there treated as i n finitely s mall He
S howed that by the use of his notation the properties of curves could
be fully expressed by e q u a ti on s Thus the equation
.
,
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,
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,
,
’
,
,
,
,
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,
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,
”
e
,
,
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,
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’
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.
,
,
,
,
-
,
,
.
,
,
,
,
’
,
.
,
.
.
.
,
.
—
/
x zx x
2
y
+
characteri zes the cycloi d
The great invention of Leibniz now m ade public by his articles in
the A cta eru di toru m m a d e little impression upon the m ass of m athe
m a t i c i an s
In Germ any no one co mprehen ded the n ew calculus
except Ts chi r nh a u s en who re mained indiff erent to it The author s
statem ents were too S hort and succinct to make the calculus generally
un derstood The first to take up the study of it were two fore i gners
the Scotch m an J ohn C r a i g and the Swiss Ja k ob (Ja mes )
B ern ou l l i
The latter wrote Leibniz a letter in 1 6 8 7 wi shing to be initiated into
the m ysteries of the n ew analysis Leibniz was then travelling abroad
so that this letter rem ained unanswered till 1 6 90 Jam es B ernoulli
1
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1
C J G e rh a rd t , Gesc hi chte der M athemati k i n Deu tschl a nd , M u n ch e n ,
.
p
.
I
59
.
.
1 8 7 7,
NEWTON TO EULER
21I
succee ded meanwhile by close application in uncovering the secrets
of the differential calculus without assistance He and his brother
John prov ed to be mathem aticians of exceptional power They app l ied
the mselves to the new science with a success and to an ex tent which
Leibniz
made Leibniz declare that it was as m uch theirs as his
carried on an ex tensive correspondence with the m as well as with other
mathem aticians In a letter to John B ernoulli he suggests a mong
other things that the integral calculus be improved by reducing i n
The integra
t eg r al s back to certain funda m ental irreducible for m s
tion of logarithm ic expressions was then studied The writings of
Leibniz contain m any innovations and anticipations of since pro m
nent methods Thus he m ade use of variable param eters laid the
foundation of a n al ysi s i n s i tu introduced in a manuscript of 1 6 7 8 the
notion of determ inants (previously used by the Japanese )in his
e ffort to simplify the expression arising in the elim ination of the u n
known quantities from a set of linear equations He resorted to the
device of breaking up certain fractions into the s u m of other fractions
for the purpose of easier integration ; he e xplicitly assu med the prin
S ingular solution
c i pl e of continuity ; he ga v e the first instance of a
and laid the foun dation to the theory of envelopes in two papers one
of which contains for the first time the term s co ordi nate and axes of
co or di n ates
He wrote on osculating curves but his paper contained
the error (pointed out by John B ernoulli but not a dmitted by Leibniz )
that an osculating circle will necessarily cut a curv e in four consecutive
points Well known is his theore m on the n th diff erential c oefli c i en t
of the product of two functions of a variable Of his m any papers o n
mechanics so m e are valuable while others contain grave errors
Leibniz introduced I n 1 6 94 the use of the word f u n cti on but not i n
the modern sense Later in that year Jakob B ernoulli used the word
in the Leibniz ian sense In the appendix to a letter to Leibniz dated
July 5 1 6 98 John B ernoulli uses the word in a m ore nearly m odern
“
sense : earum [appl i ca taru m] q u aec u n q u e fu n c ti o n e s per alias appl i
catas P Z e xpress es
In 1 7 1 8 John B ernoulli arri v es at the definitio n
of function as a quantity composed in any manner of a variable and
any constants
(O n appelle ici fonction d une grandeur variable
une quantité co mpos ée de quelque maniere que c c soit de cette gran
1
deur variable e t de constantes )
Leib n iz m ade i mportant contributions to the notation of mathe
Not only is our notation of the diff erential and integral
ma t i c s
calculus due to him but he used the S ign of equality i n writing pro
portions thus a :b = c :d I n Leibni z ia n m anuscripts occurs
for
“
2
Says
similar and
for equal and S im ilar or congruent
,
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-
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”
’
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”
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”
z
.
S ee M C an to r , op c i t , V ol I I I , 2 Ed 1 90 1 , pp 2 1 5 , 2 1 6 , 4 5 6 , 4 5 7 ; E nc ycl o
pedi c des sci ences ma thémati q u es , Tome I I , Vo l I , p p 3 5
2
S ee al so J Trop fk e ,
L ei bn i , Wer k e E d G erh ar d t , 3 F o l g e , B d V , p 1 5 3
op c i t , V ol I I , 1 90 3 , p 1 2
1
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—
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M ATHE M ATICS
A HISTORY OF
212
”
“
E B J ourdain
Leibniz himself attributed all of his ma the
ma ti cal discoveries to hi s i mprove ments in notation
B efore tracing the further develop m ent of the calculus we shall
S ketch the history of that long and bitter controversy be tw een English
and Continental m athem aticians on the invention of the calculus
The question was did Leibniz invent it independently of Newton or
was he a plagiarist ?
We must begin with the early correspondence betw een the parties
appearing in this dispute Newton had begun using his notation of
fl u xi on s in
In 1 6 6 9 I B arrow sent John Collins Newton s
tract De A n al ysi per equ ati on es etc
The first visit of Leibniz to London e xtended fro m the r 1 th of Jan
uary until M arch 1 6 7 3 He was in the habit of comm itting to writing
i mportant scientific comm unications received from others I n 1 8 90
C J Gerhardt discovered in the royal library at Hanover a sheet of
3
m anuscript with notes taken by Leibniz during this journey
They
are headed O b serv a ta P hil osophi ca in itinere Anglicano sub i n i ti um
anni 1 6 73
The shee t is divi ded by horiz ontal lines into sections
The sections giv en to C hymi c a M ec han i c a M agnetica Botanica
Anatomica M edica M iscellanea contain extens i ve m em oranda while
those devoted to m athem atics have very few notes Under Geo
m etrica he says only this : Tangentes o m niu m fi g u raru m F i g u rar um
We sus
g eom e t ri ca r u m exp li c a t i o per m o t u m p u n c ti in m oto lati
p ec t fro m this that Leibni z had read Isaac B arrow s lectures Ne w ton
is referre d to only under Optica Evidently Leibniz did not obtain a
knowle dge of fl uxi on s during this visit to London nor is it claimed
that he did by his opponents
Various letters of I Newton J Collins and others up to the b e
ginnin g of 1 6 7 6 state that Newton inv ented a method by which tan
gents coul d be drawn without the necessity of freeing their equatio ns
from irrational terms Leibniz announced in 1 6 74 to H Oldenburg
then secretary of the Royal Society that he possessed very general
analytical m ethods by which he had found theore m s of great im
portance on the quadrature of the circle by m eans of series In answer
Oldenburg stated Newton and James Gregory had also discovered
metho ds of quadratures which e xtended to the circle Leibniz de
sired to have these m ethods communicated to him ; and Newton at
the request of Oldenburg and Collins wrote to the form er the cele
b ra t ed letters of June 1 3 and October 2 4 1 6 7 6
The first contained
the Binomial Theorem and a variety of other m atters relating to
infi nite series and quadratures ; but nothing directly on the m ethod of
P
1
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,
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,
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’
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,
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,
,
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,
.
.
,
,
,
,
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.
,
,
,
,
.
P E B Jou rdai n , The N a tu r e of M athemati cs , L o n do n , p 7 1
2
J Edl e ston , Cor res ponden ce “of S i r I saac N ewt“on a nd P r of essor Cotes , L on don ,
and
F l u xi on s
C omm erc i um Ep i s toli c u m i n
1 8 50 , p
xxi ; A
D e M o rg an ,
t h e P en ny C ycl ope di a
“
3
L e i b n i i n L o n don i n S i t u ngs ber i chte der K P reuss i schen
C J G e rh ard t ,
A cademi c d Wi ssensch zu B er l i n , Feb , 1 8 9 1
1
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”
.
.
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z
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z
.
.
.
M ATHE M ATICS
A HISTORY OF
2 14
ace to a volu me of his works that the calculus di ffe ren tial i s was N ew
ton s m ethod of fi ux i o n s which had been com municated to Lei bniz
in the Oldenburg letters A review of Wallis works in the A cta
er u di tor u m for 1 6 96 re minded the reader of Newton s own ad m issio n
in the scholiu m above cited
For fifteen years Leibniz had enjoyed unchallenged the honor of
being the inventor of his calculus B ut i n 1 6 99 Fati o de Du i l l t er
—
a Swiss who had settled in England stated i n a m athe
1
6
6
(
4 17
mat i cal paper presented to the Royal Society his convictio n that
1 Newton was the first i n ventor ; addi n g that whether Leib n iz the
second in v entor had borrowed anything fro m the other he would
leave to the judg m ent of those who had see n the letters and manu
scripts of Newton This was the first distinct insinuatio n of plagiar
i sm
It would see m that the English m athematicians had for some
time been cherishing suspicions unfavorable to Leibniz A feeling
had doubtless long prevailed that Leib niz during his second visit to
Lon don in 1 6 7 6 had or m ight have see n a mong the papers of Joh n
Collins Newton s A n al ysts per a qu ati on es etc which contained ap
plications of the fi u xi ona ry m ethod but no syste matic develop ment
or e xplanation of it Leibniz certainly did see at least part of this
tract D uring the week spent in London he took note of whatever
i n terested h im a m ong the letters and papers of Collins His m e mo
randa disco v ered by C J Gerhardt i n 1 8 49 i n the Ha n over library
1
fill two sheets The one bearing on our question is headed E xcerpta
de A n al ys i per aeq ua tion es nu mero ter
ex t rac t a t u N e w t on i M s c
minoru m i n fi n i t as
The notes are very brief e xcepting those De
r es ol u ti on e cequ a ti on u rn a ecta r u rn
of which there is a n al most c o m
fl
This part was evidently n ew to him If he exa mi n ed
p l e t e copy
Newton s entire tract the other parts did not particularly i mpress
him Fro m it he see m s to have gained nothi n g pertaining to the i n
By the previous introductio n of his ow n a l
fi n i t e s imal calculus
~b
h
m
m
i
or
t
he
had
ade
greater
progress
t
h
a
n
g
y what ca me to his
knowle dge i n London Nothing m athe matical that he had recei v ed
engaged his thoughts i n the i mm ediate future for on his way back
to Holland he co mposed a lengthy dialogue on m echanical subjects
Fatio de D u i ll ie r s insinuations lighted up a fla me of discord which
a whole century was hardly suffi cie n t to e xtinguish Leibniz who
had never contested the priority of Newton s discovery and w ho
appeared to be quite satisfied with Newton s adm ission in his scholiu m
now appears for the first time i n the controversy He m ade an a n i
mated reply i n the A cta eru di tor u m and co mplained to the Royal
Society of the injustice done him
Here the affair rested for so m e ti me In the Qu adr atu re of Cu rves
published 1 7 04 for the first time a for mal expositio n of the m ethod
a n d notation of fl u xi on s was m a d e public
I n 1 7 0 5 appeared a n u m
”
L i b iz i Lo d
l c it
C J G e ha d t
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NEWTON
EULER
TO
215
favorab l e revie w of this i n the A cta erudi toru m stating that Newton
uses and al w ays has used fl u xio n s for the di ff erences of Leibniz This
was considered by Ne w ton s friends an i mputation of plagiaris m o n
th e part of their chief but this interpretation was al w ays strenuously
resisted by Leibniz J ohn K ei l l ( 1 6 7 1
professor of astrono my
at Oxford undertook with m ore z eal tha n j udgment the defe n ce of
Newton In a paper inserted in the P hi l os ophi cal Tr an sacti ons o f
1 7 0 8 he clai m ed that Ne w to n was the first inventor of fl ux i on s and
“
that the sa m e calculus was afterw ard published by Leibniz the
name and the m ode of notation being changed
Leibniz complained
to the secretary of the Royal Society of bad treat ment and requested
the interference of that body to induce K e ill to disavow the i n tentio n
of imputing frau d John K eil l was not m ade to retract his accusation ;
o n the contrary was authoriz ed by Newton and the Royal Society
to explain and d efen d his state ment This he did i n a long letter
Leibniz thereupon complained that the charge was n ow more ope n
tha n before and appealed for j ustice to the Royal Society and to
Ne w ton himself The Royal Society thus appealed to as a j udge
appointed a co mmittee which collected a nd reported upon a large
mass of d ocu m ents —m ostly letters fro m and to Ne w ton Leib n iz
Wallis Colli n s etc This report called the C ommerci u m epi s tol i cu m
appeared i n the year 1 7 1 2 and agai n i n 1 7 2 2 and 1 7 2 5 with a Recensio
prefixed and a ddi tional notes by K e ill The final co n clusion i n the
“
C ommer ci u m epi s tol i cu m was that Ne w ton was the first inventor
B ut this was not to the point The question w as not whether Ne w to n
was the first inventor but whether Leibniz had stolen the m ethod
The co mmittee had not form ally ventured to assert their belief that
Leibniz was a plagiarist I n the follo w ing se n tence they insinuated
that Leibn iz did take or m ight have taken his method fro m that of
“
Newton : And we find no m ention of his (Leib n iz s )
havi n g any other
Difi er en ti al M ethod than M ou ton s before his Letter of 2 1 s t of Ju n e
1 6 77 which was a year after a Copy of M r N ew ton s Letter of the
r o t h of D ecember
1 6 7 2 had been sent to P a ri s to be co mm unicated
to him ; and about four years after M r C ol l i n s began to co mmunicate
that Letter to his Corresponde n ts ; i n which Letter the M ethod of
Fl u xi on s was suffi ciently des c ri b d to any intelligent P erson
About 1 8 50 it was show n that what H Olde n burg se n t to Leibniz
was not Newton s letter o f D ec 1 0 1 6 7 2 but only excerpts fro m it
which o mitted Newton s m ethod of drawi n g tangents a n d coul d not
possibly convey an idea of fl u xi on s Oldenburg s letter was fou n d
a mong the Leibniz ma n uscripts i n the Royal Library at Hano v er a n d
was published by C J Gerhardt i n 1 846 1 848 1 849 and 1 8 5 and
again later
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S ee Essa ys on the Lif e an d Wor k of N ew ton b y A u g u s t u s D e M o rg a n , e di t ed , w i th
n o t es an d a pp e n d i ces , b y P hil i p E B Jo u r d a i n , C hi c a g o a n d L o n do n , 1 9 1 4
Jo ur
dain g i ves on p 1 0 2 t he b i b l io g rap hy o f t he p u b l i ca t i o n s o f N ew to n an d L e ib ni
1
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A HI STORY OF M ATHE M ATI CS
2 16
M oreover
published the C orrespondence
it beca m e known that the
of S i r I s aac N ew ton a n d P r of ess or C otes
Royal Society in 1 7 1 2 had not one but two parcels of Collins One
parcel contained letters of Ja m es Gregory and Isaac Newton s letter
“
of D ec 1 0 1 6 7 2 in full ; the other parcel which was m arked To
Leibnitz the 1 4 th of June 1 6 7 6 About M r Gregories remains
contained an abridgm ent of a part of the contents of the first parcel
with nothing but an allusion to Newton s method d escribed in his
letter of D ec 1 0 1 6 7 2 In the C ommer ci u m epi stol i cu m Newton s
letter was printed in full and no mention was m ade of the existence
“
of the second parcel that was marked To Leibnitz
Thus the
C ommer ci u m epi s tol i cu m conveyed the i mpression that Newton s u m
curtailed letter of D ec 1 0 1 6 7 2 had reached Leibniz in which fl uxi on s
“
was suffi ciently described to any intelligent person while as a
m atter of fact the method is not described at all in the letter which
Le i bniz received
Leibniz protested only i n private letters against the proceeding of
the Royal Society declaring that he would not answer an argu m ent
so weak J ohn B ernoulli in a letter to Leibniz which was published
later in an anonym ous tract is as decide dly unfair towards Newton
as the friends of the latter had bee n towar d s Leibniz John K eil l
replied and then Newton and Leibniz appear as m utual accusers in
se v eral letters addressed to third parties I n a letter dated April 9
1 716
and sent to Antonio S chi n el l a Conti ( 1 6 7 7
an Italian
priest then residing in London Leibni z agai n rem inded Newton of
the ad m ission he ha d m ade i n the scholium which he was n ow desirous
of disav owing ; Leibniz also states that he al w ays believ ed Newton
but that seeing h i m connive at accusations which he m ust have
known to be false it was natural that he (Leibniz )
should begin to
Newton did not reply to this letter but circulated som e r e
d oubt
marks am ong his friends which he publishe d i m m ediately after hearing
of the death of Leibniz Nove mber 1 4 1 7 1 6 This paper of Newto n
gives the following explanatio n pertaining to the scholiu m i n question :
He [Leibniz ] pretends that in my book of principles I allowed hi m
the in v ention of the calculus di ff eren t ial i s i n dependently of my own ;
and that to attribute this inventio n to myself is co n trary to m y
knowledge there av o w ed B ut in the paragraph there referred unto
I do not find one word to this purpose
I n the third e d ition of the
P r i n ci pi a 1 7 2 6 Newto n o m itted the scholiu m and substituted in its
place another in which the nam e of Leibniz does not appear
National pr id e and party feeling long prevented the adopt i on of
impartial opini ons in England but n ow it is generally ad m itted by
,
when J Edlesto n in
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1 8 50
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B B i o t a n d F L e fo r t s e di ti on o f t h e C ommer ci u m epi s tol i c u m,
P a r i s , 1 8 5 6 , w hi c h e xh i b i t s a l l the a l t e ra t io n s ma d e i n t h e d i ff e ren t rep r i n t s o f t hi s
’
p u b l i c a t io n an d r e p ro d u c es a l so H O l den b u rg s l e tt e r t o Le i bn i o f J u l y 2 6 , 1 6 7 6 ,
a n d o ther i mp or t an t d oc u me n t s b eari ng o n t h e c on t rov er sy
re c o mm e n d
J
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A HI STORY OF M ATHE M ATIC S
2 18
attention It was again proposed in 1 7 1 6 by Leibniz to feel the pulse
of the English mathem aticians
This may be considered as the first defiance proble m professedly
aimed at the English Newton solved it the same evening on which
it was delivered to him although he was m uch fatigued by the day s
work at the m int His solution as published was a general plan of
an investigation rather than an actual solution and was on that
account criticised by Johann B ernoulli as being of no value B rook
Taylor undertook the defence of it but ended by using very r ep re
h en si bl e language
Johann B ernoulli was not to be outdone in i n
civ ility and m ade a bitter reply Not long afterw ards Taylor sent
an open defiance to Continental m athematicians of a proble m on the
integration of a fl u xi on of co mplicated form which was known to
very few geo meters in England and supposed to be beyond the power
of their adversaries The selection was injudicious for J ohann
B ernoulli had long before explained the m ethod of this and similar
integrations I t serv ed only to display the skill and augm ent the
triu mph of the followers of Leibniz The last and m ost unskil ful
challenge was by John K eill The problem was to find the path of a
projectile in a m ed iu m which resists proportional ly to the square of
the velocity Without first m aking sure that he himself could sol v e
it K eil l bol dly chall enged Johann B ernoulli to produce a solution
The latter resol v ed the question in very short tim e not on l y for a
resistance proportional to the square but to any power of the velocity
Suspecting the weakness of the adversary he repeate dly off ered to
send his solution to a confidential person in London provided K eil l
would do the sam e K eill never m ade a reply and Johann Bernoulli
1
abused him and cruelly exulted ov er him
The explanations of the funda m ental principles of the calculus as
giv en by Newton and Leibniz lacked clearness and rigor For that
reason it m e t with opposition from several quarters In 1 6 94 B ern hard
N i eu w en tij t ( 1 6 54—
1 7 1 8)
of Holland denied the existence of differentials
of higher orders and objected to the practice of neglecting infinitely
s mall quantities These obj ections Leibniz was not able to meet
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satisfactorily
.
In his reply he said the value of
dr
_
dx
in geom etry could
be expressed as the ratio of finite quantities In the interpretation
2
of dx and dy Leibni z vacillated
At one ti m e they appear i n his
wr i tings as finite lines ; then they are called infinitely s mall quantities
a n d again
q u an ti ta tes i n as s i gn a bi l es which spring fro m q u a n ti tates
ass i gn a bi l es by the law of continuity
In this last presentation Leibni z
approached nearest to Newton
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John
P l ayfai r ,
P rog re s s of th e M at hema ti cal an d P hy si c al S ci en ce s
n
En cycl opcedi a B r i tan n i ca , 7 t h Ed , co n t i n u ed i n t h e 8 th E d b y S i r John L e sli e
’
2
C o n su l t G Vi v an ti , I l con cetto d I nfi n i tesi mo S aggi o s tor i co N uo va edi ion e
N ap ol i , 1 90 1
1
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NEWTON
EULER
TO
9
In England the principles of fl uxi on s were boldly attacked by
—
B
r
k
l
e
y
1
the eminent metaphysician in
i
s
h
o
eor
e
e
e
68
B
G
(
g
5 17
p_
a publication called the A n al yst
He argued with great acute
ness contending among other things that the fun da mental idea of
supposing a finite ratio to exist betw een terms absolutely evanescent
“
the ghosts of departed quantities as he called the m—was absurd
and unintelligible B erkeley claimed that the second a nd third
fi uxi o n s were even m ore m ysterious than the first fl u xi on
His c on
tention that no geo m etrical quantity can be exhausted by division is
“
in consonance with the claim m ade by Zeno in his dichoto my
and the claim that the actual infinite cannot be realiz ed M ost m odern
readers recogniz e these contentions as untenable B erkeley declared
as axiom atic a lemm a involving the shifting of the hypothesis : If x
receives an increm ent i where i is expressly supposed to be some
quantity then the i n crement of x divided by i is found to be n x +
"
n (u
x
If n o w you take i = 0 the hypothesi s i s s hif ted
i+
and there is a m anifest sophis m in retaining any result that was ob
t ai n ed on the supposition that i is not z ero
B erkeley s le mma fou n d
no favor a mong English m athe maticians until 1 8 0 3 when Robert
Woodhouse openly accepted it The fact that correct results are
obtaine d in the di fferential calculus by incorrect reasoning is explained
by Be rkeley o n the theory of a co mpensation of errors
This theory
was later ad v anced also by Lagrange and L N M Carnot The
publication of B erkeley s A n al ys t was the m ost spectacular m athe
ma t i cal event of the eighteenth century in England P ractically all
B ritish discussions of fl uxi on al concepts of that tim e involve issues
raised by B erkeley B erkeley s object i n writing the A n al ys t was to
show that the principles of fl u xi o n s are no clearer than those of Chris
“
t ia n i ty He referred to a n infi d el m I t h e ma t i c i an ( E d m und Halley )
1
of whom the story is told that when he j es ted concerning theological
“
questions he was repulsed by Newton with the re m ark I have
studied these things ; you have not
A friend of B erkeley when on a
be d of sickness refused spiritual consolation because the great
m athe m aticia n Halley had convinced h i m of the inconceivability of
the doctrines of Christianity This induced B erkeley to write the
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A n al yst
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—
( 1 6 84 1 750 )
Replies to the A n al yst were published by Ja mes J
of Tri nity College Cambridge under the pseudonym of P hi l al e thes
C an tab rig ie n si s
and by J ohn Wal ton of D ublin There followed
several rejoi nders J u ri n s d efence of Newton s fl uxi on s did not meet
the approval of the mathe matician B e n j ami n R o b i n s ( 1 70 7
and later in
In a Journal called the Republ i c/e of L etter s (Londo n)
the Wor k s of the L ea r n ed a long and acri monious controversy was
carried on bet w ee n Juri n and Rob i n s an d later betw ee n Juri n and
Henry P emberton ( 1 6 94
the editor of the third editio n of
u ri n
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M ach M echa n i cs
,
1 90 7 ,
—
8
pp 4 4 4 49
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A H ISTO RY O F M AT H E M ATI C S
2 20
New ton s P ri n ci pi a The question at issue was the precise m eaning
of certain passages in the writings of Ne w ton : D id Newton hold that
“
there are variables which reach their lim i ts ? Jurin answered Yes ;
“
Robins and P emberton answered No
The debate betw een Jurin
and Robins is important in the history of the theory of limits Though
holding a narrow vie w of the concept of a lim it Robins deserves credit
for rej ecting all infinitely s mall quantities and giving a logically quite
coherent presentation of fi u xion s in a pamphlet called A D i scou rs e
’
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,
c on cer ni n g
the N atu r e
an d
C er tai n ty of S i r I s aac N ew ton s M ethods of
’
M acl au r i n s Fl u xi on s , 1 7 4 2 , m ark the top
’
This and
notch of mathe m atical rigor reached during the eighteenth century
in the exposition of the calculus B oth before and after the period
of eight years 1 834—1 84 2 there existed during the eighteenth century
in Great B ritain a m ixture of Continental and British conceptions of
the new calculus a superposition of B ritish symbols and phraseology
upon the older Continental concepts Newton s notation was poor and
Le i bniz s ph i losophy of the calcul u s was poor The mix ture r epr e
sented the temporary survival of the least fit of both syste ms The
subsequent course of events was a superposition of the Leibniz ian
notation and phraseology up o n the li mit concept as developed by
Newton Jurin Robins M aclaurin D Al emb er t and later writers
In France M i chel Rol l e for a ti m e rej ected the differential calculus
and had a controversy with P V ar ig n on on the subj ect P erhaps the
m ost powerful argu m ent in favor of the new calculus was the con
sistency of the results to which it led Fam ous is D Al emh er t s advice
“
to y oung students : Allez en avant e t la foi vous vien dra
Am ong the m ost vigorous prom oters of the calculus on the Conti
nent were the B ernoullis They and Euler m ade B asel i n Swi tz erland
fa mous as the cra dle of great m athem atic i ans The fam ily of B er
n ou l li s furnished in course of a century eight m e m bers who dist i n
athem atics We subjoin the following g en ea
g ui sh ed the m selves in m
logical table
N i col au s B ern ou l l i the Father
Fl u xi on s ,
1 73 5
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—
1 6 54 1 7 0 5
J ak ob
J oh ann ,
N i col au s
N i col au s , 1 6 8 7
Dan i el
,
—7 59
1
J ohan n ,
.
1 667
—1 748
1 726
N i col au s , 1 6 95—
—
l
D ani e , 1 7 0 0 1 7 8 2
1 7 90
J ohan n , 1 7 1 0 —
1
—
8
4
4
7
1
J a k ob,
07
1
—
7 59 7 8 9
1
celebrated were the two brothers Jakob (Jam es)
and Johann
and Daniel the son of John Jakob a n d Johann were staunch
(John )
friends of G W Leibni z and worked han d in han d wi th him J ak ob
—
m
n g inter
was
born
in
B
asel
B
eco
i
J
B
e
r
o
l
l
i
1
1
0
m
a
e
s
n
u
6
)
( 54 7 5)
(
M ost
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A
222
HI STORY
OF M ATHE M ATI C S
theore m
If
where the letters are integers and
t = r + s is expanded by the bino m ial theore m then by taking n large
enough the ratio of u ( denoting the su m of the greatest term and the
to the sum of the re
n prece d ing terms and the n following ter ms )
maining term s m ay be m ade as great as we please Letting r and s
be proportional to the probability of the happe n i n g and failing of an
event in a single trial then i t corresponds to the probability that i n n t
trials the nu mber of times the event happens will lie bet w een n (r 1 )
“
and n (r + both i n clusive B ernoulli s theorem will ensure hi m a
1
permanent place in the history of the theory of probability
P ro m
i ne n t conte mporary workers on probability were M ont mort in France
and D e M oivre in England In D ece mber 1 9 1 3 the Acade my of
“
Sciences of P etrograd c el eb ra t ed the bicentenary of the law of large
nu mbers Jakob B ernoulli s A rs conj ectandi having been p ublished
at Basel i n 1 7 1 3 Of his collected works in three volu mes one was
printed in 1 7 1 3 the other t w o i n 1 744
—
B
r
J oh an n ( Jo h n ) e n o u lli ( 1 66 7 1 748)
was initiated into m athe
m at ic s by his brother He after w ards visited France where he m e t
Nicolas M alebranche Giovanni Dom enico Cassini P de _L ahi re P
For ten years he occupied the
V ar ig n on and G F de l H o spi ta l
mathe matical chair at Groni n ge n and the n succeeded his brother at
Basel He was one of the m ost e n thusiastic teachers and m ost suc
c e ss fu l original investigators of his ti me
He was a m e mber of al most
every learned society i n Europe His co n troversies were al most as
nu merous as his discoveries He was ardent in his frie n dships but
unfair m ean and violent toward all who i n curred his dislike —eve n
his own brother and son He had a bitter dispute with Jakob on the
isoperi metrical proble m Jakob convicted him of several paralogis ms
After his brother s death he atte mpted to substitute a disguised solu
tio n of the form er for an incorrect one of his own Johann ad mired
the merits of G W Leibniz and L Euler but was bli n d to those of
I Newton He i mm ensely enriched the i n tegral calculus by his labors
A mong his discoveries are the exponential calculus the line of s w iftest
descent and its beautiful relation to the path described by a ray
passing through strata of variab l e de n sity In 1 6 94 he e xplai n ed in
a letter to l H ospi tal the method of evaluating the indeterminate
’
n ou ll i s
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He treated trigonometry by the analytical
m ethod ,
studied
caustic curves and trajectories Several ti mes he was given priz es
by the Academy of Science in P aris
Of his sons N i c ol au s and D an i el were appointed professors of
mathem atics at the sam e ti me i n the Acade my of S t P etersburg The
former soon died in the prim e of life ; the latter returned to B asel i n
1 7 33 where he assu m ed the chair of experim e n tal philosophy
His
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1 To dh u n ter , Hi story
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f
o
Thear
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f
o
P r ob , p 7 7
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NEWTON TO EULER
23
first mathe matical publication was the solution of a differential equa
tion proposed by J F R i c ca t i He wrote a work on hydrodynamics
He was the first to use a suitable notation for inverse trigono metric
functions ; in 1 7 2 9 he used AS to represent arcsine ; L Euler in 1 7 3 6
1
used At for arctangent Daniel B ernoulli s investigations on prob
ability are re markable for their boldness and originality He pro
posed the theory of moral expectati on which he tho ught would give
results more i n accordance with our ordinary notio n s than the theory
of mathemati cal pr obabi l i ty He applies his m oral expec tation to the
P etersburg proble m : A throws a coi n i n the air ; if head
s o called
appears at the first thro w he is to receive a shilling fro m B if head
d oes not appear until the second thro w he is to receive 2 shillings if
head does not appear until the third throw he is to receive 4 S hillings
and so on : required the expectation of A B y the mathe matical
theory A s expectation is infinite a paradoxical result A given sum
of m oney not being o f equal i mportance to every m an account should
be taken of r el ati ve values Suppose A starts with a s um a then the
moral expectation i n the P etersburg proble m is finite according to
D aniel B ernoulli when a is finite ; it is 2 when a = o about 3 when
The P etersburg problem was discussed
a = 1 0 about 6 whe n a =
by P S Laplace S D P oisson and G Cram er D aniel B er n oulli s
m oral expectation has beco m e classic but no one ever m akes use
of it He applies the theory of probability to insurance ; to determine
the m or tality caused by s m all pox at various stages of life ; to deter
m ine the nu mber o i survivors at a given age fro m a give n nu mber of
births ; to determ ine how m uch inoculation lengthens the average
duration of life He S howed h o w the di fferential calculus could be
used in the theory of probability He and L Euler enjoyed the ho n or
of hav ing gained or shared no less tha n ten pri zes fro m the Acade my
of Sciences in P aris Once while travelling with a learned stranger
“
who asked his nam e he said I am D aniel Bernoulli
The stra n ger
could not believe that his co mpanio n actually was that great celebrity
“
and replied I am Isaac Newton
J oh ann B e rn o u l l i ( born 1 7 1 0 )
succeeded his father i n the professor
ship of m athe matics at Basel He captured three priz es (o n the cap
stan the propagatio n of light and the m agnet)
fro m the Acade my o f
Sciences at P aris N i c ol au s B e rn o u ll i (bor n 1 68 7)
held for a ti me the
mathe matical chair at P adua which Galileo had once filled He p ro v ed
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—= —
2
2
1 74 2
In
that
5 A
3 A
Joh ann
B e r n ou l l i
(born
1 7 44
)at
the age
btau Su at
of nineteen was appointed astronom er royal at B erlin and after
wards d irector of the m athe matical d epart m ent of the Aca d e m y H is
brother Ja kob took upon him self the duties of the chair of e xp e r i
,
.
G En e s t rom i n B i bl i otheca ma thema ti ca , V ol 6 , pp 3 1 9—
3 2 1 ; Vo l
2
I To dhu n ter , Hi s t of the Theor of P r ob , p 2 20
1
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1 4,
p
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78
.
A HISTORY OF M ATHE M ATIC S
24
physics at B asel previously perform ed by his uncle J akob
and later was appointed m athem atical professor in the Academy at
St P etersburg
Brief m ention will n ow be m ade of some other ma thematicians
belonging to the period of Newton Leibniz and the elder B ernoullis
a pupil of
G u illa u m e F ran coi s An toi n e l H O S pi tal ( 1 6 6 1
Johan n B ernoulli has already been m entioned as taking part i n the
challenges issued by Leibniz and the B ernoullis He helped pow er
fully i n m aking the calculus of Leibniz better known to the m ass of
m athematicians by the publication of a treatise thereon the A n al ys e
This contai n s the m ethod of find ing
des i nfi n i men t peti ts P aris 1 6 96
the li miting value of a fraction whose two terms tend to w ard z ero
at the sam e ti me due to Johann B ernoulli
Another z ealous French ad v ocate of the calculus was P i e rr e Var ig
n on ( 1 6 54
I n M em de P a ri s Ann ée M D CCIV P aris 1 7 2 2 he
follows Ja B ernoulli in the use o f polar co ordinates p and w Letting
x = p and y = l w the equations thus changed represent wholly di fferent
m“
m
=
curv es For instance the parabolas x a y beco me Fermatia n
Jo s e ph S a u ri n ( 1 6 59—1 73 7 )
solved the delicate proble m of
S pirals
how to determine the tangents at the m ultiple poi n ts of algebraic
I n 1 7 1 7 issued an ele m entary
cur v es \ Fr an cms N1col e ( 1 6 83 1 758 )
treatise o n finite differences i n which he finds the su m s of a consider
able nu m ber of interesting series He wrote also on roulettes par ti c u
Also interested i n
l a rl y S pherical epicycloids and their rectification
finite differences was P i err e Raym o n d d e M o n tmo rt ( 1 6 7 8
His chief writings on the theory of probability ; served to stim ulate
his more distinguished successor D e M oivre M ontmort gav e the
first general solution of the P roble m of P oints J e an P a u l d e G u a
—
gave the de monstration of Descartes rule of signs n ow
1
1
( 7 3 1 7 8 5)
gi v en i n books This skilful geo m eter wrote in 1 7 40 a work o n analyt
ical geo m etry the obj ect of which was to S ho w that most investiga
t ions on curves could be carried on with the analysis of D escartes quite
as easily as with the calculus He shows how to fi n d the tangents
asymptotes and various singular points of curves of all degrees and
proves by perspective that several of these points can be at infinity
M ich e l Roll e ( 1 6 5 2 —
is the author of a theore m nam ed after h im
1 7 1 9)
That theore m is not found in his Tr ai tad al gébre of 1 6 90 but occurs
in his M ethode pou r r és ou dre l es egali tez P aris
The na m e
“
—
of
Rolle s theorem was first used by M W D rob i sch ( 1 8 0 2 1 8 96 )
Leipzig in 1 834 and by Giusto B el l av i ti s in 1 846 His A l gebre contains
his m ethod of cascades
In an equation in 7)which he has trans
form ed so that its signs becom e alternately plus and minus he puts
v= x+ z and arranges the result according to the descending powers
when equated to z ero are
of x The coeffi cients of x x
m ental
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S ee F C aj o ri o n t he hi s t o ry
3 r d S , V o l I I , 1 9 1 1 , pp 3 0 0 3 1 3
1
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Theorem i n B i bl i otheca ma themati ca ,
A HI STORY OF M ATHE M ATIC S
2 26
correct T sc hi rn hau s en concluded that in the researches relating to
the properties of curves the calculus might as well be dispensed with
After the death of Leibniz there was in Germany not a S ingle mathe
mat i c ian of note Ch ri s ti an Wolf ( 1 6 7 9—
17
professor at Halle
was a mbitious to figure as successor of Leibniz but he forced the
ingenious ideas of Leibniz into a pedantic scholasticism and had the
unenviable reputation of having presented th e elements of the arith
metic algebra and analysis developed since the time of the Renais
sance in the form of Euclid —
of course only in outward for m for into
the S pirit of the m he was quite unable to penetrate ( H Hankel )
The conte mporaries and im mediate successors of Newton in Great
Britain were men of no mean merit We have reference to R Cotes
B Taylor L M aclaurin and A de M oivre We are told that at the
“
death of R o g er C ot e s ( 1 6 8 2
Newton exclaimed If Cotes had
lived we m ight have k n ow n some thi ng
I t was at the request of
Dr B entley that R Cotes undertook the publication of the se gond
edition of Ne w ton s P r i n ci pi a His m athem atical papers were pub
li shed after his death by Robert S m ith his successor in the P l u mbia n
professorship at Trinity College The title of the work H a rmoni a
M en su rar u m was suggested by the following theore m contained in it :
If on each radius vector through a fi xed point 0 there be taken a
point R such that the reciprocal of OR be the arith metic m ean of the
reciprocals of OR ] OR
then the locus of R will be a straight
line In this work progress was m ade in the application of logarithm s
and the properties of the circle to the calculus of fl u en ts To Cotes
we owe a theore m in trigono metry which depends on the form ing of
—
factors of x I In the P hi l os ophi cal Transa cti on s of London pub
li sh e d 1 7 1 4 he develops an i mportant form ula reprinted I n his H ar
mon i a M en su r aru m which I n m odern notation is i gb = log
s i n cp
Usually this form ula IS attributed to L Euler Cotes studied
“
2 = z
the curve p 0 a to which he gave the nam e lituus
Chief among
the ad mirers of Newton were B Taylor and C M aclaurin The quar
rel between English and Continental m athematicians caused them to
work quite independently of their great contemporaries across the
Channel
B roo k T aylo r ( 1 68 5—
was interested in m any branches of
1 73 1 )
learning and in the latter part of his life engaged m ainly in religious
and philosophic S peculations His principal work M ethodu s i n cre
—
men toru m di r ecta cl i n ver sa London 1 7 1 5 1 7 1 7 added a new branch
to mathem atics now called finite diff erences of which he was the
inventor He m ade many i mportant applications of it particularly
to the study of the form of m ovem ent of vibrating strings the r edu c
tion of which to mechanical principles w as first attempted by h i m
“
This work contains also Taylor s theore m and as a S pecial case
of it what is now called M ac l au ri n s Theorem
Taylor di scovered
his theorem at least three years before its appearance in print He
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NEWTON
EULER
TO
7
gave it in a letter to John M achin dated July 2 6 1 7 1 2 I ts i mportance
was not recognize d by analysts for o v er fifty years until J Lagrange
poin ted out its power His proof of it does not consider the question
of conv ergency and is quite worthless The first more rigorous proof
was given a century later by A L Cauchy Taylor gave a singular
solution of a di fferential equation and the m ethod of finding that
solution by di fferentiation of the di ff erential equation Taylor s
work contains the first correct explanation of astronom ical refraction
He wrote also a work on linear perspective a treatise which like his
other writings su ff ers for want of fulness and clearness of expression
At th e age of twenty three he gave a rem arkable solution of the prob
l em of the centre of oscillation published in 1 7 1 4
His claim to
priority was unj ustly disputed by Johann B ernoulli In the P hi l o
“
s ophi cal Tr an s acti on s Vol 3 0 1 7 1
7 Taylor applies Taylor s series
to the solution of num erical equations He assu m es that a rough
= 0 has been found
approxi mation a to a root of f (x)
Let
and x = a + s He expands
by his
theorem discards all powers of s above the second substitutes the
values of k k k and then solves for s By a repetition of this
process close approximations are secured He m akes the importa nt
observation that his method solves also equations involving radicals
and transcendental functions The first appl i cation of the Newton
Raph so n process to the solution of transcendental equations was
made by Thom as Si mp son in his Ess ays
on M a thema ti ck s
London 1 740
The earliest to suggest the m ethod of recu rri ng seri es for finding
roots was D ani el B ern ou l l i ( 1 70 0 —
who in 1 7 2 8 brought the
1 78 2)
quartic to the form
then selected arbitrarily
four numbers A B C D and a fifth E thus
also a six th by the sa me recursion formula F = aE + bD + cC + eB
and so on If the last two nu mbers thus found are M and N then
x = M + N is an appro xi mate root
D aniel B ernoulli giv es no proof
but is aware that there is not always convergence to the root This
m ethod was perfected by Leo n hard Euler in his I n tr odu cti o i n an al ysi n
i nfi ni toru m 1 748 Vol I Chap 1 7 and by Joseph Lagrange in Note
VI of hi s Res ol u ti on des equ ati ons n u méri qu es
Brook Taylor in 1 7 1 7 expressed a root of a quadratic equation in
the form of an infinite series ; for the cubic Fran goi s Nicole did sim i
l arl y in 1 7 38 and Clairaut in 1 7 46 A C Clairaut i n serted the process
i n his El emen ts d al gébre Thom as Simpson determ ine d roots by r e
version of series in 1 743 and by infinite series in 1 74 5 M arquis d e
Courtivro n ( 1 7 1 5—
also expressed the roots in the for m of i n
1 7 8 5)
1
finite series while L Euler devoted several articles to this topic
At this time the matter of convergence of the series did not receiv e
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p
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Fo r
212
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r efe ren ces see
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F C aj or i , i n C ol or ado C ol l eg e P u bl i cati on , G en e r al S e ri es 5 1 ,
.
A HI STORY OF M ATHE M ATIC S
2 28
proper att en tion except in som e rare instances James Gregory of
Edinburgh i n his Ver a ci r cu l i cl hyperbol ce qu adr atu r a
first
used the terms conv ergent and d iv ergent series while Wil l ia m
B ro u n c k er gav e an argu m ent which a m ounted to a proof of the con
vergence of his series noted above
C oli n M a c l au ri n ( 1 6 98 —
1 746 )
was elected professor of mathematics
at Aberdeen at the age of nineteen by competitiv e exam ination and
in 1 7 2 5 succeeded Jam es Gregory at th e Univ ersity of Edinburgh
He enj oyed the frien dship of Newton and i nspired by Newton s
disco v eries he published in 1 7 1 9 his Geometr i a Organi ca containing
a new and re m arkable m ode of generating co n ics know n by his nam e
“
and referring to the fact wh i ch becam e know n later as Cram er s
th
paradox that a curve of the n order is not always determ ined by
points that the number may be less A second tract De
Li n ea r u m g eometr i c a r u m pr opr i eta ti bu s 1 7 2 0 is re m arkable for the
elegance of its d em onstrations I t is based upon two theorem s : the
first is the theorem of Cotes ; the second is M acl au ri n s : If through
any point 0 a line be drawn m eeting the curve in n points and at
these points tangents be drawn and if any other line through 0 cut
the curve in R 1 R 2 etc and the system of n tangents in r l r g etc
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This and Cotes theorem are generaliz ations of
’
2
theorem s of Newton M aclaurin uses these in his treatm ent of curves
of the second and third degree culm inating in the remarkable theorem
that if a quadrangle has its v ertices and the two points of intersection
of its opposite sides upon a curv e of the third degree then the tangents
He de
d rawn at two opposite vertices cut each other on the curve
So m e
du c e d in d epen d ently B P ascal s theore m on the hexagra m
of his geom etrical results were reached independently by Wi l l i am
B r ai k en r i dge ( about 1 7 0 0 —
after
a clergym an in Edinburgh
The follow ing is known as the B ra ik en ri dg e M aclaurin theorem :
If the sides of a polygon are restricted to pass through fixed points
while all the vertices but one lie on fixed straight lines the free vertex
M ac l au ri n s m ore general
d escribes a conic section or a straight line
state ment (P hi l Tr an s 1 73 5)
is thus :If a polygon m ov e so that each
of its sides passes through a fixed point and if all its summ its excep t
one describe curv es of the degrees m n p etc respectively then the
free summ it m oves on a curve of the d egree 2 mnp
which reduces
to mn p
when the fixed points all lie on a straight line M ac
“
laurin was the first to write on pe dal curves a nam e due to Olry
T e rq u em ( 1 7 8 2
M aclauri n is the author of an A l gebra The
object of his treatise on Fl u xi on s was to found the doctrine of fl u xion s
on geom etric d emonstrations after the m anner of the ancients and
thus by r i gorous exposition answer such attacks as B erkeley s that
The Fl u xi on s contained for
th e doctrine rested on false reasoning
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HI STORY
A
2 30
O F M ATHE M ATIC S
to the hyperbola His work on the theory of probability surpasses
anything done by any other ma them atician except P S Laplace
His principal contributions are his i nvestigations respecting the
D uration of P lay his Theory of Recurring Series and his e xtension
of the value of Daniel B ernoulli s theorem by the aid of S tirling s
1
theorem
His chief works are the D octri n e of Chances 1 7 1 6 the
M i s cel l an ea A n al yti ca 1 7 30 and his papers in the P hi l os ophi cal
Tr an sacti on s
Unfortunately he did not publish the proofs of his
results in the doctrine of chances and J Lagrange more than fif ty
years later found a good exercise for his skill in supplying th e proofs
A generaliz ation of a problem first stated by C Huygens has re
“
c ei v ed the nam e of
D e M o ivr e s P roble m : Given n dice each
having f faces d eterm ine the chances of throwing any give n num ber
of points It was solved by A de M oivre P R de M ontm ort P S
Laplace and others D e M oivre also generaliz ed the P roblem on the
D uration of P lay so that it reads as follows :Suppose A has m counters
and B has n counters ; let their chances of winning in a single gam e be
as a to b; the loser in each gam e is to give a counter to his adversary :
required the probability that when or bef ore a certai n nu mber of games
has been pl ayed one of the players W ill have won all the counters of
his ad v ersary D e M o iv re s solution of this problem constitutes his
m ost substantial achieve m ent in the theory of chances He e mployed
in his researches the m ethod of ordinary finite differences or as he
called it the m ethod of recurrent series
A fam ous theory involving the notion of inverse probability was
ad vanced by Thomas B ayes I t was published in the London P hi l o
s ophi ca l Tr a n s acti on s
Vols 53 and 54 for the years 1 7 6 3 and 1 76 4
after the death of B ayes wh i ch occurred i n 1 7 6 1 These researches
originated the discussion of the probab i lities of causes as inferred
from observed e ffects a subject developed m ore fully by P S Laplace
Using m odern symbols Bayes fund am ental theore m may be sta ted
2
thus : If an event has happened p tim es and failed q ti m es the
probability that its chance at a si n gle trial lies bet w een a and b is
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x?
—
q
x
1
(
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(
1
— )d
x ?
”
x
.
A m em oir of John M ichell On the probable P arallax and M ag ni
tude of the fixed Stars in the London P hi l os ophi cal Tr an sacti on s
Vol 5 7 I for the year 1 76 7 contains the fa mous argu m ent for the
existence of design drawn from the fact of the closeness of certain
stars like the P leiad es
We m ay take the six brightest of the
P leia d es and supposing the whole num ber of those stars which are
equal in splendor to the faintest of these to be about 1 50 0 we shall
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I To dhu n t er , A H i s tor y of the M athemati cal Theor y
a n d L on don , 1 86 5 , pp 1 3 5 1 93
2
I To d hu n te r , op ci t , p 2 9 5
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P roba bili ty, C amb ri dg e
E ULER LAGRANGE AND LAP LACE
31
,
find the odds to be near
to 1 that no s ix stars out of that
nu mber scattered at rando m in the whole heavens would be w ithin
so small a distance from each other as the P leiades are
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and
Eu l er , L agr an ge,
L apl a ce
the rapid development of mathema tics during the eighteenth
century the leading part was taken not by the universities but by
the academies P articul arly prominent were the academies at B erlin
This fact is the more singular because at that time
an d P etrograd
Germany and Russia did not produce great m a t hematicians The
academ ies received t heir adornment mainly from the Swiss and
French I t was after the French Revolution that s chools gained their
ascendancy over academ ies
D uring the period from 1 7 30 to 1 8 20 S w itz erland had her L Euler ;
France her J Lagrange P S Laplace A M Legendre and G M onge
The m ediocrity of French m athem atics which marked the tim e of
Louis XIV was now foll owed by one of the very brightest periods of
all history England on the other hand which during the u np ro
d u c t i v e period in France had her Newton could n ow boas t of no great
m athem atician E xcept young Gauss Germ any had no great na m e
France now waved the m athematical sceptre M athematical studies
am ong the English and German people had su n k to the lowest ebb
Am ong them the d irection of original research was ill chosen The
form er adh ered with e xcessive partiality to ancient geo m etrical
m ethods ; the latter produced the co mbinatorial school which brought
forth nothing of great value
The labors of L Euler J Lagrange and P S Laplace lay in higher
analysi s a n d this they developed to a wonderful degree By them
analysis cam e to be completely sev ere d fro m geo m etry D uri ng the
preced i ng period the effort of m athe m aticia n s not only in England
but to som e extent eve n on the continent had been directe d to w ard
the solution of problem s clothed in geom etric garb and the results of
calc ul ation were usually reduced to geom etric form A change now
took place Euler brought about an em ancipation of the analytical
calculus from geo m etry and established it as an i n depen dent science
Lagrange and Laplace scrupulously adhered to this separati on
B ui l d ing on the broad foundation laid for higher analysis a n d m e
Euler with m atchless fertility of
c h an i c s b y Newton and Leibni z
m ind erected an elaborate structure There are few great ideas pur
sued by succee ding analysts which were not suggeste d by L Euler
or of which he d id not share the honor of invention With perhaps
less exuberance of inventi on but with m ore co mprehensive genius and
profoun der reasoning J Lagrange d evelope d the i n finitesim al calculus
and put analyt i cal m echan i cs into the for m i n which we now know i t
P S Lapla ce appl i e d the calculus a n d m echan i cs to the elaboration
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A HI STORY OF M ATH E M ATIC S
232
of the theory of universal gra vita tion and thus largely extending and
supplem enting the labors of Newton gave a full analytical discussion
of the solar system H e also wrote an epoch marking work on P rob
abil ity Am ong the analytical branches created during this period
are the calculus of Variations by Euler and Lagrange Spherical Har
m on i cs by Legendre and Laplace and Elliptic Integrals by Legendre
Comparing the growth of analysis at thi s time wi th the growth
d uri ng the time of K F Gauss A L Cauchy and recent m a the
ma ti c i an s we observe an i mportant diff erence
D uri ng th e form er
period we witness m ainly a develop ment with reference to form P lac
ing al m ost implicit confidence in results of calculation ma themati
c i an s did not always pause to discover rigorous proofs an d were thus
led to general propositions so me of which have since been found to
be true in only special cases The Combinatorial School In Germany
carried this tendency to the greatest extreme they worshipped
form alis m and paid no attention to the actual contents of form ulae
B ut in recent tim es there has been added to the dex terity in the formal
treatm ent of problems a much needed rigor of demonstration A
good example of this increased rigor is seen in the present use of i n
finite series as compared to that of Euler and of Lagrange in his earlier
works
The ostracism of geo m etry brought about by the m aster m inds of
this period could not last perm anently Indeed a n ew geom etric
school sprang into existence in France before the close of this period
J Lagrange would not perm it a single diagram to appear in his
M eca n i qu e an al yti qu e but thirteen years before his death G M onge
published his epoch m aking Géometri e descri pti ve
L e o nh ar d Eul e r ( 1 7 0 7—1 783 )
was born in B asel His father a
m inister gave him his first instruction in m athem atics and then sent
hi m to the University of B asel where he becam e a favorite pupil of
Johann B ernoulli In his nineteenth year he composed a dissertatio n
on the m asting of ships which received the second priz e from th e
French Acade m y of Sciences When Johann B ernoulli s two sons
Daniel and Nicolau s went to Russia they induced Catharine I in
1 7 2 7 to invite their friend L Euler to S t P etersburg where D aniel
in 1 733 was assigned to the chair of mathem atics I n 1 7 3 5 the solving
of an astronom ical problem proposed by the Academy for which
several e minent m athem aticians had dem anded som e m onths time
was achieved in three days by Euler with aid of improved m e thods of
his own B ut the eff ort threw h im into a fever and deprived him of
the use of his right eye With still superior m ethods this sam e problem
1
The despotism of
was solved later by K F Gauss in one hour !
Anne I caused the gentle Euler to S hrink from public a ff airs and to
d evote all his t im e to science After his call to B erlin by Frederick the
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W
.
S ar t ori u s
Wal ter sh au sen
,
Gauss ,
zu
m Gedachtn i ss , Leip z ig ,
1 8 56
.
A HISTORY OF M ATHE M ATICS
234
but also the B eta and Gamma Functions and other original investi
M ethodu s i n ven i endi l i n eas
t
i
a
on s ;
g
maxi mi mi n i mi ve pr opri etate
am ount of m athem atical genius
cu rvas
which displaying an
S eldo m rivall ed contained his researches on the calculus of v ariations
to th e invention of which Euler was led by the study of th e researches
of Johann and Jakob B ernoulli One of the earliest problem s bearing
on this subj ect was Newton s solid of revolution of least resistance
reduced by him in 1 6 8 6 to a diff erential equation (P ri n ci pi a Bk II
Sec VII P rop XXXIV Scholiu m )Johann B ernoulli s problem of
the brachistochrone sol v ed by h im in 1 6 97 and by his brother Jakob
in the same year stim ulated Euler The study of isoperim etrical
curves the brachistochrone in a resisting m ediu m and the theory of
geodesics previously treated by the elder B ernoullis and others led
to the creation of this new branch of m athem atics th e Calculus of
Variations His method was essentially geom etrical which m akes
the solution of the simpler proble m s very clear Euler s Theori a
gauden tes ,
1 744 ,
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motu u m pl an eta ru m et
Theor i a motu u m l u n ae,
metar u m,
Theor i a motu s l u n ar,
1 74 4 ,
co
1 7 53 ,
are his chief works on astronom y ;
S es
et de
1 772 ,
l ettr es a u n e pr i n cesse d A l l emagn e su r qu el qu es suj ets de P hysi q u e
P hi l os ophi e, 1 7 7 0 , was a work which enj oyed great popularity
We proceed to mention the principal innov at i ons and inv entions
“
of Euler In his I n tr odu cti o ( 1 74 8 every analytical expression in
x , i e every expression m ade up of powers , logarith m s , trigono m etric
’
.
”
)
.
”
functions etc is called a function of
Som etim es Euler used
”
another d efin ti on of function namely the relation between
and expresse d in the —plane by any curve drawn freehand libero
”
m anus duetu
In m odified form these two rival definitions are
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,
x
,
i
,
x
x
.
y
,
“
y
,
1
.
,
traceable in all later history Thus Lagrange proceeded on the i dea
in v olved in the first definition Fourier on the idea involved in the
second
Euler treated trigonometry as a branch of analysis and consistently
treated trigonom etric values as ratios The term trigonom etric
function was introduced in 1 7 7 0 by Georg S i mon K l u gel ( 1 73 9—1 8 1 2 )
2
of Halle the author of a m athematical dictionary Euler d eveloped
and system atiz ed the m ode of writing trigonom etric form ulas taking
for instance the S inus to tu s equal to 1 He simplified form ulas by
the S imple expedient of designating the angles of a triangle by A B C
and the opposite S ides by a b c respectively Only once before have
we encountered this S imple device I t was used in a pamphlet pre
pared by Ri Rawl i n s on at Oxford so m etim e between 1 6 5 5 and
This notat i on was re i ntroduce d sim u l taneously with Euler by Thomas
Simpson in England We m ay add here that in 1 734 Euler use d the
“
notation f (x)
to indicate function of x that the use of e as the
F Kl e i
El m t m th m tik v hoh S t ndp k te a s I L eip z ig
9 8 p 43 8
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M C an t o r
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en ar
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p
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un
a
V o l I V , 1 90 8 , p 4 1 3
N a tu r e, V ol 94 , 1 9 1 5 , p 6 4 2
ci t
S ee F C aj ori i n
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E UL ER LAGRANGE AND LAP LACE
235
,
symbol for the natural base of logarith m s was introduced by him in
that in 1 7 50 he used S to denote the half su m of the sides of a
“
triangle that i n 1 7 55 he introduced 2 to signify su mmation that
in 1 7 7 7 he used i for
a notation used later by K F Gauss
We pause to rem ark that in Euler s ti m e Th o ma s S i mps o n ( 1 7 1 0
an able and self taught English m athem atician for m any years
professor at the Royal M ilitary Acade m y at Woolwich and author of
several text books was active in perfecting trigonom etry as a science
His Tri gon ometry London 1 748 contains elegant proofs of two
b)
:
B)
fo rm ulas for plane triangles (a + b)
c = cos %(A
:s i n § C and (a
—B )
c = s i n %(A
: cos %C which have been ascribed to the Ger m an as
who develope d them
t ron om er K arl B r and an M al tw ei de ( 1 7 7 4
m uch later The first formula was given in di fferent notation by I
Newton in his Un i versal A ri thmeti qu e; both form ulas are given by
Fr i edr i ch Wi l hel m O ppel in
E ul er laid down the rules for the transform ation of c o—
ordinates in
space gave a m etho dic analytic treatment of plane cur v es and of
surfaces of the second order He was the first to discuss the equation
of the second degree in three variables and to class ify the surfaces
represented by it By criteria analogous to those used in the classi
fi ca ti o n of conics he obtained five species
He de v ised a m ethod of
sol v ing biquadratic equations by assu ming x =
x/ i with
the hope that it would lead him to a general solution of algebraic
equations The m ethod of elim ination by sol v ing a series of linear
equations ( invented independently by E B éz ou t)
and the m etho d of
elim ination by sym m etric functions are due to him Far reaching
are Euler s researches on logarithm s Euler d efined logarithm s as
3
exponents thus abandoning the old view of logarith m s as term s of
an arith m etic series in one to one correspondence with ter m s of a
geom etric series This union between the exponential and logar i th mic
concepts had taken place so mewhat earlier The possibility of de
fining logarith m s as e xponents had been recogniz ed by John Wallis
in 1 6 8 5 by Johann B ernoulli in 1 6 94 but not till 1 74 2 do we find a
syste matic exposition of logarith ms based on this idea I t is giv en
in the introduction to Gardi ner s Ta bl es of L oga r i thms London 1 7 4 2
“
This introd uction is collected wholly fro m the papers of W i lliam
Jones Euler s influence caused the ready adoption of the new defi n i
tion That this view of logarithms was in every way a step in advance
has been doubted by som e writers Certain it is that it i nvol v es i n
ternal diffi culties of a serious nature Euler threw a stream of light
upon th e subtle subject of the logarithm s of negativ e and im aginary
numbers In 1 7 1 2 and 1 7 1 3 this subj ect had been discussed in a
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—9 4
G En est rom , B i bl i otheca ma thema ti ca , V ol 1 4 , 1 9 1 3
2
A v B rau n m ii hl , op ci t , 2 Teil , 1 9 0 3 , p 93 ; H
mathema ti ca , 3 S , V ol 1 4 , pp 3 4 8 , 34 9
3
10 2
S ee L E ul er , I n tr odu cti o, 1 7 4 8 , C h ap V I ,
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p 81
Wi el e i t n e r i n B i bl i otheca
1
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1
,
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O F M ATHE M ATIC S
H I STO RY
A
2 36
correspondence between G W Leibniz and Johann B ernoul li L eib
n iz m aintained that since a positive logari th m corresponds to a nu mber
larger than unity and a negative logarith m to a positive number less
than unity the logarith m of 1 was not really true but i magin ary ;
hence the ratio
havi ng no logari thm is itself imaginary
M oreover if there really existed
a
logarith
m
of
then
hal
f of it
1
—
would be the logarithm of
1 a conclusion which he considered
absurd The statements of Leibniz involve a double use of th e term
imaginary :( 1 )
in the sense of non —
existent ( 2 )
in the sense of a number
of the typ e x/ T I Johann B ernoulli m aintained that
1 has a
logarithm
Since dx : x = dx : —x there results by integration
—
=
= lo
l
log (x) og
x)
the
logarith
m
ic
curve
;
y
g x has therefore two
branches symm etrical to the y= axis as has the hyperbola The corre
spo n d en c e between Leibniz and Johann B ernoulli was first published
in 1 74 5 In 1 7 1 4 Roger Cotes developed in the P hi l os ophi cal Tran s
acti on s a n i mportant theore m which was republished in his H a r mon i a
men su r aru m
In m odern notation it is i ¢ = log (cos ¢ + t si n (b)
In the exponential form it was discovered again by Euler i n 1 748
Cotes was aware of the periodicity of the t rig o n om e t ri c functions
Had he applied this idea to his formula he m ight have anticipated
Euler by m any years in show ing that the logarithm of a number has
an infinite nu mber of d ifferent values A second discussion of the
logarithm s of negative numbers took place in a correspondence b e
tween young Euler and his revered teacher Johann B ernoulli in th e
x)
years 1 7 2 7
B ernoulli argued as before that log x = l og —
Euler uncovered the diflfic ul ti es and inconsistencies of his own and
Bernoulli s views without at that tim e being able to a dv ance a
satisfactory theory He showed that Johann B ernoul li s expression
1
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4
v
i
?
Be
which is in compatible with B ernoulli s claim that log
tween 1 7 3 1 and 1 74 7 Euler m ade steady progress i n the m astery of
relat i ons involving im aginaries In a letter of Oct 1 8 1 740 to
/
\
=
=
Johann B ernoulli he stated that y 2 cos x and y e
e
’
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.
x
,
I
,
-
,
2
were both integrals of the
d iff erential
dy
=0
+
y
2
equation
22
and were
equal to each other Euler knew the corresponding expression for
B oth expressions are given by hi m in th e M i scel l an ea B er ol i n en
si n x
He gave the
s i a 1 7 4 3 and aga i n in his I n tr odu cti o 1 7 4 8 Vol I 1 0 4
value 1 7 V
57 6 3 as early as 1 7 4 6 in a letter to Chris
tian Goldbach ( 1 6 90
but makes no reference here to the i n
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1
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“
H istory
l
t
the Expo n en t i a an d L og ari hmi c C on cep t s ,
—
o
t
l
2
1
1
t
o
n
hl
V
0
i ca n M a h M
,
y,
9 3 , p p 39 4 2
2
S ee F C aj o r i i n A m M a th M onthl y, V ol 2 0 , 1 9 1 3 , pp 44 4 6
1
S ee F C aj o ri
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of
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—
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A mer
A HI STORY OF M ATHE M ATIC S
2 38
G W Leibniz nor J akob and J ohann B ernoulli had entertained any
serious doubt of the correctness of the expression
1+
—
o
r
Gu i d G a ndi ( 1 6 7 1 1 74 2 )
of P isa went so far as to conclude from
this that
In the treatment of series Leibniz ad
v an c ed a m etaphysical m ethod of proof which held sway over the
1
minds of the elder B ernoullis and e v en of Euler
The tendency of
that reasoning was to j ust ify results which seem highly absurd to
followers O f Abel and Cauchy The looseness of treatm ent can
best be seen from examples The v ery paper in which Euler cautions
against divergent series contains the proof that
.
.
,
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.
.
1
1
as follows
0
n +n
2
+
—
I
f
.
n
it
l
“
—7
hese added gi e ero
has no hesitation to write 1
0
an d no one objected to such results excepting N i c ol a u s
B ernoulli the nephew of Johann an d Jakob Strange to say Euler
finally succee d e d in converting N i c ol au s B ernoulli to his own erroneous
views At the present tim e it is diffi cult to believe that Euler S hould
—
have confi dently written si n qS—z S i n
S in 3 2
2
4 sin
but such examples aff ord str iking illustrations of the want of
0
scientific basis of certain parts of analysis at that ti m e Euler s proof
of the bino mial form ula for negative and fractional exponents which
was widely reproduced in elem entary text books of the nineteenth
century is faulty A re m arkable developm ent due to Euler is what
he nam ed the hypergeo m etric series the su mm ation of which he
observed to be depen dent upon the integration of a linear diff erential
equation of the second order but it rem aine d for K F Gauss to point
out that for special values of its letters this series represented nearly
all functions then known
Euler gave in 1 7 7 9 a series for arc tan x di fferent from the series of
Jam es Gregory which he applied to the form ula 7r = 2 0 arc tan 1+
3
8 arc tan 7 9 used for computing
The series was published in 1 7 98
Euler reached re markable results on the summ ation of the reciprocal
powers of the natural nu mbers In 1 73 6 he had found the s u m of the
2
reciprocal squares to be 7r /6 and of the reciprocal fourth powers to
In an article of 1 743 which until recently has been gen
be
2
Euler finds the su ms of the reciprocal e v en powers
er al l y ov erlooked
of the natural numbers u p to and inclu di ng the 2 6 th power Later
he S howed the connection of coeffi cients occurring in these sum s with
B ernoullian num bers due to Jakob B ernoulli
th e
Eul er d eveloped the calculus of finite differences in the first chapters
t
v
z
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Eu l er
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R R e i ff , Geschi chte der U n end l i chen Rei hen , T u b i n g en , 1 8 8 9 , p 6 8
—
2
l
o
l
i
n
t
t
S
8
1
0
t
e
c
k
e
B
i
bl
i
o
t
h
ca
m
a
h
e
m
a
i
ca
V
P S a
,
9 7 8 , pp 3 7 60
, 3
,
1
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—
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.
EUL ER LAGRANGE
AN D
,
LA P LACE
2
39
of his I nsti tu ti on es cal cu l i difieren ti al i s and then deduced the differen
tial calculus from it He established a theorem on hom ogeneous func
tions known by his name and contributed largely to the theory of
diff erential equations a subject which had received the attention of
I Newton G W Leibniz and the B ernoullis but was still unde
v el oped
A C Clairaut Alexis Fontaine des B ertins ( 1 70 5
a n d L Euler about the sam e ti me observed criteria of integrability
but Euler in addition showed how to e mploy the m to determ ine i n
t eg ra t i n g factors
The principles on which the criteria rested involve d
som e degree of obscurity Euler was the first to make a systematic
study of S ingular solutio n s of differential equatio n s of the first order
I n 1 7 3 6 1 7 56 and 1 7 6 8 he considered the two paradoxes which had
puzzled A C Clairaut : The first that a solution may be reached by
differentiation instead of integration ; the second that a singular
solution is not contained in the general solution Euler tried to es
t abl i sh an a pr i ori rule for determ ining whether a solution is containe d
in the general solution or not Stim ulated by researches of Count
de Fagnano on elliptic integrals Euler established the celebrated
addition theorem for these integrals He i n vented a new algorith m
for continued fractions which he employed in the solution of the
indeterm inate equation ax+ by= c We n ow kno w that sub tantially
the sa m e solution of this equation was given 1 0 0 0 ye ars e arlier by
the H indus Euler gave 6 2 pairs of a m icable nu mbers of which 3
pairs were previously know n : one pair had been di s covered by the
1
P ythagoreans another by F e rmat and a third by D escartes
By
giving the factors of the nu mber
when n = 5 he pointed out
that this expression di d not always represent prim es as was supposed
“
by P F e rma t He first supplied the proof to Ferm at s theore m
and to a second theorem of F erma t which states that every pri me
of the form 4n + 1 is expressible as the s u m of two squares i n one and
“
only one way A third theorem Ferm at s last th eorem that
x +y = z
has no integral solution for values of n greater than 2
was prov ed by Euler to be correct when n = 4 and n = 3 Euler di s
cov ere d four theorem s which taken together m ake out the great law
of qua dratic reciprocity a l aw independently discovered by A M
2
Legen dre
In 1 73 7 Euler showed that the s u m of the reciprocals of all prim e
nu mbers is log (l og
thereby initiating a line of research on the
distribution of prim es which is usually not carried back further tha n
3
to A M Legen dre
In 1 74 1 he wrote on partitions of nu mbers pa r ti ti o n u me roru m
In 1 7 8 2 he published a discussion of the problem of 3 6 o ffi cers of s ix
d ifferent grades and fro m six different regi ments who are to be placed
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n
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c
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e
oo
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,
S ee B i bl i otheca mathemati ca , 3 S V ol 9 p 2 6 3 ; V ol 1 4 pp 3 5 1 —
3 54
2
L eip ig ,
O sw al d B a u m g a r t U eber das Q u ad r ati sche Rcc i pr oci tats g es etz
3
G En e s tr Om in B i bl i otheca mathema ti ca , 3 S , V o l 1 3 , 1 9 1 2 , p 8 1
1
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1 885
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A HI STORY OF M ATHE M ATIC S
2 40
in a square in such a way that in each row and colu m n there are six
offi cers all of diff erent gra des as well as of diff erent regi ments Euler
thinks that no solution i s ob ta inable when the order of the square is
of the form 2 m od 4 Arthur Cayley in 1 890 reviewed what had been
written ; P A M ac M ahon solved it in 1 9 1 5 I t is called the problem
“
of the Latin squares because Euler in his notation used n lettres
latines
Euler enunciated and proved a well known theorem giving
the relation between the nu mber of vertices faces and edges of cer
tain pol yhedra which however was known to R D escartes The
powers of Euler were directed also towards the fascinating subj ect
of the theory of probability in whi ch he solved som e diffi cult
proble ms
Of no little i mportance are Euler s labors in analytical m echanics
“
Says Whewell : The person who did mos t t o give to analysis the
generality and symmetry which are now its pride was also the person
1
He worke d out
who made m echanics analytical ; I m ean Euler
the theory of the rotation of a body around a fixed point established
the general equations of m otion of a free body and the general equation
of hydrodynam ics He solved an imm ense num ber and variety of
m echanical problem s which arose in his m ind on all occasions Thus
“
on reading Virgil s l i nes The anchor drops the rushing keel is staid
he could not help inquiring what would be th e ship s m otion in such
a case About the sam e tim e as D aniel B ernoulli he published the
P r i n ci pl e of the Con s erva ti on of A r eas and defended the principle of
least action advanced by P M aup er tiu s He wrote also on tides
and on sound
Astronomy owes to Euler the m ethod of the variation of arbitrary
constants By it he attacked the problem of perturbations explain
i n g in case of two planets the secular variations of eccentric i ties
nodes etc He w as one of the first to take up with success the theory
of the m oon s m otion by giv ing approxi mate solutions to the proble m
of three bodies
He laid a sound basis for the calculation of tables
of the m oon These researches on the m oon s m otion which captured
two prizes were carried on while he was blind with the assistance of
his sons and two of his pupils His M echani ca s i ve motu s sci enti a
Vol I 1 73 6 Vol II 1 74 2 is in the language of
a n a l yti ce expos i ta
“
Lagrange the first great work in which analysis is applied to the
science of m ovem ent
P rophetic was his study of the m ovements of the earth s pole
He
showed that if the axis around which the earth rotates is not coincident
w ith the axis of figure the axis of rotation will re v ol v e about the axis
of figure in a pre d ictable period On the assu mption that the earth
is perfectly rigid he showed that the period is 30 5 days The earth
is now known to be elastic From observ ations taken in 1 884—5
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W Whew ell
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p 363
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the I ndu cti ve S ci en ces , 3 rd E d
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Vo l
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I , N ew York ,
A HI STORY OF M ATHE M ATIC S
2 42
m ission
to the Academy of Sciences In 1 754 he was m ade per manent
secretary of the French Acade my D uring the last years of his life
he was m ainly occupied with the great French encyclop aedia which
was begun by D enis D iderot and himself D Al emb er t declined i n
1 7 6 2 an invitation of Catharine II to undertake the e d ucation of her
son Frederick the Great pressed him to go to B erlin He m ade a
visit but declined a perm anent residence there In 1 743 appeared
his Tr ai té de dyn ami qu e founded upon the important general principle
bearing his nam e :The impressed forces are equivalent to the eff ective
forces D A l emb er t s principle seem s to have been recogniz ed before
h i m by A Fontaine and I n som e m easur e by Johann B ernoulli and
I Newton D Al emb er t gave it a clear m ath ematical for m and made
num erous applications of it I t enabled the la w s of m otion an d the
reasonings depending on them to be represen t ed in the m ost general
form in analytical language D Al emb er t applie d it in 1 744 in a
treatise on the equilibrium and m otion of fluids in 1 746 to a treatise
on the general causes of winds which obtained a priz e from the B erlin
Academy In both these treatises as also I n one of 1 7 4 7 d iscussing
the fa m ous proble m of vibrating Chords he was led to partial differ
He was a leader a mong the pioneers in the study of
en ti al equat i ons
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2
2
such equations
To the equation
.
of vibrating chords he gave as
7
2
33
1 $
2
2,
aris ing in the problem
general solution
—at)
the
,
1
,
,
and showed that there is only one arbitrary function if y be supposed
to vanish for x = 0 an d x = l Daniel B er n oulli starting with a par
t i c u l ar i n tegral g i ven by B rook Taylor showed that this differential
equation is satisfied by the trigonom etric series
,
.
,
,
= a S In
y
— cos
7r x
-
l
.
—
sin
+B
7rt
-
l
2 7r x
.
l
c os
2 7r t
and claimed this expression to be the m ost general solution Thus
“
Daniel B ernoulli was the first to introduce F ou ri e r s series into
physics He claim ed that his solution being compoun d ed of an i n
finite nu mber of tones and overtones of all poss i ble intensities was
a general solution of the proble m Euler den i ed its generality on
the groun d that if true the doubtful conclusion would follo w that
the above series represents any arb i trary functio n of a variable
These doubts were dispelled by J Fourier J Lagrange proceeded to
find the su m of the above ser i es but D A l emb er t obj ected to his
1
process on the ground that it involved div ergent series
A m ost beautiful result reached by D Al emb e rt with a i d of his
principle was the co mplete solution of the problem of the precess i on
of the equinoxes which had baffled the talents of the best m i nds
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.
EULER LAGRANGE AND LAP LACE
2 43
,
He sent to the F rench Academ y in 1 7 4 7 on the sa me day with A C
Clairaut a solution of the proble m of three bodies This had become
a question of universal interest to m athe maticians i n which each
v ied to outdo all others
The proble m of two bodies requiring the
d eter m ination of their m otion when they attract each other with
forces in v ersely proportional to the square of the distance betwee n
“
them had been co mp l etely sol v ed by I Newton The proble m of
three bodies asks for the motion of three bodies attracting each
other according to the law of gra v itation Thus far the co mplete
solution of this has transcen d ed the power of analysis The general
di fferential equations of m otion were stated by P S Laplace but
”
the difficulty arises i n their integration The solutions giv en at
that ti me are m erely convenient m ethods of approxi mation in special
cases when one body is the sun disturbing the m otion of the moon
around the earth or where a planet m oves under the influence of the
sun and another planet The m ost important eighteenth century
researches on the proble m of three bo dies are due to J Lagrange In
1 7 7 2 a priz e was awarded hi m by the P aris Acade m y for his E ssa i
su r l e pr obl eme des tr oi s cor ps
He shows that a co mplete solution of
the proble m requires only that we know every mo m ent the sides of
the triangle form ed by the three bodies the solution of the triangle
d epe n ding upon t w o di fferential equations of the second order and
one differential equation of the third He found particular solutions
when the triangles remain all si m ilar
In the discussion of the m eaning of negative quantities of the
fundam ental processes of the calculus of the logarith m s of co mplex
numbers and of the theory of probability D A l emb e r t paid so m e
attention to the philosophy of m athe m atics In the calculus he
favored the theory of lim its He looked upon infinity as nothing but
a limit which the finite approaches without ever reaching it His
criticisms were not al w ays happy When stude n ts were halted by
“
the logical difficulties of the calculus D Al emb e r t would say Alle z
en avant e t la foi vous viendra
He argued that when the prob
ability oi an event is v ery s mall it ought to be taken 0 A coin is to
be tossed 1 0 0 times and if head appear at the last trial and not before
100
A shall pay B 2 crow ns By the ordinary theory B S hould giv e A
which should not be argues D Al e mb e rt b e
1 cro w n at the start
cause B will certainly lose This v iew was taken also by Count de
B u ff on D Al emb ert raised other obj ections to the principles of
probability
The naturalist Comte de B ufi on ( 1 70 7
wrote an E ssai
d a r i thméti qu e mor al e 1 7 7 7
In the study o i the P etersburg proble m
he let a child toss a coi n 2 0 84 ti mes which produced 1 0 0 5 7 crowns ;
there were 1 0 6 1 gam es which produced 1 crown 494 which produce d
1
2 crowns and so o n
He was one of the first to e mphasiz e the d esir
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Fo r
re feren c es , see
I T o d hu n t e r , H i s tor y
.
f
o
Theor y
f
o
P r oba bil i ty, p 3 4 6
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.
A HI STORY OF M ATHE M ATIC S
2 44
ability of verifying the theory by actual trial He also introduced
what is called local probability by the consideration of problems
that require th e aid of geometry Some studies along thi s line had
—
h
n
r
t
t
been carried on earlier by Jo A bu hn o ( 1 6 58 1 73 5)
and Thomas
Simpson in Englan d Count de B uff on derived the probability that
a needle d ropped upon a plane ruled with equidistant parallel lines
will fall across one of the lines
The probability of the correctness of judgments determined by a
m ajority of votes was exam ined m athem atically by Jean —
A n toi n e
His general conclusions
N i col as C a r i tat de C ond or cet ( 1 74 3
are not of great im portance ; they are that voters m ust be enlightened
1
men in order to ensure our confidence in their decisions
He held
that capital punishm ent ought to be abolished on the ground that
however large the probability of the correctness of a single decision
there will be a large probability that In the course of many decisions
1
som e innocent person will be condemne d
Al e xi s C l au d e C l a ir au t ( 1 7 1 3 1 7 6 5)
was a youthful prodigy He
read G F de l H ospi tal s works on the infinitesimal calculus and on
conic sections at the age of ten In 1 7 3 1 was published his Recher ches
s u r l es c ou r bes a dou bl e cou r bu r e which he had ready for the press
when he was six teen It w as a work of remarkable elegance and se
cured his admission to the Academy of Sciences when still under legal
age In 1 73 1 he gave a proof of the theorem enunciated by I Newton
that ev ery cubic is a projection of one of five divergent parabolas
Clairaut form ed the acquaintance of P i er re L ou i s M orean de M au per
—
s
ti u ( 1 6 98 1 7
whom he acco mpanied on an expedition to Lapland
to m easure the length of a degree of the m eridian At that tim e the
shape of the earth was a subj ect of serious disagree m ent I Newton
and C H u yg en s h a d concluded from theory that the earth was flat
tened at the poles About 1 7 1 2 Jea n Domi n i qu e Cassi n i ( 1 6 2 5—1 7 1 2 )
—
m easured an arc exten d ing
and his son J acqu es C ass i n i ( 1 6 7 7 1 7 56 )
from D unkirk to P erpignan and arrived at the startling result that
the earth is elongated at the poles To decide between the conflicting
opinions m easurements were renewed M aup er ti u s earned by his
“
work in Lapland the title of earth flat ten er by disproving the
Cassinian tenet that the earth was elongated at the poles and showing
t hat Newton was right
On his return in 1 743 Clairaut published
a work Theori e de l a figu r e de l a Ter r e which was based on the results
of C M aclaurin on homogeneous ellipsoi ds I t contains a rem arkable
theorem nam ed after Clairaut that the s u m of the fractions ex
press i ng the ellipt i c i ty and the I ncrease of gravity at the pole 15 equal
to 2 —t imes the fraction expressing the centr i fugal force at the equator
the unit of force being represente d by the force of grav ity at the
equator This theorem is independent of any hypothes i s with respect
to the law of densities of the successiv e strata of the earth It em
I T dh n t
Hi t y f Theor y f P r b C h p ter 7
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1
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A HI STORY OF M ATHE M ATIC S
246
”
To the emperor s fur ther question how he attained this mastery he
“
said L i ke the celebrated P ascal by my own self
In his Cos mol ogi cal L etters he m ade som e remarkable prophecies
regarding the stellar system He entered upon plans for a m athe
mati cal symbolic logic of the nature once outlined by G W L eibniz
In m athem atics he m ade several discoveries which were ex tended
and overshadowed by his great contemporaries His first research
on pure m athem atics develope d in an infinite series the root x of the
m
equation x + px = q Since each equation of the form
m
can be reduced to x + px = q in two ways one or the other of the two
resulting series was always found to be convergent and to give a
value of x Lam bert s results stim ulated L Euler who extended the
m ethod to an equation of four term s and particularly J Lagrange
= 0 can be expressed
who found that a function of a root of a
by the series bearing his nam e In 1 7 6 1 Lambert comm un i cated to
the B erlin Academy a m em oir (published
in which he pro v es
rigorously that W is irrational I t is given in simplified form in Note IV
2
of A M Legen dre s Geometri c where the proof is exten d e d to 71
"
Lambert proved that if x is rational but not z ero then ne i ther e
nor tan x can be a rational nu mber ; s i nce tan 1r /4 = 1 it follows that
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7r
4
or
7T
cannot be
Lambert s proofs rest on the express i on
r a t Ion al .
’
1
for e as a continued fraction giv en by L Euler w ho in 1 7 3 7 had sub
2
There were at this
s t a n t i al l y shown the irrational i ty of e and e
tim e so m any circle squarers that i n 1 7 7 5 the P aris Aca de m y found it
necessary to pass a resolution that no m ore solutions on the qua drature
of the circle should be exam ined by its offi cials This resolution ap
plied also to solutions of the duplication of the cube and the trisection
of an angle The con v iction had been growing that the solution of
the squaring of the circle was impossible but an irrefutable proof
was not discovered until over a century later Lambert s Fr eya P er
s pecti ve 1 7 59 and 1 7 7 3 contains researches on descriptive geom etry
and enti tle him to the honor of being the forerunner of M onge In
his e ff ort to sim pl i fy the calculation of com etary orbits he was led
geom etrically to som e rem arkable theorem s on conics for instance
thi s : If in two ellipses having a co mm on m ajor axis we take t w o
such arcs that their chords are equal and that also the su m s of the
ra di i vectores d rawn respectively from the foci to the extrem i ti es of
these arcs are equal to each other then the sectors form ed in each
ellipse by the arc an d the two ra dii vectores are to each other as the
2
square roots of the parameters of the ell i pses
La mbert elaborate d the subject of hyperbolic functions wh i ch he
designated by si n h x cosh x etc He was however not the first to
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R C A rc hib al d i n A m M a th M on thl y, V o l
2
M C hasl es, Geschi chte der Geometr i e, 1 8 39 , p
1
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21,
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1 9 1 4,
1 83 .
p
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2 53
.
EULER LAGRANGE AND LAP LACE
247
,
introduce them into trigonometry That honor falls upon Vi n cenzo
1
Ri oca ti ( 1 7 0 7
a son of Jacopo R i cca ti
In 1 7 70 Lam bert published a 7 place table of natural logarith ms
for nu mbers 1 —1 0 0 In 1 7 78 one of his pupils J ohann K arl S chu l ze
published ex tensive tables which included the 4 8 place table of nat
ural logarithms of prim es and many other numbers up to
which had been computed by the D utch artil lery offi cer Wolf r am
A feat even more remarkable than Wolfram s was the computation
of the common logarith ms of numbers 1 —1 0 0 and of all prim es from
1 0 0 to 1 1 0 0 to 6 1 places by A br aha m S har p of Yorkshire who was
some time assistant to Flam steed at the English Royal Observatory
They were published in Sharp s Geometry I mprov d 1 7 1 7
—
o
L
a
n
d
e
n
1
1
was an English m athem atician whose
h
n
J
( 7 9 1 7 90 )
writings se rved as the starting point of investigations by L Euler
J Lagrange and A M Legendre Landen s capital discovery con
t ai n ed in a me m oir of 1 7 5 5 was that every arc of the hyperbola is
immediately rectified by m eans of two arcs of an ellipse In his
residual analysis he attem pted to obviate the m etaphysical d iffi
c u l ti es of fl u xi on s by a d opting a purely algebraic m etho d
J La
grange s C al cu l des Fon cti ons is based upon this idea Lan d en S howed
ho w the algebraic expression for the roots of a cubic equation could
be deriv ed by application of the differential and integral calculus
M ost of the tim e of this suggestive writer was S pent in the pursuits
of active l ife
Of influence in the teaching of m athe m atics in England was C ha rl es
H u tton ( 1 73 7
for many years professor at the Royal M ilitary
Academ y of Woolwich In 1 7 8 5 he published his M athemati cal Tabl es
and in 1 7 95 his M a themati cal and P hi l osophi cal D i cti on a ry the best
work of its kind that has appeared i n the English language His
El emen ts of C on i c S ecti on s 1 7 8 9 is re m arkable as being the first work
in which each equation is rendered conspicuous by being printed in
2
a separate line by itself
It is well known that the Newton Raphso n m ethod of appr oxim a
tion to the roots of nu merical equations as it was handed do w n from
the sev enteenth century lab ored un d er the defect of insecurity in
the process so that the successive corrections did not always yield
results converging to the true value of the root sought The rem oval
of this defect is usually attributed to J Fourier but he was anticipated
half a century by J Raym M ou rrai l l e i n his Tr ai té de l a r es ol u ti on
des equ a ti on s en gen er al M arseille e t P aris 1 7 6 8
M ou r r a ill e was for
fourteen years secretary of the acade m y of sciences in M arseille ; later
he becam e m ayor of the city Unlike I Newton a n d J Lagrange
M ou rrai ll e and J Fourier introduced also geo m etrical cons i d erations
M ou rrail l e concluded that security is insured if the first appro xim ation
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Vol I V ,
V ol I V ,
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1 90 8 ,
p 41 1
1 90 8 , p 46 5
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A HI STORY OF
2 48
M ATHE M ATIC S
is so selected that the curve is convex toward the axis of x for th e
interval between a and the root He shows that this condition is
1
suffi cient but not necessary
In the eighteenth century proofs were giv en of D escartes Rule of
Signs which its discoverer h ad enunciated without de monstration
G W Leibniz had pointe d out a line of proof but did not actually
—
r
ea
1
J
n
P
es
t
e
t
1
published at P aris in his
give it In 6 7 5
( 6 4 8 1 6 90 )
E l emen s des ma théma ti qu es a proof which he afterwards acknowledged
to be insuffi cient In 1 7 2 8 J ohann A n dreas S egn er ( 1 70 4—1 7 7 7 )
pub
l i sh e d at Jena a correct proof for equations ha v ing only real roots
In 1 7 56 he gav e a general d em onstration based on the consid eration
that m ult iply i ng a pol ynom ial by (x —a )increases the num ber of
var i ations by at least one Other proofs were given by Jean P au l de
I s aa c M i l n er
Fr i edr i ch Wi l hel m S tubn er
Gu a de M al ves
a
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A br aha m Gotthelf K as tn er
Gr u n er t
K F Gau ss
ber of pos i tive roots falls short
,
Wari n g
Edw ard
J
.
A
.
Gauss showed that if the n u m
of the nu mber of v ar i at i ons i t d oes so
by an even nu mber E Laguerre later extende d the rule to poly
a n d to i n
n om i al s with fractional and inco mm ensurable exponents
2
finite series
I t was established by D e Gua de M alves that the
absence of 2 m successive terms indicates 2 m im aginary roots while
the absence of 2 m+ 1 successiv e term s in di cates 2 m+ 2 or 2 m i magin
ary roots according as the two term s between which the deficiency
occurs hav e like or unl i ke signs
Edw a r d W a rin g ( 1 7 3 4—
1 7 98 )
was born in Shrewsbury studied at
M agdalene College Cambridge was senior wrangler i n 1 7 5 7 and
He published M i s
L u cas ia n professor of m athe m atics since 1 7 6 0
c el l a n ea a n al yti ca in 1 7 6 2 M edi tati on es al gebr ai cw in 1 7 7 0 P r opr i eta ti s
a l gebr ai car u m cu rva r u m in 1 7 7 2
and M edi ta ti on es a n al yti cce in 1 7 7 6
These works contain m any new results but are diffi cult of co mpre
h en s i o n on account of his brevity and obscurity of exposition
He is
said not to ha v e lectured at Cambridge his researches be i ng thought
u nsuited for presentation in the for m of lectures
He a dm itted that
he ne v er heard of any one in England outsid e of Cam bridge who had
read an d understoo d his researches
In his M edi ta ti on es al gebra i cce are som e n ew theorems on nu mber
Forem ost am ong these is a theorem di scovered by his friend John
and univ ersally known as Wilson s theorem
Wi l son
“
Waring gives the theore m known as Waring s theorem that eve ry
i nteger is either a cube or the su m of 2 3 4 5 6 7 8 or 9 cubes either
or 1 9 fourth powers ; th i s has
a fourth power or the su m of 2 3
nev er yet been fully dem onstrate d Also without proof is giv en the
theore m that e v ery even integer is the su m of two primes and every
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S ee F C aj o ri i n B i bl i otheca ma thema ti ca , 3 rd S , V ol 1 1 ,
2
Fo r re fe re n ces t o t he p u bl ic a t i o n s o f t he se w r i t e rs , see F
Coll ege P u bli cati on , G e n er al S e ri es N o 5 1 , 1 9 1 0 , p p 1 8 6 , 1 8 7
1
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132
—1 3 7
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C aj o r i i n Col or ado
O F M ATHE M ATI C S
HI STORY
A
50
points or singu lar li nes planes etc at i nfi nity occasion the with
draw a l to i nfinity of certain of t he intersection points ; and this at a
1
tim e when the nature of such S ingularities had not been dev eloped
Lou i s Ar b og as te ( 1 759—1 80 3)
of Alsace was professor of m athe
m a ti c s at S trasburg His chief work the C al cu l des Deri vati on s 1 8 0 0
gives the m ethod known by his nam e by which the successive coeffi
c i en t s of a develop m ent are derived from one another when the ex
pression is complicated A D e M organ has pointed out that the
true nature of derivation is differentiation accompanied by integration
In this book for the first time are the sym bols of operation separated
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from those of quantity
dr
The notation n for
is due to him
(E
M ari a G ae tan a Agn e s i ( 1 7 1 8—
1 7 99)
of M ilan distin g uished as a
linguist mathem atician and ph ilosopher filled the mathematical
chair at the University of B ologna duri ng her father s sickness
A g n e si was a so m na mbulist
Several tim es it happened to her that
she went to her stu dy while in the som nambulist state m ade a light
and solv ed som e problem she had left incomplete when awake In
the m orning she was surprised to find the solution carefully worked
2
out on pape r
In 1 748 she published her I n sti tu zi oni A n al i ti che
“
which was translated into English in 1 80 1 The witch of Ag n esi
“
2 = 2
or Versiera is a cubic curve x y a (a —y)
treated in Ag n esi s I n
2 ~ 2
3
=
s ti tu zi on i
but given earlier by P Ferm at in the form ( a x )
y a
The curve was discussed by Guido Grandi in his Qu adr atu r a ci rcu l i
et hyper bol ce
In two letters from Grandi to
P isa 1 7 0 3 and
Leibniz in 1 7 1 3 q rves resembling flowers are discussed ; in 1 7 2 8
Grandi published at Florence his Fl or es geometri ci He considered
curves in a plane of the type p = r sinn w and also curves on a sphere
Recent studies along this line are due to B odo Habenicht
E W Hyde
H Wi el ei tn er
The leading eighteen th century historian of m athe matics was J e a n
—
Et i e n n e M on tu cl a ( 1 7 2 5 1 7 99)
who published a H i s toi re des mathe
mati qu es in two volu m es P aris 1 7 58 A second edition of these two
volumes appeared in 1 7 99 A third volu me written by M on tu cl a
was partly printed when he died ; the rest of it was seen through the
press by the astronomer Jo s e ph J er Ome l e Fran goi s d e L al an d e
who prepared a four th volum e m ainly on the history of
( 1 73 2
4
astronomy
one of the greatest m ath e
J o s e ph L ou i s Lag r an g e ( 1 73 6
m a ti cian s of all tim es was born at Turin and d ied at P aris He was
of French extraction His father who had charge of the Sardinian
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Whi t e i n
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B u l l A m M ath S oc , V ol 1 5 , 1 909 , p 33 1
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2
L I n ter méd i a i r e des ma théma ti ci en s , Vo l 2 2 , 1 9 1 5 , p 2 4 1
3
G L ori a , E ben e C u r ven ( F S c h ii t t e , I , 1 9 1 0 , p 7 9
’
4
For d e t ai l s o n o t h e r m a t hema t i c a l hi s t o ri an s , se e S G u n t her s
C an to r , op c i t , V o l I V , 1 90 8 , pp 1 —
36
1
H S
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c hap t er
in
E UL ER LA G RAN G E
AN D
,
military
LAP LACE
chest was once wealthy but lost all he had in speculation
Lagrange considered this loss his good fortune for otherwise he might
not have made m athematics the pursuit of his life While at the
college in Turin his genius did not at once take its true bent Cicero
and Virgil at first attracted him more than Archimedes and Newton
He soon came to ad mire the geo metry of the ancients but the perusal
of a tract of E Halley roused his enthusiasm for the anal ytical method
in the d evelop m ent of which he was destined to reap undying glo ry
He now applied hi mself to mathematics and in his seventeenth year
he becam e professor of mathem atics in the royal mili ta ry academy at
Turin Without assistance or guidance he entered upon a course of
study which in two years placed him on a level with the greatest of
his contemporaries With aid of his pupils he established a society
which subsequently developed in to the Turin Academy In the first
five volum es of its transactions appear m ost of his earlier papers
At th e age of nineteen he comm unicated to L Euler a general m ethod
“
of dealin g with isoperimetrical problem s know n n ow as the Cal
cui us of Variations This comm anded Euler s liv ely ad m iration and
he courteously withh eld for a tim e fro m publication so m e researches
of his own on this subj ect so that the youthful Lagrange m ight com
Lagrange did quite
pl e t e hi s investigations and clai m t he inv ention
as m uch as Euler tow ards the creation of the Calculus of Variations
As it came from Euler it lacked an analytic foundation and this
Lagrange supplied He separated the principles of this calculus fro m
geom etric considerations by which his predecessor had derived them
Euler had assum ed as fixed the lim its of the integral i e the extrem
ities of the curve to be determ ined but Lagrange rem oved this re
striction and allowed all c o ordinates of the curve to vary at the sam e
“
tim e Euler introduced in 1 7 66 the nam e calculus of variations
and did m uch to improve this science along the li n es m arked out by
Lagrange Lagrange s investigations on the calculus of variations
were published in 1 7 6 2 1 7 7 1 1 7 88 1 7 97 1 8 0 6
Another subject engaging the attent i o n of Lagrange at Turin was
the propagation of sound In his papers on this subj ect in the M i s
cel l an ea Tau r i n en s i a the young m athe m atician appears as the critic
of I Newton and the arbiter between Euler and D Al emb e r t By
considering only the particles which are i n a straight line he reduced
the problem to the sam e partial d iff erential equa t ion that represent s
the motions of vibrating strings
Vibrating strings had been discussed by Brook Taylor J ohann
B ernoulli and his son D aniel by D Al emb e r t and L Euler In solvi ng
the partial diff erential equations D Al emb er t restricted him self to
f unctions which can be expanded by Taylor s series while Eu l er
thought that no restriction was necessary that they could be arbi
t ra ry discontinuous
The proble m was taken up wi th great skill by
Lagrange who introduce d new points of view but decide d in fav or of
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Q
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A HI STORY O F M ATHE M ATIC S
2 52
Euler
Later de Condorcet and P S Laplace stood on the side of
D A l emb er t S ince in their judgm ent so m e restriction upon the arbi
t r ary functions was necessary
From the m odern point of view
neither D A l embe r t nor Euler was wholly in the right : D Al emb er t
insisted upon the needless restriction to functions with a l imitless
number of derivativ es while Euler assum ed that the diff erential and
1
integral calculus could be applied to any arbitrary function
I t now appears that Daniel B ernoulli s claim that his solution was
a general one (a claim disputed by D A l emb er t J Lagrange and L
Euler)
was fully j ustified The problem of vibrating strings s timu
lated the growth of the theory of expansions according to trigonom etric
functions of m ultiples of the argum ent H B urkhardt has pointed
out that there was also another line of growth of this subj ect namely
the growth in connection with the problem of perturbations where
L Euler started out with the develop m ent of the reciprocal distance
of two planets according to the cosine of m ultiples of the angle b e
tween their radii v ec t o ri s
By constant application d uring nine years Lagrange at the age
of twenty S ix stood at the su mm it of European fam e B ut his intens e
studies had seriously weakened a constitution nev er robust and though
his physicians in duced him to take rest and exercise his nervous
syste m nev er fully recovered its tone and he was thenceforth subject
to fits of m elancholy
In 1 7 6 4 the French Academy proposed as the subject of a priz e
the theory of the libration of the m oon I t dem ande d an explanation
on the principle of universal grav itation why the m oon always turns
with but slight v ariations the sam e phase to the earth Lagrange
secured the priz e This success encouraged the Aca dem y to propose
f or a priz e the theory of the four satellites of Jupiter — a problem of
S ix bodies
m ore diffi cult than the one of three bo d ies previously
treated by A C Clairaut D A l emb er t and L Euler Lagrange over
cam e the diffi culties by m ethods of approximation Twenty four
years afte rwards this subj ect was carrie d further by P S Laplace
Later astrono m ical investigations of Lagrange are on com etary per
t u rb a t i on s ( 1 7 7 8 and 1 7 8
His researches
a n d on Kepler s proble m
on the problem of three bo dies has been referred to previ ously
B eing anxious to m ake the personal acquaintance of leading m athe
mat i ci an s Lagrange visited P aris where he enjoyed the stim ulating
delight of conversing with A C Clairaut D Al emb er t de Condorcet
the Abb é M ar i e and others He had planned a visit to London but
he fell dangerously ill after a dinner in P aris and was compel led to
return to Turin In 1 7 6 6 L Euler left B erlin for S t P etersburg and
he pointed out Lagrange as the only man capable of filling the place
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Fo r de t ai l s see H B u rk ha r d t s E n twi ckl u n g en n a ch os ci l l i r end en Fu n k ti on en
u n d I n tegr a ti on der D ifi er en ti a l gl ei chu n g en d cr ma themati s che n P hys i k
L ei p i g ,
Thi s i s an e xha ust i ve a n d v a l u a b l e hi s tory o f t hi s t op i c
1 90 8 , p 1 8
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A HI STORY OF M ATHE M ATIC S
2 54
self evident Other proofs of this were given by J R Argand K F
Gauss and A L Cauchy In a note to the above work Lagrange uses
Fermat s theore m and certain suggestions O f Gauss in eff ecting a com
m
algebraic
solution
of
any
bino
ial equation
l
e
t
e
p
In the B erlin M émoi res for the year 1 7 6 7 Lagrange contributed a
paper S u r l a r esol u ti on des equ ati on s nu meri qu es He explains the
separation of the real roots by substituting for x the term s of the
progression 0 D 2 D
where D m ust be less than the least dif
ference between the roots Lagrange suggested three ways of com
puting D : One way in 1 7 6 7 another in 1 7 95 and a third in 1 798 The
first depends upon the equation of the squared differences of the roots
of the giv en equation E Waring before this had derived this i m
portant equation but in 1 7 6 7 Lagrange had not yet seen Waring s
writings Lagrange fin ds equal roots by computing the highest com
m on factor between f (x)
and f (x) He proceeds to develop a new
mode of approxi m ation that by continued fractions P A Cataldi
had used these fractions in extracting square roots Lagrange enters
upon greater deta i ls in his A ddi ti on s to his paper of 1 7 6 7 Unlike
the older m ethods of approxim ation Lagrange s has no cases of
failure
Cette m éthode me laisse ce m e semble rien a d ésirer yet
though theoretically perfect it yields the root in the form of a con
t i n u ed fraction which is un d esirable in practice
While in B erlin Lagrange published several papers on the theory
of nu mbers In 1 7 6 9 he gav e a solution i n integers of indete rm inate
equations of the second d egree which resembles the Hin d u cyclic
“
me thod ; he was the first to prove in 1 7 7 1
Wilson s theore m enun
c i a t ed by an English m an
John Wilson and first published by E
Waring in his M edi ta ti on es A l gebr ai cce; he inv estigated in 1 77 5 under
f
what conditions t 2 and i 5
1 and
3 hav ing been discussed by
L Euler )
are quadratic resi dues or non resi dues of odd prim e n um
bers q ; he prov e d in 1 770 B achet de M éz i riac s theorem that every
integer is equal to the s u m of four or a less number of squares He
proved Ferm at s theorem on x + y = z for the case n = 4 also F e r
2
2= 2
m at s theorem that if a + b c then ab is not a square
In his m em oir on P yramids 1 7 73 Lagrange m ade considerable use
of determ inants of the third order and dem onstrated that the square
of a determ inant is itself a determ inant He never however dealt
explicitly and directly with determ inants ; he siIn pl y obtained acci
d enta lly identities which are now recogniz ed as relations between
determ inants
Lagrange wrote m uch on d ifferential equations Though the sub
j ec t of contemplation by the greatest m a them aticians (L Euler
D A l emb e r t A C Clairaut J Lagrange P S Laplace )yet m ore
than other branches of m athem atics do they resist the system atic
application of fi xed m etho d s a n d princ ipl es The subject of S ingular
s olutions which ha d been taken u p by P S Laplace in 1 77 1 a n d 1 7 7 4
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EULER LAGRANGE AND LAP LACE
2 55
,
was investigated by Lagrange who gave the derivation of a S ingular
solution from the general solution as well as from the diff erential
equation itself Lagrange brought to vie w the relation of singular
solutions to envelopes Nevertheless he failed to rem ove all mystery
surrounding this subtle subj ect An inconsistency in his theorem s
caused about 1 8 70 a complete reconsideration of the entire theory of
singular solutions Lagrange s treatment is given in his C al cul des
—
Fon cti ons Lessons 1 4 1 7
He generaliz ed Euler s researches on tota l
differential equations of two variables and of the nin th order ; he
gav e a solution of partial differential equatio n s of the first order (B erl i n
and spoke of their singular solutions ex
M emoi rs 1 7 7 2 and
tending their solution i n M emoi rs of 1 7 7 9 and 1 78 5 to equations of
any number of variables The M emoi r s of 1 7 7 2 and 1 7 74 w ere refi ned
—
m
m
in certain points by a young athe atician P aul Charpit
1 7 8 4)
whose m ethod of solution was first printed in Lacroix s Trai té du
cal cu l 2
Ed P aris 1 8 1 4 T II p 54 8 The discussio n on partial
differential equations of the second order carried on by D Al emb ert
Euler and Lagrange has already been referred to in our a ccoun t of
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D A l emb er t
Wh ile in B erlin ,
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Lagrange wrote the M écani qu e A nal yti qu e the
greatest of his works (P aris
From the principle of vi rtual
velocities he deduced with aid of the calculus of variations the whole
system of m echanics so elegantly and harm oniously th at it may fi tly
“
be calle d in Sir Willia m Rowan Ham ilton s words a kind of sc i en
t ifi c poem
It is a most consu mm ate example of analytic generality
“
Geom etrical figures are nowhere allowed
On n e trouvera point de
figures dans cet ouvrage (P reface )The two divisions of m echanics
—statics and dynamics—are in the first four sections of each carried
out analogously and each is prefaced by a historic S ketch of principles
Lagrange formulated the principle of least action In their orig inal
form the equations of m otio n involve the c o ordinates x y z of the
diff erent particles m or dm of the system B ut x y z are in general
not independent and Lagrange introduced in place of them any
variables 5 Il f) whatever determ ining the position of the point at
“
the time These generaliz ed c o ordinates may be taken to be inde
pendent The equations of m otion may now assu m e the form
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d dT
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or when E p
respect to f p
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75
7
(
—
dT
8?
L
“
P
are the partial diff erential
of one and the sa m e functio n
d dT
dT
d
V
—
E
ff
L
“
c oefli c i en ts
V, then
w it h
the form
E
The latter is par excel l en ce the Lagrangian form of the equations of
motion Wi th Lagrange originate d the rem ark that m echanics may
'
dt dé
.
O F M ATHE M ATI C S
HI STORY
A
2 56
be regarded as a geom etry of four dim ensions To hi m falls the honor
of the introduction of the potential into dynamics Lagrange was
anx ious to have his M écani qu e A n al yti q u e published in P aris The
work was ready for print in 1 7 86 but not till 1 7 88 could he find a
publisher and then only with the condition that after a few years
he would purchase all the unsold copies The work was edited by
A M Legendre
After the death of Frederick the Great men o f science were no
longer respected in Germ any and Lagrange accepted an inv itation
of Louis XVI to m igrate to P aris The French queen treated him
with regard and lodging was procured for h im in the Louv re B ut
he was sei z ed with a long attack of melancholy which d estroyed his
taste for m athem atics For two years his printe d copy of th e M e
—
can i qu e fresh fro m the press
the work of a quarter of a century
lay unopened on his desk Through A L Lavoisier he became i n
ter es t ed in che m istry which he found
as easy as algebra
The
disastrous crisis of the French Revolution aroused hi m again to ac
t i v i ty
About this time the young and acco mplished daughter of the
astronomer P C Le monnier took compassion on the sad lonely
L agrange and insiste d upon m arrying him Her dev otion to him
constituted the one tie to life which at the approach of death he fo u nd
it hard to break
He was m ade one of the commissioners to establish weights and
m easures having units founded on nature Lagrange strongly favored
the decim al subdiv ision Such was the m oderation of Lagrange s
character and such the universal respect for him that he was retained
as presid ent of the commission on weights and m easures e v en after
it had been pu rifi ed by the Jacobins by striking out the nam es of A L
Lav oisier P S Laplace and others Lagrange took alarm at the
fate of Lav oisier and planned to return to B erlin but at the estab
l i shme n t of the E col e N or mal e in 1 7 95 in P aris he was induced to
accept a professorship Scarcely had he tim e to eluci date the fou n da
tions of arithm etic and algebra to young pupils when the school was
closed His additions to the algebra of L Euler were prepared at
this tim e In 1 797 the E col e P ol ytechn i qu e was founded with Lagrange
as one of the professors The earliest triumph of this institution was
the restoration of Lagrange to analysis His m athe matical activity
burst out anew He brought forth the Thear i e des f on cti ons an al yti qu es
L egon s su r l e cal cu l des f on cti ons a treatise on the sa me lines
as the preceding
and the Res ol u ti on des equ ati on s n u meri qu es
which includes papers published m uch earlier ; his m em oir
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N ou vel l e méthode pou r
s er i es ,
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publi shed
r és oud r e
1 7 70 ,
l es equ ati on s l i tter al es par l e moyen des
’
gives the notation
however much earlier in a part of a
memoir
for
by
dill
dx
’
which occurs
Fran goi s Davi e t
de
HI STORY
A
2 58
O F M ATHE M ATIC S
derivatives as it was call ed has been generally abandoned He
introduced a no ta tion of hi s own but it was inconvenient and was
abandoned by h im in the second edition of his M ecani qu e in which
he used in fi n i tesim al s The primary object of the Theari e des f on cti ons
was not attained but its secondary results were far reaching I t
was a purely abstract mode of regarding functions apart from geo
m etrical or m echanical considerations
I n the further develop m ent
of higher analysis a functio n becam e the leading idea and Lagrange s
work may be regarded as the sta rti ng point of the theory of functions
as dev eloped by A L Cauchy G F B Riem ann K Weierstrass
and others
The first to doub t the rigor of Lagrange s exposition of the calculus
were A bel B u rj a ( 1 7 5 2 —1 8 1 6 )
of B erlin the two P olish m athe m aticians
and the B ohemian
H Wr on s ki and J B S n i ad ecki ( 1 7 5 6
B B ol zan o who were all me n of li m ited acquaintance and influence
I t rem ained for A L Cauchy really to in itiate the period of greater
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rIg or
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Instructive is C E P icard s characteriz ation of the tim e of La
grange : I n all this period especial ly in the second half of the eight
e en th century what str i kes us with ad m iration and is also so m ewhat
confusing is the extrem e i mportance of the applications realiz ed
while the pure theory appeared still so ill assured One perceives it
when certain questions are raised like the degree of arbitrariness i n
the integral of vibrating chord s whi ch gives place to an interm inable
and inconclusive discussion Lagrange appreciated these i n suffi c i en ci es
when he published his theory O f analytic functions where he strov e
to give a precise foundatio n to analysis One cannot too m uch
adm ire the m arvellous presentim ent he had of the rOl e which th e
functions which with him we call analytic were to play ; but we may
confess that we stand astonished before the demonstration he b e
l i ev ed to hav e given of the possibility of the develop m ent of a function
1
in Taylor s series
In the treatm ent of infinite series Lagrange displayed in his earlier
writings that laxity com m on to all m athem aticians of his time ex
c ep ti n g N i col a u s B ernoulli II and D A l emb er t
B ut his later articles
m ark the beginning of a period of greater rigor Thus in the C al cu l des
f on cti ons he gives his theorem on the lim its of Taylor s theorem La
grange s mathem atical researches extended to subj ects which have
not been mentioned here—such as probabilities finite differences
ascending continued fractions elliptic integrals Everywhere his
wonderful powers of generaliz ation and abstraction are mad e manifest
In that respect he stood without a peer but his great contemporary
P S Laplace surpasse d h i m in practical sagacity
Lagrange was
content to leave the application of his general results to othe rs and
so me of the m ost important researches of Laplace (particularly those
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f
o
A r ts
an d
S ci en c e, S t L o u i s ,
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1 90 4 ,
V ol I , p 5 0 3
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EUL ER LAGRANGE
,
LAP LACE
AN D
2 39
on the velocity of sound and on the secular acceleration of the moon)
are imp l icitly contained in Lagrange s works
Lagrange was an extrem ely m odest m an eager to avoid co n tro
v er sy and even ti mid in conversation
He spoke in tones of doubt
“
and his first words generall y were Je n e sais pas
He would never
allow his portrait to be taken and the only ones that were secured
were sketched without his knowledge by persons attending the meet
ings of the Institute
P i e rr e S i m o n Lapl a c e ( 1 749—
1 8 2 7)
was born at B eaumont en Auge
i n Norm andy
Very little is known of his early life When at the
height of his fame he was loath to speak of his boyhood spent in
pov erty His father was a s mall farm er So me rich neighbors who
recogniz ed the boy s talent assisted him in securing an education
As an extern he attended the m ilitary school i n B eaum ont where at
an early age he beca me teacher of m athe m atics At eighteen he went
to P aris arm ed with letters of recomm endation to D Al emb er t who
was then at the height of his fam e The letters rem ained unnoticed
but young Laplace undaunted wrote the great geom eter a letter on
the principles of m echanics which brought the following en thusiastic
response : You needed no introduction ; you have recommended your
self ; my support is your due
D A l emb er t secured him a p ositio n
at the Ecol e M i li tai re of P aris as professor of mathematics H is future
was n ow assured and he entered upon those profound researches
which brought hi m the title of the Newton of France
With
won derful mastery of analysis Laplace attacked the pending problems
in the application of the l aw of gravitation to celestial motions D ur
ing the succeeding fifteen years appeared most of his original contri
butions to astronom y His career was one of almost uninterrupted
prosperity In 1 7 84 he succeeded E B éz ou t as examiner to the royal
a rtillery and the following year he becam e m ember of the Academ y
of Sciences He was m ade president of the B ureau of Longitude ; he
ai ded in th e introduction of the decim al syste m and taught w ith
J Lagrange ma thematics i n the E col e N ormate When during th e
Revolution there arose a cry for the reform of everything even of
the calendar Laplace suggested the adoption of an era beginning with
the year 1 2 50 when ac c ordin g to his calculation the m ajor axis of
the earth s orbit had bee n perpendicular to the equinoctial line The
year was to begin with the vernal equinox and the z ero m eridian was
to be located east of P aris by
degrees of the centesim al division
of the quadrant for by this m eridian the beginning of his proposed
era fell at m idnight B ut the rev olutionists rej ected this schem e and
m ade the start of the n ew era coincide with the beginning of the
1
glorious French Republic
Laplace was justly adm ired throughout Europe as a most sagacious
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R u dol f
Wo l f
,
Geschi chte der A s l ron omi e, M unch e n ,
1 8 7 7,
p 334
.
.
A HI STORY OF M ATHE M ATIC S
2 60
p rofound
scientist but unh appily for his reputation he strove
not only after greatness in science but also after political honors
The political career of this e minent scientist was stained by servility
and suppleness After the 1 8 th of B rum aire the day when Napoleon
was m ade emperor Laplace s ardor for republi can principles suddenly
gave way to a great devotion to the e mperor Napoleon rewarded
this devotion by giving him the post of m inister of the interior but
“
dism issed him after six m onths for incapacity Said Napoleon La
place n e saisissait aucune question sous son véritable point de v u e ; il
cherchait des subtilit és partout n av ai t que des id ées probl emati q u e s
e t portait en fi n l e sp r i t des i n fi n im en t petits j usque dans l a dmi n i s t ra
tion
D esirous to retain his allegiance Napoleon elevated hi m to
the Senate and bestowed various other honors upon him N ever the
less he cheerfully gave his voice i n 1 8 1 4 to the dethrone ment of his
patron and hastened to tender his services to the B ourbons thereby
earning the title of m arquis This pettiness of his character is seen
in his writings The first edition of the S ysteme du monde was dedi
c a t ed to the Council of Five Hundred
To the third volu m e of the
M ecan i qu e C el es te is prefix ed a note that of all the truths contained
in the book the one most precious to the author was the declaration he
thus m ade of gratitude and devotion to the peace maker of Europe
After this outburst of affection we are surprised to find in the ed i tions
of the Theari e a n al yti qu e des probabi li tés which appeared after the
Restoration that the original dedication to the e mperor is suppresse d
Though supple and servile in politics it m ust be said that in religion
and science Laplace never m isrepresented or concealed his own con
vi c t i on s however distasteful they m ight be to others
In m athem atics
and astronomy his genius shines with a lustre excelled by few Three
—
great works did he give to the scientific world the M écani qu e C el este
the E xposi ti on du systeme du monde and the Theori e an al yti qu e des
probabi l i tes B esides these he contributed important m emoirs to the
French Academ y
We first pass in brief review his astronom ical researches In 1 773
he brought out a paper in which he proved that the m ean m otions
or m ean distances of planets are invariable or m erely subj ect to s mall
periodic changes This was the first and m ost i mportant step in his
1
attempt to establish the stability of the solar system To I Newton
and also to L Euler it had seemed doubtful whether forces so nu mer
ous so variable in position so diff erent in intensity as those in the
solar system could be capable of maintaining perm anently a con dition
of equilibriu m Newton was of the opinion that a powerful hand
m ust intervene from tim e to tim e to repair t h e derangem ents occa
s i on ed by the m utual action of the d i ff erent bodies
This paper was
the begin n i ng of a series of profound researches by J Lagrange and
an d
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Repor t,
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J A rag o
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1 8 74
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Eu l ogy
on
L ap l ac e ,
”
t ran sl a ted by B
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P ow e l l , S mi ths on i a n
A HI STORY OF M ATHE M ATI C S
262
—
of celestial m otions treating particularly of m otions of com ets of
our moon and of other satellites The fif th volum e opens with a
brief history of celestial m echanics and then gives in appendices the
results of the author s later researches The M ecani qu e C él este was
such a master piece and so co mplete that Laplace s immediate suc
The general part of
c essor s were able to add co mparatively li ttle
the work was translated into Germ an by Johann K arl B u r khardt
—
1
and
appeared
in
B
erlin
0
1 80 2
1
8
0
N a tha n i el B owdi tch
( 7 73
—
1
brought out an edition in Engli sh with an extensive com
( 773 1 83 8 )
—
m enta ry i n B oston 1 8 2 9 1 839 The M écan i qu e C el es te is not easy
reading The diffi culties lie as a rule not so m uch in the subject
itself as in the want of verbal explanation A complicated chain of
reasoning receives often no explanation whatever J B B iot who
assisted Laplace in revising the work for the press tells that he once
asked Laplace som e explanation of a passage in the book which had
been written not long before and that Laplace S pent an hour en deavor
ing to recover the reasoning which had been carelessly suppressed
”
m
m
l
i
the
re
ark
est
facile
de
voir
Notwithstanding
the
i
por
I
w th
tant researches in the work which are due to Laplace him self it
naturally contains a great d eal that is d rawn fro m his predecessors
It is in fact the organiz ed result of a century of patient toil B ut
Laplace frequently neglects properly to acknowle dge the source fro m
which he draws and lets the reader infer that theorem s and formul a:
due to a pred ecessor are really his own
We are told that when Laplace presented Napoleon with a copy
“
of the M écan i qu e Cel este the latter m ade the rem ark M Laplace
they tell m e you hav e written this large book on the syste m of the
universe and have never even m entioned its Creator
Laplace is
said to have replied bluntly Je n avais pas besoin de cette hy
o th és e—
la
This assertion taken literally is i mpious but may it
p
not have been inten d ed to conv ey a meaning som ewhat diff erent
from its literal one ? I Newton was not able to explain by his law of
gravitation all questions arising in the m echanics of the heavens
Thus being unable to show that the solar syste m was stable and
suspecting in fact that it was unstable Newton expresse d the opinion
that the special intervention from time to tim e of a powerful hand
was necessary to preserve order Now Laplace thought that he had
proved by the law of grav itation that the solar system is stable and
in that sense may be said to have felt no necessity for reference to the
Al mighty
We now proceed to researches which belong m ore properly to pure
m athem atics
Of these the m ost conspicuous are on the theory of
probability Laplace has done more towards advancing this subject
than any one other in v est i gator He publ i she d a series of papers
the m ain results of which were collecte d in his Theori e a n al yti qu e des
r obabi l i tes 1 8 1 2
i
1 8 20)
The
thir
d
e
d
tion
consists of an introduction
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EULER LAGRANGE
AN D
,
LAP LACE
2 63
and two books The introduction was published separately under
the title E ssai phi l osophi qu e su r l es pr obabi l i tés and is an ad mirable
and masterly exposition without the aid of analytical form ul a of the
principles and applications of the science The fir st book contains
the theory of generating functions which are appli ed in the second
book to the theory of probability Laplace gives in hi s work on
p robabi l i ty hi s me thod of approxi mation to th e values of definite
integrals The solution of linear diff erential equations was reduced
by him to definite i ntegrals The use of parti al d ifference equations
was introduced into the study of probability by him about the same
time as by J Lagrange One of the most importa nt parts of the
work is the application of probability to the method of lea st squares
which is shown to give the m ost probable as well as the most convem
i en t results
Laplace s work on probabili ty is very diffi cult reading particularly
the part on the method of least squares The analytical processes
“
are by no means clearl y established or free fro m error
N 0 one was
more sure of giving the result of a nalytical processes correctly and
no one ever took so little care to point out the various s ma l l con
siderations on which correctness depends (D e M organ ) Laplace s
comprehensiv e work contains all of his own researches and m uch
derived from Other writers He giv es m asterly expositions of the
P roblem of P oints of Jakob B ernoulli s theore m of the problem s ta ken
from B ayes and Count de B u ff on In this work as in his M écani qu e
C el este Laplace is not in the habit of giving due credit to writers that
1
preceded hi m A D e M organ says of Laplace : There is enough
originating from him self to m ake any reader wonder that one who
could so well aff ord to state what he had taken fro m others should
have set an example so dangerous to his own clai ms
Of Laplace s papers on the attraction of el l ipsoids the m ost im
portant is the one published in 1 7 8 5 and to a great extent reprinted
in the third volum e of the M écan i qu e C el este I t gives an exhaustive
treatment of the general problem of attraction of any ellipsoid upon
a particle S ituated outside or upon its surface Spherical harm onics
or the so called Laplace s coeffi cients constitute a pow erful analytic
engine in the theory of attraction in electricity an d m agnetism The
theory of spherical harmonics for two di mensions had been previously
given by A M Legendre Laplace failed to m ake due ack n ow l edg
m ent of this and there existed in consequence between the two
“
great men a feeling more than coldness
The potential function
V is m uch used by Laplace and is S how n by h im to satisfy the partial
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—17 7
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2
diff erential equation
5 V
hat
1
2 1
A D e M or g an , A n E ssay
p I I o f A pp e n di x I
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5 V
3
on
2
a V
?a
0.
This is know n as Laplace s
P r oba bi li ti es , L on do n ,
’
1 838
( d at e
of
P r e fac e
)
A HI STORY OF M ATHE M ATIC S
2 64
equation and was first giv en by hi m in the more complicated form
whi ch it assu m es in polar co ordinates The notion of potential was
however not introduced into analysis by L aplace The honor of
t hat achievement be l ongs to J Lagrange
Regarding Laplace s equation P E P icard said in 1 90 4 : Few
equations have been the object of so many works as this celebrated
equation The conditions at the limits may be of divers forms The
simplest case is that of the calorifi c equilibriu m of a body of which
we m ainta in the elements of the surface at given temperatures ; from
the physical point of view it may be regarded as evident that the
temperature continuous wi thin the interior since no source of heat
is there is determined when it is given at the surface A m ore general
case is that where
the temperature may be giv en on one portion
wh ile there is radiation on another portion These questions
have greatly contributed to the orientatio n of the theory of partial
diff erential equations They have called attention to types of deter
mi n at i on s of integrals which would not hav e presented themselves
1
in remaining at a point of view purely abstract
Am ong the m inor discoveries of Laplace are his m ethod of solving
equations of the second third and fourth degrees his memoir o n
singular solutions of di fferential equations his researches in finite
differences and in determ inants the establishm ent of the e xpansion
theorem in determ inants which had been previously given by A T
V an dermon de for a sp ecial case the deter m ination of the co mplete
integral of the linear differential equation of the second order In
the M écani que C él este he m ade a generali z ation of Lagrange s theorem
on the develop m ent of functions in series known as Laplace s theorem
Laplace s investigations in physics were quite extensive We m en
tion here his correction of Newton s form ula on the velocity of sound
in gases by taki ng into account the changes of elasticity due to the
heat Of compression and cold of rarefact ion ; his researches on the
theory of tides ; his m athe m atical theory of capillarity ; his explanation
of astronom ical refraction ; his formul a for m easuring heights by the
barom eter
Laplace s writings stand out in bol d contrast to those of J Lagrange
in thei r lack of elegance and symm etry Laplace looked upon mathe
ma ti cs as the tool for the solution of physical problems
The true
result being once reached he devoted little tim e to explaining the
various steps of his analysis or in polishing his work The last years
of his life were spent m ostly at Arcueil in peaceful retirem ent on a
country place where he pursued his studies with his usual vigor
until his death He was a great adm irer of L Euler and would often
“
say Lise z Euler lisez Euler c est notre m ai tre a tous
The latter part of the eighteenth century brought forth researches
on the graphic representation of im aginaries all of which remained
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Congr ess
f
o
A r ts
an d
S ci en ce, S t L ou i s ,
.
1 90 4 ,
V ol I , p 50 6
.
.
.
A HI STORY OF M ATHE M ATIC S
2 66
the Aca demy of Sciences on M arch 1 9 1 7 91 by a committee con
sisting of J C Borda J Lagrange P S Laplace G M onge de Con
dorce t
This subdivision is found in the Fran coi s Cal l et ( 1 744—1 798)
logarithm ic tables of 1 795 and other tables published in France and
Germany Nevertheless the decimal subdivision of the quadrant did
1
not then prevail
The com m ission composed of B orda Lagrange
Laplace M onge and Condorcet decided upon the ten m ill ionth part
of the ear th s quadrant as the primitive unit of length The leng th
of the second s pendulum had been under consideration but was
finally rejected because it rested upon two dissimilar elem ents
gravity and tim e In 1 799 the m easurem ent of the earth s quadrant
was completed and the m eter established as the natural unit of length
Al e xan dre Th eoph i l e Van d ermon d e ( 1 7 3 5—
1 7 96 )
studied music
during his youth in P aris and advocated the theory that all art rested
u pon one general law through which any one could becom e a c om
poser with the aid of m athem atics He was the first to giv e a con
n e c t ed and logical exposition of the theory of determ inants and may
therefore almost be regarded as the founder of that theory He and J
Lagrange originated the m ethod of co mbinations in solving equations
A dri e n M ar i e L e g e n dre ( 1 7 5 2 —
1 8 33)
was educated at the Col l ege
M azarin in P aris where he began the study of m athem atics under
Abb é Joseph F ran coi s M arie ( 1 73 8
His m athem atical genius
secured for him the position of professor of m athematics at the mili
ta ry school of P aris While there he prepared an essay on the curv e
described by projectiles thrown into resisting m edia (ballistic curve)
which captured a priz e o ff ered by the Royal Academy of B erlin In
1 7 8 0 he resigned his position in order to reserve m ore ti m e for the
study of higher mathem atics He was then m ade m ember of several
public comm issions In 1 7 95 he was elected professor at the Norm al
School and later was appointed to some m inor govern m ent positions
O w ing to his timidity and to Laplace s unfriendliness toward him but
few important public offices comm ensurate with his abili ty were
ten dered to h im
As an analyst second only to P S Lap l ace and J L agrange Legen
dre enriched mathem atics by i mportant contributions mainly on
elliptic integrals theory of numbers attraction of ellipsoids and least
squares The most im portant of Legendre s works is his Foncti ons
el l i pti q u es issued in two volu m es i n 1 8 2 5 and 1 8 2 6
He took up the
subj ect where L Euler John Landen and J Lagrange had left it
and for forty years was the only one to cultivate this new branch of
analysis until at last C G J Jacobi and N H Abel steppe d in with
2
a dm irable new discoveries Legendre imparted to the subj ect that
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pp
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For de t ai l s ,
1 38
—1 6 3
se e
R M eh m k e i n J
.
ahres b
.
z
d d M a th Verei n i g u ng , L ei p i g ,
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.
1 900 ,
.
M El i e de B e au mon t ,
S mi thson i a n Repor t, 1 8 6 7
2
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M emo i r of L eg en dre
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T r an sl a t ed by C A A l exan d e r,
.
.
EULER LAGRANGE AND LAP LACE
67
,
connection and arrange ment which belongs to an independent science
S tarting with an integral d epen ding upon the square root of a poly
nom ial of the fourth degree in x he S howed that such integrals can be
brought back to three canonical forms designated by F
and
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,
He also undertook the prodigious task of calculating tables of
arcs of the ellipse for diff erent degrees of amplitude and eccentricity
which supply the m eans of integrating a large number of diff erentials
An earlier publication which contained part of his researches on
elliptic functions was his C al cu l i n tegr al in three volum es ( 1 8 1 1 1 8 1 6
in which he treats also at length of the two classes of definite
integrals named by him E u l eri a n He tabulated the values of log
I (p)
for values of p betw een 1 and 2
One of the earliest subj ects of research was the attraction of sph e
roi ds which suggested to Legendre the function P " nam ed after him
His m em oir was presented to the Academ y of Sciences in 1 7 8 3 The
researches of C M acl aurin and J Lagrange suppose the point a t
tracted by a spheroid to be at the surface or within the spheroid but
Legendre showed that in order to d eterm ine the attraction of a
spheroid on any external point it suffi ces to cause the surface of ano ther
spheroid described upon the sam e foci to pass through that point
Other m e moirs on el lipsoids appeared later
In a paper of 1 7 88 Legen d re published criteria for distinguishing
between m axim a and m inima in the calculus of v ariations which were
show n by J Lagrange i n 1 7 97 to be i n sufli c i en t ; this matter was set
right by C G J Jacobi in 1 83 6
The two household gods to which Legendre sacrificed with ever
renewed pleasure in the silence of his closet were the elliptic functions
and the theory of numbers His researches on the latter subject
together with the nu m erous scattered fragm ents on the theory of
num bers du e to his predecessors in this line were arranged as far
as possible into a system atic whole a n d publishe d in two large quarto
volum es entitled Thear i e des n ombr es 1 8 30 B efore the publication
of this work Legendre had issued at d ivers tim es prelim inary articles
Its crowning pinnacle is the theorem of quadratic reciprocity pre
v i ou sl y in d istinctly given by L Euler without proof but for the first
1
tim e clearly enunciated an d partly proved by Legendre
While acting as one of the comm issioners to connect Greenwich
and P aris geodetically Legendre calculated the geodetic triangles in
France This furnished the occasio n of establishing form ula an d
theorem s on geodesics on the treatment of the S pherical triangle as
if it were a plane triangle by applying certain corrections to the
angles and on the m ethod of least squares published for the first
tim e by hi m without d e monstration in 1 80 6
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O B au m g ar t , Ueber
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aas
Quadrati sche
z
R eci pr oci l a l s g cs etz, L e ip ig ,
1 88 5
.
A HI STORY OF M ATHE M ATIC S
2 68
Legendre wrote an El emen ts de Géometri e 1 794 which enjoyed
great popularity being generally adopted on the Continent and in
the United States as a substitute for Eucli d This great modern rival
of Eucli d passed through num erous e ditions ; som e containing the
”
2
elements of trigonometry and a proof of the irrationality of 71 and 71
“
With prophetic vision Legendre remarks : I l est mem e probable que
l e nom bre 7r n est pas mem e comp ris dans les i r ra ti on ell e s algébriques
c est a—
di re qu il n e peut pas etre la racine dune équation algébrique
d un nombre fi n i de term es dont les c oefi c i en t s sont ra ti on el s
M uch attention was given by Legendre to the subj ect of parallel lines
In the earlier editions of the El emen ts he m ade direct appeal to the
“
senses for the correctness of the parallel axio m He then attempted
“
to dem onstrate that axiom but his proofs did not satisfy even
him self In Vol XII of the M em oirs of the Institute is a paper by
Legendre containing his last attempt at a solution of the problem
Assum ing space to be infinite he proved satisfactorily that it is i m
possible for the su m of the three angles of a triangle to exceed t w o
right angles ; and that if there be any triangle the s u m of whose angles
is two right angles then the sam e m ust be true of all triangles B ut
in the next step to S how that this s u m cannot be less than two right
angles his demonstration necessarily failed If it could be granted
that the su m of the three angles is always equal to two right angles
then the theory of parallels could be strictly d e duced
Another author who made contributions to elem entary geom etry
was the Italian L orenzo M ascheron i ( 1 750
He publ i shed his
1
Geometri a del compass o (P avia 1 7 97 P alerm o 1 90 3 ; French editions
by A M Carette appeared in 1 7 98 and 1 8 2 5 a Germ an edition by
All constructions are m ade with a pair of
J P Gruson in
co mpasses but without restriction to a fixed radius He proved that
all constructions possible with ruler and compasses are possible with
compasses alone I t was J V P oncelet who proved in 1 8 2 2 that all
such construction are possible with ruler alone if we are given a fixed
circle with its centre in the plane of construction ; A Adler of Vienna
proved in 1 8 90 that these constructions are possible with ruler alone
whose e dges are parallel or whose e dges converge in a point M asch
eron i clai m ed that constructions with co mpasses are m ore accurate
than those with a ruler Napoleon proposed to the French m athe
ma ti ci an s the problem to divide the circu m ference of a circle into
four equal parts by the co mpasses only M ascheroni does this by
applying the radius three tim es to the circu mference ; he obtains the
arcs A B B C C D ; then A D is a diameter ; the rest is obvious
E W Hobson (M ath Gazette M arch 1 1 9 1 3)
a n d others have shown
that all Euclidean constructions can be carried out by the use of
compasses alone
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Vo l
A l ist
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1 9,
of
M ascheron i
19 1 2,
p 92
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s
w r i t i ng s i s g i ve n
i n L I n ter médi ai r e des mathéma ti ci ens ,
’
A HI STORY OF M ATHE MA TIC S
2 70
tions are form ed by giving each fraction a negativ e coeffi cient in an
equation for its nu merator taken positively and for its deno minator
the su m of the positive coeffi cients preceding it if moreover unity is
added to each fraction thus formed then the largest nu mber thus
obtainable is larger than any root of the equation In 1 8 2 2 A A
Vene a French ofli cer of engineers S howed :If P is the largest negative
c oefli c i en t
and if S be the greatest coeffi cient among the positive
term s which precede the first negative term then will P + S + I be a
superior limi t
Fourier took a prominent part at his home in promoting the Revo
l u ti on Under the French Revolution the arts and sciences seem ed for
a tim e to flourish The reform ation of the weights and m easures was
planned with grandeur of concep tion The Normal School was
created in 1 795 of which Fourier becam e at first pupil then lecturer
His brilliant success secured hi m a chair in the P olytechnic School
the duties of which he afte rw ards quitted along with G M onge and
C L B erthollet to acco mpany Napoleon on his ca mpaign to Egypt
Napoleon founded the Institute of Egypt of which J Fourier becam e
secretary In E g ypt he engaged not only in scientifi c work but dis
charged i mportant poli tical functions After his return to France he
held for fourteen years the prefecture of Grenoble D uring this
period he carried on his elaborate investigations on the propagation of
heat in soli d bodies published in 1 8 2 2 in his work entitled L a Theor i e
A n al yti qu e de l a Chal eu r
This work m arks an epoch in the histo ry of
both pure and applied m athem atics I t is the source of all moder n
m ethods in m athem atical physics i nvol v ing the i n tegration of partial
diff erential equations in problem where the boundary values are
s
fixed ( boun d ary value problem s
P roble m s of this type involv e
L Euler s second definition of a function in which the relation is
not necessarily capable of being expressed analytically This concept
of a function greatly influenced P G L D irichlet The g em of
“
F ou ri er s great book is F ou ri er s series
By this research a long
controversy was brought to a close and the fact recogniz ed that any
arbitra ry function (i e any graphically given function )
of a real
variable can be represented by a trigonom etric series The first
announcement of this great discovery was m ade by Fourier in 1 80 7
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before the French Academy
bn
cos
)represents
nx
coefficients
an =
.
The trigonom etric series
the function
p(
)i
< x
S n n xdx ,
)for
qb (x
and
b7,
= a)
2 (a,,
eve ry value of
sin
x,
n x+
if
be equal to a similar
the
in
The weak point in F o u r i er s analysis lies in his failure to
prove generally that the tr i gonom etri c series actually converges to
t e g ra l
’
.
EULER LAGRANGE
AN D
,
LAP LACE
2 7:
the value of the function William Thomson (l ater L ord Kelvi n )
“
says that on M ay 1 1 840 (when he was only six teen ) I took Fourier
out of the University L ibrary ; and in a fortnight I had mastered i t
gone righ t th ro ugh it
Kelvin s whole career was in fl uenced by
“
F ouri er s work on hea t of which he said it is diffi cult to say whether
their uni q uel y original quality or their transcendant interes t or their
perennially importa nt instructiveness for physical science is most
1
to be p raised
Clerk M a xwell pronounced it a grea t ma thematical
In 1 8 2 7 F ourier succeeded P S Lapla ce as president of the
poe m
council of the P olytechnic School
About th e time of B udan and Fourier i mporta nt devices w ere
invented in I taly and England for the solution of num erical equations
The I talian scientific society in 1 80 2 off ered a gold m edal for improve
ments in the solution of such equations ; it was awarded in 1 80 4 to
P aolo R ufli n i
With aid of the calculus he develops th e t heo ry of
transforming one equation into another whose roots are all di m in ished
2
by a certain constant Then follows the m echanism for the practical
computer and here Ruffi ni has a device which is simpler than Horner s
scheme of 1 8 1 9 and practically identical with what is now known as
Horner s procedure Horner had no knowledge of Ruffi ni s m em oir
Nor did either Horner or R ufli n i kno w that their m ethod had been
given by the Chinese as early as the thirteenth century Horner s
first paper was read before the Royal Society J uly 1 1 8 1 9 and pub
l ish ed in the P hi l osophi cal Tr an s acti on s for 1 8 1 9 Horner uses L F A
A rb og aSt s derivatives
The moder n reader is surprised to find that
Horner s exposition involves very intricate reasoning which is in
m arked contrast with the S imple and ele mentary explanations found
in modern texts P erhaps this was fortunate ; a simpler treatm ent
m ight have prevented publication in the P hilosophical Transactio n s
“
As it was much de m ur was made to the insertion of the paper
The
“
elementary character of the subject said T S Davies was the
professed objection ; his recondite m ode of trea ting it was the profess ed
passport for its ad mission
A second article of Ho m er on his m ethod
was refused publication in the P hilosophical Transactions and ap
m
r ed i n 1 7 6 5 in the M a thema ti ci a n
after
the
death
of
Ho
er
a
ea
;
p
third article was printed in 1 8 30 Both Ho mer and Ruffi ni explained
their m e thods at first by higher analysis and l ater by elem en ta ry
al gebra ; both o ffered their m ethods as substitutes for the old process
of root extraction of nu mbers R ufli n i s paper was neglected and
forgotten Horner was fortunate in finding two in fluential champions
—
—
of Belfast and A
of his m ethod John Radford Young ( 1 7 99 1 88 5)
D e M organ The R uffi n i Horner process _has been used widely in
England and the United States less widely in Germ any Austria and
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S P Thomp so n L ife of Wi l l i a m Thoms on , L o n do n , 1 9 1 0 , pp 1 4 , 6 89
2 S ee F C a o r i
x i ma t i o n a n ti c ip a ted b y R u ffi n i ,
j , H o rn er s m e t hod o f app ro—
A m M a th S oc 2 d S , V ol 1 7 , 1 9 1 1 , pp 409 4 1 4
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B u ll
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A HI STORY OF M ATHE M ATIC S
2 72
I taly and not at all in France In France the Newton Raphson
1
m ethod has held al m ost undisputed sway
Before proceeding to the origin of m odern geom etry we shall speak
briefly of the introduction of higher analysis into Great B ritain This
took place during the first quarter of the last century The B ritish
began to deplore the very s mall progress that science was m aking in
England as co mpared with its racing progress on the Continent The
first Englishman to urge the study of continental writers was Robert
—
Woodhou se ( 1 7 73 1 8 2 7 )
of Caius College Cambridge In 1 8 1 3 the
Analytical Society was form ed at Cambridge This was a s mal l
club established by George P eacock John Herschel Charles Babbage
a n d a few other Ca mbridge students to pro m ote as it was hu m orously
expressed by B abbage the principles of pure D i sm that is the
Leibniz ian notation in the calculus against those of dot—
age or
of the Newtonian notation This struggle ended in th e introduction
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into Cambridge of the notation
to the exclusion of the
fl ux ion al
nota tion y This was a great step in advance not on acco unt of any
great superiority of the Leibniz ian over the Newtonian notation but
because the adoption of the form er opened up to English students
the v ast storehouses of continental discoveries Sir William Thom
son P G Tait and som e other m odern writers find it frequently con
Herschel P eacock and B abbage
v en i en t to use both notations
translated in 1 8 1 6 from the French S F Lacroix s briefer treatise
on the d i fferential and integral calculus and added in 1 8 20 two
volu mes of exa mples Lacroix s larger work the Tr ai l e du cal cu l
“
difi ér en ti el et i n tegr a l first contained the term di fferential coeffi cient
“
and definitions of definite and indefinite integrals I t was one
of the best an d most extensive works on the calculus of that tim e
“
Of the three founders of the Analytical Society P eacock afterw ards
did m ost work in pure m athem atics B abbage becam e famous for
his i n v ention of a cal culating engine superior to P ascal s It was
nev er finished owi ng to a m isunderstanding with the governm ent
John Herschel the em inent
a n d a consequent failure to secure funds
astronom er displayed his m astery ov er hi gher analysis in m em oirs
comm unicated to the Royal Society on new app l ications of m athe
mat i cal analysis and in articles contributed to c yc l op a dias on light
on m eteorology and on the history of m athe m atics In the P hilo
1
sophical Transactions of 1 8 1 3 he introduced the notation si n x
—1
2
2
for arcsi n x ar ctan x
He wrote also l og x cos x
tan x
for l og (l og x)cos (cos x) but in this notation he was anticipated
by Heinrich B u rm an n
of M annhei m a partisan of the c om
1 8 1 7)
bi n at ory analysis of C F Hindenburg in Germ any
F
ef
s
d f ther d e tai l see C l r d C l l g P bl i cati o Ge n al S e i s
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er
r e
A HI STORY OF M ATHE M ATIC S
2 74
undisputed supre macy It was reserv ed for the genius of G M onge
to bring synthetic geom etry in the foreground and to open up new
avenues of progress His Géométr i e descr i pti ve marks the beginning
of a wonderful develop m ent of modern geom etry
Of the two leading problems of descriptive geometry the one—to
—
represent by drawings geometrical m agnitudes was brought to a
—
h
m
M
high degree of perfection before t e ti e of onge ; the other to
solve proble m s on figures in space by constructions in a plan e —had
recei v ed considerable attention before his ti m e His most noteworthy
predecessor in descriptive geometry was the French man Amedee
F ra n goi s F réz i er ( 1 6 8 2
B ut it re mained for M onge to create
descriptive geo metry as a di sti n ct branch of science by imparting to
it geometric generalit y an d elegance All problem s previously treated
in a special an d uncertain m anner were referred back to a few general
principles He intro d uced the line of intersection of the horiz ontal
and the v ertical plane as the axis of projection By revolving one
plane into the other around this axis or ground line m any advantages
1
were gained
G as par d M o n g e ( 1 7 4 6 —1 8 1 8 )
was born at B eaune The constru o
tion of a plan of h i s nat i v e tow n brought the boy under the notice of
a colonel of engineers who procured for him an appointm ent in the
college of eng i neers at M ez ieres B eing of low birth he could not
recei v e a commiss i on in the army but he was permitted to enter the
annex of th e school where surveying an d drawing were taught O b
serv ing that all the O perations connected with the construction of
p l ans of fortification were conducte d by long arithm etical processes
he subs t ituted a geom etrical m ethod which the comm andant at first
refused e v en to look at ; so short was the tim e in which it could be
practised that when once exam ined i t was recei v e d with avidity
M onge d eveloped these m etho d s further and thus created his d es cr i p
tive geom etry Owing to the riv alry between the French m il i tary
schools of that time he was not perm itte d to divulge his n ew m e tho d s
to any one outsi de of this institution In 1 76 8 he was made professor of
m athem atics at M ez i eres In 1 7 80 when convers i ng with two of his
pup i ls S F Lacroix an d S F Gay de Vernon in P aris he was obliged
to say Al l that I hav e here done by calculation I could have done
with the ruler and compasses but I am not allowed to reveal these
secrets to you
B ut Lacroix set him self to exam ine what the secret
could be d iscovere d the processes an d published the m I n 1 7 95 The
m ethod was publ i shed by M onge him self in the sam e year first in
the form in which the shorthand writers took dow n his lessons give n
at the Norm al School where he had been elected professor and then
again in re vised form in th e Jou r n al des écotes n or mal es The next
—
e dition occurre d in 1 7 98 1 7 99 After an ephe m eral existence of only
four m onths the Norm al School was clo sed in 1 7 95 In the sam e year
t ll d
C h i ti
Wi
L h b hd D
G m t i c L e ip z i g
8 84 p 2 6
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.
EULER LAGRANGE
LA P LACE
AN D
,
2 75
the P olytechnic School was opened in the establishing of which
M onge took acti v e part He taught there descriptive geometry un til
his departure from France to accompany Napoleon on the E g yptian
campaign He was the first president O f the Institute of Egypt
M onge was a z ealous partisan of Napoleon and was for that reason
deprived of all his honors by Louis XVIII This and th e destruction
of the P olytechnic School preyed heavily upon his mi nd He did not
long surv i v e this insult
M onge s num erous papers were by no means confined to descriptive
geom etry His analytical discoveries are hardly less remarkable He
introduced into analytic geomet ry the m ethod ic use of th e equation
of a line He made important contributions to sur faces of the second
degree ( previously studied by C Wren and L Euler )
and discovered
between the theory of surfaces and the integration of partial d iffer
e n t i al equations a hidden relation whi ch threw new light upon both
subjects He gave the diff erential of curves of curvature esta blished
a general theory of curvature and applied it to the ellipsoid He
found that the validity of S olutions was not im paired when imaginaries
are involved a mong subsidiary quantities Usually attributed to
M onge are the centres of si militude of circles and certain theorem s
1
which were however probably kno w n to Apollonius of P erga M onge
published the following books : S tati cs 1 7 8 6 ; A ppl i cati on s de l al gebr e
The last
a l a geometri e 1 80 5 ; A ppl i ca ti on de l a n al y se a l a geometri c
tw o contain most of his m iscellaneous pap ers
M onge was an inspiring teacher an d he gathered around hi m a
large circle of pupils a mong which were C D upin F S ervoi s C J
B rian chon Hachette J B B iot and J V P once l et
J ea n B apti s te
B i ot ( 1 7 7 4
professor at the Coll ege de France in P aris cam e in
contact as a young man with Laplace Lagrange and M onge In
1 80 4 he ascended with Gay Lussac in a balloon
They proved that
the earth s magnetism is not appreciably reduced in intensity in
regions above the earth s surface B iot wrote a pop ular book on
analytical geometry and was activ e in m athem atical physics and
geodesy He had a controv ersy wi th Arago who championed A J
Fresnel s wave theory of light B iot was a man of strong individuali ty
and great influence
Ch ar l e s D u pi n ( 1 7 84
for many years professor of mechanics
in the Conservatoire des Arts e t M étiers in P aris published in 1 8 1 3
an important work on Dével oppements de geometri c in which is i ntro
d uc ed the conception of conj ugate tangents of a point of a surface
“
2
and of the indicatrix I t contains also the theorem known as D u
pin S theorem
Surfaces of the second d egree and descriptive geom
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R C A r chib al d i n A m M a th M on thl y, V o l
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2 2, 191 5,
—
1 2 ; V ol
6
pp
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G i n o L o r i a , D i e H a u ptsachl i s ten Theori en der Geometr i c
1 888 , p 49
2
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23 ,
pp
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1 59
( F S c h ut t e)L e ipz ig
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,
,
O F M ATHE M ATIC S
HI STORY
A
2 76
were successfully studied by J ean N i col as P i erre H ochette ( 1 76 9
who be ca m e p rofessor of descriptive geometry at the P oly
technic School after the departure of M onge for Rome and Eg yp t
In 1 8 2 2 he published hi s Tr ai té de g éométri e descri pti ve
D escriptive geom etry which arose as we have seen in technical
schools in France was transferred to Germany at the foundation of
technical schools there G Schreiber ( 1 7 99
professor in Karls
ruhe was the first to spread M onge s geometry in Germany by the
1
—
publica tion of a work thereon in 1 8 2 8 1 8 2 9 In the United S tates
descriptive geo m etry was introduced in 1 8 1 6 at the M ilitary Academy
in West P oint by Claude Croz et once a pupil at the P olytechnic
2
School in P aris Croz et wrote the first English work on the subj ect
—
z
r
t
La ar e N i ch o l a s M ar gu e i e C arn ot ( 1 7 53, 1 8 2 3)was born at
Nolay in B urgundy an d educated in his native province He entered
the army but continued his mathe m atical studies and wrote in 1 7 84
a work on m achines containing the earli est proof that kinetic energy
is lost in collisions of bodies Wi th the advent of the Revolution he
threw him self into politics and when coalesced Europe in 1 793
l aunched against Fr ance a m illion sol diers the gigantic task of or
g an iz i n g fourteen arm ies to m eet th e ene m y was achieved by hi m
He was banishe d in 1 7 96 for opposing Napoleon s cou p d etat The
refugee went to Geneva where he issued in 1 7 97 a work still tre
quently quoted entitled Refl exi on s s u r l a M étaphysi qu e du C al cu l
I nfi ni tes i mal
He declared hi mself as an irreconcilable enemy of
kings
After the Russian campaign he off ered to fight for France
though not for the em pire On the restoration he was exiled He
died in M agdeburg His Geométri e de posi ti on 1 80 3 and his Essay on
Tr ansvers al s 1 80 6 are i mportant con tributions to m odern geometry
While G M onge revelled m ainly in three dim ensional geom etry
Carnot co n fined him self to that of two By his e ffort to explain the
“
meaning of the negative S ign in geo m etry he established a geom etry
“
of position which however is diff erent from the Geometrie der
Lage of to day He invented a class of general theorem s on pro
j e c tiv e properties of figures which have since been pushed to great
extent by J V P oncelet M ichel Chasles and others
3 “
Thanks to C arn o t s researches says J G D arb oux
the con
c ep t i on s of the inventors of analytic geo m etry D escartes and F erma t
retook alongside th e infinitesim al calculus of Leibniz and Newton
the place they had lost yet should never have ceased to occupy With
his geom etry said Lagrange speaking of M onge this dem on of a man
wi ll m ake him self imm ortal
While in France th e school of G M onge was creating m odern
e t ry
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3
C
F
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Wi en er
,
C aj o ri ,
1 14,
ci t
p 36
Teachi ng and Hi s tor y
p
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f M athema ti cs
in U S
o
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1 17.
Congress
f
o
A r ts
a nd
S ci en ce, S t L o u i s ,
.
1 90 4 ,
V ol
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1,
p 53 5
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.
Wa shi ng ton
,
1 8 90 ,
THE NINETEENTH AND TWENTIETH CENT URIES
I n trodu cti on
N EVE R more z ealously and successfully has mathematics been
cultivated than during the nineteenth and the present centuries Nor
has progress as i n previous periods been confined to one or two
countries While the French and Swiss who during the preceding
epoch carried the torch of progress have continued to develop ma the
m a t ic s with great success from other countries whole armies of en
thu s i as t i c workers have wheeled into the front rank
Germ any awoke
from her lethargy by bringing forward K F Gauss C G J Jacobi
P
L D irichlet and hosts of m ore recent men ; Great B ritain
produced her A D e M organ G Boole W R Hamilton A Cayley
J J Sylvester besides champions who are still living ; Russia entered
the arena with her N I L ob achevs ki ; Norway with N H Abel ;
I taly with L Crem ona ; Hungary with her two B olyais ; the United
S tates with B enja m i n P eirce and J Willard Gibbs
H S White of Vassar College estim ated the annual rate of increase
in m athem atical publication from 1 8 70 to 1 90 9 and ascertained the
periods betwee n these years whe n diff erent subj ects of research r e
1
Taking the Jahrbu ch ii ber di e Fort
c ei v ed the greatest e m phasis
s chr i tte der M a themati k published since 1 8 7 1 ( founded by Carl Ohrt
m ann ( 1 83 9—
1 88 5)
of the Konigliche Realschule in B erlin and S ince
1 8 8 5 under the chief editorship of E m il Lampe of the technische
Hochschule in B erlin )and also the Revu e S emes tri el l e published since
1 8 93 ( un d er the auspices of the M athe m atical Society of Am sterda m)
he counted the nu mber of titles and in som e cases also the number of
pages fille d by the reviews of books and articles devoted to a certain
subj ect of research and reached the following approximate results :
The total annual pub licat i on double d during the forty years ;
(1)
D uring these forty years 30 % of the publication was on applied
(2)
m athem atics 2 5 % on geo m etry 2 0 % o n analysis 1 8 % on algebra
Geo m etry dominated by P l ii ck er
7 % on history and philosophy ; 3 )
his brilliant pupil Klein Clifford and Cayley doubled its rate of
production from 1 8 7 0 to 1 890 then fell off a third to regain m ost of
its loss after 1 8 99 ; Synthetic geom etry reached its m axi mum in 1 88 7
and then declined during the following twenty years ; the a mount of
anal ytic geometry al w ays exceeded that of synthetic geometry the
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H S Wh i t e , F or ty Y ears F l uc t u a t i on s i n M a t hema t i c al R esea r ch ,
N S , V o l 4 2 , 1 9 1 5 , pp 1 0 5 1 1 3
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S ci ence,
INTRODUCTION
2 79
excess being most pronounced since 1 88 7 ; (4 )
Analysis which takes
i t s rise equally fro m calculus fro m the algebra of i maginaries fro m
the intuitions and the critically refined develop ments of geom etry and
from abstract logic : the common servant and chief ruler of the other
branches of m athem atics shows a trebling in forty years reaching
“
its first m aximu m in 1 890 probably the culmination of waves set
in m otion by Weierstrass and Fuchs in B erlin by Riemann in G ot
tingen by Hermite in P aris M ittag L effi er in S tockh ol m D ini and
B rioschi in I taly ; before 1 88 7 m uch of the growth of analysis is due
to the theory of functions which reaches a maximu m about 1 8 8 7
with a sweep of the cu rve upward again after 1 900 due to the theory
of integral equations and the influence of Hilbert ; ( 5)
Algebra i n
cluding series and groups experienced during the forty years a steady
gain to 2 5 times its original output ; the part of algebra relating to
algebraic form s invariants etc reached i ts acme before 1 8 90 and
then declined most surprisin gly ; ( 6 )
D ifferential equations increased
“
in amount S lowly but steadily from 1 8 70 under the combined i n
fl u en ce of Weierstrass D arb ou x and Lie
showing a slight decline
“
in 1 886 but followed by a marked recovery and advance during the
publication of lectures by Forsyth P icard G ou rsa t and P a in l ev é ;
The m athem atical theory of electricity and m agnetis m rem ained
( 7)
less than one fourth of the whole applied m athe m atics but rose after
1 8 7 3 steadily toward one —
fourth by the labors of Clerk M axwell
W Thom son (Lord Kelvin )
and P G Tait ; ( 8)
The constant shifting
of m athematical investigation is du e partly to fashion
The progress of m athem atics has been greatly accelerated by the
organiz ation of m athem atical societies issuing regular periodicals
The leading societies are as follows : L ondon M athema ti cal S oci ety
organiz ed in 1 86 5 L a s oci eté mathémati qu e de Fr ance organiz ed in
1 8 72
Edi n bu rgh M athemati cal S oci ety organi z ed 1 883 C i rcol o ma te
mati co di P al ermo organiz ed in 1 88 4 A meri can M athemati c al S oci ety
organized in 1 888 under the nam e of N ew Y or k M athemati cal S oci ety
Deu ts che M a thema ti k er
and changed to its present nam e in
Verei n i gu ng organi z ed i n 1 8 90 I ndi a n M a themati cal S oci ety organiz e d
in 1 90 7 S oci edad M etemati ca Espaii ol a organiz ed in 1 91 1 M a themati cal
A ssoci ati on of A mer i ca organiz ed in 1 9 1 5
The nu mber of m athem atical periodicals has enorm ously increased
2
According to Felix M ii ll er there were
d uring the passed century
up to 1 70 0 only 1 7 periodicals containing m a them atical articles ;
there were in the eighteenth century 2 1 0 such p eriodicals in the
nineteenth century 950 of them
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C on su l t Thom as S F i sk e s addre ss i n B u ll A m M ath S oc , Vo l 1 1 , 1 90 5 , p 2 38
D r Fi sk e hi mself w a s a l eade r i n t h e org an i a t i on o f t he S oc i e ty
2
J a hres b d deu ts ch M a them Ver ei n i g u ng , V o l 1 2 1 90 3 , p 4 3 9 S e e al so G A
M il l er i n H i stori cal I ntroducti on to M athemati cal Li teratu r e, N ew Y ork , 1 9 1 6 ,
C haps I , I I
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A HI STORY OF M ATHE M ATIC S
2 80
A great stim ulus toward m athe matical progress have been the
international congresses of mathe maticians I n 1 889 there was held
in P aris a Congres international de bibliographie des scie nces ma th e
mat i q u es In 1 8 93 during the Colu mbian E xposition there was held
in Chicago an International M athem atical Congress B u t by c om
m on agree m ent the gathering held in 1 8 97 at Zurich Switz erland is
“
called the first international mathe matical congress
The S econd
was held in 1 900 at P aris the third in 1 90 4 at Heidelberg the fourth
in 1 90 8 at Rom e the fifth in 1 91 2 at Ca mbridge in England The
obj ect of these congresses has been to promote friendly relations to
give re v iews of the progress and present state of di fferent branches of
m athem atics and to d iscuss m atters of term inol ogy and bibliography
One of the great c o operative enterprises intended to bring the
results of m odern research in digested form before the tech nical reader
is the E n cykl opadi e der M athemati schen Wi ssen schaf ten the publica
tion of which was begun in 1 8 98 under the editorship of Wi lhel m Fr an z
M eyer of KOn in e rg P ro m inent as joint editor was H ei n ri ch B u r k
1 9 1 4)
of Zurich l ater of M unich In 1 90 4 was begun the
hardt ( 1 8 6 1 —
publication of the French revised and enlarged e dition under the
—
u
e
l
of the University of Nancy
editorship of J l s M o k ( 1 8 57 1 9 1 4)
A S regards the productiveness of modern writers Arthur Cayley
1 “
said in 1 88 3 : I t is diffi cult to give an idea of the vast e xtent of
m odern m athem atics This word extent is not the right one :I mean
—
extent crowded wi th beautiful detail not an extent of m ere uni
formity such as an objectl ess plain but of a tract of beautiful country
seen at first in the distance but which will bear to be ra mbled through
and studied in every detail of hillside and valley stream rock wood
and flower
I t is pleasant to the m athe matician to think that in his
as in no other science the achie v e ments of every age re main posses
sions forever ; new discoveries seldom disprove older tenets ; seldom
is anything lost or wasted
If it be asked wherein the utility of som e modern extensions of
mathematics lies it m ust be acknowle dged that it is at present diffi cult
to see how som e of them are ever to becom e applicable to questions
of comm on life or physical science B ut our inability to do this S hould
not be urged as an argument against the pursuit of such studies I n
the first place we know neither the day nor the hour when these
abstract develop ments will find application in the m echanic arts in
physical science or in other branches of m athematics For example
the whole subj ect of graphical statics so useful to the practical em
metri c der L age W R
g i n eer was m ade to rest upon v o n S ta u d t s Geo
Hami lton s principle of varying action has its use in astrono my ;
complex quantities general integrals and general theorem s in inte
g ra t i o n offer advantages in the study of electricity and magnetis m
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A r t hu r C ayl ey , I n au g u ral Addre ss b efo re t he B r i t i sh A s so c i at i on ,
por t, p 2 5
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1 8 83 ,
Re
A HI STORY OF M ATHE M ATI C S
2 82
of positive construction Chemistry in a modern spirit is stretching
out into m athem atical theories ; Will ard Gibbs in his m emoir on the
equilibriu m of chem ical systems has led the way ; and though his
way is a path which chem ists find strewn w i th the thorns of analysis
his work has rendered incidentally a real service in co ordinating
experim ental results belonging to physics and to chem istry A new
and generaliz ed theory of stati stics is bein g constructed and a school
has grow n up which is applying the m to biological phenom ena Its
activity howe ver has not yet me t with the sympathetic goodwill of
all the pure biologists ; and those who remember the quality of the
discussion that took place last year at Cambridge between the biom e
t ri c i an s and so m e of the biologists will agree that if the new school
should languish it w ill not be for want of the tonic of criticism
The great characteristic of m odern m athe matics is its generaliz ing
tendency Nowadays little weight is giv en to isolated theorem s
says J J Syl v ester except as a ff ording hints of an unsuspected n ew
sphere of thought like m eteorites detached from so m e undiscovered
planetary orb of speculation
In m athe m atics as in all true sciences
no subj ect is considered in itself alone but always as related to or
an outgrowth of other things The develop m ent of the notion of
continuity plays a leading part in m odern research In geom etry
the principle of continuity the idea of correspondence and the theory
of proj ection constitute the fundam ental m odern notions Continuity
asserts itself in a m ost stri king way in relation to the circular points
at infinity in a plane In algebra the m odern idea finds expression
in the theory of linear transform ations and invariants and in the
recognition of the value of ho mogeneity and symm etry
1
H F B aker said i n 1 9 1 3 that with the aid of groups a complete
theory of equations which are soluble algebraically can be given
B ut the theory of groups has other applications
The group of
interchanges am ong four quantities which leave unaltered the product
of their six diff erences is exactly sim ilar to the group of rotations of a
regular tetrahe dron whose centre is fix ed when its corners are inter
changed am ong them selves Then I m ention the historical fact that
the prob l em of ascertaining when that well known diff erential equa
tion called the hypergeom etric equation has all its solutions expressible
in finite term s as algebraic functions was first sol v ed in connection
For any linear differential equation it is
w ith a group of si milar ki nd
of primary importance to consider the group of interchanges of its
solutions when the independent variable starting from an arbitrary
point m akes all possible excursions returning to its initial value
There is howev er a theory of groups diff erent from those so far
referre d to in which the variables can change continuously ; this alone
is most extensiv e as m ay be j u dge d from one of its lesser applications
the familiar theory of the invariants of quantics M oreov er perhaps
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1
Repor t B ri ti sh A ss
’
n
( B i rming ham)
, 19 13,
L on don ,
,
19 1 4,
p 371
.
.
INTROD UCTION
the most masterly of the analytical discussions of the theory of
geom etry has been carried through as a particular application of the
theory of such groups
If the theory of groups illustrates how a unifying plan works in
mathem atics beneath the bewildering d etail the nex t m atter I refer
to well sh ows what a wealth what a grandeur of thought may spring
from what see m slight beginnings Our ordinary integral calculus is
well nigh powerless when the result of integration is not expressible
by algebraic or logarith m ic functions The attem pt to extend the
possibilities of integratio n to t he ca se when the function to be inte
grated involves the square root of a polynom ial of the fourth order
led first after many eff orts
to the theory of doubly periodic
functions To day this is m uch simpler than ordinary trigono metry
and even apart from its applications it is quite incredible that it
S hould ever again pass fro m being a m ong the treasures of civiliz e d
man Then at first in uncouth form but now clothed with delicate
beauty ca me the theory of general algebraical integrals of which the
influence is spread far and wide ; and with it all that is systematic
in the theory of plane curves and all that is associated with the con
After this cam e the theory of multiply
c ep t i on of a Riemann surface
periodic functions of any nu mber of variables which though still
ve ry far indeed fro m being complete has I have always felt a m aj esty
of conception which is unique Q uite recently the idea s evolve d in
the previous history have prompted a vast general theory of th e
classification of algebraical surfaces according to their essential prop 3
e r t i e s which is opening endless n ew vistas of thought
The nineteenth century and the beginning of the twentieth century
constitute a period when the very foundations of m athem atics have
bee n r e exam ined a n d when fundam ental principles have been worke d
1 “
out anew Says H F B aker : I t is a constantly recurring need of
science to reconsid er the exact i mplication of the term s e mploye d ;
and as num bers and functions are ine v itable in all m easurem ent the
precise m eaning of number of continuity of infinity of limit an d
so on are fundamental questions
These notions have m any
p i tfalls I may cite
the construction of a function which is
continuous at all points of a range yet possesses no definite di fferential
coeffi cient at any point Are we sure that hu man nature is the only
co n tinuous variable in the concrete world assum ing it be continuous
which can possess such a vacillating character ?
We could take
out of our life all the moments at which we can say that our age is a
certain number of years and days and fractions of day and still
have appreciably as long to live ; this woul d be true however often
to whatever exactness we named our age provided we were quick
enough in nam ing it
These inqu i ries
have been associated
also with the theory of those se ri e s w hi ch Fourier used so boldly and
H F B k er l c ci t , p 3 6 9
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A H I STO RY O F M ATHE M ATIC S
2 84
so wickedly for the conduction of h eat Like all discoverers he took
much for granted P recisely h o w m uch is the problem This proble m
has led to the precision of what is meant by a function of real variables
to the question of the uniform convergence of an infinite series as
you may see in early papers of S tokes to n ew form ulation of the
conditions of integration and of the properties of m ultiple integ rals
and so on A n d it rem ains still incompletely solved
Another case in which the suggestions of physics have caused
grave disquiet to the mathem aticians is the problem of the variation
of a definite integral N 0 one is likely to underrate the grandeur of
the aim of those who would deduce the whole physical history of the
world from the single principle of least action Everyone must be
interested in the theorem that a po t en t i al function with a gi v e n
value at the boundary of a volum e is such as to render a certain i n
B ut in that pro
t eg ral representing say the energ y a m inim u m
portion one desires to be sure that the logical processes e mployed are
free from obj ection And alas ! to deal only with one of the earliest
problem s of the subj ect though the finally sufficient con ditions for
a minim um of a si mple integral see m ed settled long ago and could
be applied for exa mple to Newton s celebrated problem of the
soli d of least resistance it has since been shown to be a general fact
that such a proble m cannot have any definite solution at all And
although the principle of Thomson and D irichlet which relates to
the potential proble m referred to was expounded by Gauss and
accepted by Riemann and re mains to day in our standard treatise
on Natural P hilosophy there can be no doubt that in the form in
which it was originally stated it proves j ust nothing Thus a n ew
inv estigation has been necessary into the foundations of the principle
There is another problem close l y connected with this subj ect to
which I would allude : the stabil i ty of the solar syste m For those
who can m ake pronounce m ents in regard to this I have a feeling of
envy ; for their m ethods as yet I have a quite other feeling The
interest of this problem alone is s ufli c i en t to j ustify the craving of
the P ure M athem atician for powerful m ethods and unexceptionable
rigour
There are others who view this struggle for absolute rigor from a
1 “
diff erent angle Horace Lamb in 1 90 4 spoke as follows : a traveller
who refuses to pass over a bridge until he has personally tested the
soundness of every part of it is not likely to go very far ; something
m ust be risked even in M athe matics
I t is notorious that even in
this real m of exact thought discovery has often been in advance of
strict logic as in the theory of i maginaries for exa mple and in the
whole provi nce of analysis of which F ou ri er s theorem is a type
2 “
Says M axi m e B Och er : There is what may perhaps be cal led the
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A dd res s b efor e S ec t i on A , B ri t i sh A ss n , i n C amb ri dg e , 1 90 4
2
M axime B ocher i n Con gr ess of A r ts a nd S ci ence, S t L o u i s , 1 90 4 , V ol I , p 4 7 2
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A HI STORY OF M ATHE M ATIC S
2 86
independently G Frege have been able to make a rather S hort
list of logical conceptions and principles upon which i t would seem
that all exact reasoning depends
B ut the validity of logical prin
c i p l e s m ust stand the test of use and on this point we may never be
sure Frege and B ertrand Russell independently built up a theory of
arith m etic each starting with apparently self evident logical prin
c ipl es
Then Russell discovers that his principles applied to a very
general kind of logical cl as s lead to an absur di ty There is evident
need of reconstruction somewhere After all are we m erely making
successive appr oxi mati ons to absolute rigor ?
1 “
A B Ke mpe s definition is as fol lows :
M athematics is th e
science by which we investigate those characteristics of any subj ect
m atter of thought which are due to the conception that it consists
of a nu mber of di ff ering and non diff ering individuals and pluralities
2
Ten years later M axim e B ocher m odified Kempe s definition thus :
“
If we have a certain class of obj ects and a certain class of relations
and if the only questions which we investigate are whether ordered
groups of t hose obj ects do or do not satisfy the relations the results
of the investigation are called m athem atics
B ocher re marks that
if we restrict ourselves to exact or deductive mathe matics then
Ke mpe s definition beco mes coextensive with B P eirce s
B ertrand Russell in his P ri n ci pl es of M a themati cs Cambridge
1 90 3
regards pure mathe matics as consisting exclusively of dedu c
“
tions by logical principles from logical pri n ciples
Another def
inition given by Russell sounds paradoxical but really expresses th e
extre m e generality and extrem e subtleness of certain parts of m o d ern
“
m athem atics : M athe matics is the subj ect in which we never kno w
3
what we are talking about nor whether what we are saying is true
Other definitions along si milar lines are due to E P apperi tz
G I tel son
and L C ou tu ra t
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S yntheti c Geometry
The confli ct between synthetic and analytic m ethods in geometry
whi ch arose near the close of the eighteenth century and the beginning
of the nineteenth has n ow com e to an end Neither S ide has com e
out victorious The greatest strength is found to lie not in the sup
pression of either but in the friendly rivalry between the two and in
Lagrange prided
the sti m ulating influence of the one upon the other
himself that in his M ecani qu e A n al yti qu e he had succeeded in avoiding
all figures ; but since his tim e m echanics has received much help from
geometry
M odern syn thetic geo metry was created by several investigators
about the same tim e I t seem ed to be the outgrow th of a desire for
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P r oceed L ondon M ath S oc , V ol 2 6 , 1 89 4 , p 1 5
2
M B oc her , op ci t , p 4 66
3
B R u s se ll i n I n ter n ati on a l M on thl y, V o l 4 , 1 90 1 , p 8 4
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SYNTHETI C G EO M ETRY
287
general methods which S hould serve as threads of Ariadne to guide
the student through the labyrinth of theorems corollaries porisms
and proble ms Synthetic geom et ry was first cultivated by G M onge
L N M Carnot and J V P oncelet in France ; it then bore rich fruits
at the hands of A F M ob i us and Jakob S teiner in Germany an d
Switz erland and was finally developed to still higher perfection by
M Chasles in France v on S ta udt in Germany and L Crem ona in
Italy
J e an Vi ctor P on c e l e t ( 1 7 88
a native of M etz took part in
the Russian campaign was abandoned as dead on the bloo dy field
of K rasn oi and taken prisoner to S ara toff D eprived there of all
books and reduced to the rem embrance of what he had learned at
the Lyceum at M etz and the P olytechnic School where he had studied
with predilection the works of G M onge L N M Carnot and C J
B r ian ch on he began to study m athem atics from its ele ments
He
entered upon original researches which afterwards m ade h im i l lus
trio n s While in prison he did for mathe matics what B unyan did for
literature —produced a m uch read work which has re mained of great
value dow n to the present tim e He returned to France in 1 8 1 4 and
in 1 8 2 2 publ i shed the work in question entit l ed Tr ai té des P ropri etés
proj ecti ves des figu r es In it he investigated the properties of figures
which remain unaltered by proj ection of the figures The projection
is not e ff ected here by parallel rays of prescribed direction as with
G M onge but by central projection Thus perspective proj ection
used before him by G D esargues B P ascal I Newton and J H
Lambert was elevated by him into a fruitful geom etric m ethod
P oncelet form ulated the so called pr i n ci pl e of con ti n u i ty which asserts
that properties of a figure which hold when the figure varies accor ding
to definite laws wi ll hold also when the figure assum es som e li miting
position
“
1 “
could not content him self with
P oncelet
S ays J G D ar b o ux
the insuffi cient resources furnished by the m etho d of projections ; to
attain i maginaries he created that famous principle of continuity
which gave birth to such long discussions between hi m and A L
Cauchy Suitably enunciated this principle is excellent and can
render great serv ice P oncelet was wrong in refusing to present it
as a S imple consequence of analysis ; and Cauchy on the other hand
was not w il ling to recogniz e that his own obj ections app l icable with
out doubt to certain transcendent fi gures were without force in the
applications m ade by the author of the Tr ai té des propri etes proj ec
ti ves
J D G erg onn e characteriz ed the principle as a valuable
instrument for the d iscovery of new truths which nevertheless di d
2
not make stringent proofs su per fluou s By this principle of geom etric
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Con gr ess of A r ts and S ci ence, S t L ou i s , 1 90 4 , V ol I , p 5 39
2
E K ot t er , Di e En twi ck el u n g der syn theti schen Geometri c
S ta vdt, L e i p i g , 1 90 1 , p 1 2 3
1
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von
M onge bi s
f
au
A HI STORY OF M ATHE M ATIC S
2 88
continuity P oncelet was led to the consideration of points and li nes
which vanish at infinity or b ec omé im aginary The inclusion of such
ideal points and lines was a gif t which p ure geom etry received from
analysis where i maginary quantities behave m uch in the sam e way
as real ones P oncelet elaborated som e i deas of D e L ahi r e F S ervoi s
—
m
D
r
o
n
n
e
into
a
regular
ethod
the me thod of r e cipr o
G
e
and J
g
cal polars
To him we owe the P rinciple of D uality as a consequence
of reciprocal polars As an independent principle it is due to G erg on n e
D a rb o u x says that the significance of the principle of duality which
“
was a little vague at first w as sufficiently cleared up by the dis
c u s s i on s which took place on this subj ect between J
D G e rg o n n e
It had the advantage of making
J V P oncelet and J P l ii ck er
correspond to a proposition another proposition of wholly diff erent
“
aspect
This was a fact essentially new To put it in evidence
G e rg o nn e invented the syste m which since has had so much success
of m e moirs printed in double colum ns with correlative propositions
in j uxtaposition (B arboux )
Jo s eph D i az G e r g onn e ( 1 7 7 1 —1 8 59)
was an offi cer of artillery then
professor of m athem atics at the l yceum in N i mes and later professor
at M ontpelli er He solved the Apollonian P roblem and claim ed
superiority of analytic m ethods over the synthetic Thereupon
P oncelet pub l ished a purely geo m etric solution G erg on n e and P once
let carried on an intense controversy on the priority of discovering
the principle of duality No doub t P oncelet entered this field earlier
wh i le G erg on n e had a deeper grasp of the principle Some geom eters
particularly C J B ri an c hon entertaine d doubts on the general val id
ity of the principle The controversy led to one new result namely
G erg on n e s considerations of the cl ass of a curve or surface as well
1
as its order P oncelet wrote m uch on applied mechanics In 1 83 8
the Faculty of Sciences was enlarged by his election to the chair of
m echanics
J G D arb ou x says that presented in opposition to analytic
geom etry the m ethods of P oncelet were not favorably r eceived by
the French analysts B ut such were their importance and their
nov elty that without delay they aroused from di v ers sides the
M any of these appeared in the A nn al es
m ost profound researches
de ma thema ti qu es published by J D G erg on n e at N i m es from 1 8 1 0
to 1 83 1 D uring over fifteen years this was the only j ournal in the
worl d devoted exclusively to m athe matical researches G erg onn e
collaborated often against their will with the authors of the m em oirs
sent hi m rewrote them and som etim es m ade them say m ore or less
than they would have wished
G e rg onn e having becom e rector
of the Academ y of M ontpel lier was forced to suspend in 1 83 1 the
publication of his journal B ut the success it had obtained the taste
for research it had contribute d to d evel op had comm enced to bear
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p
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ci t
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pp
.
1 60
—64
1
.
A
2 96
H I STORY
OF M AT H E M ATIC S
”
the greatest geometrician since the
J ak ob S te in er ( 1 7 96
time of Euclid was born in U tzen dorf in the Canton of B ern He
di d not learn to write till he was fourteen
At eighteen he became a
pupil of P estalo zz i Later he studied at Heidelberg and B erlin
When A L C rell e started in 1 8 2 6 the celebrated mathem atical
journal bearing his nam e S teiner and Abel becam e leading con
t ri b u t or s
Through the influence of C G J Jacobi and others the
chair of geom etry was founded for hi m at B erlin in 1 8 34 This posi
tion he occupied until his death which occurred after years of bad
heal th
In 1 83 2 Steiner p ublished his S ystemati sche En twi ckel u ng der A b
“
han gi gk ei t geometri scher Ges tal ten von ei n ander
in which is uncovered
th e organism by which the most diverse phenom ena (Erschei nu ngen )
in the world of space are united to each other
Here for the first
time is the principle of duality introduced at the outset This book
and von S tau dt s lay the foundation on which synthetic geo metry in
its later form rested The researches of French m athe m aticians cul
m ina ti n g in the rem arkable creations of G M onge J V P oncelet
and J D G erg on n e suggested a unification of geom etric processes
“
This work of uncovering the organis m by which the m ost different
forms in the world of space are connected with each other this ex
posing oi a s mall num ber of very simple fundamental relations in
which the sche me reveals itself by which the whole body of theore m s
can be logically and easily developed was the task which S teiner
1 “
assu m ed Says H Hankel : In the beautiful theorem that a conic
section can be generated by the intersection of two proj ective pencils
( and the dually correlated theorem referring to proj ective ranges)
J S teiner recogniz ed the fundam ental principle out of which the
innu m erable properties of these rem arkable curves follow as it were
automatically wi th pl ayful ease
Not only did he fairly complete
the theory of curves and surfaces of the second degree but he m ade
great advances in the theory of those of higher degrees
In the S ystemati sche En tw i ck el u ngen ( 1 8 3 2 )
S teiner directed atten
tion to the complete figure ob tained by joining in every poss ible way
s ix points on a c onic and showed that in this h exag ramm u m m ys t i c u m
“
“
the 6 0 P ascal lines pass three by three through 2 0 points ( Steiner
points which lie four by four upon 1 5 straight lin es P l ucker
J P l ii ck er had sharply criticiz ed S teiner for an error that had crept
into an earlier state m ent ( 1 8 2 8 )
of the last theorem Now Steiner
gave the correct state ment but without acknowledgment to P l ii ck er
Further properties of the h exag rammu m mys t i c u m are due to T P
Kirkm an A Cayley and G Sal mon The P ascal lines of three hexa
gons concur in a new point Kirk m an
There are 6 0 Kirk
m an po i nts
Correspon ding to three P ascal lines which concur in a
Steiner point there are three Kirk m an points which lie upon a straight
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H H an k e l , El emente der P r oj ecti vi schen Geometr i c,
.
1 8 75 ,
p
.
26
.
.
SYNTHETI C GEO M ETRY
2 9:
”
line Cayley line
There are 2 0 Cayley lines which pas s four by
“
four through 1 5 Sal mon points
Other new properties of the mystic
1
hexagon we re obtained in 1 8 7 7 by G Veronese an d L Cre mona
In S teiner s hands synthetic geometry m ade prodigious progress
New discoveries followed ea ch other so rapidly that he often did not
take tim e to record their demonstrations In an article in Gr el l e s
Jou r n al on A l l gemei n e Ei gen s chaf ten A l gebr ai scher C u rven he gives
without proof theorems which were declared by L O Hesse to be
“
like Fermat s theorem s riddles to the present and future genera
tions
Analytical proofs of som e of them hav e been given S ince by
others but L Crem ona finally pro v ed them all by a synthetic m ethod
S teiner discovered synthetically the two pro minent properties of a
surface of the third order ; v iz that it contains twenty seven straight
lines and a pentahedron which has the double points for its vertices
and the lines of the Hessian of the given surface for its edges This
subj ect will be discussed more fully later S teiner m ade investigations
by synthetic m ethods on m axima and minima and arri v ed at the
solution of problems which at that time surpassed the analytic power
of the calculus of variations I t will appear later that his reasoning
on this topic is not always free from criticism
2
S teiner generaliz ed M alfa tti s prob l em Gi ovann i Fr ancesco M al
—
of
the
university
of
Ferrara
in
proposed
t
h
e
a
tt
i
1
1
1
8
0
1
8
0
(
)
7
7
3
3
f
problem to cut three cylindrical holes out of a three sided pris m in
such a w ay that the cylinders and the prism have the sam e altitude
and that th e volume of the cylinders be a maximu m This problem
was reduced to another n ow generally known as M alfa tti s problem :
to inscribe three circles in a triangle so that each circle will be tangent to
two sides of the triangle and to the other two circles M alfatti gave an
analytical solution but Steiner gave without proof a construction
remarked that there were thirty two solutions generaliz ed the problem
by replacing the three lines by three circles and solved the analogous
problem for three dimensions This general problem was sol v ed
analytically by C H Schellbach ( 1 80 9—1 8 9 2 )
and A Cayley and by
R F A C l eb sch with the aid of the addition theorem of elliptic
3
functions
A simple proof of S teiner s construction was given by
A S Hart of Trinity College D ublin in 1 8 56
Of interest is S teiner s paper Ueber di e geometri schen C ons tru cti onen
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fii hr t mi ttel s
au s ge
der ger aden L i n i e
u nd ei n es
K r ei s es
f es ten
in which he shows that all quadratic constructions can be eff ected
with the aid of only a ruler provided that a fixed circle is drawn once
for all It was generally known that all linear constructions could be
eff ected by the ruler without other aids of any kind The case of
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.
,
G S al mo n , C on i c S ecti on s , 6 t h Ed , 1 8 7 9 , N o t e s , p 3 8 2
2
K a r l F i n k , A B r i ef Hi s tor y of M a thema ti cs , t ra n sl b v W W B em an
S mi t h , C h i ca g o , 1 900 , p 2 5 6
3
A Wi t t s t ei n , zu r Geschi chte des M a lf a tti s chen P robl ems , N o r dli n g e n ,
1
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and
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1 8 78 .
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A HI STORY OF M ATHE M ATI C S
292
cubic constructions calling for the determination of three unk nown
elem ents (poi nts)
was worked out in 1 86 8 by Lu dwi g H erman n K or
1 90 4 )
tu m ( 1 8 3 6 —
of B onn and Stephen S m ith of Oxford in two r e
searches which received the Steiner priz e oi the B erlin Academy ; it
was shown that if a conic (not a circle )
is given to start with then all
such constructions can be d one w ith a ruler and compasses Fran z
—
of B reslau demonstrated in 1 895 that these cubic
London ( 1 8 6 3 1 9 1 7)
constructions can be e ffected with a ruler only as soon as a fi xed
1
cubic curv e is once drawn
2
F B u tzb erg er has recently pointed out that in an unpublished
manuscript Steiner disclosed a knowledge of the principle of inver
sion as early as 1 8 2 4 In 1 84 7 Liouville called it the transformation
by reciprocal radii After S teiner this transform ation was found
in depen dently by J B el l av i ti s in 1 83 6 J W S tubbs and J R Ingram
in 1 84 2 and 1 843 and by William Tho m son (Lord Kelvin )
in 1 84 5
Steiner s researches are confined to synthetic geom etry He hated
analysis as thoroughly as J Lagrange disliked geom etry S teiner s
Gesa mmel te Wer ke were publishe d in B erlin in 1 8 8 1 and 1 88 2
—
s
e
s
1
M ich e l Ch a l ( 7 93 1 880 )
was born at Epernon entered the P oly
technic School of P aris in 1 8 1 2 engaged afterwards in business which
he later gave up that he m ight devote all his tim e to scientific pursuits
In 1 84 1 he becam e professor of geodesy and m echanics at the Ecole
“
polytechnique ; later P rofesseur de G éométrie sup érieure a la F ac u l te
He was a volu minous writer on geom etrical
des Sciences d e P aris
subjects In 1 83 7 he published his ad mirable A per gu hi stori qu e s u r
l or i gi n e et te dével oppemen t des méthodes en g eometri e containing a
history of geo metry and as an appendix a treatise sur deu x principes
g én éraux de la Science
The A per gu hi s tor i qu e is still a standard
historical work ; the appendix contains the general theory of H omog
and of duality (Reciprocity )The na me du al i ty
r aphy ( Collineation )
is due to J D G erg onn e Chasles introduced the term a nhar moni c
l tni ss and to Cli ff ord s
ra ti o corresponding to the Ger man D oppel ver ha
Chasles an d J S teiner elaborated independently the
c r oss r a ti o
m odern synthetic or proj ective geo met ry Nu m erous original m e moirs
of Chasles were published later I n the Jou rn al de l Ecol e P ol ytechni qu e
He gav e a reduction of cubics different from Newton s in this that
the fi v e cu rves from which all others can be proj ected are sym metrical
with respect t o a centre In 1 86 4 he bega n the publication in the
“
C omptes r en du s of articles in which he solves by his
method of
“
characteristics and the principle of correspon dence an immense
number of problem s He d eterm ined for instance the nu mber of
intersect i ons of two curves in a plane The m ethod of characteristics
contains the basis of enu m erati v e geo m etry
As regard s Chasles use of i maginaries J G D arboux says : Here
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J a hr es b d d M a th Ver ei n i g u n g , V ol 4 , p
2 B u l l A m M ath S oc
l 2 0 , 1 9 1 4, p 4 1 4
, Vo
1 63
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A HI STORY OF M ATHE M ATIC S
2 94
The labors of Chasles and Steiner raised synthetic geom etry to an
honored and respected position by the side of analysis
—
r
h
r
i
s
t
i
a
n
o
t
a
u
d
t
1
K ar l G eo g C
v nS
was born in Rothen
( 7 98 1 86 7 )
burg on the Tauber and at his death was professor in Erlangen His
great works are the Geometri c dcr L age N ii rn b e rg 1 84 7 and his
—
B ei tr age zu r Geometri c der L age 1 8 56 1 8 6 0
The author cut loose
from algebraic form ul a and from m etrical relations particularly the
anha rm onic ratio of J Steiner and M Chasles and then created a
geo metry of position which is a complete science in itsel f independent
of all m easure m ents He S hows that proj ective pr operties of fi g ures
have no dependence whatever on m easure ments and can be estab
“
lish ed without any m ention of the m
In his theory of throws or
Wii rfe he even gives a geo metrical definition of a number in its
rel ation to geo metry as d etermining the positio n of a point Gu stav
K ohn of the University of Vienna about 1 894 introduced the throw
as a fu ndam ental concept un derl ying the proj ective properties of a
geom etric configuration such that according to a principle of duality
of this geome try throws of figures appear in pairs of reciprocal throws ;
figures of reciprocal throws form a complete analogy to figures of
equal throws Referring to Vo l i S ta u dt s num erical c o ordinates
defined without introducing distance as a fundamental idea A N
“
Whitehead said i n 1 90 6 : The establish m ent of this result is one of
the triu mphs of m odern m athe m atical thought
The B ei tr age contains the first co mplete and general theory of
i maginary points lines and planes i n proj ective geo metry R epre
se n t a t i on of an i maginary po int is sought in the co mbination of an
in v olution with a determ inate direction both o n the real line through
the point While purely proj ective v on S taud t s m ethod is inti
mately related to the proble m of representing by actual points and
1
li n es the imaginaries of analytical geom etry Says Kotter : Staudt
“
was the first who succeeded i n subjecting the im aginary elem ents to
the fundam ental theorem of projective geom etry thus returning to
analytical geom etry the present which in the hands of geo metri
c i an s had led to the m ost beautiful results
Von S tau dt s geo metry
of position was for a long tim e disregarded m ainly no doubt because
his book is extre mely condensed An i mpulse to the study of this
subj ect was given by C u l man n who rests his graphical statics upo n
the work of von Staudt An interpreter of von S taudt was at last
found in Theodor Reye of Strassburg who wrote a Geometri c der
L ag e in 1 8 6 8
The graphic representation of the imaginaries of analytical geom
e t ry was syste m atically undertaken by C F M axi mi l i en M a r i e ( 1 8 1 9
who worked how ever o n entirely d iff erent l i nes fro m those
of v on Staudt Another in d ependent attemp t was made i n 1 8 93 by
F H Loud of C ol orado C ol l eg e
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E K ott er ,
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p
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ci t
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p
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1 23 .
,
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.
SYNTHETI C GEO M ETRY
2 95
Synthetic geo m etry was studied with m uch success by Lu i gi Cr e
who was born in P avia and becam e in 1 860 pro
m o n a ( 1 8 30
fessor of higher geo m etry in B ologna in 1 8 6 6 professor of geom etry and
graphical s tatics in M ilan i n 1 8 7 3 professor of higher m athematics
and director of the engineering school at Ro m e He was influe nced
by the writings of M Chasles later he recogniz ed v on S ta u d t as the
true founder of pure geom etry A mem oir of 1 8 66 on cubic surfaces
secured half of the Steiner priz e fro m B erlin the other half being
awarded to Ru dolf Sturm t hen of B rom berg Crem ona used the
m etho d of enu m eration with great e ffect He wrote on plane cu r ves
on surfaces on birational transform ations of plane and soli d space
The birational transform ations the simplest class of which is now
“
called the Cremona transformation proved of i mporta nce not
only in geo metry but in the analytical theory of algebraic functions
and integrals I t was developed m ore fully by M N Other and others
1
Beyond the
H S White comm ents on this subj ect as follows :
linear or proj ective transform ations of the plane there were known
—
the quadric inversions of Ludwig I mm anuel M agnus ( 1 7 90 1 86 1 )
of
B erlin changing lines into conics through three funda menta l points
and those exceptional points into singular lines to be discarded
Cre mona described at once the highest ge n eraliz ation of these trans
fo rmations one to—
one for all poi n ts of the plane except a finite set
of fundamental points He found that it m ust be m e diate d by a
net of rational curves ; any two intersecting in one variable point
and in fixed points ordinary or m ultiple which are the funda mental
poi n ts and which are the mselves t ran form ed into s i ngular rational
curves of the sa m e orders as the indices of m ultiplicity of the
points When the fundam ental points are enu m erated by classes
according to their se v eral in d ices the set of class nu mbers for the in
verse transform ation is found to be the sam e as for the direct but
usually related to d i ffere n t indices Tables of such rational nets of
l o w orders were m ade out by L Crem ona and A Cayley and a
wide n ew vista seem ed opening ( such indeed it was and is )
when si
m u l tan eou sl y three invest i gators announced that the m ost general
Cre m ona transform ation is equivalent to a succession of qua dric
transformations of M agnus s type This seem ed a cli m ax and a
set back to certain expectations
Cre mona s theory of the trans
form ation of curves a nd of the correspondence of points on cu rves
was extended by him to three di m e n sions There he showed how a
“
great variety of particular transform ations can be constructed but
anything l ike a general theory is still in the future
Ruled surfaces
surfaces of the second order space curves of the third order an d
the general th eory of surfaces received m uch attention at his hands
He was interested in map d rawi ng wh ich had engaged the attention
of R Hoo ke G M ercator J Lagrange K F Gauss and others For
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p
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24 2
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A
2 96
HI STORY
O F M ATH EM ATIC S
a one one correspondence the surface must be unicursal and this is
suffi cient L Crem ona is associated with A Cayley R F A C l eb sch
1
M N Ot her and others in the develop m ent of this theory
Cre mona s
writings were translated into Ger man by M axi mi l i an Cu rtze ( 1 8 3 7
p rofessor at the gym nasiu m in Thorn The Opera matemati che
di L u i gi Cremon a were brought a t M ilan in 1 9 1 4 and 1 9 1 5
One of the pupils of Crem ona was G i ovan ni B atti s ta G u cci a ( 1 8 55
He was born in P alermo and studied at Rome under Cremona
In 1 8 8 9 he becam e ex traordinary professor at the Un iv er si ty of P al
ermo in 1 8 94 ordinary professor He gave m uch attention to the
study of curves and surfaces He is best known as the founder in
1 8 84 of the Ci r col o ma temati c o di P al ermo and director of its Rendi
con ti
The society has becom e i nternational and has been a po w erful
stimulus for mathem atical research i n Italy
K ar l C u l man n ( 1 8 2 1
professor at the P ol yt echn i c u m in Z u
rich published an epoch m aking work on Di e gra phi s che S ta ti k Zurich
1 86 4 which has rendered graphical statics a great rival of analytical
statics B efore C u l mann B arthel emy Edou ard Cou si n cry ( 1 7 90—
1 8 51 )
a civil engineer at P aris had turned his attention to the graphi cal
2
calculus but he m ade use of perspective and not of m oder n geom etry
C u l mann is the first to undertake to present the graphical calculus
as a symm etrical whole holding the sam e relation to the new geom
e t ry that analytical m echanics does to higher analysis
He m akes
use of the polar theory of reciprocal figures as expressing the relation
between the force and the funicular polygons He deduces this rela
tion without leav ing the plane of the two figures B ut if the polygons
be regarded as proj ections of lines in space these lines m ay be treated
as reciprocal elem ents of a N ull sys tem
This was done by Cl erk
M axw el l in 1 86 4 and elaborated further by L Cremon a The grap hi
cal calculus has been applied by O M ohr of D resden to the elastic
line for continuous spans H en ry T Eddy ( 1 844
then of the
Rose P olytechnic Institute n ow of the University of M innesota gives
graphical solutions of proble m s on the m axim u m stresses in bridges
“
under concentrated loads with aid of what he calls reaction poly
gons
A standard work L a S tati qu e gr aphi qu e 1 8 74 was issued by
M aurice Lev y of P aris
D escriptive geom etry [ reduced to a science by G M onge in France
and elaborated further by his successors J N P H achette C Du pi n
—
of P aris Ju l cs de l a C ou rn er i e of P aris ]
Theodor e Ol i vi er ( 1 7 93 1 8 53 )
was soon studied also in other countries The French directed their
attention m ainly to the theory of surfaces and their curvature ; the
—
of Karls
Germ ans and Swiss through Guido Schreiber ( 1 799 1 8 7 1 )
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1
2 79
P roceedi n g s
f
o
the Roy
.
S oc
.
f
o
L on don , V ol
75 , L on don ,
1 90 5 ,
pp
.
2 77
.
B o i s , Gr aphi cal S tati cs , N ew York ,
dc N omogr a phi e, P a ri s , 1 899 , p 5
2
.
A
.
J ay d u
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1 8 75 ,
p
.
x x xn ;
M d O cag ne , Tr ai té
’
.
O F M ATHE M ATIC S
HI STORY
A
2 98
and centroid of a triangle are collinear lying on the Euler line
H C Gossard of the Unive rsity of Oklahom a showed in 1 9 1 6 that
the three Euler lines of the triangles form ed by th e Euler line and the
sides taken by twos of a given triangle form a triangle triply per
sp ec t i v e with the giv en triangle and having the sa m e Euler line
Con
“
nine point circle the
spi c u ou s a m ong the new develop m ents is the
discovery of which has been erroneously ascribed to Euler A m ong
the several independent discoverers is the English m an B enj ami n
—1 8 8 who proposed in L e b ou rn s M athema ti cal Re os i tor
B eva n
3 )
y
y
p
I 1 8 1 80 4 a theore m for proof which practically gives us the nine
po int circle The proof was supplied to the Reposi tor y I P art 1
p 1 43 by John B u tterw orth who also proposed a problem solv ed by
hi mself and John Whi tl ey; from the general tenor of which it appears
that they knew the circle in question to pass through all nine points
These nine points are explicitly m entioned by C J B ri an chon and
J V P oncelet i n G erg on n e s A n n al es of 1 8 2 1 In 1 8 2 2 K arl Wi l hel m
Feu cr bach ( 1 8 0 0
professor at the gym nasium i n Erlangen
published a pamphlet in which he arrives at the nine point circle
an d proves the theorem known by his nam e that this circle touches
the incircle and the three exc i rc l es The Germ ans call it F eu erb ach s
Circle The last independent discoverer of this re markable circle so
far as known is T S Davi es in an article of 1 8 2 7 in the P hi l osophi cal
M agazi n e II 2 9—3 1 Feuerbach s theorem was e x tended by A ndr ew
fellow of Trinity College D ublin w ho
S ea rl c H a r t ( 1 8 1 1
S howed that the circles which touch three given circles can be dis
tributed into sets of four all touched by the sam e circle
In 1 8 1 6 A u gu s t L eopold Cr el l e published in B erlin a paper dealing
wi th certain properties of plane triangl es He showed how to deter
mine a point 0 inside a triangle so that the
angles ( taken in the same order)
form ed by
the lines joining it to the vertices are equal
In the adjoining figure the three m arked angles
are equal If the construction is m ade so that
angle Q A C = Q C B = O B A then a second point
is obtained The study of these new
B Q
angles and new points led C rel l e to exclaim :
It is indeed wonderful that so s imple a figure as the triangle is
so inexhaustible in properties How m any as yet unknown proper
ties of other figures m ay there not be ! Investigations were m ad e
—
also by K ar l Fri edri ch A ndr eas J acobi ( 1 7 95 1 8 5 5)of P for ta and
som e of his pupils but after his d eath in 1 8 55 the whole
m atter was forgotten In 1 8 75 the subject was again brought befor e
the m athematical public by Henri B rocard ( 1 84 5
whose re
searches were followed up by a large num ber of investigators in France
England and Germ any Unfortunately the nam es of geom etrician s
which have been attached to certain re markable points lines and
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G EO M ETRY
SYNTH ETI C
2 99
”
circles are not always the names of the men who first studied their
properties Thus we speak of B rocard points and B rocard
angles but historical research brought out the fact in 1 8 8 4 and 1 886
that these were the points and lines which had been stu die d by A L
“
The Brocard Circle is Brocard s own
C re l l e and K F A Jacobi
creation In the triangle AB C let ( 2 and Q be the first a n d second
B rocard point
Let A be the intersection of B 9 and C Q ; B of
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and C O
and A O The circle passing through A
“
C is the B rocard circle
A B C is B rocard s first triangle
“
Another lik e triangle A B C is called B rocard s second triangle
"
"
The points A B C together with Q Q and two other points
lie in the circu mference of the B rocard circle
In 1 8 73 Emi l e L emoin e ( 1 840
the editor of l I n ter medi ai re
des mathémati ci en s called attention to a particular po i nt within a plane
triangle which has been variously called the Le moine point
sym
m edian point and Grebe point nam ed after E rn s t Wi lhel m G rebe
—
of Kassel If CD is so drawn
1
8
0
( 4 1 8 7 4)
as to m ake angles a and b equal then one
of the two lines AB and CD is the an ti
el of the other
with
reference
to
ra
l
l
C
a
p
the angle O Now OE the bisector of
AB is the medi an and OF the bisector of
the anti parallel of AB is called the sym
medi an ( abbreviated fro m symetri qu e dc l a
médi an c)The point of concurrence of the
three symm edian s in a triangle is called
—
of Univ ersity College School in Lon
er
1
2
T
u
c
k
8
after Robert
( 3 1 90 5)
—
of
M
1
8
don the symm edian point
John S turgeon ackay ( 43 1 9 1 4 )
E dinburgh has pointed out that some of the properties of this point
brought to l i ght since 1 8 73 were first d i scovered previously to that
AO
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A H I STOR Y O F M AT H EMATI CS
30 0
date The anti parallels of a triangle which pass through its sym
m edian point m eet its S ides in s ix points which lie on a circle call ed
“
“
the second Lemoine circle
The firs t Le moine circle is a special
“
“
case of a Tucker circle and concentric with the B rocard circle
“
The Tucker circles may be thus defined Let D F = FE = ED
let m oreover the follow ing pairs of lines he anti parallels to each
other : AB and ED B C and FE CA and D F ; then the six points
“
D D E E F F li e on a Tucker
circle
Vary the length of the equal
anti parallels and a family of Tucker
circles is obtained Allied to these
are the Taylor circles due to H M
Taylor of Trinity College Cambridge
“
Still different types are the M ackay
circles and the N eu b erg circles
due to Joseph Neuberg ( 1 840
of
C Luxe mburg
A syste matic treatise on
this topic Di e B r ocardschcn Gebi l de was
written by Albrecht E mm erich B erlin 1 8 91 Of the al most i n
nu m erable m ass of new theorem s on the triangle and circle a great
nu m ber is given in the Tr eati se on the Ci rcl e a nd the S phcr c Oxford
1 9 1 6 written by J L Coolidge of Harvard University
Since 1 88 8 E Le moine of P aris developed a system called g eome t
rog raphi c s for the purpose of nu m erically co mparing geo m etric con
s t r u c t i on s with respect
to their S im plicity Coolidge calls thes e
“
the best known and least undesirable tests for the s implicity
“
of a geo m etrical construction ; A E mch declares that they are
hardly of any practical value in so far as they do not indicate how to
S i mp l ify a construction or how to m ake it m ore accurate
A n ew theore m upon the circum scribed tetraedro n was propounded
in 1 8 97 by A S B ang and proved by J ob Gehrke The theore m is :
Opposite edges of a circum scribed tetraedron subtend equal angles at
the points of contact of the faces which contain them I t has bee n
the starting point for extended developments by Fran z M eyer J
1
N eub erg and H S White
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Li n k moti on
-
The generation of rectilinear motion first arose as a practical prob
l em in the design of stea m engines
A close approxim ation to such
“
m otion is the parallel m ot i on d esigned by Jam es Watt in 1 78 4 :
In a freely j ointe d qua d rilateral AB CD with the S ide AD fi xed a
point M on the side B C m oves in nearly a straight line The equa
“
tion of the curv e trace d by M so meti mes called Watt s curve
was first derived by the French engineer Fr an goi s M ari e dc P rony
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B ul l A m M ath S oc
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V ol
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1 4 , 1 90 8 ,
p
.
2 20 .
A HI STORY OF M ATHE M ATIC S
30 2
The barrister at law Alfred B ray Ke mpe of London showed in 1 8 7 6
that a link m otion can be found to describe any given algebraic curve ;
he is the author of a popular booklet H ow to dr aw a S trai ght Li n e
London 1 8 7 7 Other articles of note on this subj ect were prepared
by Sam uel Roberts Arthur Cayley W Woolsey Johnson V Lig u in e
of th e University of Odessa and G P X Koenigs of the Ecole P oly
technique I n P aris The dete rm ination of the linkage with m inim um
nu mber of pieces by which a given curve can be described is still an
unsolved problem
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—
P a r al l el L i n es , N on Eu cl idean Geometry and Geometry of
n
Di mensi on s
D uring the nineteenth century very re markable generali zations were
m ade which reach to the v ery root of two of the oldest branches
—
of m athem atics elem entary algebra and geom etry In geom etry
the axio m s have been searched to the bottom and the con
c l u si o n has bee n reached that the
defin ed b y Eucli d s axio m s
is not the only possible non m
i c t o ry space
Euclid prov ed
“
that if a straight line falling on two other straight lines m ake
(I 2 7 )
the alternate angels equal to one another the two straight lines shall
be parallel to one another
B eing unable to prove that i n every
other case the two lines are not parallel he assu med this to be true in
“
what is now generally called the 5th axio m by som e the 1 1 th or
“
the 1 2 th axio m
Simpler and m ore obvious a xiom s have been advanced as sub
“
As early as 1 6 6 3 John Wallis of O xford recommen d ed : To
s t i t u t es
any triangle another triangle as large as you please can be drawn
which is S imi lar to the giv en triangle
G S acche ri assu m ed the ex
i s t en c e of two si m ilar unequal triangles P ostulates si m ilar to Wallis
hav e been proposed also by J H Lam bert L Carnot P S Laplace
m
l
e
f
A
C
Clairaut
assu
es the existence of a rectangle ;
D
e
b
o
u
J
W Bolyai postulated that a ci rcle can be passed through any three
points not i n the sam e straight line A M Legen dre that there e xisted a
finite triangle whose angle —
s u m is two right angles J F Lorenz and
Legendre that through every point within an angle a line c a n be
drawn i n t eresc t i n g both sides C L D odgson that in any circle the
inscribed equilateral quadrangle is greater than any one of the seg
m ents which lie outside it B ut probably the S i mplest is the ass u mp
tion m ade by Joseph Fenn in his e dition of Eucli d s El emen ts Dub
—
m
m
lin 1 7 69 and again sixteen years later by Willia Ludla ( 1 7 1 8 1
vicar of Norton and adopted by John P layfair : Two straight lines
which cut one another can not both be parallel to the sam e straight
1
line
It is noteworthy that this axiom is distinctly stated in
P r oc l u s s note to Eucli d I 3 1
B ut the m ost nu m erous e ff orts to re move the supposed defect in
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T L H ea t h , The Thi r teen B ook s
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f
o
E ucl i d s El emen ts , V ol I , p
’
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2 20
.
SYNTHETI C GEO M ETRY
39 3
Euclid were a ttempts to prove the parallel postulat
ri es
of espera te but fru i tl ess en deal q r the hol d i dea
the
m i n s of sev e ral f riath éma ti c i an s tha t a geo m etry m ight be built u p
—
th e :
a
l
l
e
xio
m
While
A
M
Legendre
till
a
S
pai l
e n déa
Y Of ed tO establish the axio m by rigid proof L ob a ch evsk i brought
out a publication which assu med the contradictory of that axiom
and which was the first of a series of articles destined to clear up Oh
sc u r i t i es in the funda m ental concepts and greatly to extend the field
of geo m etry
N i ch ol au s I van ovi ch L ob a ch e vs k i ( 1 7 93—
1 8 56 )
was born at M a
karic i in N i zhn i
at Kasan and fro m 1 8 2 7
to 1 846 was professor and rector of the University of Kasan His views
on the foundation of geom etry were first set forth in a paper laid before
the physico mathem atical depart m ent of the University of Kasan in
February 1 8 2 6 This paper was never printed and was lost His
earliest publication was in the Kasan M cssengcr for 1 8 2 9 and then in
the Gel ehr te S chrif ten der Uni versi tat K asan 1 83 6 —1 83 8 under the
“
title N ew Ele ments of Geo metry w i th a complete theory of P ar
allels B eing in the Russian language the work remained unknown to
foreigners but even at hom e it attracted no notice In 1 840 he pub
l i shed a brief state m ent of his researches in B erlin under the title
Geometr i sche Un tersu chu n gen zu r Theor i e der P aral l el l i n i cn
Loba
“
c h e vsk i constructed a n
i maginary geom etry as he called it wh ich
_
“
has b eéITdé scrib ed by W K :Clifford as quite simple m erely Eu c lid
withou t the vicious assum ption
A re m arkable part of this geo m etry
is this that through a point an indefinite number of lines can be drawn
l
i
A
si
m
ilar
in a plane none of which
n the sa m e plane
n
e
i
g i v en_
_
k temfi i f geom et ry was deduced independentl y by the B olyais i n
sy
“
Hungary who called it absolute geo metry
was born in S z ekler Land
Wo lf g an g B olyai de B ol ya ( 1 7 75—
1 8 56 )
Transyl v ania After studying at Jena he went to Gottingen where
he bec ame intimate with K F Gauss the n nineteen years old Gauss
used to say that B olyai was the only man who fully understood his
views on the m etaphysics of m athem atics B olyai beca m e professor
at the Reformed College of M aros V as arhely where for forty seven
years he had for his pupils m ost of the later professors of Tran syl
vania The first publications of this remarkable genius were dramas
and poetry Clad in old tim e planter s garb he was truly original in
his private life as well as in his m ode of thinking He was extre m ely
m odest No m onu m ent said he should stand over his grave only an
apple tree i n m emory of the three apples ; the t w o of Eve and P aris
w hich m ad e hell out of earth and that of I Ne w ton which elevated
1
the earth again into the circle of heavenly bodies His son J oh ann
d
F S h mid t
Joh
A
d m L be zw ei er
g i h r M t hema t ik
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c
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Wolfg an g
1 90 1
w
)
,
B o l ya i
us
v on
e
n
un
’
ar sc
B ol ya , Gr u ner t s A rchi v,
i n B u dapest
t t
as an a rc hi ec
e
.
e
a
1 86 8 .
,
er
an n
F r an z S chmi d t
un
(182 7
A HI STORY OF M ATHE M ATIC S
3 04
B olyai
hirn sel f
was educated for the army and dist i n g uished
as a profound m athe matician an impassioned violin—
player
and an expert fencer He once accepted the challenge of thirteen
O ffi cers on condition that after each duel he m ight play a piece on his
violin and he vanquished them all
The chief m athematical work of Wolf gang B olyai appeared in
—
2
two volum es 1 83 1 833 entitled Ten tamen j u ven tu tem s tudi os am i n
i n tr odu ccndi
I t is followed by an ap
cl emen ta ma thes eos pu r a
Its twenty six pages make the
p en dix com posed by his son Johann
nam e of Johann B olyai im m ortal He published no thing else but he
l eft behind one thousand pages of manuscript
While L ob ach ev sk i enjoys priority of publication it may be that
Bolyai developed hi s system som ewhat earlier Boly ai S atisfied
hi m
—
self of the non contradictory character of his new g eom et ry on or be
fore
there Is som e doub t whether Lob achev sk i had reached this
point in 1 8 2 6 Johann B olyai s fa ther se em s to have been the only
person in Hungary who really appreciated the m erits of his son s
work For thi rty fi v e years this appendix as also L ob ach ev sk i s
researches rem ained in al most entire oblivion Finally Richard
Baltz er of the Univ ersity of Giessen in 1 86 7 cal led attention to the
won derful researches
In 1 86 6 J H oii el translated L ob achev sk i s Geometri sche Un ter
In 1 8 6 7 appeared a French translation of
s u chu n g en into French
Johann Bolyai s A ppendi x In 1 891 George B ruce Halsted then of
the University of Texas rendered these treatises easily accessible to
American readers by translations brought out under the titles of
Bolyai
s
T
h
e S c i en ce A bs ol u te of S pa ce and N L ob ach evs ki s Geo
J
metr i cal R es ear ches on the Theory of P ar al l el s of 1 840
The Russian and Hungarian mathematicians were not the only
ones to who m p a n g e ome try suggested itself A copy of the Tentamen
reached K F Gauss the elder Bolyai s form er roo mmate at G ottingen
a n d this Nestor of Ger man m athe maticians was surprised to disco v er
in it worked out what he hi mself had begun long before only to leav e
it afte r hi m I n hi s papers As early as 1 7 92 he had started on researches
of that character His letters S how that In 1 7 99 he was trying to prove
a pr i or i the reality of Euclid s syste m ; but so m e ti me within the next
thi rty years he arrived at the conclusion reached by L ob achevski
“
and Bolyai In 1 8 2 9 he wrote to F W B essel s tating that hi s con
v i c t i on that we cannot found geo metry co mpletely a pr i or i has b e
“
come if possible still firm er and that if nu mber is m erely a product
of ou r m ind S pace has also a r eal i ty beyond our m ind of whi ch w e
cannot fully foreordain the laws a pr i or i
The term n on Eu cl i dean
t
h
e
f
i
s
f
l
i m se s of
It
is
surprising
that
m
t
r y is due to Gauss
eo
e
_
r
p
g
non Euclid ean geom etry were had I n the eighteenth _centu ry Ger on
a J esuit father of M ilan i n 1 733 wrote
n i mo S a ccher i ( 1 6 6 7
'
( 1 80 2
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A H I STORY O F M AT H E M ATIC S
3 06
contemporary researches on parallel lines due to A M Legendre in
France
The researches of K F G auss N I L ob achev sk i and J Bolyai
have been considered by F Kl ein as c on s ti t u ti rig the first p eriod I n
the histo ry of non Euclidean geometry It is a period I n which the
syhth et i c m
eflI Od s of elementary g eom etry were in vogue
The
second period e mbraces the researches of G F B Riemann H Hehn
holt z S Lie and E B eltra m i and e mp l oys the m ethods of diff erential
geometry It was in 1 8 54 that Gauss heard from his pupil Riemann
a marvellous dissertation whi ch considered the foundations of g eome
try fro m a new point of view Riem ann was not fa miliar with Lo
b achev ski and Bolyai
He developed the notion of n ply extended
magni tude and the m easure relations of which a manifoldness of n
dimensions is capable on the assu mption that every line may be
measured by every other Rie mann applied his ideas to space He
taught us to disti n guish between
f infin i t e
extent Acco rdi ng to hi m we have I n our m ind a m ore general notion
of space i c a notion of non Euclidean space but we learn by axpe
r i en ce that our physical space is if not exactly at least to a high degree
of approxim ation Euclidean space Riemann S profound dissertation
was not published until 1 8 6 7 when it appeared in the Gotti ngen A b
B efore this the idea of n dim ensions had suggested itself
ha ndl u ngen
under various aspects to P tolemy J Wallis D A l emb er t J Lagrange
m
and
H
G
Grass
ann The idea of ti me as a fourth di
P
l
ii
k
e
r
c
J
men si on had occur red to D Al emb er t and Lagrange About the sa m e
ti me wi th Rie m ann s paper others were published from the pens of
This period marks the beginning
H H el mhol tz and E B el tr ami
—
of lively discussions upon this subj ect So me writers J B ell avi ti s
for e xample—were able to see in non Euclidean geo metry and n
dirn en s i on al space not hi ng but huge caricatures or diseased out
growths of mathe matics H Hel mholt z s article w as en t i tl ed That
sachen
w el che der Geometr i c zu Gr u nde l i egen 1 8 6 8 and contained
many of the ideas of Riemann Helmholt z popul ariz ed the subj ect in
lectures and in articles for various m aga zines Starting with the
idea of congruence and assuming the free m obility of a rigid body éfid
the return unchanged to its original position after rotation about an
axis he proves that the square of the line elem ent 15 a homogeneous
function of the second degree in the diff erentials
Hel mholtz s
investigations were carefully exam ined by S Lie who red uced the
Riemann Hel mholtz problem to the following for m : To determ ine
all the continuous groups in space which in a boun ded region have
the property of displacements There arose three types of groups
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S om merv ill e i s t he au thor o f a B i bli ography of n on E ucl idea n g eometr y i n cl udi ng
the theor y of para ll el s , the f ou ndati on s of geometry, and space of n di mens i on s , L o n don ,
-
19 1 1
1
.
D M Y S o mm erv il l e ,
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op ci t
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p
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1 95
.
SYNTHETIC GEO M ETRY
30 7
which characteri ze the three geo metries of Euclid of N I L obach ev
1
ski and J B olyai and of F G B Rie mann
Eu g e ni o B e l trami ( 1 83 5
born at Cremona wa s a pupil of
F Brioschi ? H e was professor at Bologna as a colleague of L Cremo n a
Betti at P av ia as a c o worker w ith F
a t P isa as an associate of E
C a sora t i and since 1 8 9 1 at Ro m e where he S pent the last years of
his career uno degli illustri maestri dell analisi in Italia
Beltrami
w rote in 1 868 a classical paper S aggi o di i n ter pr etazi on e del l a g eometri a
n on en cl i dea (Gi or n
which is a nalytical ( and lik e
di M atem
se v eral other papers S hould be m entioned elsew here were we to adhere
to a strict separation between synthesis and analysis ) He reached
the b rilliant and surprising conclu sion t hat in part t he the OremS of
non Euclidem g eome try find thei r fr eal ization u pon surfaces of con
stant neg at ive curvature H e studi ed also surfaces of constant
posi tive curvatur e and e nded with the interesting theorem t hat
the spac e o f constant positi v e curvature is contained in the space
of con sta nt negative curvat u re These researches of B eltrami H
Helmholtz and G F B Riem ann cul minated in the conclusion that
on sur faces of constant curvature we may have three geo metries
the non Eucli d ean on a surface of constant negative curvature
the spheri cal On a sur face of constant positive cur vature and the
Eucl idean geo metry on a surface of z ero cu rvature The three g e
o me t r i es do not contra dict each oth er but are me mbers of a system
a geo m etrical trinity
The ideas of hyper S pace were brilliantly ex
poun ded and popularised in England by Cl ifford
William Ki n g d on Cl iffor d ( 1 84 5—1 8 79)
was b orn at E xeter edu
c a t ed at Tri n ity College , C ambridge and fro m i 8 7 1 until his death
professor of applied mathematics in University College L ondon His
premature death left incomplete several brilliant researche s which
he had entered upon Am ong these are his paper On Cl assificati on
of L oci and his Theor y o Gr aphs
n the C a n on i cal
He
wrote
articles
O
f
Form a nd Di ss ecti on of a Ri eman n s S u rf a ce on B i qn a ter n i on s and
an incomplete work on the El emen ts of Dyn ami c He gave exact
“
meaning in dynamics to such fam iliar words as
spin
twist
“
“
sq ui rt
whi rl
The theor y of polars o f _c urves and surface s was
generaliz ed by him and by Reye H i s cl assifi c a ti on of loci 1 8 7 8
b eii In
en erab s t udy of cu rves Wa s an introduction to the study
i
o f_
e
n si on a l S pace in a di rec t ip n
ainly
proj
ective
Th
i
s
study
i
m
;
_
ni
m
has been continued since chi efly by G Veronese of P adua C Segre
of Turin E Bertini F A schi er i P Del P e zz o of Naples
Beltrami s researches on non Euclidean geom etry were followed
in 1 8 7 1 by important investigations of Felix Klein resting upon
Cayley s S i xth M emoi r on Qu a nti cs 18 59 The develop m ent of geom
e t ry in the first half of the nineteenth century had led to the separation
L i e Th ar i der T nsf ormati on sgru ppen Bd III L ei pzi g 1 893 pp 4 3 7 5 43 ;
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1 54
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A HI STORY OF M ATHE M ATIC S
3 8
0
of this science into two parts : the geomet ry of position or descriptiv e
geo metry which dealt with properties that are unaff ected by proj e c
t ion and the geo m etry of m easure m ent in w hi ch the funda mental
notions of di stance angle etc are changed by proj ection Cayley S
S i xth M emoi r brought these strictly segregated parts together again
by his definition of distance between two points The question
whether i t is not possible so to e xpress the metrical properties of
figures that they will not vary by proj ection (or linear transformation )
had been sol v ed for special projections by M Chasles J V P oncelet
and E Laguerre but it re mained for A Cayley to give a general
solution by defining the distance between two points as an arbitrary
constant m ultiplied by the logarith m of the anharmonic ratio in which
the line joining the two points is divided by the fundamental quadric
These researches applying the principles of pure projective geo metry
m ark t h e third period in the develop m ent of non Euclidean geo m etry
Enlarging upon this notion F Klein S howed the independence of
projecti v e geo m etry fro m the parallel axio m and by properly choosing
th e law of the m easure m ent of d i stance deduced fro m proj ecti v e
geom etry the spherical Euclidean and pseudospherical geom etries
nam ed by h im respectively the elliptic parabolic and hyperbolic
geom etries This suggestiv e i n vestigation was followed up by nu m er
ous writers particularly by G B attaglini of Naples E d O v i dio of
Turin R de P aolis of P isa F A sc hi e ri A Cayley F Lin d emann of
M unich E Schering of Gottingen W Story of Clark University
H S ta hl of T ubingen A Voss of M unich H om ersh am Cox A B uch
1
heim The notion of parallelis m applicable to hyperbolic S pace was
the only extension of Euclid s notion of pa rallelis m until Cli fford dis
covered i n elliptic S pace straight lines which possess m ost of the prop
e r t i e s of Euclidean parallels but di ffer fro m the m in being skew
Two
lines are right (or left )
parallel if they cut the sam e right (or left)
generators of the absolute
Later F Klein and R S Ball m ade
extensiv e contributions to the knowle dge of these lines M ore re
c en t l y E Stu dy of B o n n J L Coolidge of Har v ard University W
Vogt of Hei delberg and others hav e been studying this subject The
m e thods e m ployed hav e been those of analytic an d synthetic geom etry
2
as well as those of d iff erential geo metry and vectorial analysis The
geo metry of n dimensions was stud ied along a line mai n ly m etrical
by a host of writers a mong who m may be m entioned Simon N ew c omb
of th e Johns Hopkins Univ ersity L S chl afl i of B ern W I Stringham
—
1
of the Univ ersity of California W Killing of M unster
8
( 4 7 1 90 9)
—
T Craig of the Johns Hopkins Ru dolf Lipschitz ( 1 83 2 1 90 3 )
of B onn
R S Heath of B irm ingha m and W Kill ing in v estigate d the k i nem at i cs
Regular sol i ds in n dim ens ional space
a n d m echanics of such a S pace
—
i
were studied by Str ngham Ellery W D avis ( 1 8 5 7 1 9 1 8)of th e
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G Lo r i a , Di e ha npts a chl i chs ten Theor i en der Geometr i e,
B u l l A m M a th S oc , V ol 1 7 , 1 9 1 1 , p 3 1 5
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1 888 ,
p
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10 2.
A HI STORY OF M ATHE M ATIC S
3 10
A n al yti s ch Geometr i s che En twi ckl u n gen
in two volum es Therein he
adopted the abbreviated notation [ used before hi m I n a more restricted
way by Etienne B ob il l i er ( 1 797
professor of m echanics at
Chalons sur M arne] and avoided the tedious process of algebraic
elim ination by a geom etric consideration In the second volume the
principle of duality is form ulated analytically With h im duality and
homogeneity found e xpression already in his system of c o ordinates
The ho mogenous or trilinear system used by him is m uch the same as
the c o ordinates of A F M ob i u s I n the i d entity of analytical opera
tion and geom etric construction P l ii c k er looked for the source of his
e
e
e
ri e
contains a
roofs
The
s
t
e
m
er
n
a
l
t
i
s
c
h
n
m
t
1 83 5
S
d
A
G
o
p
y
y
complete classification of plane curves of the third order based on the
nature of the points at infinity The Theari e der A l gebrai schen Cu rven
1 8 39 contains besides an enu m eration of curves of the fourth ord er
the analytic relations betw een the ordinary singularities of plane
“
curves know n as P lucker s equations by which he was able to
“
explain P oncelet s parad ox
The discovery of these relations is
“
says A Cayley the m ost i mportant one beyond all comparison in
the entire subj ect of m odern geom etry
The four P lucker equa
tions have been e xpressed in di fferent form s Cayley studied higher
singularities of plane curves M W Haskell of the University of
California in 1 91 4 S howed from the P l ucker equations that the
m aximu m num ber of cusps possible for a curve of order m is the
( except when m is 4 and 6 i n which
g reatest integer i n m (m
case the m aximu m nu mber is 3 and
and that there is always a
self dual curve with this m aximu m num ber of cusps
Certain interrelations of the various geom etrical researches of the
first half and mi ddle of the nineteenth century are brought out by
1 “
J G D arb ou x in the following passage : While M C hasles J
Steiner and later
v on Staudt were intent on constituting a
riv al doctrine to analysis and set in so m e sort altar against altar
J D G erg onn e E B ob illi er C Sturm an d above all J P l ii ck er per
f ec te d the geom etry of R D escartes and constituted an analytic sys
t em in a m anner adequate to the disco v eries of the geo m eters
I t is
to E B ob ill i er and to J P l ii ck er that we owe the m ethod called
a br i dged n ota ti on
B ob ill i er consecrated to it so m e pages truly n ew
in the last volu mes of the A n n al es of G erg on n e P l ii ck er comm enced
to develop it in his first work soon followed by a series of works where
are established i n a ful ly conscious m anner the foundations of the
m odern analytic geom etry It is to him that we owe tangential co
ordinates trilinear c o or dinates employed with ho m ogeneous equa
tions and finally the e mploym ent of canonical form s wh ose val i dity
was recogniz ed by the m ethod so deceptive so m etim es but so fru i t
ful cal led the en u mer ati on of con s ta n ts
In Germ any J P l ii c k er s researches m e t with no favor His m ethod
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Congr ess
f
o
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A r ts
a nd
S ci ence, S t L o u i s ,
.
1 9 04 ,
Vo l 1, pp 5 4 1 , 5 4 2
.
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.
ANAL YTI C GEO M ETRY
311
was declared to be unproductive as compared with the synthetic
m ethod of J Steiner and J V P oncelet ! His relations with C G J
Jacobi were not altogether friendly S teiner once declared that he
woul d stop writing for Cr el l e s Jou rn al if P lucker continued to con
1
tribute to i t The result was that m any of P l ii c k er s researches were
published in foreign journals and that his work cam e to be better
known i n France and England than in his native country The charge
was also brought against P l ii ck er that though occupying the chair
of physics he was no physicist This induced him to relinquish
m athe m atics and for nearly twenty years to devote his energies to
physics I mportant discoveries on Fresnel s wa ve s urface m agnetis m
spectru m analysis were m ade by him B ut towards the close of his
—
—
m
life he returned to his first love
athe matics and enriched it with
n ew discoveries
By consid ering S pace as m ade up of lines he created
“
a n ew geom etry of space
Regarding a right line as a curv e i n
volving four arbitrary param eters one has the whole system of lines
“
in space B y connecting the m by a single relation he got a co mplex
“
of lines ; by connecting the m with a twofold relation he got a con
His first researches on this subj ect were laid before
g r u e n cy of lines
the Royal Society in 1 8 6 5 His further investigations thereon ap
p e ared i n 1 8 6 8 i n a posthu m ous work entitled N eu e Geometri e des
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Rau mes gegr un det
'
f
di e B etr achtu n g der ger aden Li n i e
’
Klein P l ii ck e r s analysis lacks
au
al s
Kou mel e
edited by Felix
the elegance
found in J Lagrange C G J Jacobi L O Hesse and R F A
C l e b s ch
For m any years he had not kept up with the progress of
geo m etry so that m any invest igations in his last work had already
receiv e d m ore general treatm ent on the part of others The work
co ntaine d nev ertheless m uch that was fresh a n d origi n al The theory
of complexes of the second degree left unfinished by P l ucker was
continued by Fel ix Klein who greatly e xtended and supplem ented
the ideas of —
h l S m aster
—
i
d
tt
e
s
Lu w g O o H s e ( 1 8 1 1 1 8 74)
was born at Konigsberg and
stu died at the university of his nativ e place under F W B essel C G J
Jacobi F J R i c hel o t and F Neu m ann Ha v ing taken the doctor s
d egree i n 1 84 0 he beca m e docent at K onigsberg and in 1 8 4 5 e x t ra or
di n a ry professor th ere A m ong his pupils at that ti m e were H ei n ri ch
Du r ege ( 1 8 2 1 —
of P rague Carl Neum a n n R F A C l eb sc h
1 8 93 )
G R Kirchho ff The K onigsberg period was one of great activity
for Hesse Eve ry new discovery increased his z eal for still greater
achievem ent His earliest researches were on surfaces of the second
order and were partly synthetic He solved the proble m to construct
any tenth point of such a surface when n ine points are giv en The
analogous problem for a conic had been solv ed by P ascal by m eans
of the hexagra m A difficult proble m confronting m athem at i c i ans
of this tim e was that of el im i nati on J P l ucker had seen that the
men t,
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A d D ro n k e , J
.
u l i us
P l uck er , B o nn ,
‘
1871
.
A HI STORY OF M ATHE M ATI C S
12
m ain
adv antage of his special m ethod in analytic geom etry lay in
the avoidance of algebraic elimination Hesse however showed h ow
by determ inants to make algebraic elim ination easy In his earl ier
results he was anticipate d by J J Sylvester who published his dialytic
These advances in algebra Hesse
m ethod of eli m ination i n 1 840
applied to the analytic study of curves of the third order By linear
substitutions he reduced a form of the third d egree I n three variables
to one of only four terms an d was led to an i mportant determ inant
inv ol v ing the second d i fferential coeffi cient of a form of the third
“
d egree called the
Hessian
The Hessian plays a lea d ing part
in the theory of invariants a subj ect first studied by A Cayley
Hesse sho w e d that his determ inant give s for every curve another
curv e such that the double points of the first are points on the second
M any of the
r Hessian
Sim i larly for surface s ( C r el l e
m ost i mportant theorem s o n curves of the third or der are due to
Hesse He d eterm i ne d t h e curve of the 1 4 th order which passes
through the 5 6 points of contact of the 2 8 b i tangents of a curv e of
the fourth order His great m e m oir on this subj ect ( C r ell e 1 8 55)
was published at the sam e tim e as was a paper by J Steiner treating
of the sam e subj ect
Hesse s incom e at Konigsberg had not kep t pace with his growi ng
reputation Hardly was he able to support himself and family In
1 8 5 5 he accepted a m ore lucrative position at Halle and in 1 8 5 6 one
at Heidelberg Here he rem ained until 1 8 6 8 when he accepted a
1
At Heidelberg he revised
posi tion at a technic school In M unich
and enlarged upon his previous researches and published I n 1 8 6 1 his
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Vor l es u n gen ti ber di e A n al yti s che Geometr i e des R au mes , i ns bes on der e
M ore elementary works soon followed
uber Fl a chen
2
Or dn u ng
“
While in Heidelberg he el aborated a principle , his U eb er trag u ng s
princip
Accordi ng to thi s , there corresponds to every point in a
”
.
.
.
.
.
plane a pair of points in a line and the proj ective geo metry of the
plane can be carried back to the geo metry of points in a line
The researches of Plii ck er and Hesse were continued in England
by A Cayley G Sal mon and J J Sylvester I t m ay be premised
here that amon g the early writers on analytical geom etry in England
was J ame s B oo th ( 1 80 6
whose chief results are e mbo di ed in hi s
Tr eati se on S ome N ew Geometr i cal M ethods and J a m e s M a c C u l l a g h
who was professor of natural p hi losophy at D ublin
( 1 80 9
and made som e valuable discoveries on the theory of quadrics The
influence of these m en on the progress of geom etry was insign ificant
for the interchange of scientific results between di fl er en t nat i ons w as
not so complete at that tim e as m ight have been desired In further
illustration of this we m ention that M Chasles in France elaborated
subjects whi ch had previ ously been disposed of by J Steiner in Ger
many and Steiner publ i shed researches which had been gi ven by
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G u s ta v B a u er , Ged achtn i ssr ede
f
au
Otto Hess e, M un ch en ,
1 88 2 .
3
A HI STORY OF M ATHE M ATIC S
14
had shown how elliptic functions could be advantageously
applied to M alfa t ti s problem The idea involved therein vi z the use
of higher transcendentals in the study of geom etry led hi m to his great
est di scoveries Not o nl y did he apply Abelian functions to geom etry
but con v ersely he drew geom etry into the servi ce of Abelian functions
C l eb s ch made liberal use of determ inants His study of curves and
surfaces began with the determination of the points of contact of lines
which m eet a surface in four consecutive points G Salm on had
pro v ed that these points lie on the intersection of the surface with a
—
deri v ed surface of the degree 1 1 n 2 4 but his solution was gi v en in
i nconvenient for m C l eb sc h s investigation thereon is a most beautiful
piece of analysis
The representation of one surface upon another (Fl achen abbi l du ng)
so that they have a ( 1 1 )
correspondence was thoroughly studied for
the first ti me by C l eb s c h The representation of a S phere on a plane
is an old proble m whi ch drew the attention of P tole my Gerard M er
cator J H La mbert K F Gauss J L Lagrange Its i mportance in
the construction of m aps is obvious Gauss was the first to represent
a surface upon anothe r w ith a view of more easily arriving at its
properti es J P l ucker M Chasles A Cayley thus represente d on a
plane the geom etry of quadric surfaces ; C l eb s ch a n d L Crem ona that
of cubic surfaces Other surfaces have been stu died in the same way
by recent writers particularly M ax N other of Erlangen A ng el o
of Ro m e Fel i x Kl ei n Georg H L K or ndorf er
A r men a n te ( 1 844—1 8 7 8 )
of Naples H G Z eu then of Cope nhagen
E ttor e C apor al i ( 1 8 5 5—1 8 8 6 )
A funda mental question which has as yet received only a par tial a n
swer is this :What surfaces can be represented by a ( 1 1 )
correspond
ence upon a given surface ? This and the analogous question for
curves was stu died by C l eb sch Higher correspon dences between
surfaces hav e been investigated by A Cayley and M 1 Nother I m
portant bearings upon geometry has Riemann s theory of birational
transfor m ations The theory of surfaces has been stu died by J os e ph
i
P
professor
at
the
Sorbonne
in
ar
s Jea n
Al fr e d S erre t
)
5
—
of D ublin Wi l l i a m
Gaston Da r bou x of P aris J ohn C as ey ( 1 8 2 0 1 8 9 1 )
—
—
n
r
t
r
r
h
of
D
ublin
e
i
i
h
S
b
e
1
8
2
of
H
c
c
Rober ts ( 1 8 1 7 1 88 3 )
(
9 1 8 92 )
B reslau El wi n B r u n o C hri stofi el ( 1 8 2 9
professor at Zurich
later at S trassburg Chri stoffel wrote on the theory of potential on
minim al surfaces on the so called transfor mation of Christoff el of
i sothermic surfaces on the general theory of curved surfaces His
researches on surfaces were ex tended by Ju l i u s Wei ngarten 1 83 6—1 9 1 0 )
of the University of F r ei b u rg and H a n s von M angol dt of Aachen in
A S we shall see m ore ful ly later surfaces of the fourth order
1 88 2
were investigated by E E Ku mmer and Fresnel s wave surface
stu di ed by W R Ham i lton is a part i cular case of Kummer s quartic
1
surface with sixteen double points and sixteen S ingular tangent planes
C l e b sc h
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A C ayl ey , I nau g u ral A dd ress ,
.
1 88 3 .
ANALYTIC GEO M ETRY
315
P ro m inent
in these geometric researches was J ean G as ton Da r
b o u x ( 1 84 2
He was born at Nim es found ed in 1 8 70 w i th the
—
collaboration of Gu i l l a u me J u l es H oitel ( 1 8 2 3 1 886 )
of B ordeaux and
J u l es Tan n ery the B u l l eti n des s c i en ces ma théma ti q u es et as tr on omi qu es
and was for half a cent ury conspicuous as a teacher In 1 900 he
becam e permanent secreta ry of the P aris Academy of Sciences in
which position he was succeeded after hi s death by E mil P icard B y
his researches D arb oux enriched the synthetic analytic and i n fi n
i t esi mal geo metries as well as rational m echanics and analysis
He
wrote L egons su r l a theori e gen er al e des su rf aces et l es appl i cati on s
—
t
r i u es du cal cu l i nfi n i tes i mal
é
o
m
e
P
aris
88
1
g
7 1 8 96 and L egons su r
q
l es systemes or thogon au x ct l es coor donn ees cu r vi l i gn es P aris 1 8 98
He
investigated t riply orthogonal ystem s of surfaces the deformat i on
of surfaces and rolling of applicable surfaces infi nitesimal deforma
tion spherical representation of surfaces the develop m ent of the
moving axes of co ordinates the use of i magina ry geo m etric elem ents
1
the use of isotropic cylinders and dev el opab l es ; he introduced
p entaspherical coordi nates
“
Eisenhart says : B arbou x was a strong ad vocate of the use of
i maginary el em ents in the study of geo metry He believ ed that their
use was as necessary in geometry as in analysis He has been i m
pressed by the success with which they have been e mployed in the
sol ution of the problem of m inimal surfaces Fro m the very beginning
he m ade use in his papers of the isotropic line the null sphere ( the
isotropic cone )
and the general isotropic dev elopable In his first
m emoir on orthogonal systems of surfaces he S howed that the envelope
of the surfaces of such a system when defined by a S ingle equati on
is an isotropic developable
D ar b o u x gi v es to E douard C om
the credit of being the first to apply the consi dera
b escu re ( 1 8 1 9—
P)
tions of kine matics to the study of the theory of surfaces with the
consequent use of m oving c o or dinate axes B ut to D arb o u x we ar
indebted for a realiz ation of the power of this m ethod and for its
D ar b o u x s ability was
systematic development and exposi tion
based on a rare co mbination of geo metrical fancy and analytical
power He did not sympathi z e with those who use only geo metrical
reasoning in attacking geom etrical problems nor with t hose who feel
that there is a certain vi rtue in adhering strictly to analytical proc
esses
In common with M onge he was not content with dis
but fel t that i t was equally important to make di sciples
c ov er i es
Like thi s di stinguished predecessor he developed a large group of
geometers including C Guichard G Koenigs E C o ss era t A D e
m oulin G Tzi t z ei ca and G D emar t r es Their brilliant researches
a re the best tribute to his teachi ng
P roceeding to the fuller consi deration of recent de v elop m ents w e
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A m M a th M on thl y, V o l 2 4 1 0 1 7 p 3 54
S ee L P E i sen h ar t s
C o n t r i b u t io n t o G eome t ry i n B ull A m M a th S oc , V ol 2 4 , 1 9 1 8 , p
1
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D a rb o u x
2 2 7.
’
s
A
31 6
H I STORY
or M AT H E M ATIC S
quote from H F B aker s address before the International Congress
1
held at Ca mbridge in 1 9 1 2 : The general theory of Higher P lane
Cur ves
would be impossible without the notion of the genus of a
curve The inv estigation of Abel of the nu mber of independent i n
t eg ral s in ter m s of which his integral su m s can be e xpressed m ay thus
be held to be of paramount importance for the general theory This
was further e mphasi z ed by G F B Riem ann s consideration of the no
tion of birational transformation as a fundamental principle After
thi s two streams of thought were to be seen First R F A C l eb s ch
remarked on the existence of invariants for surfaces analogous to the
genus of a p l ane curve Thi s number he defined by a double integral ;
it was to be unaltered by birational transformation of the surface
B ut
C l eb sc h S idea was carried on and developed by M Noether
also A B rill and Noe ther elaborated in a g eom etrical form the
results for plane curv es which had been ob tained with t ran s c en
dental consi derations by N H Abel and G F B Riemann Then
the geo meters of Italy took up N o ether s work with very remark
able genius and carried i t to a high pitch of perfection and clear
ness as a geome t rical theory In connection therewith there arose
the i mportant fact which does not occur in N oe ther s papers that
it is necessary to cons i der a surface as possessing two genera ; and
the nam es of A Cayley a n d H G Zeuthen S houl d be referred to at
this stag e B ut at this tim e another stream was running in France
E P icard was develop i ng the theory of Riemann integrals—S ingle
integrals not double integrals— upon a surface How long and
laborious was the task m ay be j udged fro m the fact that the publica
tion of P icard s book occupied ten years—and may even then have
seemed to m any to be an artificial and unproductive i mitation of
the theory of algebraic integrals for a curve In the light of s ub s e
quent events P icard s book appears likely to remain a permanent
landmark in the history of geometry For now the two strea m s
the purely geo metrical in Italy the transcendental in France have
united The results appear to me at least to be of the gr eatest i m
portance
The work of E P icard in question is the Thear i e des
fon cti on s al gébri q u es de deu x var i abl es i ndépendan tes which was brought
out in conj unction with Georges Si mar t between the years 1 897 and
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1 90 6
.
H F Baker proceeds to the enumeration of some indivi dual r e
s u l t s : Gui d o Caste l nuovo of Rom e has shown that the deficiency of
the characteristic series of a linear system of curves upon a surface can
not exceed the di ff erence of the two genera of the surface F ederig o
Enriques of Bologna has completed this result by show ing that for an
algebraic syste m of curves the characteristic series is complete Upon
this resul t and upon E P icard s theory of integrals of the second
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,
P r oceed of the 5 th I n ter n C on gr ess , V ol I , C a mb r i dg e , 19 1 3 , p 4 9
Fo r m o r e
d e t a il , c o n s u l t H B B a k e r i n t he P r oceed of the L on don M a th S oc , V o l 1 2 , 1 9 1 2
1
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A HI STORY OF M ATHE M ATIC S
1
3 8
”
theory of cubic surfaces This w a S don e later also by L Cre mona and
R Stur m between whom in 1 866 the S teiner pri z e was di vided
An elegant notation w as invented by Andrew Hart but the notation
which has me t with general adopti on was advanced by L S chl afli of
B er n i n 1 8 58 i t is that of the double S ix S chl afli s double six theorem
“
proved by him and by m any others S ince is as follows : Gi v en fi ve
l ines a b c d e whi ch m eet the sa m e straight line X ; then m ay any
four of the five lines be intersected by another li ne Suppose that
A B C D E are the other lines intersecting (b c d e)(c d e a )
and (a b c d)
respectively T hen A B C D E
(d e a b)(e a b c )
will all be me t by one other straight line x
L S c hl afli first considered a division of the cubic surfaces into
species in regard to the reality of the 2 7 lines His final classification
was adopted by A Cayley In 1 8 7 2 R F A C l eb sc h constructed a
“
m odel of the di agon al s u rf ace with 2 7 real lines w hi le F Klein es
t ab l i sh ed the fact that by the principle of continuity all form s of real
surfaces of the third order can be deri v ed fro m the particular surface
ha ving four conical points ; he exhibited a co mplete set of models
of cubic surfaces at the World s Fai r in C hi cago in 1 8 94 In 1 86 9
C F Geiser showed that the projection of a cubic surface from a point
upon i t on a plane of proj ection parallel t o the tangent plane at that
i c cur v e ; and that every quartic cur v e can be generate d
oint
is
a
quart
p
“
i n this way
The theory of var i eti es of the third order says A
Henderson that is to say curved geom etric form s of three dim en
sions contained in a Space of four di mensions has been the subject
of a profound m e moir by C or r ado S egr e ( 1 88 7)
of Turin The depth
and fecundity of thi s paper is evinced by the fact that a large pro
portion of the propositions upon the plane quartic and i ts bi ta ng en t s
P ascal s theore m the cubic surface and i ts 2 7 lines Ku m m er s sur
face and i ts configuration of sixteen singular points and pla n es and on
the connection between these figures are derivable fro m propositions
relating to Segre s cubic variety and the figure of si x points or spaces
fro m which i t S prings Other investigators into the properties of thi s
beautiful and i mportant locus in space of four dim ensions and som e of
its consequences a re G Castelnuovo and H W Richmond
In 1 86 9 C Jordan first proved that the group of the problem of
the trisection of hyperelliptic functions of the fi rst order is isomorphi c
with th e group of the equation of the 2 7 th degree on which the 2 7
l ines of the general surface of the thir d deg r e e depend
In 1 88 7 F
Klein sketched the e ffectiv e reduction of the one problem to the other
while H M aschk e H B urkhardt and A Witting completed the work
outlined by Klein The Galois group of the equation of t he 2 7 lines
was investigated also by L E Dickson F Kuhnen H Weber E
P ascal E K a s n e r a n d E H M oore
Surfaces of the fourth ord er ha v e been studied less thoroughly than
those of the third J S teiner worked out properties of a surface of the
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ANALYTI C GEO M ETRY
319
fourth order in 1 844 when he was on a journey to Italy ; that sur face
bea rs this na m e and later received the attention of E E Kumm er
1
In 1 8 50 Thom as Weddle remarked that the locus of the vertex of a
quadric cone passing through six given points is a quartic s urface
and not a twisted cubic as M Chasles had once stated A Cayley
gave a symm etric equation of the surface in 1 86 1 Thereupon Chasles
in 1 8 6 1 showed that the locus of the vertex of a cone whi ch di vides
six given segm ents harmonically is also a quartic surface ; thi s more
general surface was identified by Cayley with the Jacobian of four
q uadrics the Wedd l e surface corresponding to the case in which the
four quadrics ha v e S ix co mmon points P roperties of the Wed dle sur
face were stu died also by H B ateman
The plane section of a
We ddle surface is not an arbitrary quartic cur v e but one for which an
invariant vanishes Frank M orley proved that the curve contains an
6
i nfini ty of configurations B where it is cut by the lines on the sur
face
In 1 863 and 1 8 6 4 E E Ku mmer entered upon an intensive study
of surfaces of the fourth order Noted is the surface named after him
which has 1 6 nodes The various shapes it can assum e have been
stu died by Karl Rohn of Leip zig I t has received the attention of
m any m athem aticians inclu d ing A Cayley J G D a rb ou x F Klein
2
H W Richm ond O B olz a H F Baker and J I Hutchinson I t has
been known for som e tim e that F r esn el s wave surface is a case of
Kummer s S ixteen nodal quartic surface ; al so it is known that the
surface of a dynamical m e diu m possessing certain general properties
is a typ e of Ku mmer s surface which can he derived from F r esn el s
surface by m eans of a ho m ogeneous strain Ku mm er s quartic surface
as a wave surface is treated by H B atem an
The general
Ku mm er s surface appears to be the wave surface for a medium of a
purely i deal character
F R Sharpe an d C F Craig of Cornell University have studied
birational transform ations which leave the Kummer and Weddle
urfaces invariant by the application of a theory due to F Severi
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Q uintic surfaces have been investigated at intervals since
,
1 86 2 ,
principally by L Crem ona H A Schwarz A C l eb sch M N oe ther
R Sturm J G D arb ou x E Caporali A D el Re E P ascal John E
Hill and A B Basset No serious attempt has been m ade to en u mer
ate the different form s of these surfaces
Ruled surfaces with isotropic generators have been considered by
G M onge J A Serret S Lie and others L P Eisenhart of P rinceton
d etermines such a surface by the curve i n which it is c ut by a plane
a n d th e directions of the projections on t h e plane of the generators
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Camb 69 Du bl i n M a th J our , V ol 5 , 1 8 50 , p 6 9
’
2
C o n su l t R W H T H u d so n ( 1 8 7 6
K u mmer s
br i dg e , 1 90 5
1
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Qu arti c
S u rf ace, C am
.
A HI STORY OF M ATHE M ATIC S
3 20
of the surface In th is way a ruled surface of this type is determ ined
by a set of lineal elem ents in a plane d ependin g on one param eter
While the classification of cubic curves was given by I Newton
and their general theory was well under way two centuries ago the
theory of quartic curves was not pursued vigorously until the ti me
of J Steiner and L O Hesse Neglecting the class i fication of quartic
curv es due to L Euler G Cram er and E Waring n ew classifications
0
2
1
have been made either according to their genus ( Geschlecht)
3
or according to topologic considerations studied by A Cayley in
H G Zeuthen
Christian Crone ( 1 8 7 7)and others
1 86 5
J Steiner in 1 8 5 5 and L O Hesse began researches o n the 2 8 double
tangents of a general quartic ; 2 4 inflect ions were found of which
G Salm on conjectured and H G Zeuthen proved that at m ost 8 are
real An enu m eration (containing nearly 2 0 0 graphs )
of the fun da
“
m ental for m s of quartic curves whe n proj ected so as to cut the line
i nfinity the least possible nu m ber of times was given in 1 8 96 by Ruth
Gentry ( 1 8 6 2
then of B ryn M awr College
Curves of the fourth order hav e receiv ed attention for many years
M ore recently a good deal has been written on special cu rves of the
fif th order by Fran k M orley Alfred B Basset Virgil Snyder P eter
Field and others
Gino Loria of the University of Genoa who has written extensively
on the history of geom etry and the history of curves in particular
has adv anced a theory of pa n al g eb r ai c curv es wh i ch are in general
transcen d ental curves By definition a pan al g eb rai c curve m ust
satisfy a certain differential equation A book of reference on curves
was published by Gomes Tei xei r a i n 1 90 5 at M adrid under the title
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Tr a tado de l as
cu rvas es
eci a l es n ota bl es
p
.
The i nfinitesim al calculus was first applied to the determination
of the m easure of curvature of surfaces by J Lagrange L Euler and
—
Jean Baptiste M arie M eu sn i er ( 1 7 54 1 7 93 )
of P ar i s noted for hi s
m ilitary as well as scientific career M e u sn i e r s theorem relating to
curv es drawn on an arbitrary surface was extende d by S Lie a n d i n
1 90 8 by E K as n er
The researches of G M onge and E P C Dupi n
were eclipsed by the work of K F Gauss who di sposed of this difli c u l t
subj ect in a way that opened new vistas to geom etricians His treat
m ent is embo died in the Di squ i si ti on es g en er al es ci r ca su perfi ci es cu r vas
and Un tersu chu n gen uber Gegen stan de der hoher en Geodas i e of
( 1 8 2 7)
1 8 43 and 1 8 46
In 1 8 2 7 he established the idea of curvature as it is
understood to day B oth before and after the tim e of Gauss various
d efin i t i ons of curvature of a surface had been advanced by L Euler
M eu sn i er M onge an d D upin but these di d not m eet with general
a doption From Gauss measure of curvatu re flows the theorem of
J oha n n A u gu st Gru n er t ( 1 7 9 7
professor in Greifswald and
foun der in 1 8 4 1 of the A r chi v der M a themati k u nd P hysi k that the
arithmetical m ean of the radii of curvature of all norm al sections
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A HI STORY OF M ATHE M ATIC S
32 2
The m etric part of differential geom etry occupied the attention
of m athe maticians S ince the tim e of G M onge and K F Gauss and
has reached a high degree of perfection Less attention has been
giv en until recently to projectiv e diff erential geom etry particularly
to the diff erential geom etry of surfaces G H Halphen started with
the equation y=f (x)
of a curve and determ ined functions of y dy/dx
2
2
d y/dx etc which are left invariant when x and y are subj ected to
a general proj ective transformation His early formulation of the
problem is unsymm etrical and unhom ogeneou s Using a certa in
system of partial diff erential equations and the geom etrical theory
of sem i covariants E J Wilcz ynski obtained hom ogeneous form s
1
such form s being deduced later also by Halphen Wilc z ynski treats
of the proj ective diff erential geo m etry of curves and ruled surfaces
these surfaces being prerequisite for his theory of S pace curves Wil
c zyn s k i treated ruled surfaces by a system of two l i near ho m ogeneous
d iff erential equations of the second order The m etho d was extended
to fi v e d imens i onal space by E B Stou ffer of the University of Kan
?
sas
D evelopable surfaces were stu died by W W D enton of the
Univ ersity of Illinois B elonging to projective differential geom etry
are J G D arb ou x s conj ugate triple system s which are generaliz ed
notions of the orthogonal triple system s The proj ective diff erential
geometry of triple system s of surfaces of one param eter families of
S pace curves and conj ugate nets on a curved surface and allie d topics
were studied by G ab ri el M ar c u s G r e e n ( 1 891
of Harvard
University
D iff erential projective geom etry of hyperspace was greatly advanced
by C Guichard who intro d uced two elem ents dependi ng on two va
r iabl es ; they are the r es eau and the con gru en ce
D iff erential geometry
of hyperspace was greatly enriched since 1 90 6 by Corrado Segre of
Turin and by other geom eters of I taly particularly Gino Fano of
3
Turin an d Federigo Enriques of B ologna ; also by A Ranum of
Cornell C H Sisam then of Illinois and C L E M oore of the M assa
c h u se tt s Institute of Technolo g y
The use of vector analysis in difl er en t ial geom etry goes back to
H G Grassm ann and W R Ham ilton to their successors P G Tait
C M axwell C B urali Forti R Rothe and others These m en have
“
“
introduced the term s grad
div
rot
A geom etri c study of
trajectories with the aid of analytic a n d chiefly contact transform a
tions was m ade by Edward K asn er of Columb i a Univ ersity in his
“
geom etric
P rinceton Colloquiu m lectures of 1 90 9 on the differential —
aspects of dynamics
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S ee E J Wil c yn sk i i n N ew Ha ven C oll oq u i u m, 1 906 N ew H a ve n , 1 9 1 0 , p 1 5 6 ;
al s o hi s P r oj ecti ve D i_
[j er en ti a l Geometr y of C u r ves a n d R u l ed S u rf aces , L e ip ig , 1 90 6
2
B u ll A m M a th S oc , V o l 1 8 , p 444
3
S e e En r i co B o mpi a n i i n P r oceed 5 th I n ter n C on gr ess , C a mbr idg e, C a mb r i dg e ,
1 9 1 3 , V ol II , p 2 2
z
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ANAL Y
GEO M ETRY
TI C
2
3 3
A n al ysi s S i tus
”
Various researches have been brought under the head of analysis
S itus
The subj ect was first inv estigated by G W Leibni z and
was later treated by L Euler who was interested in the problem to
cross all of the seven bridges over the
P regel river at Konigsberg without passing
7
twice over any one then by K F Gauss
D §
whose theory of knots ( Vers chl i ngu ngen )
Q
has been e mployed more recently by Jo
hann B enedict Listing ( 1 80 8—1 88 2 )of Gottingen Oskar Si mony
of Vi enna F D i ng el dey of D ar mstadt and others in their topologic
studies
P G Tait was led to the study of knots by Sir Wi l lia m
Thomson s theory of vorte x ato ms Through Rev T P Kirkm an
who had stu died the properties of polyhedra Tait was led to study
knots also by the polyhedral m ethod ; he gave the n u mber of form s
of knots of the fi rst ten orders of knottiness Higher orders were
1
treated by Kirk man and C N Little Th o ma s P e n yn gt on K i rk man
—
1
8
0
was born at Bolton near M anchester D uring boyhood
6
1 8 95 )
(
he was forced to follow his father s business as dealer in cotton and
cotton waste Later he tore away entered the University of D ublin
then beca me v icar of a parish in Lancashire A S a mathe matician
he was al most entirely self taught He wrote on pl u q u at er n i on s I n
vol v ing more i mag i nar i es than i j k on group theory on m athe
“
the most
m a t i c al mne m on i cs producing what D e M organ called
“
curious crocket I ever saw on the proble m of the fifteen school
girls who walk out three abreast for seven days where it is requi red
to arran g e them daily so that no two shall walk abreast more than
once This proble m was stud i ed also by A Cayley and Sylvester
and is related to researches of J S teiner
Another unique proble m was the one on the coloring of maps first
mentioned by A F M ob i u s in 1 840 and first seriously considered by
Francis Guthrie and A D e M organ How many colors are necessary
to draw any m ap so that no two countries hav ing a line of boundary
in common S hall appear in the sam e color ? Four di ff erent colors
are found e xperimentally to be necessary and suffi cient but the proof
is diffi cult A Cayley in 1 8 7 8 declared that he had not succeeded
in obtaining a general proof Nor have the later demonstrations by
A B Kempe P G Tait P J Heawood of the University of D urham
W E S tory of Clark University and J P eterson of Copenhagen
2
re moved the difli c u l ty Tait s proof leads to the interesting con
c l u s i o n that four colors may not be suffi cient for a m ap drawn on a
Further
m ult i ply connected surface like that of an anchor r i ng
studies of maps on such surfaces and of the problem in general are
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A M ac fa rl an e , Ten B r i ti s h
2
W A hren s , Un ter hal tu n g en
1
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M o thema ti ci an s
a nd
S pi el e,
,
1 90 1 ,
1916 ;
p
p 3 40
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1 2 2.
A HI STORY OF M ATHE M ATIC S
3 24
due to O Veblen ( 1 9 1 2 )
and G D B irkhoff
On a surface of
“
genus z ero i t is not known whether or not only four colors always
s ufli c e
A S imi lar question considers the m axim u m possible number
of countries when e v ery co untry touches every other along a line
Lothar H eff t er wrote on t hi s conundr um in 1 8 9 1 and again in la t er
articles as did also A B Kempe and others In the han d s of Rie mann
the analysis S itus had for its obj ect the determi nation of what remains
unchanged under transformations brought about by a co mbination
of infinitesim al di stortions In continuation of his work Walter Dyck
of M unich wrote on the analysis S itus of three di m ensional S paces
Researches of this sort have i mportant bearings in m odern mathe
m aties particularly in connection with correspondences and di ff er
1
e n ti al equations
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—di
I ntri ns i c C o
or
nates
a reaction against the use of the arbitrary Cartesian and polar
c o or di nates there ca m e the suggestion fro m the philosophers K C F
Krause ( 1 7 8 1
A P eters ( 1 8 0 3—
1 8 76)
that m agnitudes inherent
to a cur ve be used such as s the length of arc m easured from a fixed
point a n d (0 the angle which the tangent at the end of s m akes with
a fixed tangent Wi l l i am Whew el l ( 1 7 94 1 86 6 )
of Cambridge the
author of the H i s tory of the I ndu cti ve S ci en ces 1 8 3 7 —1 8 3 8 introduced in
“
intrinsic equation and pointed out its use in study
1 8 49 the na m e
ing successive evolutes and involutes The m ethod was used by Wil
—
1
1
liam Walton ( 8 3 1 90 1 )
of Cambridge J J Sylvester in 1 8 68 J
Casey in 1 8 6 6 and others Instead of using s and 50 oth er writers
have introduced the radius of curvature p and ha v e used either s
were e mployed by
and p or go and p The c o ordinates ( (p p )
L Euler and sev eral nineteenth century m athe maticians but alto
gether the c o—
or dinates (s p)
hav e been used m ost The latter wer e
used by L Euler in 1 74 1 by Sylvestre Fran cois Lacroix ( 1 7 6 5
by Thom as Hill ( 1 8 1 8—1 8 91 who was at one tim e presiden t of Harvard
College)and in recent years especially by Ernesto Cesaro of the
University of Naples who published i n 1 8 96 his Geometr i a i n tri ns eca
whi ch was translated into Germ an in 1 90 1 by G Kowalewski un der
?
the title Vorl es u ngen uber n atitr l i che Geometr i e
Researches along
of P aris the
1 90 6 )
this line are due also to Amed ee M annh ei m ( 1 83 1 —
designer of a well known S lide rule
The application of intrinsic or natural co ordinates to surfaces is
3
less co mm on Edward K asn er said in 1 90 4 that in the theory of
surfaces natural c o—
ordinates may be introduced so as to fit into the
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S ee J H adamard , F ou r L ectu res on M a thema ti cs del i ver ed a t C ol u mbi a U n i ver s i tv
i n 1 9 1 1 , N ew Y or k , 1 9 1 5 , L ec t u re III
’
2
N a t u rl i c h e K oo r
0 m i n form a t io n i s d raw n f ro m E Wo l ffin g s a r t i c l e o n
d i n a te n i n B i bl i otheca ma thema ti ca , 3 S , V o l I , 1 9 00 , p p 1 4 2 1 5 9
3
B u l l A m M a th S oc , V o l 1 1 , 1 90 5 , p 3 0 3
1
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A HI STORY OF M ATHE M ATIC S
3 26
see m desirable to depart from our e mpirical notions so far as to allow
the ter m curve to be applied to a region more restricted definitions
of i t beco m e necessary C Jordan de manded that a curve
S hould have no double points i n the interval a S t S b
Schon
fl i es regards a cur v e as the frontier of a region 0 Veblen defines it
in terms of order and linear contin ui ty W H Young and Grace C
Young in their Theory of S ets of P oi n ts define a curve as a plane set
of points dense nowhere in the plane and bearing other restrictions
yet such that it m ay consist of a net work of arcs of Jordan curves
O ther curves of previously unheard of properties were created as
the resul t of the generaliz ation of the function concept The con
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n(
2 b
)where
= oo
n
curve represented by
t in uou s
y
=
.
n
a x
c os
,
a
is an even
integer 1 b a real positive nu mber 1 was shown by We i erstrass
to possess no tangents at any of its points when the product ab e xceeds
1
,
,
a certain l imit ; that is we have here the startling phenomenon of a
continuous function which has no derivative As Christian Wiener
explained in 1 88 1 this curve has countless oscillations withi n e v ery
finite interval An intuitively si mpler curve was inv ented by Helge
v Koch of the U niversity of S tockholm in 1 90 4 (A cta ma th Vol 30
p 1 4 5)
which is constructe d by ele m entary geom etry is con
1 90 6
ti n u ou s yet has no tangent at any of its points ; the arc between any
two of its points is infinite in length Wh i le this cur v e has been r epr e
sented analytically no such representation has yet been found for
the so called H curve of Ludwig B oltzmann
of Vienna
in M ath A n n al en Vol 50 1 8 98 which is continuous yet t an g e n tl ess
The a djoining figure shows its construction B oltzmann used it to
visualiz e theorem s in the theory of gases
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Fu nda mental P os tu l ates
The foundations of mathe matics and of geom etry in particular
received m arked attention in Italy In 1 8 8 9 G P eano took the nov el
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1
P
d u B o i s R eym o n d
r ee l l e r A rg u me n t e ,
-
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t io n e n
”
G
V e rsu c h e i n e r K l a ss i fi ca t io n d e r
rel l e ,
Vo l 7 4 ,
.
1 8 74,
p
.
29
.
w i llk u rl i c h e n F u n k
AN A LYTI C
GEO M ETRY
327
iew tha t geome tric elements are mer e thi ngs and laid down the
principle that there S hould be as few undefined symbols as po s sible
—
of Catania used o n ly two
In 1 8 97 9 his pupil M ari o P i eri ( 1 86 0—
1 90 4 )
undefined symbols for proj ective geometry and but two for metric
geometry In 1 894 P eano considered the independence of axioms
B y 1 897 the Italian m athematicians had gone so far as to make it a
postulate that points are classes These fundamental features elab
orated by the Itali an school were embodied by D avid Hilbert of
G ottingen along with important novel considerations of his own in
his famous Gr u ndl agen der Geometri e 1 8 99 _
A fourth enlarged e di tion
of thi s appeared in 1 91 3 E B Wilson says i n praise of Hilbert : The
a rc hi medean a xio m the theore m s of B P ascal and G D esargues the
analysis of segm ents and areas and a host of things are treated either
for the first time or in a new way and with consum mate ski ll We
should say that i t was in the technique rather than in the p hilosophy
1
In H i lbert s S pace of
of geometry that Hilbert created an epoch
1 8 99 are not all the points which are in our S pace but only those that
starting from two given points can be constructed with rul er and
compasses In his space remarks P oincaré there is no angle of
So in the second e di tion of the Gr u ndl agen Hil bert introduced the
assum ption of completeness which renders hi s space and ours the
same Interesting is Hilbert s treatment of non Arc hi m edean geom
e try where all hi s assumptions re main true save that of Archi m edes
and for which he created a system of non Archim edean numbers
This non Archim edean geom etry was first conceived by Gi u s eppe
Ver on es e ( 1 8 54
professor of geometry at the University of
P adua
Our common space is only a part of non Archi m edean
S pace
Non—
Archim edean theories of proportion were given in 1 90 2 by
A Kneser of B reslau and in 1 90 4 by P J M ollerup of Copenh agen
Hilbert dev oted in his Gru ndl agen a chapter to D esargues theorem
In 1 90 2 F R M oulton of C hi cago outlined a simple non D esarg u esi an
plane geometry
In th e United States George B ruce Halsted based his Ra ti on al
G eometry 1 90 4 upon Hilbert s foundations A second revised edition
appeare d in 1 90 7 One of Hilbert s pupils M ax W D ehn showed
that the om ission of the axiom of Archim e des ( Eudoxus )
gives rise
to a sem i Euclidean geom etry in which S im ilar triangles exist and their
s u m is two right angles
yet an infinity of parallels to any straight
l ine may be drawn through any given point
Systems of axiom s upon which to build proj ective geom etry were
first stu d ied more particularl y by the Ital i an school—G P eano M
P ieri Gino Fano of Turin
This subj ect received the attention also
of Theodor Vahle n of Vienna and Friedrich Schur of Strassburg
Ax ioms of descriptive geometry have been considered mainly by
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1
B u ll A m M a th S oc , V ol 1 1 ,
’
d raw n from Wil son s ar t i c l e
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a re
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1 904 ,
p 77
.
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O ur
re m a rk s o n
t h e I t al i a n
S c h ool
A
328
H I STORY
or M ATHE M ATI C S
I talian and American m athematicians and by D Hilbert The
introduction of order was achieved by G P eano by taking the cl ass
of points whi ch lie between any t w o points as the fundamental idea
by G V ail ati and later by B Russell on the fundam ental conception
of a class of rel ati ons or class of points on a straight line by O Veblen
on the study of the properties of one sin gle three term relatio n
( 1 904)
“
1
of order A N Whitehead refers to O Veblen s m ethod : This
me thod of conceiving the subject results in a notable simplification
and combines advantages from the two previous m ethods
While
D Hil bert has six undefined term s (point straight line plane between
paral lel congruent)
and tw enty one assu mptions Veblen gives only
two undefined term s (point between )
and only twelve assu mptions
However the derivation of fundamental theorems is so m ewhat
harder by Veblen s axioms R L M oore showed that any plane
—
m
satisfying Veblen s axio s I V III X I is a number plane and con
ta ins a system of continuous curves such that with reference to these
curves regarded as straight lines the plane is an ordinary Euclidean
plane
In 1 90 7 Oswald Veblen and J W Young gav e a completely i n
dependent set of assumptions for proj ective geom etry i n which points
and undefined classes of points called lines have been taken as the
undefined elements Eight of these assu mptions characteriz e general
projective S paces ; the addition of a ninth assum ption yields properly
?
projective S paces
Axiom s for line geometry based upon the line as an undefined
“
element and intersection as an undefined relation between u n
ordered pairs oi ele m ents were giv en in 1 90 1 by M P i eri of Catania
and in S im pler form in 1 91 4 by E R Hedrick and L Ingold of the
University of M issouri
Text books built upon som e such system of axio ms and possessing
great generality and scientific interest have been written by F ede rig o
Enriques of the University of B ologna in 1 8 98 and by O Veble n
and J W Young in 1 9 1 0
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Geometri c M odel s
Geom etrical m odels for advanced students began to be manu
fac t u r ed about 1 8 7 9 by the fi rm of L B rill in D ar m stadt
M any
o f the early m odels
such as Ku mm er s surface twisted cubics the
t ractrix of revolution were made under the direction of F Klei n and
Alexander von B rill Since about 1 8 90 this fi rm developed into that
of M artin Schilling ( 1 8 6 6 —1 90 8 )
of Leipz ig The catalogue of the
S i nce 1 90 5 the fi rm of
fi rm for 1 9 1 1 described som e 40 0 m odels
B G Teubn er in Leipz ig has off ered m odels designed by He rmann
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The A xi oms of Descr i pti ve Geometr y, C amb ri dg e ,
2
B ul l A m M a th S oc , V o l 1 4 , 1 90 8 , p 2 5 1
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1 90 7 ,
p
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33
0
A HI STORY OF
.
afl
=a
x
‘
(D )
(C)
y
M ATHE M ATIC S
(E )
apply
to the general exponent a and finds that (A )( B )
and (E )
are i n com
“
many m any more
pl e t e equations since the left m em bers have
v alues t han the right m em bers although the right hand values ( i nfi nite
“
in number)
are all found am ong the i nfi nite times in finite values on
the left ; that ( C )
and (D )
are complete equations for the general case
A failure to recog niz e that equation E is incomplete led Thomas
of Altona to a paradox (Grel l e s Jou rn al Vol 2
Clausen ( 1 80 1 —1 88 5)
1 8 2 7 p 2 86)
which was stated by E Catalan in 1 8 6 9 in m ore con
i where m and n are distinct integers
i =e
den sed fo rm thus : e
Raising bo th sides to the power 5 there results the absurdi ty
—
W
m
=
m
e
e
O hm introduced a notation to designate som e particular
value of a but he did not introduce especially the particular value
which is now calle d the principal value
O therwi se his treatm ent
of the general power is m a i nly that of the present t ime except of
course in the explanation of the irrational From the general power
Ohm proceeds to the general logarith m having a complex num ber
as i ts base I t is seen that the Eulerian logari thm s served as a step
la dder lea ding to the theory of the general power ; the theory of th e
general power i n turn led to a m ore general theory of logari thm s
having a com plex base
The P hi l os ophi cal Tr an s acti on s (London 1 8 2 9)
contain two articles
on general powers and logarithm s—one by John Graves the other
by John Warren of Ca mbridge Graves then a young man of 2 3
was a class fellow of William Rowan Hamilton in D ublin Grav es
becam e a noted jurist Ham ilton states that reflecting on G rav es s
ideas on im aginaries led him to the i nvention of quaternions Graves
obtains log
Thus Grav es claimed t hat gen
eral logarithm s involv e two arbitrary integers m and m instead
of simply one as given by Euler Lack of explicitness inv olved
Grav es In a discussion with A D e M organ and G P eacock the out
come of which was that Graves wi thdrew th e statem ent contained
in the title of his paper and i mplying an error in the Eulerian theory
whi le D e M organ a dm itted that if Graves desired to extend the idea
of a logari thm so as to use the base e + m i there was no error i n
volv ed In the process Sim ilar researches were carried on by A J H
Vincent at Li l le D F Gregory D e M organ W R Hamilton and
G M P agani ( 1 796—1 8 5 5)but their general logarithmic system s
in v olv ing complex numbers as bases fa i led of recognition as useful
1
m athe m atical inventions
We pause to S ketch the l ife of D e M organ
was born at M adura (M adras )
Au gu s tu s D e M org an ( 1 8 0 6—1 8 7 1 )
and educated at Tr i nity College Cam bridge For th e determ i nation
of the year of his birth (assum ed to be In the nineteenth century)
he
2
proposed the conun dru m I was x years of age in the year x
His
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For
19 1 3,
—8
r efe r e n ces a n d
pp
.
1 75
1
2.
fu ll e r de t a il s
se e
F
.
C aj or i i n A m
.
M ath M on thl y, V ol
.
.
20,
ALGEBRA
3:
scruples about the doctrines of the established church prevente d him
from procee ding to the M A degree an d from sitting for a fellowship
“
I t is said of him he ne v er voted at an election and he never visited
the House of Commons or the Tower or Westm inster Abbey
In
1 8 2 8 he beca m e professor at the newly es tablished University of
London and taught there until 1 86 7 except for five years from 1 83 1
1 835
He was the first president of the Londo n M athematical Society
whi ch was founded in 1 8 6 6 D e M organ was a unique m anly char
acter and pre—
em inent as a teacher The value of his original work lies
not so much in increasing our stock of mathem atical knowledge as in
putting it all upon a m or e logical basis He felt keenly the lack of clos e
“
reasoning in m athe matics as he received it He said once : We know
that m athem aticians care no m ore for logic than logicians for m athe
m at i c s
The two eyes of exact science are mathe m atics and logic :
the m athem atical sect puts out the logical eye the logical sect puts
out the m athe matical eye ; each believing that it can see better with
one eye than with two
D e M organ analyz ed logic mathe matically
a n d stu d ied the logical analysis of the laws sym bols and operations
of mathe m atics ; he wrote a Formal L ogi c as well as a D ou bl e A l gebr a
and corresponde d both with Sir William Ham ilton the m etaphysician
and Sir William Rowan Ham ilton the mathematician Few con
temporaries were as profoundly read in the history of m athem atics
as was D e M organ N 0 subj ect was too insignificant to receive his
“
attention The authorship of Cocker s Arithm etic and the work
of circle squarers was investigated as m inutely as was the history of
the calculus Nu merous articles of his lie scattered i n the volu mes of
“
the P enn y an d E n gl i sh C ycl opwdi as I n the article In d uction (M athe
first printe d in 1 8 38 occurs apparently for the first tim e
“
the n ame m athem atical in d uction ; it was ad opte d and populariz ed
“
by I Todhunter in his A l gebr a The term i n d uctio n h ad been used
by John Wallis in 1 6 56 in his A ri thmeti ca i nfi n i tor u m; he use d the
i nduction
known to natural science In 1 6 8 6 Jacob B ernoulli
critici sed him for using a process which was not binding logi cally and
then advanced in place of it the proof from n to n + 1 This is one of the
sev eral origins of the pr ocess of m athem atical induction From Wall i s
“
to D e M organ the term induction was used occasionally in m athe
m a t i cs a n d in a double sense ( I )
to indicate inco mplete inductions of
the kind known in n atural science ( 2 )
for the proof from n to n + 1 D e
“
M organ s m athem atical induction assigns a distinct nam e for the
“
latter process The Germ ans employ m ore com m only the nam e voll
s t an di g e Induktion
which becam e current am ong them after the use
of it by R D e d ekind in his Wa s si nd u nd w as s ol l en di e Z ahl en 1 8 8 7
D e M organ s Difi eren ti al C al cu l u s 1 84 2 is st i ll a standard work and
contains m uch that is or i ginal with the author For the En cycl opa di a
M etr opol i tan a he wrote on the Calculus of Functions ( giving principles
of symbolic reason i ng)
and on the theory of probab i lity In the Cal
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HI STORY
A
33 2
O F M ATHE M ATIC S
“
of Functions he proposes the use of the slant line or solidus
for printing fractions in the text ; this proposal w as adopted by G G
“
Stokes in
Cayley wrote Stockes I think the solidus looks
very well indeed
it woul d gi v e you a strong clai m to be P reside n t
2
of a Society for the prevention of Cruelty to P rinters
Celebrated is D e M organ s B u dget of P a radoxes London 1 8 7 2 a
second edition of which was edited by D avid E u gene S m ith i n 1 9 1 5
D e M organ published mem oirs
O n the Foun dation of Algebra
in the Tr an s of the C a mbri dge P hi l S oc 1 84 1 1 84 2 1 8 44 a n d 1 84 7
The ideas of George P eacock and D e M organ recogni ze the possi
Such al gebras
bili ty of algebras which diff er fro m ordinary a l gebra
were indee d not slow in forthcoming but like non Euclidean geo metry
som e of the m were slo w in fi n di n g rec ognitio n Th i s is true of H G
G ras sman n s G B el l a vi t i s s and B P eirce s d iscoveries but W R
Hamilton s quaternions m e t with i mmediate appreciatio n i n England
These algebras off er a geom etrical interpretation of im agi n aries
—
was born of Scotch parent s
Wil l i am R o w an H a mi l to n ( 1 80 5 1 8 6 5)
His early education carried on at ho m e was m ainly i n
i n D ublin
languages At the age of thirteen he is said to hav e been fam iliar with
as many languages as he had lived years About this ti me he cam e
across a copy of I Newton s Un i ver sal A r i thmeti c After rea ding that
he took up successively analytical geom etry the calculus Ne w ton s
P r i n ci pi a Laplace s M écan i q u e C el es te At the age of eighteen he pub
li sh ed a paper correcting a m istake in Laplace s work
I n 1 8 2 4 he
entered Trinity College D ublin and in 1 8 2 7 while he was still an
undergraduate he was appointed to the chair of astrono m y C G J
J acobi m et Ha m ilto n at the m eeting of the B ritish Association at
M anchester in 1 84 2 and a ddressing Sectio n A called Ha m ilton l e
Lagrange de votre pays
Ha milton s early papers were on optics
In 1 83 2 he predicted conical refraction a d iscovery by aid of m athe
m at ics which ranks with the disco v ery of Neptune by U J J Le
Verrier and J C Adams Then followed papers on the P r i n ci pl e of
and a general m etho d of dy n a mics ( 1 8 34
Varyi n g A cti on ( 1 8 2 7 )
He wrote also on the solution of equations of the fifth degree t he
hodograph fluctuating functions the num erical solution of diff erential
equations
The capital discovery of Ha m ilto n is his quaternions in which hi s
s tudy of algebra cul m inated
In 1 83 5 he publ i shed in the Tran sacti ons
He re
of the Royal I r i sh A cademy his Theory of Algebraic Couples
“
garded algebra as being no m ere art nor language nor primarily
c ul u s
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G G S t ok es , M a th a n d P hys P a per s V ol I C amb ri dg e , 1 880 , p V I I ; see al so
J L arm or, M emoi r a nd S ci e C orr of G G S tok es , V ol I , 1 90 7 , p 39 7
2
A n e arl i er u se o f t he so l id us i n d es ig n a t i ng f rac t io n s oc c u rs i n o n e o f t he ve ry
fi rs t t ext b oo k s p u bl i s he d i n C a l i fo rn i a , v i z , t he D efi n i c i on de l a s pr i n c i pal es
opera ci on es de a r i s méti ca b y H e n r i C a mb u s t o n , 2 6 p ag es p r i n t e d a t M o n t e re y i n
1 84 3
The s o li du s app e ar s S l ig h t ly c u r ved
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M ATHE M ATIC S
A HI STORY OF
3 34
e xtended as was predicted The change in notation made in France
by Jules Hoii el and by C A L a is a n t has been considered in England
as a wrong step but the true cause for the lack of progress is perhaps
There is indeed great doub t as to whether the
more deep seated
quaternionic product can clai m a necessary and funda mental place
in a syste m of vector analysis P hysicists clai m that there is a loss
of naturalness in taking the square of a vector to be negative
Widely different opinions have been expressed on the value of
quaternions While P G Tait was an enthusiastic champion of this
science his great friend William Thomson (Lord Kelvin )declared
that they though beautifully ingenious have been an unmixed
e v il to those who have touched the m in any way including Clerk
“
1
A Cayley writing to Tait in 1 8 74 said I ad mire the
M axwell
equation d 0 = u qdpq extre mely—i t is a grand exa mple of the pocket
Cayley ad mitted the conciseness of quaternion formulas b u t
m ap
they had to be unfolded into Cartesian form before they could be
“
m ade use of or e v en understood Cayley wrote a paper On C o or
d i n a t es versus Q uaternions in the P r oceedi n gs of the Royal Society
“
of Edinburgh Vol 2 0 to which Tait replied On the Intrinsic Nature
of the Q uaternion M ethod
In order to m eet m ore adequately the wants of physicists J W
G i bbs and A M a cf a rl a n e have each suggested an algebra of vectors
with a new notation Each gi v es a definition of his own for the
product of two vectors but in such a way that the square of a vector
is positive A third syste m of vector analysis has been used by Ol i ver
H eavi s i de in his electrical researches
What constitutes the m ost desirable notation in vector analysis
is still a m atter of dispute Chief a mong the various suggestions are
those of the A merican school started by J W Gibbs and those of the
Ger man Italian school The cleavage is not altogether along lines
“
of nationality L P ran dl of Hanover said in 1 90 4 : After long del i b
cration I have adopted the notation of Gibbs writing a b for the
inner ( scalar )and a ><b for the outer ( vector)
product If one ob
serves the rule that in a multiple product the outer product must be
taken before the inner the inner product before the scalar then one
can write with Gibbs a b><c and a b c without giving rise to doub t
2
as to the m eaning
In the following we give German Italian notations fi rst the equiv
second Inner product a Ib a b ;
al en t A merican n otation ( Gibbs )
ab Ic d ,
v ector product ab a >
<c ; ab c
<h ; also ab c a b >
2
ab c d
R M ehmke
ab
“
said in 1 90 4 : The nota t ion of the German Italian school is far pref
e r a b l e to that of Gibbs not only in logical and m etho di cal but als o
in practical respects
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S P Tho mp so n L if e of L or d K el vi n , 1 9 1 0 , p
J a hr es b d (1 M a th V er ei n i g , V o l 1 3 , p 39
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1 138.
.
ALGEB RA
33 5
In 1 8 95 P M olenbroek of The Hague and S Ki mura then at Yale
University took the first steps in the organiz ation of an Internationa l
Association for P ro mo t ing the S tudy of Q uaternions and Allied
S yste ms of M athe matics P G Tait was elected the first presi dent
but could not accept on account of failing health Al e xan d e r M ac
f ar l an e ( 1 8 5 1 —1 9 1 3)
of the University of Edinburgh later of the
University of Texas and of Lehigh University served as secretary of
the Association and was its president a t t he tim e of his death
At the international congress held in Rome in 1 90 8 a committee
was appointed on the unification of vectorial notations but at the
time of the congress held in Cambridge in 1 91 2 no defini te conclusions
had been reached
Vectorial notations were subjects of ex tended discussion in L E n
—
—
s ei gn emen t mathema ti qu e Vols
1 1 14
1 90 9 1 9 1 2 between C B urali
Forti of Turin R M arc ol o ng o of Naples G C o mb eri ac of Bourges
H C F Timerdin g of Strassburg F Klein of Gottingen E B
Wilson of Boston G P eano of Turin C G Knott of Edinburgh
Alexander M acfarlane of Chatham in Canada E Ca r allo of P aris
and E Jahnke of B erlin In America the relative values of notations
were di scussed in 1 91 6 by E B Wilson and V C P oor
We m ention two topics outside of ordinary physics in which vector
analysis has figured The generalization of A Einstein known as
the principle of relativi ty and its inte rpretation by H M inkowski
hav e opened new points of Vi ew So m e of the queer consequences of
this theory disappear when kine matics is regar ded as identical with
the geo metry of four di mensional S pace H M i nk owski and following
hi m M ax Abraha m used vector analysis in a li mited degree M i n
k ow s k i usually preferring the m atri x calculus of A Cayley
A m ore
exten ded use of vector analysis was made by Gilbert N Lewis of the
Uni v ersity of California who introduced in his ex tension to four di
me n si on s some of the original features of H G Gr assman n s syste m
“
A dyn am e is accor ding to J P lucker ( and others )
a sys t em of
forces applied to a rigid body The English and French call i t a
t or sor
In 1 899 this subject was treated by the Russian A P
K o tj el n ik o ff under the nam e of proj ective theo r y of vectors In 1 90 3
E Study of Greifswald brought out his book Geometr i e der Dyn amen
in which a line geo metry and kine matics are elaborated partly by
the use of group theory which are carried over to non Euclidean
spaces ; S tu dy clai ms for hi s system somewhat greater generality
than is found in Hamilton s quaterni ons and W K Clifford s bi
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.
H e rmann G u n th er G ras s man n ( 1 80 9—
1 8 7 7)
was born at S tettin
attended a gym nasium at hi s native plac e (where his father was
teacher of m athe matics and physics )and studied theology in Berlin
for three years His intellectual interests were v ery broa d He started
as a theologian wrote on physics co mposed texts for the stu dy of
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O F M ATHE M ATIC S
HISTORY
A
33 6
German Latin and mathematics e dited a political paper and a mi s
si o n ary paper in v esti gated phonetic laws wrote a di ctionary to the
Rig Veda translated the Rig Veda in verse harmoni z ed fol k songs
in three voices carried on successfully the regul ar work of a teacher
and brought up nine of his eleven chi ldr en—all this in addi tion to the
great m athem atical creations whi ch we are about to describe In
1 834 he succeeded J S teiner as teacher of m athematics in an industrial
school in B erlin but returned to Stettin in 1 836 to assume the duti es
of teacher of m athem atics the sciences and of religion in a school
1
there
Up to thi s tim e his knowledge of m athem atics was pretty
much confined to what he had learned fro m hi s father who had
“
“
written two books on Raum lehre and G rOssenl ehre
B ut now
he m ade his acquaintance with the works of S F Lacroi x J L La
grange and P S Laplace He noticed that Laplace s resul ts coul d
be reached in a S horter way by som e new ideas advanced in his father s
books and he proceeded to elaborate this abridged m ethod and to
apply it in the study of tides He was thus led to a new geom etric
analysis In 1 840 he had m ade considerable progress in its develop
m ent but a new book of Schleier m acher drew hi m again to theology
In 1 84 2 he resum ed m athem atical research and becom ing thoroughl y
convi nced of the i mportance of hi s new analysis decided to devote
himself to it It now becam e his ambition to secure a mathematical
chair at a university but in thi s he never succeeded In 1 844 ap
p ear ed his great classical work the L i n ea l e A u s dehn u ngsl ehr e w hi ch
was full of new and strange m atter and so general abstract and out
of fashion in its m ode of exposition that i t could hardl y have had
less influence on European m athe matics during its first twenty years
had it been published in China K F Gauss J A Grunert and A F
M ob i u s glanced over it praised it but co mplained of the strange
“
terminolog y and its philosophi sche Allgemei nh eit
Eight years
afterwards C A B re t schn ei der of Gotha was said to b e the only
An article in Gr ell e s Jou r nal in
man who had read it through
whi ch Grassm ann eclipsed the geometers of that time by constructi ng
with aid of hi s m ethod geo metrically any algebraic curve remained
again unnoticed Need we m arvel if Grassmann turned his attention
to other subjects —to Schleierm acher s philosophy to politics to
p hi lolog y ? S till articles by him continued to appear in Gr ell e s
J ou rn al and in 1 86 2 ca m e out the second part of his A usdehn u ngsl ehr e
It was intended to show better than the first part the broad scope of
the Aus dehnungslehre by c onsidering not only geometric applica
tions but by treating also of al gebraic functions infinite series and
the di fferential and integral calculus B ut the second part was no
more appreciated than the first At the age of fi fty three thi s won
der ful m an with heavy heart gave up mathe matics and directed hi s
energies to the study of Sanskrit ach i eving in philology results which
V i c tor S hl g l H m
Gr
m
L ei p z i g
8 78
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A HI STORY OF M ATHE M ATIC S
338
woode spoke of i t in D ublin as som ethi ng above and beyond 4 nions
I have not seen i t but Sir W Hamilton of E dinburgh used to say
that the greater the extension the sm aller the intention
M ultiple algebra was powerfully adv anced by B P eirce whose
theory is not g eom etrical as are those of W R Hamilton and H G
Grassmann B e n j a mi n P e i r c e ( 1 80 9—1 880 )
was born at Salem M as s
and gra duated at Harvard College having as undergraduate carried
the study of mathem atics far beyond the li m its of the college course
When N B owditch was preparing his translation and comm entary
of the M éca n i qu e C el este young P eirce helped in reading the proof
He was m ade professor at Harvard in 1 8 33 a position whi ch
S heets
he retained until hi s d eath For som e years he was in charge of the
N au ti cal A l ma n ac an d super i ntendent of the United S tates Coast
Surv ey He published a series of college tex t books on mathematics
an A n al yti cal M echa n i cs 1 8 55 and calculated together wi th Sears
C Walker of Washington the orbi t of Neptune P rofound are his
researches on L i n ear A ss oc i a ti ve A l gebr a The first of several papers
thereon was read at the first m eeting of the Am erican Association
for the Adva nce ment of Science in 1 864 Lithographed copies of a
m e moir were d istributed a m ong friends in 1 8 70 but so s mall seemed
to be the interest taken in t hi s subj ect that the m emoir was n ot
printed until 1 88 1 (A m Jou r M ath Vol IV No
P eirce works
out the m ul tiplication tables first of s i ngl e algebras then of dou bl e
algebras and so on up to sextuple m aking in all 1 6 2 algebras whi ch
he S hows to be possi ble on the consideration of sym bols A B etc
which are linear functions of a determinate nu mber of letters or units
i j k l etc with coefficients which are ordinary analytical m agni
tudes real or i maginary —the letters i j etc being such that e v ery
2
binary combinat ion i ij j i etc is equal to a linear function of the
1
letters but under the restriction of satisfying the associative law
C harl es S P ei r ce a son of B enj am in P eirce and one of the foremost
writers on m athem ati cal logic S howed that these algebras were all
defecti v e forms of qua drate a l gebras w hich he had previously dis
covered by logical analysis a n d for which he had devised a si mple
notation Of these quadrate algebras quaternions is a simple example ;
n o n i on s is another
C S P eirce showed that of all linear associative
algebras there are only three in which d ivision is unambiguous These
are ordinary S ingle algebra or di nary double algebra and quaternions
from whi ch the i maginary scalar is excluded He S howed that hi s
father s algebras are operational and m atricular Lectures on m ultiple
algebra w ere deli v ere d by J J Syl v ester at the Johns Hopk ins Uni
versity and published in various journals They treat largely of the
algebra of m atrices
Wh i le B enjam in P eirce s co mparativ e anatomy of linear algebras
was favorably receiv e d i n Englan d i t was cr i ti ci sed i n Germ any as
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A C ayl ey , A dd r es s be fore B r i t i s h A s so c i a t i on ,
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1 88 3 .
ALGEBRA
33 9
.
being vague and based on arbitrary principles of classification Ger
m an writers along this line are Eduard Study and Georg W S c heffe r s
An estimate of B P eirce s linear associative algebra was giv en in
1
who e xtends P eirce s method and shows its
1 90 2 by H E Hawkes
full power I n 1 8 98 Elie Cartan of the University of Lyon used the
characteristic equat i o n to dev elop several general theore m s ; he ex
h i bi t s the s emi si mpl e or D edekind and the ps eudo n u l or nilpotent
Su b algebras ; he shows that the structure of every algebra may be
represented by the use of d ouble units the first factor being quad
rate the second non—
quadrate Extensions of B P eirce s results were
m ade also by Henry Taber Olive C Haz lett gave a classification of
nilpotent al egb ras
As S hown abov e C S P eirce advanced this algebra by using the
m atrix theory P apers along thi s line are due to F G Frobenius and
“
J B Shaw The latter shows that the equation of an algebra de
termines its quadrate units and certain of the di rect units ; that the
other units form a nilpotent system which with the quadrates may
be reduced to certain canonical form s The algebra is thus made a
sub algebra under the algebra of the associ ati ve u n i ts used in these
canonical form s Frobenius proves that every algebra has a D ede
k ind sub a l gebra whose equation contains all factors in the equation
of the algebra This is the sem i simple algebra of Cartan He also
S howed that the re m aining units fo rm a nilpotent algebra whose units
m ay be r egu l ar i zed (J B Shaw ) M ore recently J B Shaw has
extended the general theorem s of linear associative algebras to such
algebras as have an i nfinite number of units
B esides the m atrix theory the theory of continuous g rOu ps has been
used in the study of linear associative algebra This iso morphis m
was first pointed out by H P oi n c are
the m ethod was followed
by Georg W S cheff er s who classifi ed algebras as quaternionic and
non quaternionic and worked out co mplete lists of all algebras to
order five Theodor M ol ien in 1 8 93 then I n D orpat de monstrated
“
that quaternionic algebras conta i n independent quadrates and that
q uaternionic algebras can be classified according to non quaternionic
types (J B Shaw)An ele mentary exposition of the relation between
2
l inear al gebras and continuous groups was given by L E D ickson of
“
Chicago This relation enables us to translate the concepts and
theorem s of the one subj ect into the language of the other subj ect
It not onl y doubles our total knowle dge but gives us a better insight
into either subj ect by exhibiting it fro m a new point of view
The
theory of matrices was developed as early as 1 8 58 by A Cayley in an
i mportant me moir which i n the opinion of J J Sylvester ushered in
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H E H aw k e s i n A m J ou r M ath , V ol 2 4 1 90 2 , p 8 7
We are
J B S haw s S ynops i s of L i near A ssoci a ti ve A l gebr a , Washi n g ton , D
u si n g a l so
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S haw g i v e s b i b lio g raphy
I n t ro d u c t i o n
2
B u ll A m M ath S oc , V o l 2 2 , 1 9 1 5 , p 5 3
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1 90 7 ,
A HI STORY OF M ATHE M ATIC S
3 40
the reign of Algebra the Second W K Cliff ord Sylvester H Taber
C H Chap man carried the inv estigations m uch further The origi
nator of matrices is really W R Hamilton but his theory published
in his L ectu r es on Qu ater n i ons is less general than that of Cayley
The latter m akes no reference to Ham ilton
1
The theory of determ inants was studied by H oen é Wr onski ( 1 7 7 8
1 85
a poor P olish enthusiast liv ing m ost the tim e in France whose
egotism and wearisom e style tended to attract few followers but who
m ade som e incisiv e criticis m s bearing on the philosophy of m athe
?
m a ti c s
He s tudied four S pecial form s of determinants which were
—
of B re m en and
extended by H ei n ri ch Ferdi n a n d S cher k ( 1 7 98 1 88 5)
Fer di n a nd S chw ei n s ( 1 7 8 0 —1 8 5 6 )
of Heidelberg In 1 83 8 Liouville
de monstrated a property of the special forms which were called
wronskians by Tho mas M uir in 1 88 1 D eterm inants receiv ed the
attention of Jacqu es P M B i n et ( 1 7 8 6—1 8 56 )
of P aris but the great
m aster of this subject was A L Cauchy I n a paper (J ou r de l ecol e
P ol yt IX
Cauchy developed several general theorem s He in
1 6)
t r o du c e d the nam e deter mi n a n t a ter m used by K F Gauss in 1 8 0 1 i n
the functions considered by him In 1 8 2 6 C G J Jacobi began using
this calculus and he gave brilliant proof of its power In 1 84 1 he
wrote extended m e moirs on determ inants in Gr el l e s J ou r n al which
rendere d the theory easily accessible In England the study of l inear
transform ations of quantics gave a powerful impulse A Cayley de
v el op ed skew deter m inants a n d P faffi a n s and introduced the use of
determ inant brackets or the fa m iliar pair of upright lines The m ore
general consideration of determ inants whose elem ents are form ed from
the ele m ents of giv en d eterm inants was taken up by J J Syl v ester
and especially by L Kronecker who gave an elegant theore m
(1851)
3
known by his nam e
Orthogonal determinants received the atten
2
tio n of A Cayley in 1 8 4 6 in the study of n elem ents related to each
other by
equations also of L Kronecker F B rioschi and
others M axim al values of determ inants received the attention of
and especially of J Hadamard ( 1 8 93)
who
J J Sylvester
proved that the square of a determ inant is never greater than the
norm product of the lines
—
Anton P u c h ta ( 1 8 5 1 1 90 3 )
of C z ernowitz in 1 8 7 8 and M Noether
in 1 88 0 showed that a sym m etric d eterm inant m ay be expressed as
the product of a certain nu m ber of factors linear in the elem ents
D eterm inants which are form ed from the m inors of a determinant
were investigated by J J Sylvester in 1 8 5 1 to whom we owe the
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Th o mas M u i r , The Theor y of D eter mi n a n ts i n the H i s tor i cal Or der of Devel op
men t [ V 0 ] I ] , 2 n d E d , L on don , 1 90 6 ; V ol I I , P e ri od 1 84 1 t o 1 8 6 0 , L on don , 1 9 1 1
M u i r w as S u p e r i n ten de n t G e n eral o f Ed u c a ti on i n C ape C ol ony
2
O n Wr on sk i , see J B er t ran d i n J ou rn a l des S a va n ts , 1 8 9 7 a n d i n R evu e des
’
D eu x M ond es , F e b , 1 89 7
S ee a l so L I n ter medi a i r e des M a thema ti ci ens V o l 2 3 ,
1
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—1 6 7
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19 16 ,
3
1 13 , 1 64
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18 1
—1 8 3
,
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pp
E P a s ca l , D i e D eter mi n an ten , t r a n sl by H L e i tz m a nn ,
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1 90 0 ,
p
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10 7
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A HI STORY OF M ATHE M ATIC S
34 2
The symbol E to express identity was first used by G F B
1
Riem ann
M odern higher algebra is especially occupied with the theory of
l inear transform ations
Its develop m ent is mainly the work of A
Cayley an d J J Sylv ester
Arth u r C ayl e y ( 1 8 2 1
born at Richmond i n Surrey was
educated at Trinity College Ca mbridge He ca m e out Senior Wrang
ler in 1 84 2 He then devoted som e years to the study and practice
of law While a student at the bar he went to D ublin and alongside
of G Sal mon heard W R Ham ilton s lectures on quaternions On
the foundation of the S adl e ri an professorship at Cambridge he ac
c ep t ed the o ff er of that chair thus giv ing up a profession pro m ising
wealth for a very m o dest provision but which would enable hi m to give
all his tim e to m athe matics Cayley began his m athem atical publica
tions in the C ambri dge M a themati cal Jou rn al while he was still an
un dergra d uate Som e of his m ost brilliant discoveries were made
during the time of his legal practice There is hardly any subj ect
in pure m athem atics which the genius of Cayley has not enriched but
m ost im portant is his creation of a new branch of analysis by his
theory of inv ariants Ge rms of the principle of invariants are found
in the writings of J L Lagrange K F Gauss and particularl y of
G B o ol e w h o showed in 1 8 4 1 that invariance is a property of dis
c r im i n a n t s generally and who applied it to the theory of orthogonal
substitution Cayley set him self the problem to determ ine a pri ori
what functions of the c oefli ci en ts of a giv en equation possess this prop
and found to begin with in 1 84 5 th at the so
e r ty of invariance
called hyper determinants possessed it G B oole m ade a number
of ad ditional d iscoveries Then J J Sylvester began his papers in the
C ambri dge and Du bl i n l ll a themati cal Jou rn al on the Calculus of Form s
After this d iscoveries followed in rapid succession At that t ime Cay
ley and Sylvester were both residents of London and they sti mulated
each other by frequent oral comm unications I t has often been dif
fi c u l t to d eterm ine how m uch really belongs to each
In 1 88 2 when
Syl v ester was professor at the Johns Hopkins University Cayley
lecture d there on Abelian and theta functions
Of interest is Cayley s m ethod of work A R F orsyth describes it
“
thus : When Cayley had reached his most advanced generali zations
he proceeded to establish them di rectly by som e m ethod or other
though he seldo m gave the clue by which they had first been obtained
a procee ding whi ch does not tend to make hi s papers easy reading
His literary style is direct si mple and clear His legal training had
an influence not m erely upon his m ode of ar rangement but also upon
his expression ; the result is that his papers are severe and present a
curious contrast to the luxuriant enthusiasm which pervades so many
1 80 8
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1
L K r o n e ck e r , Vor l esu n g en uber Z ahl en theor i e,
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1 90 1 ,
p 86
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.
ALGEB RA
”
papers
Curiously
343
1
of Sylvester s
Cayley took little interest in
quaternions
—
m
J a e s J o s e ph S yl ve s ter ( 1 8 1 4 1 897)
was born in London His
father s name was Abraham Joseph ; hi s eldest brother assum ed in
America the name of Sylvester and he adopted this nam e too About
the age of 1 6 he was awarded a pri ze of $500 for solving a question
?
in arrangem ents for contractors of lotteries in the United S tates
In 1 83 1 he entered S t John s College Cambridge and cam e out
Second Wrangler in 1 83 7 George Green being fourth Sylvester s
J ewish origin incapacitated hi m from taking a degree Fro m 1 83 8 to
1 8 40 he was professor of natural philosophy at what is now University
College London ; in 1 84 1 he becam e professor of mathematics at the
Uni versi ty of Virginia In a quarrel with two of his students he slightly
wounded one of the m with a m etal pointed cane whereupon b e re
turned hurriedly to England In 1 844 he se rved as an actuary ; in
1 84 6 he became a student at the Inner Te mple and was called to the
bar in 1 8 50 In 1 84 6 he beca me associated with A Cayley ; often
they walked round the Courts of Lincoln s Inn perhaps di scussing
“
the theory of invariants and Cayley ( says S ylvester ) habitually
d i scoursing pearls and rubies
Sylvester resumed mathematical
research He Cayley and Wil l iam Rowan Ham ilton entered upon
discoveries in pure mathematics that are unequalled in Great Britain
S ince the ti m e of I Newton
Sylvester made the friendship of G Sal
m on whose books contributed greatly to bring the results of Cayley
and Sylv ester withi n easier reach O f the m athematical public From
1 8 5 5 to 1 8 7 0 Sylvester was professor at the Royal M ilita ry Acade my
at Woolwich but showed no great e fficiency as an elem entary teacher
There are stories of hi s housekeeper pursuing hi m from hom e carrying
his collar and necktie Fro m 1 8 76 to 1 8 83 he was professor at the
Johns Hopkins University where he was happy in being free to teach
whatever he w ished in the way he thought best He beca m e the first
editor of the A mer i ca n Jou r n al of M a thema ti cs in 1 8 7 8 In 1 884 he
was elected to su cc eed H J S S mith in the chair of S avi l i a n professor
of geometry at Oxford a chair once occupied by Henry Briggs John
Wallis and Edmund Halley
Sylvester someti mes amused himself writing poetry His L aws of
Ver s e is a curious booklet At the reading at the P eabo d y Institute
in B altimore of hi s Rosalind poem consisting of about 400 lines all
“
rhyming with Rosalind he first read all his explanatory footnotes
so as not to interrupt the poem ; these took one hour and a half Then
he read the poem itself to the remnant of hi s au di ence
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1
P r oceed L ondon Royal S oci ety, V ol 5 8 , 1 89 5 , pp 2 3 , 2 4
’
2
H F B ak e r s B i og raphi c al N o t i ce i n The C oll ec ted M ath P a pers of J J S yl
ves ter , V ol I V , C amb r i dg e , 1 9 1 2
We h av e u sed al so P A M ac M ah’ on s n o t i c e i n
P r oceed Roya l S oc of L ondon , V o l 6 3 , 1 8 9 8 , p i x
For S yl v es t e r s ac t i v i ti es i n
B al timore , see F ab i an F r an k li n i n J ohn s H opki n s U n i v Ci r cu l ars , Ju n e , 1 89 7 ; F
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A HI STORY OF M ATHE M ATIC S
344
Sylvester s first papers were on F r esn el s optic theory 1 83 7 Two
years later he wrote on C S turm s m em orable theorem S turm
once told him that the theore m originated in the theory of the c om
pound pendulum S ti m ulated by A Cayley he made important i n
v es t i g a t i o n s on m odern algebra
He wrote on elimination on trans
form ation and canonical form s in whi ch the expression of a cubic
surface by fi v e cubes is given on the relation between the mi nor de
t er mi n a n t s of linearly equi valent quadratic functions in w hi ch the
notion of invariant factors is i mplicit whi le in 1 8 5 2 appeared the
first of hi s papers on the principles of the calc ul us of forms In a
reply that he m ade in 1 86 9 to Huxley who had claim ed that mathe
ma t i c s was a science that knows not hi ng of observation induction
in v ention and e xperi m ental verification Sylvester narrated his per
“
sonal e xperie nce : I discovered and developed the whole theory of
canonical binary form s for odd degrees and as far as yet m ade out
for ev en degrees too at one eveni ng S itting with a decanter of port
wine to sustain nature s flagging energies in a back o ffi ce in Lincoln s
Inn Fields The work was done and well done but at the usual
cost of racking thought—a brain on fire and feet feeling or feelingless
as if plunged in an ice pai l Tha t n i ght w e sl ept n o mor e
His reply
to Hux ley is interesting reading and bears strongly on the qualities
of m ental activi t y involved in m athe mati cal re search I n 1 8 59 he
gave lectures on partitions not published until 1 8 97 He wrote on
partitions again in Baltimore In 1 8 6 4 followed his famous proof
of Newton s rule A certain fundam ental theorem in invariants
which had form ed the basis of an i mportant section of A Cayley s
work but had resi sted proof for a quarter of a century was demon
Noteworthy are hi s m emoirs on
s t ra t e d by Sylvester i n B alt i m ore
C h eb i ch e v s m ethod concerning the total i ty of pri m e nu m bers wi thi n
certain li mits and h i s latent roots of matrices His researches on
in v ariants theory of equations m ultiple algebra theory of numbers
linkages probability constitute i mportant contributions to m athe
m a ti c s His final stu di es entered upon after his return to Oxford
were on reciprocants or functions of d iff erential coeffi ci ents whose
for m is unaltered by certain linear transformations of the variables
and a generali z ation of the theory of concom itants In 1 9 1 1 G
1
Gree nh ill told reminiscently how Sylvester got everybody interested
“
in rec iprocants now clean forgotten ; One day after Sylvester
was noti ced walking alone a ddressing the sky aski ng i t : Ar e R ec ipr o
cants Bosh ? B erry of K i ng s says the Reciprocant is all Bosh !
There was no reply and Sylvester hi m self was t iring of the subj ect
and so B erry escape d a castigation B ut recently I had occasi on
from the Aeronautical point of view to work out the theory of a Vortex
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C aj ori , Tea chi n g
pp
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1
2 6 1- 2 7 2
a nd
H i s tor y of M a thema ti cs i n the U n i ted S ta tes ,
.
M al hemati cal G a zette V o l
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6,
19 1 2,
p
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108
.
Washi n g ton
,
1 8 90 ,
A
34 6
O F M ATHE M ATIC S
HI STORY
—
1 8 74 )
analysis Wi th Abb é
he founded in
(
A mong his pupils
1 8 5 8 the A n n al i di ma tema ti ca pu r a ed appl i ca ta
1
at P avi a were L Crem ona and E B eltrami V Volterra narrates
how F B rioschi in 1 8 5 8 with two other young Ita l ians E n r i co B etti
later professor at the University of P isa and F el i ce
( 1 8 23
later professor at the University of P avia
C as or a ti ( 1 8 3 5
started on a j ourney to enter into relations w ith the forem ost m athe
The scientific existence of Italy
ma ti c i an s of France and Ger many
as a nation dates from this j ourney
It is to the teaching labors
and dev otion of these three to their influence in the organi z ation of
to the friendly scientific relations that they insti
a d v anced stu dies
t u t e d between I taly and foreign countries that the existence of a
school of analysts in Italy is due
In Germ any the early theory of invariants as developed by Cayley
Syl v ester and Sal mon In England Hermite In France and F B rioschi
in Italy did not draw attention until 1 8 58 when S i e g f ri e d H e inr i ch
—
l
1
of the technical high school in B erlin pointed
Ar on h o d ( 1 8 9 1 884 )
out that Hesse s theory of ternary cubic form s of 1 844 involved i n
variants by which that theory could be roun d ed out F G Eisenstei n
and J Steiner had also given early publication to isolated develop
m ents invol v ing the i n v a r i an tal i dea
In 1 8 6 3 A ronh ol d gave a
system atic and general expos i tion of invariant theory
He a n d C l eb sc h use d a notation of their own the symbolical notation
d i ff erent fro m Cayley s which was use d in the further dev elop m ents
of the theory in Germ any Great d ev elop m ents were started about
1 86 8 when R F A C l eb sc h and P Gordan wrote on t ypes of binary
form s L Kronecker and E B Christo ff el on bilinear form s F Klein
and S Lie on the invariant theory connecte d with any group of li near
was born at Erlangen and
substitutions P a ul G ord an ( 1 8 3 7 1 9 1 2 )
becam e professor there He produced papers on finite groups par
t ic u l a rl y on the sim ple group of ord er 1 6 8 and its associated cu r ve
3
3
3 =
v e m ent is the proof of the
x
0
His
best
known
achie
x
z
z
+
+
y
y
existence of a complete system of concom itants for any giv en binary
2
form While C l eb sc h aim ed In his researches to devise m ethods by
which he could stu dy the relationships bet w een i nvarian tal form s
( F or m en v er w an dt schaf t)the chief ai m of A ron h ol d was to examine
3
the equivalence or the linear transform ation of one form into another
In v estigations along this line are due to E B Christoff el who showed
that the nu mber of arbitrary param eters containe d in the substitution
coefficients equals that of the absolute invariants of the form K
Weierstrass who gave a general treatm ent of the equivalence of two
l i ni
Tor to
B ar n a ba
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1 80 8
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B u l l A m M ath S oc , Vol 7 , 1 900 , p 6 0
2
N a tu r e, V o l 90 , 1 9 1 3 , p 5 9 7
3
We a r e u si n g F ran M eyer , B eri cht u be r den g eg en w a rt ig en S t an d de r I n
v a r i an t en the o r i e
i n t he J a hr es b d d M ath Ver ei n i g u n g , V o l I , 1 8 90 9 1 ,
S ee p 99
pp 7 9 2 9 2
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ALGEB RA
34 7
linear system s of bilinear and quadratic form s L Kronecker who ex
ten ded the researches O i Weierstrass and had a controv ersy with C
i n 1 8 74 gave
Jordan on certain d iscordant results J G D ar b oux w ho~
a general and elegant derivation of theore ms due to K Weierstrass
and L Kronecker G F r oben iu s who applied the transform ation of
“
bilinear form s to P faff s problem : To determ ine when two give n
linear di fferential expressions of n term s can be converted one into
the other by subj ecting the variables to general point transfor mations
The study of invariance of quadratic and bilinea r form s from the
stand point of group theory was pursued by H Werner
S
and W Killing
Finite binary groups were exa m
Lie ( 1 88 5)
i n ed by H A Schwarz
and Felix Klein Schwarz is led to the
“
problem to find all spherical triangles whose symm etric repetitions
on the surface of a S phere give rise to a finite number of spherical
triangles d iffering i n position and deduces the form s belonging
thereto With out a knowledge of what Schwarz and W R Ha m ilton
had d one Klein was l ed to a deter m inatio n of the finite binary linear
groups and their form s Representing transformations as m otions
and adopting Riem an n s interpretation of a com plex variable on a
S pherical surface F Klein sets up the groups of those rotations which
bring the five regular solids into coincidence with them selves and the
accompanying form s The tetrahedron octahedron and icosahedron
lead respectiv ely to 1 2 2 4 and 6 0 rotations ; the groups in question
were stud ied by Klein The icosahedral group led to an icosahedral
equation which stan ds in intim ate relation with the general equation
of the fifth degree Klein m a d e the icosahedron the centre of his theory
of the quintic as given in his Vorl esu n gen uber das I k osaeder u nd di e
A uflbsu n g der Gl ei chu ngen f u nf ten Gr ades Leip z ig 1 88 4
Finite substitution groups and their form s as related to linear
d i ff erential equations were invest i gated by R Fuchs (Cr el l e 6 6 6 8 )
in 1 86 6 and later If the equation has on l y algebraic integrals then
the group is finite and con v ersely F u c hs s researches on this topic
were continued by C Jordan F Klein and F B rioschi Finite ternary
and higher groups have been studied in connection with invariants by
F Klein who in 1 8 8 7 m ad e two such groups the basis for the solution
of general equations of the six th and seventh degrees In 1 88 6 F N
Cole under the guidance of Kle i n had treated the sextic equation in
the A m Jou r of M ath Vol 8 The second group use d by Klein was
stu died with reference to the 1 40 lines in space to which it lea d s by
H M aschke in 1 8 90
The relationship of i nv arian tal form s the study of which was
initiated by A Cayley and J J Sylvester receiv ed S ince 1 8 6 8 em
phasis in the writin gs of R F A C l eb sch a n d P Gordan Gordan
proved in C r el l e Vol 6 9 the finiteness of the system for a S ingle
“
binary form This is known as Gord an s theorem
Even in the
later si mplified fo rm s the proof of it is inv olv ed but the theorem
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348
.
H I STORY
A
O F M ATHE M ATI C S
yields practical m ethods to determ ine the existing system s G
P eano in 1 8 8 1 generaliz ed the theorem and applied it to the
cor
respondences represented by certain double binary form s In 1 890
D Hilbert by using only rational processes dem onstrated the fi n i t e
ness of the syste m of invariants arising from a given series of any form s
in n variables A m odification of this proof which has so m e advantages
was given by W E Story of Clarke University Hilbert s research
bears on the nu mber of relations called syzygies a subj ect treated b e
fore this tim e by A Cayley C Herm ite F Brioschi C Stephanos
of Athens J Hamm ond E Stroh and P A M ac M ahon
The symbolic notatio n i n the theory of invariants introduced by
S H A ron h ol d and R F A C l eb sc h was developed further by P Gor
d a n E Stroh and E S tu dy i n Germ any English wr i ters endeavored
to m ake the expressions in the theory of form s intuitively evi dent by
graphic representat i on as when Sylvester in 1 8 7 8 uses the atomic
theory a n idea appl i ed further by W K Cli ffor d The sym bolic
m ethod in the theory of inv ariants has been used by P A M ac M ah on
in the arti cle Algebra in the ele v enth e d ition of the E n cycl opaedi a
B r i tan n i ca and by J H Grace and A Young in their A l gebr a of I n
var i a n ts
Ca mbridge 1 90 3 Using a method in C Jordan s great
m e m oirs on invariants these authors are led to novel results notably
“
to an exact for m ula for the m axim um order of an irreducible c o
variant of a syste m of binary form s
A co mplete syz ygetic theo ry
of the absolute orthogonal conco mitants of binary quantics was c on
structed by E dwin B Elliott of Oxford by a m ethod that is not sym
bolie while P A M ac M ah on i n 1 90 5 employs a symbolic calculus i n
volving im aginary u mbr ae for simi lar purposes While the theory
of invar i ants has played an important r ole in m odern algebra and
analytic proj ective geom etry attention has been directed also to its
e mploym ent in the theory of nu mbers Along this line are the r e
searches o i L E D ickson in the M adi s on C ol l oqu i u m of 1 9 1 3
The establis hm ent of criteria by m eans of which the irreducibility
of expressions in a given dom ain m ay be ascertained has been i nv es
t ig a t e d by F T v Schubert
K F Gauss L Kronecker F W
P S chOn eman n F G M Eisenstein R D e dekind G Floquet L
K o nig sb e rg e r E Netto O P erron M B auer W D u m as and H
Blumberg The theore m of S chOn eman n and Eisenstein declares that
—
with integral coefficients is
c
if the polynom ial x + c 1 x +
f—
2
such that a prim e p divides every coeffi cient c l
but p does
c
not go into c then the polynomial is irred u cible in the domain of
rational num bers This theorem may be regarded as the nucleus of
the work of the later authors Floquet and K On ig sb erg er do not
l i mit the m selves to polynom ials but consider also linear ho m ogeneous
d ifferential expr e ssions
Blum berg gi v es a general theore m which
1
practically includes all earlier results as special cases
—
F o b i bli og p hy ee T
A m M th S o
V ol
6
7
9
pp 5 7 544
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in
HI STORY
A
3 50
O F M ATHE M ATIC S
in his book Refl ess i on i i n tor n o all a s ol u zi on e del l equazi oni
al gebr ai che was substantially the sam e as that gi v en later by P i err e
1
but only the second par t of Wan tz el s
L au r en t Wa n tzel
(18 14
simplified proof rese mbles R u fli n i s ; the first part is m odelled after
Abel s Wan t z el by the way deserves credi t for having given the
first rigorous proofs (L i ou vi l l e Vol 2 1 83 7 p
of the im
possibility of the trisection of any gi v en angle by m eans of ruler and
co mpasses and of avoiding the irr educible case in the algebraic
solution of irreducible cubic equations Wan t z el was r épétiteur at
the P olytechnic School in P aris As a student he excelled both in
“
mathe matics and languages Saint Venant said of him : Ordinarily
he worked evenings not lying down until late ; then he read and took
only a few hours of troubled sleep making alternately wrong use of
coffee and opium and taking his m eals at irregular hours until he
was m arried He put u nlimited tr ust in hi s constitution very strong
by nature which he taunted at pleasure by all sorts of abuse He
brought sadness to those who mourn his prem ature death
R u ffi ni s researches on equations are remarkable as containing
“
?
anticipations of the algebraic theory of groups
per
R u fli n i s
“
m utation corresponds to our ter m group
He di vided groups into
“
“
S i mple
and comple x and the latter into intransitive transitive
i mprimitive and transitiv e primitive groups He established the
“
i mportant theorem for which the nam e R u flfini s theorem has
3
been suggested that a group does not necessarily have a subgroup
whose order is an arbitrary divisor of the order of the group The
collected works of R u ffi ni are p ublished under the auspices of the
Circolo M atematico di P alermo ; the first vol u m e appeared in 1 9 1 5
with notes by Ettore B or tol ot t i of B ologna A transcendental solu
tion of the quintic invol v ing elliptic integrals was given by Ch Her
m ite (C ompt R end 1 8 58 1 86 5
After Hermite s first publica
tion L Kronecker in 1 8 58 in a letter to Hermite gave a second
solution in whi ch was ob tained a simple resolvent of the S ixth degree
Abel s proof that higher equations cannot al w ays be solved alge
b r a i c all y led to the i nquiry as to what equations of a given degree
can be solved by ra dicals S uch equations are the ones discussed by
K F Gauss in considering the division of the circle Abel advanced
one step further by proving that an irreducible equation can always
be solved in ra di cals if of two of its roots the one can be expressed
rationally in ter m s O f the other provided that the degr ee of the equa
tion is prime ; if it is not prime then the solution depends upon that
of equations of lower degree Through geometrical considerati ons
’
1 813
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E B or t ol ot t i , I nfl u en za , e t c , 1 9 0 2 , p 2 6 Wan tzel s p roo f i s g i v e n in N ou vell es
A n n al es M a thémati q u es , V o l 4 , 1 8 4 5 , p p 5 7 6 5 S ee a l so V o l 2 , pp 1 1 7—
1 27
Th e
’
s eco n d p a r t o f Wa n t ze l s p ro o f, i n v o l v i ng s u b s t i t u t i on t he o r y , i s re p rod u ced i n J A
’
S e rre t s A l gebr e s u pér i eur e
2
H B u rk h ard t , i n Z ei tschr f M a themati k u P hys i k , S u p pl , 1 8 9 2
3
G A M i l l e r , I n B i bl i otheca mathema ti ca , 3 F V o l 1 0 , 1 90 9 1 9 1 0 , p 3 1 8
1
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ALGEBRA
35 1
L O Hesse cam e upon algebraically solvable equations of the ninth
degree not included in the previous groups The subject was power
fully advanced i n P aris by the youthful Evar i s te Gal oi s ( 1 8 1 1 1 8 3
He was born at B ourg l afl f eifi e near P aris He b egan to exhi bit
m ost ex traor di nary m athematical geni us after his fifteenth year
His was a short sad and pestered life He was twice refused a d
m i t t an c e to the Ecole P olytechnique on account of inability to m eet
the ( to hi m)
trivial demands of e xam iners who failed to recog ni z e
his geni us He entered the Ecole Nor male in 1 8 2 9 then an inferior
school P roud and arrogant and unable to see the need of the cus
t o mary detailed e xplanations hi s career in that school was not s mooth
Drawn into the turm oil of the revolution of 1 830 he was forced to
leave the Ecole Normale After se v era l m onths spent in prison he
was killed I n a duel over a love aff air Ordinary tex t books he di s
posed of as rapidly as one would a novel He read J L Lagrange s
m emoirs on equations also writings of A M Legendre C G J
Jacobi and N H Abel As early as the seventeenth year he reached
resul ts of the highest i mportance Two m emoirs presented to the
Acade my of Sciences were lost A brief paper on equations in the
B ul l eti n de Feru ssa c 1 8 30 Vol XIII p 4 2 8 gives results which seem
to be applications of a general theory The night before the duel he
wrote his scientific testament in the for m of a letter to Auguste Cheva
lier containing a statement of the mathem atical results he had reached
and ask ing that the letter b e published that Jacobi or Gauss pass
j udg ment not on their correctness but on their importance
Two
m emoirs found a mong his papers were published by J Liou v ille in
1 84 6
Further m anuscrip ts were published by J Tannery at P aris
in 1 90 8 As a rule Galois did not fully prove his theore ms It was
only with difficulty that Liouv ille was able to penetrate into Galois
ideas Several comm entators worked on the task of filling out the
lac u n ae in Galois e xposition Galois was the first to use the word
“
group in a techni cal sense in 1 830 He div ided groups into simple
and compound and observed that there is no si mple group of any
“
composite order less than 6 0 The word group was used by A
Cayley i n 1 8 54 by T F Kirk man and J J Sylvester in
Galois
proved the i mportan t theore m that every inv ariant subgroup gives
rise to a quotient group which exhibits many funda mental properties
of the group He showed that to each algebraic equation corresponds
a group of substitutions which reflects the essential character of the
equation In a paper published in 1 84 6 he established the beautiful
theorem : In order that an irreducible equation of prim e degree be
solvable b y r adicals it is necessary and suff i cient that all its roots be
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A nn a l es de l ecol e n or mal e s u pér i eu r e, 3 S , V ol X II I ,
’
1 896
S ee al so E P i c a rd , O eu vr es ma th d Eva r i s te Gal oi s , P a ri s , 1 8 9 7 ; J P i e rp o n t ,
B u ll eti n A m M a th S oc , 2 S , V o l I V , 1 8 9 8 , pp 3 3 2 —
3 40
2
G A M ill er in A m M a th M on thl y, V o l
18
, 191 3, p
1
See l i fe by P aul D u pu y
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XX
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A
3 52
O F M ATHE M ATIC S
HI STORY
rational in any two of them Galois use of substitution groups to
determine the algebraic solvability of e quations and N H Abel s
so m ewhat earl i er use of these groups to prove that gen er al equations
of degrees higher than the fourth cannot be solved by radicals fur
ni shed strong incentives to the vigorous cultivation of group theory
It was A L Cauchy who entered this field next To Galois are due
also so m e valuable results in relation to another set of equations
presenting them selves in the theory of elliptic functions vi z the
m odular equations To Cauchy has been given the credit of being
1
the founder of the theory of groups of fini te order even though funda
m ental results had been prev iously reached by J L Lagrange P ietro
Abbati ( 1 7 86
P R u ffi n i N H Abel and Galois
Cauchy s
first publication was in 1 8 1 5 when he proved the theorem that the
number of distinct values of a non sym m etric function of degree n
cannot b e less than the largest prim e that divides n without b ecom
ing equal to 2 Cauchy s great researches on groups appeared in hi s
Exer c i s es d a n a l ys e et de phys i q u e ma théma ti qu e 1 8 44 and in articles
—
in the P aris C omptes Ren du s 1 84 5 1 84 6 He did not use the term
group but he uses (x y z u v w )
and other devices to denote sub
“
s t i t u t i on s uses the ter m s
cyclic substitution
order of a su b s ti tu
“
“
“
tion
identical substitution
transposition
transitive
in
transit ive
In 1 844 he prov ed the fundam ental theore m ( stated
but not pro v ed by E Galo i s )
whic h is known as Cauchy s theore m :
E v ery group whose order is divisible by a given prim e nu mber p m ust
contain at least one subgroup of order p This theore m was later
extended by L Sylow A L Cauchy was the first to enu merate the
o i d er s of the possible groups w hose degrees do not e xceed si x but thi s
enum eration was inco mplete At ti mes he fixed attention on prop
er t i e s of groups without i mm ediate concern as regards applications
a n d thereby took the first steps towar d the consideration of abstract
groups In 1 84 6 J Liouville m ade E Galois researches bett er known
by publication of two m anuscripts At least as early as 1 848 J A
Serret taught group theory in P aris In 1 8 5 2 Enrico B etti of the
Uni v ers i ty of P isa published in the A n n al i of B Tor tol i ni the first
rigorous exposition of Galois theory of equations that m ade the
theo ry intelligible to the g eneral public The first account of it gi v en
in a text book on algebra is in the third edition of J A S erret s A l ge
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br e ,
1 86 6
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In England the earliest studies in group theory are due to Arthur
Cayley and William R Ham ilton In 1 8 54 A Cayley published a
paper in the P hi l os ophi cal M agazi ne which is usually accepted as
foun ding the theory of abstract groups although the idea of abstract
groups occurs earl ier in the papers of A L Cauchy and Cayley s
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O u r accou n t of C a u c h y s res earc h es o n g rou p s i s dr aw n fro m th e ar ti c l e o f G A
—
—
M il l e r i n Bi bl i otheca W
l a thern a ti ca , V o l
1 9 0 9 1 9 1 0 , pp
1
3 7 3 2 9 , a n d t ha t o f
1 9 1 3 , pp 1 4 1 1 48
,
Jo sephi n e E B u rn s i n A m M ath M onthl y, V ol
’
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O F M ATHE M ATIC S
HI STORY
A
3 54
substitutions of n letters which are co mmutative wi th every sub s ti tu
tion of a regular group G on the sam e n letters constitute a group
which is S im ilar to G To Jordan is due the funda mental concept of
class of a substitution group and he prov ed the constancy of the
factors of composition He also proved that there is a finite nu mber
of primitiv e groups whose class is a given nu mber greater than 3
an d that the necessa ry and s uff i cient condition that a group be solvable
1
is that its factors of co mposition are pri me numbers
P ro minent
a mong C Jordan s pupils is Edmon t M oi l l et ( 1 8 6 5
editor of
L I n ter medi ai r e des mathémati ci en s who has m ade ex tensive c on tri b u
tions
In Germ any L Kronecker and R D edekind were the earliest to
becom e acquainted with the Galois theory Kronecker refers to it
in an article publ i shed in 1 8 53 i n the B eri chte O f the B erlin Acade my
D edekind lectured on it in G Ot ti n g en in 1 8 58 In 1 8 79—1 8 80 E Netto
gave lectures in Strassburg His S u bs ti tu ti on s theori e 1 88 2 was trans
—
e
by
i
u
s
e
B
a
t
t
a
i
n
i
1
2
lated into Italian in 1 8 8 5
G
of the
( 8 6 1 8 94)
pp
gl
Un iv ersity of Rom e and into Engl ish in 1 8 9 2 by F N C ol e then at
Ann Arbor The book placed the subj ect within easier reach of the
m athem atical public
In 1 8 6 2 —1 8 63 Lu dw i g S ylo w ( 1 83 2 —1 9 1 8)
gave lectures on sub s ti tu
tion groups in Christiania Norway which were attended by S oph u s
Lie Extending a theorem given nearly thirty years earlier by A L
“
Cauch y Sylo w obtained th e theorem known as Sylow s theorem
m“
m
Every group whose order is d ivisible by p but not by p
p being a
prim e nu mber contains 1 + hp subgroups of order pm About twenty
years later this theore m was e xten ded still further by Georg Fr oben i u s
—
of the Univ ersity of B erlin to the e ff ect that the number
( 1 8 4 9 1 91 7 )
of subgroups is k p+ 1 k being an integer even when the order of the
group is d ivisible by a higher po w er of p than pm Sophu s Lie took
a very im portant step by the explicit application of the group concept
to new dom ains and the creatio n of the theory of continuous groups
2
—
M a ri u s S Oph u s L i e ( 1 84 2 1 8 99)was born in N or dfj or dei de i n Nor
way In 1 8 59 he entered the Univer sity of Christiania but not until
1 8 6 8 di d this slowly developing youth d isplay m arked interest i n
m athe m atics The writings of J V P oncelet and J P l ucker awakened
—
8
his genius In the winter of 1 6 9 18 7 0 he m et Felix Klein in B erli n
and they published so m e papers of j oint authorship The sum mer of
1 8 70 they were together i n P aris where they were in close touch w i th
C Jordan and J G D arb o ux I t was then that Lie discovered his
contact transform ation which changes the straight li nes of ordinary
space over into spheres This l ed h i m to a general theory of trans
form ation At the outbreak oi the Franco P russian war F Klei n
—
l
t
m
t
i
S
V
X 99 9 p 3 3
i B i bli th
m h
G A M il l
3
—
6
t
l
I
6
F E g l i B i bl i th
t
h
m
i
S
V
m
4 ; M N o th er
9
pp
3
—
l
i M th A
V l 5 3 pp
4
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n n a en ,
eca
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ec a
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ca ,
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10 ,
20
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ALGEBRA
3 55
left P aris ; S Lie started to travel afoot through France into Italy
but was arrested as a spy and imprisoned for a month until D arb oux
was able to secure his release In 1 8 7 2 he was elected professor at the
Univ ersity of Christiania with all his time a v ailable for research In
—
1 8 7 1 1 8 7 2 he entered upon the study of partial di fferential equations
of the first order and in 1 8 7 3 he arrived at the theory of transforma
tion groups according to which finite continuous groups are applied
to infinitesim al transform ations He considered a very general and
important kind of transform ations called contact transfor mations
and their application in the theory of partial di ffere n tial equations
of the first and second orders As his group theory a n d theories of
integration m e t with no appreciation he ret urne d in 1 8 7 6 to the study
of geo metry—minim al surfaces the class ification of surfaces according
to the transformation group of their geodetic lines The starting of
a n ew journal the A rchi v f or M athemati k og N a tu rvi den s k ab in 1 8 7 6
enabled h im to publish his results promptly G H H al ph e n s pub
l ic a t i on s of 1 8 8 2 on di ff erential in v ariants induced Lie to direct a t
tention to his own earlier researches and their greater generality In
1 88 4 Friedrich Engel was induced by F Klein and A M ayer to go to
Christiania to assist Lie in the preparation of a t r ea ti se the Theor i e
der Tr an sf or ma ti on s gr u ppen 1 8 8 8 —
1 8 93
Lie accep ted in 1 8 8 6 a
professorship at the University of Leip zig In 1 88 9—1 8 90 over work
led to insomnia and depression of spirits While he soon recovered
his power for work he ever afterwards was over sensitiv e and mis
trustful o f his best friends With the aid of Engel he published in
1 8 9 1 a m em oir on the theory O f i n finite continuous transfor m ation
groups In 1 8 98 he returned to No rway where he died the following
year Lie s lectures on Difler en ti al gl ei chu n gen given in Leip z ig were
brought out in book form by his pupil Georg S cheffer s in 1 8 9 1 In
1 8 95 F
Klei n declared that Lie and H P oincar é were the two most
activ e m athe matical inv estigators of the day The following quota
tion from an article written by Lie in 1 8 95 indicates h ow his whole
“
1
soul was perm eated by the group concept :
In this century the
concepts kno w n as substitution and substitution group transforma
tion and transform ation group operation and operation group
inv ariant differential invariant and differential para meter appear
continually m ore clearly as the most important concepts of m athe
m atic s
Wh ile the curve as the representation of a function of a
single variable has been the m ost important object of m athe matical
inv estigation for nearly two centuries fro m D escartes while on the
other hand the concept of transform ation first appeared in this
century as an e xpedient i n the study of curves and surfaces there
has gradually d eveloped i n the last deca des a general theory of trans
formations whose elem ents are represented by the transform ation
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B eri chte d K oen i gl S aechs Gesel l schaf t,
M ath M on thl y, V ol I I I , 1 89 6 , p 2 96
1
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1 89 5 ;
t ran sl a t e d b y G A
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M ill er i n A m
O F M ATHE M ATIC S
HI STORY
A
3 56
”
its elf while the series of transformations in particular the transforma
tion groups constitute the obj ect
In close association with S Lie in the advancement of group theory
and its applications was F elix K l ein ( 1 849
He was born at
D usseldo r f in P russia and secured hi s doctorate at B onn in 1 868 After
studying in P aris he becam e privat—
docent a t G ottingen in 1 8 7 1
professor at Erlangen in 1 8 7 2 at the Technical High School in M u
nich in 1 8 7 5 at Leip z ig in 1 88 0 and at G ottingen in 1 8 8 6 He has been
active not only in the advance men t of various branches of ma them a t
ics but also in work of organi z ation Fa mous for laying out lines of
research is hi s Erlangen paper of 1 8 7 2 Vergl ei chende B etr achtu ngen
uber n eu er e g eometr i s che For s chu n gen
He beca me member of the com
mission on the publication of the E n cykl opci di e der mathemati s chen
Wi ss en s chaf ten and editor of the fourth vol um e on m echani cs also
editor of M a thema ti sche A n n al en 1 8 7 7 and in 1 90 8 president of the
International Co mmission on the Teachi ng of M athe matics A S an
inspiring lecturer on m athe matics he has wielded a wide influence
upon German and American students About 1 91 2 he was forced by
ill health to discontinue his lectures at G Ot ti ng en but in 1 9 1 4 the ex
c i t em e n t of the war roused h i m to acti v ity m uch as J La g range was
aroused at the outbreak of the French Revolution and Klein res um ed
lecturing He has constantly e mphasi z ed the importance of both
schools of m athe m atical thought nam ely the intuitional school and
the sc hool that rests everythi ng on abstract logic In his opinion
“
the intuitive g rasp and the logical treatment should not exclude
but should supplem ent each other
S Lie s m ethod of treating differential invariants was further i n
—
v e s t i g a t ed by K Z oraw ski in A cta M a th
Vol XVI 1 8 9 2 1 8 93 In
1 90 2 C N Haskins deter m ine d the nu mber of functionally i n dep e n d
ent in v ariants of any order while A R Forsyth obtained the invariants
for ordinary Eucli dean S pace Diff erential param eters have been
1
inv estigated by J Ed mund Wright of Bryn M awr College
Lie s
theory of inv ariants of finite continuous groups was attacked on logi
cal grounds by E S tudy of Bonn in 1 90 8 The vali di ty of this e riti
c i sm was partly ad mitted by F Engel
Another m ethod of treating di ff erential invariants originally due
t o E B C hr i s tofl el has been called by G Ricci and T Levi —
Civita
of P adua covariant derivation (M athema ti sche A n n al en Vol 54
2
A thi rd method was intro d uce d by H M aschke who used a
s ymbolis m S i mi lar to that for algebraic invariants
—
Henry W S tager published i n 1 9 1 6 A S yl ow Factor Tabl e f or the fi rst
Tw el ve Thou s a n d N u mbers : For every nu mber up to 1 200 the d ivisors
of the for m p(kp+ 1 )
are given where p is a pri me greater than 2 and
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We
a re u s i n g
J E Wrig ht
’
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s
'
f Quadr ati c Difier en ti al
I n var i an ts
o
5 8
pp
-
2
Tra n s A m M ath S oc
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,
Vo l
.
1,
1 90 0 ,
pp
.
197
—4
20
.
For ms ,
1 90 8 ,
O F M ATHE M ATI C S
HI STORY
A
3 58
L E Dickson said in 1 900 : When a problem has been exhibited
in g roup phraseology the possibility of a solution of a certain char
acter or the e xact nature O f its inherent di fli c u l ti es is deter mined by a
study of the g roup of the proble m
As the che mist analyz es
a compound to d etermine the ulti mate elements composing it so the
group theorist deco mposes the group of a given problem into a chain
of si mple groups
M uch labor has been e xpended in the de
termination oi simple groups For continuous groups of a fini t e
number of para meters the problem has been completely solved by
W Killing and E J Cartan
with the result that all such simple
groups aside fro m five isolated ones belong to the systems investi
gated by S ophu s Lie vi z the general proj ective group the p ro
j e c ti v e g roup of a linear co mplex and the projective group leaving
invariant a non degenerate surface of the second order The cor
responding proble m for infinite continuous groups re mains to be
sol v ed With regard to finite si mple groups the problem has been
2
3
attacked I n two directions O H older F N C ol e W B urn side
G H Ling and G A M iller have shown that the only S i mple groups
of co mposite orders less than 2 000 are the previously known S imple
groups of orders 6 0
On the other hand
50 4
var i ous infinite syste ms of finite si mple groups have been determ ined
The cyclic groups of pri me orders and the al ternating group of n
letters (n > 4 )
ha v e long been recogni z ed as S im ple groups The other
known syste ms of fi nite S i mple groups have been discovered in the
study of linear groups Four syste m s were found by C Jordan
in his study of the general linear the abelian
( Tr a i té des s u bsti tu ti on s )
and the two hypoab el i a n groups the field of reference being the set
of resi dues of integers with respect to a prim e m odulus p Generaliz a
"
i
t ons m ay be made by e mploying the Galois field of order p (desig
n a t e d GF
co mposed of the p Galois co mplexes formed wi th a
root of a congruence of degree n irreducible m odulo p Groups of
linear substitutions in a Galois field were studied by E B etti E
M ath i eu and C Jordan ; but the structure of such groups has been
determined only in the past decade The si mplicity of the group of
unary linear fractional substitutions in a Galois field was first proved
by E H M oore (B ul l eti n A m M a th S oc D ec 1 8 93)
and shortly
afte r w ard by W B urnside The co mplete generaliz ation of C Jor
dan s four syste m s of S i mple groups and the determination of three
new triply i nfi n i t e syste ms have been made by the writer (i e by
1
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S ee L E D i ck s on i n C ompte
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pp
1 90 2 ,
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2 25 ,
2 26
du I I
vendu
.
Con gr i n ter n
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,
P ar i s ,
1 900
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P a ri s ,
.
O H ol de r p ro v e d i n M a th A n n al en , 1 89 2 , t h a t th e r e a re o n l y t w o S i mp l e g ro u p s
o f c o mpos i t e o r d e r l ess t h an 200 , v iz , t hose o f o r de r 6 0 a n d 1 6 8
3
F N C o l e In A m J ou r M a th , 1 8 9 3 fo u n d t ha t t h e i c co u l d b e o n l y t hre e su c h
660
g ro u ps b e t w ee n o rd e rs 2 00 a n d 6 6 1 , v iz , o f o rd e rs
4
W B u rn side s ho w ed t ha t t here w as on l y on e Si mp l e g rou p of co mposi te orde r
b e t w een 6 6 1 an d 1 09 2
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AL GEBRA
3 59
L E Dickson in
Aside fro m the cyclic and alternating groups
the known syste ms of finite si mple groups have been derived as quo
tient groups in the series of co mposition of certain linear groups
M iss I M S c ho t t en fe l s of C hicago showed that it is possible to con
struct two si mple groups of the sam e order
The determination of the s m allest degree ( the class of any of the
non identical substitutions of pri mitive groups whi ch do not include
the alternating group was taken up by C Jordan and has been called
“
Jordan s problem
It was continued by Alfred B ocher t of B reslau
and E M aillet B oc her t proved in 1 8 92 : If a substitution group of
deg ree n does not include the alternating group and is more than
S i mply transitive its C l ass e x ceeds i n —
r
if i t is m ore than d oubly
transitive its class e xceeds 3n 1 and if i t is more than triply transi
—
n
ti v e its class is not less than % 1 E M aillet showed that when
the degree of a primitive group is less than 2 0 2 its class cannot be ob
t ai n ed by di mi n i shing the degree by unity u nl ess the degree is a power
of a prime n umber In 1 900 W B urnside proved that eve ry transitive
per muta tion group in p symbols p being prime is either solvable
or doubly tra nsiti ve
As regards linear groups G A M iller wrote in 1 8 99 as follows
The linear groups are of e xtreme i m
portance on account of their
numerous direct applications Eve r y group of a finite order can
clearly be represented in many ways as a linear substitution group
since the ordinary substitution (perm utation )groups are merely
very special cases of the linear groups The general question of rep
resenting such a group with the least nu mber of variables seems to
be far from a co mplete solution I t is closely related to that of de
t ermi n i ng all th e linear groups of a finite order that can be represented
w i th a small number of variables Klein was the first to determ ine all
the finite binary groups (in 1 8 7 5)
while the ternary ones were con
s i der e d independently by C Jordan ( 1 880 )
and H V a l en ti n er
The latter discovered the i mportant group of order 3 60 which was
o mitted by Jordan and has recently been proved (by A Wi man of
S i mply iso m orphic to the alter n ating group of degree 6
Lund )
H
M aschk e has considered m any quaternary groups and established
in particular a complete for m syste m of the quaternary group of
—
r
1
8
0
linear
substitutions
i
n
i
h
a
s
H
e
c
M
c
h
k
e
1
8
was born
( 53 1 90 8 )
5 4
in Breslau studied in B erlin under K Weierstrass E E Kumm er
and L Kronecker later in G ottingen under H A Schwarz J B
L i sting and F Klein He entered upon the study of group theory
un der Klein In 1 8 9 1 he came to the United States worked a year
with the Weston Electric C o then a cc ep t ed a place at the University
of Chicago
Linear groups of fin i te order first treated by Feli x Klein were
later used by h i m i n the extens i on of the Galo i s theory of algebraic
equat i ons as seen in his I k os aeder A S state d above Klein s de
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A HI STORY OF MATHE M ATIC S
36 0
termination of the linear groups in two variables was followed by
groups in three variables developed by C Jordan and H V al en tin er
and by groups of any num ber of variables treated by C
Jordan Sp ecial linear gr oups in four variables were di scussed by E
and G Bagnera of P alermo
The complete
G ou r sa t ( 1 88 9)
determination of the groups in four variables aside from intransi ti v
and m onomial types was carried t hrough by H F B l i chfel dt of L e
1
land S tanford University Says B l i chfel dt : There are in the main
four distinct principles employed in the determination of the groups
the original geo metrical process of Klein
in 2 3 or 4 variables :( a )
the processes lea di ng to a diophantine equation which may be
(b )
approached analytically ( C Jordan
or geom etrically (H Valen
tiner G B agnera H H M itchell )
a process involving the relativ e
; (c )
geo metrical properties of transform ati ons whi ch represent hom ologies
and like forms (H V al en ti n er G B agnera H H M itchell
(d)
a process developed from the properties of the m ultipliers of the trans
form ations which are roots of unity (H F B li chfel dt ) A new prin
c i pl e has been added recently by L Bieberbach though i t had already
been use d by H V al en ti n er in a certain form
Independent of
these principles stands the theory of group characteristics of whi ch
G F rob en i u s is the discoverer
There is a m arked diff erence between finite groups of even and of
?
o dd order
A S W B urn side points out the latter admit no self
in v erse irreducible representation except the identical one ; all irre
G A
du c i bl e groups of o dd or d er in 3 5 or 7 sym bols are soluble
M iller pro v ed in 1 90 1 that no group of odd order with a conj ugate
set of operations containing fewer than 50 m e mbers could be simple
W B urnside prov ed In 1 90 1 that transitive groups of odd order whose
degree I S less than 1 00 are soluble H L Rietz I n 1 90 4 extended thi s
last result to groups whose degrees are less than 2 43 W B urnside
has shown that the num ber of prime factors in the order of a simple
is a lower
group of odd order cannot be less than 7 and that
li mit for the order of a group of odd degree if S im ple These results
suggest that perhaps simple groups of odd order do not exist Recent
researches on groups m ainly abstract groups are due to L E Dickson
Le Vavasseur M P o tr on L I Neikirk G Frobenius H Hilton
3
A Wiman J A de Ség u i er H W Kuhn A Loew y H F B l i chfel dt
W A M anning and m any others E x tensive researches on abstract
groups have been carried on by G A M iller of the University of
Illinois In 1 9 1 4 he showed for instance that a n on abelian group
can have an abelian group of iso morphism s by proving the exi stence
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We
,
—
74 1 7 7
pp
2
W B u rn si d e
1
H
u si n g
a re
.
F
.
B l i ch fel d t
,
’
s
Fi n i te C oll i n eati on Gr ou ps , C hi cag o,
1 9 1 7,
.
Theor y of Gr ou ps of Fi n i te Or der , 2 Ed C a m b r i dg e , 1 9 1 1 , p 5 0 3
’
3
C o n s u l t G A M i l l e r s Thi rd R e po r t o n R e c e n t P ro g ress i n t h e T heo ry of
G r o u p s o f F i n i t e O r de r i n B u l l A m M a th S oc , V ol 1 4 , 1 90 7 , p 1 24
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A HI STORY OF M ATHE M ATIC S
36 2
then the exact number of positive roots is ascertainable fro m the
v ariations of S ign in the series
M ichel Fekete and Georg P OIya
?
—
both of B udapest use
x)
for the sam e purpose
The theo ry of equations commanded the attention of L e opol d
He was born in Liegnitz near Breslau
Kr on e ck e r ( 1 8 2 3
studi ed at the gymnasi u m of his native town under Kummer later
in B erlin under C G J Jacobi J Steiner and P Dirichl et then in
B reslau again under E E Ku mm er Though for elev en years after
1 8 44 engaged in business and th e care of his estates he did not neglect
In 1 8 55 he went to B erlin
m athematics and his fam e grew apace
where he began to lecture at the Univers ity in 1 8 6 1 He was a very
stim ulating and interesting lecturer K umrn er K Weiers trass and
L Kronecker constitute the triumv i rate of the second mathematical
school in B erlin This school e mphasi z ed severe rigor in dem onstra
tions L Kronecker dwelt intensely upon arithm etization which
repressed as far as possible all space representations and rested solely
upon the concept of nu mber particularly the positive integer He
displayed m anysided talen t and extraor di nary ability to penetrate
2 “
new fields of thought
B ut says G Frobenius
conspicuous as
his achiev ements are in the different fields of num ber research he
does not quite reach up to A L Cauchy and C G J Jacobi in an al y
SiS
nor to B Riemann and Weierstrass in function theo ry nor to
Dirichl et and Kumm er in nu mber theory
Kronecker s papers on
algebra the theory of equations and elliptic functions proved to be
diffi cult reading A m ore co mplete and simplified exp osition of hi s
results was giv en by R D edekind and H Weber
A mong the finest
3 “
of Kronecker s achieve ments says Fine
were the connections
whi ch he established among the various di sciplines in which he worked :
notably that between the theory of quadratic form s of negative deter
minant and elliptic functions through the S ingu lar m oduli which give
rise to the complex m ultiplication of the elliptic functions and that
between the theory of n u mbers and algebra by hi s arithm etical
theory of the algebraic equation
He held to the view that the theory
of fractional and irrational numbers could be built upon the integral
“
numbers alone
Die gan z e z ahl said he schuf der liebe Gott
alles U eb rig e ist M enschenwerk
Later he even denied the existence
of irrational numbers He once paradoxically remarked to Linde
“
m ann : Of what use is your beautiful research on the num ber i r ?
Why cogitate over such problem s when really there are no irrational
nu mbers whatever ?
In 1 8 90—1 8 9 1 L Kronecker developed a theory of the algebraic equa
tion with num erical coeffi cients whi ch he di d not live to publish
From notes of Kronecker s lectures H B Fine of P rinceton prepared
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B u l l A m M ath S oc , V o l 2 0 , 1 9 1 3 , p 2 0
2
G F ro b e n i u s Ged ii chtn i ssrede a uf L eopol d K r on eck er , B e rli n ,
3
B ul l A m M a th S oc , V ol I , 1 89 2 , p 1 7 5
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1 893 ,
p
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1.
AL GEB RA
3 63
”
?
an address in 1 91 3 giving Kronecker s unpublished results
All
“
who have read Kronecker s later writings says Fine are familiar
with his contention that the theory of the algebraic equation in its
final for m m ust be based solely on the rational integer algebraic
nu mbers being excluded and only such relations and operations being
admi tted as can be expressed in finite ter ms by means of rational
numbers and therefore ulti mately by means of integers These lec
tures of 1 890 —
9 1 are chi efly concern ed with the develop ment of such
a theory and in partic u lar with the proof of two theore ms which
therein take the place of the fundam ental theorem of algebra as
commonly stated
’
’
,
“
,
,
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,
.
S ol u ti on
f
o
N u meri cal E qu ati on s
a native of Geneva
Switz erland and the successor of P oisson in the chair of mechanics
at the Sorbonne published in 1 8 2 9 hi s celebrated theorem determining
the nu mber and situation of real roots of an equation co mprise d
between given limits D e M organ has said that this theore m i s the
co mplete theor eti cal solution of a di fficul ty upon whi ch energies of
e v ery order have been employed since the ti m e of Descartes
S turm
e xplains in that article that he enjoyed the privi lege of readi ng Four
i er s researches whi le they were still in m anuscript and that his own
discov ery was the result of the close study of the princi ples set forth
by Fo urier In 1 8 2 9 S turm published no proof P roofs were given
—
in 1 830 by Andreas von E ttinghausen ( 1 7 96 1 8 7 8 )of Vienna in
1 8 3 2 by Charles C ho q u e t e t M athi as M ayer in their Alg ebre and in
According to J M C D uham el S turm s
1 8 3 5 by S tur m h i m self
discovery was not the result of observation but of a well ord ered
l i ne of thought as to the kind of function that woul d m eet the r e
“
According to J J Syl v ester the theore m stared him
q u ir em en t s
in the face in the midst of som e mechanical investigati ons
( Sturm)
connected with the motion of compound pendulum s
D uh amel an d
Syl v ester both state that they received their informat i on from S turm
di rectly Yet their statements do not agree P erhaps both statem ents
are correct but represent diff erent stages in the evolution of the di s
?
c ov e ry in Stur m s m ind
By the theorem of S turm one can ascertain the nu mber of comple x
roots but not their location That limitation was re moved in a bril
liant research by another great French man A L Cauc hy He di s
covered in 1 8 3 1 a general theorem whi ch reveals the number of roots
whether real or co mplex which lie withi n a given contour This
theorem makes heavier demands upon the m athem atical attainm ents
J acqu es Char l es Fr an coi s S tu r m
( 1 80 3
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B u ll A m M ath S oc , V ol 2 0 , 1 9 1 4 , p 339
“
2
C o n su l t a l so M B och e r ,
T he p u b l i s h e d
S t u rm o n a l g eb ra i c an d di f e ren ti al Eq u at ion s
1 9 1 2 , pp 1 1 8
1
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—
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”
i
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an d
n
u n p u b l i sh e d
B u ll
.
Wo rk
of
A m M ath S oc
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.
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,
C h a rl es
V ol 1 8 ,
.
A
3 64
O F M ATHE M ATIC S
HI STORY
of the reader and for that reason has not the celebrity of Sturm s
theore m B ut it enlisted the lively interest of men like S turm J
Liou v ille and F M oigno
A remarkable article was published in 1 8 2 6 by Germi n al Dandel i n
—
in the m em oirs of the Academy of Sciences of Brussels
( 1 7 94 1 84 7 )
He gave the conditions under which the Newton Raphson m ethod of
approxim ation can be used with security In this part of his research
he was anticipated by both M ou rrail l e and J Fourier In another
part of his paper ( the second supplem ent )
he is more fortunate ; there
he describes a new and m asterly device for approximating to the roots
O f an equation
which constitutes an anticipatio n of the fa mous
m ethod of C H Gr affe
We m ust add here that the fundamental
i d ea of G raff e s m ethod is found even earlier in the M i scel l anea
a n al yti ca
1 76 2
of Edward Waring If a root lies between a and b
—b < 1 and a is on the convex side of the curve then D an del in
a
puts x = a + y an d transform s the equation into one whose root y is
small He then m ultiplies f (y)
by f ( y)
and O b tains upon writing
2=
z an equation of the sam e degree as the original one but whose
y
=0
roots are the squares of the roots of the equation f (y)
He remarks
that this transform ation m ay be repeated so as to get the fourth
eighth an d higher powers whereby the m oduli of the powers of the
roots div erge suffi ciently to m ake the transform ed equation separable
into as m any polygons as there are roots of distinct m oduli He ex
plains how the real and i maginary roots can be obtained D an del in s
research h ad the misfortune of being buried in the ponderous tom es
of a royal acade my Only accidentally d id we co m e upon this antici
pa t i o n of the m ethod of C H Gr affe Later the Acade m y of Sciences
of B erli n off ered a priz e for the inventio n of a practical m etho d of
computing im aginary roots The priz e was awarded to C a r l H e i nri ch
G r afi e ( 1 7 99
professor of m athematics in Zurich for his paper
published i n 1 83 7 i n Zurich entitled D i e A ufl osu n g der hoher en
n u mer i s chen Gl ei chu n gen
This contains the fa mous Gr affe m ethod
to which reference has been m ade Gr affe proceeds from the sam e
principle as did M or i tz A br aham S ter n
of G ottingen in
the m ethod of recurrent series and as did D an del in By the process
of involution to higher and higher powers the s maller roots are caused
to vanish i n comparison to the larger The law by which the n ew
equations are constructed is excee dingly S imple If for example
the coeffi cient of the fourth term of the given equation is a 3 then
the corresponding coefli c i en t of the first transform ed equation is
—
—
2as
In the com putation of the new coeff icients
a 2 2 a 2a 4+ 2 a 1a
Graff e uses logarithm s By this re markable m ethod all the roots
both real an d i maginary are found S im ultaneously without the
necessity of determ ining beforehand the nu mber of real roots and
the locat i on of each root The d iscussion of the case of equal i maginary
roots o mitte d by Graffe was taken up by the astrono mer J F Encke
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A HI STORY OF M ATHE M ATIC S
3 66
Italian mathem aticians have placed the determination of real
a n d I m aginary roots of nu merical equations by the m ethod of infinite
series w i thin reach of the practical co mputer The methods them
sel v es indicate the num ber of real and i maginary roots so that one
can dispense with the application of Stu rm s theorem here just as
easily as one can in the D a n del i n Gra ffe m ethod Considerable at
tention has been given to the solution of special types—trinom ial
equations— b y G D an del i n
K F Gauss ( 1 840
J
Lord John M L a ren
The last three used
B e l l a v i ti s
logarithm s of su m s and diff erences which were first suggested by
“
G Z Leonelli in 1 8 0 2 and are often called Gaussian logarithm s
The extension of the Gaussian m ethod to q uadrino mials w as under
taken by S G u n delfi n g e r i n 1 884 and 1 88 5 Carl F aerb er in 1 889 and
Alfre d Wiener in 1 8 8 6 The extension of the Gaussian method to
any equation was taken up by R M eh mke professor in D arm stadt
who published in 1 8 89 a logarith mic graphic m ethod of solving nu
m erical equations an d in 1 8 9 1 a m ore nearly arith metical method of
solution by logarith m s The m ethod is essentially a mixture of the
Newton R aphs on m etho d a n d the regula falsi as regards its the oretical
basis Well known is R M ehm k e s article on m ethods of co mputation
in the En cykl opadi e der mathemati schen Wi ss enschaf ten Vol 1 p 938
an d
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M agi c S qu ar es
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C ombi n a tory A n al ysi s
The latter part of the nineteenth century witnesses a revival of
interest in m ethods of constructing m agic squares Chief among the
writers on this subj ect are J Horner
S M D rach
Th H armu th
W W R B all
E M aillet
E M
E M c C lin toc k
M agic
L a q u i er e ( 1 88 0 )E Lucas
squares of the diabolic typ e as Lucas calls the m are designated
by M c C l in tock These and S imilar form s are called
p an di ag o n a l
Nasik squares by A H Frost An interesting book M agi c S qu ar es
Chicago 1 90 8 was prepared by the A merican electrical
a n d C u bes
engineer W S Andrews S till m ore recen t is the C ombi n atory A n al
ys i s Vol I Ca mbridge 1 9 1 5 Vol II 1 9 1 6 by P A M a c M ahon
“
which touches the subj ect of magic squares Says M a cM ahon : In
fact the whole subj ect of M agic Squares and connected arrange
ments of nu mbers appears at fir st S ight to occupy a position w hi ch is
completely isolated fro m other departm ents of pure mathe matics
The Object of Chapters II and III is to establish connecting lin ks
where none previously existed This is accomplished by selecting
a certain diff erential operation and a certain algebraical fun ction
I p VIII
“
The P robl em e des Rencontres
can be discussed in the sa me
The reader will be familiar with the old question of the
m anner
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1
En cycl opedi c des
s ci ences
mathém F I , Vol
.
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.
2 , 1 90 6 ,
pp 6 7 7 5
.
.
ANALYSIS
36 7
letters and en v elopes A given nu mber of letters are written to di f
f eren t persons and the envelopes correctly addressed but the letters
are placed at rando m in the envelopes The question is to find the
probability that not one letter is put into the right envelope The
enu meration connected with this probability question is the first
step that must be taken in the solution of the famous problem of the
Latin Square I p IX
The proble m of the L atin Square : The q uestion is to place n dif
2
feren t letters a b c
in each row of a square of n co mpartmen ts
in such wise that one letter being in each compart ment each colu mn
involves the whole of the letters The num ber of arrangemen ts is
required The question is famous because from the tim e of Euler
to that of Cayley inclusive its solution was regarded as being beyond
the powers of mathematical analysis It is solved without difficul ty
by the method of diff erential O perators of whi ch we are speaking
In fact it is one of the simplest exa mples of the method whi ch is S hewn
to b e c apabl e of solvin g questions of a much m ore recondi te charac
l
ter
The ex tension of the prin ciple of magic squares of the plane to thr ee
dimensional S pace has commanded the attention of many M ost
successful in t hi s field were the Austrian Jesui t Adam A daman du s
K oc han sk y
the French man Josef Sauveur
the Germans
Th Hugel ( 1 8 59)
and Hermann Scheffl er
In Vol II M ajor M ac M ahon gives a re markable group of identi ties
discovered by S Ra manujan of Cambridge which have applications
in the partitions of numbers but have not yet been established by
rigorous demonstration
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A n al ysi s
Under this head we find i t convenien t to consider the subj ects of
the diff erential and integral calculus the calculus of variations i n
finite series probability di fferential equations and integral equations
An early representative of the critical and philosophical school of
mathematicians of the nineteenth centu r y was B e rna r d B ol zan o
professor of the philosophy of religion at P rague In
( 1 78 1
1 8 1 6 he gave a proof of the bino mial for m ula and e xhibited clear
notions on the convergence of series He held advanced views on
variables continui ty and li mits He was a forerun ner of G Cantor
Noteworthy is his posthumous tract P ar adoxi en des Un endl i chen
e di ted by his pupil Fr P i ihon sk y B ol zano s writings
( P reface
were overlooked by mathematicians until H Hankel called attention
“
“
to them
H e has everything says Hankel that can place hi m
in thi s respect [notions on infinite series] on the same level with Cauchy
only not the art pecul iar to the French of re fi ning their ideas and
communicating them in the most appropriate and taking manner
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P A
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M acM aho n , Combi natory A n al ys i s , Vol I , C ambri dg e ,
.
1 9 15 ,
p ix
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.
A HI STORY OF M ATHE M ATI C S
368
”
So it ca me about that B ol z ano re mained unknown and was soon for
gotten
H A Schwarz in 1 8 7 2 looked upon B ol zano as the inventor
of a line of reasoning further dev eloped by K Weierstrass In 1 8 8 1
O Stol z d eclared that all of B ol z ano s writings are remarkable
inasm uch as they start with an unbiassed and acute criticism of the
1
contributions of the older literature
A reformer of our science who was e minently successful in reachi ng
the ear of his contemporaries was Cauchy
2
Au gu s ti n Lo u i s C au chy ( 1 7 8 9—
1 8 5 7)
was born in P aris a nd r e
c ei v e d his early education fro m his father
J Lagrange and P S
Laplace with whom the father cam e in frequent contact foretold
the future greatness of the young boy At the Ecole Centrale du
P a n theon he excelled In ancient classical studies
In 1 80 5 he entered
the Ecole P olytechnique and two years later the Ecole des P onts et
C h au S S ées
Cauchy left for Cherbourg in 1 8 1 0 in the capacity of
engineer Laplace s M ecan i qu e C el este and Lagrange s F on cti on s
A n al yti q u es were a mong hi s book co mpani ons there
Considerations
of health induced hi m to return to P aris after three years Yielding
to the persuasions of Lagrange and Laplace he renounced engineerin g
in favor of pure science We find him next holding a professorship at
the Ecole P olytechnique On the exp ul sion of Charles X and the
accession to the throne of Louis P hilippe in 1 8 30 Cauchy being
exceedingly conscientious found hi mself unable to take the oath de
man ded of hi m B eing in consequence depri v ed of his positions he
went into voluntary exile At F ri b ou rg in Switzerland Cauchy r e
sum ed his stu dies and in 1 83 1 was induced by the king of P iedm ont
to accept the chair of mathe m atical physics especially created for hi m
at the Un iversity of Turin In 1 8 33 he obeyed the call of hi s exiled
king Charles X to undertake the education of a grandson the D uk e
of Bordeaux This gav e Cauchy an opportunity to visit various parts
of Europe and to learn how e x tensively his works were being read
Charles X bestowed upon hi m the title of B aron On his return to
P aris in 1 8 3 8 a chair in the Coll ege de France was o ffered to hi m
but the oath dem anded of hi m prevented his acceptance He was
nominated m e mber of the B ureau of Longitude but declared ineligible
by the ruling power D uring the political events of 1 848 the oath was
suspended and Cauchy at last became professor at the P olytechnic
School On the establishment of the second e mpire the oath was r e
instated but Cauchy and D F J Arago were exempt from it Cauchy
was a man of great piety and in two of his publications staunchly de
fended the Jesuits
Cauchy was a prolific and profound mathem atician By a prompt
publication of his results and the preparation of standard text books
he exercised a m ore imm ediate an d benefic i al influence upon the great
C s l t H B e g ma
D
P hil
d B lz
H ll e 9 9
phi h W k B
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C A Va l son , L a Vi e cl l es trava ux du B ar on C a uchy, P ari s,
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a nos ,
1 86 8 .
a
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0
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O F M ATHE M ATIC S
HI STORY
A
3 76
wi th the indefinite equations in order to determ ine co mpletely
the m axi m a and m inima of m u l tiple integrals was awarded a priz e by
the French Acade my in 1 84 5 honorable m ention being made of a
paper by C E D elaunay P F S arr u s s m ethod was S implified by
—
2
of P avia at
A L Cauchy In 1 8 5 Gaspare M ainardi ( 1 800 1 8 7 9)
tempted to exhibit a new method of discri minating ma xim a and
minim a and ex tended C G J Jacobi s theorem to double integrals
M ainardi and F B rioschi showed the value of determ ina n ts i n ex
h i b i ti n g the terms of the second variation
In 1 86 1 I s a ac Todhu n ter
—
1
8
2
0
1 8 84 )
of St J ohn s College Cambridge published his valuable
(
work on the H i s tory of the P r ogr ess of the C al cu l u s of Vari ati on s which
contains researches of his own I n 1 8 66 he published a m ost important
research develop ing the theory of discontinuous solutions (discussed
in particular cases by A M Legendre )
and doing for this subj ect what
P F Sarrus had done for m ultiple integrals
The following are the more important older authors of systematic
treatises on the calculus of variations and the dates of publication :
Robert Woodhouse Fellow of Caius College Ca mbridge 1 8 1 0 ;
Richard Abba t t in London 1 83 7 ; J ohn Hewitt Jell e tt ( 1 8 1 7
once P rovost of Trinity College D ublin 1 8 50 ; Georg Wilhehn S trauch
of Aargau in Switz erland 1 849 ; F r an goi s M oigno ( 1 8 0 4
(181 1
—
of the
z
1
8
2
of P aris and Lorent Leonard Lindel of ( 7 1 90 8)
1 884 )
University of Helsingfors in 1 8 6 1 ; Lewis B uff ett Carll in 1 8 8 1
Carll ( 1 8 44
was a blind m athe matician graduated at C 0
lumbia College in 1 8 70 and i n 1 8 9 1 —
1 8 92 was assistant in m athe m atics
there
That of all plane curves of given length the circle includes a maxi
m u m area a n d of all closed surfaces of given area the sphere encloses
a maxim u m volum e are theore m s considered by Archimedes and
Z en odoru s but not prove d r i gorously for two thousand years until
K Weierstrass and H A Schwarz Jakob Steiner thought he had
proved the theore m for the circle On a closed plane curve different
from a circle four non cyclic points can be selected The quadrilateral
ob tained by successively joining the S ides has its area increased when
it is so deform ed ( the lunes being kept rigid )
that its vertices are
cyclic Hence the total area is increased and the circle has the m a xi
m um area Oskar P erron of Tubingen pointed out in 1 91 3 by an ex
“
a mple the fallacy of this proof :Let us prove that 1 is the largest of
all positive integers N 0 such integer larger than 1 can be the m axi
mum for the reason that its square is larger than itself Hence I
“
m ust be the m axi m u m S teiner s proof does not prove that am ong
all closed plane curves of given length there exists one whose area is a
1
K Weierstrass gav e a simple general existence theorem
maxim u m
applicable to the extrem es of c ontinuous ( stetige )
functions The max
imal property of the S phere was first prov ed r igorousl y I n 1 884 by
bin ed
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B l asc hk e
In
J a hr es b d deutsch M ath Ver ei n i g
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2 4,
1 9 1 5,
p
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1 95
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ANALYSIS
37 1
H A Schwarz by the aid of results reached by K Weierstrass in the
calculus of v ariations Another proof based on geo m etrical theorems
was giv en i n 1 90 1 by Hermann M inkow ski
The subj ect of m inimal surfaces which had received theattention of
J Lagrange A M Legendre K F Gauss and G M onge in later
ti me co mmanded the special attention of H A Schwarz The blind
physicist of the University of Gand Joseph P l ateau ( 1 8 0 1
in
1 8 7 3 described a way of presenting these surfaces to the eye by means
of soap bubbles m ade of glycerine water Soap bubbles tend to be
co me as thick as possible at eve ry point of their surface hence to make
their surfaces as sm all as possible M ore recent papers on m inim al
surfaces are by H a r ri s H ancock of the University of Cincinnati
Ernst P ascal of the University of P avia expressed him self In 1 8 97
1 “
on the calculus of variations as fOl l ow s : It m ay be said that this de
v e l opm en t [ the finding of the di ff erential e u a t i o S which the u n known
q
functions in a problem must satisfy] closes with J Lagrange for the
later an alysts turned their attention chiefly to the other m ore dif
fi c u l t proble m s of this calculus
The proble m is finally disposed of
if one considers the S i mplicity of the form ulas which arise ; wholly dif
f e ren t is this m atter if one considers the subj ect fro m the standpoint
of rigor of deriv ation of the formulas and the ex tension of the dom ain
of the problem s to which these form ulas are applicable This last
is what has been do ne for some years I t has been found necessary to
prov e c ertain theorem s which underlie those formulas and which the
first workers looked upon as axioms which they are not
This n ew
fiel d was first entered by I Todhunter M Ostrogradski C G J
Jacobi J B ertrand P du B ois Reymond G Erd mann R F A
C l eb sc h but the I ncisive researches which m ark a turn i ng point in the
history O f the subj ect are due to K Weierstrass As an illustration
of Weierstrass s m ethod of c ommu n i c a ti ing m any of his m athem atical
2 “
results to others we quote the following fro m O Bolz a : U n for tu
n a t el y they [ results on the calculus of variations ] were given by Weier
strass only in his lectures [ since
and thus became know n only
very slowly to the general m athem atical public
Weierstrass s
results and m ethods may at present be considered as generally known
partly through dissertations and other publications of his pupils
partly through A K n eser s L ehrbu ch der Vari ati on s r echn u n g (B raun
schweig
partly through sets of notes
of
which a great number are in circulation and copies of which are ac
cessible to every one in the library of the M athematische Verein
at B erlin and i n the M athe matische Lesez imm er at G ottingen
Under these circu mstances I have not hesitated to mak e use of Wei er
strass s lectures j ust as if they had been published i n print
Weier
strass applied modern require ments of rigor to the calculus of varia
l Di V i ti
A S h pp L i pz ig
h
899 p 5
E P
g ub e s
C hi g
O B l z a L ct res on the C a l cul
9 4 pp ix xi
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asc a
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ar a
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O F M ATHE M ATIC S
HI STORY
A
372
tions in the study of the first and second v ariation Not only did he
giv e rigorous proofs for the first three necessary conditions and for the
sufficiency of these con d itions for the s o called weak extremum
but he also extended the theory of the first and second variation to
th e case where the cu rves under consideration are given in para m eter
representation He discov ered the fourth necessary con dition and a
s uffi ciency proof for a so called strong extremu m which gave for
the first tim e a co mplete solution by m eans of a new m ethod based 0 11
1
the so called Weierstras s s construction
Under the stimu lus of
Weierstrass new dev elop ments were m ade by A Kneser then of
D orpat whose theory is based on the extension of certain theore ms
on geodesics to extrem als in general and b y D avid Hilbert of Gif t
tingen who gave an a pri or i existence proof for an extre m um of a
—
definite integral a d iscovery of far reaching i mportance not only for
the Calculus of Variations but also for the theory of differential
equations and the theory of functions (O B olz a) In 190 9 Bol z a
published an enlarged German edition of his calculus of variations
including the results of Gustav v Escherich of Vienna the Hilbert
m ethod of proving Lagrange s rule of m ultipliers (m ultiplikator regel )
and the J W Lindeberg of Helsingfors treatm ent of the isoperim etric
proble m About the sam e tim e appeared J Hadamard s C al cu l des
va r i a ti on s r ecu el l i es par M
J a cqu es H adama rd
Fr echet P aris 1 9 1 0
was born at Versailles is editor of the A n n al es sci en tifi qu es
( 1 86 5
de l ecol e n or mal e su pér i eu r e
In 1 9 1 2 he was appointed professor of
mathe matical analysis at the Ecol e P olytechnique of P aris as successor
to Cam ille Jordan In the above m entioned book he regards the cal
“
c u l u s of variations as a part of a new and broader
functional calculus
along the lines followed also by V Volterra in his functions of lines
This functional calculus was initiated by M aurice Fr echet of the Uni
versity of P oitiers in France The authors include also researches by
W F Osgood Other prominent researches on the calculus of varia
tions are due to J G D arb oux E G ou rsa t E Zerm elo H A Schwarz
H Hahn and to the Am ericans H Hancock G A Bliss E R Hedrick
A L U nderhill M ax M ason Bliss and M ason systematically ex
tended the Weierstrassian theory of the calculus of variations to
problems in S pace
In 1 8 58 D avi d B i er ens de H aan ( 1 8 2 2—1 8 95)
of Leiden published his
A rev ision an d the consid eration of the
Ta bl es d I n tegral es D efi n i es
underlying theory appeared in 1 8 6 2 It contained 8 339 form u las
A critical examination of the latter m ade by E W Sheldon in 1 91 2
“
showed that it was re markably free fro m error when one i mposes
proper li mitations upon constants and functions not stated by Haan
The lectures on definite integrals delivered by P G L Dirichlet
in 1 8 58 were elaborated into a standard work in 1 8 7 1 by Gustav
Ferdi nand M eyer of M u nich
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1
Thi s
su mmary
i s t ak e n fro m O B olz a,
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p
o
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ci t
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,
P r efac e
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A
3 74
O F M ATHE M ATIC S
HI STORY
Abel His letter to his friend B M H ol mb oe ( 1 8 2 6 )
contains severe
criticisms It is very in ter es t ing reading even to m odern students
In his de monstration of the binomial t heorem he established the
theore m that if two series and their product series are all convergent
then the product series will converge towards the product of the
sum s of the two given series Thi s remarkable resul t would dispose
of the whole problem of m ultiplication of series if we had a universal
practical criterion of convergency for se mi convergent series Since
we do not possess such a criterion theorems have been recently e s
t abli shed by A P r in g sh ei m of M unich and A V O S S now of M un ich
which remove in certain cases the necessity of applying tests of con
vergency to the product series by the application of tests to easier
related expressions A P ri n g shei m reaches the following interesting
conclusions : The product of two conditiona l ly convergent series can
never converge absolutely but a conditionally convergent series or
even a divergent series mu ltiplied by an absolutely convergent series
may yield an absolutely convergent product
The researches of N H Abel and A L Cauchy caused a considerable
stir We are told that after a scientific m eeting in which Cauchy
had presented hi s first researches on series P S Laplace hastened
hom e and re mained there in seclusion un til he had exam ined the
series in hi s M ecan i qu e C el es te Luckily every one was found to be
convergen t ! We m ust not conclude however that the new ideas
at once displaced the old On the contrary the new views were
generally accepted only after a long struggle A S late as 1 844 A D e
“
M organ began a paper on divergent series i n t hi s style : I believe
it w ill be generally admitted that the headi ng of this paper describes
the only subj ect yet remaining of an elem entary character on whi ch
a serious schi sm exists a mong m athematicians as to the absolute
correctness or incorrectness of results
First in ti me in the evolution of m ore delicate criteria of convergenc e
and div ergence co m e the researches of Josef Ludwig Raabe ( 1 80 1
of Zurich in Cr el l e Vol IX ; then follow those of A D e M organ
1 8 59)
as given in his calculus A D e M organ established the logarithm ic
criteria whi ch were di scovered in part independently by J B ertrand
The form s of these criteria as given by J B ertrand and by Ossian
B onnet are m ore convenien t than D e M organ s It appears from
N H Abel s posthum ous papers that he had anticipated the above
nam ed w riters in establishing logarithmi c criteria It was the O pin
ion of B onnet that the logarithmic criteria never fail ; but P D u
Bois—
Reymond and A P rin g shei m have each discovered series dem on
s t r ab l y convergent in whi ch these criteria fail to deter mi ne the con
vergence The criteria thus far alluded to have been called by P ring
s h ei m s peci al criteria because they all depend upon a co mparison of
"
n
t
h
the
term of the series with special functions a n n (log
etc
A mong the first to suggest gen er a l criteria and to consider the subject
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AN ALYSIS
375
fro m a still wider poin t of view cul mi nating in a reg ul ar mathematical
theory was E E Kummer He established a theorem yiel di ng a test
consisting of two parts the first part of whi ch was afterwards found
to be superfluous The study of general criteria was con tinued by
—
of P isa P D u B ois Reymond G Kohn of
Ulisse D ini ( 1 84 5 1 91 8)
Vienna and A P r in g sheim D u B ois Reymond di v ides criteria into
two classes : criteria of the fir st ki nd and criteria of the s econd ki nd ac
th term and
cordi ng as the general n th term or the ratio of the (n + 1 )
the n th term is m ade the basis of research E E Kummer s is a
criterion of the second kind A criterion of the first ki nd analogous
to this was invent ed by A P r in g sheim From the general criteria
established by D u B ois Reym ond and P r in g sheim respectivel y all
the S pecial criteria can be derived The theo ry of P rin g sh eim is very
complete and o ff ers in addi tion to the criteria of the fir st ki nd and
second kind entirely new criteria of a thi r d ki nd and also generaliz ed
cri teria of the second kind w hi ch apply however only to series with
never increasing term s Those of the thi rd ki nd rest m ain ly on the
consideration of the lim it of the diff erence either of consecutive terms
or of their reciprocals In the generali z ed criteria of the second ki nd
he does not consider the ratio of two consecutive ter ms but the ratio
of any two terms however far apart and deduces among others two
criteria previously given by Gustav Kohn and W Ermak off re spec
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t iv el y
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It is a strange vicissitude that divergent series which early in the
nineteenth century were supposed to have been banished once for
all from rigorous m athem atics S hould at its close be invited to return
In 1 88 6 T J Stieltjes and H P oincar é S howed the im portance to
analysis of the asym ptotic series at that tim e employed in astronom y
alone In other fields of research G H Hal phen E N Laguerre and
T J S ti el tj es have encountered particular examples in which a whole
series being divergent the corresponding continued fraction was
conv ergent In 1 8 94 H P ad e now of B ordeaux established the possi
b ili ty of defining in certain cases a function by an entire divergent
series This subj ect was taken up also by J Hadam ard in 1 892
C E Fabry in 1 896 and M Servant in 1 8 99 Researches on div ergent
s eries have been carried on also by H P o i ncar é E B orel T J S t i el tj es
E Ces aro W B Ford of M ichigan and R D Carm ichael of Illi
—
e
e
nois Thomas J ean S ti l tj s ( 1 8 56 1 8 94)was born in Zwolle in
Holl a nd cam e in 1 88 2 under the influence of Ch He rm ite beca m e a
French citiz en and later received a professorship at the University
of Toulouse Stieltj es was interested not only in divergent and con
di t i on all y convergent series but also in G F B Rie m ann s C function
and the theory of num bers
1
Difficult questions arose in the study of Fourier s series
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A rn o l d S a ch se , Vers u ch ei n er Ges chi chte der Dar s tel l un g wi ll k i trl i cher Fn u k
ti on en ei ner var i a bl en d urch tr i g on ometr i sche Rei hen , G ot t i n g en , 1 8 79
1
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A
376
HI STORY
-
or M ATHE M ATIC S
Cauchy was the first who felt the necessity of inquiring into its con
vergence B ut his m ode of proceeding was found by P G L D irichl et
to be unsatisfactory Dirichlet m ade the first thorough researches
on this subj ect (Gr el l e Vol IV) They culm inate in the result that
whenever the function does not becom e infinite does not hav e an
infi nite nu mber of discontinuities and does not possess an infinite
nu mber of m axi ma and m inim a the n F ou ri er s series converges toward
the value of that function at all places except points of discontinuity
an d there it converges toward the m ean of the two boundary values
L S chl afli of Bern and P D u B oiS Rey mond expressed d oubts as to
the correctness of the m ean value which were however not well
founded D i ri chl e t s conditions are sufli c i en t but not necessary
Ru dolf L i ps chi tz ( 1 8 3 2
of B onn proved that F ou r i er s series
still represents the functio n when the nu mber of discontinuities is
infi n ite a n d established a condition on which it represents a function
ha v ing an infinite nu mber of m axi ma and m i n ima D i ri chl e t s belief
that all continuous functions c an be represented by Fourier s series
at all po i nts was S hared by G F B R i emann and H Hankel but
was prove d to be false by D u Bois Reymond and H A Schwarz
A Hurwitz S howed how to express the product of t w o Ordi n ary Fourier
series i n the form of another Fourier series W W K ii s t ermann
solved the analogous proble m for double Fourier series i n which a
relation i n v ol v ing Fourier constants figures vitally For functions
of a single var iable an analogous relation is due to M A P arseval
and was proved by h im under certain restrictions on the rfat u re of
convergence of the Fourier series involved In 1 8 93 de la Vall ée
P oussin gave a proof requiri n g m erely that t h e functio n a n d its square
be integrable A Hurwitz in 1 90 3 gave further develop m ents M ore
recently the subj ect has com man d ed general interest through the r e
?
searches of Frigyes R i esz and Ernst Fischer (R ie sz Fischer theore m)
R i emann inquired what properties a function m ust have so that
there may be a trigonom etric series which whene v er it is convergent
converges toward the value of the fu n ction He found necessary
and s u ffi cient con ditions for this They do not decide however
whether such a series actually represents the fu n ction or not Rie
m ann rejecte d Cauchy s definition of a definite integral on account of
its arbitrar i ness gave a n ew definition a n d then inquired when a
function has an integral His researches brought to light the fact
that continuous functions need not always have a d iff erential co effi
cient B ut this property which was shown by K Weierstrass to b e
long to large classes of functions was not found necessarily to exclude
them from being represented by Fourier s series D oubts on so m e of
the conclusions about F ou r i er 1s ser i es were thrown by the observation
m ade by Weierstrass that the integral of a n infinite series can be
shown to be equal to the s u m of the integrals of the separate term s
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S u mm a r y t ak e n fro m B u ll A m M ath S oc
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V ol
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2 2 , 1 91 5,
p 6
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A HI STORY OF M ATHE M ATIC S
3 78
1
through the use of illustrations with urns that were exactl y
alike and contained black and white balls in different n umbers and
different ratios the observed event being the drawin g of a white ball
fro m any one of the urns For the sam e purpose Johannes von Kr ies
in his P ri nzi pi en der Wahrs chei nl i chk ei tsr echnu ng Freiburg i B 1 886
used as an illustration six equal t u b es of which one had the
S ign
on one S ide another had it on two S ides a third on three S ides and
finally the six th on all six sides All other sides were marked w ith a 0
Nevertheless obj ections to certain applications by B ayes Theore m
hav e been raised by the D anish actuary J B in g in the Tidsskrif t f or
M atemati k 1 8 79 by Joseph B ertrand in his C al cu l des probabi li tes
P aris 1 8 8 9 by Thorwald Nicolai Thiele ( 1 8 3 8—
1 91 0)
of the observa
tory at Cope nhagen in a work published at Copen hagen in 1 88 9 ( an
English edition of which appeared under the title Theory of Observa
ti on s London 1 90 3)
by George Chrystal ( 1 8 5 1 —1 9 1 1 )
of the Univer
?
A S recently as 1 90 8 the Danish
and others
s i ty of E d inburgh
philosophic writer Kro m an has com e out in defence of B ayes Thus
it appears that as yet no unanim ity of j udgm ent has been reached
in this m atter In determ ining the probability of alternative c auses
deduced from observed events there is often need of evidence other
than that which is afforded by the observed ev ent By in v erse prob
ability som e logicians ha v e explained induction For example if a
m an who has never heard of the tides were to go to the shore of the
Atlantic Ocean and witness on m successiv e days the rise of the sea
t hen says A dol phe Qu etel et of the observatory at B russels he would
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be entitled to conclude that there was a
p robability
equal to
m+ 1
m+ 2
that the sea would rise next day P utting m = o it is seen that this
view rests upon the unwarrantable assu mption that the probability
of a tota lly unknow n ev ent is 3 or that of all theories proposed for
investigation one half are true Willia m Stanley Jevons ( 1 83 5—1 88 2 )
in his P ri n ci pl es of S ci en ce founds induction upo n the theory of inverse
probability and F Y Edgeworth also accepts it in his M athemati cal
“
D aniel B ernoulli s moral expectation which was elab
P s ychi cs
orated also by Laplace has received little attention from more recent
French writers B ertrand e mphasiz es its im practicabil ity ; P oincar é
3
in hi s C al cu l des probabi l i tes P aris 1 8 96 disposes of it in a few words
The only noteworthy recent addi tion to probability is the subject
of local probability developed by several Engli sh and a few Amer
ican and French m athem aticians G L L B uff on s needl e problem
is the earliest i mportant problem on local probability ; it received the
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Encycl ope di a M otr e p I I , 1 84 5
’
2
We ar e u s i n g Em an u el C z u b er s E ntwi ckel u n g der Wahrschei n l i chkei tstheori e in
t he J a hr es b d deu ts ch M a thema ti k er Ver ei n i g u n g 1 8 99 , p p 9 3 —
1 0 5 ; a l so A rne
F i she r , The M a thema ti ca l Theor y of P r oba bi l i ti es , N e w Y o rk , 1 9 1 5 , pp 5 4 5 6
3
E C z u b er , op c i t , p 1 2 1
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ANAL YSIS
3 79
consideration of P S Laplace of E mile B arbier in the years 1 8 6 0
—
of the m il i tary school
and 1 8 8 2 of M organ W Crofton ( 1 8 2 6 1 9 1 5)
at Woolwi ch who in 1 86 8 contributed a paper to the London P hi l
“
P rob
os ophi cal Tr a ns acti on s Vol 1 5 8 and in 1 8 8 5 wrote the article
ability i n the En cycl opedi a B r i ttan n i ca ninth edition The name
“
local probability is due to Crofton Through considerations of
local probability he was led to the evaluation of certain defi nite in
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t eg ral s
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Noteworthy is J J Sylvester s four point problem : To find the
probability that four points taken at random within a given boun dary
shall form a re entrant quadrilateral Local probability was studied
in England also by A R Clarke H M c C oll S Watson J Wol s t en
hol me W S Woolhouse ; in France also by C Jordan and E Le moine ;
in A merica by E B Seitz Ric h collections of problem s on local prob
ability have been published by E manuel C zu b er of Vienna in his
Geometr i sche Wa hr s chei n l i chk ei ten u n d M i ttel w erte Leip z ig 1 8 84 and
by G B M Zerr in the E du cati on al Ti mes Vol 55 1 8 91 pp 1 3 7
The fundamen t al concepts of local probability have recei ved the
1 92
?
—
special attention of Ernesto Ces aro ( 1 8 59 1 90 6 )
of Naples
Criticism s occasionall y passed upon the prin ciples of probability
and lack of confidence in theoretical results have induced several
scientists to take up the e xpe r imental side which had been e mphasi z ed
by G L L B u ffon Trials of this sort were m ade by A D e M organ
W S J evons L A J Quetelet E C zu b e r R Wolf and S howed a
remarkably close agreemen t with theory In B u ffon s needle prob
l em the theoretical probability involves
This and S i milar e xpres
2
sions have been used for the empirical determination of
Attempts
to place the theory of probabil i ty on a purely empirical basis were
m ade by John S tuart M il l
John Venn ( 1 8 34
and
G Chrystal M ill s induction m etho d was put on a sounder basis
by A A C hu pr off in a brochure D i e S ta ti s ti k al s Wi s s en s chaft Em
p i r i c al m etho d s have co mmanded the attention of another Russian
v B or tk i evi cz
In 1 83 5 and 1 83 6 the P aris Academ y was led by S D P oisson s
researches to discuss the topic whether questions of m orality coul d be
treated by the theory of probability M H Navier argued on the
affi rmative while L P oi n sot and Ch D upin denied the applicability
“
as une sorte d ab erra t i on de l espr i t ; they declared the theo ry
applicable only to cases where a separation and counting of the cases
or events was possible John Stuart M ill opposed it ; Joseph L F
B ertrand ( 1 8 2 2
p rofessor at the Coll ege de France in P aris
?
and J v Kries are a mong more recent writers on thi s topic
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S e e Encycl opédi e des sci en ces ma th I , 2 0
p 23
2
E C zu b e r , op ci t , p p 88—
91
3
C o n s u l t E C z u b e r , op c i t , p 1 4 1 ; J S M i ll , S ys tem
Ed , 1 884 , C h ap 1 8 , pp 3 7 9—
3 8 7 ; J v K r i e s , o p c i t , pp
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L og i c , N e w Y o r k , 8 t h
2 53
—2 59
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A H I STORY O F M ATH E M ATIC S
3 80
Among the various applications of probability the one relating to
v erdicts of j uries decisions of courts a n d results of elections is specially
interesting This subj ect was studied by M arqui s de Condorcet
P S Laplace and S D P oisson
To e xhi bit Laplace s m ethod of
determ in i ng the worth of candidates by co mbining the votes M W
Crofton e mploys the fortuitous division of a straight line Thi s in
volves however an a pr i or i distribution Of values covering evenly
the whole range fro m 0 to 1 00 E xperience S hows that the normal
law of error exhibits a m ore correct distribution On this point Karl
?
P earson produced a m ost i mportan t research
He took a random
sample of n individuals from a population of N me mbers and derived
an e xpression for the average d i fference in character between the pth
and the (p+ 1 )
th indi v idual when the sa mple is arranged in order of
H L M oore of Columbia University
mag nitude of the character
h as atte mpted to trace P earson s theory in the statistics relating to
the e ffi ciency of wages (Econ omi c J ou r n al D ec
Early statistical stu dy was carried on under the nam e of political
arithmetic by such writers as Captain John Graunt of London ( 1 6 6 2 )
Application of the
and J P S ussmil ch a P russian clergyman
theo ry of probability to statistics was made by Edmund Halley
Jakob B ernoulli A D e M oivre L Euler P S Laplace and S D
The establishm en t of offi cial statistical societies and statis
P oisson
tical off ices was largely due to the influence o f the B elgian astronomer
and statistician A d olph e Q u e te l e t ( 1 7 96 —1 8 74)
of the observatory
“
“
at B russels the foun d er of m odern statistics
Q u et el et s average
i n who m
all processes correspond to the average results
man
“
obtained for society who could be considered as a type of the
beautiful has given rise to m uch critical discussion by Harold
“
A de Foville in his homo
Westergaard
J B ertillon
of 1 90 7 Joseph Jacob s in his the M iddle Am erican
m e dius
2
and the M ean Engl i shm an
Qu e t el e t s visit in England led
to the organi z at i on in 1 8 3 3 of the statistical section of the B ritish
Association for the Advance m en t of Science and in 1 83 5 of the S tatis
tical Society of London Soon after in 1 8 3 9 was formed the A mer
ican Stat i st i cal Society Qu e t el e t s best researches on the application
of probability to the physical and social sciences are given in a series
of letters to the duk e of Saxe Coburg and Gotha L ettr es su r l a theor i e
“
He laid e mphasis on the law of
des pr oba bi l i tés B russels 1 84 6
large numbers whi ch was advanced also by the Frenchm an S D
the Scandi
P oisson and d iscussed by the Germ an W Lexis
n avi an s H Westergaard and Carl Charlier and the Russian P a n u ti
f
—
h
e
bi
h
i
i
h
C
c
e
v
1
8
2
of the University of P etrograd To
L w ow c
1 1 8 94 )
(
C h eb i c hev we owe also an interesting proble m :A proper fraction being
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1 “
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N o te
on
S ee F ran
19 13;
P 3 74
z
A
Z Iz ek
’
s
—
B i ometr i ca , V ol I , pp 3 90 399
S ta ti s ti cal A vera ges , t ra n sl b y W M P e rson s , N ew Y or k ,
Fra n c i S C al to n
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P ro bl e m ,
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A HI STORY OF M ATHE M ATIC S
382
”
general and adequate m athematical m ethods for the analysis of
“
biological statist i cs To hi m are due the term s mode
standard
“
dev iat i on and coe fficient of variation
B efore hi m the normal
curve of errors had been used exclusi v ely to d escribe the distribution
of chance events This curve is symm etrical but natural phenomena
som eti m es indicate an asymm etrical di stribution
Accordin gly
P earson in his C on tr i bu ti on s to the Theory of E vol u ti on 1 8 99 de
v el op e d skew frequency curves
About 1 8 90 the Georg M endel law
of inheritance beca m e generally known and caused some m odification
in the application of statistics to heredity Such a readj ustm ent was
?
eff ected by the D an ish botanist W Johannsen
The first study of the most advantageo u s combinations of data
of observation is due to Roger Cotes in the appendix to hi s H armoni a
men s u r ar u m 1 7 2 2 where he assi g ns weights to the observations
The use of the arithmetic mean was advocated by Thom as Simpson
“
in a paper An atte mpt to S how the advantage arising by taking the
2
m ean of a nu mber of obser v ations in practical astrono my
also by
J Lagrange in 1 7 73 and by D aniel B ernoulli i n 1 778 The first
printed statement of the principle of least squares was made in 1 80 6
by A M Legendre without dem onstration K F Gauss had used
it still earlier but d id not publish it until 1 80 9 The first deduction
of the law of probability of error that appeared in prin t was given in
1 8 0 8 by Robert A dra i n in the A n al ys t a journal published by him self
in P hiladelphi a Of the earlier proofs given of this law perhaps the
m ost satisfactory is that of P S Laplace K F Gauss gave two
proofs The first rests upon the assumption that the arithm etic
m ean of the observations is the most probable value Atte mpts to
prove this assu mption have been m ade by Laplace J F Encke
A D e M organ
G V Schiaparelli E J S tone
and A
Ferrero
Valid criticis m s upon so m e of these investigations were
3
passed by J W L Glaisher The foundi ng of the Gaussian proba
bil i ty law upon the nature of the ob served errors was atte mpted by
F W B essel
G H L Hagen
P G
J F Encke
Tait
and M W Crofton
That the arithm etic mean
taken as the m ost probable value is not under all circumstances
co mpatible with the Gaussian probabilty law has been S how n by
Joseph B ertrand I n his C al cu l des pr obabi l i tés
and by others
The developm ent of the theory of least squares along practical lines
is due m ainly to K F Gauss J F Encke P A Hansen Th Gallo
5
Simon
way J B ienayme J B ertrand A Ferrero P P i zzetti
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P u b A m S tat A ss n , N S , Vo l X I V , r 9 1 4 , p 4 5
2
M i s cell a n eou s Tr acts , Lo n don , 1 7 5 7
3
L and A s tr S oc M em 3 9 , 1 8 7 2 , p 7 5 ; for f u r t her re fe ren ces , see Cykl opadi e d
M ath Wi ss , I D 2 , p 7 7 2
1
C o n s u l t E L D o dd , P ro b a bi l i ty o f t he A ri t h me t i c M ean , e t c , A n n a l s of
M athema ti cs , 2 S , V o l 1 4 , 1 9 1 3 , p 1 8 6
5
E C z u b er , op ci t , p 1 79
1
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Qu ar t
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ANALYSIS
83
”
generali z ed theory of the
Newco mb of Washington ad v anced a
1
co mbination of observations so as to obtain the best resul t ,
when
large errors arise more frequently than is allowed by the Gaussi an
probability law The sa m e subj ect was treated by R Lehmann
Filhes in A str on omi sche N achr i chten , 1 88 7
.
.
.
A criterion for the r ejection of doub tful observations was given
by B enjamin P eirce of Harvard It was accepted by the Am erican
astrono mers B A Gould ( 1 8 2 4
W Chauvenet ( 1 8 2 0
and J Winlock ( 1 8 2 6
but was criticised by the English as
t r o n omer G B Airy
The prev ailing feeling has been that there
e xists no theoretical basis upon which such criterion ca n be rightly
established
The application of probability to epidemi olog y was first consider ed
by Daniel B ernoulli and has more recently comm anded the atten
tion of the English statisticians William Farr ( 1 80 7
John
B rownlee Karl P earson and Sir Ronald Ross P earson studied
norm al and abnorm al frequency curves Such cur ves have been
fitted to epide mics by J B rown lee in 1 90 6 S M Greenwood in 1 9 1 1
and 1 91 3 a n d Sir Ronald Ross in
So me interest attaches to the d iscussion of whist from the stand
point of the theory of probability as is con tained in William P ole s
P hil osophy of Whi s t New York and London 1 88 3
The problem
is a genera liz ation of the game of trei z e or recontre treated
by P ierre R de M ontmort in 1 7 0 8
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Difier en ti al E qu ati on s
Difler en ce Equ ati on s
-
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’
Criteria for distingui shing between singular solutions and particular
solutions of diff erential equations of the first order were ad v anced
by A M Legen d re S D P o i sson S F Lacroix A L Cauchy and
G Boole After J Lagrange the c d i scrim inant relation commanded
the attention of J ea n M ar i e C on s tan t D u hamel ( 1 7 97—1 8 7 2 )
of P aris
C L M H Navier and others B ut the entire theory of S ingular
solutions was r e investigated about 1 8 70 along new paths by J G
D arb ou x A Cayl ey E C Catalan F C a sora t i and others
The
geometric side of the subj ect was considered more m inutely and the
cases were e xplained in which Lagrange s m ethod does not yield
S ingular solutions
E v en these researches were not altogether satis
factory as they did not furnish necessary and suffi cient conditions
for singular solutions which depend on the d iff erential equation alone
and not in anyw ay upon the general solution Return ing to m ore
purely analytical considerat i ons and building on w ork of Ch Briot
and J C Bouquet of 1 8 56 Carl Sch m idt of Giessen in 1 8 84 H B
of
Fine of P rinceton in 1 890 and M eyer Hamburger ( 1 8 38—1 90 3)
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A m J ou r M a th , V ol 8 , 1 886 , p 3 43
2
Gou l d A s tr J ou r , I I 1 8 5 2
3
N atur e, V o l 9 7 , 1 9 1 6 , p 2 43
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A HI STORY OF M ATHE M ATIC S
3 84
Berlin brought the problem to fi n al solutions Acti v e in this line
?
were also John M uller Hill and A R F or syth
The first scientific tr eatm ent of partial diff erential equations was
gi v en by J Lagrange and P S Laplace These equations were i n
v es ti g a t d i n m ore recent ti m e by G
M
onge
P
f
f
F
fa
C
G
J
J
e
Jacobi E m ile B our ( 1 83 1 —1 8 66 )
of P aris A Weiler R F A C l eb sch
A N K ork i n e of S t P etersburg G B oole A M eyer A L Cauchy
J A Serret S ophu s Lie and others In 1 8 73 their r e seaches on
partial diff erential equations of the fi rst order were presented i n
text book form by P aul M ansion of the University of Gand P ro
c eedi n g to the consi d eration of som e detail we re m ark that the keen
researches of Jo h ann Fri e dri c h Pf aff ( 1 7 6 5 1 8 2 5)
m arked a decided
ad v ance He was an intim ate friend of K F Gauss at G ottingen
Afterwards he was with the astronom er J E B ode Later he beca m e
professor at Hel mst adt then at Halle B y a peculiar m ethod P faff
found the general integration of partial d iff erential equations of the
first order for any number of variables Starting from the theory of
ordina ry diff erential equations of the first order in n variables he
gives first their general integration and then consid ers the integra
tion of the partial di fferential equations as a particular case of the
form er assu ming howe v er as known the general integration of
differential equations of any order between two variables His r e
“
searches led C G J Jacobi to introduce the n a m e P fafli an prob
l em
From the connection observ ed by W R Hamilton between
a syste m of or dinary d ifferential equations ( in analytical m echanics)
and a partial differential equation C G J Jacobi drew the conclu
sion that of the series of syste m s whose successive integration P faff s
m ethod de m an d ed all but the first syste m were entirely superfluous
R F A C l eb s ch cons idered P fa ff s proble m fro m a n ew point of vie w
and reduced it to syste m s of si multaneous linear partial di ff erential
equations which can be estab l ished in dependently of each other with
out any integration Jacobi m aterially ad v ance d the theory of dif
The problem to determ ine u n
f er e n t i al equations of the first order
known funct i ons i n such a way that an i n tegral containing these func
tions and their d iff erential coefficients in a prescribed m anner S hall
reach a m axim u m or m ini mu m value dem an ds in the first place
the van i shing of the first variation of the integral This conditio n
l eads to differential equations the integration of which determ ines th e
functions To ascertain whether the value is a m axim u m or a mini
m u m the second variation m ust be e xa mi ne d This leads to new and
difli c u l t d i fferential equations the integration of which for the simpler
cases was ingeniously deduced by C G J Jacobi from the integra
tion of the d ifferent ial equations of the first variation Jacobi s
soluti on was perfected by L O Hesse while R F A C l eb sch extended
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1
We
A bit
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h av e u sed S R o t h en b e r g ,
Gesek d M a th Wi ss en s ch
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G eschi ch t e
der
( M C an tor , H ef t
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)
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s i n g u l are n
XX
,
z
L os u n g e n
3 L e ip i g ,
.
1 90 8 .
in
A HI STORY OF M ATHE M ATIC S
38 6
prim itiv e is obtained by the use of processes that so met imes are frag
m entary in theory usual ly are tentative i n practice an d nearly always
are indirect in the sense that they are compounded of a nu mber of
formal operations ha v ing no organic relation with the primitive
In such circu mstances
is the prim itive co mpletely compr eh en
sive of all the integrals belonging to the equation ? A M Amp ere i n
1 8 1 5 propounded a broad definition of a general integral —
one i n
which the only relations which subsist a mong the variables and the
derivatives of the dependent variable and which are free from the
arbitrary elem ents in the integral are constituted by the diff erential
equation itsel f and by equations deduced from it by di fferentiation
This definition is inco mplete on various grounds E G ou r sa t gave i n
1 8 98 a S im ple instance to sho w that an integral satisfying all of A m
p ere s requirem ents was not general A second definition of a general
integral was given in 1 88 9 by J G B arboux based on A L Cauchy s
existence theore m : An integral is general whe n the arbitrary ele
m ents which it contains can be S peciali z ed in such a way as to provide
the integral established in that theorem This definition according
to A R F or syth calls for a m ore caref u l discussio n of obvious and
latent singularities
There are three principal m ethods of procee ding to the c on s t ru c
tion of an integral of partial differe n tial equations of the second order
which lead to success i n special cases One m ethod given by P S
Laplace in 1 77 7 applies to linear equations with two in dependent
variables I t can be used for equations of order higher than the
second I t has been developed by J G D a rb ou x and V G I msh e ne t
ski 1 8 7 2 A second method or iginated by A M Amp ere while
general in S pirit and in form depends upon in dividual skill unassisted
by critical tests Later researches along th i s line are due to E B orel
and E T Whittaker
A third method is due to J G
( 1 8 95)
D a rb ou x and includes according to A R Forsyth s classificatio n
the earlier work of M onge and G B oole A S first given by J G Dar
boux in 1 8 70 it applied only to the case of t w o in dependent variables
but it has been extended to equations of m ore t ha n t w o independent
variables and orders higher than the second ; it is not universally
“
“
e ffective
Such then says Forsyth are the principal m ethods
hitherto devised for the formal integration of partial equations of the
second order They have been discussed by m any m athe maticians
and they have been subj ected to frequent mo d i fi cations in details :
but th e substance of the processes remains unaltered
Instances are known in ordinary linear equations when the primitive s
can be expressed by definite integrals or by m eans of asym ptotic
e xpansions the theory of which owes m uch to H P oincar é Such
instances within the region of partial equations are due to E Borel
G F B Riem ann had re m arked i n 1 8 57 that functions expressed
by K F Gauss hypergeometric series F ( a 8 y x)which satisfy
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ANALYSIS
38 7
a homogeneous linear diff erential equation of the second order with
rational coeffi cients m ight be utilized in the solution of any lin ear dif
?
fere n ti al equation
Another mode of solving such equations was due
to Cauchy and was e x tended by C A A B riot and J C B ouquet)
and
consisted in the development into power series The fertility of the
conceptions of G F B Riemann and A L Cauchy with regard to
diff erential equations is attested by the researches to which they have
given rise on the part of L azaru s Fu ch s ( 1 833—
1 90 2 )
of B erlin Fuchs
was born in M oschin near P osen and became professor at the Uni
versity of B erlin in 1 884 In 1 8 6 5 L Fuchs co mbined the two m ethods
in the study of linear differential equations : One method using power
series as elaborated by A L Cauchy C A A B riot and J C Bou
quet ; the other m ethod using the hypergeometric series as had been
done b y_G F B Riemann B y this union Fuchs initiated a new
?
theory of linear di ff erential equations
Cauchy s develop m ent into
power series together with the calcul des limites aff orded e xistence
theore ms which are essentially the same in nature as those relating to
diff erential equations in general The S ingular points of the linear
diff erential equation received attention also from G F rob en i u s in
B ocher in 1 90 1 A second approach to
1 8 74 G P eano in 1 8 8 9 M
existence theore m s was by successive appro xi m ation
first used in
1 86 4 b y J Caqu é then by L Fuchs in 1 8 7 0 and later by H P oin
care and G P eano A third line by interpolation is originally due to
A L Cauchy and received special attention from V Volterra in 1 8 8 7
The general theory of linear diff erential equations received the atten
tion of L Fuchs and of a large number of workers including C Jordan
V Volterra and L S chl e sin g er Singular places where the solutions
are not indeterm inate were investigated by J Tanne ry L Schlesinger
G J Wallenberg and m any others Ludwig Wilhel m Thome ( 1 84 1
of the University of Greifswald discovered in 1 8 7 7 what he
1 91 0 )
called normal integrals Divergent series which formally satisfy dif
fer en t i al equations first noticed by C A A Briot and J C B ouquet
in 1 8 56 were first seriously considered by H P oincaré in 1 88 5 who
pointed out that such series may represent certain solutions a symp
Asymptotic representations have been examined by A
t o t i c al l y
and A Hamburger
Kneser
E P icard
J Horn
“
A special type of linear diff erential equation the F u ch si an
type with coe ffi cients that are single valued (eindeutig)and the
solutions of which hav e no points of in determinateness was investi
gated by Fuchs and i t was found that the coe ffi cients of such an equa
tion are rational functions of x S tudies based on analogies of linear
differential equations with algebraic equati ons first undertaken by
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L S ch e si n g e r , En twi ck el u n g d Theor i e d l i n ear en Di ffer en ti al
g l e i chu n g en s ei t 1 8 6 5 , L e i p i g a n d B e r l i n , 1 90 9
2
We a re u si n g h e r e a re po r t b y L S chl esi n g e r i n J ahr es b d d M ath Ver ei n i g u n g ,
—
l
1
o
1
1
0
8 , 9 9, pp
V
33 2 6 0
1
are u si n g
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A HI STORY OF M ATHE M ATIC S
388
N H Abel J Liouville and C G J Jacobi were pursued later by P
Appell
by E P icard w ho worked under the influence of S
Lie s theory of transform ation groups and by an army of workers in
France England Germ any and the United S tates The consideration
of differential invariants enters here Lamé s differential equation
considered by hi m in 1 8 5 7 was taken up by Ch Herm ite in 1 8 77 and
soon after in stil l m ore generali z ed form by L Fuchs F Brioschi
E P icard G M M ittag L effi er and F Klein
The anal ogies of linear di ff eren tial equations wi th algebraic f unc
tions problem s of inversion and uniformi z ation as well as questions
in v olving group theory received the attention of the analysts of the
second half of the century
The theory of invariants associated with linear di ff erential equations
as developed by Halphen and by A R Forsyth is closely connected
with the theory of functions and of groups Endeavors have thus
been made to deter mine the nature of the fun ction defined by a di f
f er en t i al equation fro m the d iff erential equation itself and not from
any analytical expression of the fun ction O btained first by sol v in g
Instead of studying the properties of the
t h e di ff erential equation
integrals of a di fferential equation for all the values of the variable
investigators at first contented the m selves with the study of the prop
The nature of the i ntegrals
e r t i e s in the v i cinity of a gi v en point
at S ingular points and at ordinary points is entirely di ff erent Char l es
A u gu ste A l ber t B r i ot ( 1 8 1 7—
1 88 2 )
and J ean Cl aude B ou qu et ( 1 8 1 9—1 88 5)
both of P aris stu di ed the case when near a singular poin t the di f
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fe r en t i al
,
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Q
equations take the form (x —x o)
dx
L Fuchs gave
.
the develop ment in series of the integrals for the particular case of
linear equati ons H P oincar é di d the sam e for the case when the
equations are not linear as also for partial d ifferential equati ons of
the first order The develop m ents for or di na ry points were giv en
M adam e
by A L Cauchy and S ophi e K oval evs k i ( 1 8 50
K ov al ev ski was born at M oscow was a pupil of K Weierstrass and
beca me professor of Analysis at Stockhol m
—
H e nri P o in c ar é ( 1 8 54 1 91 2 )
w a s born at Nancy and co mm enced
hi s stu dies at the Lyc ée there While taking high rank as a student
he di d not di splay exeep ti on al precocity He atten d ed the Ecole
P olytec hnique and the Ecole Nationale Sup érieure des M ines in P aris
receiv ing his doctorate from the University of P aris in 1 8 7 9 He b e
cam e instructor in m athem atical analysis at the Univ ersity of Caen
In 1 8 8 1 he occupied the chair of physical and experim ental m echanics
at the Sorbonne later the chair of mathem atical physics and after
the death of F Tisseran d the chair of m athem atical astronomy and
celestial mechanics Al though he di d not reach old age he published
numerous books and more than 1 500 m em oirs P robably neither
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HI STORY OF M ATHE M ATIC S
A
0
39
'
Every day I seated myself at my work table and S pent an
hour or two there trying a great m any co mbinations but I arrived at
no result One night when contrary to my custom I had taken black
coff ee and I could not S leep ideas surged up in crowds I felt them as
they struck against one another until two of the m stuck together so
to S peak to form a stable combination B y morning I h ad established
the existence of a class of fuchsian functions those whi ch are derive d
from the hyp ergeo metric series I had m erely to put the res ults in
1
S hape which only took a few hours
P oincaré enriched the theory of integrals
The attempt to express
i ntegrals by develop m ents that are always convergent and not li mi ted
to particular points in a plane necessitates the introduction of new
transcendents for the old functions per mit the integration of only a
s mall nu mber of differential equations H P oincaré t ried this plan
with linear equations which were then the best kn own having been
studied in the vicinity of gi v en points by L Fuchs L W Tho me G
Frobenius H A Schwar z F Klein and G H Halphen C on fi n ing
hi mself to those with rational algebraical c o efli ci en t s H P oincaré was
able to integrate the m by the use of functions na m ed by h im Fu ch
“
?
s i an s
He divided these equations into families
If the integral
of such an equation be subj ected to a certain transformation the
resul t will be the integral of an equation belonging to the sa me fam ily
The new transcendents have a great analogy to elliptic functions ;
while the region of the latter may be divided into parallelogra ms each
representing a grou p the form er m ay be divi d ed into c urvi linear
polygons so that the knowledge of the function inside of one polygo n
carries with it the knowledge of i t inside the others Thus H P oin
car e arrives at what he calls Fu chs i a n gr ou ps He found m oreover
that F u chsi an functions can be expressed as the ratio of two trans
in the sa m e way that elliptic functi ons can
c e n den t s ( theta fuchsians )
be If instead of linear substitutions with real coe ffi c i ents as e m
ployed in the above groups i m aginary coeffi cients be used then d is
continuous groups are ob tained which he called K l ei n i a ns The ex
tension to non linear equations of the m ethod thus applied to linear
equations was begun by L Fuchs and H P oincar é
M uch interest attaches to the deter mination of those linear d i ffer
e n t ial equations which c a n be integrated by si mpler functions such
as algebraic ell iptic or Abelian This has been studied by C Jordan
P Appell of P aris and H P o i n car e
P a u l Appe l l ( 1 8 5 5
was born in Strassburg After the an
n e xa t i o n of Alsace to Ger m any in 1 8 7 1
he e mi grate d to Nancy to
escape Germ an citiz enship Later he stu died in P aris and in 1 88 6
n o r an t
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H P o in c aré , The Fou n dati ons of S ci en ce , t ran sl by G B H al s t ed , The S c i e n ce
P ress N ew Yo rk a n d G a rr i so n , N Y , 1 9 1 3 , p 3 8 7
2
H en r i P o i n c a ré, N oti ce s u r l es Tr a va u x S ci en tifi q u es de H en r i P oi ncare, P a ri s ,
1
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1 886 ,
p 9
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ANALYSIS
39 1
becam e professor of mechanics there His researches are in analysis
function theory infinitesimal geo metry and rational m echanics
Whe ther an ordinary diff ere n tial equation has one or more solutions
which satisfy certain terminal or boundary conditio n s and if so what
the character of these solutions is has received rene w ed attention
th e last quarter century by the consideration of finer and m ore re m ote
?
questions
E xistence theore ms oscillation properties asymptotic
expressions develop m ent theore ms have been studied by D avid Hil
bert of G ottingen M axim e B ocher of Harvard M ax M ason of the
University of Wisconsin M auro P icone of Turin R M E M ises of
Strassburg H Weyl of G ottingen and especially by George D B irk
hoff of Harvard Integral equations have bee n used to so me extent
“
in boundary proble ms of one dim e n sion ; this m ethod would see m
however to be chiefly valuable in the cases of two or m ore dim ensions
where many of the S implest questions are still to be treated
A standard text book on Differ en ti al Equ ati on s inclu d ing original
matter on integrating factors S ingular solutions and especially on
symbolical m ethods was prepared i n 1 8 59 by G B oole
A Tr eati se on L i n ear Difi er en ti al Equ ati on s ( 1 889)
was brought out
by Thomas Craig of the Johns Hopkins University He chose the
algebraic m ethod of presentatio n followed by Ch Herm ite and H
P oincar é instead of the geom etric method preferred by F Klein and
H A Schwarz A notable work the Tr ai te d A n al yse 1 8 9 1 —1 896
was published by Emile P icard of P aris the interest of which was made
to centre in the subj ect of di fferential equations A second editio n
has appeared
Simple d iff erence equations or finite diff erences were studied by
eighteenth century mathem aticians When in 1 8 8 2 H P oincar é de
v el oped the novel notion of asy m ptotic representation he applied it
to linear diff erence equations In recent years a n ew type of proble m
has arise n in connection with the m I t looks n ow as if the continuity
of nature which has been for so long assu med to exist were a fictio n
“
and as if discontinuities represented the realities
I t see ms al most
certain that electricity is done up i n pellets to which we have give n
the name of electrons That heat co mes i n quanta also see ms prob
2
able
M uch of theory based on the assu mption of continuity may
be found to be m ere approximation Hom ogeneous linear diff erence
equations not intimately bound up with continuity were take n up i n
depen dently by invest i gators widely apart In 1 90 9 Niels Eri k N or
lund of the University of Lund I n S w eden Henri G al b ru n of l Ecole
Norm ale I n P aris and i n 1 9 1 1 R D Carm ichael of the University of
Illinois entered this field of research Carm ichael used a m ethod of
successive approxi m atio n and an extension of a contour integral due to
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See
hi st or i cal s u mmary b y M axi me B ech e r i n P r oceed
gr es s, C ambr i dge, 1 9 1 2 , V o l I , C a mb r idg e , 1 9 1 3 , p 1 6 3
2
R D C ar mi c h ae l I n S ci ence , N S V ol 4 5 , 1 9 1 7 p 4 7 2
1
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f
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the 5 th
i n ter n
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Cou
HI STORY
A
39 2
O F M ATHE M ATIC S
C Guichard G D B irkho ff of Harvard m ade important c on t rib u
tions showing the existence of certain intermediate solutions and of the
principal solutions The asymptotic form of these so l utions is de
t erm i n ed by hi m throughout the co mplex plane
The extension to
non homogeneous equations of results reached for ho mogeneous ones
?
has been made by K P Williams of th e U niversity of Indiana
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I n tegral Equ ati on s , I n teg r o difi er en ti al Equ a ti on s , Gen er al A n al ysi s ,
Fu n c ti on al C al cu l u s
-
The mathem atical perplexities which led to the invention of integral
2
equation s were stated by J Hadamard in 1 9 1 1 as follows : Those
problem s ( such as D i ri chl e t s)
exercised the sagacity of geom etricians
and were the obj ect of a great deal of i mportant and well know n work
through the whole of the nineteenth cen tury The very variety of
”
ingenious m ethods applied S howed that the question did not cease to
preserve its rather m ysterious character Only in the last years of
the century were we ab l e to treat it with som e clearness and under
stand its true nature
Let us therefore inquire by what d evice
this new vie w of D i ri c hl e t s problem was ob tqi n ed I ts peculiar and
most re markable feature consists in the fact that the partial differential
equation is put aside and replaced by a new sort of equation namely
the integral equation This new m ethod makes the m atter as clear
as it was form erly obscure In many circu mstances i n m odern analysis
contrary to the usual point of view the operation of integration proves
a m uch simpler one than the operation of derivation An exa mple of
thi s is given by integral equations where the unknown function is
written under such signs of integration and not of d iff erentiation The
typ e of equation which is thus obtained is m uch easier to treat than
the partial differential equation The type of integral equations
correspon ding to the plane D irichlet problem is
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d>(x)
K (W )
dy
Mr )
where (b is the unk now n function of x in the interval (A B )
f and K
are known functions and A is a known param eter The equations
of the elliptic type in m any d im en S Ion al space give sim ilar integral
equations containing however m ultiple inte5rals and several in de
pendent variables B efore the introduction of e q uations of the abo v e
type each step in the stu dy of elliptic par t ial diff e ren t ial equations
seemed to bring wi th it new diflficu l ti es
[ B ut] an equation such
as ( I )
gives al l the required results at once and for all the pos
sible types of such problems
P reviously in the calculation of
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Tr an s A m M a th S oc , V ol 1 4 , 1 9 1 3 , p 2 0 9
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J H adama rd , Four L ectu r es on M a l hemati cs del i ver ed
1 9 1 1 , N ew Y o r k , 1 9 1 5 , pp 1 2 1 5
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at
Col u mbi a Un i vers i ty i n
A HI STORY OF M ATHE M ATIC S
3 94
was born in B oston and graduated
1 91 8
M ax i me B é ch er ( 1 8 6 7—
)
at Harvard in 1 888 Af ter three years of study at Gottingen he r e
turned to Harvard where he was successively instructor assistant
professor and professor of m athem atics He was president of the
Am erican M athem atical Society in 1 90 9—1 9 1 0 Among his works are
Rei hen en twi ck el u n gen der P oten ti al theor i e 1 8 9 1 enlarged in 1 894 and
L egon s su r l es M ethodes de S tu rm contai n ing the author s lectures de
liv ered at the Sorbonne in 1 9 1 3—1 9 1 4
1
A Voss in 1 91 3 stresses the value of integral equations thus :
In the last ten years
the theory of integral equations has
attained extraordinary importance because through them problem s
in the theory of diff erential equations m ay be solved whi ch previously
coul d be disposed of only in S pecial cases We abstain from sketchi ng
their theory which m akes use of infini te determinants that belong
to linear equations with an infinite num ber of unkn owns of quadratic
forms with infinitely many variables and which has succeeded in
throwing new l ight upon the great problems of p ure and applied
m athe matics especially of m athe m atical physics
I mportant advances along the line of a general analysis and its
application to a generali z ation of the theory of linear integral equa
tions hav e been m ade since 1 90 6 by E H M oore of the University
?
of Chica go Fro m the e xisten ce of analogies in different theories he
infers the existence of a general theory co mprising the analogous
“
theories as special cases He proceeds to a unification resul ting
first fro m the recent generaliz ation of the concept of independent
v ariable effected by passing fro m the consideration of variables defin ed
for all points in a giv en interval to that of variables defined for all
points in any giv en set of points lying in the range of the variable
secon dl y fro m the consideration of functions of an infin ite as well
as a fin ite n umber of variables and thirdly from a still fui ther gen
“
e ral i zat i on which leads hi m to fun ctions of a
general variable
E H M oore s general theo ry includes as special cases the theories of
E I Fredh ol m D Hilbert and E Schmi dt G D B irkhoff in 1 9 1 1
3
presented the following birds eye View of recent m ovem ents : Since
the researches of G W Hill V Volterra and E I F r edh ol m in the
di rection of extended linear system s of equations m athematics has
been in the way of great develop ment That attitude of mind which
conceives of the function as a generali z ed point of the method of
successive approxi mation as a Taylor s expansion in a function va
r i abl e of the calc ul us of variations as a li miting form of the ordinary
a lgebraic proble m of m axi ma and m ini ma is now c rystalli z ing into a
new branch of mathematics under the leadership of S P incherle J
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A Vo ss , U eber das Wesen d M ath , 1 9 1 3 , p 6 3
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S ee P roceed 5 th I n ter n C on gr ess of M a thema ti ci a n s , C amb ridg e ,
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B u ll A m
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M ath S oc , V o l
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p 415
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19 13 ,
V ol I ,
.
AN ALY SI S
395
H adamard D Hilbert E H M oore and others For thi s field
defined as
P rofessor M oore proposes the term General Analysis
the theory of syste ms of classes of functions functional O perations
etc i nvolving at least one general variable on a general range He
has fixed attention on the most abstract aspect of this field by con
The nearest
s ideri n g functions of an absolutely general variable
approach to a si milar investigation is due to M Fr éch e t (P aris thesis
who res tricts hi mself to variables for which the notion of a
limiting value is valid
Researches along the li ne of E H M oore s
General Analysis are due to A D P itcher of Adelbert College and
E W C hi ttenden of the University of Illinois In his General
Analysis M oore d efines co mplete independence of postulates
whi ch has received the further attention of E V Huntin gton R D
B eetle L L D ines and M G Gaba
V Volterra di scusses integro —
differential equations which involve
not only the unknown functions under signs of integration but also
the unknown functions themselves and their derivatives and shows
their use in m athematical physics G C E v ans of the Rice Institute
e xtended A L Cauchy s existence theore m for partial differential
equations to integro differential equations of the static type in
which the variables of differentiation are different from those of i n
t eg r a ti on
M ixed linear integral equations have been discussed by
W A Hurwitz of Cornell University
The S tu dy of integral equations and the theory of point sets has led
to the developm ent of a body of theory called functional calculus
One part of this is the theory of the functions of a line As early as
1 88 7 Vito Volterra of the University of Ro m e developed the funda
mental theory of what he called functions depen ding on other func
tions and functions of curves Any quantity which depends for its
value on the arc of a curve as a whole is called a func tion of the line
The relationships o f functions depending on other functions are
called fon c t i on el l es by J Hadamard in his L egons su r l e cal cu l des
1 91 0
var i a ti on s
and f unctionals by English writers Functional
equations and syste m s of functional equations have received the atten
tion of Griffi th C Evans of the Rice Institute Luigi Sinigallia of
P avia Giovanni L T C Giorgi of Ro me A R Schweit z er of Chicago
Eric H Neville of Cambridge and others Neville solves the race
course pu zzle o i cov ering a circle by a set of five circular discs Says
1
G B M athews : We must express our regret that English math
e ma t i c s is so predo m inantly analytical
Cannot som e one for in
stance give u S a truly geometrical theory of J V P oncelet s poristic
polygons or of von S t au dt s thread constructions for c on i coi ds ? In
“
the theory of functional equations a single equation or a syste m of
equations expressing som e property is taken as the definition of a
class of fun ctions whose characterist i cs particular as well as collective
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Nature, V ol 9 7 ,
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1916,
p 39 8
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.
A HISTORY OF M ATHE M ATICS
396
are to be developed as an outco m e of the equations (E B Van
V leck )
An i mportant generali z at i on of Fourier series has been made
and we have a great class of expansions in the so—
called orthogonal
and b ior thog on al functions arising in the study of diff erential and
integral equations In the field of diff erential equations the most
irnpor tan t class of these functions was first defined in a general and
explicit m anner (i n 1 90 7)
by
G D B irkhoff of Har vard Univer
si ty ; and their leading fun da mental properties were developed by
1
In boun dary value problems of diff erential equations which
hi m
are not self adjoi n t b i or thog on al system s of functions play the sam e
Anna J
r Ol e as the orthogonal syste m s do in the self adjoint case
P ell established theorem s for b i o r th og on al syste m s analogous to those
of F Ri esz and E Fischer for or thogonal systems
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Theori es
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I r r ati on al s
and
Theory
f
f
The new non m etrical theories of the irrational were called forth
“
by th e demands for greater rigor The use of the word quantity
as a geo metrical m agnitude without reference to number and also
as a nu mber w hi ch m easu res som e magnitude was disconcerting e s
p ec i al l y as there existed no safe ground for the assu mption that the
same rul es of operation applied to both The metrical view of n umber
in volved the entire theory of measurem ent whi ch assu med greater
In a t
di fli cul t i es w i th the advent of the non Euclidean geo metries
tempts to construct arithm etical theories of number irrational n um
bers were a sour ce of trouble It was not satisfactory to operate with
i r rational numbe r s as if they were rational What are irrational
numbers ? Considerable atten tion was paid to the definition of them
as limits of certain sequences of rational nu mbers A L Cauchy in
“
an irrational number is the
hi s C ou r s d An al ys e 1 8 2 1 p 4 says
l imit of div erse fractions w hich furnish m ore and m ore approxim ate
P robably Cauchy was satisfied of the e xistence of
values of it
irrationals on geometric gr ounds If not his exposition was a rea
soning in a circle To make this plain suppose we have a develop
m ent of rational n umbers and we desire to define l i mi t and also i rr a
“
With Cauchy we may say that when the successive
ti on al n u mber
val ues attributed to a variable a pproach a fi xed value indefinitely
so as to end by di ff ering from it as l ittle as is wished this fix ed value
is called the limi t of all the others
Since we are still confined to
the field of rational numbers this limit if not rational is non existent
and fictitious If now we endeavor to define irrational number as a
limit we encoun ter a break down in our logical develo pment It
becam e desirable to define irrational number arithmetically w i thout
reference to limits This was achieved independently and at al mos t
o
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A ggr ega tes
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R D C armi ch ael i n S ci en ce, N S
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V ol 4 5 ,
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1 9 1 7,
p 471
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O F M ATHE M ATIC S
HI STORY
A
3 98
has been placed upon a basis en tirely independent of measu r
able m agnitude and pure analysis is regarded as a schem e whi ch deals
with number only and has per s e no conce rn with m easurable quan
tity Analysis thus placed upon an arithmetical basis is characteri z ed
by the rej ection of all appeals to our S pecial intuitions of S pace tim e
and m otion in support of the possibil i ty of its operations ( E W
Hobson )The arithm eti zation of m athem atics whi ch was in progress
during the en tire nineteenth century but m ainly durin g the ti me of
Ch M eray L Kronecker and K Weierstrass , was characteriz ed by
1 “
E W Hobson in 1 90 2 in the following terms : In some of the text
books in common use in this co u ntry the symbol 0 is still used as if
it denoted a n u mber and one in all respects on a par with the fin ite
numbers The foun dations of the integral calculus are treated as if
Riemann had never lived and worked The order in which doubl e
li mits are taken is treated as imm aterial and in m any other respects
the critical results of the last century are ignored ”
The theory of exact m easurement in the domain of the ideal ob
j ec t S of abstract geometry is not i mmedi ately derivable from intui tion
but is now usually regarded as requi ring for its develop m ent a previ ous
independent investigation of the nature and relations of number
The relations of n umber having been developed on an independent
basis the scheme is applied by the help of the principle of congruency
or other equivalent principle to the representation of extensive or
intensive m agnitude
This complete separation of the notion
of number especially fractional number from that of m agnitude
involves no doub t a reversal of the hi stori cal and psychological
or ders
The ex treme arithm etiz ing school of which perhaps
L Kronecker was the founder ascribes reality whatever that may
m ean to integral nu mbers onl y and regards fractional numbers as
possessing only a derivative character and as being introduced only
for convenience of notation The ideal of this school is that every
theorem of analysis S hould be interpretable as giving a rela tion b e
tween integral numbers only ”
“
The true ground of the difli cu lti es of the older analysis as regards
the existence of limits and In relation to the application to measu r
able quantity lies in its inadequate concept i on of the domain of
number i n accordance with whi ch the only num bers really defin ed
were rational numbers This inadequacy has now been removed by
means of a purely arithm etical definition of irrational n umbers by
means of which the continuum of real numbers has been set up as
the domain of the independent variable in ordinary analysis Thi s
—
definition has been given in the main in three form s one by E Heine
and G Cantor the second by R D edekind and the third by K Weier
strass Of these the first two are the simplest for working pu rposes
an d are essentially equivalent to one another ; the diff erence between
ti on al ,
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P r oceed L ondon
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M ath S oc , V ol 3 5 , 1 90 2 , pp
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1 1 7 - 1 39 ; se e
p
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1 1 8.
ANALYSIS
399
them is that while D edekind defines an irrational number by means
of a section of all the rational numbers in the Heine Canto r form of
definition a selected convergent aggregate of such num bers is em
ployed The essential change introduced by thi s definition of irra
t i on al numbers is that for the sche me of rational n u mbers a new
scheme of numbers is substituted in which each num ber rational or
irrational is defined and can be exhibited in an indefinitely great
number of ways by means of a convergent aggregate of rational
n u mbers
By thi s conception of the do main of n u mber the root
d iffi culty of the older analysis as to the e xistence of a lim it is turned
each number of the continuum being really defi ned in such a way that
it itself exhibits the li mit of certain classes of convergent sequences
It S ho ul d be observed that the criterion for the convergence of an
aggregate is of such a character that no use is made in it of i nfi n i t esi
mals definite fin ite numbers alone being used in the tests The old
attempts to prove the e xistence of limits of convergent aggregates
were in default of a previ ous arithmetical definition of irrational
num ber doom ed to inevitable failure
In such applications of
analysis—as for example the rectification of a cu rv c the length of
the curve is defined by the aggr egate formed by the lengths of a proper
sequence of inscribed polygons
In case the aggregate is not
convergent the curve is regarded as not r ec t ifi abl e
“
It has in fact been shown that many of the properties of functions
such as continuity diff erentiability are capable of precise definition
when the domain of the variable is not a continuum prov ided how
ever that domain is perfect ; this has appeared clearly in the course
of recent investigations of the properties of non dense perfect agg r e
gates and of fun ctions of a variable whose domain is such an agg re
gate
In 1 91 2 P hilip E B Jourdain of Fleet near London characteriz ed
1 “
Dedekind s
theories of the irrational substantially as follows :
theory had not for its obj ect to prove the existence of irrationals :
it S howed the necessity as D edekind thought for the mathematician
to create them In the idea of the creation of numbers Dedekind
was followed by O S tol z ; but H Weber and M P asch showed how
the supposition of this creation could be avoided : H Weber defined
real numbers as sections (S chni tte)
in the series of rationals ; M P asch
as the segments which generate these sections In
(like B Russell )
K Weierstrass theory irrationals were defined as classes of rationals
Hence B Russell s obj ections ( stated in his P r i nc i pl es of M athemati cs
Cambridge 1 90 3 p 2 8 2 )
do not hold against it nor does Russell
se em to credi t Weierstrass and Cantor with the avoidance of quite
the contradiction that they did av oid The real objection to Weier
theory and one of the objections to G Cantor s theory is
s trass
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P E B Jou rda i n ,
I ntern ati on al C ongress
1
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O n I soid R el a t ion s an d The ori e s o f I rra t i o n al N u m b e r
o M athemati ci a n s , C am b r id e , 1 1 2
f
g
9
.
in
M ATHE M ATIC S
A HI STORY OF
400
that equality has to be r e—
defi n e d In the v arious arithm etical theories
of irrational num bers there are three tendencies : ( a )
the n umber is
de fined as a logical entity—a class or an operation
as w ith K
Weierstrass H Weber M P asch B Russell M P ieri ; (b )
it is
created or more frankly postulated as with R D edekind O
Stol z G P eano and Ch M eray ; ( c )
it is defi ned as a S ig n ( for what
is left indeterm inate )as with E Heine G Cantor H Thomae A
In the geom etrical theories as with P aul du Bois
P r in g sh e im
Reymond a real nu mber is a sign for a length In B Russell s theory
it appears to be equally legitimate to define a real number in various
way s
The theory of aggregates (M engenlehre th éorie des ense mbles
theory of sets)
owes its d e v elop m ent to the en d eavor to clarify the
concepts of in depen dent var iable and of funct i on Form erly th e
notion of an independent variable reste d on the nai ve concept of the
geom etric continuu m N o w the in dependent variable is restricted
to som e aggregate of values or points selected out of the continu um
The term function was destined to receive various definitions J
Fourier advanced the theorem that an arbitrary functio n can be
represented by a trigono metric series P G L D irichlet looked upon
the general functio n al concept as equivalent to any arbitrary table of
values When G F B Riem ann gav e an exa mple of a function ex
pressed analytically which was d i scontinuous a t each rational point
the nee d of a m ore comprehensive theory becam e evident The first
atte mpts to m eet the n ew needs were m ade by Herm ann Hankel and
l
P aul d u B ois Re ym ond
The Al l gemei n e Fu n k ti on en theor i e of du
Bois—
Reym on d brilliantly sets forth the proble ms in philosophical
form but it remained for Georg Cantor to advance and develop the
necessary ideas invol v ing a treat m ent of infinite aggregates E v en
though th e infinite h ad been the subj ect of philosophic conte mplation
for m ore than two thousan d years G Cantor hesitated for ten years
before placing his i deas before the m athematical public The theory
of aggregates sprang into being as a science when G Cantor intro
?
G Cantor began his
du c e d the notion of enu m erable aggregates
publications in 1 8 70 ; i n 1 8 8 3 he publi shed his Gru ndl agen ei n er al l
I n 1 8 95 and 1 8 97 appeared in
gemei n en M a n n i chfal ti gk ei ts l ehr e
M athemati s che A n n al en his B ei tr age zu r B egr undu ng der tran sfi n i ten
2
These researches have played a m ost conspicuous r Ol e
M en gen l ehr e
not only I n the march of m athematics toward logical exactitude but
also in the real m of philosophy
G Cantor s theory of the continuu m was used by P Tannery in
1 8 8 5 in the search for a profounder vie w of Zeno s argum ents against
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A S ch oen fli es , En twi ck el u n g der M en genl ehr e u n d i hr er A n w end un g en , gemei n s a m
n zi t H a n s H a hn her a u s g eg eben , L e i p ig u B e r l i n , 1 9 1 3 p 2
2
T ran s l a t ed i n t o En g li sh b y P hi l ip E B Jo u r d ai n an d p u bl i s hed b y t he O p e n
C o u r t P u b l C o , C hi c ag o , 1 9 1 5
1
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A
2
46
OF M ATHE M ATIC S
H I STORY
in 1 90 6 relates to the aggregate of decim al fractions between 0 and 1
which can be defined by a finite num ber of words ; a new dec imal frac
tion can be defined which is not included in the previous ones
B ertrand Russell discovered another paradox given in his P ri n
ci pl es of M a thema ti cs 1 90 3 pp 3 6 4—
which is stated by
368 10 1
P hilip E B Jourdain thus : If w is the class of all those term s x such
that x is not a m em ber of x then if w is a m ember of w it is plain that
w is n o t a m ember of w ; while if w is not a m e mber of w it is equally
1
p l ain that w is a member of w
These paradoxes are closely allied
to th e Epi m enides pu zzle : Epim eni des was a Cretan who said that
all Cretans were liars Hence if his statem ent was true he was a li ar
H P oin care and B Russell attribute the paradoxes to the open and
“
clan destine use of the word all
The difli cul ty lies in the definition
“
of the word M enge
Noteworthy among the attempts to place the theory of aggregates
u pon a foundation t hat w i ll exclude the para doxes and antino m ies that
had arisen was the formulation in 1 90 7 of seven restricting axioms
by E Zerm elo in M a th A n n al en 6 5 p 2 6 1
Ju l i u s K an i g ( 1 8 4 9
the Hungarian mathe m atician in bi
N eu e Gru n dl agen der L ogi k A r i thmeti k u nd M en gen l ehr e 1 9 1 4 speaks
of E Z erm el o s axiom of selection (Auswahlaxiom)
as being reall y
a l ogical assu mption not an axiom in the old sense whose freedom
from contradiction m ust be dem onstrated along with the other
axiom s He ta kes pains to steer clear of the antinom ies of B Russell
and C B urali Forti For a discussion of the logical and philosophical
questions invol v ed i n the theory of aggregates consult the second
edition of E B orel s L econs su r l a théori e des f on cti ons P aris 1 9 1 4
note IV wh i ch giv es letters written by J Hadamard E Borel H
Lebesgue R B a i re touching the val idity of Z e rm el o s d em onstration
that the l inear continuu m is well ord ered A set of axiom s of ordinal
m a g nitu d e was given by A B F ri zell in 1 9 1 2 at the Ca mbridge
Congress
In the treatm ent of the infinite there are two schools Georg Cantor
prov ed that the con t inuum is not denum erable ; J A Richard con
tending that no m a thematical entity e xists that is not defin able in a
finite num ber of w ords argued that the continuum is denum erable
H P oincar é claim ed that this contradiction is not real since J A
?
3
Richard e mploys a non predicative definition
H Poincaré in
di scussing the logic of the infinite states that according to the first
school the pragm atists the infinite flows out of the fi nite ; there is an
infinite because there is an infinity of possible finite things Accord
ing to the second school the Cantorians the infinite precedes the
finite ; the finite is obtained by cutting off a small piece of the infinite
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P E B Jo u rdai n , C on tr i bu ti on s , C hi cag o , 1 9 1 5 , p
2
B u l l A m M a th S oc , V o l 1 7 , 1 9 1 1 , p 1 93
3
11
S c i en ti a , V o l 1 2 , 1 9 1 2 , p p 1 —
1
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206 .
ANALYSIS
40 3
For pragmatists a theore m has no meaning unless it can be verified ;
they reject indirect proofs of e xistence ; hence they reply to E Zermelo
who proves that S pace can be converted into a well ordered aggregate
: Fine convert it !
We cannot carry out this
(wohl geordnete M enge)
transformation because the number of operators is infinite For
Cantorians m athematical thi ngs exist independently of man who may
think about them ; for them cardinal number is no myste ry On the
other hand pragma t ists are not sure that any aggregate has a cardinal
number and when t hey say that the M ach tig k ei t of the continuum
i s not that of the whole nu mbers they mean simply that it is i m
possible to set up a correspondence between these two aggregates
which co u ld not be destroyed by the creation of new points in S pace
If mathe maticians are ordinaril y agreed a mong themselves it is
because of confirm ations which pass fi nal j udgment In the logic of
infinity there are no confirmations
E J Brouwer of the University of Amsterdam expressing views
of G M an n ou ry said in 1 91 2 that to the psychologist belongs the
“
task of e xplaining why we are averse to the S O called contradictory
systems in whi ch the negative as well as the positive of certain propo
s i t i on s are valid
that the i ntuitionist recogni zes only the ex i stence
“
of denum erable sets and can never feel assured of the exactness of
mathematical theory by such guarantees as the proof of its being
non contradictory the possibility of defining its concepts by a finite
number of words or the practical certa i nty that it will never lead to
1
misun derstandi ng in hu m an relations
A B Fri z ell showed in
1 9 1 4 that the field of denu m erably infinite processes is not a closed
—
domain a concept which the i ntuitionist refuses to recogni z e but
“
which need not disturb an intuitionist who cuts loose from the prin
M ore recent ten dencies of research In this
c ipi um c on t r adi c t i on i s
2 “
field are described by E H M oore : From the linear continuum
with its infinite var i ety of functions and corresponding singularities
G Cantor developed his theory of cl ass es of poi n ts (P u n ktmengen l ehr e)
with the notions : li mit point derived class closed class perfect class
with
etc and hi s theory of cl asses i n gen eral (allgem eine M engenl ehre )
These
th e n o t i on s :cardinal n u mber ordi nal number order type etc
theories of G Cantor are perm eating M odern M athematics Thus
there is a theory of functions on point sets in particular on perfect
point sets and on more general order types while the arithm etic
of cardinal numbers and the algebra and function theory of ordinal
numbers are under development
“
Less techni cal generaliz ations or analogues of fun ctions of the
continuous real variable occur throu ghou t t h e various doctrines and
applications of analysis A function of several variables is a function
of a S ingle m ultipartite variable ; a distribution of potential or a field
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1
L E
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2
N ew
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J B rou w er i n B u ll A m M a th S oc V ol 2 0
Haven Col l oq ui u m, 1 906 , N ew H a v en 1 9 1 0 ,
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19 13,
p
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2-
pp
4
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A HI STORY OF M ATHE M ATIC S
40 4
of force is a function of position on a cuve or surface or region ; the
value of the definite integral of the Calculus of Variations is a func
tion of the variable function entering the definite integral ; a curvilinear
integral is a function of the path of integration ; a functional operation
is a fun ction of the argum ent f u nction or functions ; etc etc
A mul tipartite variable itself is a fu nction of the variable index
of the part Thus a finite sequence : x 1 ;
x
of real numbers is a
=1
i
function x of the inde x i V i z
n ) S i mi larly
(
;
an i n finite sequence :x l ;
of real num bers is a functi on x
=1
of the ind ex n v i z
u
Accordingly n fold
(
; 2;
algebra and the theory of sequences and of series are embraced in
the theory of functions
“
A S apart fro m the determ ination and e x tension of notions and
theories in analogy with simpler notions and theories there is the
e xtension by direct generaliz ation The Cantor m ove ment is in this
direction Finite generali zation from the case u = 1 to the case n = u
occurs throughout Analysis as for instance in the theory of func
tions of several independent variables The theory of functions of a
1
denu merable in fi nity of variables is another step in thi s directi on
We notice a more general theory dating fro m the year 1 90 6 Recog
n i z i ng the funda m ental r ole played by the notion l i mi t el emen t (n u m
ber point fun ction curve etc )
in the various S pecial doctrines M
F réch e t has given with e xtensive applications an abstract generaliza
ti on of a considerable part of Cantor s theory of classes of points and
of the theory of continuous functions on classes of points F réchet
considers a general C l a S S P of elements p with the notion l i mi t defined
for sequences of ele ments The nature of the elem ents p is not S peci
fi ed ; the notion l i mi t is not expli citly defined ; it is postulated as de
fined subj ect to specified conditions For particular applications
explicit d efinitions satisfying the conditions are given
The
fun ctions considered are either functions u of variables p of Specified
character or functions p on ranges P with postulated features : e g
l i mi t; di stan ce; el emen t of con den sa ti on ; conn ecti on of specified char
acter
E H M oore s own form of general analysis of 1 90 6 considers
f u n cti on s p of a g en er al vari abl e p on a gen er al r ange P where this
g en er al e mbraces every well defi n ed particul ar case of variable and
range
Early in the develop ment of the theory of point sets it was pro
posed to associate with them nu mbers that are analogous to those
?
representing lengths areas volumes On account of the great arbi
t r ar i n ess of thi s procedure several di fferent definitions of such n u m
bers have been given The earliest were given in 1 8 8 2 by H Hankel
and A Harnack Another definition due to G Cantor ( 1 8 84)
was
generali z ed by H M inkowski in 1 900 M ore precise measures were
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D H i l be r t , 1 90 6 , 1 909
2
En cycl opédi e des sci ences m a théma ti q u es , To me I I , Vol
1
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I
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1 91 2,
p
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1 50 .
.
HI STORY
A
46 6
O F M ATHE M ATI C S
admits a finite number of discontinuities but
( Riemann s integral )
’
an infinite nu mber only under certain narrow restrictions A totally
discontinuous function—for exa mple one equal to z ero in the rational
points whi ch are everyw here dense in the interval of integration
and equal to 1 in the rational points which are likewise everyw here
dens e—is not integrable a la Riemann The restriction became a
very hampering one when mathe maticians began to reali z e that the
analytic world in whi ch theorem s are deducible does not consist
m erely of hi ghl y ci v ili z ed and continuous functions In 1 90 2 Lebesg ue
with great penetration fram ed a new integral which is identical with
the integral of Riemann when the latter is applicable but is immeasur
ably more comprehensive It will for instance include the totally
discontinuous function above mentioned This new integral of
Lebesgue is proving itself a wonderful tool I might co mpare it with
a m odern Krupp g un so easily does it penetrate barriers which before
were i mpregnable
Instructive is also the description of this m o v e
1 “
Volterra has pointed out in the
m ent as given by G A Bliss :
introductory chapter of his L egons s u r l es f on cti ons des l ign es
the rapid develop m en t which is taking place in our notions of infinite
processes e xamples of which are the definite integral limit the solu
tion of integral equations and the transition fro m functions of a
finite num ber of variables to functions of lines In the field of i n
t eg ra ti on the classical integral of Rie mann perfected by D arb oux
was such a convenien t and perfect instrum ent that it impressed itself
for a long time upon the m athematical public as being somethi ng
unique and final The advent of the integrals of T J S tieltj es and H
Lebesgue has shaken the co mplacency of m athe maticians in thi s
respect and with the theory of linear integral equations has given
the S ignal for a r e examination and extension of many of the types of
processes which Volterra calls passing from the finite to the infinite
It shoul d be noted that the Lebesgue integral is only one of the evi
den c e s of this restlessness in the particular do main of the integration
theory Other new defin itions of an integral have been dev ised by
S tieltjes W H Young J P ierpont E Hellinger J Radon M
F réch e t E H M oore and others The definitions of Lebesgue
Young an d P ierpont and those of S tieltjes and Hellinger form two
rather well defined and distinct types while that of Radon is a gen
eral i z a t i on of the integrals of both Lebesgue and S tieltjes
Th e
e fforts of Frechet and M oore have been directed toward definitions
valid on m ore general ranges than sets of points of a line or higher
spaces and whi ch include the others for S pecial cases of these ra nges
Lebesgue and H Hahn with the help of somewhat complicated
transformations have shown that the integrals of S tieltj es and Hel
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G A B l i ss , I n t e g ral s o f L eb e sg u e B u ll A m M ath S oc , V ol 2 4 , 1 9 1 7 , pp 1
S ee al so T H H i l debran d t , l oc c i t , V ol 2 4 , pp 1 1 3 1 44 , w ho g i ves b i bli og ra
47
phy
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ANALYSIS
40 7
linger are expressible as Lebesgue integrals
Van Vleck has
remarked that a Lebesgue integral is expressible as one of S tieltjes
by a transfor mation much S i mpler than that used by Lebesgue for
the opposite purpose and the S tieltj es integral so obtained is readily
expressible in terms of a Riemann integral
Further more the
S tieltj es integral see ms distinctly better suited than that of Lebesgue
to certain types of questions as is we