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ON DECIMALS
FIRST BOOK
SAN FORD
By VERA
The Lincoln
School
The history of the decimal fraction has two distinct sections :
the evolution of the idea of decimals, and the evolution of a con
venient symbolism.
The idea might, indeed, be traced to the
The
of the Arabic notation.
of
the
very beginning
place-value
as
witness
the
not
has
been
standardized
Eng
today,
symbolism
lish 3 14, the French 3,14 and our 3.14. Yet, strangely enough,
in complete
the theory of operations with decimals appeared
?orm in the first work devoted wholly to the subject, La Disme
(1585) by Simon Stevin of Bruges.
It is the purpose of this sketch to show, by excerpts from La
Disme, what Stevin did with decimals and what uses he antici
pated men would make of them. Since his introduction, defini
tions and symbols give a clue to the way in which he came to
invent these numbers, a reconstruction of his line ?f thought is
into
pertinent to the case, even though this may degenerate
one
been
he
what
have
had
thinks
he
would
thought
merely
its proper setting, there must be
Stevin. And to give La Disme
added a note on computations with fractions in the sixteenth
century.
The devices that might
tion fall into two groups
the decimal arrangement
involve symbols that are
have developed
into the decimal frac
: those which depend specifically upon
of the Arabic notation, and those which
happy accidents or actual forerunners
of the decimal point, according to one's interpretation of their
use.
In the first group was a scheme for finding the nth root
a
of
number.
This may be expressed
in modern
symbols as
?
? /a?
n\/?*10fcw
_Jl-:
Its value lies in the fact that it enables one
10*
to find a root to any required degree of accuracy and that frac
In a similar way,
tions enter only in the last step of the work.
on
a
interest was computed
100,000 times too large,
principal
were
found from a circle of radius
and trigonometric functions
321
322
10,000,000.
THE MATHEMATICS
Symbols
TEACHER
typical of the second group are the follow
ing: in dividing by multiples of 10, 100, 1,000,Pellos (1492)
used a period to mark
dend as if to indicate
bly to the remainder;
interest table much as
off one, two or. three places in the divi
that digits to the right belonged inevita
and Rudolff
(1530) printed a compound
we would save, that a bar (/) acts as a
decimal point.
Several other writers used symbols of this sort,
but no one before Stevin explained what they actually signified.
Any one of these devices might have been the beginning of the
decimal fraction, but a careful study of La Disme
indicates that
Stevin approached
the subject from a different standpoint?
that of adapting to everyday use and to the current number sys
tem a method
devised by scholars of an earlier time. Stevin was
to appreciate
well qualified
the learning of the past and the
He lived in the Netherlands
difficulties of his contemporaries.
in the days of the great struggle against Spain.
His ability in
mathematics and physics, and his skill in applying his knowledge
to military and civil matters, won for him the post of adviser to
of Nassau,
the Silent.
Maurice
the son ofWilliam
Thus Stevin
was in close touch with the affairs of a country which neces
sity had driven to seize every aid that science could give, and he
was collaborator with a prince who realized this necessity most
keenly.
In all such work, numerical computations must have played
a large part, and although methods such as are noted above were
in use for special cases, there was no satisfactory general scheme
for dealing with fractions.
in
fractions were difficult to use and unsatisfactory
were
in
extremes
to
carried
which
results.
The
they
expressing
an effort for accuracy is shown in the case where Stevin gives
2SS-ift?ftWftrm- f?r tne present value of an annuity of 54 lb.
Stevin wTas not alone in
yearly for 6 years, interest at 12%.
Common
He makes
occasional
attempts to
using large denominators.
as
reach significant figures in his results,
in computing his in
as
terest tables where he counts
1 "for
it is greater
than one-half," but in La Disme he considers that the 0.00067
mal
No.
A
discussion
Fraction,
5.
of these
by D. E.
devices
Smith,
is given
Teachers
in The
College
Invention
Bulletin,
of the Deci
Series
.First
323
THE FIRST BOOK ON DECIMALS
in 301.17167
is of no
account,
and
he writes
his
result
as
301.171.
Stevin's decimals seem to have evolved from the sexagesimal
fractions of the scientists.
Sexagesimals were really a series of
denominate numbers in which a unit was 60 minutes
(primes),
a minute was 60 seconds, a second was 60 thirds and so on.1
and the
These numbers were inherited from the Babylonians
Greeks, and they are still used in our divisions of the hour and
the angle. The advantages of sexagesimals
in expressing results
are obvious.
Two such numbers could be readily compared.
of
fractions could be expressed exactly in sexagesimals
Many
the first order, and others could be given in combinations of
In one case, for instance, an ap
minutes, seconds and thirds.
an
root
of
proximate
equation is carried to the sexagesimal of
the tenth order. These numbers had grave disadvantages,
how
ever. They were of little use if one wished to extract roots, and
or division of two sexagesimals was difficult.
the multiplication
It is not surprising that Stevin felt obliged to write a careful
to the work that was to revolutionize all computa
tions.
One can imagine the, scepticism that would
greet a
of
octavo
36
labeled
with
that
ambitious
pur
pages
pamphlet
pose, and Stevin's attempts to reconcile the size of his book to
introduction
its sub-title, and his desire to clear himself from implications of
undue boasting make his introduction fully as interesting as his
in Flemish
first appeared
theory. As for the book itself?it
same
under the title La Thiende
In
the
year, a
(Leyden, 1585).
to
French
translation La Disme was printed as an appendix
In the next half century, the work
Stevin's
l'Arithm?tique.
went through at least six editions, one of them a translation into
English (1608).1
Let us suppose for the moment that we have never heard of
and that we are among the people to whom Stevin dedi
In this case his introduction may sound like
cated La Disme.
a fairy tale that we wish were true, but that we fear is not. We
decimals
shall be interested, however, to learn what it is that makes the
author at once so confident and so modest.
Any theory that will
1Pare minata
etc.
?ecunda,
prima? para minata
1 The
was made
sec
in this article
translation
from Girard's
quoted
ond edition
Oeuvre?
de Simon
of Les
Stevin
Math?matiques
(Leyden,
1634).
324
THE MATHEMATICS
TEACHER
take such hold gf a man is worth considering, so let us begin La
des Practiques''?we
which Stevin called "La Practique
it
the
call
"super-method."
might
Disme
Teaching
formed by
LA
how all Computations
alone without
Integers
Written
first
in Flemish*
DISME
that are met
in Business
the aid of Fractions.
and
now
done
may
be per
into French
by
Simon
Stevin
of Bruges
To
of Tapestry,
Measurers
Astrologers,
Surveyors,
Gaugers2,
meters3
in general,
and
to all Merchants.
Mint-Masters,
Simon
Stevin
sends Greeting.
Stereo
the small
contrasts
size of this book with
Any one who
your great
honorable
sirs to whom
it is dedicated,
think my
ness, my most
may
idea absurd,
bears
if he imagines
that the size of this volume
especially
the same ratio to human
that its usefulness
has to men of your
ignorance
the extreme
but, in so doing, he will have
outstanding
compared
ability;
com
not be done.
terms of the proportion
which
Let
him rather
may
the third term with
the fourth.
pare
But what
is it that is here propounded?
invention?
Some wonderful
that it hardly
deserves
the name
that, but a thing so simple
Scarcely
for it is as
if some
lout chanced
upon
stupid
invention;
country
great
treasure
without
if anyone
in the finding.
thinks
any skill
But,
using
in explaining
I am boasting
the usefulness
of La Disme,
of my
that,
cleverness
in devising
neither
it. he shows without
that he has
doubt
nor
the
to
the
intelligence
things
distinguish
simple
judgment
or
is jealous
from
he
common
that
the
else
of
difficult,
good.
I
to
However
mention
the
usefulness
this may
shall
not
fail
of
this
be,
even
in
the
But
face
this
of
.man's
since
the
calumny.
thing
empty
an unknown
has found by chance
its
mariner
who
declare
all
isle, may
to the king as. for instance,
its having
riches
beautiful
fruits, pleasant
its being
to him as con
minerals,
etc., without
plains,
precious
reputed
so may
I speak
of this invention,
usefulness
ceit;
freely of the great
a usefulness
than I think any one of you anticipates,
without
greater
on my achievements.
constantly
priding
myself
of
Stevin here enumerates cases which show the usefulness
number in the work of the astronomer, the surveyor, the mint
He
of
and the merchant.
stresses the difficulties
master,
with sexagesimals and with denominate numbers
manipulations
and speaks of the almost inevitable errors in calculation
that
vitiate excellent work.
Stevin
claims that La Disme
teaches how
these computations may be performed by whole numbers and he
says that work tedious even to a skilled computer may now be
accomplished with the same ease as in reckoning with counters.
an
men whose
was
business
the measuring
of wine-casks,
2Gaugers:
in connection
with
the excise
important
duties,, and necessitated
thing
time.
in
barrels
and
the
the
the
the
size
of
of
divergence
great
by
shape
well
in the American
into the last century.
arithmetics
lingered
Gauging
3 Stereometers:
men who measured
Stevin
later
volumes
of all sorts.
is stereometry,
to show
is at great
all gauging
pains
that, although
all stereometry
is not gauging.
325
THE FIRST BOOK ON DECIMALS
force of the comparison is apparent only when one reflects
in Stevin's time wTas a device used prin
that counter-reckoning
little
skilled in the finer points of working
cipally by people
with Arabic numerals?its
greatest value was in addition and
reason for presenting
this
Then comes Stevin's
subtraction.
to the public :
The
be
which
If by these means,
time may
be saved
otherwise
would
as well
as disputes,
be avoided,
lawsuits,
mistakes,
may
lost, if work
I willingly
submit
and
other mischances
thereto,
commonly
joined
to your
the point
that
Dieme
?a
consideration.
raise
Someone
may
seem good
which
at first sight are of no effect when
inventions
many
one wishes'
in a
to use
new methods,
them, and as often happens,
good
few minor
are worthless
No
in more
ones.
such
cases,
important
doubt
in this instance,
exists
this method
for we have
to expert
shown
in Holland
and
surveyors
abandoned
which
the devices
they
they have
had
to lighten
invented
use
of their computations
and now
the work
this one
to their great
The
same
satisfaction
will
come
satisfaction.
to each, of you, my most honorable
done.
do as they have
sirs, who will
The" compactness of the book implies no lack of formality.
The Argument
outlines the topics to be treated under Defini
tions and Operations
and promises that "At the end of this dis
cussion there will be added an Appendix
setting forth the use of
in real problems."
La Disme
How modern
it all is! Motiva
tion that would whet anyone's
interest, a thorough preliminary
testing out which we are assured was successful, and, as a climax,
real problems!
We come then to the actual theory.
Definition
I
is a kind of arithmetic
on the idea of the progres
based
sion by tens, making
use of the ordinary
Arabic
in which
numerals,
any
number
be written
and
may
all
that are met
by which
computations
in business
be performed
may
without
the aid
alone
of
by integers
fractions.
La
Disme
Explanation
Let
the number
one
one
thousand
hundred
in
eleven
be written
Arabic
numerals
as 1111, in which
form it appears
that each
1 is the
tenth par.t of the next higher
in the number
Similarly,
figure.
2378,
each unit of the 8 is the tenth part of each unit of the 7, and so for all
the others.
But
it is convenient
since
that the things which
we
study
have
and
since
this type of computation
names,
is based
upon
solely
the idea of the progression
or
tens
is
our
as
seen
later
in
disme,1
by
we
of
discussion,
this treatise
as La Disme,
may
speak
and
properly
we
see that by it we may
shall
all the computations
we meet
perform
in business
without
the aid of fractions.
coin
l Disme:
later
the word
was
contracted
"tithe,"
has
this connection
with
the first decimals.
into
dime,
so
our
326
THE MATHEMATICS
Definition
A whole
number
is called
TEACHER
II
the Commencement
Definition
and
has
the
symbol
?.
III
The
tenth part of a unit of the commencement
a Prime
is called
and has
the symbol
and the tenth part of a unit of prime
a
is called
0,
Second
and has the symbol
for each
tenth part of the unit
Similarly
@.
of the next higher
figure.
Explanation
Thus
is 3 primes
7 seconds,
3 thirds, 9 fourths,
and
3(5) 90
3q
7?
so we might
continue
It is evident
from the definition
that
indefinitely.
the latter numbers
are 3/10, 7/100, 3/1000, 9/10000,
and
that this num
ber is 3739/10000.
Likewise
has
the value
8 9/10, 3/100,
8?
90
3@
7?
so for other numbers.
And
We must
realize
also
7/1000, or 8 937/1000.
that in La Diurne we use no fractions
and
that the number
under
each
the commencement,
never
except
symbol;
exceeds
the 9. For
instance,
we do not write
but 80
for this has the same value.
70
12?,
2@ instead,
Stevin 's choice of names and symbols for these numbers is ex
in the first pages of l'Arithm?tique, which was pub
plained
lished in the same volume as La Bisme.
Here, he calls the terms
of a geometric progression by the ordinals : prime, second, third,
. . . with the signs ?, ?, ?
. . . giving as examples ?2 ?4
In
?3
?9
?27.
VAstronomie, he uses these symbols in
?8,
In
both
and sexagesimals.
decimals
writing
Arithm?tique,
he extends this to the case where the ratio is an unknown quan
tity. Thus ?, ?, ? stand for 0C,0C, oc . The commencement, as
in l'Arithm?tique, was an integer or an irrational num
ber used in algebraic computation.
The symbol ? was to enter
only when the commencement was an abstract number. Denomi
nate numbers in Stevin's other works and in the Appendix
to
La Bisme, appear as 1 hour 3? 5?, 5 degrees 4? 18?, 2790
defined
The origin of the symbol for the commencement
verges 5? 9?.
has no connection with the zero exponent. It is probably a direct
consequence of the fact that the mediaeval writers left a blank
space
These
between
the units
writers
had
suitable
and
denoted
or
abbreviations
the first place of sexagesimals.
the orders of the fractions by
sometimes
either
by numerals,
some
The
mark.
by
distinguishing
above or designated
?
for a degree* first appeared
in print in 1586.
In
symbol
uses
notes
same
Stevin
that
Bombelli
the
l'Arithm?tique,
symbols
with the exception of the ?.
[These quantities were written by
Bombelli
,
,
(1572) as
.] One concludes that Stevin bor
written
rowed his symbolism directly
from Bombelli,
but as the latter
327
THE FIRST BOOK ON DECIMALS
to the ?,
felt no need for a symbol corresponding
forced to invent one for himself, and the zero was
the logical outcome, as it fitted with the rules which he laid
and division of the numbers of La
down for the multiplication
Yet he never seems to have connected these symbols
Disme.
writer had
Stevin was
with
the
exponents
of
the
tens
in the donominators
they
!
replaced
In 1525, Finaeus
had unwittingly provided a precedent for
use
same
of
Stevin's
the
symbols for decimals and sexagesimals,
as 5.36, i. e.,
by writing the product of 8 by 42 (minutes?)
(1616) used both the comma and
Beyer
?ff. "Subsequently,
the ', notation for decimals.
Napier used ', ", etc., %inhis Rab
dologia
(1617) but in his Constructio he shifts to the comma
which Burgi had used in 1592. Dr. Glaisher1 gives the opinion
that Stevin used his array of symbols to make the numbers of
La Disme more like those to which his contemporaries were ac
customed.
But one point should be mentioned
In La
here.
=
Geometrie, expressions such as these occur : 1875?
|X |
f
and 7 X | = 280?.
It is possible th?t this omission of the signs
was made by the editor Girard;
but by 1625, the date of the
publication of his first edition of Stevin's works, Girard would
have had ample opportunity to see the advantage of the comma
which
was
Had
he been making
using.
these
influences
would have led
changes
symbolism,
him to retain the ? and drop the ? or the ?.
if
Similarly,
Stevin was
influenced by the bar of Rudolff
etc., we would
have expected him to do likewise. But Stevin appears to have
thought of his 1875 as 1875 thirds rather than as 1.875.
separatrix
in Stevin's
Burgi
Stevin's
I and his choice of names
explanation of Definition
bear out our hypothesis that his work with decimals resulted
from an effort to devise a system of numbers analogous to sexa
of the Arabic notation would
gesimals in which the place-value
function. His retention of the symbol of the last number, rather
than that of the commencement, seems to indicate that he was
not guided by the sporadic inventions of Rudolff and the rest.
Yet, at first sight, these appear to be the more
of ancestors for the decimal fraction !
l Logarithms
and
Computation,
Napier
Tercentenary
probable
Volume
group
p.
77.
THE MATHEMATICS
328
TEACHER
Operations with the numbers of La Disme are treated in four
and Di
Addition,
Subtraction,
Multiplication
propositions:
vision. Each contains a formal statement of what is given and
of what is to be proved.
Careful reference is made to the cor
The rigid
responding work with integers in l'Arithm?tique.
uniformity of these propositions permits us to judge their style
from a single example, and their content mar be summarized
without including the rather tedious proofs which are identical
in method with
Proposition
Given
three
875?
7?
8?
To
Required:
Construction:.
figure,
adding
the proof
in Proposition
I.
I. To add numbers
of La Disme.
numbers
of La Disme,
27?
4?
8?
7?,
2?.
find their sum.
in order as
the numbers
Arrange
them in the usual manner
of adding
37@
6
7?
5?,
in the accompanying
integers.
??@?
2 7 8 4 7
3 7 6 7 5
8. 7 5 7 8 2
9 4 1 3 0 4
This
of l'Arithm?tique)
the sum 941304,
(by the first problem
gives
as the symbols
the numbers
is
above
show,
which,
941?
0?
4?.
3?
And
this is the sum required.
Proof:
the
third
number
definition
of
this
the
book,
given
By
27?
or 27
the
is 27 8/10, 4/100, 7/1000
Similarly,
847/1000.
8?
4?
7?
37? 6? 7? 5(?) is 37 675/1000and the 875? 7? 8? 2?
27 847/1000,
These
three numbers
37 675/1000,
to the tenth problem
of l'Arithm?tique
cording
this same value,
has
and
is,
3?
4?
0?
941?
was
which
to be shown.
Conclusion:
been given
of La
numbers
Having
was
to be done.
found their sum, which
Note:
in question,
some
If, in the numbers
fill its place with a zero.
order be lacking,
For
the second
number
and
5?
7? where
8?
6?
5?
insert
as the
take
and
prime,
0?,
5?
7?
0?
as before.
875
is 875 782/1000.
added
782/1000,
941 304/1000,
give
the true
therefore,
Disme
to add, we
ac
but
sum
have
of the natural
figure
in the numbers
example,
lacks a figure of order
and add
number
given
@?@
8 5 6
5 0 7
13
6 3
Stevin has three ways of writing these numbers, the symbols
may be placed above or below the numbers or after the digit
which they qualify.
It should be noticed that Stevin shifts deci
mals to common fractions, never vice versa. And although he
uses a zero to fill a gap in his decimals, he feels no obligation to
329
THE FIRST BOOK ON DECIMALS
begin with the prime.
writes 3? 7? 8?.
In Proposition
IV,
for
instance,
he
the symbol of the last number of the
In multiplication,
is found by adding the symbols of the last terms of
and multiplier.
Stevin gives as proofs the case of
multiplicand
to the rule,
2? multiplied
3?.
The
by
product, according
and
should be 6?.
The values of the given numbers are
*s identi
and the value of the supposed product is -, ^ >Du^ ^is
cal with the product of 2? by 3?, which was to be shown.
product
is
In division, the symbol of the last term of the quotient
found by subtracting the symbol of the last term of the divisor
If the numbers of the divisor should
from that of the dividend.
be of higher order than those of the dividend, zeros must be
added
to the latter.
It sometimes
that the quotient
cannot
be expressed
by whole
happens
in the case
that
as
it appears
of
numbers,
divided
Here,
by 3?.
4?
the quotient
in
will
be
one-third
threes with
always
infinitely many
In such a case, we may approach
addition.
as near
to the real quotient
as the problem
that
and omit the remainder.
It is true indeed
requires
or
is the exact result, but in La Diente,
3?
3%?,
13? 3?
3?
13?
3%?
we propose
to use whole
we notice
numbers
that in
only, and, moreover,
one does not take account
business
of the thousandth
part of an ounce
or of a grain.
as
such
Omissions
these
are made
by the principal
and arirhmeticians
even in computations
geometers
of great
consequence.
and Jehan
de Montioyal,
for instance,
did not make
up their
Ptolemy
tablesi with
the utmost accuracy
num
that could be reached with mixed
is more
of these tables,
bers, for, in view of the purpose
approximation
useful
than perfection
the
square roots may be extracted by first making
Finally,
last symbol an even number by the addition of a zero if neces
The symbol of the
sary, then getting the root as with integers.
last number of the root will be one-half that of the given num
ber. "Similarly,
for all other roots/'
The Appendix
that
proposes to show how all computations
arise in business may be performed by the numbers of La Disme.
Its greatest interest to us is the treatment of units of area and
to less
volume^ and the extension of the numbers of La Disme
obvious applications.
We may safely guess that the sections of
1The
tables
of Ptolemy
to which
Stevin
had access
were
probably
those
in the translation
of the Almagest
made
and Regi
by Peurbach
omontanus
(Stevin's
in the 15th century.
tables
In these
Montroyal)
from a circle of radius
trigonometric
functions were
computed
10,000,000.
THE MATHEMATICS
330
TEACHER
the least interest to the readers of the sixteenth century are of
the greatest interest to us today, and conversely.
Article
One
of Surveying
Of the Computations
are used
in surveying,
the verge2
the numbers
of La DIsme
or
the commencement,
and
it is divided
into ten equal
parts
if smaller
each prime
is divided
units are re
into seconds,
and,
primes,
so on so far as may
the seconds
into thirds, and
be necessary.
quired,
For
the purposes
are sufficiently
into seconds
of surveying,
the divisions
that require
as in the measuring
greater
accuracy
small, but in matters
of lead roofs, one may
do
need to use thirds.
surveyors,
however,
Many
not use the verge, but a chain of three, four or five verges,
and a cross
staff* with
its shaft marked
with
in five or six pieds
their doigts.
These
the same practice
men may
follow
five or six
here, substituting
re
with
use
their seconds.
without
should
these marks
primes
They
to the number
in that
of pieds
and doigts
that the verge
contains
gard
and add,
and divide
numbers
the resulting
locality,
subtract,
multiply
...
as in the preceding
To find the number
of pieds and
examples.
in 5 primes
9 seconds
in one of the examples
doigts
(the result
cited)
look on the other
side of the verge
to see how many
and doigts
pieds
a
with
match
the surveyor
must
do but
them; but this is
thing which
i. e., at the end of the account
to the proprietaries
which
he gives
once,
and often not then, as the majority
of them think
it useless
to mention
the smaller
units.
When
is called
This last illustrates a consequence of the tremendous diverg
ence in units of measure resulting from the feudal conditions of
Mediaeval
Europe, where each petty ruler ordained such meas
ures as he saw fit?standards
that might or might not bear
to
other
relation
of the same name in other
units
recognizable
samt
of
the
parts
country.
Stevin's
surveying
area of a
drawn
to
In all of
in
examples of the use of the numbers of La Disme
consist of adding and subtracting areas, finding the
rectangle, and determining where a line should be
cut a rectangle of given area from a given rectangle.
is
this work, the prime of a unit of square measure
one-tenth of the unit itself.
Article Two advises that the aune of the measurer
be divided into tenths as was done with the verge
of tapestry
of the sur
veyor.
2 The
a unit of both
linear and
and
the
measure,
verge was
square
word was
also used
for the surveyor's
rod.
*
a
is
at
cross-staff
of wood
mounted
its midpoint
per
piece
Th^
to other posi
to a shaft and
free to move
this shaft
pendicular
along
tions parallel
to the first one.
ad
To use this instrument,
the observer
it so that the lines of sight from the end of the staff to the tip of
justs
the cross-piece
coincide
with
the endpoints
of the lines to be measured.
are
Distances
from one measurement
and
similar
computed
triangles.
When
neither
is com
end of the required
line is accessible,
the distance
from
other.
uted
at known
from each
two observations
distances
he pied and doigt were
to our foot and inch.
units approximately
equal
THE FIRST BOOK ON DECIMALS
Article Three deals with
that he has made his
writing to master gaugers,
He
is almost unintelligible.
says
331
the measurement
of casks.
Stevin
demonstration brief because he is
not to apprentices.
As a result il
gives great emphasis to the division
of the measuring
rod by points spaced according to the square
roots of 0.1, 0.2, 0.3, . . . with intermediate points correspond
ing to the square roots of 0.11, 0.12, . . . This division is made
by finding the mean proportional between the unit and its half
and the mean propor
gives a point answering to y0.5,
tional between this segment and its fifth part gives V0.1.
The method of using this is not explained, but it is evident that
the diameters measured
by this rod would have the same ratio
as the areas of their respective sections when measured by the
which
divided rod.
Four discusses the numbers of La Dis me in volume
measurement.
The illustrative problem is to find the volume of
a rectangular
column of dimensions
2?
2?
3?
4?,
2?,
3? 5?.
The volume is 1? 8? 0? 4? 8? 0?.
decimally
Article
Xote:
is ignorant
Someone
who
of the fundamentals
of stereometry,
for it is such a man
that we are addressing
the
now, may wonder
why
volume
is but
of the above
column
than
more
etc., for it contains
10,
180 cubes of sides
that the cubic verge
He
should
is not 10
realize
10.
but 1000 cubes
the prime
of side
is
unit
of the volume
Similarly,
1?.
j00 cubes
of side
been
had
how many
the question
If, however,
1?.
of side
are
cubes
in the above
have
been
column,
the result would
10
to conform
altered
to this requirement,
in m1??d that each prime
bearing
of volume
is 100 cubic
units
and
that each
second
is 10 cubic
primes,
primes.
Thus, Stevin makes his units proceed strictly in accordance
with the decimal scheme.
In this way he, forestalls the objec
tion to the metric system that its linear, square, and volume
units follow the progressions by tens, hundreds, and thousands
respectively, but he sacrifices the simple connection between the
submultiples of the units of the three systems. The prime and
second of his units of area are squares whose sides are V0.1
and V0.01
; the prime and second of his units of volume are cubes
and
whose edges are ^0.1
^/0.01. This is startling, for we
think at once of the difficulty of teaching it to an immature
person, but it would be more convenient than the metric idea if
we, like Stevin, felt a need of labeling each of our subdivisions
and wrote 2 cubic decimeters 55 cubic centimeters instead of
2055 cm*.
THE MATHEMATICS
332
TEACHER
Article Five explains that the division of the angle into min
utes and seconds was devised so that astronomers might work
"
because 60
with iritegers. The sexagesimal
scheme was used
"
is commensurate with many whole numbers.
Paying all due
reverence to the past, Stevin maintains
that the decimal pro
is even more
He says that he contemplates
tables
publishing
using the decimal division of the
a
on the
and
ends
he
with
characteristic
degree,
peroration
beauties of the Flemish language in which these tables are to be
written. As a matter of fact, the tables which Stevin published
gression
convenient.
astronomical
Father Bosnians
(La Thiende,
(1608) were on the old basis.
on
this
two grounds?Stevin
did not
Louvain,
1920) explains
van
as
in
is
did
fortunate
Ceulen
(which
rejoice
computations
for the world, for the impatience which may have cost us a set
of tables, gave us La Disme)
; and secondly, large errors would
in shifting instrument readings to the decimal system for
computation, and in then changing the results back again.
1
Mer
of Mint-Masters,
Article Six,
'On the Computations
chants, and in general of all States," begins with the thesis that
all measures may be divided decimally, and that the largest unit
of each denomination should be the commencement.
In the case
arise
of money,
the limit for the divisions would be the first sub-unit
that is less than the smallest coin. Instead of half pound, ounce,
and half ounce weights, the smaller weights should be the 5,
3, 2, 1 of each order, and the names prime, second, third, etc.,
should be retained because of the aid which they offer in com
of the decimal di
Stevin illustrates the advantage
putation.
marc of gold is
vision of money by a problem of exchange?1
worth 36 lb. 5? 3?, how much is 8 marcs 3? 5? 4? worth?
The value of a simple method of treating such a problem is ob
vious when w? consider that the value of the marc varied in the
different cities, and that the number of the smaller units in a
marc
varied
also.
that
of all the common
We
rules of arithmetic
might
examples
give
as the rules of partnership,
to business
etc.,
exchange,
pertain
interest,
out by integers
to show how
be carried
how
alone
and also
they may
be performed
of counters,
but as
they may
by the easy manipulation
we
them
from the preceding,
be deduced
shall not elaborate
these may
here.
show by comparison
with
We
with
frac
might
vexing
problems
num
tions the great
difference
in ease between
with
ordinary
working
the numbers
of La Dlwme, but we omit this in the interest
bers and with
of brevity.
333
THE FIRST BOOK ON DECIMALS
between
the sixth article
we must
of one difference
speak
Finally,
individual
that any
may make
and
the five preceding"
articles,
namely
in
but this is not the case
in the five articles
set forth
the divisions
as good
and
be accepted
the results must
the last where
by everyone
it would
of the decimal
usefulness
In view of the great
division,
lawful.
into
this put
if the people
would
be a praiseworthy
urge having
thing
of measures,
to the common
divisions
effect so that, in addition
weights,
declare
the decimal
division
the state would
that now exist,
and money
use
to the end that he who wished
of the large units
might
legitimate,
is newly
that
if all money
also
It would
this cause
them.
furthery
etc.
thirds,
on this system
of primes,
be based
coined
should
seconds,
so soon as we might wish, we have
But
if this is not put into operation
that
for it is certain
the consolation
that it will
be of use
to posterity,
like men
not be neglect
if men of the future are
of the past,
they will
an advantage.
it is not the most
ful of so great
discourag
Secondly,
labor
free themselves
from such great
that men may
ing thing to know
the sixth article may
not go into
at any hour they wish.
though
Lastly,
use
the five preceding
effect
for some
individuals
time,
may
always
as it is clear
in operation.
that some are already
articles
The
end
of
the Appendix.
Stevin's scheme for the decimal division of measures
that of the metric system in three respects :
1. He
did not reach the idea of a universal,
is unlike
invarient
standard.
2. All of his units proceed according
tens with the disadvantage
noted above.
to the progression
by
3. He avoids the multiple units by taking the large unit as
for a
the commencement, but, in Article Four, he provides
in
the
to
the
unit according
change
exigencies of the problem.
These are minor matters, however, in comparison with the
idea itself!
use
that
the decimal
The
individuals
may
suggestion
use
our
to
of
the deci
of measures
mind
divisions
brings
on
the
mile
the
and
the
divided
speedometer
milepost,
mally
the
and
of
divided
foot
the
surveyor's tape,
grinding
decimally
of piston heads to the thousandth of an inch. From a scientist
of Stevin's
his device;
anticipate
type, we would expect a rigorous demonstration of
from a person of his prominence in affairs, we would
an application
to real problems. But who would have
man
to write on decimals would have
that the first
imagined
had the vision to see decimal weights and measures,
and that,
pending the adoption by the state of such a system, he should
recommend the very compromises used by individuals
today!