"y^^t^p^r-
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f.
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"v
I
ARITHMETIC
DECIMAL
PERFECT:
MADE
m
T
R,
O
E
H
of Infinite Decimals
Management
y
DISPLAYED,
of the Arithmetic of Circulating
Dodrine
Being the Whole
and Curious ExamNumbers, explainedby many New
fc?^. Of all
in
Substraction,
Addition,
pies
made
which the laft Age was entirely
ignorant,but now
With
the
meaneft
Familiar
and
to
Capacity.
Eafy
Demonftrations
proper
hitherto
Manner
by
Unattempted,
at
leaft not
Improvements made
the very
which
Firft
IS
therein
Attempt
in
a
ft-cfixcd.
by its feveral Authors, from,
down
N
D
the Prefcnt Time.
to
annexed
Tables
Large
A
A
to
compfcatthe Whole..
K
I^X,
APPEND
containing
^
The
;
Publilhed
1ntroductton(fhewingthe Progrefiand
Historical
With
or
Author*
any
To
An
illuftrate the Whole
to
of the Five
Arithmetic
Decimal
Fradlions,as
Primary Rules
commonly Taught*
in
*;
By JOHN
MARSH,
in the
and Accomptant,
Writing-Mafter,
of
Sarutoi.
City
LONDON::
f
Printed for the^ A U
and
and
H
O
R ; and
Sold
Edwa
rd
Eastow^
by
Collins, Bookfellersin 5"r"" ; and John
in Ludgate-Stretty
Knapton,
London.
17+2.
Benjamin
Paul
T
r-.v
'":^
"
'"'
'
^^
*..
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t
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I.
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V.
f
i
T
TO
RIGHT
HE
HONOURABLE
,
HENRY
"
PEMBROKE
One
of
RL
A
and
His
of
MONTGOMERT,
mod
Majesty's
Honourable
Privy-Coancil, ^c*
AND
High
Lord
As
a
Steward
of
the
Cxtf
great Encoafager of Arts
This
of NEW-SA"t^M"
and
Sciences^
TREATISE,
OP
III with
and
all
Perfed*
made
Arithmetic
Decimal
Humility^ moft
humbly
I'refented
Dedicated,
Br,
My
LORD,
Twr
tarifk^i
nuji iMIicnt Servant,
I
MAk$H.
r-it
"
""
-
f
...
J
1
,
/*.
..
"
"
.
1
-
"
*
-:
^
^J.
\o
I
THE
PREFACE.
t
A
IH
V
Book
lowing
it
were
with
at
Way
of
the
from
juftly excufed
none
fo
not
all.
that
niaking
have
to
I
long
'd
chofe
here
be
may
more
very
deed
in-
Cuftom,
have
made
by the
Readers
place
A*
And
I (hould
to
in
with
gratify fome
to
Infinite
them
Preface.
comply
However,
Prefacing, I
of
I think
a
follow-
concerning
Inclination,
own
my
much
the
in
Management
Calculations,
ticbmetical
and
Introdu(flion
faid
and
Decimals,
than
the
in
E
the
following
Remarks.
ti
Other
univerial
The
Fractions
of
Kind
Application of
befpeaks
Wherefore
do.
biy
(hall
be
making
of
^ving
of
recommending
knowing
2.
Place
no
any
myfelf
to
moft
if I
the
make
PerfoD,
( by
Reader
who
is
Way
(hall
take
following
ignoraut
the
A
of
under,
managing
the
z
the
Pains
Arithmetic
)
Necel^
perfedly
them.
in
Liberty
;
I
indeed
or
abfolute
Obfervation
of
that
long Harangues
the
lies
of
more
pof!i-
believe,
general Ufefulnefs,
my
I
to
all
thematics,
Ma-
can
fpare myfelf
Trouble
exa"t
however,
But
to
the
Mixt
of
Encomium
inclined
Arithmetician
every
the
of
am
their
on
before
Superlative Excellency,
readily excufed,
of
fity, that
I
Branch
every
Periods
laboured
all the
than
to
their
Decimal
viz.
of
this
That
Infinite
PREFACE.
Decimals,
Dite
faid
be
can
underfland
to
Decimal
its AfTiftperfedtlywell ; becaufe without
mud
of
his
Operations
ance
generally be
Error
confiderable
the
and
when
imperfe(5t,
too^
very
For Inftance ;
be deals with
largeNumbers.
Arithmetic
the
Let
Refult
fuppofe the following Finite
us
Mixt
Number^
given to be multiplied by ^06, where
96,75, was
Place
would
of Huainfinitelyrepeat fi'om ue
viz.
6
dredths
of
Multiplicand
is
little by
dud.
And
fuch
fmall
For,
uniefs
carefiil
fift of
to
the
Eight
in
every
more
or
will
hinl, if he
how
very
imperfeift. Whereas
ftm"t
Vv
adl
how
him
in
a
'
very
to
narrow
I
mon
Com-
is
5,805;
6,45 : So
^645^ which
^ro-
will arife froQi
great
tl^
may
largeNumbers*
!
Decimals
common
be
Approximate Fador.tb
conin
its
Fraftiotfal
Figures deep
be
very
confiderable.
pleafe, make
Figures deep, yet aftqr all his,labour
be
in
of its Common
an.
Pra"titioner
thence
is
be
Error
above,
with
Part, the 'Error
let
Ninth
as
Multiplier
which
would
deal
make
give
in
their ProduA
Product
exadt
one
Numbers^
we
to
Former
juft the
be, when
Defeft
and
PraSitioner
content
if fo confiderable
the
,06, ftom
Expreilions,
of. the
Defed
as
every
its Mathematical
the
toa
be
Finite
two
whereas
that
almoft
would
Integral Number
96^
as
diminitive
that
Decimals
with
is fo little
is fo
in appearance
believe
readiily
as
that the
in Confideration
Now
the
Unit.
an
them
the
And
deed
in-
of
Millions
Refult
the
"v^
following Sheets will
find the Refult mathen^tic^Uy
Compafs.
'
3.
inex-^
Who
I"
Who
3.
R
E
C
Fj^
tbeM^e,
E.
fuch
amoi^g
defire
as
"fteeim"df Com}"kat Aiithmetkk^^
wodkl
longer igni^Mt df the Arithmetic
Which
Decimafte?
ibe
teadct
dofQbt,m\\
vary ea^y and
of
^ny
^ot
And
fomiifar
C^^iiy :
FiwEtim"
only, withmt
Yfttg^kr
Complext AlgebraicalTheorem
comrtion
the
to
Author
with
will
of
Account
meet
the
in
the
to
turn
I
do
to
the
hanng
for
not
every
Principles
recoarfe
to
its AfTiftance.
expeft,
to
cording
ac-
the
Particulars
of
Body
his
Work,
fo,
of this Book,
Contents
he
which
he
may
whoever
will
there
of the Order
Succefiion
or
very ample Account
the fcveril Parts of the whole
Compofition to
find
of
a
j
Which
therefore, to avoid
I muft
Pfdlixity,
here
refer
Reader.
my
5.
for
As
my
Stile,I
have
endeavoured
of
plain and uniform, as the Nature
will give leave.
And
for the Redundancies
;i8
Readers
here
are
turn
out
to
make
the
difierent
it
Subje"5^
( or if my
rather chuie to call them
Tautologies) which
and there to be met
with, I hope they will
an
Advantage to the iiiere Engli/h Scholar.
I have
For
of
In""ite
of
even
upon
Reader
tinue
con-
of Prefaces, that the
general Cuftom
ifomewhere in this Place give hiai a
fhould
fuccind
too
for the
natural
being
It
4r
now
foBovring Traft,
be
to
for many
Years
experienced,that
Rules have been
delivered in the leaft
Didtion,
there
Youth
in
general
have
where
Variety
made
tlie
qilickeftProgreis.
Wherefore,
Ufe
of
to
with
as
the moft
ufe
as
few
plaincommon
I
Compofition for
defign'd my
I have
Illiterate,
Variations
Senfe
in
been
the
mfore
the
ful
care-
Exprefiionas agreeably
I well
could.
6. And
PREFACE.
thimi^ioatthe
that
I hope
laftly,
t. And
Whole
Penifal any foch
If upon
ihould appear, I (hall be very thankful
to that Peribn
Such
of thenn.
who
(hall be fo kind as to apprise me
there
Errata
fmall
avoid
To
the
|N
^
5
II.
Ex,
P.
fir
the
,003973,17
Nombct
^^
fir ^li?
39.
1. 3*
9.
the
firft Place in Dedmab;
P.
rtad
P.
P.
been
P.
1.
6x.
1.
1.
Place
more
10.
fir
12'
12.
the
lemovt
7.
r.
Ex.
tlie
3.
Speck
P.
59*
P.
65.
"
It.
,6 by ,8.
multiply
one
In
fir Expref-
14.
(13456788
r.
h
70.
fet
r.
he per*
Gentlemen.
is*
i" 'i.
$""
|. the left,/or 11345678
53.
,00307117.
ibo"ld"na?e
r.
IJ1500
P.
.
,,.
before
following Errata*..
for Gentleman
'"
=sS7547x6.
%t,
16, i7*/^s=^754753
to iti next
Fi^re
place a Speck over
^
I*.15.
^4.
Expieflioft.
r.
Iroffl
h
1.
corredt the
to
PSage tjI. fine
Intndnftiony
Bookj
iooB
is defired
Book^
P.
Books
Mifconftru^lion, the Reader,
any
ufes the
and
Pen
but
I doubt
not
eipecially,
unprejudicedReader wiU candidlyexcufe
every
correA.
and
both
attend
commcmiy
as
in Numerical
Prefs^
that
oiaterial Faults.
no
are
P.
77.
Left-hand.
towaidsthe
*
P.
1. 20)
80.
fir i%fOO
P.
3io,o.
n
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the
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p.
1; 8.
1 10.
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fir 5,7945
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X714185.
THE.
THE
CONT
ENTS.
INTRODUCriON,
nP//"
H
C
Defimtions and
fVbai
1
2.
A
?"
I.
PropqfiimSf
RepiUnd
a
.
Page
Circulate
w
ibid.
is^
ibid.
Single or Compound^
eitber
Repetends are
i
ibid,
fnat
aSifigU Rcpetend is^
4*
fUM
a
5.
Repetendsin general are
6.
What
a
Pure
Repetend iSf
ibid.
fFbat
a
Mxt
one^
ibid,
3.
7.
8.
Compound
the Finite
^at
one^
z
eitber Pun
Circulati^
and
or
ibid.
Mbtt^
Parts
are
Mwt
of any
Repetendy
9.
How
find
to
a
3
Vulgar
Fra^ion
EfuHvalent 40
any
ghets
Repetendy
10.
How
5
multiply any given Number
to
by
Number
any
tfg^s
mofi compendioujlyy
11.
How
"
12.
do the fame
to
ded
to
Fra^ion
13.
I4"
1
6*
FraaUms
any
is to bead^
given Number
Produffy
its
of amy
the Value
Shews
159
when
JborteftMethod
fbe
10
Shew
are
of
how
a
to
Repetend when
the
or
find the EquivalentVulgar
Repetendy
Miut
when
9
i x
cfs
1
,9
Denominators
with
E^ivalent Repetends arey
( b )
or
without
or
9,
"c.
3
16
Vulgar
of any
what
their
0%
x6,
17
ij.tbe
CONTENTS.
The
17.
Method
of transforming a given Repetend (0 another
Equivalent one^ having the fame or a greater Number
their fever al Defnonftraof Places of Figures^ wilb
t ions
Repetends
t8.
19.
20.
2
1
Similar
IVhat
a
Dijftmilarone
How
to
fFbat
How
one
transform
Diffimilar^
24
ibid.
is^
ibid.
is^
two
or
mere
DijftmilarRepetends tOf
ibid.
Multiple
leaftCommon
to
two
or
Nume-
more
iSy
to
25
leaftCommon
the
find
to
make
Similar
MultipU
to
two
or
mont
ibid.
given Numbers
How
or
onesy
the
bers
24.4
Similar
a
Similar
83.
either
are
What
.
22.
J
Finite
a
or
Number
Determinate
bccme
CirculatingExprejfton^
C
H
A
P.
U.
3a
CHAP.
III.
Sukra^i"n^
25.
lbt"
4X
to
The
i
multiplyany Single or Compound Repetend by
100,
or
26.
^
2^
of Circulatesy
M(Ution
9
1
J
or
"c.
1000,
48
mofi Cempendiouj^dethod of dividing
any Number
Number
of 9%
any propofed
e
H
A
10,.
P.
by
50
IV.
1
"
ft
56.
Multiplicationr
.
C
H
A
^e
mojl Compendious Way
by
V.
bb
'
DiviftoHt
27.
p.
any
Number
of multiplying
any 'Repetend
of 9V,
105.
CHAP.
CONTENTS.
CHAP.
VI.
tLedulfiotit
I20
CHAP.
and
InvoUHim
8.
a
ff^iU
lis
19.
13*
ibid.
is^
KnUy
ibid.
"
.
Evolution
Powers
Jll
31.
Emriutimit
Imobitiott
What.
30,
YUi
.
is^
dboBoe
145
the
firftare
either
Rational
or
rational^
Ir-
ibid.
32
23*
And
PFbat
Rational
ff^^t
Irrational
Powers
Iqjtty^TbeT^lis
Powers
with
ibid.
are^
"
their
ibid*
are^
Explanation and Ufe^
1
54
^c.
"
\
*
"
(ba)
"
Jh
Jin
EXPLANATION
and
the
of
made
Abbreviations
Character
in
ufe of
tbefolhwing
Sheets.
Kames.
Characters.
Equal
=
Significations.
The
to.
Equality
of
Mark
As
"
s.
L.
I
Plus,
*)-
of Addition.
Mark
The
More.
or
20.
=
9-1-6=15;
Minus,
r
"
Lefi.
or
6
more
or
9
read
lefs 6
Mark
The
is
of
As
9x6
equal
Diykled
^^
Mark
The
by
to
9-^6
1
J
6
is
nus
mi-
9
equal
=54
to
3,
read
\
hj
into 6,
or
54.
of
Divifion.
read
equal
diyided
9
to
As
-|)
(or thus
"
15.
on.
Multiplicati-
^ multiplied
IS
to
Subtradion.
6=35
"
or
Multiplied by.
of
plus
9
areequal
Mark
The
As
X
read
As
i
=
by
"
.
a
Squire
^
v^ 36
As
Root.
Square
to
then
HeiKe
by
go,
9,
is
to
whofe
equal
to
^^
?
=
Produft
25
divided
is
"
add
by
7,
read
and
d
=
Root
read
5
of
36
is
the
equal
6.
thus
that
;
Sum
2
multiplied
divided
90.
ABBRE^
by
ABBREVIATIONS.
Nunf
for Numerator.
"
Denominator.
Denom^'/^r
".
S.
E.
V.
C,
P.
In
making
avoid
And
V.for Equivalent Single
F.
for Equivalent
for Common
many
ufe
and
by it alio
we
or
of
the
Fradion
above
Produfb.
Symbols
frequent Repetitions
have
Fra"ions.
Fraftion.
Vulgar
Firft
or
this farther
Ad
of
or
the
vantase,
Charafter^,
fame
we
Words.
viz.
of
com-
Thirds
Subjed in, or nearly wiA,
two
prifing the whole
that
the
verbal
would
of the Paper
Way
neoefl"rily require.
A
CA*
IS
L
T
"
F
O
of
Names
as
T
H
E
of the
many
Subscribers
made
Arithmetic
Decimal
As
in
came
"
to
Time
be
to
Perfect,
inferted.
*
A
*
JLMm
"
"^
"
'
"
'
t
"
1
of Seend^ Gent.
Awdry
AMbrofe
Williain
Mr.
Sai^ouei Allen
J^ir.
^he
Reverend
The
Rev.
Andrews
of tU
A.
Abbot,
Mr.
Dock,
Portfmouth.
Andrews.
Dummer
Mr.
of Ain.l;"rofl"Hry^
Writing-
M.
Fellow
ofBaliol College Oxon.
j
Samuel
Mr.
Biggs of
Bcnnct
Thomas
James
Brohcir
M.
Honourable
Gent.
EJqi
Gent.
of Lower
D.
Bowles,
Thomas
Right
of Southton.
Balkerville
Thomas
the
Park.
of Salthrop,
Bartlett of Sarum,
P.
Mr.
the
Biggs of Woodford,
Henry
to
,
of Pembroke.
Earl
the
Gardener
Brown
James
Mr.
D.
Richardfon,
Re3or
EJqi
of Brackley
in
Northamp-
tonfliire.
Reverend
The
Balguy,
Mr.
M.
A.
Fellow
of
St.
John^s
CollegeCambridge.
Bryant of
Mr.
William
The
Reverend
Mr.
St.
BarfoFd,
Johrfs CollegeCambridge.
A.
B.
Fellow
of King's College
Cambridge.
John
Mr.
Banfon
John
Barker,
John
Thomas
F.
R.
M.B.
of Trinity Hall, Cambridge.
of the Clofe of Sarum.
Batt, M.
B.
of Baliol
College^Oxon
5
and
S.
Mr.
'-^
of i^e^ 8cc.'
T
s
Majifi^s
his
in
Pdrtfmouth.-
Dock.
Arthur
I
Builder
Bucknall, AJJifiant
Thomas
Mr.
L
of Baliol College^Oxon.
Batt, "/^;
7be
Reverend
Mr\
Thomai
The
Reverend
The
Reverend
Blake
Mr.
Bacon
of Siirum.
of Sarum.
George Bowditch.
Mr.
William
Mr.
A.
Bowles,
Fellow
B.
of New
College,Oxon.
She
in tfnyoer^
Bradley, Prof^J/orof-AfiroHomy
M-.
Revered
ftty College^Oxon.
The
Rev.
Biifs, Profefor 'ofGeomfry'^mUniverfiiy Got-
Mr.
legeyOxon*
^
PTriiing-MaJTerin Oko".
College^Okon.
Baylcs of Pembroke
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y
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THE
thf
I"]
THE
*".'"
INTRODUCTION.
Perfon,
EVERY
in Decimal
converfant
Fraftidm^
that in the turning of a Vulgar
Fradlion
into a: Decimal
Fradion, it is very rare
that the Quotient is finifhed of without
leaving any Re^
mainder.
muft
likewife
they muft
And
Quotient) which
compleat Decimal
Figure,
fame
obfenred,
have
tb^ Quotient
turns
out
a
infinitum^the fame Figure, or
nitely repeated in the Quotient.
preflionsas thefe,
are
therefore
fe"t, determinate, compleat,
'"
But
Age
either
in
on
infi-^
be
Decimal
Ex-
,
called
or
the
continued
were
fuch
And
or
fomewhere
occur
interminate, indeter-
in ContradiftinAion
infinite Decimals,
or
find
often
I^igures,would
ad
minate^
perfe6t"determinate,
that, if the Divifion
and
in the
notice, that
taken
Ezpreflion, they
Figures, continue to
or
;
not
have
finite Decimal
to
a
per-*
Expreflion.
in Arithmetical
Managennent of infinite Decimals
Operations, by Addition, SabftraAion, "c. the
fhiUis pubDo"or
Yc^ars after) in which
^and many
to
as
the
liflied his
in the Year
Hijtoryof Algebra^ which was
1685^
For
of:
the
Dr.
who
the
Hiwas
entirelyignorant
wrote
ftory of Decimals, is wholly filent therein, .not giving ib
much
as
6fr.
but
a
Hint
at
a
Method
by Approximation
made,
Improvements lately*
or
Difference,
whatfoever,
not
the
Produft,
furd
how
to
add
or
fubftraA
only. Whereas
now,
it is poflibleto give
or
Quotient of any
by
^
Roots, in Decimal
them"
the
the
Sum,
Fradions
Expreflionsmathematically
exadt.
B
As
INTRODUCTION.
ii
I intend
As
of
this Introduction
learned
Dofbor's
bb
comaim
which
to
here.firft
ilemarki
curious
to
tranfcribe
part of
that' ^
of Algebra
I^ary
Account
Time, in the M a"
titl they latelyarofe to their
Decimahi
leave
I b^
prefentPerfe"ion,
Hiftorical
an
Time
of infinite
nagcment
the
from
made
Pn^refs,
the
as
on
from
Chap. 89,
rcjpeating
i^eciitials*
The
having treated,
Dr.
of
ences,
the
of
method
Exhauftions,
that of
^f iiifinitts^
whickdepcMftiM
of infinite
the Method
(funded
DiVifion
tinued
;
This
(but
on
and
Exliauftibn^
)h
Species, is
cf
nMich
Fra"ions
to
Decimals
ikme
Quotient:
As
I
"=0,5
:
:
Y"Ofift5
^
:
:
"
.20
^S,
aso.4
be
prime
iiBterfaiinittfe
: As
1
27
Number
(fo reduced)
"than
""560,3353
1
"
*
;
:
Which
he
being firft
(or Divithan
2
"nd
compounded
of
5) the Quotient will
or
f^c.
=:o,i
"
5
Fraction
mily then, wlien (tlie
happens,
reduced to the fmalleft Terms) the Denominator
for) is com|)ouHdedof no other prime Numbers
is compounded.)
which
10
5, (of
'6at if the Denominator
1
:
4
="
and
then
in
ro
H5
-^=0,15
ctids
'
=Ot"
"
5
3
1
t
any other
Nacort
IbmetioKS
which
:
a
4
the
uniVerfai) with that (in Ktirtibers)^f reducing
X
ssO)t5
alTocf
oondnual
or
"
"
as
Chapter made the following%un^
Decimals,
circulating
upon
Divifidn
determinate
a
Arithmetic
in the above
hath
tnore
comfmon
the
and
Set*
A|)ptxMrimaktabs
the "ane
ftiticiptei)
arifingpriocipdly froAa
in Species mfinicejy
""tniftrbii of Roots
con-
Obfervations
ous
Serine
mathematical
other
among
"
"=#,"6666 ti'r.
"
=
3
37
In
i NT
In wlueh
if
Order
two
or
(as in-^
fooner, it doth
Places
as
the
are
in
}
"
,
)
#"""""""
Unici
of
SgnniQiitft
"
b"t
alwav!*
return
to
Ditj^e
(^c^ For
142^57
7
the Remainder
Repetition
a
in fo tfimf
in tht Divifor.
btgia
=99142857
"
"
circu^
vad
ntum,
fometime
:
iU
GoQcinnicy*thM
agiun
ften in
leaft
at
Numbo*
Forinftance,
dQ
before
as
fi^k Fwirc, ( mwm
more,
N:
it jiec alwtyt dm
one
not
UCTIO
D
time, the Numbers
late in the "ime
of
dioe
Caft
after fome
of
It 0
-
vifor
being
and
thereforei,
Place,
needs
7,
2,
leaft,if
at
return
before,
Time
returning as before, the
Quotient muft alfo return
The
of
Number
late, is
"
never
nwre
not
of
Part
one
:
fame
of
the Remainders
And
fame
the
muft
Remainder
Figures
or
|
feventH
in
the
fo onward.
Figurestherefore whicfc dp tjioscircy
of tlnitsin the DU
than the Number
But
one.
fuch
tinies, it is
many
Number^
or
only
fbm/e leflcr Number
an
alj-
whidi
L
."
know
to
firft reduced
when
its
to
( reduced
"tf that
tliat in the
it
aliquotpa^rtof if. '^
ah
And
lefi than
So
Figure
and
;
:
always be
r
vifi"r, wanting
quot
6
or
3, 4, ^
not
fecond
a
mvft
this
happens, die
fmalle^ T^^^.^" ^^
Fradion
^^^
l^eing
DenominiLtor
) Fraftion, being farther re(3uced,by dir
^ (.the Comtpicnte of 19 ) as .o(^aj^
If then
it come
it can
to be 9^'9^, !rao eff. ( confijpbing
:
only of the Figure 9 repeated,) bf an aliquotpsn ofliicS
the Fjgure| of 0 in fuch Nufn-*
as arc
^timber v Sq many
viding
it
ber which
by
2
and
firftoc^urreti?,
fo many
are
thejPiguresof 'fich
Circulation.
Thus,
be
if the
Divifor
9,3,6 (=2x3,)
the Circulation
is of
12
or
Denominator
(=2x2x3,)
fingleFigures *,
B
2
of the Fraction
15
"5fr.
(=5x3,)
becaiife in 9,
that
Fi-
gure
INTRODUCTION.
W
is but
gure
and
6^12,
( the
written
once
15
3
an
^
;
5,
".
of
Components
"
is
aliquot Part of
by Multiplicationsof 3 by 2, or
made
are
and
;
)
10.
"
.....
-
If99"""2^
"r.
by
of
one
them
diough
pertain to
by
or
a,
thefe alfo be
of
1
(
25
of 12,
of
an
times
;
or, is
written
If
of
(which
21,
times
is not
of 3x7
Compound
of
(which
will
is
6
an
;
with
doth
13
6
for
( the
accurately
wherein
is fix
9
deligned by
Or
a
3
Number:
) it is
aliquotPart
an
whereof
one
it is of
;
999999,
Number
prime
a
places ; but
aliquot Part
terminate
3
Circulation
the
repeated.)
yet is not
divides
999999.
it
) becaufe
i^
"
only
not
(=2x27,)
the
1
3,
fingleFigures. )
999,27,54
If
or
thefe afu
admit
by
) ^^
Figures.
Figures, ( which
becaufe
) be
9,
99 ) becaufe
even
aliquot part of
( but not of any
fewer
9
( but
" ~^^37
is 13
mention
therefore
and
;
reduced
three
which
divide,
Figure
( fo
CX27, ) 27t74"
=r
like realon,'
is
half
Rank
of
is denoted
99
aliquot Parts of
Multiplicaliopsof
here
not
aliquotParts
by Couples,
Divifor
If the
( tdo
:
(=2X3^,j
becaufe
are
ii,33"
made
by
iare
5
the former
Circulation
and
^
66
1")
Figures,
two
"r.
22^55,66,
and
;
written
twice
9
(=5^1
33,55
is of
Circulation
the
99
(=2x11,)
of
of
20:=
2
becaufe
thus,
of
1
i,)
"
is
21
7 requires(as before)
Circulation
only of i
Hx
a
a
culation
Cir-
place,
6,*) this ((ixtimes repeated)
Revolution
of
6
places.
r
"
the like
And
Ciro^Iadon of
Circulations
6, ) three 6f thelc
Ni^mbcr^y,,which
the
So
259=7x37
which
3 places (
of
;
is of
becaufe
is alfo
this, will end
like in.other
becaufe
9f77r=7xii;
places, ( which
2
an
6
an
will terminate
with
for
Qne
places.
requiresbut
aliquot Part of
with
requiring but a.
aliquot part pjf
1 1
is alfo
37
One
Circulation
a
6
;
of that for
)
7.
two
of
lations
CircuAnd
the
Cafes.
.-t
-.
.
,
^
'
\
'But
1
INTRODUCTION.
VI
therefore
not
Fradions
;
fingty,tillthe 3d Place of
Figures, of
fignificant
operate
all the
when
So
Quotient are fpeht.
of
15=5x3,)
"
'*
i)
28 "fr.
And
^-sMs
) becaufe I do
confidered
But
the
and
~tH
by
more
0,8035714285714
=
on
particularly
remember
not
to
found
it fi"
other.
any
) which thus appears
Divifion, ( the fame
the Dr.
Concmnity ( continues
of
a
Quotient
again returning in a continual
be
this, ( fays the
that 1 have
in the interminate
Numbers
not
is, (becanfe of
That
the like in other Gifes.
infilled the
I have
Dr.
And
3f^
11=0,803571428571428
fc?f.
5^=8*7
firft
^r.
=0,13333
5
the
"^. that is (becauie
0,1 3333
And
=ro,4.
"
"
Decimal
expeded
in
like Manner
Circulation
;
is
)
of
the Extra6tion
in
Roots, ( (quare,cubic,or of higher Powers.
the
bird
Koot
Decimal
Uke
as
may
of the numeral
Recurrence
As
we
) For though
by Approximation in
continued
Yet
Farts, infiBitely
:
iin Divifion
that
be
have
we
Figuresin the
Which
14^1 3564-*
yet hinders
admitted
Ax)proximation may be Cifely
v[^=i"4
this
and
",
0,33333
if fo
C^r.
to equal"
infinitely,
3
Thus
a"y
Men
fame
ends,
on
in las Time.
began
to
the
Order"
had.
continued,
fuppofedinfinitely
pofed to equalthe Root of that furd Number
tice
therein
not
this
Subject,as
not
but
in prac^
mult be liip*
;
as
truly as
.
"
great
a
Mathematician
as
J doi^iw^
4Jpon wbpfe cS^feRvsitions,
and at
fpeculate,
laft
to
contrive
thod
Me-
a
to apply the Dodruie
( unthousht of then ) how
Circulates to Arithmetical
Operations.
N.
1^
A
i
INTRODUCTION.
When
r. B.
I
"
to
come
diis Book
in
treat
concerning
and
whether
of Roots,
Evolution, or die Eztraftion
of iquarc, cubic, or of higher Powers, I (hall there
txhibk
ExampIcs,^ feomttiKlyimdonal
Involution,
will have,
which
rtrarnittg
in
in their
a
ftall attb mOkt
be
the iame
Kooc^
Divifion.
a
it evident,
C^eampUs might
fim,
of
that
Root
triM
and
be
contft
as
Number
isomI
of
of inch
I
ienfiblc
any
any
rtvf
am
Examples fhould
Ihall happen, its
fuch
iiich
madieaMtical
a
I
Place
infinite Number
Expicffion, as
an
agam
that
m
\ though
prodisced
confift
then
Numbers,
Numbers
And
an
iUdom
it may
be but
diat
in FiraAice, yet whenetw
(Aoor
iurd
or
Circulation, as a^ppearsin the in-
continual
Quotient
cenniaate
"
"It
Anfwer^
exaft
of any
is the Root
nu
whatever^
"
The
Aiithor which
next
fionaUy
the
xiataon
fame
Subjed
in bis Sympfit Pidmarionm
Year
1706, pag. 104,
IngmiouB Mr. Jcm^ who
#"^, publiflicdin the
concifeiy,and diat but
had largelyexplained before.
thors,
Guide^
ders
themfelves
xxmtent
that
fo much
as
intimate
Numbers
id
page
only
Numbers
fome
Part
a
will
with
of
and
^,
informing
but
arithmetical
cites
re-
105,
Dr.
the
^arJ^
Mr.
ciioilaae,
Ma^
what
his
in
other
An*
"hetr
Rea"-
none
of applying
Poflibility
a
any
txx",
Gentleman,
MtUbiiMiHcuufs
dkn^
ung
"f
above
the
fince
And
and occaHands,
my
the
with Dr. WdtUs^
was
to
came
of
fuch
them
circula-
Operatioos,but by
way
Approximation only.
fiift Author
The
can
learn )
Numbers
Mr.
inthe
who^ippear'd
in hts
Bot
1709:
tip ; for aftsr
rim Kingdom,
J
could
muft
I
a
very
and
not
a
^
as
I
and^appliedcinrukting
the
Reverend
Operations, was
4f Decimml
I
Ymsr
or
Book,
S^^m
for
in fViUic,
arithmetical
to
ArMM,
followingSubjeft," fe
the
on
krilkmsdcjpublifiied
liippofeabout
leave
its awe
Dale
the
Years
for others
lyolt,
to
fill
careful
of
Cn"|ufry in various Pam
for a Sight of that
fong Expe"atkm
Wheve*
fo h^py
it.
fs w" obcam
fore
viii
f
N
I
R
O
Readers
be
muft
my
Performance
this Author's
fore
and
Cunnj
from
Mr.
D
I
10
N.
with
fuch Informations
ihall
tranfcribe
content
as
T
C
U
fiom
of
Mr.
Malcolm.
m
The
former
Method
this
though
of
ceflary to be known,
fufficientlyhandled
Mr.
verend
fuch
manages
interminate
continuallyrepeated
Fa"tors
will
as
(being,
as
much
to
as
I
remarks
this. That
the
Re*
Arithmetic^
of Decimal
S^em
hadi
exttuit
fingle Digit
in Multiplicationuieth only fuch
fingleRepetend in the Produftg
a
as
Mr.
have
a
unwilling ib
Cunn^
and
Repetitions )
compound
PraAitioner
work
to
without
in Divi-
"xaftnc"
5, 6.
Preface^ page
latter Gentleman,
The
abfolulely ne*
hitherto
fuppofe, continues
the
Cuntfs
Vide
produce
is
Treatife
Decimals
but
y
mention
leaves
fion
his
in
Brovon^
FradUons
ufing
yet no
it : And
Preface^ obfenres, tha^
his
in
Gendeman,
Preface, remarks. That
his
in
probably the firft,as he iias himfelf obJenred,
"x)nfidered this curious SubjeA of circuhas diftindtly
lating
has
He
Decimals.
) given us
( fays Mr. Malcolm
of
but
without
fundamental
Demonftratiit,
Theory
fFaUis
Dr.
who
the
is
has
nor
;
of
mianaging
And
Mr.
but
meddled
he
on
the
infinite Decimals
fingleCafe
of
Vide
pleatlyneither.
pradlical Part,
Way
Operations.
in arithmetical
in his Decimal
Brown^
one
with
Arithmetic,
Pra"ice,
the
MalcoMs
and
or
handled
has
that
Preface^ page
not
com*
ii"
"
in my
However,
brings
no
fmall
probably
for
The
the
Arithmetic
of ibe DoSrine
7 14.
curious
1
to
Wherein
mation
firft,though little.Inti-
that
to
circulatingNumbers*
of
Reverend
his Peiformance,
was
it
apply
firft Book
Ingenious Mr.
\
Reputation
it
thinking how
Arithmetic
the
towards
this
Opinion
that
of
entitled,
of FraHions^
he
hath
Escamplesy both
A
which
Treatife
a
new
firft
exhibited
in
Hands,
to
my
Circulates, is
Cunn^
that fet others
;
upon
univerfally.
more
came
Gentleman
and
treats
of
by the
wrote
compleat Treatife
publiihed
in
the
the Arithmetic
of
fingleand compound
Year
many
circulating
Decimals,
INTRODUCTION.
Decimalsy
mixc
and
pure
ix
being
;
the firftgreat
Work
on
Subjeft.
the
this Gentleman
had
And
faid he
firft
ftrations
what
done,
defigned to do,
his Examples, I
viz.
in
have
to
Preface
his
he
Demon*
given
perfuaded we (hould then
have had no
Book
this Subjed very foon, or at leaft
new
on
for one:
had
occaGon
But, unhappy for the young
no
! He
Learner
wrote
concifely, and in a way not
very
to be comprehended
cafily
by any. Nay, the great Mr. Maito
colntyin the Preface, page
am
his
to
1 1
.
(in my
parable
incom-
Opinion)
NewSyftetn of Arithmetic ^^xMxfhtd in the year 1730,
does
ftick
not
his Rules
of
view
this further
Rules,
be
indebted
it muft
qualifiedto
have
is evident
from
I
it coft
frankly own
never
them.
or
very
However,
in
himfelf
to
ufeful Hints.
Cunn
that Mr.
me
Reafons
the
let Mr.
not
Mr.
Fields^
tributed
That
ferved
delivering
was
every
way
SubjeA in a. clearer Light,
curious
his many
Examples } but what
fay.
doing it I cannot
Subjed ;
acknowledge, in
Friend
the
of
Way
acknowledges
two
or
out
fet his whole
this valuable
to
one
far
as
adds, I muft
could
one
of
he
exprefs
Co
litde Pains, fbme
no
of
the feveral
Years
Methods
made
by this Gentleman,
I muft
as
C""/r's
Malcolm
be allowed,
difcover
to
ago,
ufe of
Mr.
Reafon
the
for
from
prevented him
And
by
him
to
little lower
a
themfelvcs
Preface, Mr.
fame
Indeed
as
And
chofen
has
fet the Reafon
to
as
Effcdlof
into
led
be
hardly
the
That
Cunn
Mr.
manner,
a
poffible.
as
obfcrve
thefe
fuch
in
that
fay,
to
no
his
and
Hints
his
the Memories
for
thefe
the fole
Improver of
He
is (b ingenuous and grateful
:
Preface, that he cannot
forgethis
Difcovery
Figures.
to
pafs, as
Ftavellj Schoolmafter
Robert
whole
Cunn
their
of
of
in
St. Gile/s
particularMethods
the
the above
Gentlemen
joint Labours,
C
and
Nature
and
Laws
may
remain
had
of
in
con"*
culating
cir-
be prein
high
Efteeoi
INTRGjpiUCriON.
X
Eftecm
the
to
I
J
Mankind^
among
ufeful
be
Admirers.
For
of
the
^nce
with
Error
brought
Mr.
in
"
*'
"
**
**
"
"
came
the
which
are
the
of
are
C"if"*s
Mr.
by themfelves
*"
into
"
I
the
:
lieved
;
fo that
Rules
the
BoNC^.i
which
not
of
of them,
Speculationsmade
.may
the
Words
and
:
Divifion,
(hould
have
therefore
Vid"
I
"
Effeft
Rules,
That
hardly, be led
pv
very
confequently into the Way
it will be the
more
eafily benor
given,
for this Reafon
faid )
"
upon
might otherwife
I
this
be
faw
this
I acknow-
Improver
Pradice,
fuppofed
"
o"
only, that I
whom
C^^e fyfl: great
Effe"l
the
are
before
this Siibje6l,
bethought ungrateful-toone
"
*"
thefe
never,
I have
upon
this further
obferve
delivering
I-.mention
ledge the. firft Ai^thor
whom
I muft
could
one
Reafon
that
But
Way
chofefl
have
Malcolm
**
either of
own
Vulgar Fradion
equal to )
for
though the Demonftraought to be as fimple and
Rule
the
omitted,
""
"'
from
in his
of ( or
finding the finite Value
circulatingDecimal
any
eafy as pofSbJe.
***
that
Infinite Decimals
more
^'
""
Book
next
wherein
;
Multiplication
.
"
one
complex and difficult Parts) Mr.
fo eafilyfollowed
beDirections
Cunn's
are
not
are
j and
for the Memory
than
I
harder
the Method
fides much
have
chofen^ which
depends all upon the eafy and natural Explication of one
fingle Propofition; viz. the
(which
tions
"
fee Im-
proceed.
to
the
of
take
deals
tic
Arithme-
But
my
Do"lrine
different
of
Rules
Decimal
now
Hands
whole
fomewhat
"
**
cimal
De-
Science,
recited, was
to
th^
treaty
in
That
metic
Arith-
he
daily
not
and
Perfe(5):iont
Syftem, above
before- mentioned,
Authors
the
when
too,
Art
every
its utmoft
manner
a
tQ
of their
underftand
to
did
one
affirm, that
Subjefb, that
great Author
^^
almoft
to
U"
MalcoMs
this
on
if
And
*"
in
made
iliould be tempted
was
faid
confiderable
very
large N^imbers
provementf
ignorant of
be
can
continue
one
perfedly well ; becaufe without its Afliftof his Operations mull generally be imper-
Refult
the
is
who
Decimals,
Infinite
Arithmetic
fe"5l,and
Perfon,
no
long as Numbers
of
hearty Wifh
as
is the
World,
Mr.
from
to
have
"
th. Prefaa.
"
bor-
^"
borrowed
^"
knowledge
Malcolm*
I
all that
deduced
or
him
owe
or
one
JPrefacey
page
s
criO
DU
RO
NT
I
fay ;
yet I do acufeful Hints."
Vide
I
two
are
we
and
12.
greatlyobliged to this
Proponcion, widi many others
Indeed
xi
N.
for fet-
Gentleman
SubjeA, in
will carefullylook
into
fo clear a
Light ; but whoever
Mr.
Cum*s
Examples, both in Multiplicationand Divithat Propofitionalfb^
lion, will find that he well underftbod
remarks
or
explains it. I
though he no where particularly
muft
confels 'tis hard, nay almoft
impoflible,for a young
find out
ufe
Learner
to
why or wherefore Mr. Cunn made
of other
adequate figural Exprefllons inftead of thofe b^
firft propofed.
But I ihall in fome
places in this Tfa"fc
ting that
have
occafion
an
laft Gentlemen
two
who
Author
The
wrote
this
on
Aberdeen
at
j"ro"eedswholly
remark
to
of
tor
his
be careful
what
wrote
Book
and
Book,
the neitt
to
now
Foot
before
(for the Sake
amifs
in Chap.
but
d"
pafics into
of
the
Arithnietit
o^As
fi'eely
that he
MalcolrH^ I hate nothing
of Ml*.
;
it
he therein
as
(rfthe
treats
1734,
young
Iiere
ad^ife
fecond
a
Learner
the
Edi-^
Edition,
)
to
to
toitedb
ocightto bd all nev(^
anil th"J Isriflt.
Th6
except its Rule, the firft Eiiample
in general well anfwers
itt Title, w^
Aphin^ eaff^
PraHkalr
Fraffidns, h"tb Vulvar and
compkat Syftem.of
is
Decimal
cbe
Year
his Method
on
proceed
Subjeft, viz.
But
the
on
(hall
thoie
on
ff^tigbtjA. M.
Writinglikewife in his Treatijeof Frac*
who
;
Decimals.
Infinite
Remarks
more
Aleicander
tionsy publifliedin the
of
few
a
therefore
5
ingenious
Mafter
make
to
this
on
\
and
I
Snbjeft of
The
next
Hands,
was
know
Fractions
Book
a
on
very
xvii.
if there
not
of
whidi
be
t
better
Bek)k
Oil
its ^itt.
this*
laborious
which
SofbjeA,
aftd
curibus
came*
to
t^f
Performanc"i
of
of Cbicbefter^
erttkled, A ne^y
Benjamin Martin
com*
and
pUaty
univerfalSyftem or B^dy of Decimal Arithmetic^
fo
late
Where
the Author
in his Preface,
printed
as
1735.
of lufinite Decimals,
%caking of his Management
lays.
Mr,
The
Foundatibn*
on
whict
I have
C
2
bailc diis
Soperftruftune
if
INTRODUCTION.
xii
is Mr.
But
that
and
it
CirculatingKumbers
having
great Mailer
Decimal
of
fmall, but learned^ Treatifc of the Doftrine
C"""*s
in
great Meafure
be of ^rvice
a
might
clofe and
lay it
of
out
laid
the
this
to
Remark.
Foundation
the
vulgar Ken,
Students^
young
to their View.
open
more
with
aod
;
deep,
I
thought
litde
a
dif-
to
had therein
heartilywiih that Mr. Alartin
this Subject, for the SAke
been
of the
more
copious on
Student.
In my
than
Opinion no body better qualified
young
made
it exceeding eafy and
familiar to
himfelf, to have
I could
And
of
this attempt
In which, how
rity:
to
Capacity
common
every
others
of
I have
Not
Decimals
that
in all bability
proftill in Obfcu-
fuccecded,
but
be
far,
I
I take
muft
leave
this Gentleman's
far, the beft
very
publifhed, or
was
ever
done,
remained
that
altogether, to
Book,
Syftem
well
he
had
had
mine
determine^
to
which
*,
perhaps
fee
expefl to
; containing their
Arithmeticallyor Geometricallyto
Applications whether
in the various
the beft Advantage
all ufeful Knowledge
to
Arts, Trades, and Bufincfs of Life : In (hort, it is impofof this his truly
in Commendation
fible to
fay too much
that this
End
the
The
of
hope
can
curious
and
new
Age
Syftem
this Book
laft Book
Subjed,
is Mr.
or
I
have
wherein
Vulgar and
fiiUyexplained;
with
manner
is
of
valuable
a
whole
Decimal,
both
Method
of
DoArines
in which
and
managing
Mr.
and
in
Numbers
fet in
in
I wilh
a
feen
at
the four
this
on
compendiousSyftem
1738;
Fractions,
clear Light, and
year
and
followed
he hath
its
treats
the
Infinite Decimals,
Martin^
Book,
are
and
new
printed
of praSical Arithmetic^
the
feen, that
PardonV
William
be
it is advertifed.
where
which
will beft
its Contents
:
Mr.
Ckn9f%
after
much
primary Rules.
ingeniousAuthor
the
It
a
ftii-
Pains.
But I b^
Encouragement for his uncommon
leave in this Place, with all due Refpeft to the Author, to
table
point out
of
to
his Book,
him
an
page
hafiyor
171.
inaccurate
where
Aflertion
treatingof
the
vulgar Fra"bions, in their producing
in their Quotients^ he
repeating Decimals
fome
in the
Body
Property cwf
fuch
there
and
fuch
affirms.
That
INTRODUCTION.
"iv
And
h^ljj
I
tife
prevailed upon
Subjeft, after fo many
I differ from
As
to
itfelf
on
metical
them
in the
firft, I thought
the
Subjedt,
be
fiich Students
cially to
by them
learned
think
feveral
the
already on
him
fome
a
count,
Ac-
Trea-
new
Authors.
And
particularlywherein
thereof.
entirely by
wrote
being mixt with other
more
acceptable Book,
the
who
write
Treatife
a
without
would
Rules,
to
Management
the
might reafona^
give him
me
alfo inform
I fhould
That
idly,
this Place
in
ft. What
this
on
I fhould
that
bly expeft
Reader
Tfie
conclude
to
Books
they have
Branches
common
arith*
efpeenough
of Arithmetic.
"
And
and
I
hope
ferve
as
Key
a
here, which
aflure
that
Title
Page,
Book
declare
Uiefulnefs
hh
for
me
:
prove
your
In
the fecond
well
*Tis
but
larger
Service
be
to
Ihould
how
which,
will
heartilyat
Book
my
Subjeft.
I have
that
Reader,
my
the
as
found.
made
every
I have
a
well
it my
its Bulk
as
who
Place,
things
laftly,I
utmoft
anfwer
performed,
little one,
than
many
And
way
that
Authors
with
meet
will
Difliculties
of
will
myfelf,
mine
any
elfe
where
no
in
Reader
my
are
this
that
feeming
the
with
on
me
the whole,
all
meet
I flatter
Foundation,
fome
open
may
before
gone
upon
here
to
Learners
young
hav^
without
not
firft Place
In Ae
the fecond.
to
as
1
yet
fuch
;
deavour
Ento
let
hope
as
my
School
in
Santm,
its Author.
s^^JLrw..740.
"
John
\/r
Marsh.
DECl-
my-
its.
it is,
9
From
its
(O
DECIMAL
ARITHMETIC
MADE
SPr.
PERFECT,
^m^t^titmm^ti^mm^it^iJmmmmimtitUL^A^thmt^am^^mmmm^mmm^m^
I.
CHAP.
T
A
WH
preflionis, hath been
going
other
for
Infinite, or
an
Definitions
the
already (hewn
Introdudtion.
and
And
Managcmeht
each
the
DifiTerence, or
or
they occur
Figures i. 2. 3.
( as
which
will foon
any
or
The
of
thefe, and
we
Produ^,
are
The
4. 5. Csfr.
be
all other
many
known
kinds
readily give
can
of
their
jQuetient ; I (hall
or
of
in Point
Pkce,
Ufe
Order
or
and
to
) prefix
of
Advantage
appear.
Definitionsand
1.
there
as
Ex-
in the fore*
Propofitionsneceflaryto
of
circulating
Exprefiions,before
Sum,
Decimal
Circulating
Propofitions.
Figures) continually repeating in
Numerical
late,
Expreffion, is called a Repetend, or Circuthe firftFigure
) And
( for they are fynonimous Terms.
Repetend,
(or Figures) of either is called the Given
Figure, (of
Circulate.
Repetends
2.
3.
A
are
either
Single or Compound*
fingleRepetend i^ that,
wliich
confifts of
GV.
7777
in the Qiotierit
Or
:
4444
repeat infinkeljr
in
the
would
Quotient:
infinitely
repeat
continuallycirculating:
where
8 would
repeat
As
one
gure
Fi-
where
7 would
(^c. where
4
Qr
^c.
8^8-8
in the (Quotient.
infinitely
4,
A
or
compound
A
4.
of
353535
repeat
either
Pure
6.
O""
1958158
1
belong
i^c.
nitely
infi-
repeat
in
Integral
Or
Repetend,
13587
general
are
,007474
6rr.
Ot
,0303
"5?r. Or
,0036700367
Numbers
Or
only
Point.
are
Aj
6?^.
486,486486,
i^c.
,370370
have
or
Decimal
Or
6?r. Or
concerned.
6fr.
Or,i2i95i2i95
(^c.
,00384615384615
6?^.
Decimal
where
Or
Places
concerned.
Repetends
Mixt
7.
tient
Quo-
Figure
fignificant
no
the
^^-
,0666
are
the
to
and
them
where
have
as
Or
only
As
Quotient.
iuch
45t4545
"r.
5
,3636
Oras
the
Csfr. where
1358713587
the
are
what
betwixt
^^-
3"333
8
but
o*s
or
o
a
would
007
circulating Expreflions
or
Repetends
Figures,
or
two
mixt.
or
pure
infinitely in
repeat
where
Or
infinitelyin
Repetends,
5.
(^c.
007007007
in the Quotient:
would
would
35
Or
:
of
Figures continually circulating:
places
^^' where
more
confifts
that, which
is
Repetcnd
fuch
are
as
have
Figures prefixt before the
exhibited
of which
are
Examples
fbme
gure
fignificantFi-
Circulation
or
in
the
begins ; the
three following
Cafes.
I.
CASE
Where
Examples^
Circulation
Thus
begins
;
"?c.
or
,5333
only
are
,263434
Decimal
(Sc.
CASE
or
Places
before
,62057845784
the
Gfr,
II.
"
Examples^
Decimsd
and
Thus,
Or
tfr.
are
Places,
before
36,777
2,4777
or
Where
iSc.
6fr.
or
159,10695757
either
or
the
Integral, or
Circulation
3,842842
52,38444
both
Integral
begins ;
"fr.
6ff.
or
10,5473587358
i^c.
CASE
(3)
CASE
III,
the Repetends begin
Examples^ Where
Part with Integralsbefore them.
Thus
"?r.
57,777
Parts,
viz.
As
,26
form
Repetend,
Finite
a
in the
,5 the
or
Mixt
Every
8.
and
and
of
Integral
6547*4^747 ^^"
or
a
Parts
make
their fevend
following manner
are
,03
commodi-
moft
are
the
to
Circulating Parts
which
the
of
Parts
Finite
federal
oufly diftinguiihedafter
two
:
concerned
not
are
the
Both
,0005784.
CirculatingPart
Cafe I. preceding. There
are
their
and
Circulate, confifts of
or
which
the ,620
Circulate,
Repetends;
9O034
Fart
Examples
the
fc?r.
329,494
the
"c.
87444,444
or
or
in
;
viz.
1 03
26 -J0034
5-
{^0005784
620
Obfcrve
as
to
there
prefix
Point.
For
in the
the
36, the
159,1069,
their
^007358
Both
after the
the
the Finite
feveral
and
which
and
2,4
the
Given
it and
late,
Circuthe
cimal
De-
is manifeft.
,2634
,26 l-,oo34"
of
Examples
3,
are
5
before
Figures betwixt
of
Reafon
The
j5-1-j03=,53
Again,
And
Places
are
o's
many
as
the
(^c.
Cafi II. preceding : Ther6
^^^
the
52,38 the 10,54
of
Parts
their feveral
Circulating Parts
are
Repetends:
,7 ,842 ,07 ,004
,000057.
Parts
are
moft
following manner
36-1-7
3-[.
842
24
5238
commodloufly diftinguifhed
viz.
i
"
f
\ 07
I 004
1054-007358
Here
being
no
Decimal
Places before their Circulations
are
no
begin, therefore
o's
there
prefixt to their
Pares*
given Circulating
1591069-1-000057
NoU^
(4)
"
Notiy
of
Number
The
fitly (hew
kte, moft
Given
of Decimal
Number
the
each
prefixtto
o*s
Circa-
Places
in
Expreflion6201- 0005784
And
note
de,620 l-,ooo5784=,6205784.
24-! 07
their feveral Finite Parrs
denote
As
:
the
And
2,4-|-,07=2,47.
i59io69-[-oooo57
denoD^
159,106957 fcff.
15931069-1-,000057=
"3)
lailly,ii^
And
the
Cafe III. preceding :
of
Examples
moft
Finite and
Circulating Parts are
after the following manner
diftinguilhed
Their
As
329,494
6547,47
fc?r.
thus
tfr.
thus
Here
concerned
not
feveral
7,0.
to
Finite
7000-I-945000.
the
57000.
their
form
or
their feveral
and
\
5
Integral Numbrrs^
Circulates, are
their
Parts
Circulating
are
945,000.
in
muft
it, care
be
taken
that
Single or Compound
Given
Places
the
Parts
6500-I 4700
Examples of this laft Form, where the
fore
begins in the Integral Part, with Integralsbe-
Note^ That
each
make
47,00.
9,40.
Repetend
6500.
320.
50.
I 940
320
^^^^
6?^-
57945*945
viz.
diftinguilhedthus 50-|-70
fc?f. is
57,77
;
commodioufly
in
concerned
the
we
annex
as
many
o's
ta
tegral
Repetend, as there are InCirculacingPart preceding
Point.
Decimal
tht Finke
md
Neceffity of righdy diftinguillihig
Cir":ulat",wilt
CirculatingParts, as above, of any Mixt
and by"
appear by
The
I
am
very
ia the laft
xs^a
fenfible that the
Form,
the
with
wouJd
the
not
two
might
Dtcimal
Ptirts
Circulatinsg
Repetends, which
hy TransfomHUioH be
Point
and
then
made
their
be
perplex tlie yowg
together, therefore I omit
Loaroer
with
teo
occur
gin
be-
to
Finite
diftinguiAed "fter the
above.
fiicasRplesin CafiB.ll.
wookl
firft
;
imy
many
and
manner
But
Rules
it.
And
I
(5)
And
avoid
to
the Given
down
Cbmpoand,
Repetend
thaii
more
^ke )
Uuftration
TroaMe
the
for
the
future of
Circulate, whether
or
iometimes
( except
once
(hail henceforward
we
for
II-
each
diftinguifli
by
placing a Period over the firft Figure, or
laftFigures of the given Repetetxd.
the
As
7
and
s
thus
8
Expreffion 7777
the
as
i
"r.
thus
diftinguiflied
thus
"r.
Expreflion4444
be
wiU
the
"s?r. thus
4
and
1
"^.
88SS
fingleRepetends.
^f'
Expreflion 3S35i5
G07
and
;
thus
Proceed
we
"
35
"^r.
13587x3587
Repetends.
Compound
the firft and
over
"
And
writing
Single or
j
Md
thus
0(^^607
13587
5
as
now,
"
9.
Finite
the
find
To
of any
Value
preflion
CirculatingEx-
:
OR,
How
to
or
or
find
a
Vulgar
Frafkion
petend
equivalent to any ReSingle or Cctnpdlind, Pure
Circulate, whether
Mist.
C
Of
jt
Pure
S
L
B
Circulates.
RULE.
"
TRThen
t^
it is
equal to
given Circulate
as
as
Hs
there
are
Dedivm)
Expreflion
a
a
Vulgar Fradlion
is
^
Places
o*s anneatt
many
D^imal
PoioL
and
of
as
whofe
in the
Figures
there happen
z
to
Cifciiltfe^
then
Numerator
wiH
its Denominator
D
Pure
gvfvn
be
as
is the
many
9*5
Circulate, with
be o's betwixt
it and
ExampUU
(6)
Examples.
(0
^'
=-| (3)
05
.
(4) "oo8
^9) 0053=^.CO)
'053=fo-
75^436
^nj
=
|-.
(,2) ,03=
^
3367
.
.0745
,'003367
V
,
.
.345
'
'
"
r^
(14)
=
99
9999990000
=
^.
=
And
(")
"
,5
.000;=^-^. (6) ,57=1(7)
(5)
gf
=
i. (2)
,3=
(15)
0001=:
.
9999
999999
II.
CASE
Of
Circulates.
Pure
RULE.
of
Expreffion is a Pure Circulate confifting
Integral Figures, then its Finite Value" or Equivalent
be
its Numerator
to
Vulgar Fraftion, is found by making
the
When
are
o's anncxt
many
Figuresin the Given
Circulate, with
the Given
IntegralPfeces
of
will
its Denominator
to
as
be
Circulate
and
j
hath
9's as the Circulate
many
as
it, as there
places of Figures.
Examples.
99
99
94.000
343^=*^-
(4)
*,
'
(5)46.58="
.
^
99
465800
"
"
,,.
".(6)347.347
99999
999
435782000000
^
'
999999
KB,
(8)
f
"*"
(2) "475
'
^
o
_
.348573+
,
fir. (3)
a?53^^ its E.
34^'"9^!"-\0005734
"
"
"
"
99E
ggo
"
^"^'
"'
'
=
"=
"
-
"^^
S. F.
5999000
(4) 36,7
Be,
?^^
=
^^' E.
its
==
5. F.
i.
E-
..
quivalcntSingle Fraaion*
"
(6) 20.046
fif.
'"^*
=
^^o
"^^^
8347 X999-V 00046
"*.
its E.
8 3
1
1
9
1
34
__
(7) 8.3"740"
=
^9^000
99SOOO
S. F.
II.
CASE
Of
Circulates.
ASxt
RULE.
'When
the
Expreffionit
a
Mizt
Circulate, which
Integral Place, then its Finite Value,
in this manner
is foand
fcnt Vulgar Fraftion,
and
down
its Finite Part, (found by Jtt. 8. )
Figures
9^*,as there are Wace"of
by as
or
in fomc
:
begtot
Ecpiirar
Firft
fet
multiply
it
in the Qprca
many
Rcpctend
Part,
of
the
j
( found
the Produft
to
as
above
) and
icquiz"d Fn^n.
of which
that
And
Sum
add
its
Circulating
(hall be the
for its Deoom'.
Num'.
uke
as
*many
(9j
g\
many
as
the
Given
this Fra"ion
and
Ex.
(0
Circulate
ihall be
its
hath
Places
Figures i
Equivalent Single Fra"ion.
50^J:r
5'7^77
^c.
of
=
5"". its E. S. F.
=
9
9
JE.S.F.
".
(3) W5.8475"*
".
13Z51!S?
=
its E.
""2^^"2^i"
=
S. F.
9999
"
"
-"*"
(4)
g^^^
*"""
57945*945
=*
'
"
'
S. F"
its E.
999
;r*
"Jt.
its E.
^-
^r'k
(5)
"7"x"*"9-V4""o
"^:.^
07444*44
^^
""V
V^*
-^
'''^
I-
I
m*
""
787000
^
11
I
i"
S. F.
(6) 5794^7^94^
^5794100000.^^3
^"^'=
^-
!^^ ^-^^^^^^^
p.
9^99
"
(
In
thefe and
the
foregoingCafes
t
"
liave exhibited
Eit-
that cah, I think, pof*
all the Varieties
amples, by which
their
fibly happen in Pradlice, may
readily be reduced
to
Equivaknt
Vulgar Fraftions,
172"r
(
Tie
And
is
to
eqaal
divide
turns
to
Proof
am
I chofe
felves,
to
a
Pure,
fendble
to
fet
fear the
for
I Ihould
But
a
Vulgar Expreflion.
Vulgar
Equivalent Expreffions ;
lower
down
firft offered
they
as
Operations (hould
now
proceed
fliew
to
Expeditious Method,
Vulgar
have
too
appear
them*
complex
Pra"itioner.
young
more
Quotient
of the above-found
that many
them
if the
then
Circulate, you
Mixt
the exa"t
to
muft
Circulate, you
and
or
have
reduce
would
Mixt
or
founds
Fradllon
Equivalent Vulgar
its Denom'.
by
that you
very
Proof.
Pure,
Given
the
out
Fradions
but
Given
its
its Num'.
certain
I
Chat the
prove
)
*o
FraAion
to
how
Mixt
any
a
eafy,
more
to
Grculate
find
as
well
as
a
Equivalent
the
Expreflion.
previoufly to this, it is neceflary
that
ihew
I here
a
Method.
lo.
I
ft. How
Number
of
to
9*s in
a
multiply
very
any
5674
be
given
be
to
Number
by
any
Compals.
narrow
Example
A8" kt
Given
i"
multiplied by
99.
Operation.
Here
Therefore
567400
Subft.
will leave
5674
is
100
the
times
5674
given Multiplicand,
and
it
561725=5674x99.
Example
)
(""
Example
let
Again,
be
715892
2.
gi^en
be
to
multipliedby 9999*
Operation.
Here
Therefore
Subft.
will leave
7158204108
be
475050
it
715892x9999.
=
given
715892
given Multiplicand, and
the
715892
Example
L"t
times
is ioooo
7i5892oocx"
be
to
3.
multipliedby
999999.
Operation.
Here
is
475050000000
Therefore
Subft.
the
475050
will leave 47505952495"
From
that
to
of
it
given Multiplifand, and
47505o'"999999-
=
preceding Examples it is very manifeft,
ber
multiply any Numerical
Expreffion by what Numnexing
9's you pleafe,'cis moft readily done : ift. By anof
then,
9*s, and
from
given Multiplicand, their
Produft
required.
the exadt
if either,
that
Obferve,
Decimal,
perationwould
or
have
Finite Mixc
been
the
the Method
of
Increafed,
will
Difference
the Fadors
had
Expreffion, yet
iame;
off the Fraftional
mark'd
to
a
both
or
Part
tiplier
Mul-
the
as
it thus
the
fubftraf):
Finite
given Multiplicand,
the
o*s to
many
confifts
have
475050
the three
as
a
times
loooooo
be
been
the O-
only then you. muft.
in
the
Produd,
cording
ac-
Decimals.
common
(2dly,)
z
I
But
let
.
added
to
the
us
fuppoie fome
Produdt,
where
Number
the
were
required to
Multiplier confifts
of
be
9*s.
only.
"
Example
Example
Let
715892
whofe
be
%vttn
it
is
Produft
Number
fought
i"
be
multiplied by 99999"
required to add
^ere
5746:
to
to
the
?
RULE.
In fudi
fore
Cafe"
a
when
direded,
it wiU
be
Number
add
the
( which
in
this
Example
71589205746:
candy
and
from
then
Multiplicandis
the
715892O0000:
Number
is
and
) and
it will
fobftraA
its Diffcfrence
will
the
be
to
the
be^
as
thk
to
required
5746
which
inereafed
be
then
ereafed
in-
added"
become
Multipli^
given
Number
foqgbt*
Opefatnn.
71589205746
Sul^..
Multiplicand inereafed,
the
^
dire6ted
"
715892
Axifw.
given
the
Number
ExantpU
Let
whofc
875492
Produft
Number
b^
given
its
required
to
be
to
laft
%
the
71588489854
as
Multiplicand
;.
fought*.
^.
multiplied by 9999991
add
896^:47 : ^ere
t"
the
fought ?
Operation.
Subft
J ^7549^*9^547
*^
Multiplicand inereafed,
the
Number
"c.
hMAi
Afifw.
875492021055
^
"
.
,
"
,
Example
Let
Produft
57
be
It 11
fought,
-
"
given
to
be
laeqttiredto
3.
htuklplied by
add
49^00
:
to
gggg^^
S^ere
the
whofe
Nudiber
fougbs?
OperatioHi,
*i.
OperaHM.
X
""kfl.
the
"
i
57
Anfw.
the Number
5749443
If thele
Examples
then
Learner,
well
are
underftood
folIow3 in the next
what
(horter
A
fought.
how
Method,
of
fhew,
to
Mist
any
young
noeci no
wHl
find the Finite
to
Equivalent Vulgar Fradtion,
the
the
by
Artidp
Therefore I fhall proceed
lUuftrations.
12.
Multiplicandincteafqj, "!?^.
given MuUiplicaad J
tbe
5749500
^"
Value,
Circulate
or
or
Repecend.
RULE.
the Mixt
From
Circulate
before
ftand
which
its Di0ereace
Note
^
ft.If the Circulate
I
xhe Numerator,
begins next
its Difference,
muft
Denominator
Places
Figure
of
Figure or
the
with
Figures,
Circulate, and
foU
the three
Obfervations.
lowing
then
fubftraft the
firft
the
above,
as
be
as
the Decimal
below
is the
Numerator
Point,
and
;
its
9*s as the Circulate confifts of
maey
Figures.
of
ft asy PJac^, Qqt Aext^ Ibut
l^ecjmiajPlace, annef
as
many
2iflly,If the Circulate bc^os
any
o*s
to
firft
above
*
the
9's
9 or
Places of
the
for
Figures between
the
its
Denomin^or^
Decimal
Point
as
and
muft
are
-the
be
as
direAed.
annex
as
.as
l^igere of
;
Circulate
be^ns
If th^ Circulate
directed,
Point
abovefaid
as
of the Circulsittip*
lt% Numerator
Figure
3dly,
then
the
elfe, below
where
and
o*s
many
there
die
are
^iven
to
IntegralFigures,
the
found,
its Numerator
Integral
Places
between
GrcuTat^,inctufive
its Denominator
confifts of
in
Places
muft
of
E
be
as
^
many
tbC
above
as
the
firft
Decihial
9V
^s
tl^
Figures,
2
What
t '4)
What
faid above
been
hath
will be beft arorehended
the
following Examples, taken,
the
preceding Examples
Learner
how
than
of Mixt
for the
moft
part,
Circulates, that
by
from
the
tentive
at-
might eafilyfee, by comparing both
much
more
expeditious this lad Method
gether,
to-
is
that.
Examples for Obfervaticni.
Here
(i) s^y
sj
Therefore
5=52.
"
is
"
its
E.
S. F*
9
{ue.)
Equivalent
"
"
Here
(2) 100,47.
5217
Single Fradion.
10047
100
"
=
994.7.
Therefore
s. F.
j3 jjg E
.
99
"
"
Here
(3) 347.584-
347^37
i5 ij3 "
347584"
fore
There-
347=347*37-
s. F.
999
(4) 3,842.
its E.
Here
3842
"
Therefore---^is
3=3839.
S. F.
Here
(5) a7"54753-
^254753
i3 jjj ^
fore
There-
2754753"27=2754753.
s, F.
99999
1.
Examples for bhfervation
25
*
Here"7
(i) ,27.
"
2=25.
isits""S.
Therefore"
"
(a) "476E.S.
"
Here
r
475"4=471.
Therefore
F.
471
-^
is its
F.
(3)
"
*",
"i
"
r
(i8)
Examples.
7
"
8
"
.
-^-
As
rt
6
"
And
ooo8.
==,
990
9990
99000
-^
And
;oooo6.
5te
"
Anjd
,007.
And
,0035.
=
99999000
^
And
,00000475.
=
"
9990
and
,0000075843
/^
T^
=s
fo
9999990000
on,
7*8
Notey ThcExprcffion"
=
And
,075
=:"ooo8.
90
hath been affirmed in the laft three
what
Now
9000
Articles, might all
preceding
eafilyproved by Divifion in common
be
DecimaJ^ only.
other curious Remarks
which might be
many
this Subjeftconcerningthe Properties
of Vulgar
There
made
are
on
FraAions
which
;
ver,
if he
trom
the learned
attentive Learner
an
the
reflefts on
carefully
with
fingleRepetend might
A
17.
pound Repetend.
or
,4444
Or
It
Thus
,44444
might
,44
fo
and
be made
"
be made
,4
to
the
be made
might
as
be made
proceed
Article.
next
a
Pure
,44
or
Com,444
or
"
"
"
"
Introdudtion,
fTdllis ; therefore I (hall not
Dr.
farther therein, but fhall haften
any
eafilydifco-
gether
foregoingExamples, to-
tranfcribed in my
I have
what
will
on.
Mixt
a
or
Repetend.
"
*
Or
,444
^4444
"
"
or
Thus
.
,4444444
,4
might
.
and
fo on,
as
Neceffitynaay require.
In
fine, any
given Repetend,
either Pure
changed^
Number
into
of
or
another.
Places, or
Mixt,
whether
may
be
Singleor
pound,
Com-
transformed, or
Repetend, confidingof
the
of
Places
a
greater Number
of
fame
of
Figures
('9)
Fjffurcs
and
pleafurc\
at
will retain
the fame
New
yet, ftill each
with its firft given
Value
Expreflion
circulating
Expreflion.
RULE.
Write
the Given
down
for its Transformation
petend
and
;
of
Places
many
as
Repetend
then
often
as
mark
Figures,
for
off
as
is
as
neceflary
New
a
Re*
required.
are
"
Thus
the
transformed
be
Expreflion ,4 might
"
into "44
or
"
"
C^c.
"444
above.
as
^
"
"
"
"
""o
or
,5757
transformed
into
"
"'
*575757
be
might
Expreflion ,57,
the
And
""
^C'
*57575757
"
it
R
O
be transformed
might
into ,57575
into
or
and
^5757575757
fo
"
into 95757
into )575or
on,
or
Neccflity
as
require.
may
"
"
Expreflion 93863 might
the
And
"
"
"
be
transformed
"
"
into
"
.
,386363
into
; or
"
or
;
"
or
into
the
fome
t^c.
as
with,
is above
or
equal to
of my
,44
"
1
"
the
"
not
is of
the
"
; or
,444
Therclfore, for their
demonftrate.
Expreflion 94^944^444"
may
"
"
to
or
"
(ift^) That
Readers
a0erted..
prebcnfion^I ihall here
^^or into
"?^.
CirculatingExpreflion ,4
"
Value
,38636363
"
,38636363636
probablethat
is very
perceive that
eafily
fame
into
"
,38636363;
It
,38636363
*
to
better
,444
Ap-
"
,4444=
"
F
"
As
( ao)
"
L
When
of
Nmnerafior
the
the
the other
of
the
one
Exprtffionyas
of this.
that is to the Denominator
of
Denominator
the
as
theone
of
NomcnMr
of the
Numerator
Or
Exprefllon is to
other ExprelfioQ
its Denominator.
to
For
7 is
then
the Numerator
its Denominator,
is
equal to
28
7
words,
Expreflionsare
12
^"i^
-
Evpreffion is to
in ether
=
J.
of difierent
3
aa
M
M
E
Fradions
two
ether"
the
S757
G?^-
y57S75757
eaeh
"
Exprtffion 957=^5757=^57
that the
alfo
As
If ^
Exam^e^
28
to
or
;
Aod
when
ProduA
of
is
3
equat to
ars
to
7,
as
Means
is
3; is
to
x2,
as
28.
to
thus
four Nunobers.aMr^
the
,
is
12
thm
~-
Proportbnal,
equal to the Produ^
then the
of
the
Extreams.
For
And
equals
ai"
the
PvoduAs
the Denominator
then,
Hence
moreeafe,
two
of
the
the other
by
can
the
might iboner,
we
whether
two,
any
toeqiial.
Method
common
Expreflions
of
NumctKor
of diflferqatExpreffions are
wo
different
the onie btf
will alfo be equaL
alternately
by this Method
determine,
of
Fra"ions
of
the Means.
of the Extreams.
the Produ"i
when
Wherefore
of
the ProduA
12x7=84
3x28=84
of
or
and
with
tions
Frac-
more.
each atbss^ tbaa
fomctimes
many
and
rtltf tedious E"Mfions"
t
I
come
now
to
prore
(by ^r/.9.).Iam
44
444
to
4444
that
pEove
99
999
9999
,4s::,44=,4ft44
that
44444
_
"
the
Thaei"
Expreflions
444444
_
99999
Af^
"
=
/;"",_
999999
equal
(a.
equal
to
each other, and
)
confequentlyequal
them"
among
ielires.
#
xfty Becaofe
the
And
4^999=444"^9=399^
And
4""9999=4444^9=3999^
And
4"^99999=44444""9=399f9^
And
4""999999=444444^9=399999^-
forafmuch
Now,
it is manifeft
bove
are
quently equal
the
Value
ieyeral
each
each
to
one
each
of
nately
alter-
fore
other, there-
Cxpredjons
a-
di"renc
other, and
Which
themfelves.
among
of the other
with
compared
that
equal in
Denominator
the
the Numerator
of
the Produfb
as
ExpreflSon by
equal, when
are
one
=396
4"^99 =44x9
was
to
confe-
de"
be
monftrated*
pemonftration will
alio prove
^'"
Expreffioa t57=i575?'=!=*5757S7="S7S?$7^?
is that
^7
^7
^,..
^
99999999
999999
9999
And
57 57''99-
57^9999-
57575757
^75757
^
^
99
For
57^999999-575757^99
adly.
It remains
that I alfo ^ove
"?tf. Or
=:,444=:"4444
=,44444444
"r.
dut
Exptelfion"4=,44
equalto j444a="4444-fei444'H4"
however
or
die
thus
varied.
"
likewUe
So
"
"
""
thus varied.
iwwcver
That
that
"
"
the
of
like Method
The
thac
".
,57575=,5757575757
the
"
"
"
Ezpreflion ,57=:,575=,5757=
"
^e.
however
or
F
2
thus
varied.
Denuni'
C")
DemonfiraticH.
Now
Art.
(by
) the Ezprefliont4
9.
,4;=*-?
And
9
4
=
=
"
.909
.
9
4000
,
And
,4444
=
4
=
-
"
-
9000
where
And
F.
!
400
,444
itsE.S.
"
=
90
And
=
9
Expreflions are eqaal to
Expreffion, they muft be equal each one
themfelves.
other, conlequently equal among
the
and
each
two,
or
more
one
fame
;
Therefore
"
to
.
which
"r.
,4=,44rr,444=^4444
to
was
be
demonftrated.
"
"
I
Again,
fay
C^c.
,44444444
that
"444=
,4=
however
or
"
"
"
"4444=
"
"4444444
thus varied.
Demtmftratkn.
A
the
Now
Expreflions ,4
==
before.
as
"
9
"
"
,444-^-^
And
990
""
.
"
'1
j
9
4400
4
9900
9
.
.
*
*
444000
"
A
And
J
,4444444
=1
122
=3.
999000
'
.
-
\
"
And
,44444444
9
44444000
"
J
=3
4
iiZrt
s*
4
"
99999000.
"_ 5
And
in)
where
And
two
which
to
one
"
"
"
^4444444=
,4444=
,44444444
be demonftrated.
to
was
"
,444=8
,4=
equal
.
"
"
Therefore
"c.
Expreifionsare
more
Expreflion "f
die iatne
and
or
"
"
jdly.
And
laftly
I
fay
,
that the
".*""-"
"
Expicflion,57=,
575=
,'
"
^^-w
"5757=*57575-"5757575757
however
thus varied.
Demot^rattM.
( by Art.
Now
57
::
ExprcffionyS7
9. ) the
"
=*
~
:,
i"
EAF.
99
\.
And
ygys
=s
"
990
And
g^
=^-=H
.5757
9900
And
g^
"Z222 =6^?
,57575
="
99000
where
And
the
two
or
9^
Expreflions are
more
Expreffion, they mul^
fame
"
"
other, confequently equal
equal to
be equal
each
one
themfelves.
among
one
and
to
each
-
"
"
"
Therefore
which
"c.
From
to
the
of
of
,
"
be
demonftrated.
foregoing Denlptiftrationsit
any
transformed
ing
"
y57=y575'=^575?=,S7575=yS75^375757
was
ift, That
be
"
the fame
or
Repetend/or
Given
changed
Number
into another
;of.Places,
manffeft!.^
li
'
of
^
a
confift-
ber
greater Num-
Places of figures /at plcAfur?
j .^d
ftanding
fuch
TcarisTorn^atiop^
"
Circulateji'ii
Repetend,
or
"
notWrthj^eit',
new' Exgfefiic^will
"
*
"
retain
( u)
its firftGiven
with
the iatne Value
tetain
CirculatingEx"
praflion.
.-"
2dly, Hence
ptncioqa
Ufe
of
"C
and
8!
19.
"
end
Q
topre
be
Repetends ipigbl be nade
Neocffitymay
requim. The
|"efi
in
feen
that by .fuch Transfer-
too,
together,ta
Addition
and
Subllra"ion,
proceed,
tQ
either Similar
Repetendsare
Similar
of
leam
may
which^will
But
1
tw^
4Qy
begin
to
we
Like
or
DUIimilar.
or
Repetends
Rcpatcndi do begin
their feveral
firitFigures
fuch, whoie
are
Point
before or after the Decimal
) and
( whether
of Piaces of Fig^ves.
fiitof d"e ianM Number
Diffimilar
20.
Figures of
their feveral
before
Place, ( whether
of
confift
not
Repetends are fuch,
Repetends do not begin
after the Decimal
w
the fame
con-
firft
in the fame
Point
Places of
of
Number
do
whofe
Unlike
or
Place,
fame
in the
)
or
do
though
Figures, al-
they do begin in the 6ufm Place.
How
21.
to
to
Similar
trinsferiR-tWQ
ift, Make
all the
then
the Place
from
found
prefledin
or
tbey
$n
as
togethers
many
all made
convenient
to
take
their
Sum,
the feweft
Number
of
to
Poiot.
2dly,
let each
Places
to
ftands
which
of
of tbv
Figures,
beg^n tpgedber,
feveral Numbers
of
Kepetends.
have
order
by
b^ins,
of the
lyiultiple
in all the Given
^.J9. It is moft
Multiple, in
of
begin togedier, (by
to
the Decimal
all end
confift
where
One
below
them
b the leaft Common
Pbces
that
or
make
Repetends
Given
a9
to
above
".
Repetends
where
Transformation)
And
tends
Repe-
Repetenas.
AUL
lowdS:, either
Diffmilar
siore
or
their leaft Common
or
Difi!erence,ex*
Places of
Figures*
Some
{a")
"
I take
N.B,
down
the
;"
us
abere, until I nudce the
Divifor 5.
"
*
Example
Find
the leaft Common
2.
Multiple
to
2
.
4 and
5.
Divifors
Btvifors
2
2.4.5
4
2
1.2.5
4
"
5
-5
^i^.
leaft
5
is their
20
Common
Multiple.
"
20
Example
Find
and
the leaft Common
3.
Multiple to
2
.
4
6
"
.
xo
:
12
15.
Anjiv.
Here
is their leaft Common
60
follow
transformed
or
Examples of
changed into Similar
the
Multiple.
Diifimilar
oks"
Repetends
^
.
Example
{
)
"7
Example
Difliniilar
made
i.
Similar.
Here
they all are made
begin together;
"4
CO
'^57.
where
that
gins,
which
k"wdft
from
One
"577
,083
.083
beftands
the
cimal
De-
Point"
Example
Diflimtlar
made
2.
Similar.
pHcre
77
is
2
thejr
Common
"
.
.54
f54
leaft
Multiple.
-"
/"
Example
DilHmilar
made
3.
Similar.
"
"475
"
1 Here
they
made
all
are
begin
to
"
.3*4
"
"
"4;547547
together,
,32424242
bove
"
"
"
and
*59595959
leaft
.32.7
.32777777
"
as
diredted
6
is
a*
\
their
Common
Multiple.
"
"IIIIII|I
Rxamph
( "8)
Similar.
made
DiSiinilar
all
they are
made
to bqgin
Here
475"39
475.39
"7"5
"87.58
"
I-
"
as
direded
*"
above
and
3
is
"
mon
leaft Com-
their
543"54
543*
Mther,
to-
Multiple*
EpcampU
5.
made
Diffimilar
Similar.
57.777777777777777
8,49
8,498498498498498
7.5647
7,564764764764764
,803574857485748
,8035748
Here
12
is their kaft Common
Example
Multiple.
6.
24.
A
y
( *9
Finire,
A
24.
preflion,
require
may
not
fo low
loweft
when
in
it,
to
where
as
Order
the
as
that
;
Finite
One
Determinate
made
be
may
o's
many
or
of
Similar
a
that
Repetend
one
Place.
But
Order
of
begin
together, immediately
the
Finite
gether,
as
Varieties,
Addition
Eroreffion
before
which
of
the
;
and
directed.
I
fhall
with
muft
Naitf
exhibit
Single Afid Compounid
the
all
In
among
be
ftands
aU
fiirfto"
19
mt^e
p
motexod
1b
made
Wt
this Ctft
than
loweft
be
cnd.totfi
there
aw
a-
the.Exaiiqdwirf*
CircwatDi*.
4
"
Ga
that
lower
or
muft
fbuids
obfenre,
to
low,
as
Rqpetends
reaches
begins, which
careful
as
Example
Bxpreflion
begins, which
Repetend
that
then
be
reaches
Expreflion
in the
Finite
the
Ex-
by annexing
Rmetends
is, when
Decimal
or
Grculate,
Given
the
where
Place,
Number,
I
"
"
CHAP.
( 30)
CHAP.
Addttim
IF
Circulates
of
Repetrxlds given
Art.
Similar, (by
Repttends.
or
for Sit^U Circulates.
Rule
the
11
Diilimilar, make
are
Example
the Rigbt-
cotififts of
Single Oircu'ates only, add up
by 9'fl,and place the Overplus, if any,
its bottom,
for a Single Circulate
o at
;
hatDd Cokimn
4f
nothing,
Place
a
for
car^y One
every
add
and
\
ufual in Addition
fiibfcribed
2dly,
Units
many
in that Column
as
where
and
id's
as
Figures
ibught.
Circulates.
confifts of
Example
there
or,
"next
the
if any,
by
Decimals, and the
up all the Columns,
this Caurion, viz. to
lo's, with
as
the
to
Columns,
for Compound
When
Column
diat
add
Circulates
add
in
(halt be the Total
bottom
at
found
9
up the other
of Common
Rule
But
all
if the
) ift. Then,
21.
dtem
when
made
Compound
Similar, by
the
Right-hand Column
lo's, ( mentally * found
are
all the Circulates
do
or
made
are
}
to
tom
begin together; and then the Figures fubfcribed at the botof the Circulatii^ Columns
Ihall be the Circulate
by lo's, as above
fought I and add up the other Columns
direded.
all
Make
Author
means
^
Bv
mentallr
tal Cowmns,
are
be found
may
carry
to
the
found,
thus
Right-hand
its definiM
t)"t
vou
Column
add
add
previooily
(the
cimals,
De-
as
up
many
the feve-
are
to
in
1
:
where
the Repetends
below
ia the Pbces
how
oider
in
to
io*s
dif^over
together,
begin
many
mental
that
Addition
elTe
fometimct
that Column,
by
;
wnich
all made
for Compound Repetends"
(i.e. the Repetends) Conterminous,
Similar) and then add as in Common
only
'
Rule
Cunn'j
Mr.
Unit
ihort of tut Tnith
to
the firftRight hand
do
or
may
you
Column.
Units
(3'
Units
there
as
lo's in that Column
are
atbrefaid
then
and
begin t(^ether9
all
)
Repetehds
Figures fubfcribed
laft of
the firft and
fliall be
Columns
the
the
where
to
the
the
Repe-
tend"
Mr.
of
this Rule
Cir/m's
Mr.
fome
a
general Rule
bring out the true
it is not
univcrlallygood for
he gi^es the followingRule.
Cales, yet
in page
and
is infufficient for
it will
fays, that though
and
ii.) complains that
( in his Preface, pag.
Malcolm
479.
1
in
Anlwer
all Cafes
\
Repetends ) all Similar, then take
the Slum of the Repetends upon
a
feparatePaper, and dithe
Tide It by a Number
confiftingall of 9*5, as many
as
of the
Number
of Places in the Repetend \ the Remainder
is the Repetend ^of the Sum,
be fet under
the
Divifion
to
if it has not as
Figures added, with o*s on the left Hand,
the Repetends ; the Quote is to be carried
Places
as
many
Make
to
them
the
the
(u
Column,
next
and
Cafes,
Prafticc,
my
their
divide
oiF Places
I
indeed
And
76, and
fage
the
lowing
bis oWn
I
of
api
Rule
own
as
by
mine
*,
left Trouble
:
and*!
for all
think it eaO^r for
for
'
Figure$, fcf^.
of
fee but
3d
2d
'
'
can
in
Examples,
two
viz. the
4th
In
throughout Mr. Cunn%
lying before me ) where
Milkke in adding them by fol"
page 77,
Edition
now
Rute
Opinion
in
is univerfally
good
s
too
polliblyniake
he could
done
'
the
( it is
Book,
Addition
by adding up the
Caution^ as aboyefdifefted,you mentally
Sum
bf as niany 9*s, ^ the* Kepetend
giving
as
"kepetendswith
confifts
Malcolm-
is that of
fo
the
^
of Mr.
Rule
the reft of
and
R^les^
common
This
the
e.
:.
a
but
he
as
that Mr.
avdded
happily
Cutm
ftridl a. manner,
as
himfelf
( have
it in both,
underftood
his
cabtiouflyword*
edn^ne.
N
I
vras
Learner
wiUing
might
And
to
exhibit
chule
which
the Reafon
why
Mr.
he
we
Makolm*%
found
Ihould
Method^ihat
mo"E
divide
eafy
for
tht
tice.
Prac-
their Sum
by
as^
(3*
as
)
9's, as the Repetends confiftsof Places of FigurcSt
who
Aiall
attentive
Reader,
to
plainly appear
every
many
muft
reBedt
on
hath
what
been
faid
concerning
Equivalent Vulgar Fradtion
finding an
I ihall now
Exprcfllon, ^c.
proceed
2,6
Sum
I fhall
pie,
make
Sum
Sum
2,00
thiougbqut Addition,
except
fuch CirculateS|
ufe of
as
of
Method
Circulating
give Examples.
to
to
the
any
1,2500
in
the
=
1,25
Jaft Exam*
offer themfelve$
in my
might have a Proof "2f
the Principlesof Vulgar
his ]"camples from
Fraftibns,
will be fo many
ufefiil,a$ weU as eafy, Illuftntioog
which
I fhall have this peculiar Adand by which
to the Whole
;
^f fewer Words,
which
otherwife
of making v^ci
vantage^
bewould
be unavoidably nece0ary, and fo fwell this Book
its intended Bulk.
yofli4
Tables
annexed
;
that
the iUanier
Blufiratms tf tbtf^r^dug Exampks.
"
1
J.
i.
=
"
1 =.""
la
(3*)
^
":
^
^
^'
',
(4-)
.
Difllmilar
made
0?
Similar.
M
;.!
5.^8
5,68888
":
-
4.1^75
*"
-^"
loweft
"
'
Sum
That
is
where
as
in
Order-
of
?*** *" "J""fore
*?"
^
14,85555
^^^^
Hfpctend governs
this
r.
low,
One
Rcpetend
begins,which fiands
J
4"?a75o
'
fo
that
.\
'.
T=z
the Finite Exorcf-
not
C
^
1
I
4")Here
"
-
,
14,85
ExampJe,
they
muft
in
where
4II txgin
.
together:which
"
"
"
ift
"
Variety.
the
is,
'
-
^1
iv
Be.
L.
-"J
(5.)
O
;^^iO?milar;^^,^^ made
^^
"-
^--
Similar.
iHere
'
17,488/
4^05
Exprcf-
(^
4,050
'
".anl^;
reaches
V
.
o,
I
1 1
1
y
fi^.^nr
fionf'4,o5
as
where
that
One
\Jow, as
RepetendbqginSywhich
-ftand"
of
21,65c;
Sum
'.^^
That
Finite
the
"
I7"48
o,
";
is
21,65 ccMapleat
Plaice
^^^
.
thtMjfo^c
5
this Exam-
m
P'f""f^^!^^y
begin
which
I
in Order
^^"V^ Expreffion
governs
ail
_
loweft
.c
\ csUl
^
^^^
together
the
:
2d
Variety.
'
r*.
-:
\^.^
r
1
"*.
(6.)
(35)
1
Eit.
Diifitnilar
made
(6.)
Similar.
Here
5*3^
5*3^666
7,916
7jgi666
teaches
5*5625
lower, than where that
"
33^33333
^
1,5
,
Repetend b^ins^
which
ftands
^^^
^f Pl^ce
loweft
...^^A
1,50000
,
SjS^^S
Expref**
^lOn
One
33"3
Finite
the
f^^
I
5^56250
thcrc-
pj^j^
^^^
"^^
preffion governs
"^
"
;
Transformation
the laft
the
: as
."
*
in
in
Example.
^
53"^79i^
been
I have
the
careful
more
exhiUt
the laft
3 Ex*
eiven
be added
to
to
Finite
where
Expreflions are
with
Circulating Expreflions, becaufe 1 have not feen it (b
before me
cautioufly exprefledin any Author
alfo to
; and
the
Learner's
committing any Miftake, when fuch
prevent
amples,
in Pradlice.
fliall occur
Varieties
Illuftratumsof the foregoingExamples.
In
Example
7
2
.
A
-"
4
.
Their
Equivalent Vulgar Fradions
17^
I
=
"
In JEx.
They
2.
^
arc
+*
12
to
which
add
their
will be
their Total
3.
^
They'
^
^
+*
compleat.
'^
^
=^^
=
15
24
Integral Numbers
1,34,6
'^^*"
4320
(viz.) ^-Yyj^ and
84,3416.
'
In "".
=2,225
p-
12960
24
15
12
are
28831
'
'
3
i.
-^
+
arc
9
"^12^
+
|-4- LJ f^fl?
6
=
"
48
^
31104
"
,71527,
to
which
add
their
H
Intq;ral Numbers
{viz.)
In
( 36 )
1.85,
=
'
to
which
add
their Total
In
En.
their
48
lateral
will be
s.
They
'
Numbers
and
which
arc
add
will
their Total
34560
(viz. ) 4-^5-1
\
"
120
"
=
9
their
Int^al
be 21,65.
^
"
-
8100
Numbers
=965 ^
6.
2if2-.
r=
a
They
,679 1 6,
to
-1--
-^
are-
trhicfa add
their
com.
( viz. ) 4-1-17s
"
laEx.
4" and
14985.
45
pleat,to
"6
45
-"--!-
4
=
r
IntegralNumbers
"
(viz.) 5'\-t4-53+7
Examples
wherein
are
-l-5"and
their Total
Compound
to be
will be
Extends or
53,67916.
Circulates pven
added.
JBuftratiutt
(37)
of theforegring
niuftrations
Examples,
In
Example
.
They
In"x.2.
Equivalent Vulgar
Their
i
are-+i?
'
whicli
add
their Total
their
will be
=1^=2
21
13
11-.
1911
Integral Numbers^
27
arc
'
7
to
i
+
Fractions
viz.
1911
20
^5,
and
=27,00732^.
-"
'
^
'"*
1911
-rf 5
C
11.
"
Compound
Of Dii/ifiilar
-Bf.
made
Diflimilar
(I.)
Similar.
,
1
^
"
"
,15625
Circulates.
^
C
562 500
Finite Ex-
the
Here
preflioh
governs
iT^f^iiS
the
Transformation,
as
"2I
,
I
,36837"
made
latcs.
Similar.
"
"
'^He^e
,025000000/
the
Finite
Ex-
preffiongorenis,as
"
.4594"9459r
"459
V
"
Circu-
,
,625
"
Variety
Single
i^
*
'
2d
^o
^
Sum
DIQUnilar
the
m
"
.
"
,461538461
"46i538
Sam
'"
**"
^
^*'"'
pie.
J
1,545997920
H
2
-E*.
(3-)
f 38 )
^.
(3.)
Diffimikr
Simikr.
nude
"""
.777777
"7
"
"
"454545
"
"
,814
,8148X4
Sum
2,047138
Sum
341*5919989
Ex,
Diffimilar
"
"
(5.)
made
Similar.
Here
"
34.09
the
134*09090909
"
97,26
nite
Fi-
Expref"
fion
gyj26666666
not
9*08333333
reaches
fo
"e.
low,
fore
Thereit is
'"5
the
"
as
ift Vari-
"
0,81481481
0,814
Sum
J
ety.
242*75572390
Jllufirations
(39)
lUufiratims ef
In
32
ExampU
forgoing Examples.
Equivalent Vulgar Fractions
Thdr
are-
"05"
33
In
i.
the
"".
They
2.
f
art
i?
+
'
8
+
||i|=
=
~
'
37
3848
13
".S459979"-
In
",.
Tl"y^
3.
^
"Z-
^
+
H
+
'
f^
=
*
2673
27
II
9
=
2,047138.
^
XT,
In
Ex.
They'
4.
^
2,5919989,
to
34-1724-24-162,
\'"
'
whkh
and
add
thdr
Ex.
They'
5.
^
187722
will
JL
be
'
vt}M:h
I
r+:r:
'
12
add
22
I
+
'
15
vw.
34i*59i9989"
I
h~+-::
_j^^^^^2^^^ jjj
Z^^Z
79365
13
Int^ral Numbers,
Total
are
205714
=
"
^
II
their
II
__
10
,
+"
'
15
I
In
1
"\
,
are
37
=
8
14
".L
2
their
'
27
Integral
106920
Numbers,
viz.
"
"
and
i-l-9-l-97-l-x34"
their Total
will be
24""75572a90-
\
Example
6.
Eumph
Sum
fet my
I have
but
the
applied
whole
diem
Examples
Gyration
to
Money,
above
would
or
69929497012285
as
have
Weights,
abftrad
been
to
the
Numbers
fame,
Time,
iiires, "r.'
CHAP.
or
had
Mea-
;
I
(40
(J.)
Minuend
7,16
Subtrahend
3,26
That
compleat
is 3,9
"
Remainder
3,90
lUuftratms of tbeforegmg Examples.
Ex.
I
.
ExprcOcd
whole
=1:6,083 And
6"
Vulgar Fradtions
in
Rcpetend
is 7
of
"
i
"
=
"
its Subtrahend
is
12
]efler"than the Repetend of its Minuend.
Ex.
Is
2"
10
4"
15
of
its
ed )
Minuend,
where
to
which
of
Reafon
in
And
whofe
3
is greater,
added
was
i
its Subtrahend,
=6^66.
"
IS
its Subtrahoid
of
Repetend
6
=
before
than
the
Repetend
in the Rule
( as
direft-
the Subduftion
began.
The
For do but
is manifeft.
all.fuchCafes
the 3 and 6 above continued infinitely
below
; then
of the Right-hand Figure 6 from
after the firftSubdudion
imagine
its
oppo0te
3,
Or
6.
to
muft
we
"
i
exprels myfelf
6
thus, ttke
carry
to
the
next
in other
ci
from
Equivalent
o
I cannot,
"
9
Left-hand
but take
9,
"
gure
Fi-
Words
12
from
9
and
"
9
o
the Remainder
will be
=6,
"
as
above.
9
"
3. Is 7 2-
Ex.
For
the
of
thdr
3"
"
Repetend
of
its Minuend"
Remainder
muft
"
3
"=3"9compleat
its Subtrahend
then
be
as
above.
is
equalto the Repetend
the
confequently
Repetend of
o.
CASE
(43)
CASE
Of Difimlar
II.
Singk CircakUt,
Examples,
(I.)
Minuend
Subtrahend
5,6458^
Remainder
*
I
(2.)
Diflimilar
Minuend
made
ii"8i25
11,81250
r
"
Subtrahend
Similar.
2"7
'
"
"
"^"7mr
-
,
Retiiaioder.
^03472"
M
(3.)
Diffimilar
mad"( ^wpjiar.
A
..
:,
\
Jb
.
From
"
5/"$
5.0333
'
"
Take
3*041^
3,0416
9
Remainder
i"99i6
(44
r
(44)
(4.)
DlffitiiflJir
iikdc^mikr.
'
Toot
Tom
.
From
110^6
Take
110,66666
.*..."..
9^^45i3
94r,i4583
'"''':'
Rimi.
1^,52083
..Illiiftratms
of the foregoingExamples.
"
7
Ex.
I.
Exprefled
Vulgar Fradions
in
13-^
is
1
7"
"
=
"
3.1s $t"ml"
"Af.
lb
I
:
19
:
JBc. 4.
Is
=
10
510;:^
w
Take
lbs
:o
:
lo
Remainder
1,9916.
from'ni"Ti""'i3C.'^
i
Q^
9
165"
O*
3
Take
16
94
Tuns
Tuns
10
2
C^*-
C.'*' 3
t-
02
lb
Q;;;' 18
i8
lb io-0"-
10
"
0"-
Remainder
=16,52083
Tuns.
3
ExampUi
(4$);
vibtttin"n
ExmfUi
CmmU
R^ttnds
Cimlafes
or
h
to
MtraUti,
'"
'.
.
\\
C
Similar
^/
.
^S
t
E
Compound
,
From
Circula^,
From
^
t
t
I*
'
1,3571448
Take
:.,,.^ftg7'42
,,
JlluftratiMs
if
the
foregoingExamples.
Expreffed in Vulgar FraAionsis
JE"f. 1.
fiuyi'
7^,;.^
,
3,81
Take.
"!"
*
3
'^^^^
"
"
"
i
==
"
"
-
,.
II
6Ex,
Is
2^
I
.
.
="=,5714181
"
,
r
".
"
"
II.
.:Q4SE
,
0/ DlJ!milar Compound Circulates.
"
'^'ifflfiftl"3itnilar.
Diffim^r
"
"
Tal^_JL"857i
;i^2f?ii..:?,
fv^fiSsTt^'-'T
.fr.*"l
,R"n.
2
,9285714
""iV4MM"irir-
I
a
(J.)
f46)
(2.)
jMde
Djffimiltf
ttom
34f479"fi
^
Take
Similar.
34.479166
17,681818
Rem.
16,797348
(3).
Diffimilar
"
From
Similan
made
"
"
"
Take
"
4,6i9i"476.
49619047
"
1.9545454
1,954.
Rem.
Rem.
2,6645021
,5721153846^
(".)
(47)
Diflimilar
Similar.
made
"
From
10,5
Take
"
10,500
g^ii
3,4^
"
Hem.
79O45
Examples.
JUuJfraiiims of tbeforegtnng
M9(.
io
Exprefled
I.
Fradtionsis
Vulgar
"
C'^-
6"
z
"
T
^
e^"*
C*^
2
=
in other
Or
C^-
85714
~2,Q2
"
word^
14
*+
is
it
from
C"**
6
Remainder
is
(^'
2
C^
2
became
leis
3.
Is
than
4
"
"
I,
wasi
direfted.
than
given
the
given
of
the
the
Minuend.
"
and
2,6045021
,
.
by
402
Repetend
Repetend
before
of
Repetend
,"
=
-^
the
added
=2
"
22
X
Transformation
greater,
I
^^^^^
307
21
where
=1^,797348
Repetend
the
'
2
above
the
T
""
IB
'
'
528
Transformation
16
Q^
2
P
-
"
22
13
ys
=s
by
that
16
^
48
Subtrahend
42
17"
"
C^*-
3
loHb*-
x^
34-"
obferve
take
15
Q[^
3
22
Ex.i.h
8
the
of
of
the
Subduftion
the
Subtrahend
Minuend
:
bc;gan,
came
before
whereas
was.
(+8)
I,,ol._3i=7
Sr.6.
^,=7.045
4
N.
tory before
other
in
Decimal
Truth,
there
exhibits
to
its Difference
above,
Anfwer
my
is
it,among
Differenee
true
And
the
pretty
near
-|- by Approximation
7,04545
Method
give
so
fFallis'^ Hif-
Dr.
give it its
mathematically exafb.
impofllble
as
Way,
viz.
from
{Chap. 8.) who
the Praditiooer
direds
I took
Example
nnmed^
Examples,
a
the
Uft
This
B.
the
whereas,
:
exprelTed
Dr.
in
by
Decimal
a
For fince his time
there
is
mathematically cxaft.
in the Management
made
fo confiderable
an
Improvement
Thoufands
of Examples
that many
of Decimal
Fradions,
we
now
can
might be produced, to each of which
give
Way,
the
Anfwer
to
true
Labour
with much
And
littletrouble.
Exa"flnefs, with
mathematical
a
\t
is
very
poffibleto
find
.
and
out
cxprefs
Fradions
25.
Before
proceed
I
I Ihatl
fhew
by
a
how
moft
or
viz.
multiply
or
100,
new,
to
any
acctrratcly.
any
eafy,
and
Multiplication,
Sngl^, or
"r.
1000,
divide
howco
a
Way,
Rule,
next
to
Method,
after
9%
the
10,
a Decimal
in
Surds)
to
firft (hew
of
Number
Anfwer
(not
Circulate
Compound
any
true
whatfoevcr
I (haU
2dly,
the
a
And^
MTvmfaierby
cooipoftdiotis
Manner.
"
rft, Lee
the
multiplied by
Produds
Single Circular;
10,
will be
as
or
ibo,
or
,4
1000,
(
^
-f^
t"c.
j be givoi .tp
^
the'fcvci?!
And
follows,
"
.
*
'
.-x4*io
{
That
X
12195
is
loooooooo
vtz.
X
"
to
thos
is
12195121,
500000000
z.
=
":
41
=
"
4^
1
"?if.
12195121,55
the fame
After
be
equal
loooooooo
5
J
proved,
^
)
50
proved
be
to
Manner
Produd
other
each
might
alio
true.
Secondly.
"
I
26.
to
"
divide
to
now
come
Number
a
(hew
by
a
any
Compendious
propoCed Number
Example
Let
Method,
of
how
9'$.
(i 4)
be
5749^22148887874482278975
by 99999.
given
be
to
vided
di-
RULE.
the
Separate
of
there
where
the LefcJiand.
at
having
Columns,
( fupplying its Defed
each
in
Figures
End
diftind
into
underneath)
beginning
Dividend
is
no
Kepetend )
9's in its Divifor, ( which
are
in
5
as
wich
are
Places
man/
as
( as
o's
at
the
the Number
the above
of
Example: )
and
Figures in the firft Column
(57498)
and
their
place them under the Figures in the 2d Column,
in the 3d Column,
Sum
( 79646 ) pllce under the Figjires
take
then
and
5
the
Figures, plaoe under
of
Places
as
below
laft, place,under
the
)
their Sum
and
gd Columns,
fon
becaufc
( 168433.)
Sum
their
fo continue
and
Examples,
unto
the lad
to
it conflfts of
the
do,
as
Column
:
you
then
they ftand iplaced,by io*s,
Quotient required.
in the
4th and
for the fame
(213255)
the 5th and
as
the
Figures
than
more
Rea-
4th Columns,
fee in the
add
and
as
following
the
up
(
their Sum
bers,
Num-
ihall be
t
*
"
"
N.
B.
The
of
uhder
Figjures
the Divifioni
or
the
laft Column
they arc
part of
are
the
the Remainder
Quotient,
being
(SI
)
tbe Dirtdcnd
when
being a Declmftl Greoltte
Integersonly, as in this Example.
eonfifts of
Optratum.
88787
79646
22148
574^
57498
] 44822
68433
13255
2
Quotient
68435
79647
57498
13257
92232
92232
is
Quotient
57498796476843513257
99999
"
Or
I hare
Thus
"
5749879^476843513257,92232
its
obtained
Remainder,
and
Quotient
or
Circulate, by Simple
Compound
Quotient as a Mixt
Addition
only, the moft eafy of all Operations : and to do
fee it required but 23 Figures in the iK^ole Work
it you
;
ufual
of
Method
the
and
bewhereas
Divifion,
common
by
its
fides
a
Reafon
The
there
next
want
fliould
Sum
Right-hand
ceflity for
of
why
2
placed, is, becaufe
which
more
Thoufands,
a
true
it
would
238 Figures to obtain
lels than
not
Attention,
careful
under
the
Figures,
be
Remainder,
the
5
in the
or
a
as
to
above
juft Mixt
K
laft
that Column
added
placed
Praditioner
the Anfwer.
Column
is
is 292230,
the
Figures in
there being no
if any
but
;
the Units, in the
Column,
muft
of
Sum
been
have
the
coil
;
Place
elfe
of
we
the
ne-
ioo*s
(hould
Quotient.
Example
Exampk
Divide
(2.)
by
979891
9999
9798
91,00
9798
I
9988
is 97
Quotient
~
Or,
"
97,9988
-
9999
The
been
needlefi
have
to
repeated,
annexM
#
in
998^ would
Circulate
the
to
placed,
o*s
mope
been
have
it would
therefore
more
laft Column,
the
in
been
there
Columns,
more
oft
as
placed
are
Had,
had
have
to
have
Dfefcft.
the
fupply
order
,o's above
two
o*s.
""
E^cample (3.)
Divide
(^c. where
3501,23022
by
2
would
peat,
infinitelyre-
99.
OperoHoHi
22
35
59
3
Quot.
In
35"
this
find
where
and
I found
36
59
61
Example
about
it
as
the
above
84
I
06
placed
Quotient
marked
28
22
50
at
would
72
95
17
in
pleafure,
produce
39
a
61
84
order
to
Circulate
off.
The
5
(53)
WW.
-22
"J,
why
R"ifon
The
bccaufe
in
the J
placed
the
found
3
the
(thfi Sum
be- the iuiri oftfie
would
I
phaei
259-1-22=281
=303
Therefore
I
of
next
in, the
laft
Column.
that Column
lower
Place
of
)
Column
loo's.
"
as
above.
ExampU
by 99Sf99^^
62952937047
Divide
(4.)
Operation.
629529
6z9r52,9
Quot.
which
37o47"o
is
62953
999999'
Example
Divide
for
compleat
(5.)
by
123353332211
its Quotient,
999;
OperdthH.
123
Qiot.
Which
is
123
333
332
211
123
456
788
456
788,999
compleiat^fofits*Quotient.
123456789
QQO
For
12345678
^^^
is
equal
to
it.
999
K2
Example
Exampk
Divide
(6L)
by
92609907390
999999.
Operatm.
926099
07390,0
926099
92669,9
Quot.
is
Wjhich
92610
999999
compIeaCc
Example
Divide
2465529966,9
(7.)
by
999999^
Operatm.
^4655^
9966,90
OOOQOO
243242
I
2465,53
Quot.
"
Which
wrought
readily Use
"
2465,5324
IS
have
I
^43243
243243
the
that
this
Example
324
in its
large^
at
Quotient
that
you
may
infinitelyre*
would
feat.
What
JV. B.
confifts
Places
In
we
are
of
of
the
lefler
a
Figqres
order
to
often
eafieft Manner
of
the
of
be,
of
or
its Divifon
wkh
find
would
Number,
the
out
obliged
confifts
ceflary that
Quotient
to
Places
Learner
divide
of
See
by
as
Figures
fhould
doing it, as
the
Produft
true
;
be
above
when
the
in
X4"
15,
i6^
Multiplication^
9*!$as the
many
therefore
it was
acqu^nted
:
of
Number
fanM
Art
Dividend
by
which
with
he
culate
Cirne-
the
might
wkk
( 5S^ ))
greateftExpedition find the Quote of any DiviPlaces or
of
hundred
fion by 9*8, if required, to an
more
Figures, with very little Trouble.
with
the
%^
HirefoBows
Blufirationif
geitigCompenims
ift. Let
Method
the
an
Compendious
the
fere*
multiplied by 9999
be
Vide
Way.
in
Viz.
DivifieHf
S^^jS^AI be giiren to
in the moft
ufei
Art.
xo"
Operaium^
its Prodttd.
58471099305^
dious
Divifion
As
of this is itsProof
the Converie
Now
2dly.
by Compenf*
\
divide
W
58471099305J
9999-
Oferatm.,;
Divifor
I 5847
9999
I
j
1099
^^
I
which
is
5847 6946: jBl^na.9999
equal
to
The
Dividend
6946
iH7
Quotient
3053
For
58476947.
5847^94^
=
the
Rem.-^?
'
9999
18
9999
equal to
x
bc-58476946
this
WherefiEM^Vhc
Unity.
or
'
1- I
=
58476947^
^
exaft
Quotient
muft
above.
Illuftr^oii,T-am
per"aded every attentive
Reader
will, by Infpeftion only, more
eafilyperceive the
of dividing by
Rationale for the Compendious Method
any
From
Number
tion that
of
can
g^\
than
be
offered
he
would
by any
verbal
Demonftra-^
himc
CHAP.
("set )
"
H
C
V
A
.'
P.
A
IV.
General Rule for all Cafes.
.
"
.
.
.
Multiplicand and Multiplier to their EquivalentSingle Fku^ionfi ; then proceed according
the Rule
to
prefcrib'd in Multiplication of Vulgar
Educe
R
the
and
Fra"ions,
in
compleat
its
by
Numeratoc
difcover
Frai^tiop^Arifing will
the
you
have
be
cduki
the
wifh;
Quotient
to
content
its
in
you
for
you
yoa
its
till you
or
then
happeo
when
your
divide
the
fp foon
think
purpofe,
and
you
riiay
give. it s^the ^P^dpft approxjiinately
.
i
"
'
"
\
oi
.
'
'
^
"
"
.
.
1
Multiply ,4jij,.$
bjr"
^1
\
you
have
theCe
ceafe,
may
enough
near
of
the Produft
remain,
o
Quotient,
if neither
Produ(5t,(ought. But
as
until
be
if
And
DenocuMf^^
Circulate
a
Fra^ion.
Vulgar
a
^
and
ijb,/TSktir Ecm""Icii^Singk:Fi:aaipn8afe^
I
And
i^
X
~
9
I
=/?^i.K
Ac.
Produd
compleac
in
sk
9
_
which
being divided as above directed,
Vulgar Fradion
;
will produce fpr its QuQCiept 1507. which isia.Finils Iqtcgpal rroduft.
t
Example
9\
^"999938P
"
688888
I
999999
"
t
I
I
i 20
firft
the
Quotient
688888
68,8888888888
which
the
is 68,8
the
fecond
Quotient
Produdt,
68
or
1
Example
"
"
Multiply
(5.)
by
57HB*9A5
That
E.
,ft. Their
by
57945,
F.
S.
is.
"
"
Multiply
57"7
sy,
^^!H22?
^^
and
arc
999
57888000
And
^5io.^30'oi7foooo
999
Gompleat
duoed
to
in
9
Vulgar
a
3347987198
'^
999
5
Fradtion,
which
p^"^
^
9
Ezpreflion
by Cultellation,
as
tbelaft
is beft
re-
Example.
Operation.
9
I
999
1
3010
I
760000
334
4^4
333
798
"^"^
"
is
firft
Quotient
"
7998
"
which
the
o
the
fecond
Quotient
"
3347987,
the
Produft.
Example
" $9
)
Eitati^k (6.)
Mulriplf 14,857142
ift. Their
E.
bf 7,0714285
'^ill"
S. F.
??Z"ii"5
and
arc
999999""
999999
'"5o6ioi43g745"o
And
their Produa
is
Compleat
in
^^
a
Fradion
vulgar
follows
}
9999990
999999
is reduced
bj
Cultelladon
as
:
io5o5i
I
which
014367
I 0506 I
0506
4520,00
I
19428
119,428 57*428
105061 224489
I
571428I571428 571428
367345938773'5"o2oi
2'
i|
2
'.105,061224489795918367346938775510204081632
continued
Dividend
571428
08 1 629
571428
653057
4
3
653061
-Quotientcontinued
Here
the
done
at
duft
turns
i)ove.
But
Ezpence
out
a
we
an
Unit
to
be
of
Quotient
a
very
trae
few
Compound
mixt
may
105,0612244897
of
the
have
we
224489
be
content
to
which
wants
not
to
Figures deep,
54
Figures
%
Circulate,
take
the
the
the
and
as
Pro-
markM
Produd:
a-
thus
;
Part
exaft
Exampli
i"t")
k
"
"
"
*.
ift. Their 4E. JS.
H^
"re
"
^
ahU
^9
And
*
-
9W
^""t"'ifu.
ti
their "Prbauft cdiripreat
m
=:
"
99
999^9
99
.
.
.
Vulgar Fradion,
tellation
which
Expreflion is
..,.-,"".,
by CtU'
.
reduced
beu
follows.
as
I
99
oo
94r77.
ft
95,72
999
the firft Qu
72
1 '95." '272 \1^7
957
272
95^
229
2
I
rfki
,0958
230
""
which
is
"^^*.
rtiM
"09582^
958 230
It
i
ii
their
t
J*
the fecond
Quote.
"v
true
Produd.
feciog Aat the divi'diag
by*ftnyNumber
hath beeh e^ght^ ( in Art. ^6. ) is much
eafier
readilydone, than any other Divifion* whatever,
of 9^5, a"
Now
dividing by
i,
or
10,
or
100,
fePf.'or
by
'and
2,
more
except
t)r^2o"
the
t)r
made
are
Ifay, feeing that fuch Divifions
now
fo very eafy, from thence
ting
then appears the A d'Vintagcttftoin Vulgar Fraftions
the Compleat Produfts
continue
as
20c,
"?^.
firft occur,
without
fions in a Vulgar Way,
they
always have
9
or
reducing them
becaufe
9*s" with
or
in
to
fuch
without
0%
lower
Cafes
Exprefthey
for their
will
De-
nominators.
nominators,
as
fee
you
in
federal
the
Vulgar
Produfts
above.
of the fafegoing EMmfks.
Bluftratlons
Ex.
^i^
f
A?.
2.
Vulgar
a
Way
ia 45
"
x
^
a;
complcat.
507
1
=
in
Expneflbd
I.
Is 9
j"
"
g^
=a
"
coajpkaL
=4"2
*
^7
38480
2
"--"
T
T.
I3. s 65
"x.
"*.
4.
x;
--
Is 4-
X
15
35
Is
5-
=475"05"
^^
=
-
57945
^^57
1H4880000
=
r"
37
"
=*68,8.
7
f!
_
_
-E*.
=
-
=
~
9
333
I^
=105,0612244
"
t
6.
"x.
Is 14
X7"
'
"
^
=
-^
-^
t
X
-
=,095823.
=
"
37
tff.
"^
9S
tij
7
Ex.7,'
"
407
Rule
for all Ciifes^ cbe Ptiodbfta of an)r
By the General
^ven CirculatingExpreflions are very eafllyand readilyotK
tained
s
that
and
any burden
the ProduQ:
too
with
Iktle
or
no
trouble, and
without
the
than
is necefl"ry to ind
more
Memory,
fince
of two
tbs
Vulgar Fradlions
j ef|"eriaUy
of finding the Equivalent Single FraAion
Methods
to
any
of
that
with
CirculatingExprefflon,together
dividing by
any Number
made
is now
to
of
9*5, ( the coiAant
Differs
in
fuch
Cafes)
beyond Expectation eafy,
L
2
Howe*
(62)
However,
Produds
there
as
nude
ufe of
this Place
in
Number
;
the
upon
the
I Ihall add
therein, I
of
one
their
Obfervations
fuch
(hall look
whoever
other
and
him
willinglymake
too
that
Whole,
finding
Cunnj
followed
have
more
becaufe
and
Mr.
the learned
fince, who
Authors
great
Riall
by
for
Methods
other
are
into Mr.
Cunf^
Examples of Multiplicationfor the future, may from mine
of the
the
Reafons
various
Methods,
eafily underftand
which
that Author
was
pleas'dto make ufe of to give their
Produds.
true
in order
But
ceffityof diftributingthe
thereunto, I
Exampjes
in
under
am
the Ne-
Rule
this
into
3
Varieties, viz.
Variety
Where
Examples^
1.
confifl
Mi^iltiplicands
the
Circulates, either Pure
Compound
Finite
are
and
Mixt,
or
of
Single or
their Multipliers
Expreffions,either Integral, Decimal
or
Mixt*.
CASE
I.
Examples having SingleCirculates.
RULE.
When
next
fubfcribe
for
a
other
then
and
make
Single
the
Circulate
Figures
in
Single Circulate,
the
fo ffaall you
9
before
add
you
their feveral
Be
taught
in
:
careful
Multiplicand,
obtain
their
their Sum
to
;
the
mark
of
Multiplication
as
true
carry
in its
if any
in its Product
Circulates
and
found
Overplus,
confifts of two
Multi|"lier
or
of
them
each
with
proceed
in Addition
N'B.
mark
a
for every
Place
and
r
the
multiply
you
Left-hand
and
;
Produd,
or
if
and
none
proceed
a
with
o,
the
plication
Multi-
in Common
Produd.
its
to
i
But
when
(ignificantFigures,
its firil Figure v
as
particular Produds
together,
to end
taught
together,as was
fhall be the Product
fought.
off
more
with
its
Fiadional
Part,
as
Decimals.
The
is
(63)
The
Juftnefs of
well-known
the above
Multiplicationis
Truth, viz. That
from
is manifeft
Rule
Addition,
"
this
manifold
a
ft
Samples*
(I.)
(4.)
(3.)
I5"4
"925
(7-)
(8.)
W
36
4
2
Which
it 3,9
8-
34.99990
3^90
Finite.
Which
is 34*9999
3
2,0
Which
Finite.
it z Finite
^
(9.)
(10.)
45.^
.4583
,0625
160,5
22916
8027
33
1370
91666:
112388
.
8750000
1
nhkh
507.0
is 1507
Finite
,02864583
.
280,972
Sbiftratwu
(64)
"x.
In
It
I.
IS
X
"
=
"
=,08.
"
10
9
90
'^^
9"
10
90
*
^.4.
It is 15
-it
^
X
==
"
9
=,926.
900
lOQ
V
Ic is
-E^. 5.
4
"
X
Ex.6.
I%"3888
x2^
IQ
9
";
;;;:
--
=3,p
compkat.
90
7Jt,,.9"=3i49^_
^^'^^-'^
$0
1000
90000
'
_
^^.
7.
It
IS
^.
^^^^
=
=8,88
-
eV^ where
8 would
iUfioi^lyrepeat.
t
"
r
iif. 9.
It is
v'dc Ex.
4*~-i!33
x.wiv
the Gcnc-
inlRulc.
"".
**.
10.
II,
It is
It
IS
^x ^ =^
j6o
~
X
I
i
=.^"8d^^
5=
.
3.
^ =280,972.
(66)
(6.)
(50
(7)
*
"
""""
tbt
^
Sluftrationt
Itis-^ x^=^=2,45.
.2.
1
II
II
189
67
"x.
3.
Ex. !.4.
"*.
5.
,
^s=--*^=27comples"t.
Itis3-x
6.
It is 4
It is
iz6
-""^
Itisx
41
jooo
22
^
X
-
41
"*.
ExampUs.
Iti.^^xl=^=:,05;
In"..i.
""
fortgmg
f-
X
=,003073
5^8
=
-|-
"
*
=3,402439.
164
1?
=
17,
41000
4
Z.
-
=
=,75
corapleat.
In
I'ia'l
lUs
"af
thus
l^lciplier,
the
'\"^
Ig aU
i^u^d
Take
:
Mult^lifand)
of
g's,
a^
write
i
^uA
Mai-
kini, Mr.
the
as
niapy
down
tw
Repetend.
for fevcry
JtialProduitsr
,^%6 Repetendft SlmilaP,
'
f^t
far
as
lui%-
the
as
in
fought,
Common
firft
;
whidi
Rule.
cafity perceive,
Multipliers, it would
may
__
[jrueProdoft
"lery,the
S
that
too
whereas
j
fame
Puipolc
with
Multiplicand
little
with
or
that
'
you
mentally
"rding as
Figurcs,
^%
are
the
divide
Repe-
Cs'f.
Finite
Expref-
^
'e^tSingleor
Compound
CASE
( 68
A
C
)
L
E.
S
.
Exotics
Ul
R
CircuTateV i
Compound
Multiplication of Common
Produd
by
coniifts
Pix)dudt
i
with
this
laft
Single^
Pure
a
nThole
rLM'xht
Decimals
0*5
many
as
of
Places
of
'9*s divide
E.
the^ Multiplier is
ift, WhcH
Circulates.
Pure
bawig
Figures
Operation
then
;
annexed
the
alike
be
in
as
its
Circulate
Nuofber
fo ftiall its Qaotient
Product,
Pure
a
multiply
as
with
and
y
or
be
the
of
tnic
required*
'
'
O/thus,
2dly5. Find
the
,tben'with, its Numerator
,a$
in
Common
nuUtiply the
Decio^als
")enominator"
Sihsle
MultipJkr^jiEquivalent
that
;
Multiplicand,
Given
Produdb
ihall
Quotient
the
and
and
^
FraAion
be
divide
the
its.
by
Product
required.
In
N^B,
riety, wl)ere the
firft Dirjcftion
ufed,
I
there
(hall
Examples^
Operation
every
the
Muklplier
Given
is followed
fecond
explain
wrought
D
the
in this and
botk
is made
; But
where
ire Aibn
takes
Above
according
Rules
to
a
the
next
ufe of, there
new
Va-
the
Multiplier is
place.
by
both
the
two
following.
Directions..
Bj^
( ^^ )
Dire^ion.. Example(i.)
fy ff/e^rfi
,
^
,6 bjr,8
Multiply
"
^
~
.
Optra$icn,
,6
,
,48 the firftor
Prodoa
Common
5
"M*i
,4,8 the C. P. multipliedby
10
:
t^mmm^mm
will
give
for its
,53
Quotient
1
which
isthe
Pra
true
du"^ required^
i
"
Multiply ,875 by ,36
Operation.
,875
,36
^
" ""
"
2025
"
,
the firft or.Common
,3 1500
the
31,500
And
99
1
Produd
:
9
-M
C. P.
multipliedby
lOO:
31,50
3"
will
give
for
,3181
"
"
.
^
its
Quotient,
Produfl:
M
2
which
required.
is its true
'
^^
^
(
)
70
Examples hy tbefemd
fBe famt
DireSfm.
*
Multiply ,6 by
equalto
,8 is
Note
8
8 is its New
Therefore
8
-
"
"
Multiplier.
I 4""
9
its true
,53
FroduA.
Therefore
is its New
36
,36
Note
Multiply ,875 by ,36
|-
=
Multiplier.
2025
99
I 31.
"
"
81
,31
8
1
its tnjie Produft, which
b ,31 8.^
Bluftr(rtions%
Ex.
Now
1..IS"
=,1"
X"
its Produd:
compleat
^
a
Fraftion
Vulgar
9
\
whofe
90,
Numerator
and
Denominator
8
A.
being
divided
by
Quotient is ^S2^
And
Ex.
2.
10,
Exprcffionwill
inrthe
^
Is
the
i^
^
minator
Vulgar FraAion
being divided by
^-2.
2-2P
590
"
a
whofe
become
whofe.
"
Operation.
3i5oo
X
1000
pleat in
in
'^
10
.^
=
99
;
p^odua
9900a
Numerator
whole
100,
the
com^
and
will
")(pre(fion
Quotient is ,318,
as
in the
Deno-^
become
Operation^.
99
Compare
^
(?"
-
Cbtnparrthefeand
the
)
Uloftntia"s
following
with
their
Examples, and you will eafilyfee'lheRwfonof
my
1 have done binder Cajc i.
penning the Rules in the manner
this Variety.
m
feveral
Atore
Examples.
.
Multiply"oi5625
by
9
,8
V.
.
Ptodufl:
Common
1 ,125000
,0x38 True
.
multiplied
by
10^
Produa.
"
k
r
(isV
C4.)
M.
7,875
By
9
V
.
,3
C. P.
I 23,625
X
M.
2,54
oy
,03
9
10
1 ,762
C. P^
X
la
t.
Phjduft
2,62 5 True
Finite.
,0846 True Produft^
(7^"
{6.)
^
M.
M.
540
636
"
by _j7
9.
1 3780
420
"'y. i
.
C. p.
True
X
lo
Produa
9
Finite.
I 190^0
2 1 20
C. P.
True
X
10
Produd-
Fi-
nitc.
Slttfiratiottt
n
( 7a"
Enamfhs,
i tbefartffiMg
MhpnHm
Istt
64
J"fa,
^
Is
".4.
X
=
"
9
5
^
?-
X
7
=,0138.
-r-
576
=
=z,6a5
"
89
iwmplcat.
72
9000
90
100
I
199
Ex.
Is
7.
-^
X"
M
Obfcrve
be
the
,3
complcat.
=2120
"
"
9
whtn
9^r
Multiplier,
is.the
its
its*Maltiplicand,whether
of
"
=
this be
3
Circulating Expreffion:
For
,3
"
the
Produft
true
333
1000
^c.
times,
or
a
But
"
when
.
^003
the;
Finite
"
or
6?r. then
,0903
the
Multiplicand wIUBe
its
3000
300
30
or
fc?^. of
6n'
"
or
a
will
I
==
"
Multiplierh either ,03
i-or*
^
=
Produft
true
:
Afad
when
then- the
(^c.-of
"
the
the
of
is
Milltiplier
10
times,
or
Multiplicandwill
either 3,
100
be
or
33,
.times,
its-true
or
Pro-
doft.
(8.)
(7+)
do.)
M.
19448
loo
"
by
"
"38o954
38896200
97240500
175032900
'555848000
58344300
1 7408791259120
999999
C.
o
P^
i 000000
X
I 740879
True
which
is
'
Finite.
7408800
Exam
of thefot^w^
niuftrations
999
999
I
no$96o950o
29268
500875
fits.
.
^*
'
_
__
^
99999000
99999
1000
"
"
110,597200.
Is
10.
74308792591200
g5o952
^"0448100
,
Ex.
"
X
=S
^"-"
r
-^
"
^
999999
999999
1
_
compleat.
7408800
the great Number
From
of
fuaded
in
what
he
farther need
any
may
no
even
very
I fliall omit
Practitioner
phaTashe
the
to
Reader,
common
of
my
ciples
the Prin-
upon
whoever,
Affiftance for
readilyfupply himfelf.
And
I
am
pernow
can
but
more,
I
wifli I
Wherefore
having given fo many.
and
leave it to the
exhibiting any more,
might efcapc Cenfure
^
lUuftrations
Vulgar Fraftions, .already exhibited,
that
fkand
of
apply
for
them
to
fuch
himfelf fliallthink may
of
the
following Exam-
requireit,
CASE
)
(75
n.
E
AS
c
Circulates.
Examples having Mixt
"
L
RU
E,
m
but
Repetend
confifts of
the
Multiplier is
the
When
Places, then
more
given Circulate,
Produft
of
the
Produd
of
the
ift.
Find
remainingFigures
given Circulate
fought.
be the Produft
its Finite
Figures, and
the
add
the la"
to
14,26
5
I
,
,^
P. oF
C.
285,450==
io6
ilo
X
^/r
31,716
95150
=
Trucl^.
=
P.
.
of 4:
=
P.
47575
=
P. of
6787',^66
I
f
"
is
4
'J
"
(Which
10:
1
*
6787,36
'
"
"
i
of, ,a;
J90300
^
'
'
-
'
f
^66:
cff
{?
%
"
Find
the
Refult
Examples^
by
in
:
its
Part
Produffc
2dly,
which
;
and
:
true
and
laft directed;
as
having
Grculate,
Mixt
of
Plaoes
2
or
i
a
of
the
true
fhall
( 76 )
by
6i"o3
"*"
22,50180
True
=
of
"=P.
6750540
4050324
P.
1,0:
P. of 60
=
of ,oj:
:
41200,79580
which
is
Finite*
41200,7958
(3.)
f
J
M*
487965
"
5,06
by
I
9
C.
:^
292,590
32,510
=
^438250
=
P.
of 906
P. of
True
X
,06
la
1
Pf Sfi^v
P.
2470,760
'^
(4-)
by
'
70,4*
*
i^-^
202905
162324
""
99
I
5
"
C. p. of ,45
s=
X
too
I
18 4"4
28
40
67
I
"
59
o
o
=
"
28
59
I
=
"i5 90
True
P.of
P. of
70
,45
:
:
"
True
Produft.
Obrcrvc
n
(
)
77
that in fijch Operatioiis
as thofe it is very necefthe
Produft
of the given Circulate
ofF
mark
true
Obfenre
faryto
DiftinAion, as above; that being a
to placethe next
ProduA, for want
its Decimal
with
where
Guide
which
The
of
arife.
might ealily
Error
an
tain
cer-
fdlowing Examples are wroiightaccording to
this Variety,
two
the 2d, under
Direction
(50
M.
Note
,28125
the
Expreflion4,36 =i^
99
Subft.
the Finite Part
4
New
the
will leave 432
of
Multiplier
the
given Circu-
late^s E. S. F.
56250
84375
X
;99
I
12500
12
Here
[00
00
1,5
27
i
its New
"
"
1,2
2
the Fraaidnal
off from
Part*is
marked
the
MultipUcandt and
Multiplier*
"
7 27
"
True
Produd
"
which is 1,227.
For
as
much
as
the
4,3^isequalto
Expreflion
^^^,
99
is manifeft that the Operattonin all fuch Cafes muft be
it
as
above.
N
2
(2.)
(78)
(2.)
"
M.
the
Note
Siooo
"
Exprefllon 24^925
^490
k
=
999'
"
"
24"925
Subft.
24
will leave
the
24901
Multiplier,(^c.
New
8icx"o
7290000
324000
*
162000
t-
""
tamm^
1201
999
201
which
I
Firft Produa
698
100
201
899
9
899 9,99
9
Produfb
:.
Finite;
is 2019000
mud
True
t
obferve
here
to
Reader,
my
that
That
moft
com*-^
Q)mpound:
Circulates under this 2d Variety^is more
eafilyand quicker
folved by changing the Multiplicand for the Multiplier,"
under
with the Operation as direded
n cpnfrik by proceeding
monly
Cafe
2.
Example,
every
Example,
and
for
As
Variety,I.
Inftance
Ihall fee the
we
;. let
Advantage
""
M.
in
happen
may
us
take
in.lb
the laft.
doing.
I
24,925^
by
81
000
I
00
jf
the
00
00
r
o*s
Learner's
Miftake
.
.
place the
in
thus,
to
prevent
making
any.
oflTthcL
marking
FraiftiooalRart,.
24925925.
199407407+.
2018999,999
,
"
_^^
which
is 2019000
^
compleat;
Thus^
(
Thus
where
make
this 2d
the leaft Trouble.
in ordicr
Variety,
ufe
made
perations
the
of
Mr.
by
Multiplicand
Circulates
Compound
with
Produft
true
( by changing
yo'j may
Multiplier,
)
79
to
given ) find the
are
I
But
obliged
was
of
explain fome
Cunny
for the
other
and
the
to
O-
Authors
fince him.
Variety
3,
ExampleSjVThtrc.hoth Multiplicand and MultiplierconPure
fift of Circulates, either
or
Single or Compound,
Mixt.
RULE.
multiply
then
by
.Produft
Places
as
9's,
many
Figures ;
of
in
taught
was
as
Multiplier is
Given
the
ift, Wbent
as
Multiplier
Giv^n
Quotient fhall
the
divide
and
Variety, i,
the
Circulat)e,
Pure
a
be
the
true
its
hath
Produft
required.
2dly, But
the Given
when
Mixt
Multiplieris a
Circulate,
with
and' then
its
Equi^^alent Single Fradion,
minator,
direded
by its Denomultiply as above
Numerator,
j and
Produ""t ^ the Quotient (hall be the.
that
divide
firft find
true
its
Produdt
required.
^
a,
I 6"9i40
,7682
=
C.
True
P.
X
10
Produfl^
9
""
I 360,40
=
C,E.
X
ion
40,04*
(30
(8o)
9
I 74"337
C. P.
=
X
10
:
9
| 12,672
"
"
very
uncertain
may
FaAors
the
we
muft
very
Figure
often
approaching as
Produd
confifi: of
of Circulate
Produft, betore
Common
therefore
the
of
x
"
10
:
]
1,40802469135
kind
what
becaufe
Produft,
the
both
When
C. P.
"
8125975308641
Obferve,
=
or
the
will arife in the
Figures which
dividing by
in fuch
near
Circulates, it is
Oifes
be
the Truth,
repeat in
or
9
content
as
true
9's i
with
Neoeffity
require.
19,218
Prodiift.
True
192,18 True
Froduft.
"MiV
(7.)
M.
5,3
by
6
Note
6,="
9
Therefore
$
I
fo
39,00
35,
is its New
Multiplier.^
Firft Pjoduft:
TnieProduft,
(8.)
(
)
82
(".)
"
"
M.
i
by
7. under
Ex.
Set
351
the General
Role.
"
,27
82
,095
30
True
95
Produd
,095823,
(12.)
"
M.
m
by
57945"945
57"7
"
k.
That
5"o
Note
57945"
by
Therefore
"
"
."
M.
"
the
Exprei"oo $y,
=
"
9
S7*
is the Kew
520
Multiplier
"
"
II58918
28972972
.9
I
Firft Produd:
891
30 131
"
"
334798^,9
TrucProduft
"
"
.
which
is
.
33479874
^/^
That
1
( 83 )
to inform
nothing might be wantingperfectly
ihall
confift
the
of
next
Praftidoner,
two
Examples
fuch
that
when
fhall
Decimal
Expreflions
occur,
;
any
That
may not be at
don in the true
Lofe how
a
off the Decimal
mark
to
the
low
he
Diftinc-
Produft.
approximate
or
Us)
9
"
"
Note
MultiplyfOiy
,02
=
"
99
"
"
"
*"r
.03
Theivfore
3
is the New
:
Multiplier
"
"
,081 FirftProdud:
Si^
1 ,08
81
10
08
98
08
1-1
"
"
,0008
19
00
t
which
is
08
19
00
True
Produd:
"
,000819
(14.)
"
M.
Note
,027
by
die
"
27
Expreffion
,027=-!-
,027
Therefore
27 is the New
"
Multiplier;
"
189
0540
,729 Pirft Produft.
999
I
"729
729
729
729
458 187
I
729
2
App.P. ,000730460 189
729
729
"c,
645
2
919
649
3y (dc, the
wluch I flull exhibit in Involution,Chap. 7.
O
whole
of
( 84)
"
In the
in their
ftindion
of
Laws
j
it
carefullyobfervc,
Decimal
being according
Decimals
in Common
Divifion
the Learner
have
Produfts
true
prefixedtwo
I fct the
o*s, before
latter three
the
in
and
I
above
Examples
ft of the two
I
*
to
which
:
prevent
o\
Di-
to
the
I would
Miftakes.
(15.)
"
M.
"
"
Note
14,857142
by
9999990
70
the New
70714215
=
70214215
7,0714285
Subft.
7,0714285
Multiplier^
74285714
"
"
I4857I428
"
"
2971428571
"
"
59428571428
148571428571
^399999999999
00000000
.
.
.
"
"
^^39999999999999
"
"
1050611194,285714
its
See
true
Firft
Produft.
Produfir, ExampU
6.
and
with
under
the General
Rule.
I (hall conclude
this
Variety,
by exhibitingthe Operations ( after my
three
Examples^ from Mr. Cunn^ pag.
doubt
not
but that
it
Multiplication,;^
) of the laft
And
82, 83.
J
manner
by thefe Examples^and thofe preceding,.
the
(85
Ac
Capacity,
mcaneft
readily be able
give
to
that
which
Methods,
with
Author
leare
little
Attention,
Reafon
( or
any
other
find
the
true
the
moft
may
for all the
various
fince
him)
Pkx1u"
of
was
any
Expreffion 4,297="^
999
"
4,2J7
Subft.
^ill
to
Note
3,145
"
by
very
aflSgn a
or
ufe of,
pleafed to make
Circulating "xpreffion$"
M.
)
4
.the New
4293
Multiplier
9435
"
"
283090
"
"
629090
12581818
"
999
\ 135
13,5
"
I
03,4
363
636
135
169
532
53^
169
169
02
Firft Produft
:
TrueProdud.
Mj
(86)
M.
the
Note
2,172
'
I
A
1
I
Exprdlioo 111,98'=:
1870
999
Suhft.
II
Therefore
is the
111870
culation
"
"
"
"
the Cir-
Multiplier, becaufe
begins in the Place
New
of Units.
"
00
1
52090
J
pjj^^ ^
^,j ^
the
j^yeM
making any tniftake
marking off the Fraftionat
Learner^
1738181
in
"
"
g^u
Part.
2172727
2172727Z
217272727
Firft Ptoduft
062,1999
I 243
999
:
1305
243
2
""
"
"
^43"
If. B.
"
Becaufe
in
Produ^.
True
3p6
306
the
the
Produft
common
Circulate
is
"
99,
Place
I
wrought
the
Mr.
therefore
true
thereby
it
at
Produft
Cunn
making
large" to
would
judicioufly added
the
2
let the
turn
out
become
Reader
a
to
i
3
:
fee that
but
the
I
either
next
have
way
the lannie.
Multiply
(8?)
Multiply
by
481,7652
481764800
481,7652
Subft.
will
Note
21485,314
"999999"
4
leave
the New
481764800
the
"
*
Multiplier,bccaufe
Circulation
begips in
the Place
"
of
Tens.
000
"
"
0000
"
"
17188251451
"
*
85941257257
1289118858858
15039720020020
2
14853
143
143 14
1718825145145145
10350868153572,772
"M^i"
-"""
^"
Firft
Froduft,
which
for
"
"
I take
conveniently dividing by 999999,
down
neath.
under-
as
-
.
103508
72,7727
727727I727727
727
103508
785043
512770
240497
968 fcfr^
968227
69,
2i
I
103508
78,5044
IF it be defired
is left out,
Placet
of
6?r.
68j535
512772
240499
FraAion"
to
find
the Vahie
obferve,
that
of the laft
by omitting as many
Repetend of the Multiplierdid
Figures, as
the
of that
which
confifk
(88
"confift
Places, theq the
6
of, viz,
)
be
will
Produft
Approximate
which
10350878,50445127722409996;
of
wants
the
772^
^^^
Produft
true
9
999
^f^^
Unit
in the
laft
Place, the
999999
Place
of
3 Nines.
after the
6
.
772
Now
Exprefllon
the
^
^^^
is
"
equal
to
999999
21Z
i, and
"
is in the
6
the
i8th
below
Place
Unity.
^
998999oai
approximate
the
therefore
in the
Prodoa
true
Produft
Quantity
is defeftive
?
-51i
of
"
"
of
Part
Unit, the Value
an
being
One
the
one
,
998999001
trillionth
of
'
of
the
Fradlion
left
out.
of
Operations
that
take
will
Whoever
the
with
mine
firft Products
their
Pains
thofc
to
of
widely
thefe laft three
compare
Mr, Cunn^
will perceive
differ
in
their
Fradlional
Parts.
Reafon,
The
as
follows
as
He
:
as
I take
muft
Decimal
have
confidered
Expreffions, viz.
iL21
confidered
it, ( for he aflignsnone
as
j
his
the ieveral
New
his firft he
muft
'
iecond
) muft
tipliers
Mulhave
^^
as
^999
be
;
and
his laft
^999
481,764800
as
"999999
And
his
he
omitted
the
End
having
the
of
thus confidered
o*s in the
Decimals
laft
neither
them,
two
is the Reafon
Multipliers.
increafe
nor
For
diminilh
why
o's at
the
Produft.
.Alio
(90)
C
V.
P.
A
H
Divifion of Circulates.
A
R
the
Dividend
and
the Divifor
Educe
for all Cafes.
Rule
General
Equivalent
the Rule
Single Fraftions, then proceed according
of Vulgar Fradions
*, and
prefcribedin Divilion
in a
Fraftion
arifingwill be the Quotient compleat
to
until
Denominator,
remain,
o
Quotient, you have
fought. But if neither
wi(h, you
Divide
And
"^
Fra"ion
=
"
4.
^
being divided,
which
;
are
Quotient compleat
*^
~g
be
may
i.
Equivalent Vulgar Fractions
ift.Their
will
by
19,1
as
have
you
and
Quotient near enough for your purpofe,
to give it as the Quotient approximately.
content
the
Example
tient
Quo-
true
think
you
late
Circu-
a
fo foon
happen
thofe
of
ceafe, when
may
the
then obtained
in its
could
difcover
*tillyou
or
by its
its Numerator
divide
if you
And
Vulgar Fra"ion.
you
their
to
above
as
and
in
a
^
gar
Vul-
diitded^
its Quotient 4,7.
producefor
Example
Divide
2.
by ,6.
115,4
^^
'
ift, Their
S.
E.
F.
and
are
10
9
ji
And
'039
-^
J
"
^
X0300
.
-:-
9
Vulgar Fraction
"
""
,
the
^
.
Quotient compleat
in
a
54
'^
;
-^
;
"
which
is equalto
192,407.
Example
)
(9"
ExamfU
2470,76 by 5,06.
Divide
ift. Their
3*
E.
are^^l^^
and
S, F.
j
And!17"7i^45i^f!13p4p^Quotient
pleat
in
rra"bion
Vulgar
a
ipAich
i
is
"qual "o
com-
4^^^5
rkicep
Example
4"
/
"
Divide
ift. Their
579,6 by
E.
,243.
^^
S.P.
and
are
^^
i7S??^
213
:=
-
10
j
999
"o
Alifl
^
tte
"^jo"ntcoinpl"t
in
^430
999
.
a
Volgar FraAion
*,
which
"
Wuldply
by
is
equal to agSijS
".
.
Finite.
"
,24
J
the Diyifor
2^%'fithe
QuotieM.
i*"M**"MM"M
"
"
W5
4864
"
"
^94594
729729
4864864
519*5999
which
is
579,6
the Dividend
P
as
above.
Example
( 9aJ
Zpcamfk
"
"
Divide,
ift, Their
1
E.
5.
"
67 by
,75.
^
S. F.
are
"
and
-
999
^""
68
167
J
.
IK070
which
}
,
is
Quotient compleat
^.
eqoaita
536830948595654478007419
by
101*4,97717
./.
E.
S.
F.
6.
23,41^.
122137^740
are
23"4i40o
^^
"
99000
^'
Qaottent
Fraftion
2627
1002
'/^
v^r
whofe
5
"
is
which
-
1
99999
627260
"
^^'"PJ^t
,;,yJs6oZoo
is equal
a
25666242713301
^221
'
Ift, Their
m
1838897*7.
Example
Divide
.
****"
"Z-^
67932
~
90
Vulgar Fradion
"
"
7:Zr.'^'Z:
909
j
90
in
Vulgar
a
10432,428287432538411^^1.
.
I fet the Anfwer
him,
may
in
As
often
be
After
Rule
thus, that he^ wjiofi?
proceed to find its Rqpetendv
Cuftom
above
manner
readily folved.
of
other
the
the
might
more
Examples,
Vancnes, viz.
all the
However,
of whofe
foregoinggeneralMethod.
l-carner"s
*^
-
.
to
Examples
can
occur
other
the
Me
Operations chiefly depend
And
in order
for
the
fliall diftribute all
in this Rule, into
three
ready Apprchenfion,
which
in this
compAr with
Authors, I fhall here exhibit
thods, the Foundation
on
prompts
"
-
Multiplication,fo here in Divifion,we muft
vcrv
content
to
give the Quotient approximately.
the
be
Curiofit/
I
Variety
(93
)
Variety (i,)
dcnd
Pure
Divide
as
Operation
to
oft,
is
Divifor
Finite
Expreffion, and its Divieither Single or
confifts of a Circulate
Compound,
Mixt, obfcrve the following Rule :
or
the
When
*till the
a
Decimals
in Common
apply
the
Quotient
given
turns
;
Circulate
out
a
but
be careful
in the
Circulate
:
in
the
Dividend
fo
if that does
or
be content
happen foToon as you could wilh, you
may
Quodent approximately. For indeed in many
to give your
in Pra"ice,
where
the
occur
Examples that may
except
its Dividend
is a fingleDigit, and
confifts of a
Divifor
not
chofen
Produft
with
of its
one
lingleCirculate, on fome
Faftors, thp Operation will frequentlyprove very tedious^
if
you
are
determined
to
find
its
Quotient.
circulating
Examples.
(2)
ip6 I ,925
Quote
'
Quote
P
2
15^
(4.)
ii95S"
(5.)
f94)
(";)
(5.)
Divide
581
by
Diride
9i
585,42
by
,7.
,71585,424242
81581,81
IM
Quote
Qooje
72,72
wMcIiia
S96,3"e54
70,
(7-)
Divide
7 1 4"*5^i4tt*5f
4,857142
by 7.
I4*857*4"857i42857i4a85y
L42857142
0^^,693877551020408163265^612244897959183673+
I took
of
down
the three
applyingthe given
Quotient turned
out
a
laft
Examples
Circulate
',
whichi
for
the
Conreniency
repeated until
the
Circulate^
(?^)
)
(95
96,378 by 58
Divide
"
"
58 196,378
1,(6)696179
~
"
383
348
1,6616961^
/Mvert
348
matdy.
'^
"
6^
.
"
"
t
T.
5"4
557
35"
348
J
03
58
.' I
457
406
5"8
ttj64
.5*
Thai,
S"mi2e
1
.H*,
to
"!""
"*"""? En-lfl^r!!*
MaltiplicMion, jjhere he
wll
meet
mmiMiiT
( 96)
divided
by
Quotient.
is the
Divifion
For
this is of
as
its Fadtors, will
of
one
the
give
Proof
bcft
other
of
for its
tion,
Multiplica-
that.
.
.
"
t
...
"
'
.
"
.
-
^
Variety (z.).
the Divifor
When
confifts
of
Cir(!ulate,either
a
Single
Pure
ics Dividend
of
Mixt, and
or
Compound,
the followix)g
nite "xpre(Iion" obferve
Dircftions.
or
Find'
the
Diyiror*sEquivalent.Single
Fradion
Denoniihator, ( which
tii this Variety and
without
o*s )
will always be 9 or p's, with
or
the Dividend
IntegralNumber,
as an
multiply
with
its
next
dered
divide
this Produ6k
red
that
by
the
Divifor^sNumerator,
the
Integral Number,
fought.
an
as
in
the
confi\
and
confide-
Quotient arifingfliallbe
I.
CASE
Circulates^ either Singleor Compound.
Pure
Of
then
5
,
Fi-.
a
^
Examples.
(I.)
(2.)
Divide
5664 by ,8,
Divide
^^
746,3 by
.
8
5
"
'
Here
,8 I 5664
^
-
is-the
-
^5
Here
) 746,3
I 50976
New
is the
S. F.
Dividend.
6372 Finite.
Quotp
"
Div^ifor's
^
Divifor'sE.
S. F.
8
,5.
5
| 6yi6^y
Qf
New
Dividends
Finite.
i343"34
'
"f
I here
fhem
ihanner
'
'
beg
the Rule
of
leave
to
above,
Vulgar
"-
thofe
Examples, and with
by working both Examples after the
IHuflrate
Fractions.
do
( 5"
)
hE.
RU
more
Mulriply the Dividend
by a, cutting,off one
Right-hand Figure in the Pl-^jkt,whfch"Biib*W your new
di.^ftde
then
Dividend
the Qspticnt will be
..and^
"s "l^,
;
juft.
1
t;
Let
mBHim
us
^laft Example,
my
Di^dc 746,3
by
\t
work
and
by
\A
\5.
"i#i.^"'*=^' 746,3
Multiply by
'
'
9
*"
1 ^yi^^?
Dlrt^"S5
1343,34
is the fame
Which
NcwDiiridcnd
=*
Quote
True
=
as
mine.
ParacJn^ /""
Air.
Quotient*
fxj
given KVifiir:bfe'i Single R4t"eRiid" apd 7pur
terminatft NumBfr,
multiply the Dividend
by
Dividend
a
Pro4pa
As,
by the^^vcn Divifor.
,^, and d"yide.l*a(t
If your
.
Divide746,3 by
qphe^Dtvidend
"fi.
^7^,3
"iJiv^or=,54^67i,"67
1543
He"e,
inff
m
it is manifcft that
Wo'
"
34
=
True
Quotient.
by inultiplyingby
,9, accordDividend
in
Pardon^ inftead of 9, the New
MiBft^s
t"ctfnRb thfe fime With Mr.
'dbe'i^atibfi
duft with
M
..
DTvIiSend.
New
=
Mr.
to
aid
mV'dWidKJg'9V'5,
three
.
^.
Multiply by/
..
...
either.
different
"
miiie,
Hence
Rules,
fec6hafc
rtot
and
the laA*
i5'"t^'O*'^^'
-,
IrtieW)-
though dfey tna"y fceih to
puzzle the
by that.means,
tfel,'
y^
ttefr #eas
HWd'maVcm^tee'Riuriaftm^
and
feriie
young
fiu"fe "afe bne
mWWr.
(99)
Method
My
being
Fradlionsy
Method
But
when
and
wholly
built
partly
that of
on
that
on
New
of
Principlesof
Deciilials
and
gar
Vultheir
"
Decimals.
.
obferve^that in either of
the
the
partly upon
Dividend
is
the three Methods
found,
the
Infinite
aforegoing,
Divifor
in
a Finite
Operation then becomes
Expreflion; elfe the
of Circulates ought to
of multiplying and fubilra"ting
Laws
be obferved, as underneath
where
the fame
Example is
the
,
wrought
at
large.
"
,5
"
I 746,30
in
Infinitum.
By this Operation you may perceive that there is an Infinite
and fi)continues
to be
ProduSlfdr every InfiniteRemainder
vamjhes into o, the Univerfal Symrepeated until the Infinity
bole or Char aSer for Infinity.
j
CL
(3.)
(
)
lOO
(3)
.7
(40
1 68,743
I "o"45
.7
9
7
9
j 6iS,687
New
"
Quote
Div^-
"
the
Liberty
New
Div**
"
8",38385jfi44,
I (hall take
| 94,05
7
here
to
,0006
"
Quote
13,43571428.
remind
the
Learner,
=
9000
6
,006
=
900
f
Their
That
the
federal
.
|^
Expreffion
^E.
S. F.
9
V
00000
6666,
=
9
and
t^
Let
J
propofe the following Queries,
742,85
be
given
to
be divided
by
viz.
each
of the Infinite
above.
"]q"reflions
(5-)
(
)
""""
(5)
the Dividend
I 742,85
xft, 9OO06
the Denominator
9000
6
the
j 6685650,00
J
1
True
14275,
New
as
above
Dividend
Quotient.
(60
6666
^th,
the Dividend
I 742,85
the Denominator
9
"0000
1 1
14275
exhibited
the
,
I have
amples, leaving
the
the New
I 6665^6$
True
Dividend
Quotient*
Operationsof
intermediate
the
above
at
the firft and
as
ones
laft Ejt"
Ettrcife
an
for
Learner.
(7)
Divide
6794
'
5,18*
5^80
U
5,18
Now
by
=
.
999
6794000
6794
5180
I 6787206
3E
IHrtdoid
K
999
1
1310,2714285
(rue
Quotient.
Ct2
It
(
the
Divifions
ncceflary to cxprefs fuch
It is not
caufc
)
I02
is the
Operation
fame,
large ;
Divifion
in
as
at
of
be-
mon
Com-
Decimals.
fame
The
of
would
Decimals,
the
wrought wholly upon
Example,
be
Principles
undcmcatb.
as
5"i8
6794
6794
5,18
1 6787,206
Dividend
=
x
,999
;
1310,2714285
Quotient
true
then
And
prefied thus
as
above.
under
the Rule
to
make
it
as
this 2d
with
Divifor's
the
its Denominator,
preffion,multiply
by
Mixt
a
the
^c.
confidered
Dividend
Numerator,
be
that
Exampl/{i.)
By
Divide
56,097
579^^75
Fraftion
a
ex-
;
then
Decimal
this Produ"b
confidered
be
IntegralExpreflion,'if the
Quotient arifing (hall
as
and
;
Expreflion, if the Divifbr
Finite
be
2.
Equivalent Single
Divifor*s
Given
the
the
fhould
univerfaK
RULE
Fmd
Variety
mixt,
Divifor
or
be
as
Ex-
divide
a
Finite
confidered
as
Integral,and
fought.
tins
laft Rule.
by 56,097.
5794,87500000
5794^75
5^y"97 I 579'^y^^705'^5
6ff.
103,29994565217
An
approximate Quotient.
=
Dividend
x,99999
"j
( 103
!tbe fame
)
the
Example hy
firftRule.
Now
^"^?"-
56.097
-
99999
5794,87500000
5794875
5609700
I 579481705,125
which
by Infpeftiononly we
the fame
Quotient as before.
Thus
I have
by, that
exhibited
Circulates, whether
Ways
two
Of
the
having Decimal
Exprefllon, if
all
Divifors
produce
Examples
arc
Pure
Circulates.
confifts
Divifor
work
in
it, and
the
of
its
Examples
is the fame
the Rule
Way,
work
muft
;
11.
Mxt
Places
you
to
99999
Single or Compound.
CASE
ift, When
x
plainlyfee
can
Variety, whofc
this
fall under
Dividend
=
with
a
Mixt
Circulate,
Dividend
Finite
a
wholly
in
a
mal
Deci-
the laft"
Examples
'
(I.)
Divide
5"o6
Subft.
50
2470,76 by 5,06.
2470,76
247076
4,56 I 2223^684
487,65
True
Quotient
Finite.
=
I omit
Dividend
the
x,99
Operation
the ReafoD
betbre
large,for
given*
at
(2.)
(
)
"04-
(2.)
"
Divide
by
2019000
2019000
24*925
Subft.
.
2019600
24
^'
"
249925*
*
I
24,901
fiividend ^099
2007981,000
81000
Integral.
Quotient Finite and
True
(3.)
6794,75
Divide
by 753^658.
^794*7500000)
Subft.
679475
7
M"
"
1
753*^51
67:94,6820525=^1
Diiridettd x"99999
^80^5306 fifc.
Quorieflt%
approxiiMt^
9,oi
An
confifts
the Divifof
2dly, When
having IntegralPlaces only,
and
of
Mixt
a
its Dividend
a
Circulate,
Finite
Ex-
prefljon. As,
(4-)
698,4 by 6347,
Divide
FtA
the kft
Rule.
6347 1h^i4
6984
Subft.
TUmt
1***^^
^34
19
i
I
M
I
^9797016
K
Dividend
x,99^
,110030216
an
approximate Quotient.
Fari"ty
( io6)
Examples of
Let
multiplied by
be
5,7
Singk
Mxt
or
9
99
tf^.
999
S"177
5.77
Subft.
57
Produft
or
Operation.
Operation.
Subft.
Circulates.
5,7x9
52,0=
57
Produa
572,0
$^7^99
=
Operation.
Operation.
5.77777
SuWJ:.
p.
Subft.
57
5772,0
P-
5',1^9SS
=
SI
57772.0
=
Sy7^9999
Operation.
Operation.
5.777777
I
Subft.
P-
and
S7777^*"^=
fo
Subft.
57
on
for any
5^7^99999
P-
(ingleCirculate
57
577777^'"'o"
5^7^999999
whatever.
Example
(
Examples of
"
Let
)
'07
Compouud Circulates*
Mxt
"
6^75
be
multipliedby
or
9
99
or
"r.
999
Operation.
Subft.
Bys obferving the
"
"
6,75""9
=
"
"
"
P.
6975x99
.669,00 =
E.
Subtraftion
"
"
60,81
Produft
of
Laws
6750,8
"
6^75^999
1 =
"""*"
Opert^tat,
"
Operatim,
t
"
Subft.
675
Subft.
"
6*7575757
6,757575
675
9
P.
67569,00=6,75x9999
P.
675750,8
"
"
1
w
WW
like
in
So
4792,5x9
manner
"
Produd
:
And
If any
pofe
"007"
Pure
wUl
give
its
"
Finite
IntegralNumber,
Single Circulate
its Produft
for
43133,3
47^2,5x99^=474466,6.And
A
r:;47978oo.
"
=6,75x99999
is
^4.
multipliedby
will be the fame
R
4792,5x999
9,
Figure
as
Finite
fupin
the
i"8
(
the
Place
Left-hand
next
( viz. ) ,07.
will be
9*s, its Prodaft
Left-hand
repeated in the next
of
the Number
j^'sm
faftie
the
two
to
)
Places
:
If
multiplied by
Figure Finite twice
fo on,
And
ing
accord-
Multiplier.
the
"
Examples of
As
,007
X
99
And
,77.
=
SingleCirculates.
Pure
X
,007
999
And
=
7,77.
=
777"77'
"
"
,007
X9999
fi"on.
likewife
So
And
77,77.
=
6
x
9
60
=
Pure
Examples of
Let
be
9875142
Op. by
"
"
"
Subft.
P.
7,876285
or
Op.
"
"
"
"
Subft.
875142
P.
86,639144
"
875142.
874,267732.
.^miam^tmmmmmmmmk
Op. by 99999.
"
"
,8751428751
"
"
,875142875.
by 9999.
"
"c^
999
Op. by 999..
mmmmmmm^m^mmmm^
kMBMiMiMM^ai^
Subft.
99,
99.
"
Subft.
875142
P.
or
9
,87514287
,8751428
^^
Circulates.
Compound
Op. by
9.
99999
"r.
hiultipliedby
"
"
X
,007
"
,87514287514
"
^75^4''^
"
Subft.
"
875142
^mm^
P.
8750.553608
P.
87513,412371
Qg.
(
op. by
)
109
999999-
,875142875142
Subft"
875142
P.
Finite.
875142,
if the fcveral given Circulating ExprerThat
Obfcrve,
been all* Integral
fions 'in the
preceding Examples had
Finite
Integrals,either
Figures 2^
you
Circulating, having
or
been
fame
the
fee in the fev^ral Prodads.
that
Rule
Exprefllons by
Number
any
9's ;
)
are
before
As,
their
alfo
( which
obtained
by obferving
be
given
to
60000
in fuch
Cafes
the
all Integrals
are
fame
multiplied by
be
to
Op. by 99,
666666
66666
P,
that
Laws
as
prefcribed.
let 6S"6
6666
fuppofe
us
"
"
Produds
Op. by 9.
Subft.
Let
or
Numbers,
of
9*5.
gral
87444, or 579467, 6?f. being all Inteber
were
given to be multiplied by any Num-
6666^
as
ot
"
"
fuch
and
alfo have
would
nothing might be wanting to compleat this
of compendioufly
multiplying any kind of circulating
But
are
Produdb
their leveral
then
Numbers,
Subft.
Finite.
or
99
Op. by 999*
Subft.
Finite.
6666
P. 6660000
Finite,
on.
R
or
6666666
6666
P. 660000
9
z
And
(
And
87444
be
Op. by
9.
let
)
no
be
given to
multipliedby
9
or
o"
99
i^e"
999
Of. by
8744444
874444
Sobft.
87444
P.
787000
99.
87444
P.
Finite
Finite,
8657000
and
loon.
"
And
999
let
"
"
be
579467
given
be
to
multipliedby
9
or
99
(mt
"^.
Op. by
9.
Note
5794679,
5794^7
Subft.
obferving
"
579^^7=^5794^1'
"
521^211,-579467x9.
p.
Op. by
Op.
99.
"
P-
by 999.
"
"
"
57946794
Subft.
of Subcra"ioa^
the Laws
"
579^^79^6
"
Subft.
57*^467
573^73^^*
P.
579467
57888S478,
m^"^^
Op. by
S999-
5794679467
579467
F.
5794100000
Finite, and
""
on..
Aftdi
/Ift
)
"
let
laftl/y
And
"
be
10124,97717
girco to
be
multiplied
by 99999Operalbn.
"
"
10124,9771717171
1012497717
P.
fo
tipliersyas
fo
off
Decimal
as
the
be
multiplied by
I
well
the
of
Here
if
all the
they had
laft "x"
would
or
out
the
Prolixity,
Remarks
more
many
I fhall
make
but
Circulating Exprefiion
any
is
9's, as the given Circulate conlifts of
Multiple thereof, there the Proany
a
Finite
it would
an
Number
the
many
turns
avoid
to
Improvement,
foregoing Products,
as
after
And
as
perfuaded
am
if
"
ProdudVs
where
That
Figures,
always
duift
but,
:
viz.
of
and
Multiplier in the
Expreflion, its Product
Decimal
a
preceding
made
this one,
Places
}
10124,8759219454..
From
might
Numbers
Mul-
hare
muft
Expreffion", then we
Places
of Figures in their feveof
Decimal
Mukipfiers confifted
"
been
hare
preceding
Icveral
for Inftance, if the
been
had
ample
the
as
As
Places.
more
many
Produ^,
lal
the
Integral
many
many
mark'd
confidercd
1 hare
Obferve^
been
1012487592:, 19454
Ejrpreflion.
be
for
an
entertainingExercife,
the Learner
by dividing
compendious
Manner
them
of
to
prove
by
their
fome
*
as^
of
pliers
Multi-
dividing, by
any
9'a.
follow
the
Examples which fal^ under this Variety..
work
them
wholly iiva Decimal
Way,
as
you
arc
bbferve
the DiredHons
following Examples,
under
laft Rule
in
Variety 2^
to
which
I refer the Reader.
(
)
""
(I)
Divide
8,724
by
,5.
8,7244
8724
New
New
Divifor ,5 17,8520
Finite*
Quotient 15,704
459,68 by
Divide
Dividend
7*
7l459,6M
45968
New
Divifor
7
New
| 4i3"7^
59,10285714.
Quotient
(3.)
Divide
78,048
,08,
78,0487
^8
"
"
Sttbft.
by
o
7 8048
,c8 1 70,2439
Quotient
New
Dividend
878,04.
(4.)
(
)
"4
(6.)
"
Divide
"
"
630,54 by
'"
"
630,545
4444.4
I
Subft.
4444t
4444
*
*
1
63054
"
"
4000,0
Quotient
Dividend
New
I 567,490
,141872
(7.)
"
Divide
"
"
7623,37 by
666666666,
"
r
"
"66666666^
666666666
Subft.
"
7623,373
762337
"
,0
"
Quotient
Thus
Digit
the
you
have
/)OOOi
a
Method
New
Dividend
"
1435^6
how
infinitelyrepeated, whether
Integralor Decimal
"
1 6861,036
to
it
divide
b^ins
by any
any
fingle
where
b
Places.
(8.)
v^
(
)
"5
("")
"
Divide
"
243.306306
111,98
-i^^^^k
^^3306
II
'"
C
S
laft three
^^^
been
New
1,87 1243,063000
"
The
Figures
"'ght have
omittSl.
"
"
1 1
"
243,306 by 111,98.
Dividend
22374
2,172
19323
Quotient.
11187
81360
78309
1
I
}"
^ i, Afimtum.
30510
",
22374
J
8136
(9.)
Divide
"35"
,095823 by
,351.
.095823095
095823
"35i'
.09572727a
New
Dividend
702
,27
Quotient
2552
2457
.
**
^^fi^**^'
f
095
(10.)
( 116)
("o.)
"
"
Divide
,167 by
,1671
*7S
"
"
167
7
"
,22
1
25
,75,
Dividead
New
,681,1504
136
tSc.
144
136
85
6%
"7o
136
344
340
^/.
4
,22:125 ^^.
Example 5. in the General
of
found
on
by carrying,
Quotient
Figures of
the ^cw
the
i^
Rule
the
Circulate
i
true
which
Quotient,
alfo
Dirifion, and
under
might here be
applying the
alternately.
504
(II.)
"
120,54
46,21
Subft.
by 46,2 J
12054
4
Quotient 2^60829346092
by carrying on the Diviiion,
the New
.
120,54545
New
46,171 120,42490
of
"
Circulate
90
Dividend
"s?r.
and
approximately, found
applying of the Figures
alternately.
(12.)
)
("'7
"
,4681
,5552 33"i by
Divide
"
,4681 ,595238095
Subft"
*
*
1
5952^80
46)
_
"589^857i4
New
'Qootiene 1537139233934
Dividend
"c.
afupnudmately.
(13.)
"
Divide
"
8,63 by ,07317.
in
S
a
h^nitim.
"m"!^
ii8
(
"
Examples of Integrals.
(H)
"
"
Divide
3347987.987
57945
Subft.
57
57888
^7 57945"
33479^7*
987
3347
I 3344640,
Dividend,
New
289440
cr
"
'
a
450240
57*
405216
Quotient
the
Mulripli-
being ^g^g.
"
)
(
,
:
"
45024
Obfervc,
Number
of
its Divifor,
then
its New
I chufe
Proof
to
"
the
Places
of
does
Dividend
to
the
exhibit
tn
InJtHttum.
S,
)"
confifts of the fame
given Dividend
Figures in its Circulate with thofe of
confift
of fonie aliquotPart
thereof,
will turn
Finite Expreflion.
out
a
When
or
^
the
preceding
following Example^
becaufe
it is
a
one.
e
('5)
(
)
"9
US')
33479^7
"
"
"
"
Divide
^7
"
49
I leave
Before
of
two
with
very
J
this
Variety, I cannot
help taking Notice
particularthings, viz, Firft, that I never
met
Example
an
57-
(
in all the Authors
I
have
the Circulate
its Dividend
in
Subjeft ) where
requiredin Subftrading it to carry one
in my
hand
Column,
as
preceding Examples
6th, iich, iirhy 13th, and ]i5th,which. I
purpofc.
feen
this
on
mation
by Transforits Right-
to
viz, the
4th,.
contrived
oa
^
And
2dly,
Example,
fewer
a
due
of
1 1,
Attention
not
Figures than
and
13,
in the
that I
remember
the Circulate
where
Places
Examples
I dp
except
ever
in its Dividend
of
that
One,
Propofer,
is
its
and
with
met
confifted
Divifpr^
as
that, 'for
an
of
in my
want
wrought falfely.
CHAP.
(rf
(
)
120
VI.
CHAP.
ReduStton
Circulatu.
of
I.
CAS^E
Time,
or
Mcafurcs,
Money, Weights
their Equivalent DeciiiKkl
ExpreiTions, or
reduce
TO
,
i^c.
to
it, I know
near
(byDiviflon)
Firft, Reduce
of
there
it, whether
above
Example er
noc
This
2dly,
to
given
this
Decimal
the
to
Example
D"
S
4
12
^
Io
1
13
3"
of
Quotient add
the
if there
be
of
next
be
that
Species
Number
any.
And
ding
higher Species, ad-
Number
of
that
Species
fo
and
proceed^
any ;
the Integralfought.
(u)
'Q0"
:
3
the
to
Decimal
of
L.
a
Steriii^
0:;-
I 11,75
I 1 3,979
jinfw.
it
:
Decimal
of that
the
if there
Example,
arrive
Reduce
the
tfielowcft
Number
the
to
the
to
any
Qaotient found,
this
until you
be
Example,
reduce
laft Sum
in the
to
;
Species in the
that
thau
thcNumberof
ETample,.
the
in
Species, given
in the
Method
readier
a
Rule.
following
next
not
D.
1
with
6 5. with
,"989583
of
D.
II
a
1
added
3^. added
L,
to
to
,75 D.
"979
1
the
6^.
ift"^
the 2d Q^
Sterling.
ja
I
(
)
^2^
Example (4.^
*
Pint!.
Gall*.
Reduce
63
r
39
Gallons.
the
to
7
'
of
Decimal
'
r,;
an
Hogfhcad
of
"
I 7, Pints
8
^ 9
63 -"
^7\
Gallons
I 39*875
"
:
4*4305
Anfwer
963293 6507
of
Hogfliead
an
.
Example (5:)
"
Reduce
Zodiac
of
17
"
"
/
:
44
:
to
19
Decimal
the
of
a
Sign
of
the
30.
Seconds
60
I 19
60
I 44,316
30
1 17.73861
Primes
"
Anfwer
,59128703
Degrees
"
of
a
Sign
of the
2^iac.
Example
(
)
i"3
Exampli (6.)
*
Reduce
of
dmal
*^"
It
7
:
:o
a
it
Duodecimal
Fradion
to
the
De-
Foot.
a
Thirds
12
I
12
1 ,916 Seconds
II
Pri
I 7,07638
12
"
"
^njw.
,589699074
CASE
of
Foot.
a
II.
Of Reduffion.
CirculatingExpreffionto its loweft
poffibleEquivalent Vulgar Fraftton.
How
reduce
to
any
RULE.
Find
and
12,
of
Equivalent Single Fradion, as taught in
this new
widi
Expreffion proceed, as in the
its
of
Reduftion
Vulgar
Fractions
So
Ihall you
thod
Metain
ob-
Fradtion.
Equivalent Vulgar
its lowed
:
jfrf.
Example (i.)
Reduce
,571428
ift, ,571428
^
being
will
produce
"
its loweft
^li^
,
999999
by the
its loweft
Equivalent Vulgar
its E.
=
reduced
fion
to
Method
V.
F.
^
of
Which
Exprcf.
^
Vulgar Fraflions,
Equivalent Vulgar
T
tion.
Frac-
Fraftion.
Example
( I*!-)
r
^anfte
^^
ift, ,3863
(2.)
V. F.
it""qiii(ralene
=
**
^
9900
^-
And
'-^ its \tm^
^
=
9900
". V.
P.
44
EDtamptt (3.)
"
"
Reduce
3,642*571
its loweft
to
^
ift, 3,6428571
^^^
Equivaknt
its
=
V.
F.
Equivalent V.
F.
999999"
3"l2!5i5
And
1%
=
omitted
(killed in
Pcrfon
of
finding
V."
F.
E.
14
9999990
I have
loweft
their
Operations
Fradions
Vulgar
lai^ge^becaiife every
at
muft
Mcafure
the greateftcommon
know
to
the Method
given
two
any
Numbersv
CASE
III.
Of Reduaion.
How
which
to
expreflesfomc
which
Time,
fiftd the
to
Value
ktibwn
refers, whether
drMcafures, i^c.
it
of
any
Part
it
or
be
CirculatingDecimal,
Parts
to-
of
that
Money,
Integer,
Weights,
RULE.
given Expreflion, ( according to the Laws
of Units
of Circulating Numbers
) by the Number
tained
con-
Multiply
in the
which
the
the
next
given
lower
Denomination
Expreflion refers
;
of
and
fo
that
Species,
proceed
to
to
mul-
tiply
tiply by
its lowed
ral Parts
its
lower
next
Parts
:
until
Denominations,
the fpwxjcalProduds
and
come
you
fhall be
to
the feve-
required.
"
"_
i/, O/CoiN..
Example,
Reduce
,8739583
to
(i.)
known
the
Parts of
a
X. Sterlings
20
12
D'
^.
Reduce
5j750OO
a
'3
r
3,00
,59920634
to
Parts
known
the
of
a
Guinea
Sterling.
21
ii984i2"98
7
S.
s.
'^?^^'
12,58333333
D*
^^-7
12
2). 7,000
T
a
Efcampk
(
126
)
Example (3.)
"
Reduce
".
to
"49074
the known
Parts of
a
Moidore
of
zyS,
27
the
tiplier
Operations,obferve, When
any Mulis compofed
of two
1 2
Digits, either
greater than
of i2's, and
o*s } or
without
fome
other
with
or
Digits,
o's \ then multiply by the Multiplieither with or without
ers
and the laft Refult fhall
alternately,
compofed Numbers
be the Product
fought.
But
to
contraft
ExampU
959920634
Reduce
7
to
(2.) refumed.
the
known
Parts
of
Guinea
a
Sterling.
4" 19444444
3
S.
12,583
iz
D.
S.
"
D.
Jf^er
X2
:
7
7"ooo
Example
( "a7
)
,
Example (3.) njumei.
"
"
Reduce
the known
to
949074
of
Parts
9
I
Moidorc
a
S.
%y
S.
Anfwer
In
all Refults
Repetend
the
latter
of
a
only
where
the
as
of
)5
to
Places
of
Figures,
other
retain
above*
Example
Reduce
D.
13:3
confifts of (bme
Repetend
fewer Number
of
the known
Parts
(4.)
of
a
L,
Sterling.
4
^^
1,3
Example
'( laS.)
Example
Reduce
,8984375
the
to
(5.)
of
Parts
known
of
Mark
a
5.13:4.
-""
"
"
"""-
"
-^^"""
"
12^^53125=^
9
C.
P.
X
10.
"
,29947916
26953125
8984375
11,97916666
S.
S.
^^
Anfwer
*
1
D.
iz
:
ii"
-
4
D.
11,75000
4.
-
^.
3,00
2''6'"Q^
W
Example
E
H
I O
T
S.
(i.)
"
""
,
Reduce
,571428
^
to
Parts
the known
Anfwer
of
ii
a
Tun
:
i
Aver-
:
20
4
^
1,714285
7_
of
4
fi)|20,o^
jl^f^cr
II
a
Tun.
:
3:17:11
exaft.
( I30
)
Example (3.)
Reduce
^8958 3
to
the known
Parts
of lb
Troy;
12
O"-
10,75000
20
Oss.
'
D^
Jnfwer
15,00
Example
"
Reduce
,9772
Dwt.
:
10
15
(4.)
"
to
the known
Parts
of
a
15
Troy,
12
Ozs"
jfnfiff.II
Thus
Decimal
two
or
Rule,
Dwt.
:
14
Gxaas,
:
13,09
Parts of
proceed to find the known
Expreffion given. I Ihall therefore propofe
and
with
conclude
them
three Examples more,
leaving their Operations to the Practice of
you
may
any
but
this
the
Learner.
S^fyj
(
0/Time;
3"%
"
Heduce
)
I^I
"
"
,9285714
"
'
"
the. known
to
oF
Parts
a
Year
of
365,25 Days.
Anfwer
,
339
:
many
and
Feet
2
25,71428.
:
Measures.
of
is ,972
Inches
Yard.
a
Inches*
Feet*
Anfw"r
51
:
3
d^bly^ Of
How
Secoads.
Mm.
Hft"
Dtyt.
:
'
ii"
ft
t
How
many
Poles,
13
:
of
many
a
of
Sign
the
Anfaoer
30
,34469
of
a
Furlong.
i.
Mot
ion.
Seconds,
Degrees, Minutes,
is
,5912^703
Zodiac.
17
Seeonli"
Ir^utet.
Degrea.
Note,
:
4
Sthly^ Of
How
is
Foot*
Yards.
Poles.
Anfwer
Feet
Yards,
"
44
:
Degrees nuke
V
one
:
.
19..
Sign
of the Zodiac.
CHAl".
"
(
)
i8"
VII.
CHAP.
Involution
Of
Evolution of Circulating
and
Numbers.
i/, Of
pBfiniHoHand
As
28.
For
IS
^
^
I
I
E
H
Involution.
continued
intoitftif
of
Mulriplication
IffuoWion^ or
is caHed
railing the
Example :
HMh.
feveral
any
the
of that
Powers
Quantitf
Quantity.
29. If a Quantity be multipliedby t^If, the
called its Square, or 2d Power
its i ft Pdirw"
1
being
the
itfelf.
given Quantity
multiplied by !t" firft Power,
CiAe,
or
3d t^ower.
by its ift Power,
Thus
4th Power.
you
pleafeof
And
the
that Produ"
you
That
or
Root"
being
is caUed
its
beirigmultiplied
is called its Biquadrate,or
Pov^er
3d
raife what
to
on
given Quantity,
any
Produft
Power
Produ"
the
proceed
may
2d
of
manner
whether
er
Pow-
Finite,
or
Circulate.
A
TABLE
of InfinihSquares proceeding
from thefeveral
InfinUeEicprejions
from
Vheir
"
"
Now
,1
,2
"
=
9O49382716
"
"
"
"
,4,x,4
"
=
"
"
"
"
X
"
,4
t
"
"
"
,8x,8
,9X,9
=
"
*7^*7
"
"
"
,6
"
"
"i9753c864
,308641975.
,5x^5=:
96
"
"
"oi2345679
"
"
Powers.
"
x,.2
,2
2d
Inctkfive.
=;
"
"
And
"
"
x,x
,1
"
Squaresor
"
".
to ^^
yi
,604938271
"
=
"
"
,790123456
"
=
1,0
Nottr
(
"
"
As
Nctij
"
90123456799
Root
of
,1
of
,3.
that the
as
Square
deroonftrated
is
4th
"
"
is the
"
"
of
Square
,4,
"
"
is ,6, therelbre
,4
is the
"
is
9197530864
of
,9 is
equal to
Unity,
or
i
is evi-
continued
is equal to i,
,9 infinitely
And
the Square of i, is
( in Art* 13O
That
hence.
from
dent
,012345679,
of ,6.
the 4th Power
And
of
Root
Square
die
and
the
"
93,197530864
"
and
,1,
"
"
And
of
Square
therefore
is ,3,
"
Power
is the
"
"
Square
)
'33
*
"
.
I,
for
And
100
Square
therrfore the
;
the
of
Square
Square of
Square
of
is equal
to
the
to
of
Square
equal to
is
99,
equal
is
the
eqpal tp
,0001
and
,000001,
of
Square
%
i.
is
and
the
And
:
fo
Q"
dw
on.
equal
,09
equal
is
9,
ioooo
is^qualjtoio6ooo6i"artd
999,
,009
is alfo
^
iaine Reafon
the
fo likewife
And
of
to
Square
,01
$
the
of ,0009
fo on:
1
"
-
.
And
are
as
the
feverally i
2d,
or
3d, 4th, 5th, or
Unity
fo likewife
",
6th
Powers,
the
fcfr. of
2d, 3d, 4th, 5di,
"
or
6th
Powers,
f^e.
of
,9
U
arc
2
fevewlly
i
i
pr
Unity.
Here
)
f 134
Sera
TJB
folldws".
ibe
frm
LE-.of
InfiniteCuieSi proceedings
the
from
feveral htfinite EpcpreJJions^
,1
t$
y^,Inr
clufive.
Their
"
"
Now
,i
"
"
"
Cubes
or.
3^ Powert.
"
,00137174211248285^2435939
,itx,ixii'=
6433470507544581618655692
7297668038408779149519890
the
260631
of
I..
,2-
Cub^
or
3d
Power
,1.
.
And
,oi09759368998^8257"875ijr
,2Xi2X,2=
14677640603566.52949245541
"
,
838^344307^170233196159122;
"
2
^
the
085048
.a?'*3"'"3
.3:
^
"o37,
=
Cube
of
of
,3,
the Cube
,2..
.
,4?",4X"4.=="p87:f5H95ia89026o63iooi37-
,4
"'
'
K
124828532235939643;34
17421
7050754458i6i865569272976"
'
^
,5
"
"
the
680384
=
,5x,5"""5
Cube
of
,4..
""7i4677.64o6o3566529492455
4183813443072702331961591
2208504801097393689986282.
"
"
the
578875
,
,
"
"
,6
,6x,6x,6
,7
.7"""7'".7
"
=
=
Cube
,5.
"
"
,296
of
the Cube.of
,6.
,4705075445816186556927297
6680384087791495198902606
31001
37 1742
396433
II
the Cube
248285322359,
of ,7*
Roots
f
*^
....
"
f
"8
,
,8x,8x,8
"
)
125
,7023319615912208504801097
3936899862825788751714677
6406035665294924554183813.
=
^
"S*
K
^
the
44.3072
o
^
"
"9^99^i9
"9
f"
"
"
"
t
For
much
as
as
"s?r. as
,00137
,1,
oP
Root
Square
of ,9.
above, is the 3d Power
"
the
,8.
"
the Cube
ijO
=
"
and
of
Cube
"
is ,3^
,1
of
"
therefore
fcfr.
,00137
"
is-the 6th
of ,3.
Power
"
And
and
,08779
as
the
"
fc?^.
above,
as
of
Root
Square
is the
,4, is
3d
,6, therefore
of ,4,,
Power,
(sfc. is
,08779
m
fhould
Whoever
"
of ,6,
Power
6th
the
be
"""""
,j
inclined
raife
to
the
4tK Powers
of
"
,2
,4,
,8, will
,7 and
,5
find
that
each
CfrculatingEx-"
Places of Figures deep. And
in
of CompoundCirculates, we
muft
take an
approximate Power, inftead
prefCon will confift of 729
raifingthe 2d or 3d Powers
frequentlybe
content
of
one
the
exa"
Hundreds,
or
However^,3,
of ,6,
and
to
which
;
Thoufands
fome
as
will
the
to
are
ad,
be
often
very
of
of
fome
Figures,deep.
3d, 4thj
found
confift
with
and
5th
little or
Powers
no
of
trouble, I
dhfdfe in this place to exhibit their Operations at large ^thac
the
him, might know
Curio"y. fhould
lie, wbofe
prompt
Method
fhorteft
"""
)):,?
how
raife the
like
or
higher
Powers
of
"
""""
,4 ,5^^,7
to
or
,8,
or
of
any
Compound
Repetends,
lit.
{ "36)
f
ift.
Let
"3 be
given
to
be
involved
to
its
5th
Power;
Opiration^
1,0
its
,1
Sqiure,
2d
or
Power.
*3
1
.3
its
,037
Cube,
or
3d
Power.
"3
I, III
1012345679
its Biquadrate, or
4th Power.
I"035
91*037037037
,004115226337448559670781893
5di
or
ics ^orfidid,
Power.
I
.
And
duft
if this laft Rcfult
divided
foyoa
inclined
may
by
9,
the
proceed on
multipliedby
be
Quotient will
to
raife what
be
and
,3,
its 6th
Power
of
its Pro-
Power,
,3
-
you
and
arc
to"
Obfervc
the
4th Power
of
,3
is the 2d
Power
of
,
1.
2dly,
( 13*
)
JExample (i.)
the
Find
of
Square
,36.
Operation.
Note
.J6
36
,36
=
"99
2l8
"
"
1090
1 13
99
09
22
13
09
3"
I
I
09
09
09
40
49*58 67 7685
09
09
09
09
09
09
94-03
12
I
22
.13
40
31
67 76 85
58
49
95
04.
Square is the circulatingExpreflion, as inark*d
Its
13
22
above.
"
"
After
be
to
will
like
the
Method
the
,0330578512396694214876
be
found
be
to
we
as
:
"
918
And
the
will be found
Square
of ,72
,5289256198347107438016.
otherwife
Or
Forafmuch
of
Square
thus
already
have
:
obtained
the
Square
of
'
"
"
"
,36,
and
therefore
that
the
the
( that is
"
"
"
x
Square
of
"
is the
one
half of
"
,36,
^ of the Square of
"
4
"
"
the
=
"
22
4
^3)5will give
"
Expreffion ,18
y
"
,1.8.
And
( 139
t
And
the
as
"
Expreffion ,72
,36, therefore
times
4
)
is twice
( that
is
2x2
"
"
,36
will
the
give
Square
obfcrve. There
For
Similar
Integral,Mixt
former
The
Expreffions,and
Expreffion ,36
the 3
of
or
or
4
as
of
of
be
to
be
Finite
as
( the
between
or
arifingfhall
5
.
confidercd
,36,
or
"
as
its
-
or
"
"
.
or
-V-
Multiple of
any
the
of
The
.^lultiplication
:
Square required,
'
"
-7-
of
fonu:
.
aliquot Put
.6?^. I .firft
"
Gf^;. and
"
;
with
of
^
fquare the
it divide
the
69
4
t
Infinite
as
.
Square
.
or
3
2
"
"
their
6
.
be the
the
.
lar
Simi-
ones.
Square of
tlve Laws
to
if I wanted
their
and
Powers
.
But
ference
Dif-
Figures arifing in
to
are
tween
befubfifting
I firft fquare
7 times "c.
7 (^c. and with it multiplythe Square
4
.
6
or
5
3
Square
their Roots,
is found
the
want
the
Harmony
and
indeed
I
,j6, according
Produft
fame
as
the latter
then, if
Hence
of
'
Fraftional, Finite
or
that
"
of Places
excepted )
Powers
Roots.
is the
Number
4)
Times
,72.
Infinite Powers
of the
feveral
of
2
=
'
"
or
"
of
of Divifion : the
,36, according to the Laws
Square
arifing Ihall be the Square required. And
(Rodent
not
the
but
the
only
aliquotParts,
aliquaatParts thereof alio
it would
might be taken too ; but frequently
prove a tedious
I
Operation
i
fuch
thofe
i
"
29
"
in
JV*^, The
all
8^-
"
13
fame
higher Similar
II
6?r*
"
13
as
5
^
as
7
"
17
Or
fuch Expreffions
*^
19
6?r,
17
Harmony
fubfifis in Similar
Powers
i
X
only thoe
the
Cubes, and
Multiples,or
Aliquot
(
Aliquot Parts,
fions required.
muft
be
I40
)
cubed,
^c.
der
of
Square
{viz.)
"
obtain
the
Expref-
Examples.
Atore
The
to
)i4%857
is the
,02040816
"(.
circoladng Exprefllon unSee
the
Expreflion
at
49
large in
the Table.
,285714
4
"
"
9
And
16
.42
"-
times
the lame
is the
Eiq"refllon
857
S^- of ""
,571428
"
"
"7i4285
36J
"857i4"
"
The
Square
of
"
is
,54
,2975206611570247933884.
"
"
iTfaeSqiwpof
"
The
Square
"
,360
of
is
"
'
'
"
'
"
,129859589319048778508237
967697427156886616346075
805535264994724454183913.
.643373102832562299202175
"
"
t
14S1
i"
TKe
2
10940670400.
"
Sqqar*."rf.i,^3is .40495867768595O4i3a23i
o
*
*
i
^
.
"
-^".
Find
r
"
\
"
)
("4X
Fmd
the
Square
oF ,i6.
Jttfie^,0^75
,o"7
Find
Square of 8,3.
the
Multiply 8,3
by
Subft.
8,3
8
j^
New
Multiplier^
416
5833
9
1 625,0
Anfwer 69,41
69,4
X
2
Find
(
)
"4a
"
Find
the
Square
of
"
54963.
"
"
Multiply 54563
bf 54.63
Subft.
54
New
5409
"
Muldplier.
"
7"
491
OQO
"
a
"
185454
273x8181
LJ
99
I
*9
1
09LWo90909'o909)09
28, 0.9 09 09 0909
'I
'9 84 12 a I '303948
15
I
I
i"
I
I
57^75^*493
I
ll l| ll
II
02
2I
1
lUo 29
2i
21
2i
2.
2985,13223140.4958677685950413223140
^"
I
"
"
M^"
"
f^i^"
Anftver^ Its Square is the Compound
outk'd
"^l^l
MiM
"
HI
"
I
"
I
Circulate^
"
as^
above.
Find:
(
Find
Multiply
the
Square of
,027.
Note
,027
)
"43
the
32.
Exprcflion,027
=
999
by
,027
Therefore
is the New
27
Multiplier.
189
"
"
0540
,729
I'll
I yy^B
599
f
I
r
"fr"
729' 729|729'729
^c.
187 916)645
374
4l
5
31
J
'729729
729
458
460
189
649
108-8385682980^77
5748721694667640613586559532505478451424397370,
730
"ooo
919
379
the
34331628926223520.81811541271;
Obfcrve,
That
6th
Power,
eafy
to
let
6fr. of
the
Depreientthe
fuppofing its
higher
that
or
too
Hand
the
Root,
or
of
ift Power
begin
to
lA
or
the
in
littl"
a
like Root,
the
^27-..
2x1, 3d,. 4th, 5th, or
ift Power,
it is very
Expreffion, with
fame
of
next
Alteration,.
Power,, by
Place, either
the
Places
Integral or Decimal
the Decimal
Diftindlion
by only removing
in
lower,
four
Places
of
Figures,
towards
and
v
either-
the
right
:.
Or
elfe
by- prefixing00,
Decimal
the
Expreffion is
after that
J^owfr,,
to
what
cither
manner
or
a
you
or
000,
before
Diftindion
the
And
have
any
Powers
three,,or
two,
if you
Square
2d,
the
or
Degree higher
Whole,
3d,
remove
may
oc
coco's,
or
and
placing.
according
4Ch Power^
its
Root,
as.
(slc^.
or
rib
bwenyou^pleafe*.
For.
(
Indance
For
which
of
Root,
or
Units,
of
Place
Power,
Now
Unit.
dn
ft
I
Tens,
Squares will
be
thus
in the Place
of
Units
and
And
of
of
of
and
,004
"
likewife
So
Root,
or
I
ft
Tens
s
to
begin
Tens
of
6
or
feveral
in the
Thou-
muft
Squares
be
"
and
,00004
repeat
of
Place
the Root
their
to
^c.
Thoufandths,
"r.
feveral
their
begin
to
Hundreds,
"
"
6?r.
}
in the
begin
to
in the
have
or
Unit,
an
6
Root
4444,
to
Hundredths,
fandths
and
-,
the contrary,
on
Place
for the
44,
;
Root
Hundreds,
or
in the Place
444444,
the
have
to
or
of ,6
Square, or 2d Power
begins at the Place of Tenths
the
is
,4
-,
)
'4-4
"f^.
,0000004
"
"
,037
is the Cube,
or
begins
the
Power,
at
of
3d Power
,3
;
of Tenths
Place
which
of
an
"
Unit.
Now
Units,
or
"
and
And
the
Tens,
"
"
be 037,
of
"
fo
or
lower
as
Powers
th"s
make
to
at
their
have
the
Root
will
we
Cubes
muft
be
"
,000000000037.
may
Roots
of Thoufandths
Tens
"
in
begin
3 to
or
feveral
their
and
manner
of
"f^.
"
,000000037
after
of
Many
(^c.
"
and
,000037
And
Unit,
Place
"
Thoufandths,
or
in the
their feveral Cubes
037037037,
to
contrary,
Hundreths,
an
"c.
"
and
begin
to
3
Hundreds,
037037,
of
"
or
the Root
"
the
on
Place
have
to
to
ers,
proceed with any Powbegin any where, higher
Plcafure, regard being
had
to
be
made
the different
terations
Al-
their different Periods.
more
I have
Obfervations
here
might
concerning
exhibited, and their Roots, "r.
but I
the
am
perfuaded
Method
The
)
extrafting
the
with
fame
is the
Powers
of
t 146
that
of
of
Roots
Circulating
Numerical
other
Powers
being taken in the Difpofition of the feveral Periods
applying them alternately( like as in Divifion ) to each
by
care
Refolvend
long
as
It is befide
in
therefore
mu"k
Subje"b"
than
intended
my
this Place
Canons,
that
the Procels
as
refer
the
And
I
the
to
new
is continued*
Brevity
for
;
;
the
lay down
to
of
Powers
other
Books
Refolution
Reader
confult
to
fuppofe I
Ingenious Mr.
fend
cannot
Ward'i
Rules,
him
to
a
or
I
:
on
ter,
bet-
Maihemati'-
Tmtng
ttaifs Guide.
XAMP
E
L
ES
in the
Square
Root.
"
"
"
(i.)
the
What's
It would
needlefs
be
Reader
large, my
Square
of
exhibit
of
to
Extradions
cxprels their Preparations
with
,012345679
?
the feveral
being fuppofed
the Method
with
to
Root
Operations at
be throughly acquainted
therefore
I (hall only
;
their Roots
as
follows.
Tbej^ven Refolvend Prepared.
,0123456790
(2.) What*s
6?r.
the
(,ijiii
Square
Root
iSc.
of
its Root.
,4
?
Preparation.
*.
,
"
"
t
fSc, i y666 (^c.
,444444
(3.) What's
the
Square
its Root.
Root
of
,132231
t^c f
Vide
Preparation.
,I3?23
1404958
iSc.
(,363636
6fr.
its Root.
(4.)
(4.) What*"
the Cube
tlM
)
H7
{
Sfloc of ,0011717
Squirs
^f-
Vide
of i.
"
"
^r.
,001371742112
t3c. it9 Root.
(,037037
\
"
(5.) Wha^s
fogt
the
Square
of
Root
ti"985S"56?^
^
Vide
140.
Preparotm.
"
G?f.
,129859589319
"
"ff. its Root.
(,360360
"
the
(6.) What's
Square
Root
of
"
,027
?
Preparatim.
"
"
"
"
fSc.
,02702702
the
(7.) What's
"
(,1666 6ff. its Root.
Square
Root
of
,0204081
(ie ?
Vide
in the Table,
49
Prepantitu.
,020408163265
(8) What's
tbe
6f"^.
(.142857
S^oaic Root
of
ejftf.its Root.
69,4?
Preparotm.
"
"
"
,69,444444
6ff. (
8,333 Cj?f. its Root.
(9.) What's
F^^
tht
(9O What's
page
( 1+8
)
ScfiArtRoot
of
2985,132
(^c?
Vide
142.
Preparation.
"
"
6?^.
,1985,1 3"a3H^
E
XAMT
LES
(i.) What*sthc
^ahle
of
(54"6363
Cube
the
in
Root
Cs?f. its Root.
of
Cuhe
Root.
,00137171
t^ct
Vide
Cubes.
Preparation.
"
^
"
"*?r. (jiiii
,001371742112
the Cube
(2.) What's
Root
of
fc?^. its Root.
,037
?
Preparation.
"
"
"
tff.
,037037037037
(3.)
^able
the
What's
of
( ,3333
Cube
Root
^f-
of
Jts Roo'"
,70233196 ^c
f
Vide
Cubes.
Preparation.
"
"
"
,702331961591
^c,
(,JJ888 6?f. its Root.
"
XJ
Mr
(
LES
XAMP
E
in the
)
'49
Biquadrati R)ot^
or
Siiuar"
fyuared
Root.
the
(i.) What's
Tabic
Vide
Biquadrate
of
Root
"o
12345679?
of Squares.
RULE.
Flrft
and
the
extraft
then
Square
Square Root
the
the
of
Root
given Refolvend
;
will be itsi
Biquadrate
of its Root
required.
Root
Preparation
"
"
"
"
,0123456790
"
"
^".
6f^.
(2.) Wnat*s
the
Root
Preparation
,1975308641
t^i.
"
,44444444
its
of
Biquadrate Root*
Ofr?
,197530864
i.
fef^. its firft Root.
(,44444
Preparation
"
2.
6f^.
(,33333
Biquadrate
its firft Root.
^c.
(jiiiii
Preparation
,iiiiiliiii
1.
2.
t
C^c.
( ^6666
C^c. ks
Biquadrate Root.
"
Y
2
"
"
EXAM^
(ISO
EXAMPLES
In the
it^ tbe sib
the
(t.) Whu*s
page 136.
)
bavEicamplts
SittftSdXMt, tr
Power
gheH
to
find Us Root,
Surlblid Rootb!
"t
,00411
Vide
?
Pftparathn.
"
^0041
What's
(2.)
Vide
"
"
"
6ff.
152*6337448
page
the
(,333
^e,
Rooc
of
Surfolid
M
Root.
fcfr ?
,13168724
137.
Prepantion.
"
"
"
"
i^.
^31687242798353
EXAMPLES
Root
in
i
or
tbe
( ,665 Cs'r. its Surfolid
Square CiAed^
baving
Exav^ks
tbe 6tb
or
Power
Cube
ghen
Rooc.
Squared
to
find its
Ami.
RULE,
Firft Extrad
and
then
Squared
die
Crfie
the
Root
R"M
Root
sf
of
dw
Refolvend
will be
Root
m
die Cabt
Sqoand
the
$
Qibe
f^t f Vide
of ,00137
Root
Cubes.
Preparation
i.
"
,00137x742x12482853
Square
given
required^
(i.)wimp's
f^of
Square
6fr*
"
("037037037
(^c.
its
Kooc.
Pripanakn.
( IS'
)
i
PreparatioA
"
"
m
"^.
^37037037
(2,) What's
the
ks
ii".
( ,333
Squared koot
Cube
Preparation
,087791495198902606
Squared
Squared
Root.
of ^oZyygi^^
(^c f
"
"
( 5666 "r.
"r.
only add
to give an
naft
extrad
the
Root,
(^c.
of
its Roots
the
Truth,
plying
the
2.
For
from
%
or
Rational
PowetS
Oabe,
which
or
;
we
if it
mirft
a
Approximate
approach
om-
matical
Mathe-
;
to
SurfoUd
I
Anfwer
nearer
be
I
required
were
Biquadrate, or
will
and
;
already defined.
Irrational Power
an
Root.
only
infiead of
Anfwer,
inftance
,142857
to give an
but
Squared
for their Roots
Approximate
Square,
its Cube
hatll been
are,
this. That
one.
content
of
tretsed
Powers
Irrational
be
(*fc. its
"
I have
muft
^1^6ig62^6
(
Root.
,296296296
what
i.
fcff,
Preparatum
tent
Cube
Tdbli of Cubes.
Vide
fhall
%.
iky,
we
for each
and
nearer
according as each Procefs, by continually ap*
and
lower
given Circulate, is carried down
lower.
though
And
yet
near
we
may,
any
pofliblycome
cannot
by carrying
the Truth,
leis than
we
on
that its Defeft
the
Work,
(hall be
at
its juftRoot"
attain the
as
little,
or
Root
fo
indeed
aflignableDifference*
CON-
)
( ija
CONCLUSION.
AM
I
Men
to
of
in each
exhibited
be
me
with
their
feldom
very
latter, whilft
turn
out
by
ingenuous
ally
when
figned
he
to
Perfons
of
Part,
ficient
leave
of
for
it to
inclined
what
their
chufc
I have
that
weakeft
of
to
with
former
to
the
they will generally
even
it
Writing
Head
more
;
was
efpecichiefly de-
believe
and
Science,
that
the
would
have
their
:
own
To
tions,
Calculadefirous
yet
lefs than
exhibited,
rejeftat
Numerical
in
perfefl:Information
and
confiderate
every
Capacities.
this
here
that
my
clear
a
though ignorant
am
they mud
careful Attention
readily excufe
will
fhall refleft
the
it, I
becaufe
amples
Ex-
of
ples
prevalent Examdeed
inPrecepts. For Youth
a
myfclf,
Reader
inform
to
than
Multitude
more
and
proper
of the
Multitude
give
a
much
Pupils,
I flatter
and
learn
how
-,
Apology
no
ready pradical Arithmeticians.
Wherefore
As
Chapter
fully convinced,
are
for the
Profeffion
own
my
make
need
I
perfuaded
thoroughly
fuch
one
fourth
been
therefore
Difcrccion.
TABLES
to
fufI
(
DENOMINATORS.
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NUME-
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2
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20
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DENOMINATORS.
35
34
22
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37
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162
38
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NUME-
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THE
and
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F
H
T
USE
E
TABLES.
Foregoing
EXPLANATION.
rbeir
.
E
H
T
Tables
the
exhibits
Part
divided
are
",
lecond
The
fuch
the
to
"
firft
Exprcffions
vhofe
2
which
Circulates
run
-of
U-
an
for
50
by Infpeftiononly.
found
are
Part,
Tabular
only
the
Dtximai
The
'
V
nit inclufivc
from
end)
they
^
(except
Parts.
*wo
Equivalent
for all Fraftiolns,
dceip before
into
of
fake
the
from
Numbers
exhibits
Brevity,
6ff.
"
to
with
"
,
5^
51
their feveral
Equivalent Decimal
5i
99
Expreffions.
"
-
..
,
neir
Let
it
be
requiredto
USE.
find
the
Decimal
Fra6lion
the
of the Tables
equal
"^
Firft find
in that
then
its Denominator
Column
in the
A-
4
.
A
on
right againft
fide Column,'
its Equivalent Decimal
8
you
Top
7, found
will find
the
among
^875
\
j
merators
Nu-
which
Fradtion.
And
is
like
the
after
And
)
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(
will
you
manner
find
that
=
"
7
And
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I
"
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II
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=
"
And
J
And
16
'
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=
And
;
=
I
^c.
And
"
=
"
fo on.
the Decimals
Where
Obferve,
-;25
,058823
=
"
And
"5?^.
,020408
And
-,
"
I
And
J
i^
,48
=
33
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21
y
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14
20
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deep,
run
at
as
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17
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I
I
1
I
fcfr. I there
only
find the Decimals
multiplying
the
anfwering
is
to
any
by. the
them
of
either
myfelf with placing
"
correfpondingDecimals
onci^ who
but any
5
contented
...''.
31
34
23 29
in their feveral Columns
of them
19
to
each
inclined^
mi^ht eafily*
of their
Multiples, by
Numoratqrs
of
their
,
given Parts, according
hers.
inftance
For
Firft find in the
",
the
to
let the
Laws
bf
Decimal
of
the Decimal
Tables
Kuracirculating
of
be required.
~
;
"
then
multi-
7
ply
that
by
3
;
correfpondingto
Tables
4
5
\y
17
17
7
17
17
Let
would
the
in
.
5
.bfe cbc
6
.
7
,
8
.
.
or
9
10
6?^.
Equivalent Dccicnals
to
10
9
17
4
-.
-^
17
the Decimal
of
--
*
"
be
required.
49
ic the Tables
Decimal
found
Number,
26
Again,
Equivalent
.
tbc ieveral Refults
8
the
if that
And
-
n^ultipliedby
was
6
(hall be
its Refult
the Decimal
equal to
Firft find
y.
-
}
then
mulciply
that
49
Cc
b/
(i8o)
by 36
its Refult
;
refponding
fhall
^.
to
the
be
Which
Equivalent
DecimaF
Infinite Decimal
cor-
will
alfo be
49
6
Square,
the
Id
or
26-
Of
RoM
Square
of
Power
,857142
6
=
"
}
^
"
and"
7
For
"
c=
'
,857142*
^
"^^
7
-
.
Commodious
It is moft
be
exprefled
in the
Aliquot
foregoing Tables,
we
its EquivalentDecimal
ceadlLy^obtain
find
you
from
Thus
;
T''
is the
-"
"
1.14
the
firft find
wherefotc
5
"
thence:
T
Here
114
of
fome
of
Fra"ion.
-^-
be
valent
Equi-
if it
pears
ap-
Part
can
7
Fraftion
the reduced
Let
giveri^
its
it is fi) reduced^
is any
given Fradion
Fraction
Vulgar Fraftion
before
when
And
one.
that the
one
let the
to
in its leaft Terms,,
Decimal
the
.
6
"
49
=
Decimal
Zl
Expreffionfdrr
'
57
"
r
1?-
9
tken the
--V
of
-^
fliall*be
that
the
Decimal
equal
tQ*
7
.
"A"
-
114
IT2
FfaAibn
the reduced
Let
Ag^;
**^
be
It
~^.
Here
^55
the
of
"
Expreifiont
5^
5
.
.
.
for
2S5,
firft find the Decimal
wherefore
;
"
is.,
"
by
"
the ^Tables
.
then
i
^
the
5th Part
of that IhaU
be.:
51
the
Decimal
the
given Numerator)
mal
equal
equal
1
to
"
to
"
-
;
which
being multipliedby
this laft Refult
fiiall be
the
i
j^^
Deci--^
1 2
".
Hoflce
i8i
(
)
m
then,
Hence
that
evident
viz.
predion
to
Aliquot
Part
^
the
how
of
-
any
or
find
to
Fraftion,
of
Parts
the
that
Ufe
of
this
farther
Fraction
it
Tables,
rs
Advantage,
Ex-
Equivalent Decimal
is either a Mutiple,
one
any
the
from
or
the
an
to
"
the forc-
of
inclufive, by the Afliftancc
Unit
an
obtained
have
we
Method
a
fhewing
by
the
99
,
Tables.
going
It may
perhaps be obgeAed by fome. That forafmuch
large Circulates are not eaiilymanaged in Arithmetical
perations, therefore I might
pf particularlyentering of
That
there
of
by Way
form
within
EUTay^
Vulgar
their
Circulates, and
the
FraAions
of
Compafe
there
at
large
to
Tables,
that die
in
with
deal
having
;
many
Trouble
I
anfwer.
I know
of, but
Places
many
O-
of
gures
Fi-
require to compleat
may
my
readily perceive,
commodious
how
the
fuch,
that
Rule,
determine
to
myfelf
To
them.
univerfal
no
fome
them
more
being
faved
have
as
found
I
thofe
willing
was
PraAitkxier
Cafes,
which
when
or
fall
to
hibit
ex-
might
the
it is
moft
Approximates.
"
I take
the
add
the
following Table,
think
it
be
acceptable to fuch Perfons
may
very
the Intereft of any
m^ft corredly compute
given
Money,
particularly large Sunis, for any Number
And
for many
Liberty
other
to
Reafons^
Cc
2
that
mi^bt
be
becaufe
as
would
Sum
of
I
("
Daya.
afligned.
J
TABLE
"
f
B
TA
LE
of
a
for the readyfining the exa"l Decimal
Tear
equal to any Nuniber of Days^ "c.
Dlay5,
I
)
"""W*
MMMiilB
A
182
Days.
Days.
=,002739726
10
,02739726
=
20
=,005479452
05479452
=
082
=1,008219178
30
=
4
=,010958904
40
=
19
I
78
5
"
=,01369863
,82191780
60=
1,00000000
J
I-
13698630
50=
=,016438356
300=
"
of
a
of
a
Year=,25
"
"
6
,54794520
3^5^
10958904
"
200=
"
"
"
,27397260
"
a
3
100=
"
"
2
Parts
1^438356
I
"
7
=r,0l9l7"062
70=
=,021917808
80
"
19
I
78062
2I9I7808
=
3
=,024657534
The
If the
of
Number
Days
thus:
is
can
be
exadly found:
their exad
Decimal
Day$)
a)fi):lbundag^inft them
by. Infpedtion only.
ift the
arp
of this Table
USE
:=^9';5
24657534
90=
propofcd
ofa Year
"
"
9
=,5
"
"
8
Year
2
But
Table,
the
when
there
at
Parts
muft
according
given
View,
one
be
as
Parts
(under
both
then
collefted
the
Number
out
Days
They, and
of the
given Number
of
Table
cannot
at
their
twice,
be
found
Decimal
or
tMce
requires.
As
i
( -83
for
As
Decimal
Example
of
Parts
a
Days
20a
Then
=
*
^
^24657534^
90=
'299
T Add
"547945205
"
J
find
to
the
t
=
Hence
Suppofe it were
required
Year
equal to 299 Days.
"
\
^,
:
)
=
,024657534
,819178082
*
C
C
^
thefe
ther
Parts
according
I^ws
Laws
togeto
tha
nfCir^iknncr
of
Circulating
Decimab.
the Decimal
Parts
required*.
"OTiB"ia
ANf:
( "84
N
A
E
P
P
A
)
D
N
I X,
CONTAINING
Aridiiiietic
^be
Fraftions
Decimal
in
the Fivt
cf
R
primary
l
u
commonly Taught,
as
^"""li^F^-"*""
CHAP.
A
into
Ten
equal
Parts
divided
more
that
by
fuppofed
or
loooo,
I
into
each
lO,
fuppofed
of
thofc
in fuch
or
Equal
iQOOoths,
loooths,
oaths,
and
is
Subdivifion,
t?r.
looooo,
Unit
defcending
divided
be
to
fo
Unit
an
the
Decimal
continual
a
is when
and
of
Parts
or
equal Parts,
;
i*^
I.
Part
a
Fraftion
Decimal
A
be
is
Fra"ion
"
e
into
may
or
looo,
called
loooooths
Ten
Unit
or
Parts,
be
Progreffion,
the
loo,
to
loths^
Parts
of
an
Unit.
Fraftion
Decimal
A
Numbers,
whole
from
Figure,
Thus
called
,5
of
jHundredths
fandths
5
Unit
an
Jionths
an
Tenths
Unit;
an
Thoufandths
of
now
the
Figures, exprqffing
or
of
frequently diftinguiflied
before
the
by prefixing a Comma
is
Unit.
and
j
of
an
Unit
of
an
and
,00703
;
and
Decimal
Unit
,596
;
Fraftion.
and
called
called
,000055
,04
called
5g6
O
703
called
4
Thoue
^^
dred
Hun-
Mil-
i86
(
the firft Place,
idlyy That
the
the fccond
the
I
perfiiadedthat
am
even
feveral
the
being
increafe
Numbers
fo do
increafe
Places
the
decreafe, in
or
the
be
with
whole
of
j^c.
in
5
and
Unity.
foon
convince
Places
Whole
ot
portion,
decuple, or tenfold ProDecimal
Expreflions likewife
a
tenfold
or
Fradions,
Operations
the Learner
is
,5
fo on,
and
"
of
;
and
Unit,
an
will
Proportion.
in every
muft
in
Refpeft
Sgbtradiclh,
Addition,
carefullyobferye,
equal
,
=:
-
-
That
=
-
z=z
".
thus
however
viz.
~^
to
10
-^-
Unit
an
of
Unit
an
Numbers.
Exprefljon
the
as
the
have
I would
of
the
as
decuple,
a
Decimal
Finite
fame
of
of
wards
to-
naturally follow, that all the Operations
DiviSubtrafbion, Multiplication, and
in Addition,
fion, of
that
Unity
it will
then
Hence
Tenths
littleRefledlion
a
decreafe
or
of
Subdivifions
Decimal
Reader,
common
every
Place
Thoufandths
of
of
the Place
of Hundreths
Place, the Place
third
fo on,
the Place
Place,
from
is the
Right-hand,
)
100
lOQO
varied.
lOOOQ
Or
Expreflion
the
as
,04
is
equal
^
to
-^^=
"
ICO
-
"
-
fo on,
and
'"
=
"
,
however
1 coo
thus
varied,
100000
I 0000
So
from
hence
Expreflion
it neither
it is
with
Again,
annex
you
increafcs
manifcft, that if
any
decreafes
nor
Regard
10
to
100
to
Number
the
Whole
any
of
Value
o's
mal
given Deciut
Plealure,
thereof.
Numbers.
1000
10000
.
As
^^
I
r=
=:r
"
10
"
100
"
=z
1000
"
(^c,
IQOOp
Or
"
i
"*"?)
200
20
20000
2000
^
^
Or
as
2
=
=
"
^
ai 50
-
^=
"
^552.
=
=
lOOOO
1000
100
likewife is it manifeft, that if
So
to
maf
any
yeu" amies
betwwn
DiftinAlon
ikaAd thmK
win
ftitfcentkraeof
Amuc: Vaiueae
appear
Dr.
their DedNumber
before/
^
.
Ad"aficag"oi^ amieadttg^^s* at pleaiivs^
and
ia
^ven Integral
with
Hmhmgal
...
TheUfe
any
of 0%
Number
^31
^^^
lOOOO
Number
Ae
500000
500oa^
=
10
"
cw.
xooo
100
10
Or
=
"
Siibtraftiba
ahdrl)i4PifiM.
XPtf/2rxremarks,
fedlytre^^d of
(whidi ^
thb
(ti^ jif firft Author
who
Subge^,waf 5r'fi|^;;i
S^nus^
calls
profef-
in
Trea-
a
BkcimdU) iiibjoinedto his
Arithmetic, publilhed in French^ and printed at Lejden^
Planiiffs Printing^Houfe)in the Year 1*585t
"in Cbriftiifber
firft written
ia lUftcbj (and perhaps had
which
be ^d
that
into
and
softer tranflateSP
in
language)
publilhed
tile
French^ zpd
^hap^ c^
fo
Dtjm
pubUflied it,
of
This artificial
Way
;Uoity,
can
be
never
;be
more
J
much
expeA
and
Time,
foever
or
out
one
hope
to
to
One
but
of
compleat Arithmetician
mjgbt heartilywiih for,
fee
^ft^ljlilhment,Jhcrefpre
lb
let
JDd
happy,
us
won-
Arithmetic
think, that three Months
for
Farts of
hoyr
^Weights,
Dedmally fubdi-
a
to
or
the various
thus
were
Time
fufficijent
turn
Part,
all the "ules. incident
inclined
am
than
*Capacity,to
iiow
expvefliiwany
all
thraugb,if ^iniverfally
Coins, Meafures,
^ided.
^i^e Kfiory ^f JBgAra^^
^gply f^gied. Fc^
too
quick would
decfujU^T
Jbe gone
or
and
a
would
tolerable
in.
yet we
fo uniform
proceedto
CHAP.
But
not
canas
{ "8")
II.
CHAP.
RULE.
"
careful
BE
under
fandths,
the
as
marked
as
under
and
underneath
Tenths,
Thoufandchs
Hundredths
c^
beIow"
Whole
will
l
And
Numbeh
then
be
the
;
Total
Thou-
under
proceed
whether*
Simple,
Expreffions,
given
Addition
off
place
Tenths
HundmUhs,
"r".as
feveral
in
to
its
to
or
add
up
Mixt"
Reifult,when
fought.
CHAP.
( "89
)
"
C
P.
HA
Ill
'
t
i
It
(
"
RU
L
"..
1
;'.
A
Tenths
PLACE
iSc.
Hundredths,
under
Tenths,
Hundredths
under
before
taught. Then
proceed
whether
Simple, or Mixt,
giren Expfeffions,
to
fubtraft
as
in Subtradion
the
of
as
Whole
Numbers
the.Rcfult
i
will be
fought.
the Difference
V;
"ii..*
v'^/"'
i^'ju
\
,\
c\i-
"
^
E^camplesr
Tards.
Id.
From
,534
From
Take
,396
Take
,138
Diff.
Diff.
'
^^,475965
From
39,4715
Take
,9975
SA7^4^5'
^
'
8,794765
t"i'fib55^7673
^
"'
'
\
Tards.
"
-
'^
_
IV.
CHAP.
ACTION.
MULTIPLIC
RULE.
C E
PJLin OWhole
E
with
D
both
FadQp^
Wbrjr BLcfpeftas
in
-! ?.
j.U!
Numbers.
.,
And
to
determine
the Value
of
thejJPcodiift,obferve
following Dircflions.
*
ift, Mark
Part
yen
in both
off"
in the
as
the
...
Places
many
Product,
as
of
there
Figures for
are
Decimal
the
tional
Frac-
Places
gi-
Fadors.
D
d
2
zdly^ But
)
( *^
1
"Myj
But
Places
ib many
Places
many
when,
o*s
to
of
often
may
ha^oen,there
iidt Srbduft,
J]lgures.!m
as
the Whole
And
then
EzprcBon
let the
are
there
given in. both Iia"im, be careful
impef^eft9r0(l6", 'asr is
the 'firjl
fiipplythe Defeat
bdore
as
it
to
are
not
cimal
De-
prefix
fuffident
as.
ca
DiftinAton
Decimal
fitttfc Produa,
as
in the
jd^
4th, and 5th Examples following..
1
1
1
C
HA
P.
(
C
)
"3"
H
R
A
DIVI
V.
SION,
t
RULE.
PROCEED
in
And
lumbers.
tiett^osUbr^a
the
the
detcrniine
to
following
1*, WhcniShc"ivifor,
Mixt, or
^or
Azxi its given
"ire
to
be
fo
die^tdTQltki^he^otient
Ncccffity
of
fa'be
Whole
cffllhe
y^M^
Q^o^
'Ihtegral,.
of Figures-
Places
more
carefbl
to
oiaf/capCKO^
o*s
annex
continurthe
to
as
Vakfc
the
"^htttier
Dividend,
in
as
Dtfeftmi^
Simple,, confilts
l^mdend,.
the
Operation,
plea-
at
Operation
oifcar
until
4tbe Truth
i
as^
ma,jr, require
%3fy4 IVfiuffeoff
in thfc
QgotimtMas-many-pficcs
of
Fi-
.
Pra"fi5nal
^gdres fef the
ufed
Places
Blaces
Places
in
in
Number
the
of
in the
Drvifor
the
Psrt,
Decimal
is the
Excefs
of
more
than
the
Decimate
of
Decimal;
Dividend,
That
:
Quotient
as
4s, the Number
Bivifoc^
and
nuift
be
the
equal
ta
cimal
De-
the
lifed in the IDividend.
Places
.
When
there
3^9
zxt
the Quotient to mark
off
prefix a fufficient Number
to fupply the Defeft^ and
not
Koces
of
Jm^ms
enough
in
.
before
the
fj^r-^
l^ftionflil
efo'ts
then
to
the
f^m,
imperfeft Quotient*,
fet the Decimal
whole
Expreflion, for the Quotient,
3d) 4th9,8th,.and pdi Examples following,^
(i.)
ywimiift:
Diftin"lion.
as
in
(1.)
"
7-
1 4Sg5"347.i-
Qjjote
695,0495-^
r6
1_^7"4^'
Qjiote. 175,894a
thcj
"
(
)
192
(4.)
(3.)
8
I
Quote
"
I
I ,0017544
1,2
50472
,001462
Quote
,018809
(5.)
Divide
,475
by
(6.)
Divide
,0012
I ,4750000
,0012
9S^te
I
,9
Qyotc
355*833 4-
^
by
,,9
1,000000
I,;
1
11
4-
1
VPM^iaMWii^^*
(8)
(70
4.9
D.
1 479*47585
4
by
57"49
441
Q;
I 4,o"ooooO
57,49
97."522^384
(^c.
34494
Qi
.o%57
455060
"f.
(9-)
Divide
48 I
i
by 48
1,000000
96
^uote
"\-
,020833
49
t??f
"
.
CHAP.
To
reduce
Vulgar Fraaion
Fra^OD,
it.
pr near
a
VI.
to
its
'K/
'
Equivalent
mal
Deci-
^f^f-M
(
^
(4.) Reduce
to
)
"94
Fraftioii.
Decimal
a
040
640
I 31OQQOQO0
,0046875
^j!^
Finite.
,0046875
""
-^
(5.) Reduce
"
Decimal
a
Fraftion.
II
-4|^"ir
"ppitiKiinaJ"ljF. J8tttwhew
in* its Qootioit;,
FepQac.iimniteI]r
"^^^
."6.) Reduce
to
"
a
Decimal
f raiEfcipn.
-^
21
21
^* woi^fd
z'
.
I 4"ooQaoa
.
21
.,
i
90476
-fc?4?.
""^*""""^
"r.
190
y*j/5wr
,190476
appcoximatelx..
%t
wh"re
,190474
,
would
repeat
r^
(7.) Reduce
,a
to
a
kfr"^tient.
'Decimal
Fraaion.:
t"ecim"l- Exprefl!en equd
Sterlingbeing the Integer.
/words,
L.
in
infinitely
find
the
^6^
x)thcr
to-jQae Fai;thin"
9^
"sfr.
40
"?r.
"
Anfiver ,oawa4j(tf^ appniasoiatdf.
repeat
in
I 1,0000099^
""""""""^iM-w^w^
"ooio4^6
Or,
Infinitelyfrom
.Quotient, if the
the
Divifion
Place
^m^
But
of
wb^re
^
would
loooooooths
continued
on
in its
ad
In-
)i
( 1^5
be
It may
an
agreeableAmuTement
K^a-
o^my
fome
to
,
'
they
when
ders*
Decimals^
Infinite
being
"Coi04i6,
acquainted
are
find
to
that
Divilbr
the
the
with
the
Decimal
Expreffion
of
given Number
Quotient their Equivalent
io
any
Sterlings will give in its
of Farthings matliematicallyexaft
Number
Founcb
diac the fitme
the contrary,
tipUed by iwy given
Number
might be cqmpofed
all the Intermediate known
^plyingit according
5,
g^^. inclwve
indeed,
But,
fiftance
the
Addition
of
Refpeft,
to
anfwer
to
and
on
^^
that from
Parti
th$ Laws
the
large
accurate
an
i
Decim?^ Oxpreifion, being mul*
of Farthings^will giv6 in its
Piwiu^thdr.equivalfQt
Numb^
.1 mi)0: farther obfefve,
of
Management
above
Table
Decimal
of
a
of
Cirwlatn
""
Exprefllon
mul-
Sterling,by
l^
as
s"
Ihort
fblj^Tving
of
Table,
by
6n|y, wil|"
Ciitulatea
the
in
t
.
V
a"
Af*
ever|r.
the frmf ^rpofe.
.1
'.
^"
9^^
.
.""
of
JfA'
'