Isaac Newton.

The method of fluxions and infinite series : with its application to the geometry of curve-lines online

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third Term, and fo on : So as finally to leave - , or a, for the

Aggregate of the whole Series. And here it is likewile to be obferv'd,
that we may flop whenever we pleafe, and yet the Equation will be
good, provided we take in the Supplement, or a due part of the next
Term. And this will always obtain, whatever the nature of the
Series may be, or whether it be converging or diverging. If the
Series be diverging, or if the Terms continually increafe in value,
then there is a neceflity of taking in that Supplement, to preferve
.the integrity of the Equation. But if the Series be converging, or
if the Terms continually decreafe in any compound Ratio, and there-
fore finally vanifh or approach to nothing ; the Supplement may be
fafely neglected, as vanishing alfb, and any number of Terms may-
be taken, the more the better, as an Approximation to the Qium-
tity a. And thus from a due confederation of this fictitious Series,
the nature of all converging or diverging Series may eafily be appre-
hended. Diverging Series indeed, unlefs when the afore-mention'd
increafing Supplement can be affign'd and taken in, will be of no
feivice. And this Supplement, in Series that commonly occur, will
be generally fo entangled and complicated with the Coefficients of
the Terms of the Scries, that altho* it is always to be understood.,
neverthelef?, ii is often impoffible to be extricated and affign'd.
But however, converging Series will always be of excellent ufe, as
Affording a convenient Approximation to the quantity required, when
it cannot be othei wile exhibited. In thefe the Supplement aforefaid,

tho'



and INFINIT E SERIES. 153

tho' generally inextricable and unnflignable, yet continually decreafes
along with the Terms of the Series, and finally becomes lefs than any
aflignable Quantity.

The. lame Quantity may often be exhibited or exprefs'd by feveral
converging Scries ; but that Series is to be mod edeem'd that has the
greateft Rate of Convergency. The foregoing Series will converge
fo much the fader, cteteris paribus, as p is lefs than q y or as the

Fraction - is lefs than Unity. For if it be equal to, or greater than

Unity, it may become a diverging Series, and will diverge fo much
the fader, as p is greater than q. The Coefficients will contribute
little or nothing to this Convergency or Divergency, if they are
fuppos'd to increafe or decreafe (as is generally the cafe) rather in a
fimple and Arithmetical, than a compound and Geometrical Propor-
tion. To make fome Edimate of the Rate of Convergency in this
Series, and by analogy in any other of this kind, let k and / re-
prefent two Terms indefinitely, which immediately fucceed each
other in the progrefTion of the Coefficients of the Multiplier a -+-
bx -if ex* -f-^x 3 , &c. and let the number n reprefent the order or
place of k. Then any Term of the Series indefinitely may be repre-

fented by -f- l'-Jf-~*- where the Sign mud be -+- or , accor-

?"
ding as n is an odd or an even Number. Thus if == i, then

k = a, 1 = 1', and the firft Term will be -f- *_LlL^Z . ]f ==2j

then & = />, l = c, and the fecond Term will be c ^~p. And
fo of the red. Alib if m be the next Teim in the aforefaid pro-

grefTion after /, then -f- -^~lp"~ l -f- ^ 7 /." will be any two fuc-

?" ?"

cefiive Terms in the fame Series. Now in order to a due Conver-
gency, the former Term abfolutely confider'd, that is fetting afide
the Signs, mould be as much greater than the fucceeding Term, as

conveniently may be. Let us fuppoie therefore that JL^Jp-i j s

i"

greater than ' ^p", or ( dividing all by the common factor c" } \

r " ~^ ' t" '

that ^ + /f? is greater than ^ - , or ( multiplying both by pq, )

that Ipq -f- krf is greater than nip* +- Ipq, or (taking away the com-
mon IpqJ that kf is greater than //.y, 1 , or (by a farther Diviiion,)

that - x is greater than unity ; and as much greater as may be.
fl X This



7%e Method of FLUXIONS,

This will take effeft on a double account ; firft, the greater k is in
refpecl: of ;;;, and fecondly, the greater 5* is in refpect of p\ Now
in the Multiplier a -\-bx -f- ex* -\-dx>, &c. if the Coefficients a, b,
r, &c. are in any decreafing ProgreiTion, then k will be greater than
/, which is greater than m ; fo that a fortiori k will be greater than
m. Alfo if q be greater than p, and therefore (in a duplicate ratio)
j* will be greater than /*. So that (cater is faribus) the degree of
Convergency is here to be eftimated, from, the Rate according to
w hich the Coefficients a, b, c, &c. continually decreafe, compounded
with the Ratio, (or rather its duplicate,) according to which q fhall
be fuppos'd to be greater than />.

/ n

The fame things obtaining as before, the Term .j_ A will be





i



what was call'd the Supplement of the Series. For if the Series be
continued to a number of Terms denominated by n, then inftead of
all the reft of the Terms in itifinitutn, we may introduce this Sup-
plement, and then we fhall have the accurate value of a, inftead of
an approximation to that value. Here the firft Sign is to be taken
if n is an odd number, and the other when it is even. Thus if

n= i, and confequently k=a, and /= <, we fhall have

* == a. Or if == 2, and /= c, then b lX e tll x t +
q ill

c\i . f 7 j .i bb-^-a-j ff-> r f-a p <{$ -4- cq

L - -a. Or if n = 3, /= a, then J-If _L_L_I x - -4- -
f i 1 i q

x ^ =.$. And fo on. Here the taking in of the Supple-
ment always compleats the value of a, and makes it perfect,
whether the Series be converging or diverging ; which will always
be the beft way of proceeding, when that Supplement can readily
be known. But as this rarely happens, in fuch infinite Series as ge-
nerally occur, we muft have recourfe to infinite converging Series,
wherein this Supplement, as well as the Terms of the Series, are
infinitely diminifh'd ; and therefore after a competent number of
them are collected, the reft may be all neglected in infinitum.

From this general Series, the better to aflift the Imagination, we
will defcend to a few particular Inftances of converging Series in
pure Numbers. Let the Coefficients a, />, c-, d, &c. be expounded by

,, , | ; < , to, refpectively ; then ** _ * x ^ + ^ x

^ ^c! or L^_fl^ x ^ H _-2^ x ^_^-+5ix / 4, &C.
5 ' (XC< J ' 27 r.x; ? 7 3 x 4? f 4x55. 5 3'

'. r . That the Series hence arifmg may converge, make/ lefs

than



a?:d IN FINITE SERIES. 155

than q in any given ratio, fuppofe - = ~, or /> = i, q = 2, then

A |.x|H-4^x^ T V x -J., &c. = i. That is, this Series of
Fractions, which is computed by Binary Arithmetick, or by the
Reciprocals of the Powers of Two, if infinitely continued will
finally be equal to Unity. Or if we defire to flop at thefe four
Terms, and inftead of the reft ad infinitum if we would introduce
the Supplement which is equivalent to them, and which is here
known to be j x T y, or T V, we Hull have 4 | -+- - T ^ -f-
T 'o- = i, as is eafy to prove. Or let the fame Coemdents be ex-
pounded by i, |, -i, i, -f, &c. then it will be - - -+-

f 4iz^ 1 f = 4f / & Thu Series m ehhei .

1 3 X 47 J* 4 X 5? i 3

be continued infinitely, or may be fum'd after any number of Terms

i, _ n

exprefs'd by ;?, by introducing the Supplement ; ~ infteadof all

H-IXJ*

the reft. Or more particularly, if we make (jr= $p, then -2 _f.
7-^-. -+- - - -f ( ^ . &c. = i, v/hich is a Number

6x5! liXjS 20X^4 30X;;!'

exprefs'd by Quinary Arithmetick. And this is eafily reduced to the
Decimal Scale, by writing ~ for -f, and reducing the Coefficients ;
for then it will become 0,99999, &c. = i. Now if we take thefe
five Terms, together with the Supplement, we mall have exadly

-f- r 11 - + - 12 - -f- - + -~ 4- ^-, = i. Again, if

2x5 6x,i 12x5} 20x54 30x5' 6x;

we make here 77= ioo/^, we fhall have the Series

JJ "

^^- 6 >c -i- + 40 ~ 9 x 9 -f- < co -': x 27
- x 3 iccoo 3 X 4 i oooooo 4 X 5 locoocooo

which converges very fa ft. And if we would reduce this to the re-
gular Decimal Scale of Arithmetick, (which is always fuppos'd to be
done, before any particular Problem can be faid to be coinplcatly
folved,) we muit let the Terms, when decimally reduced, orderly
under one another, that their Amount or Aggregate may be tlifco-
ver'd ; and then they will ftand as in the Margin. Here the Ag-
gregate of the firfc five Terms is 0,99999999595, 0,985
which is a near Approximation to the Amount of the
whole infinite Series, or to Unity. And if, for proof-

lake, we add to this the Supplement _ +/ ' = 1L ,-

+ , ,/' ' 5 '" |OJ

= 0,00000000405, the wh< . be Unity exaclly.

X 2 There




T f6 The Method of FLUXIONS,

3 *f

There are alfo other Methods of forming converging Series, whe-
ther general or particular, which fhall approximate to a known quan-
tity, and therefore will be very proper to explain the nature of Con-
vergency, and to mew how the Supplement is to be introduced, when
it can be done, in order to make the Series finite ; which of late
has been call'd the Summing of a Series. Let A, B, C, D, E, &c.
and a, />, c, d, e, &c. be any two Progrcffions of Terms, of which
A is to be exprefs'd by a Series, either finite or infinite, compos'd
of itfelf and the other Terms. Suppofe therefore the firft Term of
the Series to be a, and that p is the fupplement to the value of a.

Then is A = a -}-/>, or p = ~ a . As this is the whole Supple-
ment, in order to form a Series, I fhall only take fuch a part of it
as is denominated by the Fraction - , and put q for the fecond Sup-
plement. That is, I will afiimie - = (p=) - -XTJ -\-q, or

/A a b \ A a E b .. .... .,

q f xi R=7 ~~B~ x ' Again, as 1S 1S the whole

value of the Supplement q> I fhall only aflume fuch a part of it as is de-
nominated by the Fradion > and for the next Supplement put r.

/A a

orr = ( - x
Now as this is the whole
value of the Supplement r, I only afTume fuch a part of it as is denominated
by the Fraction - , and for the next Supplement put s. That is, ~

Bl> Cc A a B /; Cc, A a

x -7 x = ( r = ) - x - x -rr-a -+- s, or s = -77- x

^ I ^ ' D ^ U Ij

B /; Cc 7 A a B '> Cc T>d A j /- c

x x i TJ r- x r- x 77 x . And lo on as far

as we pleafe. So that at lafr. we have the value of A.'=a-\-p,
where the Supplement p = - ~l)-\-q, where the fecond Supple-

A a B b A a E l> Cc ,

inent q == g x TT-C -}- r, where r = g x ^~ x -]y 4- s,

A B b C c D d

where s = '^- x -7- x -rr x r-e-\- 1. And fo on ad tnfinitum.

D (*. U H,

_,. r 11 A A a. A a B /; A a Eb C c ,

That is finally A = a -+- b .+- x ^-c -\ x -7 x -jj-

A a 7, b Cc D d c \ -a r^ TT\ -O Of*

-\- x TV- x -jj- x -J7- e, Kc. where A, B, C, D, E, ere. and a y

b, r, d, e, 6cc. may be any two Progreffions of Numbers whatever,
whether regular or defultory, afcending or defcending. And when

it



. .

= (?=) -g- x



x rr- x




and INFINITE SERIES. 157

it happens in thefe Progreffions, that either A = a, or B=^, or
___ 5cc. then the Series terminates of itfelf, and exhibits the
vilue of A in a finite number of Terms : But in other cafes it ap-
proximates indefinitely to the value of A. But in the cafe of an
infinite Approximation, the faid Progreffions ought to proceed re-
"ularlv, according to feme Hated Law. Here it will be eafy to ob-
fcrvc," that if 1C and k are put to reprefent any two Terms indefi-
nitely in the aforefaid Progreffions, whofe places are denoted by the
number ;/, and if L and / are the Terms immediately following ;
then the Term in the Series denoted by n -f- i will be form'd from

(v /-

the preceding Term, by multiplying it by -^ /. As if n = i,
K = A, k = a, L = B, l=b, and the fecond Term will



i /t j ', A T f-* 17 L t"1~ipn TC - - L - "R k - " u

DC ** 1 r> " H * '

A a, B A a B f>

I z c, and the third Term will be jr-^* ~7cT r == ~tT x ~Tr~ r;
and fo of the reft. And whenever it fhall happen that L =/, then the
Series will ftop at this Term, and proceed no farther. And the
Series approximates fo much the fafter, catcris paribus, as the
Numbers A, B, C, D, &c. and a, b, c, d, &c. approach nearer to
each other refpedively.

Now to give fome Examples in pure Numbers. Let A, B,C, D,
&c. = 2, 2, 2, 2, &c. and a, b, c, d, &c. = i, i, i, i, &c- then we
fhall have 2 = i -h 1 H- + T +* -V> & c - And fo always, when
the given Progreffions are Ranks of equals, the Series will be a
G<~ metrical Progrefnon. If we would have this Progieffion ftop at
the next Term, we may either fuppofe the firft given Progreilion
to be 2, 2, 2, 2, 2, i, or the fecond to be i, i, i, i, i, 2, 'tis all
one which. For in either cafe we mall have L= /, that is F ==/,

TC ^ p*

and therefore the laft Term muft be multiply'dby - , or = i.

Then the Progreffion or Series becomes 2 = I +T-r-ir~+"T + Tci
+-TT- Again, 'if A, B, C, D, &c. = 5, 5, 5, 5, &c. and a, b, c, d,
&c. =4, 4> 4, 4, &c - then 5 = 4 H- T + T + T + TTT -H *-TT, & c -
or ^. = H- T ' T -i- -4 T + T!T> &c Or if A, B, C, D, &c. = 4,
4, 4, 4, 6cc. and </, *, f , d, &c. = 5, 5, 5, 5, &c. then 4=5
i -4- -fV - - *- S T H- ^Tr, &c. If A, B, C, D, &c. = 5, 5, 5, 5, &c.
and tf, ^, c, d, &c. = 6, 7, 8, 9, &c. then 5 = 6 T7-f-4-Xy8
^-xf x-f-y -h -^- x-fx AX 10, &c. If we would have the Series
ftop here, or if we v/oiud find one more Term, or Supplement,
which fhculd be equivalent to all the reft ad inftnitum y (which in-
deed



Method of FLUXIONS,

deed might be deiirable here, and in fuch cafes as this, becaufe of thc-
llow Convergency, or rather Divergency of the Series,) fuppofe F==/j

and therefore ~ - = ""7^ ^ T mu ^ be niultiply'd by the la ft

Term. So that the Series becomes 5 = 6 1.7 -f- .1. x -^-S - - f x ..

^ 3 n I * v * ^ ' v 4 TO ' v '* v 3 ' v * f Tf A R CD for - >

X T9 T X T X T X T 10 T X T X> T x TJ 1 r **> ^ ^> ^> (XU 2 >

3, 4, 5, &c. and < - , b, c, d, &c. = i, 2, 3, 4, &c. then 2 = 1-4-
T-+^x^ 3 +|x^xi4-|-|x^xix^5, &c. If A,B,C,D,6cc.
=^ i, 2, 3, 4, &c. and ^, b, c, d, &c. = 2, 3,4, 5, &c. then i =

- 13 + T x|4 T XT;<i5 H- T *T X T X T 6 > &c - And from
this general Series may infinite other particular Series be eafily de-
rived, which fliali perpetually converge to given Quantities ; the chief
ufe of which Speculation, I think, will be, to iliew us the nature
of Convergency in general.

There are many other fuch like general Series that may be readily
form'd, which mall converge to a given Number. As if I would
confliucl a Series that flrali converge to Unity, I fet down i, toge-
ther with a Rank of Fractions, both negative and affirmative, as
here follows.

'* - - - &c
I - """""'"'



-h


a

' A


b


c

- c


e r


A-\-a


-+


Ab Ba


-J-


ti ^/> L De-Ed c _


A


AB


BC DE ' C * ]



Then proceeding obliquely, I collect the Terms of each Series toge-
ther, by adding the two nrit, then the two fecond, and fo on. So
that' the whole Series thus conftrudled muft neceflarily be equal to
Unity ; which alfo is manifeft by a bare Infpeclion of the Series.
From this Series it is eafy to defcend to any number of particular
Cafes. As if we make A, B, C, D, &c. = 2, 3, 4, 5, 6cc. and a, b,
i, &c. then A J- ^ __l___i_ 6 ,

&c. And fo in all



.= . , .

other Cafes. The Series will flop at a finite number of Terms,
whenfoever you omit to take in the firft part of the Numerator of
any Term. As here | -JL ? _ -1- ^ -1 - .^ = ,.

Laftly, to conftru6t one more Series of this kind, which mail
converge to Unity ; I fet down i, with a Rank of Fractions along

with



and INFINITE SERIES, 159

with it, both affirmative and negative, iiich as are feen here below ;
which being added together obliquely as before, will produce the
following Series.



i 4-


a f
A ~*~
a


ab


(T


"+" A BCD ~

/>:</


abcJe /,
hrvrr


AB
a!>


"t" ABC

abt


abctle

N~C


A


AB


AB^


' ABCDE' C ' *


A a


l-b


?4


_~- L^_


D rf ,


E c , ,

I .._ , fjhrii ATP T


A






*" AB^DE^' ~ C> J<



This Series may be made to flop at any finite number of Terms,
if you omit to take in the latter part of the Binomial in any Term.
Or you may derive particular Series from it, which fhall have any
Rate of Convergency.

For an Example of this Series, make A, B, C, D, &c. = 3, 3,
3, 3,6cc. and a, b, c,d y &c. = i, i, i, i, &c. then y4-f -+-TV + TT>
&c. = i, or JL 4- 4- -\ 4- T V, &c. = . And whenever A, B,
C, &c. and a, b, f, &c. are Ranks of Equals, the Series will be a
Geometrical PiogrefTion.

Again, make A, B, C, D, &c. = 2, 3, 4, 5, &c. and a, b, c, d, &c.

= i, i, i, i, &c. then i-4- 7^ 4- 7777; + r x 3 x 4 x 5 + 2x3x4x5x6
&c. = i. Or in a finite number of Terms T + T+ 77^ + 2X 3 x S

_i I = i. And the like may be obferved of others in an

2x3x4x5

infinite variety.

And thus having prepared the way for what follows, by explain-
ing the nature of infinite Series in general, by difcovering their origin
and manner of convergency, and by fhewing their connexion with
cur common Arithmetick ; I mall now return to our Author's Me-
thods of Oj , or to the Reduction of compound Quantities
to fuch infinite Series.

SECT. II. The Resolution of fimph Equations, or pure
Powers, by I?ifihi.'d Szries.




3, 4. ' | ^HE Author begins his Reduction of compound (

tit ; -, to an equivalent infinite Series of fmiple Tc-ms,
fir ft by fhevr: j; how the Piocefs may be peiform'd in Divifion.
Now in his Example the manner of the Operation is thus, in imi-



taton



j6o *fi>e Method of FLUXIONS,

tation of the ufual praxis of Divifion in Numbers. In order to ob-
tain the Quotient of aa divided by b -f- x, or to relblve the com-
pound Fraction T|T- into a Series of fimple Terms, firft find the
Quotient of aa divided by l> } the firft Term of the Divifor. This
is ^ , which write in the Quote. Then multiply the Divifor by

this Term, and fet the Product aa -h ^ under the Dividend, from

whence it muft be fubtracted, and will leave the Remainder ~ .

Then to find the next Term (or Figure) of the Quotient, divide
the Remainder by the firft Term of the Divifor, or by b, and put

the Quotient "~ for the fecond Term of the Quote. Multiply

the Divifor by this fecond Term, and the Product ^ ^r
fet orderly under the laft Remainder ; from whence it muft be fub-
tracted, to find the new Remainder -h "-^- . Then to find the

bo

next Term of the Quotient, you are to proceed with th-is new
Remainder as with the former ; and fo on in infimtum. The Qup-

r . a* K* c c*x* a*x3 c , - *

tient therefore is j -+- ^- , &c. (or -j into i

? .+- ^ ^ , 6cc.) So that by this Operation the Number or



Quantity ^ , (or a 1 x^-t-*!" 1 ) is reduced from that Scale in
Arithmetick whofe Root is b -+ x, to an equivalent Number, the
Root of whofe Scale, (or whofe converging quantity) is . And
this Number, or infinite Series thus found, will converge fo much
the fafter to the truth, as b is greater than x.

To- apply this, by way of illustration, to an inftance or two in
common Numbers. Suppofe we had the Fraction |, and would
jeduce it from the feptenary Scale, in which it now appears, to an
equivalent Series, that mall converge by the Powers of 6. Then

, we (hall have j = ^ ^ ; and therefore in the foregoing general

\ Fraction -^- , make a-=. i, b = 6, and #==1, and the Series

b -"j~ x

will become f ~ + ^ ^, &c. which will be equivalent to
Y. Or if we would reduce it to a Series converging by the Powers
of 8, becaufe f= ~ , make a= i, ^=8, and .v = i,

then



and IN FINITE SERIES. j6r

then ~ = T + ~* -+- & -+- ^ > &- c - which Series will converge fafter
than the former. Or if we would reduce it to the common Denary (or
Decimal) Scale, becaufe f -~r- , niake a= i, l> = 10, and

x= 3 ; then 7 = -rV -4- -4-0- -+- Wo-o- -f- -o-Vo-o- + TO-O-^O-S-J <* c '
= 0,1428, &c. as may be eafily collected. And hence we may
obferve, that this or any other Fraction maybe reduced a great va-
riety of ways to infinite Series ; but that Series will converge iafteft
to the truth, in which b mall be greateft in refpect of x. But that
Series will be mod eafily reduced to the common Arithmetic^,
which converges by the Powers of 10, or its Multiples. If we
mould here refolve 7 into the parts 3 -f- 4, or 2+5, or i -f- 6,
&c. inftead of converging we mould have diverging Series, or : fuch
as require a Supplement to be taken in.

And we may here farther obferve, that as in .Divifion of com-
mon Numbers, we may flop the procefs of Divifion whenever we
pleafe, and inftead of all the reft of the Figures (or Terms) ad in-
finituniy we may write the Remainder as a Numerator, and the
'Divifor as the Denominator of a Fraction, which Fraction will be
the Supplement to the Quotient : fo the fame will obtain in the
Divifion of Species. Thus in the prefent Example, if we will flop

at the firft Term of the Quotient, we mall have -^- = "~ a ^L. .

^ ' b + X o b X /; |- x

Or if we will ft op at the fecond Term, then - r\. = j "-~r -f-
Or if we will flop at the third Term, then ^- = ^
_ ^- x . And fo in the fucceeding Terms, in which
thefe Supplements may always be introduced, to make the Quotient
compleat. This Obfervation will be found of good ufe in fome of
the following Speculations, when a complicate Fraction is not to
be intirely refolved, but only to be deprefs'd, or to be reduced to a
fimpler and more commodious form.

Or we may hence change Divifion into Multiplication. For hav-
ing found the firft Term of the Quotient, and its Supplement, or

aa ta aax i i *' -i K /lit

the Equation ^ = - -^ x -, multiplying it by ? , we fhall
have -^- = T~^- , fo that fubftituting this value of

IldVC i * 3- a '




ant Ml aa aa aaX

_ffL_ in the firft Equation, it will become ^ = y -^ -f-

. a>A '*- where the two firft Terms of the Quotient are now known.

Y Multiply



162 The. Method of FLUXIONS*

Multiply this by ^ , and it will become



* L - , which being fubfthuted in the laft Equation, it will become

aa ra aav fi^.v 1 a*** a' i x*' i .1 c r- n

r =. - - - -4 - - - 1- -. r- . where the four nrlt

t-^-x b b* b* I* iS+i*X '

Terms of the Quotient are now known. Again, multiply this

,-, . , A.4 rf 5 .v4 fl-.v4 a*x* a*x 6 '

Equation by ^ , and it will become ^7^ x = * - - JT+ ~




-p r- -, r 8 - , which being fubftituted in the laft Equation s

... , aa a* a z .v a*x* 7 .<3 a**4 a 1 x !

it will become - - = 4 - 1 f p- 4-

i 6 17 i V 8

fyi- ^- -4 5T- , where eight of the firft Terms are now

hi b t9-^-6x

known. And fo every fucceeding Operation will double the num-
ber of Terms, that were before found in the Quotient.

This method of Reduction may be thus very conveniently imi-
tated in Numbers, or we may thus change Divifion into Multipli-
cation. Suppofe (for inftance). I would find the Reciprocal of the
Prime Number 29, or the value of the Fraction T ' T . m Decimal
Numbers. I divide 1,0000, Gfc. by 29, in the common way, fo
far as to find two or three of the firft Figures, or till the Remainder be-
comes a fingle Figure, and then I afliime the Supplement to compleat
the Quotient. Thus I mail have T ~ =. 0,03448^ for the compleat
Quotient, which Equation if I multiply by the Numerator 8, it will
give ^ = 0,275844^., or rather ^.==0,27586^. I fubftitute
this initead of the Fraction in the firft Equation, and I (hall have
^=1:0,0344827586^. Again, I multiply this Equation by 6,
and it will give T * 7 = o, 2068965517^, and then by Subftitution T ' 7 ==
0,03448275862068965517^. Again, I multiply this Equation by 7,
anditbecomes T 7 ? =o,24i3793io3448275862oi|-,andthenbySubfti-



where every Operation will at leaft double the number of Figures
found by the preceding Operation. And this will be an eafy Expe-
dient for converting Divifion into Multiplication in all Cafes. For
the Reciprocal of the Divifor being thus found, it may be multi-
ply'd into the Dividend to produce the Quotient.

. . , c , , aa aa n*x -** **S

Now as it is here found, that j =7 77 -+ ~jr Z7~>

&c. which Series will converge when b is greater than A* ; fo when
it happens to be otherwife, or when x is greater than b, that the
Powers of x may be in the Denominators we muft have recourfe to

the



and INFINITE SERIES, 163

the other Cafe of Divifion, in which we fhall find -^-^ = ^
i _j_ a ^- "^ , &c. and where the Divifion is perform 'd as

before.

5, 6. In thefe Examples of our Author, the Procefs of Divifion



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