Isaac Newton.

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

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(for the exercife of the Learner) may be thus exhibited :



o x i +o

AT 1 .V4



+7 *-



Now in order to a due Convergency, in each of thefe Examples,
we muft fuppofe x to be lefs than Unity; and if x be greater than

Unity, we muft invert the Terms, and then we fhall have l

XX "^ 1

i i

1 I I I c *

= ^ ^ + 7* &c -



ii



*/*

7, 8, 9, io. This Notation of Powers and Roots by integral and
fractional, affirmative and negative, general and particular Indices,
was certainly a .very happy Thought, and an admirable Improve-
ment of Analyticks, by which the practice is render'd eafy, regular,
and univeifal. It was chiefly owing to our Author, at leaft he car-
.ried on the Analogy, and made it more general. A Learner fhould
be well acquainted with this Notation, and the Rules of its feveral
Operations fhould be very familiar to him, or otherwife he will often
find himfelf involved in difficulties. I fhall not enter into any far-
ther difcuffion of it here, as not properly belonging to this place,
or fubject, but rather to the vulgar Algebra.

1 1. The Author proceeds to the Extraction of the Roots of pure
Equations, which he thus performs, in imitation of the ufual Pro^
cefs in Numbers. To extract the Square-root of aa +- xx ; firft the
Root of aa is a, which muft be put in the Quote. Then the Square
of this, or aa, being fubtradted from the given Power, leaves -+-xx
for a Refolvend. Divide this by twice the Root, or 2a, which is

Y 2 th



164 ?$ Method' of FL u X r or N s,

the firft part of the Divifor, and the Quotient muft be made the
fecond Term of the Root, as alfo the fecond Term of the Divifor.
Multiply the Divifor thus compleated, or -za -J- x ~ , by the fecond

Term of the Root, and the Produft xx + muft be fubtrafted
from the Refolvend. This will leave , for a new Refolvend,

4-"

which being divided by the firft Term of the double Root, or 2tf,

. A

will give j for the third Term of the Root. Twice the Root

before found, with this Term added to it, or 2a -+- ^ -^ , be-
ins multiply 'd by this Term, the Product ^- 1- muft

4a* 8^4 640''

be fubtrafted from the laft Refolvend, and the Remainder -f-

. B

will be a new Refolvend, to be proceeded with as before,






for finding the next Term of the Root ; and fo on as far as you
pleafe. So that we (hall have \/ ' aa -+-xx = a+ '- _ -' _i_ ~

1 T.a oa* io*



It is eafy to obferve from hence, that in the Operation every new
Column will give a new Term in the Quote or Root; and therefore
no more Columns need be form'd than it is intended there mall be
Terms in the Root. Or when any number of Terms are thus ex-
traded, as many more may be found by Divifion only. Thus hav-
ing; found the three firft Terms of the Root a -f- , by

2a fcu3 " J

v^ v4

their double -za -\ , dividing the third Remainder or Re-
folvend -\- 7^: , the three firft Terms of the Quotient

in. 4 04*. ^* l6&^

c* 8 7x l

; H ',- will be the three fucceeding Terms of the Root.

1 2 Oil ' 2 COrt* fj

The Series a -f- ^i H TT* > ^ c< t ^ us f unt l f r the fquare-

root of the irrational quantity aa -f- xx, is to be understood in the
following manner. In order to a due convergency a is to be (iippos'd

greater than x, that the Root or converging quantity - may be leis

than Unity, and that a may be a near approximation to the fquare-
root required. But as this is too little, it is enereafed by the fmall

quantity , which now makes it too big. Then by the next

Operation



and INFINITE SERIES. 165

Operation it is diminim'd by the ftill fmaller quantity ^; which
diminution being too much, it is again encreas'd by the very fmall
quantity -7 - r , which makes it too great, in order to be farther di-

minifli'd by the next Term. And thus it proceeds in infinitum, the
Augmentations and Diminutions continually correcting one another,
till at lalt ihey become inconfiderable, and till the Series (fo far con-
tinued) is a lufficiemly near Approximation to the Root required.

12. Wh-ii a is Ids than x, the order of the Terms muft be in-
verted, 01 ihe fquare-root of xx -+- aa muft be extracted as before;

in which cafe it will be x -+- -f-. , &c. And in this Series

5 '



2X



the converging quantity, or the Root of the Scale, will be -. Thefe

two Scries are by no means to be understood as the two different Roots
of the quantity aa -+- xx -, for each of the two Series will exhibit thofe
two Roots, by only changing the Signs. But they are accommodated
to the two Caf s of Convergency, according as a or x may happen to
be the greater quantity.

I (halt here refclve the foregoing Quantity after another manner,
the better to prepare the way lor what is to follow. Suppofe then
yv=.cni-\- xx, where we may fi'-d the value of the Root y by the
f 11 ,wir ;.j Proccfy ; yy = aa -+- XX= (\f)' = rf-f-/) aa^-zap -\-pp-,



or zap -+- pp = xx = (If p = + q} xx + zaq -{- ~
- qq; or 2rf?-J- ^ -H^= - = (if ?=== _^



+ r > or



rr = '-, . - 6 - (if r = + j) &c. which Procefs may

oi,' O-|t. 1 VU* J

be thus explain 'd in wo-ds.

In order to find V ua - xx, or the Root y of this Equation
yy-=aa-\-xx, iuppofc 1 y = ^-f-/', wheie a is to be undeiftood as
a pretty near Approxii: arion to the value of _y, (the nearer the bet-
ter,) and p is the lnv.,11 Supplement to that, or the quantity which
makes it compleat. Then by Subftitution is deiivcd the fir It Sun-
plementiil Lqu^i'oa zap -+-//; = xx, whole Root/; is to bt fou:,d.
INOW as 2uJ> is n:iich bigger than ff, (lor za is bigger than the Sup-

plement/,) v;c fh;.!l have nearly p - , or at leaft ve (hall have

exactly ;- = : ; -f- -', fuppofmg q to reprefent the fecoiid Supple-

ment



j66 *ft>e Method of FLUXIONS,
ment of the Root. Then by Subftitution zaq -+- ^q -4-^= =
^1 will be the fecond Supplemental Equation, whofe Root q is the
fecond Supplement. Therefore q will be a little quantity, and qq
much lefs, fo -that we mall have nearly q= g - 3 , or accurately
q =. ^ -f- r, if r be made the third Supplement to the Root.
And therefore zar -f- r r -f- r* = f- will be the

U 4^ ou*r L^,"

third Supplemental Equation, whofe Root is r. And thus we may
go on as far as we pleafe, to form Refidual or Supplemental Equa-
tions, whofe Roots will continually grow lefs and lefs, and there-
fore will make nearer and nearer Approaches to the Root y, to which
they always converge. For y =5= a -{-/>, where p is the Root of this

Equation zap - pp-=xx. Or y =: a~\- -+-g, where q is the






Root of this Equation zaq -\ -q-\-qq-=z - ^ . Or y ; a -f-
* -. -f- r. where r is the Root of this Equation zar -f- r

Ztt oa> a

~ I rr-=. -~ ~. And fo on. The "Refolution of any one

of thefe Quadratick Equations, in the ordinary way, will give the
refpeclive Supplement, which will compleat the value of y.

I took notice before, upon the Article of Divifion, of what may
be call'd a Comparifon of Quotients; or that one Quotient may be
exhibited by the help .of another, together v/ith a Series of known
or iimple Terms. Here we have an Inftance of a like 'Comparifon
of Roots; or that the Root of one Equation may be exprels'd by
the Root of another, together with a Series of known or fimple
Terms, which will hold good in all Equations whatever. And to
carry on the Analogy, we mall hereafter find a like Comparifon of
Fluents ; where one Fluent, (fuppofe, for inftance, a Curvilinear
Area,) will be exprefs'd by another Fluent, together with a Series
of fimple Terms. This I thought fit to infinuate here, by way of
anticipation, that I might mew the conftant uniformity and har-
mony of Nature, in thefe Speculations, when they are duly and re-
gularly purfued.

But I mall here give, ex abundanti, another Method for this, and
fuch kind of Extractions, tho' perhaps it may more properly be-
long to the Refolution of Affected Equations, which is foon to fol-
low ; however it may ferve as an Introduction to their Solution.

j The



and INFINITE SERIES. 167

The firft Refidual or Supplemental Equation in the foregoing Pro-
cefs was 2ap -\-pp-=. xx, which may be refolved in this manner.

Bccaufe />= -^-, it will be by Divilion p = - -{ -f- ^

' za + t y 3 " za Aa*

** ! x*tA

,-^7 + -^ , &c. Divide all the Terms of this Series (except the
fir ft) by p, and then multiply them by the whole Series, or by the
value of />, and you will have p = - + ' 3 -^ -f-

ia 8*' 8*4 3Z

^ -g , 6cc. where the two firft Terms are clear'd of />. Divide all the

Terms of this Series, except the two firft, by />, and multiply them
by the value of />, or by the firft Series, and you will have a Series
for p in which the three firft Terms are clear'd of p. And by re-
peating the Operation, you may clear as many Terms of p as you

pleafe. So that at laft you will have p = ~ -+- , 7^

-+- ^~, &c. which will give the fame value of y as before.

13, 14, 15, 16, 17, 18. The feveral Roots of thefe Examples, and
of all other pure Powers, whether they are Binomials, Trinomials,
or any other Multinomials, may be extracted by purfuing the Me-
thod of the foregoing Procefs, or by imitating the like Praxes in
Numbers. But they may be perform'd much more readily by gene-
ral Theorems computed for that purpofe. And as there will be fre-
quent occalion, in the enfuing Treatiie, for certain general Opera-
tions to be perform'd with infinite Series, fuch as Multiplication,
Divilion, railing of Powers, and extracting of Roots ; 1 mall here
derive fomc Theorems for thofe purpofes.

I. Let A H- B 4- C + D -+- E, &c. P-f-Q^-R-f-S-t-T, &c. and
a _l__j_^_{_j\.4_ g) &c. reprefent the Terms of three feveral Series
refpedlively, and let A-|-B-{-C-f-D-|-E, &c. into P+Q-t-R-f-S+T,
&c. = a, -\- /B -{- y -i- <f~ -\- e, &c. Then by the known Rules of
Multip'ication, by which every Term of one Factor is to be multi-
ply'd into every Term of the other, it will be = AP, /3 = AQ^-j-
BP, 7=AR-i-BQ^-CP, ^z^AS-i-BR-i-LQH-DP, g=AT-f-
BS + CR-t-DQ^-4- E'P ; and fo on. Then by Subftitution it will be

. x 1- 4- "^.-t-K -f- o -t- i7ov. = AP +BP -i-Cf'+DP-f-E,-, <3c.



And



1 68 'The Method of FLUXIONS,

And this will be a ready Theorem for the Multiplication of any
infinite Series into each other 5 as in the following Example.

(A) (B) (C) (D) (E) (P) (QJ (R) (S) (T)

X* *J A'4 ,, . x 1

afr*+ + & + &> &c - mto*-fx-f- -
&>+X^+i*?,rb +~, $cc, =^+t^+tf^ 1

X* A 4

JL/rv _ v* a -P.

= , t* A ^^TT 1 ** V- 1 '- _

9# \2a"

+**'+ +7|?

i! * 4 _

7 a i \a '*

.3*

-*- 9 ^

And fo in all other cafes.

II. From the fame Equations above we fhall have A = -.



.-DQ.-CR-BS-AT ^ And then by Subftitution ^

i

^(A + B-J-C+D + E, &c. =) S + a



p p

will ferve commodioufly for the Divifion of one infinite Series by
another. Here for conveniency-fake the Capitals A, B, C, D, &c.
are retained in the Theorem, to denote the firft, fecond, third, fourth,
&c. Terms of the Series refpedively.

M (0)
Thus, for Example, if we would divide the Series #* _f. .ax -+.

(>) (/) (t) ( p J (QJ W (S) (ij

ii x * _}_ ^-ii^-_{_ . 2 ' " z , &c. by the Series a+^x-i- -f- ~^. , &c.

the Quotient will . be a -f- - a *"~ T * -f- fx*



, &c. Or reftoring the Values of
A, B, C, D, &c. which reprefent the feveral Terms as they /land in
order, the Quotient will become a f # + _i_ .11 & r

5 7 z *^ g a 3 ' Ut ^'

And after the fame manner in all other Examples.

HI.



and INFINITE SERIES. 169

III. In the laft Theorem make .r=r, /3 = o, o>=o, ^ = 0,60:.

.

then



^_

V. l' p ~~F~ ~~p

DQ+CR+BS+AT ^ &( , whkh Theorcm win readl]y find th

cal of any infinite Series. Here A, B, C, D, &c. denote the feveral
Terms of the Series in order, as before.

( p ) (QJ
Thus if we would know the Reciprocal of the Series a-\- f.v-{-

<R) (S) (T)

_|_ ^ 4- ^ , &c. we fhall have by Subftitution I t_i _



&c. And reftoring the Values of A, B, C, D, &c. it will be *-
- - - - h ^~- > &c. for the Reciprocal required.

la 1 - 12^ 84 720*'



^- 2. l ,. x + f ..- A< ..{ i f f . = i + f.v + i*+i<, &c. And
fa of others.

IV. In the firft Theorem if we make P=A, Q^==B, R = C,
S=D, &c. that is, if we make both to be the fame Series ; we mail have

A+B+C+D+E+F+G7&^ I * tf= A+ zAB + zAC+ zAD + 2 AE + zAF + zAG.tff.

+ B 1 + zBC + 2 BD + zBE + aBF
+ L* + zCD+ zCE
+ D*

which will be a Theorem for finding the Square of any infinite
Series.



Fv i -'_



-_ - .

aa Sa'^lba 5 iz8a 7 256^ 4^* ga^iea* I zSafl" 1 " 25!., .'<>



64 8 S i 2a s






i t 1 x' L txl A-4



- *^c x 1 bx* o

Ex. 3. - - H - &c.

J 2 ^a ^3 '



u i TTTI



.7/4 4
64*4



Ex. 4 . _l H _H_l^- Ii ^-, I *_ ii._fil , 30."

2 J 8 2434" - 9<?4 >



v.



Method of FLUXIONS,
V. In this laft Theorem, if we make A*= P, aAB = Q v , 2 AC



_f- B 1 =R, 2AD -+- 2BC = S, 2 AE -+- 2BD -f- C 1 = T, &c. we

O R .- R * S "* BC

fhallhave A = P^ B = ^-, C==-^- , D = -^- , E==
T ~ 2BD ~ C - , &c. Or p + Q + K-hS+TH-U, &c. | ^ = pi



iA

Q R B z S 2RC T 2BD C^ U 2 BE ^CD

-4- r -4~ 1 -4- -4- - &c

zA ^^ 2A zA 2A 2 A ' <xu

By this Theorem the Square-root of any infinite Series may eafily be
extracted. Here A, B, C, D, &c. will reprelent the feveral Terms of
the Series as they are in fucceffion.

^1 ^i ~ i^- _i_ fli a4



Ex 2^1- o

"'~~



VI. Becaufeit is by the fourth Theorem a -{[email protected] - y-\-<f<-t-t,&tc. |*
= , a 4-2a/3-f- 2a^ + 2a^H- 2ae, &c. in the third Theorem for



P, Q^ R, S, T, &c. write a 1 , 2a/3, 2> + j8S 2a^ -f- 2/3y, 2ag-
/ i , &c. refpedively. Then



X A




And this will be a Theorem for finding the Reciprocal of the Square
of any infinite Series. Here A, B, C, D, &c. ftill denote the Terms
of the Series in their order.

VII. If in the firft Theorem for P, Q^ R, S, &c. we write
A*, 2AB, 2AC + B 4 , 2AD -H 2BC, &c. refpedively, (that is
A+B+C+D,&c.| 1 3 byTheor.4.)wemallhaveA+B+C+D+E+F 3 6cc.| 5 .
= A s -i- 3A*B + sAB 1 -h sA*D -j- 3AC 1 -f- 360, &c.

6ABC+ 36^0 + 3BD
B' + 6ABD-f- 6ACD
- 6ABE



t

which will readily give the Cube of any infinite Series.
"



v9 A- 11

r. 13 ^ X '



*' *'* ^ yjf^ ^^

* *" " "T~ 2* 11 " " 15 3



Ex.



and INFINITE SERIES, 171



Ex.2. t* 1 -i~

VIII. In the laft Theorem, if we make A 3 =P, 3A*B O ,
'+.3A'C = R, B'-f-6ABC-|-3A 1 D = S, &c. then A=PT,

Q_ R 3 AB* _ S-6ABC Bi

B = p: , C = ?A x , U = - j^ - , fisc. that is
, Sec. I i^K + +l + ^



root of any infinite Series may be extracted. Here alfo A, B, C, D,
&c. will reprefent the Terms as they ftand in order.

T? x' 1 8* 15 7*"* 1 7 _ _* x s ;** IPX'* ^

-~"I" I ~ - z' I ~ ^ + 8i 8 24 3a "'



Ex. 2. f* 4 -h T ' 7 A; 7 H- T | T x 8 , 6cc. l^ =t**-t- r ' T ** H-Trr^ 4 , &c.

IX. Becaufe it is by the feventh Theorem a + -f- y -\- , &c. j J

a* + 3a i /3 -f- 3 a/3 1 -f- /3 5 , &c. in the third Theorem for P,



R, S, T, &c. write ', 3j8, 3/S i -f-Sa 1 ^ /3 } -f- 6a/3>-f-
3'fr &c. refpeflively ; then



This Theorem will give the Reciprocal of the Cube of any infinite
Series ; where A, B, C, D, &c. ftand for the Terms in order.

X. Laftly, in the firft Theorem if we make P=A ; , Q 4 ==3A 1 B,

> &c. we {hall have



A+B-f-C-i-D, &c. I 4 =A^H-4A s B-{-6A 1 B 1 -|-4ABs&c. which



will be a Theorem for finding the Biquadrate of any infinite Series.

And thus we might proceed to find particular Theorems for any
other Powers or Roots of any infinite Series, or for their Recipro-
cals, or any fractional Powers compounded of thefe ; all which will
be found very convenient to have at hand, continued to a competent
number of Terms, in order to facilitate the following Operations.
Or it may be fufticient to lay before you the elegant and general
Theorem, contrived for this purpofe, by that fkilful Mathematician,
and my good Friend, the ingenious Mr. A. De Mo'rore, which was
firft publifh'd in the Philofophical Tranfa&ions, N 230, and which
will readily perform all thefe Operations.

Z 2 Or



172 The Method of FLUXIONS,

Or we may have recourfe to a kind of Mechanical Artifice, by
which all the foregoing Operations may be perform'd in a very eafy
and general manner, as here follows.

When two infinite Series are to be multiply 'd together, in order
to find a third which is to be their Product, call one of them the
Multiplicand, and the other the Multiplier. Write dawn upon your
Paper the Terms of the Multiplicand, with their Signs, in a defcend-
ing order, fo that the Terms may be at equal diftances, and juft
under one another. This you may call your fixt or right-tand Paper.
Prepare another Paper, at the right-hand Edge of which write down
the Terms of the Multiplier, with their proper Signs, in an afcend-
ing Order, fo that the Terms may be at the fame equal diftances
from each other as in the Multiplicand, and juft over one another.
This you may call your moveable or left-hand Paper. Apply your
movenble Paper to your fixt Paper, fo that the firft. Term of your
Multiplier may ftand over-againft the firft Term of your Multipli-
cand. Multiply thefe together, and write down the Product in its
place, for the firft Term of the Product required. Move your move-
abie Paper a ftep lower, fo that two of the firft Terms of the Mul-
tiplier may ftand over-againft two of the firft Terms of the Multi-
plicand. Find the two Produces, by multiplying each pair of the
Terms together, that ftand over-againft one another ; abbreviate
them if it may be done, and- fet down the Refult for the fecond
Term of the Product required. Move your moveable Paper a ftep
lower, fo that three of the firft Terms of the Multiplier may ftand
over-againft three of the firft Terms of the Multiplicand. Find the
three Products, by multiplying each pair of the Terms together that
ftand over-againft one another j abbreviate them, and fet down the
Refult for the third Term of the Product. And proceed in the lame
manner to find the fourth, ana all the following Terms.

I ihall iiluftrate this Method by an Example of two Series, taken
from the common Scale of Denary 01 Decimal Arithmetick ; which
will equally explain the Procefs in all other infinite Series whatever.

Let the Numbers to be multiply 'd be 37,528936, &c. and
528,73041, &c. which, by fupplying X or 10 where it is under-
ftood, will become the Series 3X -\- jX -+- jX-'-f- aX-'-f- 8X-*
_j_ 9 X-44- 3X-5H- 6X- &c. and 5 X* -f- aX 4- 8X -j- 7 X- +
3X~ l -t- oX- J -+-4X-4-f- iX-s, &c. and call the firft the Multipli-
cand, and the fecohd the Multiplier. Thefe being difpofed as is
prefcribed, will ftand as follows.

Multiplier,



and INFINITE SERIES.



Multiplier,



-H4X-+

-f-oX-'



8X



Multiplicand

?X


Product

Tr Y3


iX*

- - - oX 3


3 A


*5 A


8X l


^X- 1


-i- 6^^f


. - 4.X


2X-




T**


- 8X s


- - -1 . T 1 1 Y x


AV-i








?X~ 5


i i-8X *


-8X s


-i- 6X- 5


1 1JO^\.

~* ^ ^ ~* ~ ' t" 2 O I ^\. ^ ^

r^j /*


c.



Now the firft Term of the rrioveable Paper, or Multiplier, being
apply'd to the firft Term of the Multiplicand, will give jX 1 x 3X
= i5X 3 for the firft Term of the Product. Then the' two firft
Terms of each being apply'd together, they will give jX a xyX
-f- 2X x 3X = 4-iX 1 for the fecond Term of the Product. Then
the three firft Terms of each being apply'd together, they will give
5X 1 x5X-'-t-2X x7X -f- 8X x 3X = 63X for the third Term
of the Product. And fo on. So that the Product required will be
, 5 X + 4IX 1 -H 63X H- oyX -f- i42X-f- 133%.-*+. I38X-3
-i-2OiX~ 4 , &c. Now this will be a Number in the Decimal Scale
of Arithmetick, becaufe X = 10. But in that Scale, when it is re-
gular, the Coefficients muft always be affirmative Integers, lefs than
the Root 10 j and therefore to reduce thefe to fuel), fet them orderly
under one another, as is done here, and beginning at the loweft, col-
lect them as they ftand, by adding up each Column. The reafon of
which is this. Becaufe aoiX~4 == aoX" *-f- iX~4, we muft fet
down iX~ 4 , and add 2oX~ 5 to the line above,- Then becaufe 2oX~ 3
H- i38X~s= i58X-*=i5X- ir 4-8X-* ) we muft fet down
and add i ^X.~- to the line above. Then becaufe i^X~^-f- i
= i48X-*=-i4X-'+ 8X~ l , we muft let down 8X- S , and add
i4X~' to the line above. And fo we muft' proceed through the
whole Number. So that at lift we (hall find the Product to be iX 4
_!_ 9 X 3 H- 8X * -+- 4X -f- 2X -f- 6X + 8X- 1 -f- 8X-J , &c. Or
by fuppreffing X, or 10, and leaving it to be fuppiv'a by the Ima-
gination, the Product required wil' be 19842,688, &c.

When one ihfinite Series is to be divided by another, wiite down
the Terms of the Dividend, wkh , eir \ >^ er Signs, in a defccnd-
ing order, fo that the Tunis may be at equal diftances, and juil nn-

dcr



174 t ^ }e Method of FLUXIONS,

der one another. This is your fixt or right-hand Paper. Prepare
another Paper, at the right-hand Edge of which write down the
Terms of the Divifor in an afcending order, with all their Signs
changed except the firft, fo that the Terms may be at the fame equal
distances as before, and jufl over one another. This will be your
moveable or left-hand Paper. Apply your moveable Paper to your
fixt Paper, fo that the firft Term of the Divifor may be over-againft
the firft Term of the Dividend. Divide the firft Term of the Di-
vidend by the firft Term of the Divifor, and fet down the Quotient
over-againft them to the right-hand, for the firft Term of the Quo-
tient required. Move your moveable Paper a ftep lower, fo that
two of the firft Terms of the Divifor may be over-againft two of
the firft Terms of the Dividend. Colleft the fecond Term of the
Dividend, together with the Product of the firft Term of the Quo-
tient now found, multiply'd by the Terms over-againft it in the left-
hand Paper ; thefe divided by the firft Term of the Divifor will be
the fecond Term of the Quotient required. Move your moveable
Paper a ftep lower, fo that three of the firft Terms of the Divifor
may ftand over-againft three of the firft Terms of the Dividend.
Collecl the third Term of the Dividend, together with the two Pro-
duds of the two firft Terms of the Quotient now found, each be-
ing multiply'd into the Term over-againft it, in the left-hand Paper.
Thefe divided by the firft Term of the Divifor will be the third
Term of the Quotient required. Move your moveable Paper a ftep
lower, fo that four of the firft Terms of the Divifor may ftand over-
againft four of the firft Terms of the Dividend. Collecl: the fourth
Term of the Dividend, together with the three Products of the three
firft Terms of the Quotient now found, each being multiply'd by
the Term over-againft it in the left-hand Paper. Thefe divided by
the firft Term of the Divifor will be the fourth Term of the Quo-
tient required. And fo on to find the fifth, and the fucceeding
Terms.

For an Example let it be propofed to divide the infinite Series

I2IA-5 28|X4 ., , 1C' 1

.a* + tax 4- x 1 H- ^ + 7^1 , &c. by the Series a 4- f x
_]_ -j_ L -+- -^ , &c. Thefe being difpofed as is prefcribed,
will ftand as here follows.



Divifor,



and INFINITE SERIES.



175



Divifor,

; fr




Dividend
tf


^___


Quotient


X4




-4- itfA:


-f-y/z.V ,7V ~<7V


*z

1 v


5*3

X*

~4^

**




-f- tt-v 1

( I 2ix3


4-tt** + f ^ f x 1 = -f- f x*

121*3 AT* *3 ^J A j


J' V

x l

-*-?=
^3


3

_ * V




izbca
281*4


*~I26oa lOa ""ga 43 7,1
f 28l.v4 ( A4 A 4 *4 A 4 . A4


7 *

*4


*






' I2boa'
&C.


*~126o*~1 I 4 l , ; * 1 ,2s 1 ra' ~t~ 9 a*

&c.


+ 9..

&c.



Here if we apply the firft Term of the Divifor a, to the firft
Term of the Dividend a 1 , by Divifion we fhall have a for the firft
Term of the Quotient. Then applying the two firft Terms of the.-
Divifor to the two firft Terms of the JDividend, we fhall have ^ax
to be colledled with the Produdl a x f AT, or ax, which will
make -^ax -, and this divided by a, the firft Term of the Divifor,
will give x for the fecond Term of the Quotient. And fo of
the other Terms ; and in like manner for all other Examples.

When an infinite Series is to be raifed to any Power, or when
any Root of it is to be extradled, it may be perform'd in all cafes
by a like Artifice. Prepare your fixt or right-hand Paper, by wri-
ting down the natural Numbers o, i, 2, 3, 4, &c. juft under one an-
other at equal diftances, referving places to the right-hand for the
feveral Terms of the Power or Root, as they fhall be found. The



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