Isaac Newton.

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

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Online LibraryIsaac NewtonThe method of fluxions and infinite series : with its application to the geometry of curve-lines → online text (page 16 of 30)
Font size firft Term of which Series may be immediately known from the firft
Term of the given Series, and from the given Index of the Power
or Root, whether that Index be an Integer or a Fraclion, affirmative
or negative ; and that Term therefore may be fet down in its place, .
over-againft the firft Number o. Prepare your moveable or left-
hand Paper, by writing down, towards the edge of the Paper at the
right-hand, all the Terms of the given Series, except the firft, over
one another in order, at the fame diftances as the Numbers in the
other Paper. After which, nearer the edge of the Paper, write juft
over one another, fiift the Index of the Power or Root to be found,
then its double, then its triple, and fo the reft of its multiples,
with the negative Sign after each, as far as the Terms of the Series
extend. And alfo the firft Term of the given Series may be wrote
below. Thus will the moveable Paper be prepared. Thefe multi-
ples, together with the following negative Signs, and the Numbers

7^^ Method of FLUXIONS,

> ij 2 ' 3-4> ^ c - on tne otner Paper, when they meet together, will
compkat the numeral Coefficients. Apply therefore the fecond Term
.of the move-able Paper to the uppertnoft Term of the fixt Paper,
;ind the Product made by the continual Multiplication of the three
Factors thatftand in a lin-e over-againft one another, [which are the
fecond Term of the given Series, the numeral Coefficient, (here the
given Index,) and the firft Term of the Series already found,] di-
vided by the firft Term of the given Series, will be the fecond Term
of the Series required, which is to be let down in its place over-
againft I. Move the moveable Paper a ftep lower, and the two
Produces made by the multiplication of the Factors that ftand over-
-againft one another, (in which, and elfewhere, care muft be had to
take the numeral Coefficients compleat,) divided by twice the firft
Term of the given Series, v/ill be the third Term of the Series re-
quired, which is to be fet down in its place over-againft 2. Move
the moveable, Paper a ftep lower, and the three Products made by
the multiplication of the Factors that ftand over-againft one another,
divided by thrice the firft Term of the given Series, will be the
fourth Term of the Series required. And fo you may proceed to
find the next, and the fubfcquent Terms.

It may not be amifs to give one general Example of this Reduc-
tion, which will comprehend all particular Cafes. If the Series az
_l_ b^ _j_ c&' -+-dz*, ,&c. be given, of which we are to find any
Power, or to extract any Root; let the Index of this Pov>er or Root
be m. Then prepare the moveable or left-hand Paper as you fee
below, where the Terms of the given Scries are fet over one another
in order, at the edge of the Paper, and at equal diftances. Alfo
after every Term is put a full point, as a Mark of Multiplication,
and after every one, (except the firft or loweft) are put the feveral
Multiples of the Index, as m, zm, pn, 40;, &c. with the negative
Sign after them. Likewife a vinculum may be undei flood to
be placed over them, to connect them with the other parts of the
numeral Coefficients, which are on the other Paper, and which
make them compleat. Alfo the firft Term of the given Series is
feparated from the reft by a line, to denote its being a Divifor, or
the Denominator of a Fraction. And thus is the moveable Paper
prepared.

To prepare the fixt or right-hand Paper, write down the natu-
ral Numbers o, i, 2, 3, 4, &c. under one another, at the fame equal
diftances as the Terms in the other Paper, with a Point after them
as a Mark of Multiplication ; and over-againft the firft 1 erm o

write

and INFINITE SERIES.

write a*"z m for the firft Term of the Series required. The reft ot
the Terms are to be wrote down orderly under this, as they (hall be
found, which will be in this manner. To the firft Term o in the
fixt Paper apply the fecond Term of the moveable Paper, and they

will then exhibit this Fraction *-* m ~~ - " z , which being reduced

as,. I

to this aw* < ~ t &s*+ I , muft be fet down in its place, for the fecond
Term of the Series required. Move the moveable Paper a ftep lower,
and you will have this Fraction exhibited + cz*. 2m o. a a z m

az. 2

which being reduced will become mu m - l c-{- mx "LL a m - b* xz m +'~,

to be put down for the third Term of the Series required. Bring
down the moveable Paper a ftep lower, and you will have the
Fraction -f- dz,*. yn o. a m z n
.+- cz*.

bz?. m

ma

*c -+- m x

L a m - l b 3 -

az. 3

for the fourth Term of the Series required. And in the fame man-
ner are all the reft of the Terms to be found.

Moveable
Paper, &c.

*. m

az.

Fixt Paper

o.
i.

2.

- a^i-o* -f- ma m ~*c x z"

-3 . m x - -x- -a m *l>>+mx.'- ' a m 1 6c+Ma m l dxz"

J 7. 1 T

N. B. This Operation will produce Mr. De Moivre's Theorem
mentioned before, the Inveftigation of which may be feen in the
place there quoted, and fhall be exhibited here in due time and
place. And this therefore will fufficiently prove the truth of the
prefent Procefs. In particular Examples this Method will be found
very eafy and practicable.

A a But

178 The Method of FLUXIONS,

But now to mew fomething of the ufe of thefe Theorems, and
jit the fame time to prepare the way for the Solution of Affected and
Fluxional Equations; we will here make a kind of retrofpect, and
refume our Author's Examples of fimple Extractions, beginning
with Divifion itfelf, which we fhall perform after a different and an
eafier manner.

Thus to divide aa by b -f- x, or to refolve the Fraction
into a Series of fimple Terms ; make r^=y, or by -f- xy - ,
Now to find the quantity y difpofe the Terms of this Equation after
this manner + *-J J = a 1 , and proceed in the Refolution as you fee
is done here.

I a*x a*.* 1 a^J a * x t

=** -T-* - T* 7T + -77- > &C..

+ xy\ h-r TT + -75 ,

C

IT + -JT- > OCC.

Here by the difpofition of the Terms a*- is made the firft Term
of the Series belonging (or equivalent) to by, and therefore dividing
by b, will be the firfl Term of the Series equivalent to y, as is fet

a^x

down below. Then will + be the firft Term of the Series
-4- xy, which is therefore fet down over-againft it; as alfo it is fet
down over-againft by, but with a contrary Sign, to be the fecond

Term of that Series. Then will a ~ be the fecond Term of y t

to be fet down in its place, which will give a -^- for the fe-
cond Term of -f- xy ; and this with a contrary Sign muit be fet down
for the third Term of by. Then will + ~- be the third Term of

y, and therefore + ~ will be the third Term of 4- xy, which
with a contrary Sign mufl be made the fourth Term of by, and there-
fore '~ will be the fourth Term of y. And fo on for ever.

Now the Rationale of this Procefs, and of all that will here fol-
low of the fame kind, may be manifeft from thefe Confiderations.
The unknown Terms of the Equation, or thofe wherein y is found,
are (by the Hypothecs) equal to the known Term aa. And each of

thofe

a?id IN FINITE SERIES. 170

thofe unknown Terms is refolved into its equivalent Series, the Ag-
gregate of which muft (till be equal to the fame known Term aa ;
(or perhaps Terms.) Therefore all the fubfidiary and adventitious
Terms, which are introduced into the Equation to aflift the Solution,
(or the Supplemental Terms,) muft mutually deftroy one another.
Or we may refolve the fame Equation in the following manner :

a* la* k*a* Ha* .

y = - - -4 - - - , &c.

^ A" .V X A;4

Here a 1 is made the firft Term of -+- xv, and therefore muft

" x

be put down for the firft Term of y. This will give + for the
firft Term of by, which with a contrary Sign muft be the fecond
Term of -+- xy, and therefore - ~ muft be put down for the fe-

cond Term of y. Then will ^ be the fecond Term of by y
which with a contrary Sign will be the third Term of -|- xy, and
therefore + - - will be the third Term of y. And fo on. There-

fore the Fraction propofed is refolved into the fame two Series as
were found above.

If the Fraction : were given to be refolved, make -

1 + * ' + -V"

v t or y -+- x l y=. i, the Refolution of which Equation is little
rrxpre than writing down the Terms, in the manner following :

y = i x+x xx,&cc. y 7 - - (-x- 1 *-4-|_x-

. +x*y 3 = i x-'-+x- x~ & ,

, ccc. +x*y 3 = i x-'-+x- x~ & , &c.

Here in the firft Paradigm, as i is made the firft Term of y, fo
will x 1 be the firft Term of x*y, and therefore x*- will be the
fecond Term of y, and therefore x* will be the fecond Term of
x*y, and therefore -+- x* will be third Term of y ; &c. Alfo in the
fecond Paradigm, as i is made the firft Term of x*y, fb will -f- x~'-
be the firft Term of y, and therefore x~- will be the fecond
Term of x*y, or x~* will be the fecond Term of y ; &c.

A a 2 To

180 tte Method of FLUXIONS,

i 3.

To refolve the compound Fraction . zx ~* - into fimple Terms,
i i

2.y a * 13 i

make 7 =y, or 2** # v = y 4- AT^ ^xy, which E-

I+* 1 3*

quation may be thus refolved :

= 2A^ * X^

1 3** + 34^* 73**, &c.
34* 3 } &c.
39**, &c.

Place the Terms of the Equation, in which the unknown quan-
tity y is found, in a regular defcending order, and the known Terms
above, as you fee is done here. Then bring down zx^ to be the firfl
Term of y, which will give -f- 2x for the firfl Term of the Series
4- x*y, which mufl be wrote with a contrary Sign for the fecond
Term of y. Then will the fecond Term of 4- x^y be 2x%, and
the firfl Term of the Series 3*7 will be 6x^, which together
make SAT*. And this with a contrary Sign would have been wrote
for the third Term of y, had not the Term x* been above, which
reduces it to 4- jx J * for the third Term of y. Then will 4- yx*
be the third Term of 4- x*y, and 4- 6x* will be the fecond Term
of 3fly, which being collected with a contrary Sign, will make
1 3** for the fourth Term of y ; and fo on, as in the Paradigm.

If we would refolve this Fraction, or this Equation, fo as to ac-
commodate it to the other cafe of convergency, we may invert the
Terms, and proceed thus :

O W f V 1 *- -i-

3-v / x

y = f X' -f- 7 4f *

1, &c.

ft*- 1 ' , 6cc.

Bring down AT* to be the firfl Term of 3-vy, whence ~\- ^
will be the firfl Term of y, to be fet down in its place. Then the

firfl

and INFINITE SERIES. 181

firft Term of +- x^y will be -f- f x, which with a contrary Sign
will be the fecond Term of 3*?, and therefore -+- f will be the
fecond Term of y. Then the fecond Term of -+- x^y will be -f- f#s
and the firft Term of y being -+- f x*, thefe two collected with a
contrary Sign would have made *.#* for the third Term of 3*}',
had not the Term +- zx' z been prefent above. Therefore uniting
thefe, we fhall have -f- x* for the third Term of 3*7, which

will g lve ?j-x~* f r the third Term of y. Then will the third
Term of -+- xh be if, and the fecond Term of y being -+- -%,
thefe two collected with a contrary Sign will make -f- if for the
fourth Term of T,xy, and therefore TT*"" 1 will be l ^ e fourth
Term of y -, and fo on.

And thus much for Divifion ; now to go on to the Author's pure
or fimple Extractions.

To find the Square-root of aa -f- xx, or to extract the Root y of
this Equation yy = aa-{- xx ; make y = a -+-/>, then we fhall have
by Subftitution zap -f- pp = xx, of which affected Quadratick Equa-
tion we may thus extract the Root p. Difpofe the Terms in this
manner zap-^= xx, the unknown Terms in a defcending order oa

H-/AJ
one fide, and the known Term or Terms on the other fide of the

Equation, and proceed in the Extraction as is here directed.

-) *4 * 5x8 7 *'

**/==* - - + s74 5i + H53.

+ -\.__ + ^ _^i + ^!_^:,

"J J 4* 8*4 640* 12Sa 8 '

x* A4 * 6 CA 9 7* 10 .,

* -I- , h -t-f-t &C.

f za 8a l \6a ! izSa 1 25O'

By this Difpofition of the Terms, x 1 is made the firft Term of

x

the Series belonging to zap ; then we fhall have for the firft
Term of the Series p, as here fet down underneath. Therefore

will be the firft Term of the Series *, to be put down in its
4474

place over-againft p 1 . Then, by what is obferved before, it muft
be put down with a contrary Sign as the fecond Term of zap,

which will make the fecond Term of/> to be - ^ . Having there-
fore

77jt2 Method of F L n v i o !-; s,-

/

fore the two firft Terms of P = *- ~, we fhall liave, (by any
of the foregoing Methods for finding the Square of an infinite Se-
ries,) the two firft Terms of p 1 = ~ . which la ft Term

AfCi 3 - #4 4 '

irmft be wrote with a contrary Sign, as the third Term of zap.
Therefore the third Term of * is ^ , and the third Term of p*

* ' L

zap - - -f- zax za* -+-

- * * ,&c.

(by the aforefaid Methods) will be -~ } which is to be wrote with
a contrary Sign, as the fourth Term of zap. Then the fourth

8

Term of p will be -||_, and therefore the fourth Term of/* is
""" 7IsI ' which is to be wrote with a contrary Sign for the fifth
Term of zap. This will give 2^- for the fifth Term of p . and fo

2^O'

we may proceed in the Extraction as far as we pleafe.

Or we may difpofe the Terms of the Supplemental Equation thus :

> J a* c

~ x * ^ ' &c '

, &c.

A X 3 y

Here * a is made the firft Term of the Series/ 4 , and therefore x,
(or elfe x,) will be the firft Term of p. Then zax will be the
firft Term of zap> and therefore zax will be the fecond Term of
p"- . So that becaufe /> 1 = # a 2rfx, 6cc. by extracting the Square-root
of this Series by any of the foregoiug Methods, it will be found
/ x a, &c. or a will be the fecond Term of the Root />.
Therefore the fecond Term of zap will be 2<2% which muft be
wrote with a contrary Sign for the third Term of/ 1 , and thence (by

Extraction) the third Term of / will be - . This will make the
.third Term of zap to be , which makes the fourth Term of/ 4

to be - , and therefore (by Extraction) o will be the fourth Term
of/. This makes the fourth Term of zap to be o, as alfo of / z .
Then ^ will be the fifth Term of/. Then the fifth Term of

I zap

and INFINITE SERIES, 183

zap will be . , which will make the fixth Term of />* to be

f 4* 5

.ll ; and therefore o will be the fixth Term of p, &c.

Here the Terms will be alternately deficient ; fo that in the given
Equation yy = aa -\- xx, the Root will be y = a -f- x a -f- "->

&c. that is y = x -f- ^- } -h -^- s , &c. which is the fame as
if we fhould change the order of the Terms, or if we fhould change
a into x, and x into a.

If we would extradl the Square-root of aa xx, or find the
Root y of the Equation yy = aa xx ; make y = a -f- p, as be-
fore ; then zap -f-/* x = x*-, which may be refolved as in the fol-

J _ .v4 X 6 ^.V 8

f I" 4fl z 8a4 64,1

f *4 X' 6 <;* 8

+ t> 1 \ 1- ; H- ^~; -f- f - -{-
r J 4 84 6^.6

^ "4 JC CX "7 X c

J* . ,^_ __ k~^_ i . . ^^^^ - ^^ ^ J, cSCC*

Here if we mould attempt to make x 1 the firft Term of -J-/ 1 ,
we mould have ^/ x 1 , or x^/ i, for the fi rfl Term of/ ; which
being rnpoflible, fliews no Series can be form'd from that Suppofi-
tion.

To find the Square-root of # xx, or the Root y in this Equa-
tion yy = x xx, make y = x^ + p, then x -+- zx^p -f- p 1 = x
xx, or zx^p -+- /* = - x*, which may be refolved after this
manner :

The Terms being rightly difpofed, make x* the firft
of zx^p; then will x* be the firft Term of p. Therefore
~\- px 3 will be the firft Term of / a , which is alfo to be wrote with
a contrary Sign for the fecoiid Term of 2x'-p, which will give f A - *
for the lecond Term of p. Then (by fquaring) the fecond Term of
^ will be i^ 4 , which will give - i* 4 for the fecond Term of

184 ffi? Method of FLUXIONS,

zx^p, and therefore -V^ for the third Term of p ; and fq op.
Therefore in this Equation it will be y=z x'* f A-'" f x* rV*''"*
&c.

So to extract the Root y of this Equation yy =.aa-\-bx xx t
make y = a-{-p } then zap -+- p* = bx xx, which may be thus
refolved.

=fa X * + *fi, & C .
L'-x' 1

tx x* Ixl

Make bx the firft Term of zap; then will l ~- be the firft Term

20.

of p. Therefore the firft Term of p 1 will be -+- b ^ , which is
alfo to be wrote with a contrary Sign, fo that the fecond Term of

zap will be -< x* ^- , which will make the fecond Term of

p to be * gjr ' Then by fquaring, the fecond Term of/*

will be 7 *-* -g^ 5 which muft be wrote with a contrary Sign

for the third Term of zap. This will give the third Term of p
as in the Example; and fo on. Therefore the Square-root of the

Quantity a^ -f- bx xx will be a -+ ^ ~ -f. -^ _,

Alfo if we would extract the Square-root of < _ a * , we may ex-

tract the Roots of the Numerator, and likewife of the Denomi-
nator, and then divide one Series by the other, as before ; but more

dire.ctly thus. Make _**! = yy, or i -{- ax 1 = yy b*x*y*.

Suppofe y = i -f- p, then ax* = zp -+-p z bx*- zbx*-p bx*-p*,
which Suppplemental Equation may be thus refolved.

zp

and INFINITE SERIES,

185

bx^p* _

ab ^ab 1 , &c.

~a^b t &c.

+TT*\

, &c.

Make ax 1 -f- bx* the firfl Term of 2/>, then will frf.v l -f- f &v
"be the firfl Term of /. Therefore abx* b*x* will be the firfl
Term of 2bx*p, and ^a*x* -f- -^abx* -f- -^bx* will be the firfl
Term of/*. Thefe being collected, and their Signs changed, muil
be made the fecond Term of 2/, which will give abx* -f- |J*A
%a*x* for the fecond Term of/. Then the fecond Term of 2bx*p
will be -^ab^x 6 l>*x 6 -f- ^a i bx 6 > and the fecond Term of p*
(by fquaring) will be found f a l bx 6 +- \ab*x 6 j-a''X 6 -{- \$frx 6 , and
the firfl Term of bx*p l will be ^a^bx 6 -^ab'-x 6 f^'AT 4 ;
which being collected and the Signs changed, will make the third
Term of 2p, half which will be the third Term of p ; and fo on as
far as you pleafe.

And thus if we were to extract the Cube-root of a* 4- x*, or the
Root y of this Equation 7' = a 3 4- tf 3 j make y = a -f-/, then by
Subflitution a 3 -f- 3d 1 / -f- ^ap 1 -+- p> = & -+- x*, or 3 i / -f
= A: S , which fupplemental Equation may be thus refolved.

243 l

B b

The

Method of FLUXIONS,

The Terms being difpos'd in order, the firft Term of the Series
<ia*p will be #', which will make the firft Term ofp to be *. Thiss

will make the firft Term of/ 1 to be ^ . And this will make the

firfl Term of ^ap* to be ^ , which with a contrary Sign muft be
the fecond Term of 3/z*/>, and therefore the fecond Term ofp will
be r . Then (by fquaring) the fecond Term of ^ap 1 will be

. ff! and (by cubing) the firft Term of *"= win be fi . Thefe

Oa* 27<:< 6

r y9

being collected make , which with a contrary Sign muft be
the third Term of ^a^p, and therefore the third Term of p will be
_j_ ill . Then by fquaring, the third Term of ^ap* will be -

__ .

and by cubing, the fecond Term of/ 3 will be ^^, which being
collected will make y^-j > anc ^ therefore the fourth Term of-^^p
will be ^T, and the fourth Term of p will be - * 11 - . And

8j<i' ' 2 43 a

fo on;'

Arid thus may the Roots of all pare Equations- be extracted, but
in a more direct and fimple manner by the foregoing Theorems.
All that is here intended, is, to prepare the way for the Refolution
of affected Equations, both in Numbers and Species, as alfo of
Fluxional Equations, in- which this Method will be found to be of
very extenfive ufe. And firfl we mall proceed with our Author to
the Solution of numerical affected Equations.

SECT. II L The Refolution of Nttmeral AffeSted Equations*

W as to the Refolution of affected Equations, and firft
in Numbers ; our Author very juftly complains, that be-
fore his time the exegefa numcroja, or the Doctrine of the Solution
of affected Equations in Numbers, was very intricate, defective, and
inartificial. What had been done by Vieta, Harriot, and Oughtred
in this" matter, tho' very laudable Attempts for the time, yet how-
ever was extremely perplex'd and operofe. So that he had good rea-
fon to reject their Methods, efpecially as he has fubftituted a much
better in their room. They -affected too great accuracy in purfuing

exact

and INFINITE SERIES. 187

exact Roots, which led them into tedious perplexities ; but he knew
very well, that legitimate Approximations would proceed much more
regularly and expeditioufly, and would anfwer the fame intention
much better.

20, 21, 22. His Method may be eafily apprehended from this one
Inftance, as it is contain'd in his Diagram, and the Explanation of
it. Yet for farther Illuftration Lfhall venture to give a fhort rationale
of it. When a Numeral Equation is propos'd to be refolved, he
takes as near an Approximation to the Root as can be readily and
conveniently obtain'd. And this may always be had, either by the
known Method of Limits, or by a Linear or Mechanical Conitruc-
tion, or by a few eafy trials and fuppofitions. If this be greater or
lefs than the Root, the Excefs or Defect, indifferently call'd the Sup-
plement, may be reprefented by p, and the affumed Approximation,
together with this Supplement, are to be fubftituted in the given.
Equation inftead of the Root. By this means, (expunging what will
be fuperfluous,) a Supplemental Equation will be form'd, whole Root
is now p, which will confift of the Powers of the affumed Approxima-
tion orderly defcending, involved with the Powers of the Supplement
regularly afcending, on both which accounts the Terms will be con-
tinually decreafmg, in a decuple ratio or falter, if the affumed Ap-
proximation be -fuppos'd to be at leaft ten times greater than the
Supplement. Therefore to find a new Approximation, which fhall
nearly exhauft the Supplement p, it will be fufficient to retain only
the two firft Terms of this Equation, and to feek the Value ofp from
the refulting fimple Equation. [Or fometimes the three firft Terms
may be retain'd, and the Value of p may be more accurately found
from the refulting Quadratick Equation; Sec.] This new Approxi-
mation, together with a new Supplement g, muft be fuhftituted in-
itead of p in this laft fupplemental Equation, in order to form a
fecond, whofe Root will be q. And the fame things may be obferved
of this fecond fupplemental Equation as of the firft; and its Root, or
an Approximation to it, may be difcover'd after the fame manner. And
thus the Root of the given Equation may be profecuted as far as
we pleafe, by finding new iiipplemental Equations, the Root of every
one of which will be a correction to the preceding Supplement.

So in the prefent Example jy 3 2y 5 = o, 'tis eaiy to perceive,
that y = 2 fere ; for 2x2x2 2x2 = 4, which mould make 5.
Therefore let p be the Supplement of the Root, and it will be y =
2. -{-/>, and therefore by fubftitution i -f- lop -+- 6p* -\-p= = Q.
As p is here fuppos'd to be much lefs than the Approximation 2,

B b 2 ty

i88 The Method of FLUXIONS,

by this fubftitution an Equation will be form'd, in which the Terma
will gradually decreafe, and Ib much the fafter, cateris parities, as
2 is greater than p. So taking the two firft Terms, i -f- io/>=o,
fere, or p =. T x _. fere ; or affuming a fecond Supplement q, 'tis
p = T 'o- -h ? accurately. This being fubftituted for p in the laft
Equation, it becomes o, 6 1 -+- 1 1,237 + 6>3?* 4- <f = o, which is
a new Supplemental Equation, in which all the Terms are farther
deprefs'd, and in which the Supplement q will be much lefs than the
former Supplement p. Therefore it is 0,61 -f- 1 1,23.^ = o, ym?,.

or q= f^e, or q-=. 0,0054-)^ accurate, by afluming

r for the third Supplement. This being fubftituted will give
0,000541554- 11,162;-, &c. =o, and therefore r-= '^^-^

= 0,00004852, &c. So that at laft/=2 -{-^> = &c. or_y =
2,09455148, &c.

And thus our Author's Method proceeds, for finding the Roots of
affedted Equations in Numbers. Long after this was wrote, Mr. Rapb-
Jon publifh'd his Analyfis Mquationum imiverjalis, containing a Me-
thod for the Solution of Numeral Equations, not very much diffe-
rent from this of our Author, as may appear by the following Com
parifon.

To find the Root of the Equation y* zy = 5, Mr. Rapbfon
would proceed thus. His firft Approximation he calls g, which he
takes as near the true Root as he can, and makes the Supplement x, fo
that he has_y==g-+Ar. Then by Subftitution < g 3 -f-3^ 1 Ar+3^x a -f-x 3 =5,

2 2

or if g=2, 'tis iOAr-f-6.v* -4- x* = i, to determine the Supple-
ment x. This being fuppofed fmall, its Powers may be rejected,

Online LibraryIsaac NewtonThe method of fluxions and infinite series : with its application to the geometry of curve-lines → online text (page 16 of 30)