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 20 of 30)
Font size will make ay = *-+-,, and therefore y = * ^ & c

4" ' 4^.4 '

Then xy = * -4- - 4 , &c. and 77 = 4 , &c. which

t" 4~

collected will deftroy each other, and therefore ay = * * -f- o,
&c. and confequently y = * * -j- o, &c. &c.

But there is another cafe of external Terms, which will be dif-
cover'd by the Ruler, and which will give A*x"- m a Ax m - o,
whence m = o, and A =a. Here the Progrefiion of the Indices
will be o, i, 2, &c. fo that ;> == A -j- B* -h Cx 1 , &c. will be the
form of the Series. And if this Root be profecuted by any of the

Methods

and INFINITE SERIES.

223

Methods taught before, it will be found y = a + x -f- _,_

g, &c.

Now in the given Equation, becaufe the infinite Series a -\- x -+-

1 * vj v4

-4- ji 4- ^r y &c. is a Geometrical ProgrcfTion, and therefore is

a

equal to - , as may be proved by Divifion ; if we fubftitute this,

the Equation will become _>* - y -f- ^ - o> And if we ex-
tract the fquare-root in the ordinary way, it will give y r=
a -~^ a ...4+ -x * or ^ exa R ootj And if this Radical

ia~ lax

be refolved, and then divided by this Denominator, the fame two
Series will arife as before, for the two Roots of this Equation. And
this fufficiently verifies the whole Procefs.

43. In Series that are very remarkable, and of general ufe, the
Law of Continuation (if not obvious) mould be always affign'd, when
that can be conveniently done; which renders a Series ftill more ufe-
ful and elegant. This may commonly be difcover'd in the Compu-
tation, by attending to the formation of the Coefficients, efpecially
if we put Letters to reprefent them, and thereby keep them as general
as may be, defcending to particulars by degrees. In the Logarithmic
Series, for instance, z=y {y 1 -f- .lys y4, &c. the Law of
Confecution is very obvious, fo that any Term, tho' ever fo remote,
may eafily be aflign'd at pleafure. For if we put T to reprefent any
Term indefinitely, whcfe order in the Series is exprefs'd by the na-
tural Number , then will T = + -j", where the Sign muft be

4- or according as m is an odd or an even Number. So that the
hundredth Term is L-y l , the next is -j-_J_^ 101 , &c. In the
Reverie of this Series, or y = z,~\- f-s* +- fa; 3 -+- -y^a 4 -4- TTo-^S
&c. the Law of Continuation is thus. Let T reprefent any Term
indefinitely, whofe order in the Series is exprefs'd by m ; then is

m

T= ^p- , which Series in the Denominator mud: be con-
tinued to as many Terms as there are Units in m. Or if c ftands
for the Coefficient of the Term immediately preceding, then is T=

_ <y m

m

Again, in the Series y = z fz"> -+- T TO-S' ToV^ 27 +
_ rT I TT _r' > &c. (by which the Relation between the Circular Arch
a*nd its light Sine is exprefs'd,) the Law of Continuation will be thus.

224 *?%* Method of FLUXIONS,

If T be any Term of the Series, whofe order is exprefs'd by w, and

_ _im I

if c be the Coefficient immediately before ; then T = f ". ,

zm I x zm 2

And in the Reverie of this Series, or z. = y -f- ^v 3 -f- -?)' -f- T A T y~
_f-_|^- T r 9 , cc. the Law of Confecution will be thus. If T repre-
fents any Term, the Index of whole place in the Series is ;//, and if

c be the preceding Coefficient j then T = "" . . ,,i-i_ And

2/11 I X 2 2

the like of others.

44, 45, 46. If we would perform thefe Extractions after a more
Indefinite and general manner, we may proceed thus. Let the given
Equation be v*_ -}- a\v -\- rf.vv 2^x-=o, _ z<jj + fl r , + j5
the Terms of which Ihould be difpofed as + *, * Q
in the Margin. Suppofe y = l> -}- p, where _ * ; * * y
/; is to be conceived as a near Approximation

to the Root y, and p as its fmall Supplement. When this is fubfti-
tuted, the Equation will ftand as it

does here. Now becaufe .v and f> -^ ? + a ~P I + ^'f" + f=~\
are both fmall quantities, the moil _i_ '/ 3 ^ " f *
confiderable quantities are at the be- + /* + */. a s ^ =

ginning of the Equation, from _ x * '

minifliing, both downwards and towards the right-hand ; as oug;ht
always to be fuppos'd, when the Terms of an Equation are dilpos'd
according to a double Arithmetical Scale. And becaufe inftead of
one unknown quantity _v, we have here introduced two, If and />,
we may determine one of them b, as the neceffity of the Relblution
iliall require. To remove therefore the moft confiderable Quantities
out of the Equation, and to leave only a Supplemental Equation,
whofe Root is/>; we may put 6* -+- a*b 2^ = o, which Equa-
tion will determine b, and which therefore henceforward we are to
look upon as known. And for brevity fake, if we put a 1 -+- 3^*
c, we mail have the Equation in the Margin.

Now here, becaufe the two initialTerms

-f- cp -+- abx are the moft con fiderable of \. aox+axp * ?

the Equation, which might be removed, if * r =

for the nrft Approximation to^ we fiiould

afiume ^ , and the refulting Supplemental Equation would be de-
prefs'd lower ; therefore make p = _f- q, and by fubftitution we

-flwll have this Equation following.

Or

and INFINITE SERIES. 225

Or in this Equation, if + J7 . ^+ 3*?*

e make ^ = ^,

3

*

.e.

-+- i =/; itwillaffume a ^ > + **? *

' 1 V

this form.

< 3

x*

Here becaufe the Terms to be next removed are-f- cq -j-^-x^we may
put y = -x l +- r, and by Sub- * +<? + 3V -f? 3 l
ftitution we fhall have another +? _ 1^-^* |
Supplemental Equation, which ? ^

will be farther deprefs'd, and fo
on as far as we pleafe. Therefore *

we mall have the Root y = < a -x - x* Sec. where b will be

c c

the Root of this Equation b* -+. a*b za* = o, c = a 1 -f- 3^*

Or by another Method of Solution, if in this Equation we affumc
(as before) y = A -\-Bx -}- Cx* +- Dx 3 , &c. and fubflitute this in
the Equation, to determine the general Coefficients, we fhall have

. a\ e c -ja' a c , . ,

y = A - -x -t- x*-\ - - ,.- - x"' t &c. wherein A is the

Root of the Equation AS -f. a* A art 5 = o, and c 1 = 3 A* + a*.

47. All Equations cannot be thus immediately refolved, or their
Roots cannot always be exhibited by an Arithmetical Scale, whofe
Root is one of the Quantities in the given Equation. But to per-
form the Analyfis it is fometimes required, that a new Symbol or
Quantity fhould be introduced into the Equation, by the Powers
of which the Root to be extracted may be exprefs'd in a converg-
ing Series. And the Relation between this new Symbol, and the
Quantities of the Equation, mu ft be exhibited by another Equation.
Thus if it were propofed to extradl the Root y of this Equation,
x = fi-\-y 4/ 1 -Hy}' 3 ^_}' 4 , &c. it would be in vain to expedt,
that it might be exprefs'd by the fimple Powers of either x or a.
For the Series itfelf fuppofes, in order to its converging, that y is
fome fmall Number lefs than Unity ; but x and a are under no fuch
limitations. And therefore a Series, compofed of the afcendiag
Powers of x, may be a diverging Series. It is therefore neceflary to
introduce a new Symbol, which mall alfo be fmall, that a Series

G g form'd

4-

226 Ibe Method of FLUXIONS,

form'd of its Powers may converge to y. Now it is plain, that x
ami rf, tho' ever fo great, muft always be near each other, becaufe
their difference y y* } &.c. is a (mall quantity. Aflame therefore
the Equation x a = z, and z will be a fmall quantity as required;
and being introduced inftead of x a, will give z-= y y* -f-
^y* -ly 4 , &c. whofe Root being extracted will be y = z->t-^^
-j-y.2 3 4-TV s4 > ^ cc> as before.

48. Thus if we had the Equation _> i3 -f-j* -h_y x 3 = o, to find
the Root y ; we might have a Series for y compofed of the afcending
Powers of x, which would converge if x were a fmall quantity, lels
than Unity, but would diverge in contrary Circumftances. Suppo-
fing then that x was known to be a large Quantity ; in this cafe the
Author's Expedient is this. Making & the Reciprocal of x, or fup-

pofing the Equation x= l - , inftead of x he introduces z into the

Equation, by which means he obtains a converging Series, confining
of the Powers of z afcending in the Numerators, that is in reality,
of the Powers of x afcending in the Denominators. This he does,
to keep within the Cafe he propofed to himfelf ; but in the Method
here purfued, there is no occafion to have recourfe to this Expedient,
it being an indifferent matter, whether the Powers of the converg-
ing quantity afcend in the Numerators or the Denominators.
Thus in the given Equation y> 4-j 5 - 4- jy * ?

'

king y = Ax m , 6cc.) A**'" 4- A 1 *"" 4- Ax m * , &c.?

j v* .3

by applying the Ruler we mall have the exterior Terms A 3 A* A' 5
= o, or m=. i, and A = i. Alfo the refulting Number or Index
is 3. The next Term to which the Ruler approaches will give 2/11,
or 2; the laft m, or i. But 3, 2, i, make a defcending Progreffion,
of which the common difference is i. Therefore the form of the
Root will be y = Ax 4- B 4- Cx" 1 4- DAT"* , &c. which we may
thus extract.

+y '

Becaufe > )' 3 =A l} ,&c.it will be _y=x,&c.and therefore _y 1 =A- 1 , &c.
which will make y* = * #*, &c. and (by Extraction)/ = * -^
&c. Then (by fquaring)^* = * ~x } &c. which with A* below,
and changing the Sign, makes j 3 = * * ~x } &c. and therefore

y

and INFINITE SERIES. 227

v = * * }*~", &c. Then ;* = * * .1, &c. and y = * ,
&c. which together, changing the Sign, make y> = * * * -4- ,
&c. and ;'=*** + T V*~S &c. Then y- * * -f- 44*-',
Sec. and _>' = * * *""'> &c. and therefore 75 _ ^ # ,. ^ _^_ ^.^-^
&c. and _v = * * * * -f- -pT x ~ 3 > & c -

Now as this Series is accommodated to the cafe of convergency
when x is a large Quantity, fo we may derive another Series from
hence, which will be accommodated to the cafe when .v is a fmall
quantity. For the Ruler will direct us to the external Terms Ax*
x 5 = o, whence m= 3, and A= i ; and the refulting Num-
ber is 3. The next Term will give 2m, or 6 ; and the lair, is 3*77,
or 9. But 3, 6, 9 will form an afcending Progremon, of which the
common difference is 3. Therefore v =Ax'' -+- Ex 6 -t-Cy 9 , &c.
will be the form of the Series in this cafe, which may be thus
derived.

y -} = x> x 6 -+- # * 4*" -f- 14*' 8 , &c.
h x s 2X> +- 3AT 11 2x" 7A- 8 , &c.

Here becaufe_)' = Ar 3 , &c. it will bej>*=x 6 , &c. and therefore
v = * x 6 , &c. Then y- = * 2x, &c. and ^5=^9, &c.
and therefore y = * * H- .V, &c. Then y- = * * -j- 3^'*, 6cc. and
y 3 = * S-^ 11 ) &c. and therefore7= * * * -f. o, &c.

The Expedient of the Ruler will indicate a third cafe of external
Terms, which may be try'd alfo. For we may put A=x= m -{- A*x*' K
-f- Ax m = o, whence m = o, and the Number refulting from the
other Term is 3. Therefore 3 will be the common difference of
the Progrelfion, and the form of the Root will be _y= A -{- Bx' -{-
Cx 6 , &c. But the Equation A 5 -f- A a + A = o, will give A = o,
which will reduce this to the former Series. And the other two
Roots of the Equation will be impofftble.

If the Equation of this Example jy 3 -f- y* -{- r x"' o be
multiply'd by the factor y i, we mall have the Equation y* y

X~'y -f- x' = o, or r+ * # y * ) ...

- t C=o, which when re-

- A'_) - AT J

'folved, will only afford the fame Series for the Root y as before.

49. This Equation \* x\y l -h xy 1 + 2f' zy -+- i = o, when
reduced to the form of a double Arithmetical Scale, will ftand as in

the Margin.

C g 2 Now

2 2 8 The Method of FLUXIONS,

Now the finl Cafe of external _> * +2."- 2>-f i
Terms, fhewn by the Ruler, in _ *- L *

order for an afcending Series, will Or making y _ Axm> fc

make A'.**" 1 _|- 2 A^ i " 2 A*" M ^ m # zAljc
-+- i = o, or ;;/ = o ; where the tm j ri

refulting; Number is alfo o. The \4-s

i 11-1 A 1 * ^

fecond is zm -h i, or i ; the third
zm-}- 2, or 2. Therefore the Arithmetical Progreffion will be o,
i, 2, whofe common difference is i ; and confequently it will be
v == A -f- Bx -+ Cx 1 -+- Dx*, &c. But the Equation A +- zA*
"_ zA -H i o, which mould give the Value of the firft Coeffi-
cient, will fupply us with none but impoffible Roots ; fo that y,
the Root of this Equation, cannot be exprefs'd by an Arithmetical
Scale whofe Root is x, or by an afcending Series that converges by
the Powers of x, when x is a fmall quantity.

As for defcending Series, there are two cafes to be try'd ; firft the
Ruler will give us A**** A 1 AT im + l = o, whence ^m = zm -f- 2, or
fff i } and A=+ i. The Number arifing is 4; the next will
be zm -f- i, or 3 ; the next 2w, or 2 ; the next m, or i ; the laft o.
But the Arithmetical Progreffion 4, 3, 2, i, o, has^ i for its common
difference, and therefore the form of the Series will be y = Ax -+
B 4- CAT"', 8cc. But to extract this Series by our ufual Method, it
will be beft to reduce the Equation to this form, / x* 4- x -+- z
_ _ 2 y~ l 4- y~* = o, and then to proceed thus :

-_ x i x 2 -f- ZX~* | A: ~ 1 > & c -

h A;~ I , &C.

97' 77 c

Becaufe jy = x* x 2, &c. 'tis therefore (by Extradlion)
y _ x JL %x~ l , &c. Then (by Divifion) zy~* = zx-*,
&c. fo that y = * * * -f- 2*- 1 , &c. and (by Extradion) y = * * *
_j_ -VAT-% &c. Then zy~ l = * -f- -i-^" 1 ? &c - and y~* == *"*,
&c.' which being united with a contrary fign, make^ 1 =* * * *
T A '~ I > & c> ant ^ therefore by Extraction y = ****- 4-i-s- v ~ 3

.

In the other cafe of a defcending Series we mall have the Equation

, A 1 *""^ -f- i =o, whence zm +- z = o, or m = i, and

A i . The Number hence arifing is o ; the next will be zm + r,

Cjf

and INFINITE SERIES. 229

or i ; the next 2//v, or 2 -, and the laft 4w, or 4. But
the Numbers o, I, 2, 4, will be found in a defcending Arith-
metical Progrelfion, the common difference of which is i. There-
fore the form of the Root is y = A.x~' -+- Bx~'- -+- Cx~*, &c. and
the Terms of the Equation mufl be thus difpofed for Refolution.

- - - 2X - I-f- A*"" 1

Here becaufe it is y~- = x 1 , &c. it will be by Extraction of the
Square-root y~ l =x, &c. and by finding the Reciprocal, y = x~',
&c. Then becaufe zy~ l = 2X, &c. this with a contrary Sign,
and collected with x above, will make_y 1 = * -{- x, &c. which
(by Extraction) makes y~ I = * -+ i, &c. and by taking the Reci-
procal, /== * ^^~ i , 6cc. Then becaufe zy~* = * i, &c.
this with a contrary fign, and collected with 2 above, will make
y~* = * * i, 8ec. and therefore (by Extraction) y~ l = * *
4*"" , &c. and (by Divifion) jy = * * -f- ^x~ 3 y &cc. Then becaufe
2y~' = * * -}- -^"S ^ w iH be y l = # * * %x~*y &c. and
j- 1 = * * * 4* 1 " 1 , &c - and >'=*** -V*~ J > & c - Then
becaufe 2y~' = * * * -f- -fA;- 1 , &c. and /* = x~~-, &c. thefe

collected with a contrary fign will make y~ z = * * * * V- v ~%
&cc. and y~' = * * * * -V*~S & c - an ^ 7 = ** ** -f- rlT*" 4 *
&c.

Thefe are the two defcending Series, which may be derived for
the Root of this Equation, and which will converge by the Powers
of x, when it is a large quantity. But if x mould happen to be
fmall, then in order to obtain a converging Series, we much change
the Root of the Scale. As if it were known that x differs but little
from 2, we may conveniently put z for that fmall difference, or
we rmy aflame the Equation .v 2 = &. That is, irulead of x
in this Equation fubftitute 2 + 2, and we mall have a new Equa-
tion y* - zy* ^zy* 2y + I = o, which will appear as in
the Margin.

Here

230 The Method of FLUXIONS,

Here to have an afcending Se- .'* * * 2; 4- '

ries, we muft put A+z*? zAz'" ~ ?*' > = l>

+ 1=0, whence m = o, and Or k .

* _ T-l__ -KT 1 1

A4., 4

A = i. The Number hence A 4 ,

arifing is o ; the next is 2/H-i,
or i ; and the laft 2m -f- 2, or 2.
But o, 1,2, are in an afcending
Progreffion, whole common difference is i. Therefore the form of
the Series is y = A -f- B;s -f- Cs a -+- D;s 3 , 6cc. And if the Root y
be extracted by any of the foregoing Methods, it will be found y =.
i -+ - iz -^s 1 , 6cc. Alfo we may hence find two defcending Se-
ries, which would converge by the Root of the Scale z, if it were
a large quantity.

50, 51. Our Author has here opened a large field for the Solution
of thefe Equations, by Shewing, that the indeterminate quantity, or
what we call the Root of the Scale, or the converging quantity,
may be changed a great variety of ways, and thence new Series will
be derived for the Root of die Equation, which in different circum-
/tances will converge differently, fo that the moft commodious for
the preSent occafion may always be chofe. And when one Series
does not fufEciently converge, we may be able to change it for an-
other that (hall converge falter. But that we may not be left to
uncertain interpretations of the indeterminate quantity, or be obliged
to make Suppositions at random j he gives us this Rule for finding
initial Approximations, that may come at once pretty near the Root
required, and therefore the Series will converge apace to it. Which
Rule amounts to this: We are to find what quantities, when fub-
ftituted for the indefinite Species in the propofed Equation, will
make it divifibk by the radical Species, increaSed or diminished by
another quantity, or by the radical Species alone. The fmall diffe-
rence that will be found between any one of thofe quantities, and
the indeterminate quantity of the Equation, may be introduced
inftead of that indeterminate quantity, as a convenient Root of the
Scale, by which the Series is to converge.

Thus ; f the Equation propofed be y= -f- axy -f- cSy x* 2#
= o, and if for x we here Substitute #, we Shall have the Terms
_y 3 -f- 2a i y 3^*, which are divisible by y a, the Quotient be-
ing y* -h ay -f- 3*2*. Therefore we may fuppofe, by the foregoing
Rule, that a x = & is but a fmall quantity, or inftead of x we
may Substitute a z in the propoied Equation, which will then
become y* -f- 2a*y azy -\- y-z 3"* -t- z= 2# 5 = o. A

Series

and INF j NITE SERIES, 231

Series derived from hence, compofcd of the afcending Powers of z 9
mull converge faft, crtfcris parifats, becaule the Root of the Scale
z is a (mail quantity.

Or in the fame Equation, if for x we fubftitute a, we fliall
have the Terms \* a 3 , which are divifible by y a, the Quo-
tient being y* -4- ay -f- a*. Therefore we may fuppofe the diffe-
rence between a and .v to be but little, or that -a x = z is
a fmall quantity, and therefore in (lead of .v we may fubftitute its-
equal a z in the given Equation. This will then become
r 3 azy -f- T,a l z -f- 303* a* = o, where the Root y will con-
verge by the Powers of the fmall quantity z.

Or if for x we fubftitute za, we lhall have the Terms _>' 3
a*? -4- 6^ 3 , which are divilible by y-\- za, the Quotient being _)*

zay-i- 3rt x . Wherefore we may fuppofe there is but a fmall dif-
ference between za and x, or that za x =z is a fmall
quantity ; and therefore infread of x we may introduce its equal

za z into the Equation, which will then become jv* a'-y
azy -4- 6a> -f- iza*z -f- 6az* -f-s } = o.

Laftly, if for x we fubftitute z~*a, we fliall have the Terms
jy 3 z^a'-y -4-tf*y, which are divifible by y, the Radical Species alone.
Wherefore we may fuppofe there is but a fmall difference between

z^'a and x, or that z^a - x = z is a fmall quantity ; and
therefore inflead of x we may fubfthute its equal 2?a z, which
will reduce the Equation to y* -f- i ^z x a"y azy -+- 3^4 x a^z

-\- 3^2 x az 1 -f- Z' = o, wherein the Series for the Root y may
converge by the Powers of the fmall quantity z.

But the reafon of this Operation ftill remains to be inquired into,
which I mall endeavour to explain from the prefent Example. In
the Equation y~> -\- axy -f- a*y x 3 za* =o, the indeterminate
quantity x, of its own nature, muft be fufceptible of all poffible
Values ; at leaft, if it had any limitations, they would be fhew'd by
impoflible Roots. Among other values, it will receive thefe, a, a,
- r - za, z~*a, 6cc. in which cafes the Equation would become y*
+- za*y 30* = o, 7 ; a 1 = o, y* a 1 y -4- 6a* = o, _y 3
2^a*y -f- a'-y =. o, &cc. refpedtively. Now as thefe Equations admit
of jull Roots, as appears by their being divifible by y -f- or an-
other quantity, and the laft by y alone; fo that in the Refolution,
the whole Equation (in thofe cafes), would be immediately exhaufted :
And in other cafes, when x does not much recede from one of thofe

Values,

232 The Method of FLUXIONS,

Values, the Equation would be nearly exhaufted. Therefore the
introducing of z, which is the fmall difference between x and any
pne of thofe Values, muft deprefs the Equation ; and z itfelf mull
be a convenient quantity to be made the Root of the Scale, or the
converging Quantity.

I (hall give the Solution of one of the Equations of thefe Exam-
ples, which mall be this, _y 3 azy -f- y-z -4- ^az* a* = o, or

Here becaufe _>' 5 = # ; , &c. it will be y = a, &c. Then azy
fl2r, 6cc. which muft be wrote again with a contrary fign, and
united with 3^*2 above, to make y* = * 2a*z, &c. and
therefore y ==. * -f-*' & c - Then s>' = * -f- ^az 1 , 6cc. and

V = * * , &c. Then

* " 3 a

ssz # * -f- .Is 3 , &c. and _)'* = *** J-.ss, &c. and y = * * *

217Z 5 <>

, OCC.

The Author hints at many other ways of deriving a variety of
Series from the fame Equation ; as when we fuppofe the afore-men-
tion'd difference z to be indefinitely great, and from that Suppofition
we find Series, in which the Powers of z (hall afcend in the Deno-
minators. This Cafe we have all along purfued indifcriminately with
the other Cafe, in which the Powers of the converging quantity
afcend in the Numerators, and therefore we need add nothing here
about it. Another Expedient is, to affume for the converging quantity
fome other quantity of the Equation, which then may be confider'd
as indeterminate. So here, for inftance, we may change a into x,
and x into a. Or laftly, to affume any Relation at pleafure, (fup-
pofe x = az -f- bz\ x = ~- , x 3 J^ 5 &c.) between the in-
determinate quantity of the Equation x, and the quantity z we
would introduce into its room, by which new equivalent Equations
may be form'd, and then their Roots may be extracted. And after-
wards the value of z may be exprefs'd by x } by means of the af-
fumed Equation.

52. The

and INFINITE SERIES. 233

52, The Author here, in a fummary way, gives us a Rationale of
his whole Method of Extractions, proving a priori, that the Series
thus form'd, and continued in infinitum, will then be the juft Roots
of the propofed Equation. And if they are only continued to a
competent number of Terms, (the more the better,) yet then will
they be a very near Approximation to the juft and compleat Roots.
For, when an Equation is propofed to be refolved, as near an Ap-
proach is made to the Root, iuppofe y, as can be had in a lingle
Term, compofed of the quantities given by the Equation ; and be*.
caufe there is a Remainder, a Relidual or Secondary Equation is
thence form'd, whole Root p is the Supplement to the Root of the
given Equation, whatever that may be. Then as near an approach
is made to /, as can be done by a lingle Term, and a new Relidual
Equation is form'd from the Remainder, wherein the Root q is the
Supplement to p. And by proceeding thus, the Relidual Equations
are continually deprefs'd, and the Supplements grow perpetually lels
and lefs, till the Terms at laft are lefs than any affignable quantities.
We may illuftrate this by a familiar Example, taken from the ufual
Method of Divifion of Decimal Fractions. At every Operation we
put as large a Figure in the Quotient, as the Dividend and Divifor
will permit, fo as to leave the leaft Remainder poflible. Then this
Remainder (applies the place of a new Dividend, which we are to
exhauft as far as can be done by one Figure, and therefore we put
the greateft number we can for the next Figure of the Quotient,
and thereby leave the leaft Remainder we can. And fo we go on,
either till the whole Dividend is exhaufted, if that can bz done, or
till we have obtain'd a fufficient Approximation in decimal places or
figures. And the fame way of Argumentation, that proves our Au-
thor's Method of Extraction, may ealily be apply'd to the other
ways of Analylis that are here found.

53, 54. Here it is feafonably obferved, that tho' the indefinite

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