Isaac Newton.

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

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Font size be 3. The la ft are zm -f- i, 5>-f- i, of which each is i. So that
here #2 = 0, and the Numbers arifing are 5, 3, i, which form a
decreafing Arithmetical Progreffion, the common difference of which
is 2. And if there had been more Parallelograms, any how difpofed,
their Numbers would have been comprehended by this Arithmetical
Progreffion, or at leaft it might have been interpolated with other
Terms, fo as to comprehend them all, however promifcuoufly and
irregularly they might have been taken.

Thus fecondly, if the Ruler be apply'd to the two external Pa-
rallelograms 5/72+ 5 and 6m-}- 3, and if thefe Numbers be made
equal, we mail have m = 2, and the Numbers themfelves will be
each ic. The three next Numbers which the Ruler .will arrive at

will

20O The Method of FLUXIONS,

will be each 11, and the two laft will be ^ach 5. But the Num-
bers 15, n> 5. will be comprehended in the decreafing Arithmetical
Progreffion 15, 13, 1 1, 9, 7, 5, whofe common difference is 2.

Thirdly, if the Ruler be apply'd to the two external Parallelograms
6m -f- 3 and 5*0-4-1, and if thefe Numbers be made equal, we fhall
have tn = 2, and the Numbers will be each 9. The two next
Numbers that the Ruler will arrive at will be each 5, the next
will be 3, the next i, and the laft -+- i. All which will be
comprehended in the afcending Arithmetical Progreffion 9, 7,

5, 3, i, -+- i, whofe common difference is 2.
Fourthly, if the Ruler be apply'd to the two loweft and external

Parallelograms 2m-\-i and 5/77 -+- i, and if they be made equal,
we fhall have again m = o, fo that each of thefe Numbers will be i .
The next three Numbers that the Ruler will approach to, will each
be 3, and the laft 5. But the Numbers i, 3, 5, will be compre-
hended in an afcending Arithmetical Progreffion, whofe common
difference is 2.

Fifthly, if the Ruler be apply'd to the two external Parallelograms
in -f- 3 and 2m +- i, and if thefe Numbers be made equal, we fhall
have m = 2, and the Numbers themfelves will be each 5. The
three next Numbers that the Ruler will approach to will each be 1 1,
and the two next will be each 15. But the Numbers 5, 1 1, 15, will
be comprehended in the afcending Arithmetical Progreffion 5, 7, 9,
II, 13, 15, of which the common difference is 2.

Laftly, if the Ruler be apply'd to the two external Parallelograms
pn -f- 5 and m-\- 3, and if thefe Numbers be made equal, we fhali
have m=. I, and the Numbers themfelves will each be 2. The
next Number to which the Ruler approaches will be o, the two next
are each i, the next 3, the laft 4. All which Numbers
will be found in the defcending Arithmetical Progreffion 2, I, p,

i, 2, 3, 4, whofe common difference is i. And thefe
fix are all the poffible cafes of external Terms.

Now to find the Arithmetical Progreffion, in which all thefe re-
fulting Terms fhall be comprehended ; find their differences, and the
greateft common Divifor of thofe differences fhall be the common
difference of the Progreffion. Thus in the fifth cafe before, the refulting
Numbers were 5, 1 1,15, whofe differences are 6, 4, and their greateft
common Divifor is 2. Therefore 2 will be the common difference of
the Arithmetical Progreffion, which will include all the refulting
Numbers 5, n, 15, without any fuperfluous Terms. But the .ap-
plication of all this will be beft apprehended from the Examples that
are to follow. 30

and INFINITE SERIES. 201

30. We have before given the form of this Equation, y< \$xy*

_j_ I!y4 ja*x 1 )* +- 6<? 3 .Y 5 4-^Ar* = o, when the Terms are dif-

pofed according to a double or combined Arithmetical Scale, in or-
der to its Solution. Or obferving the fame difpofition of the Terms,
they may be inferted in their refpedive Parallelograms, as the Table
requires. Or rather, it may be fufficient to tabulate the feveral In-
dices of A; only, when they are derived as follows. Let Ax" repre-
fent the firft Term of the Series to be form'd for y, as before, or let
y=;Ax'", &c. Then by fubftituting this for y in the given Equa-
tion, we fhall have A 6 .\- 6m \$A s x\$ m + l -+- -^xv+s 7*A a .v tIB + -f.

6fl 3 x J -f-^.x'4, &c. = o. Thefe Indices of AT, when felected from
the general Table, with their refpective Parallelograms, will ftand
thus:

4

3

4w-h3

2/W-J-2

5/W-f- I

6m

Here if we would have an afcending Series for the Root y y we
may apply the Ruler to the three external Terms 3, 2/;;-f- 2, 6/w,
which being made equal to each other, will give ; = - > an d each
of the Numbers will be 3. The Ruler in its parallel motion will
next arrive at \$m -+- i, or 37; then at 4; then at 4^-1-3, or 5;
which Numbers will be comprehended in the Arithmetical Progref-
fion 3, 37, 4, 47, 5, whofe common difference is f. This there-
fore will be the common difference of the Progreffion of the Indices,
in the Series to be derived for y. So that now we intirely know the
form of the Series, which will refuk from this Cafe. For if A, B,
C, D, &c. be put to reprefent the feveral Coefficients of the Series in
order, and as the firft Index m is found to be 7, and the common
difference of the afcending Series is allo 7, we ihall have here j =

A^ H- Bx -{- CV-H- DAT% &c.

As to the Value of the firft Coefficient A, this is found by putting
the initial or external Terms of the Parallelogram equal to nothing.

D d This

2 02 Tfo Method of FLUXIONS,

This here will give the Equation A 6 7rt*A* -j- 6<z 5 = o, which,
has thefe fix Roots, A = ,/tf, A ^/2a, A=v/ 3*7,.
of which the two laft are impoffible, and to be rejected. Of the
others any one may be taken for A, according as we would profecute
this or that Root of the Equation.

Now that this is a legitimate Method for rinding the firft Ap-
proximation Ax m , may appear from confidering, that when the
Terms of the Equation are thus ranged, according to a double Arith-
metical Scale, the initial or external Terms, (each Cafe in its turn,)
become the moil confiderable of the Series, and the reft continually
decreafe, or become of lefs and lefs value, according as they recede
more and more from thofe initial Terms. Confequently they may
be all rejected, as leaft confiderable, which will make thofe initial
or external Terms to be (nearly) equal to nothing ; which Suppofi-
tion gives the Value of A, or of Ax n , for the fir ft Approximation,
And this Suppofition is afterwards regularly purfued in the fubfe-
quent Operations, and proper Supplements are found, by means of
which the remaining Terms of the Root are extracted.

We may try here likewife, if we can obtain a defcending Series
for the Root y, by applying the Ruler to the two external Terms
^ m _j_ y and 6m ; which being made equal to each other, will give
m =T> an d hence each of the Numbers will be 9. The Ruler in
its motion will next arrive at \$m-\- i, or 8f. Then at zm -f- 2, or
5. Then at 4. And laftly at 3. But thefe Numbers 9, 8f, 5, 4,
3, will be comprehended in an Arithmetical Progreffion, of which
the common difference is i. So that the form of the Series here
will be y =A.v* -f- Ex -+- Cx^ -f- D^, &c. But if we put the two
external Terms equal to nothing, in order to obtain the firft Ap-

A4 I

proximation, we mail have A 6 + =o, or A 1 -f- - = o, which

will afford none but impoffible Roots. So that we can have no ini-
tial Approximation from this fuppofition, and confequently no
Series.

But laftly, to try the third and laft cafe of external Parallelograms,
we may apply the Ruler to 4 and 4^2-4-3, which being made equal,
will give m = -, and each of the Numbers will be 4. The next
Number will be 3 ; the next 2m -\- 2, or 2| ; the next 50* -{- i, or
27; the laft will be 6m, or if. But the Numbers 4, 3, af, 27,
if, will all be found in a decreafing Arithmetical Progreffion, whofe
common difference will be f . So that Ax* + Bx H- Cx~* -+- Dx~s
6cc. may reprefent the form of this Series, if the circumftances of

the

and INFINITE SERIES.

203

the Coefficients will allow of an Approximation from hence. But
if we make the initial Terms equal to nothing, we mall have

a,

-\- b* o, which will give none but impoflible Roots. So that
we can have no initial Approximation from hence, and confequemly
no Series for the Root in this form.

3 i. The Equation y s by 1 -+- qbx* # ; =o, when the Terms
are difpofed according to a double Arithmetical Scale, will have the
form as was (hewn before ; from whence it may be known, what
cafes of external Terms there are to be try'd, and what will be the
circumftances of the feveral Series for the Root y, which may be
derived from hence. Or otherwiie more explicitely thus. Putting
Ax 1 " for the firft Term of the Series y, this Equation will become
by Subftitution A'A.-?" M 1 * 1 " -f- gbx* x*, 6cc. = o. So that
if we take thefe Indices of x out of the general Table, they will
ftand as in the following Diagram.

Now in order to have an afcending
Series for y, we may apply the Ruler to
the two external Parallelograms 2 and
2W, which therefore being made equal, will
give m - i, and each of the Numbers
will be 2. The Ruler then in its parallel

progreis will firft come to 3, and then to yn, or 5. But the Num-
bers 2, 3, 5, are all contain'd in an afcending Arithmetical Progrefiion,
whofe common difference is i . Therefore the form of the Series
will here be ;' = AA; -f- B* 1 -f-(*', &c. And to determine the
firft Coefficient A, we fhall have the Equation bfcx 1 -f- qbx* - - o,
or A a = 9, that is A = + 3. So that either 4-3*, or 3^ may
be the initial Approximation, according as we intend to extract the
affirmative or the negative Root.

We mall have another cafe of external Terms, and perhaps an-
other afcending Series for_y, by applying the Ruler to the Parallelo-
grams 2; and 5;^, which Numbers being made equal, will g;ive
m =zo. (For by the way, when we put 2;= 5/77, we are not at
liberty to argue by Diviiion, that 2=5, becaufe this would bring
us to an absurdity. And the laws of Argumentation require, that no
Abfurdities muft be admitted, but when they are inevitable, and
when they are of ufe to mew the falfity of fome Supposition. We
fliould therefore here argue by Subtraction, thus: Becanfe cm ^t>i t
then 5//f 2:>i = o, or pn = o, and therefore m = o. This Cau-
tion I thought the more necellary, becaufe I have obferved f >mc,

D d 2 who

204 "The Method of FLUXIONS,

who would lay the blame of their own Abfurdities upon the Analy-
tical Art. But thefe Abfurdities are not to be imputed to the Art,
but rather to the unikilfulnef of the Artift, who thus abfurdly ap-
plies the Principles of his Art.) Having therefore 777. = o, we {hall
alfo have the Numbers 2/77. = 577*' = o. The Ruler in its parallel
motion will next arrive at 2 ; and then at 3. But the Numbers o,
2, 3, will be comprehended in the Arithmetical Progreffion o, i, 2, 3,
whofe common difference is i. Therefore y = A -+- Ex -+- CAT*, &c.
will be the form of this Series. Now from the exterior Terms A*
bA* = o, or A 3 = by or A = fi, we {hall have the firft Term
of the Series.

There is another cafe of external Terms to be try'd, which poffi-
bly may afford a defcending Series for y. For applying the Ruler to
the Parallelograms 3 and 5777, and making thefe equal, we (hall have
7/7=4, an d ea ch of thefe Numbers will be 3. Then the Ruler
will come to 2 ; and laftly 2777, or - But the Numbers 3, 2, if,
will be comprehended in a defcending Progreffion, whofe common
difference is f. Therefore the form of the Series will be y = Ax^
_f. BA" T -|- CA^ -f- D, &c. And the external Terms A r .v 3 A: 3 = o
will give A= i for the firft Coefficient. Now as the two former'
cafes will each give a converging Series for y in this Equation, when
.v is lefs than Unity ; fo this cafe will afford us a Series when x is
greater than Unity ; which will converge fo much the fafter, the
greater x is fuppofed to be.

32. We have already feen the form of this Equation y> -\-axy -f-
aay A? 3 2# 3 =o, when the Terms are difpofed according to a
double Arithmetical Scale. And if we take the fictitious quantity
Ax* to reprefent the firft Approximation to the Root ;', we {hall
have by fubftitution A'X= m -f- aAx m + l -+- a'-Ax" A' 3 2^ 3 , Sec,
= o. Thefe Terms, or at leaft thefe Indices of x, being felecled
out of the general Table, will appear thus.

Now to obtain an afcending Series for the
Root y, we may apply the Ruler to the three
external Terms o, 777, 3777, which being made
equal, will give m = o. Therefore thefe
Numbers are each o. In the next place the
Ruler will come to 777.4- i, or i ; and laftly
to 3. But the Numbers o, i, 3, are contain'd in the Arithmetical
Progreffion o, i, 2, 3, whofe common difference is i. Therefore
the form of the Root is y=. A -+- Ex -{-Cx 1 -+- Dx>, 6cc. Now
if the Equation A 3 + a 1 A <2a' =o, (which is derived from the

initial

and INFINITE SERIES.

205

initial Term?,) is divided by the factor A 1 -f- ah. ~t- 2a*, it will give
the Quotient A a = o, or A=.a for the initial Term of the
Root^y.

If we would alfo derive a defcending Series for this Equation, we
may apply the Ruler to the external Parallelograms 3, yn, which
being made equal to each other, will give m = i ; alio thefe Num-
bers will each be 3. Then the Ruler will approach to m-\- i, or 2 ;
then to //;, or i ; laftly to o. But the Numbers 3, 2, i, o, are a de-
creafing Arithmetical Progreflion, of which the common difference
is i. So that the form of the Series will here be y=Ax -+- B -+-
CA, ' -f- Dx~- , &c. And the Equation form'd by the external Terms
will be A 3 x 3 .v 3 = o, or A= i.

33. The form of the Equation x*)' s y+X} 1 c'x 3 - -f- c 7 = o,
as exprefs'd by a combined Arithmetical Scale, we have already feen,
which will eafily mew us all the varieties of external Terms, with
their other Circumftances. But for farther illuftration, putting A,v ra for
the firft Term of the Root y, we ("hall have by fubftitution A t x^ m + l
36 - A I ,v i " I + I c'x* -+- c\ &c. =o. Thefe Indices of x being
tabulated, will ftand thus.

Now to have an afcending
Series, we mufl apply the
Ruler to the two external
Terms o and yn -\- 2, which
m .*-, and the two Numbers anting will be each o. The next
Number that the Ruler arrives at is zm + r, or .J. ; and the la ft is 2.
But the Numbers o, i, 2, will be found in an afcending Arithmeti-
cal Progreffion, whofe common difference is -i-. Therefore y =. Ax~
_l_ B.v '> -f- C -f- D.x^, &c. will be the form of the Root. To deter-
mine the firft Coefficient A, we fhall have from the exterior Terms
A'-f-6- 7 = o, which will give A = y^c 7 = c'\ Therefore
the firft Term or Approximation to the Root will be y ==. J/-^ ,

&c.

We may try if we can obtain a defcending Series, by applying
the Ruler to the two external Parallelograms, whofe Numbers are 2
and 5;-f-2, which being made equal, will give ;;; = o, and thefe
Numbers will each be 2. The Ruler will next arrive at 2///-J- i, or
i ; and laftly at o. But the Numbers 2, i, o, form a de Icon cling
ProgreiTion, whofe common difference is i. So that die form of the
Series will here be y = A -f. B,v + Cv- J , &c, And putting the

initial

2

; +z

ZOT-J- I

O

206 The Method of FLUXIONS,

initial Terms equal to nothing, as they ftand in the Equation, we
ihall have A'* 1 c*x* = o, or A = <r, for the firft Approximation
to the Root. And this Series will be accommodated to the cafe of Con-
vergency, when x is greater than c -, as the other Series is accommo-
dated to the other cafe, when x is lefs than c.

34. If the propofed Equation be 8z, 6 f> -\- a^y* 27^ = o,
it may be thus refolved without any preparation. When reduced to

our form, it will ftand thus, 8z 6 / 3 -}-az 6 \* * * 1 ,,

y J f=o; and by

* 27^9 3 *

putting_y=A B ',&:c.it willbecome 8A*z*" ! + 6 +aA 1 z*' m +' s * * &c.7

* 27^3 *

The firft cafe of external Terms will give \$A*z* m + s 27.^' = o,
whence 3/^-1-6 = 0, or m=s 2. Thefe Indices or Numbers
therefore will be each o ; and the other 2/-f- 6 will be 2. But 0,2,
will be in an afcending Arithmetical Progreffion, of which the com-
mon difference is 2. So that the form of the Series will be y=. Az~~-
-|- B -h Cs. 1 -+- Dz*, &c. And bccaufe 8A' = 27^9, or 2A=3^3,
it will be A = J-0 3 . Therefore the firft Term or Approximation to

the Root will be 3 -^-

2

2. *

But another cafe of external Terms will give aA*-z~- mJ c 6
= o, whence 2w-f-6 = o, or /;; = 3. Thefe Indices or Num-
bers therefore will be each o j and the other yn -+- 6 will be 3.
But o, 3, will be found in a defcending Arithmetical P/ogrefiion,
whofe common difference is 3 . So that the form of the Series will
be y = Az~* -f- Ez~ 6 -f- Cs-' , ccc. And becaufe ^A 1 = 27^',
J tis A = + 3v/3 x^ 4 > f r ^ ^''^ Coefficient.

Laftly, there is another cafe of external Terms, which may pom"-
bly afford us a defcending Series, by making SA*z3 a + 6 -f- aA*z"- m ^~ 6
=: o ; whence m = o. And the Numbers will be each equal to 6 ;
the other Number, or Index of z, is o. But 6, o, will be in a
defcending Arithmetical Progreffion, of which the common difference
is 6. Therefore the form of the Series will be _y= A -f- Ez~ 6 -f-
Oc- 11 , &c. Alib becaufe 8A -+- a A 1 = o, it is A = {a for the
firft Coefficient.

I fhall produce one Example more, in order to fhew what variety
of Series may be derived from the Root in fome Equations; as alib
to fhew all the cafes, and all the varieties that can be derived, in the
prefent ftate of the Equation. Let us therefore affume this Equation,

1,vl 3^ a /. ClI I 6 fl\*

y* - _ + x z - _ + - _ _ _j_ _ _ _ .+. ^ = o, or

rather y 3 a~ 1 y i x l -\- x> a>y~- x 3 - -+- a\) 3 a \y~ z A. 1 -}- a 6 x~ s
~ i -+- a= = o. Which if we make }' = A.\ m , &c. and

difpofe

and INFINITE SERIES. 207

difpofe the Terms according to a combined Arithmetical Progref-
fion, will appear thus :

* *

***
.*x" m +** *

Now here it is plain by the difpofition of the Terms, that the
Ruler can be apply'd eight times, and no oftner, or that there are
eight cafes of external Terms to be try'd, each of which may give
a Series for the Root, if the Coefficients will allow it, of which four
will be afcending, and four defcending. And firft for the four cafes
of afcending Series, in which the Root will converge by the afcend-
ing Powers of x ; and afterwards for the other four cafes, when the
Series converges by the defcending Powers of x.

I. Apply the Ruler, or, (which is the fame thing,) afTume the
Equation a s A~=x~^ a"' A- 1 *- 1 " 1 -* = o, which will give 3/77

= 2in 2, or 7/7= 2; alfo A=^. The Number refulting

from thefe Indices is 6. But the Pailer in its parallel motion will
next come to the Index 3 . then to zm-{- 2, or 2 ; then
to o ; then to zm 2, or 2 ; then to 3 ; and laftly to 3/7; and 2/774- 2,
or 6. But the Numbers 6, 3, 2, o, 2, 3, 6, are in an af-
cending Arithmetical Progrellion, of which the common difference
is i ; and therefore the form of the Series will be y = Ax 1 - - Bx*

-f- C.v, &c. and its firft Term will be - .

a

II. Affume the Equation a 6 x~ l a''A t x - i ==z o, which will
give 3 = zm 2, or m = f } alfo A = a*. The Num-
ber refulting hence is 3 ; the next will be 37/7, or iJL ; the
next 2/72 2, or i ; the next o ; the next 2/>-f- 2, or j ;
the next 3/7;, or i ; the two laft zm 4- 2 and 3, are each 3. But
the Numbers 3, i, j, o, i, i|, 3, will be found in an
afcending Arithmetical Progreffion, of which the common difference
is f ; and therefore the form of the Series will be y = Ax^ +- Bx +-

+ Dx% &c. and its firft Term will be + ^/ax.

III.

208 7?je Method of FLUXIONS,

III. Aflame the Equation a 6 x~* a* A. 1 .* 11 ""- = o, which will
give 3 = 2?/7 2, or;;;= f; alfo A = + a*. The Num-
ber refulting is 3 ; the next 3;?;, or if ; the next 2m 2,
or i ; the next o ; the next 2m -+- 2, or i ; the next 3z, or
if; the two laft 3 and 2m -f- 2, which are each 3. But the
Numbers 3, if, i, o, i, if, 3, will be all comprehended
in an afcending Arithmetical Progreiiion, of which the common dif-
ference is f ; and therefore the form of the Series will be y - A.y~
~h B -f- Cx* -f- Dx, &c. and the firft Term will be a*x~' f , or

"v/;-

IV. Affume the Equation A 3 A: 3 ^'A 1 *-'*- 2 = o, which will
give 3 = 2; 2, or ;/z = 2; alfo A = a*. The Number
refulting is 6 ; the next will be 3 ; the next 2m -{-2, or 2 ;
the next o; the next 2m 2, or 2 ; the next 3 ; the two laft

3/tf and 2#?4-2, each of which is 6. But the Numbers 6,

3, 2, o, 2, 3, 6, belong to an afcending Arithmetical Progref-
fion, of which the common difference is i. Therefore the form of
the Series will be y = Ax~- +- Bx~' -+- C -f- Dx, &c. and its firft

Term will be ^

The four defending Series are thus derived.

I. Afllune the Equation Au 3 a-'A 1 x"- m + l o, which will

give 3;;z = 2/w -4- 2, or #2 = 2; alfo A = - . The Number re-
fulting is 6 ; the next will be 3 ; the next 2m 2, or 2 ; the next
O; the next 2;;z-f-2, or < 25 the next 3; the two laft

3/72 and 2m 2, each of which is 6. But the Numbers
6, 3, 2, o, 2, 3, 6, belong to a defcending Arithmetical Pro-
greflion, of which the common difference is i. Therefore the form
of the Series will be/ = Ax* -i- Ex ~f- C -f- D.*- 1 , &c. and the firft

X S,

Term will be .

a

II. Affume the Equation x* a~ l A i x im ~ Jri = o, which will give
2m -+- 2 = 3, or ;;:= f ; alfo A = + a*. The Number refulting
is 3 ; the next wi: be 3;^, or if; the next 2;/z-f-2, or i ; the
next o ; the next ,/ 2, or i ; the next yn, or if; the
two laft 3 and 2m 2 are each 3. But the Numbers 3,
if, i, o, i, - if, 3, belong to a defcending Arithmetical
ProgreiTon, of which the common difference is i. Therefore the
form of the Series will be_)' = Ax^-i-Ex+Cx~^-{- DAT*', &c. and
the firft Term will be + ^/ax.

III.

gve 3

and INFINITE SERIES. 209

III. Aflume the Equation x 5 <7A-** *+ = o, which will'
= 2 w H- 2, or TW = f ; alfo A = + a*'. The Num-
ber refulting from hence is 3 ; the next will be 3;;?, or if ; the
next 2m -+-2, or i ; the next o ; the next 2m 2, or i ;
the next 3777, or if ; the two laft 3 and 2m 2, each of
whichare 3. But the Numbers 3, if, i, o, i, if, 3, are
comprehended in a defcending Arithmetical Progreflion, of which
the common difference is f . Therefore the form of the Series will
bcy=Ax~*-t-Bx~'-i-Cx~~ l -l-Dx- % & c - an d the firft Term will
be + a*x~* or + a - .

IV. Laftly, aflume the Equation a 6 A-ix~i m rfA.- I
which will give 3;;; = 2m -f- 2, or m = 2 ; alfo A ==='#.
The Number refulting is 6 ; the next will be 3 ; the next zm 2,
or 2 ; the next o ; the next 2m -f- 2, or 2 ; the next 3 ; the
two next 3#; and 2m 2, are each 6. But the Numbers 6, 3,
2, o, 2, 3,' 6, belong to a defcending Arithmetical Progref-
iion, of which the -common difference is r. Therefore the form of
the Series will be/=A < x 1 H-BA 3 4-Cx 4-t-Dx- 5 , &c. and the firft

rn '

Term is .

And this may fuffice in all Equations of this kind, for finding
the farms of the feveral Series, and their firft Approximations. Now
we muft proceed to their farther Refolution, or to the Method of
finding all the reft of the Terms fucceffively, no

.SECT. V. The Refolution of Affe&ed Specious Equations,
firofecuted by various Methods of Analyfis.

35. TTT ITHERTO it has been fhewn, when an Equation is
~J_ propofed, in order to find its Root, how the Terms of the
Equation are to be difpoied in a two-fold regular fucceffion/fo as
thereby to find the initial Approximations, and the feveral forms of
the Scries in all their various circumftances. Now the Author pro-
ceeds in like manner to difcover the fubfequent Terms of the Series,
which may be done with much eafe and certainty, when the form
of the Series is known. For this end he finds Refidual or Supple-
mental Equations, in a regular fuccefTion alfo, the Roots of which
are a continued Series of Supplements to the Root required. In
every one of which Supplemental Equations the Approximation is

E e found,

2io The Method of FLUXIONS,

found, by rejecting the more remote or lefs confiderable Terms, and-
fo reducing it to a fimple Equation, which will give a near Value
of the Root. And thus the whole affair is reduced to a kind of
Comparifon of the Roots of Equations, as has been hinted already.
The Root of an Equation is nearly found, and its Supplement, which,
ihculd make it compleat, is the Root of an inferior Equation > the Sup-

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