Isaac Newton.

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

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of the Fowler,) till it finally coincides with it; the Lines FG = Y,




and IK = X, will continually decreafe, and by decreafing will ap-
proach nearer and nearer to the Ratio of the Velocities at G and K,
and will finally vanifh at the fame time, and in the proportion of
thofc Velocities, that is, in the Ratio of GN to KN. Confequently
in the Equation now form'd, if we fuppofe Y and X to decreafe
continually, and at laft to vanifh, that we may obtain their ultimate
Ratio ; we mail thereby obtain the Ratio of GN to KN. But when
Y and X vanifh, or when the point F coincides with G, and I with
H, then it will be EG = y, and HK = .x'; fo that we fhall have
y : x :: GM : KN. And hence we mall obtain a Fluxional Equa-
tion, which will always exhibit the relation of the Fluxions, or Ve-
locities, belonging to the given Algebraical or Fluential Equation.

Thus, for Example, if EF=j', and HI = x, and the indefinite
Lines y and A: are fuppofed to increafe at fuch a rate, as that their
relation may always be exprefs'd by this Equation x 1 ax* + axy
y* = o ; then making FG=Y, and IK = X, by fubftituting
y -f- Y for j, and x -+- X for x, and reducing the Equation that will
arife, (fee before, pag, 255.) we fhall have ^x"-X -f- 3#X Z -f- X 3 -
zaxX aX 1 -f- axY -\- aXj -+- rfXY 3y*Y 3jyY* Y ! = o,
which may be thus exprefs'd in an Analogy, Y : X :: 3** 2 ax
.+. ay +- 3tfX -h X 1 aX : ^ ax aX -+- 37 Y -+- Y*. This
Analogy, when Y and X are vanishing quantities, or their ultimate
Ratio, will become Y : X : : 3** ^ax -f- ay : 3^* ax. And
becaufe it is then Y : X :: GN : KN :: v : x, it will be y : x ::
3X 1 zax -+- ay : 3^* ax. Which gives the proportion of the
Fluxions. And the like in all other cafes. Q^. E. I.

We might alfo lay a foundation for thefe Speculations in the fol-
lowing manner. Let
ABCDEF, 6cc. be the
Periphery of a Polygon,
or any part of it, and
let the Sides AB, BC,
CD, DE, &c. be of any
magnitude whatever.
In the fame Plane, and
at any diftance, draw
the two parallel Lines
/6, and bf\ to which
continue the right Lines
AB4/3, BCcy,

DEes, &c. meeting the parallels as in the figure, Now if we fup-

N n 2 pofe

276 7%e Method of FLUXIONS,

pofe two moving points, or bodies, to be at $ and b, and to move
in the fame time to y and c, with any equable Velocities ; thofe
Velocities will be to each other as @y and be, that is, becaufe of the
parallels, as /3B and bE. Let them fet out again from y and c,
and arrive at the fame time at ^ and d, with any equable Velocities ;
thole Velocities will be as yfr and cd, that is, as yC and cC. Let
them depart again from and d, and arrive in the fame time at g
and e, with any equable Velocities ; thofe Velocities will be as S-t
and de, that is, as J^D and dD. And it will be the fame thing every
where, how many foever, and how fmall foever, the Sides of the
Polygon may be. Let their number be increafed, and their magni-
tude be diminim'd in infinitum, and then the Periphery of the Poly-
gon will continually approach towards a Curve-line, to which the
Lines AB^/3, ECcy, CDd, &c. will become Tangents -, as alfo the
Motions may be conceived to degenerate into fuch as are accelerated
or retarded continually. Then in any two points, fuppofe and d,
where the defcribing points are found at the fame time, their Velo-
cities (or Fluxions) will be as the Segments of the refpeclive Tan-
gents cTD and dD ; and the Lines /3^ and bd, intercepted by any
two Tangents J>D and /SB, will be the contemporaneous Lines, or
Fluents. Now from the nature of the Curve being given, or from
the property of its Tangents, the contemporaneous Lines may be
found, or the relation of the Fluents. And vice versa, from the
Rate of flowing being given, the correfponding Curve may be found.





O R,

The Relation of the Fluxions being given, to

o & 7

find the Relation of the Fluents.

SECT. I. A particular Solution ; with a preparation for
the general Solution, by 'which it is diftribitted into-
three Cafes.

E are now come to the Solution of the Author's fe-
cond fundamental Problem, borrow'd from the Science
of Rational Mechanicks : Which is, from the Velo-
cities of the Motion at all times given, to find the
quantities of the Spaces defcribed ; or to find the Fluents from the
given Fluxions. In difcuffing which important Problem, there will
be occafion to expatiate fome thing more at large. And firft it may
not be amifs to take notice, that in the Science of Computation all
the Operations are of two kinds, either Compolitive or Refolutative.
The Compolitive or Synthetic Operations proceed neceffarily and di-
rectly, in computing their feveral qit(?fita> and not tentatively or by
way of tryal. Such are Addition, Multiplication, Railing of Powers,
and taking of Fluxions. But the Refolutative or Analytical Opera-
tions, as Subtraction, Divifion, Extraction of Roots, and finding of
Fluents, are forced to proceed indirectly and tentatively, by long
deductions, to arrive at their feveral qutefita ; and fuppofe or require
the contrary Synthetic Operations, to prove and confirm every llep
of the Procefs. The Compofitive Operations, always when the
data are finite and terminated, and often when they are interminate

i or

The Method of FLUXIONS,

or infinite, will produce finite conclufions ; whereas very often in
the Refolutative Operations, tho' the data are in finite Terms, yet
the quafita cannot be obtain'd without an infinite Series of Terms.
Of this we mall fee frequent Inftances in the fubfequent Operation,
of returning to the Fluents from the Fluxions given.

The Author's particular Solution of this Problem extends to fuch
<afes only, wherein the Fluxional Equation propofed either has been,
or at leafl might have been, derived from fome finite Algebraical
Equation, which is now required. Here all the necefTary Terms
being prefent, and no more than what are neceflary, it will not be
difficult, by a Procefs juft contrary to the former, to return back
again to the original Equation, But it will moft commonly happen,
either if we aflume a Fluxional Equation at pleafure, or if we arrive
at one as the refult of fome Calculation, that fuch an Equation is
to be refolved, as could not be derived from any previous finite Al-
gebraical Equation, but will have Terms either redundant or defi-
cient ; and confequently the Algebraic Equation required, or its
Root, mufl be had by Approximation only, or by an infinite Series.
In all which cafes we mult have recourfe to the general Solution of
this Problem, which we fhall find afterwards.

The Precepts for this particular Solution are thefe. (i.) All fuch
Terms of the given Equation as are multiply 'd (fuppofe) by x, muft
be difpofed according to the Powers of x, or muft be made a Num-
ber belonging to the Arithmetical Scale whofe Root is x. (2.) Then
they muft be divided by A-, and multiply'd by x ; or x muft be
changed into A', by expunging the point. (3.) And laftly, the
Terms muft be feverally divided by the Progreilion of the Indices
of the Powers of x, or by fome other Arithmetical ProgrerTion, as
need mail require. And the fame things muft be repeated for every
one of the flowing quantities in the given Equation.

Thus in the Equation $xx- zaxx -f- axy . ^yy- -f- ajx -Q^
the Terms -^xx 1 zaxx -\-axy by expunging the points become
^x'' zax*- -+- axy, which divided by the Progreffion of the Indi-
ces 3, 2, I, reflectively, will give A' 5 ax* -+- axy. Alfo the Terms
3.X)' a * -+- ayx by expunging the points become 3j 3 * -f- ayx,
which divided by the Progreffion of the Indices 3, 2, i, refpectively,
will give y> * -+- ayx. The aggregate of thefe, neglecting the
redundant Term ayx, is x* ax* -\- axy _}" = o, the Equation
required. Where it muft be noted, that every Term, which occurs
more than once, mult be accounted a redundant Term.



So if the propoied Equation were m -f- ^yxx* m-\- 2(jyxx 1 -f-
;// -+- i ay* xx m} 4 -v n-\- $xyy> -\-n-\- lax^yy -+- nx+y nax>y
=. o, whatever values the general Numbers m and n may acquire ;
if thofe Terms in which x is found are reduced to the Scale whofe
Root is x, they will ftand thus : m -+- yyxx' m -+- zayx*. 1 -+
m-\-\ay*xx my**-, or expunging the points they will become
m -+- %yx+ m -f- Ziivx* +- m -+- \ay-x 1 m\*x. Thefe being di-
vided refpedtively by the Arithmetical Progreffion m -f- 3, m-\-2,.
m-\- i, m, will give the Terms yx+ ayx 1 -f- ay'-x 1 y+x. Alio
the Terms in which y is found ; being reduced to the Scale whofc

Root isy, will ftand thus : n -4- ^xyy* * +- n -+ iax*jy-{- nx*y;


or expunging the points they will become n -\~^ x ^ * -^~ n ~^~ iaxi )"
+- nx+y. Thefe being divided reipeclively by the Arithmetical Pro-

grefTion ^-{-3, ?i-{- 2, ?z-|-i, ;;, will give the Terms xy* -\-
ax*}' 1 -)- x+y ax*y. But thefe Terms, being the fame as the former,
mull all be confider'd as redundant, and therefore are to be rejected.
So that yx* ayx* -f- ay i x i y^x=o ) or dividing by yx, the
Equation x* ax 1 -\-ayx y* = o will arife as before.

Thus if we had this Fluxional Equation mayxx~ l m -+- 2xx
nx*yy~* -+- ;z-f- \ay =. o, to find the Fluential Equation to which
it belongs ; the Terms mayxx~ I * m -f- 2xx, by expunging the
points, and dividing by the Terms of the Progreffion m, m-\- 1, w-t-2,
will give the Terms ay x*. Alfo the Terms nx^yf 1 -+- n-\-iay,
by expunging the points, and dividing by n, n-\- i, will give the
Terms x 1 -f- ay. Now as thefe are the fame as the former, they
are to be efteem'd as redundant, and the Equation required will be
ay x 1 = o. And when the given Fluxional Equation is a gene-
ral one, and adapted to all the forms of the Fluential Equation, as
is the cafe of the two laft Examples ; then all the Terms ariling
from the fecond Operation will be always redundant, fo that it will
be fufficient to make only one Operation.

Thus if the given Equation were ^.yy 1 -f- z 3 yy~ J -f- 2yxx 3:32*
H- 6}'z.z 2cyz = o, in which there are found three flowing quan-
tities j the only Term in which x is found is 2yxx, in which ex-
punging the point, and then dividing by the Index 2, it will be-
come^* 1 . Then the Terms in which y is found are 4^*4- z*yy~~ l t
which expunging the points become ^ # * 4-s 3 , and dividing


280 72k Method of FLUXIONS,

by the Progreffion 2, i, o, i, give the Terms aj 5 s ; . Laftly
the Terms in which z is found are yzz* -J- 6yzz zcyz, which
expunging the points become 32;"' -f- 6yz* 29-2, and dividing
by the Progreffion 3, 2, i, give the Terms & + T.yz 1 zcyz.
Now if we collect thefe Terms, and omit the redundant Term z*,
we mall have yx z -+- 2y> z"' -f- yz 1 2cyz = o for the Equa-
tion required.

3, 4. But thefe deductions are not to be too much rely'd upon,
till they are verify 'd by a proof; and we have here a fure method
of proof, whether we have proceeded rightly or not, in returning
from the relation of the Fluxions to the relation of the Fluents. For
every refolutative Operation mould be proved by its contrary com-
pofitive Operation. So if the Fluxional Equation xx <xy xy-\-
ny'= o were given, to return to the Equation involving the Fluents ;
by the foregoing Rule we fliall firft have the Terms xx xy, which
by expunging the points will become x* .vy, and dividing by the
Progreffion 2, i, will give the Terms ^x 1 xy. Alfo the Terms, or
rather Term, xy -+- ay, by expunging the points will become xy.
-+- ay, which are only to be divided by Unity. So that leaving out the
redundant Term xy, we fhall have the Fluential Equation x l xy
-+- ay == o. Now if we take the Fluxions of this Equation, we
iliall find by the foregoing Problem xx xy xy -+- ay =o, which
being the fame as the Equation given, we are to conclude our work is
true. But if either of the Fluxional Equations xx xy -f- ay =o,
or xx xy -f- ay = o had been propofed, tho' by purfuing the
foregoing method we fhould arrive at the Equation x* xy-\-ay
= o, for the relation of the Fluents ; yet as this conclulion would
not fland the teft of this proof, we muft reject it as erroneous, and
have recourfe to the following general Method ; which will give the
value of y in either of thofe Equations by an infinite Series, and
therefore for ufe and practice will be the moil commodious So-

5. As Velocities can be compared only with Velocities, and all
other quantities with others of the fame Species only ; therefore in
every Term of an Equation, the Fluxions muft always afcend to the
lame number of Dimenfions, that the homogeneity may not be de-
ftroy'd. Whenever it happens otherwife, 'tis becaufe fome Fluxion;
taken for Unity, is there underftood, and therefore muft be fupply'd
when occafion requires. The Equation xz -+- xyx az'-x* = o, by
making z=i, may become -x -f- xyx ax*==o > and like wile
vice versa. And as this Equation virtually involves three variable



quantities, it will require another Equation, either Fluential or
Fluxionai, for a compleat determination, as has been already ob-
ferved. So as the Equation yx = xyy, by putting x = i becomes
yx=yy; in like manner this Equation requires and fuppofes the

6, 7, 8, 9, 10, II. Here we are taught fome ufeful Reductions, in
order to prepare the Equation for Solution. As when the Equation
contains only two flowing Quantities with their Fluxions, the ratio
of the Fluxions may always be reduced to fimple Algebraic Terms.
The Antecedent of the Ratio, or its Fluent, will be the quantity to
be extracted ; and the Confequent, for the greater fimplicity, may
be made Unity. Thus the Equation zx +- 2xx yx y = o is

reduced to this, y - = 2 -+ 2X y, or making x=i, 'tis y = 2
_^_ 2 x y. So the Equation ya yx xa -f- xx xy = o, ma-
king x= i, will become y = ( a ~^+ y = i -f- jdb = ) i +
_!_ 2L _j_ f!? _|_ ^ , &c. by Divilion. But we may apply the par-
ticular Solution to this Example, by which we mail have {x 1 xy
__ ## _{_ tfy = o, and thence y =."- ^~- . Thus the Equation
yjr = xy-{-xxxx, making x=i, becomes yy =y -+- xx, and ex-
tracting the fquare-root, 'tis y = -i \/ -j- xx = ~ the Series
2.-4-X 1 x*-{-2X 6 5X 8 -f- I4x', &c. that is, either y = i -j-
x * x 4 -{-2x 6 jx 8 -f- I4* 10 , &c. or y = x 1 -f- x 4 zx 6
_j_ rx 8 I4-.V 10 , &c. Again, the Equation y> -+-axx* i y-{-a 1 x 1 y
X 3x 3 2x'tf ? =o, putting x= i, becomes _y 3 -\-axy -\-ay x
2<7 3 - o. Now an affected Cubic Equation of this form has
been refolved before, (pag. 1 2.) by which we mail have y = a ^x +

xx iji* ? ^9^ 4 c,

6^ ~*~ uz" 1 " ' 16384^5 '

12. For the fake of perfpicuity, and to fix the Imagination, our
Author here introduces a diftinction of Fluents and Fluxions into
Relate and Correlate. The Correlate is that flowing Quantity which
he fuppofes to flow equably, which is given, or may be arTumed,
at any point of time, as the known meafure or ftandard, to which
the Relate Quantity may be always compared. It may therefore
very properly denote Time ; and its Velocity or Fluxion, being an
uniform and conftant quantity, may be made the Fluxionai Unit,
or the known meafure of the Fluxion (or of the rate of flowing) of
the Relate Quantity. The Relate Quantity, (or Quantities if ieve-

O o ral

2 8 2 The Method of FLUXIONS,

ral are concern'd,) is that which is fuppos'd to flow inequably, with;

any degrees of acceleration or retardation ; and ts inequability may

be meafured, or reduced as it were to equability, by conihntly com-

paring it with its correfponding Correlate or equable Quantity. This

therefore is the Quantity to be found by the Proble'm, or whofe

Root is to be extracted from the given Equation. And it may be

conceived as a Space defcribed by the inequable Velocity of a Body

or Point in motion, while the equable Quantity, or the Correlate,

reprefents or meaiures the time of defcription. This may be illu-

ftrated by our common Mathematical Tables, of Logarithms, Sines,

Tangents, Secants, &c. In the Table of Logarithms, for inflance,

the Numbers are the Correlate Quantity, as proceeding equably, or

by equal differences, while their Logarithms, as a Relate Quantity,

proceed inequably and by unequal differences. And this refemblance

would more nearly obtain, if w r e mould fuppofe infinite other Num-

bers and their Logarithms to be interpolated, (if that infinite Num-

ber be every where the fame,) fo as that in a manner they may be-

come continuous. So the Arches or Angles may be confider'd as

the Correlate Quantity, becaule they proceed by equal differences,

while the Sines, Tangents, Secants, &c. are as fo many Relate Quan-

tities, whofe rate of increafe is exhibited by the Tables.

13, 14, 15, 16, 17. This Diflribution of Equations into Orders,
or Gaffes, according to the number of the flowing Quantities and
their Fluxions, tho' it be not of abfolute neceflity for the Solution,
may yet ferve to make it more expedite and methodical, and may
fupply us with convenient places to reft at.

SECT. II . Solution of the Jirft Cafe of Equations.

18, 19, 20, 21, 22, 23. r ~|~^HE firft Cafe of Equations is, wherr

-i. the Quantity ? , or what fupplies

its place, can always te found in Terms compofed of the Powers
of x, and known Quantities or Numbers.. Thefc Terms are to be
multiply'd by x, and to be divided by the Index of .v in each Term,,
which will then exhibit the Value of jr. Thus in the Lquationj/ a = .xi/

-+- x l x*, it has been found that ~ = i -t-x* x* -f- 2x s $x* -f-

I4* 10 , &cc. Therefore - =^-4- x* x'-f- 2,v 7 5^' -t- 14*'*,

&c. and confequently y = x + jX* -fx 1 -f- ^x" 1 J-x 9 -j-l^A" 3 ,
&c. as may ealily be proved by the direct Method.



But this, and the like Equations, may be refolved more readily
by a Method form'd in imitation of fome of the foregoing Analyfes,
after this manner. In the given Equation make x = i ; then it
will bej)* =j/-l-.v*, which is thus refolved :


y 4 J

= X* -+- X* 2X S -f- pC 9 , &C.

y*- $ .V 4 -f- 2X & 5X 9 , &C.

Make AT* the firft Term of y ; then will x 4 be the firft Term
of j/ 1 , which is to be put with a contrary Sign for the fecond
Term of y. Then by fquaring, -f- 2X 6 will be the fecond Term
of j/, and 2x* will be the third Term of y. Therefore

5# 8 will be the third Term of j/, and -f- 5*" will be the
fourth Term of y ; and fo on. Therefore taking the Fluents, y =

I..V 5 -+- -fx* ix 7 -f-4-x, &c. which will be one Root of the
Equation. And if we fubtradt this from x, we (hall have y = x -+-
x 3 ^.v* -f- -i-A; 7 AX', &e. for the other Root.

So if - v = a 4-r -4- r h -^ > & c - that is, if ^ = ax

f 04^ 5 I 2* A'

c , # I?IA.'4 .

6? li ' &c " then v=^->^4- + ^, &c.

* A jj-T ^

-^yii: yx i ya &c then Y j. v s .



==f c *. If 4 = -, = , or ^ - **.

* ex? x c

then _y=^ f .

Laftly, if '- v = ~, or ' - = ^ = ^v ; dividing by the In-
dex o, it will be y = a - , or y is infinite. That this Expreffion,
or value of y, mufl be infinite, is very plain. For as o is a vanim-
ing quantity, or lefs than any affignable quantity, its Reciprocal -

or muft be bigger than any affignable quantity, that is, in-

O o 2

284 The Method of FLUXIONS,

Now that this quantity ought to be infinite, may be thus proved.
In the Equation 4 = - x , let AB reprefent the conftant quantity a,
and in CE let a point move equably from C towards E, and de-
fcribe the Line CDE, of Avhich let any indefinite part CD be x,
and its equable Velocity in D, (and every where elfe,) is reprefented

A o, E






T> -p.


1 - - -

J. &

by x. Alfo let a point move from a diftant point c along the Line
cde, with an inequable Velocity, and let the Line defcribed in the
fame time, or the indefinite part of it cd, be call'd y y and let the

Velocity in d be call'd y. The Equation 4- = - muft always ob-
tain, whatever the contemporaneous values of x and^ may be; or
in the whole Motion the conftant Line AB (a) muft be to the variable
Line CD (x), as the Velocity in d (y) is to the Velocity in D (x).
But at the beginning of the Motion, or when CD (x) was indefi-
nitely little, as the ratio of AB to CD was then greater than any

aflignable ratio, fo alfo was the ratio 4 of the Velocities, or the

Velocity y was infinitely greater than the Velocity x. But an infi-
nite Velocity muft defcribe an infinite Space in a finite time, or the
point c is at an infinite diftance from the point d, that is, y is an
infinite quantity.

24, 25. But to avoid fuch infinite ExpreiTions, from whence we
can conclude nothing ; we are at liberty to change the initial points
of the Fluents, by which their Rate of flowing, (the only thing to
be here regarded,) will not at all be affected. Thus in the foregoing
Figure, we fuppofed the points D and d to be fuch, as limited the
contemporaneous Fluents, or in which the two defcribing points
were found at the fame time. Let F and f be any other two fuch
points, and then the finite Line CF = b will be contemporaneous
to, or will correspond with, the infinite Line cf=c ; and FD,
which may be made the new .v, will correfpond to fd t which wiH

be the new y. So that in the given Equation - === - , inftead of


x we may write b +- x, and we fhall have ~ = - , and then by

r i i -r-v r vx f ax \ ax "X 1

Multiplication and Divifion it is -4- = ( :. = J -r -f.

x V*-}-* / b t l

~ -77 , &c. and therefore }'= ^- "- \ -f- ~ ~, 6cc.

2.6. So if ~ = - -J- 3 ' xx, becaufe of the Term -' , which
would give an infinite value for ^, we may write j -f- x inftead of
X, and we fhall then have - = ~ -4-2 zx xx, or y =

X 1 | X X

-^ 1- 2X 2X 1 x"', or by Divifion y - x - = 4x 4x* -f- x j

-f- zx s , &c. and therefore y=.^x 2x* -+- ^.x 3 |x 4 -f-

^x r , &c.

Or the Equation y ~ = -^^ +- z zx x 1 , that is y -f- xy
= 4 jx 1 x j , may be thus refolved :

y^ = 4 * 3* 11 A: J

^" 4X -J- 4X 1 AT 3 -j- 2X 4 , &C,

H- xyj h 4^ 4x* 4- x 3 2x 4 , 6cc.

y = 4 4.x -f- AT* 2x 3 -f- 2x 4 , &c.

T = 4.V 2X"' -{- ^X3 _ i X 4 _|_ x * t & c .

Make 4 the firft Term of j, then 4x will be the firfl Term of
xy, and confequently 4* will be the fecond Term of j. Then
4x a will be the fecond Term of .vy, and therefore -(- 4x* 3x fc ,
or x*, will be the third Term of_y ; and fo on.

27. So if -. = AT~^ -f- x~ J x'~, becaufe of the Term x~'
change x into i x, then == -.' -+- s/ i x. But

X yi , v I X

by the foregoing; Methods of Reduction 'tis = i -f- x -+- x*

* I X

-+- x 5 , 6cc. and v/i x = I 4-^ r-^ 4 -rr x ^ &c. a"d

Therefore collecting thefe according to their Signs, 'tis 4- i 4-

2.v-|- ix 1 -t- T^-x 3 , &c. that is-^ =x4-2x* -f- |x 3 + ^x 4 , &c.
and therefore y = x 4- x a -f- ix s 4- ^x 4 , &c.

28. So if the given Equation were == ~

X i. * ~^ ii^^C -!- 3t"A* ~ ~" X% "

. '^ - ; change the beginning of x. that is. inftead of x write

t A | '


286 7$ Method of FLUXIONS,

- y f3 c*-X yx

c x, then - = Al = c">x~* c l x- 1 , or ^ = c*x~* .
c*x- 1 , and therefore _>' = ^c=x-~ -\-c*x~ l .

SECT. III. Solution of the fecond Cafe of Equations.

2 9> 3- TT^Quations belonging to this fecond cafe are thofe,
M^ wherein the two Fluents and their Fluxions, fuppofe
x and y, x and j, or any Powers of them, are promifcuoufly in-
volved. As our Author's Analyfes are very intelligible, and fee'm to
want but little explication, I mall endeavour to refolve his Examples
in fomething an eafier and fimpler manner, than is done here ; by
applying to them his own artifice of the Parallelogram, when need-
ful, or the properties of a combined Arithmetical Progreffion in piano,
as explain'd before : As alfo the Methods before made ufe of, in the
Solution of afTeclcd Equations.

31. The Equation yax xxy aax = o by a due Reduction

.becomes ~ = ~ -+- "- , in which, becaufe of the Term - there
is occafion for a Tranfmutation, or to change the beginning of the
Correlate Quantity x. ArTurning therefore the conftant quantity b,

we may put 4- = ^ -f- -^ , whence by Divifion will be had

y v a ax ax z ax* e i i -.-, . ,

-j == -^ -I- y -+- -ji 77 > &c ' which Equation is then

prepared for the Author's Method of Solution.

But without this previous Reduction to an infinite Series, and the
Reiblution of an infinite Equation confequent thereon, we may
perform the Solution thus, in a general manner. The given Equa-

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