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 27 of 30)
Font size tion than dividing the whole by j*, that one of the Terms may be
clear'd from the Relate Quantity ; which will reduce it yy~^ ^
c= 3r, of which the Refolution may be thus :

3x -f- f X - -f- T ' T * 3 -|- ^rx* + -~-x s , &c.

X * -V^ 3 TTTT^ 4 TT4<T*'> &C '

y = f x 6 -+ T '^ 7 -4- TTY*"> &c -

Make jJ;y"~^ = 3#, .&c. or taking the Fluents, %y~' = |jc% 6co.

y^ = f x 1 , &c. or y = fAr 6 , &c. And becaufe y\$ = fx*,
&c. it will be jj)T~7 ==*_{- f^ 1 , &c. and therefore ^ = ^ 4-
^.x 5 , &c. and y^ = * -f- ^x l , &c. and by cubing y= * 4- T V x7 >
&c. Then becaufe y"' = * rV^'S & c - 'tis ji/y""^ = * * -f- -V v? >
&c.and therefore 37' = * # -+- T ' T * 4 , &c. and ;-j = * * + T Tr- v ' 4 >
&c. and by cubing ;= * * 4- T TT- v8 ) & c - And * on -

Qjl 2 53-

or

-oo 7%e Method of FLUXIONS,

53. Laftly, in the Equation y = zy^ -+- x ty*, orjj/y i== zx -4-
xx*, afTuming c for a conftant quantity, whofe Fluxion therefore
is o, and taking the Fluents, it will be 2V*= 2c -f- 2x -f- ~x'^, or
y*=c -+- x -f- -i-x'. Then by fquaring, _> = c 1 -+- 2cx -f- A-* -f.

-icx* + !** -+ f^ 3 - Here the Root _y may receive as many diffe-
rent values, while x remains the fame, as c can be interpreted diffe-
rent ways. Make c = o, then y = x 1 -+- -ix* -+- loc*.

The Author is pleas'd here to make an Excufe for his being fo
minute and particular, in dilcuffing matters which, as he fays, will
but feldom come into practice ; but I think any Apology of this
kind is needlefs, and we cannot be too minute, when the perfec-
tion of a Method is concern'd. We are rather much obliged to him
for giving us his whole Method, for applying it to all the cafes that
may happen, and for obviating every difficulty that may arife. The
ufe of thefe Extractions is certainly very exteniive ; for there are no
Problems in the inverfe Method of Fluxions, and efpecially fuch
as are to be anfwer'd by infinite Series, but what may be reduced to
fuch Fluxional Equations, and may therefore receive their Solutions
from hence. But this will appear more fully hereafter.

SECT. IV. Solution of the third Cafe of Equations, with
fame neceffary Demonftrations.

54. TT* O R the more methodical Solution of what our Author
calls a moft troublejbme and difficult Problem, (and furely
the Inverfe Method of Fluxions, in its full extent, deferves to be
call'd fuch a Problem,) he has before diftributed it into three Cafes.
The firft Cafe, in which two Fluxions and only one flowing Quan-
tity occur in the given Equation, he has difpatch'd without much
difficulty, by the affiftance of his Method of infinite Series. The
fecond Cafe, in which two flowing Quantities and their Fluxions
are any how involved in the given Equation, even with the fame
affiftance is flill an operofe Problem, but yet is difculs'd in all its
varieties, by a fufficient number of appofite Examples. The third
Cafe, in which occur more than two Fluxions with their Fluents,
is here very artfully managed, and all the difficulties of it are re-
duced to the other two Cafes. For if the Equation involves (for
inftance) three Fluxions, with fome or all of their Fluents, another
Equation ought to be given by the Queftion, in order to a full De-

terminationj

and INFINITE SERIES. 301

termination, as has been already argued in another place; or if not,
the Queftion is left indetermined, and then another Equation may
be affumed ad libitum, fuch as will afford a proper Solution to the
Queftion. And the reft of the work will only require the two
former Cafes, with fome common Algebraic Reductions, as we fhall
fee in the Author's Example.

55. Now to confider the Author's Example, belonging to this
third Cafe of finding Fluents from their Fluxions given, or when
there are more than two variable Quantities, and their Fluxions, ei-
ther exprefs'd or underftood in the given Equation. This Example
is zx z 4- yx = o, in which becaufe there are three Fluxions A-,
y, and z, (and therefore virtually three Fluents x, y, and z,) and
but one Equation given ; I may affume (for inftance) x=y, whence
x =JK, and by fubftitution zy z -\-yy = o, and therefore zy
& + T)'* = Now as here are only two Equations x y== o
and zy z-\-^y l =o, the Quantities x, y, and z are ftill variable
Quantities, and fufceptible of infinite values, as they ought to be.
Indeed a third Equation may be had, as zx z-\-x* = o; but
as this is only derived from the other two, it brings no new limi-
tation with it, but leaves the quantities ftill flowing and indetermi-
nate quantities. Thus if I mould affume zy=a-\-z for the fc-
cond Equation, then zy=z, and by fubftitution zx zjr-k-yx=;o,

or y = j^ - = x -f- .Ixv -f- ^x'-x, &c. and therefore y = x -+- ix 1

H-TT# S J & c - which two Equations are a compleat Determination.
Again, if we affume with the Author x=j s , and thence x=Z)y t
we mall have by fubftitution <\.yy z -^-yy 1 = o, and thence zy 1
z -+- ^ = o, which two Equations are a fufficient Determina-
tion. We may indeed have a third, zx z -+- ^x^ = o ; but as
this is included in the other two, and introduces no new limitation,
the quantities will ftill remain fluent. And thus an infinite variety
of fecond Equations may be aflumed, tho 1 it is always convenient,
that the affumed Equation fliould be as fimple as may be. Yet fome
caution muft be ufed in the choice, that it may not introduce fuch
a limitation, as fhall be inconfiftent with the Solution. Thus if I
fhould affume zx z= o for the fecond Equation, I mould have
zx z = o to be fubftituted, which would make yx = o, and
therefore would afford no Solution of the Equation.

'Tis eafy to extend this reafoning to Equations, that involve four
or more Fluxions, and their flowing Quantities , but it would be
needlefs here to multiply Examples. And thus our Author has com-
pleatly folved this Cafe alfo, which at firft view might appear for-
midable

302 7%e Method of FLUXIONS,

midable enough, by reducing all its difficulties to the two former
Cafes.

56, 57. The Author's way of demonstrating the Inverfe Method
of Fluxions is Short, but fatisfactory enough. We have argued elfe-
where, that from the Fluents given to find the Fluxions, is a direct
and fynthetical Operation ; and on the contrary, from the Fluxions
given to find the Fluents, is indirect and analytical. And in the
order of nature Synthefis mould always precede Analyfis, or Com-
pofidon mould go before Refolution. But the Terms Synthefis and
Analyfis are often ufed in a vague fenfe, and taken only relatively,
as in this place. For the direct Method of Fluxions being already
demonftrated fynthetically, the Author declines (for the reafons he
gives) to demonstrate the Inverfe Method fynthetically alfo, that is,
primarily, and independently of the direct Method. He contents
himfelf to prove it analytically, that is, by fuppofing the direct Me-
thod, as fufficiently demonstrated already, and Shewing the neceSTary
connexion between this and the inverSe Method. And this will al-
ways be a full proof of the truth of the conclufions, as Multiplica-
tion is a good proof of Division. Thus in the firlt Example we
found, that if the given Equation is y -f- xy y=^x x 1 I,
we Shall have the Root y=x x 1 -j- f x 3 -* -f- ^.x* -^r* 6 ,
cc. To prove the truth of which conclufion, we may hence find,
by the direct Method, _y = i 2x -{-x 1 .i* 3 -f-f#* T T X', &c.
and then fubStitute theSe two Series in the given Equation, as follows;

y - - - - f_ X _ jf 4- ]_ X 1 _ . X < _J_ _>_ X , __ _^ X 6

_{_ X y - - - - - 1_ X - __ . #3 _j_ 3.4 __ ^f _{_ _?_ X 6 5

y - r -f. 2 X A-* -{- ^ ^ + -rX' ^X 6

x 1

Now by collecting thefe Series, we mall find the refult to pro-
duce the given Equation, and therefore the preceding Operation will
be fufticiently proved.

58. In this and the fubfequent paragraphs, our Author comes to
open and explain fome of the chief My Steries of Fluxions and Fluents,
and to give us a Key for the clearer apprehenfion of their nature
and properties. Therefore for the Learners better instruction, I Shall
not think much to inquire fomething more circumstantially into this
matter. In order to which let us conceive any number of right
Lines, AE, ae t as, &c. indefinitely extended both ways, along which
a Body, or a defcribing Point, may be fuppofed to move in each

Line,

and INFINITE SERIES. 303

Line, from the left-hand towards the right, according to any Law
or Rate of Acceleration or Retardation whatever. Now the Motion
of every one of thefe Points, at all times, is to be eftimated by its
diftance from fome fixt point in the fame Line ; and any fuch Points
may be chofen for this purpole, in each Line, fuppoie B, I), /3, in
which all the Bodies have been, are, or will be, in the fame Mo-
ment of Time, from whence to compute their contemporaneous
Augments, Differences, or flowing Quantities. Thefe Fluents may
be conceived as negative before the Body arrives at that point, as
nothing when in it, and as affirmative when they are got beyond it.
In the rlrft Line AE, whole Fluent we denominate by x, we may
luppofe the Body to move uniformly, or with any equable Velocity ;
then may the Fluent x, or the Line which is continually defcribed,

A B C J> :E

a,

, * /? 9" c/^ 8

2 ! 1-1 II

reprefent Time, or {land for the Correlate Quantity, to which the
feveral Relate Quantities are to be constantly refer'd and compared.
For in the fecond Line ae, whofe Fluent we call y, if we fuppofe
the Body to move with a Motion continually accelerated or retarded,
according to any conftant Rate or Law, (which Law is exprefs'd by
any Equation compos'd of x and y and known quantities j) then
will there always be contemporaneous parts or augments, defcribed
in the two Lines, which parts will make the whole Fluents to be
contemporaneous alfo, and accommodate themfelves to the Equation
in all its Circumftances. So that whatever value is afiumed for the
Correlate x, the correfponding or contemporaneous value of the Re-
late y may be known from the Equation, and vice versa. Or from
the Time being given, here represented by x, the Space represented
by y may always be known. The Origin (as we may call it) of the
Fluent x is mark'd by the point B, and the Origin of the Fluent y
by the point b. If the Bodies at the fame time are found in A and
, then will the contemporaneous Fluents be BA and ba. If
at the fame time, as was fuppofed, they are found in their refpec-
tive Origins B and , then will each Fluent be nothing. If at the
fame time they are found in ^ and c, then will their Fluents be
-1- BC and -\-bc. And the like of all other points, in which the
i moving

304 The Method of FLUXIONS,

moving Bodies either have been, or fliall be found, at the fame
time.

As to the Origins of thefe Fluents, or the points from whence we
begin to compute them, (for tho' they muft be conceived to be variable
and indetermined in refpedt of one of their Limits, where the de-
fcribing points are at prefent, yet they are fixt and determined as to
their other Limit, which is their Origin,) tho' before w appointed
the Origin of each Fluent to be in B and b, yet it is not of abfolute
neceffity that they mould begin together, or at the fame Moment of
Time. All that is neceflary is this, that the Motions may continue
as before, or that they may obferve the fame rate of flowing, and
have the fame contemporaneous Increments or Decrements, which
will not be at all affected by changing the beginnings of the Fluents.
The Origins of the Fluents are intirely arbitrary things, and we
may remove them to what other points we pleafe. If we remove
them from B and b to A and c, for inftance, the contemporaneous
Lines will ftill be AB and ab, BC and be, &c. tho' they will change
their names. Inftead of AB we fhall have o, inftead of B or o
we fliall have -+- AB, inftead of -+- BC we fliall have -f- AC ; &c.
So inftead of ab we fliall have ac -{-be, inftead of b or o we
fliall have be, inftead of-f- /Wwe fliall have -+- bc + cd, &c. That
is, in the Equation which determines the general Law of flowing
or increafing, we may always increafe or diminifh x, or y y or both,
by any given quantity, as occafion may require, and yet the Equa-
tion that arifes will ftill exprefs the rate of flowing ; which is all that
is neceffary here. Of the ufe and conveniency of which Reduction
we have feen feveral in fiances before.

If there be a third Line a.e, defcribed in like manner, whofe
Fluent may be z, having its parts correfponding with the others, as
a/3, &y, y, & c - there muft be another Equation, either given or
aflumed, to afcertain the rate of flowing, or the relation of z to the
Correlate x. Or it will be the fame thing, if in the two Equations
the Fluents x, y, Z, are any how promifcuoufly involved. For thefe
two Equations will limit and determine the Law of flowing in each
Line. And we may likewife remove the Origin of the Fluent z
to what point we pleafe of the Line a. And fo if there were more
Lines, or more Fluents.

59. To exemplify what has been faid by an eafy inftance. Thus
inftead of the Equation y=xxy, we may aflume y = xy -+- xxy,
where the Origin of x is changed, or x is diminifli'd by Unity ; for
j -J - x is fubftituted inftead of x, The lawfulnefs of which Re-
duction

and INFINITE SERIES.

305

duftion may be thus proved from the Principles of Analyticks. Make
x i -\-z, whence x=z, which (hews, that xand2 flow or increale
alike. Subftitute thefe infteadof x and x in the Equation^':=xxy, and
it will become y = zy -+- zzy. This differs in nothing elle from
the afTumed Equation y = xy -f- xxy, only that the Symbol x is
changed into the Symbol z, which can make no real change in the
argumentation. So that we may as well retain the dime Symbols
as were given at firft, and, becaufe z-=x- - i, we may as well
fuppofe x to be diminiih'd by Unity.

60, 6 1. The Equation expreffing the Relation of the Fluents will
at all times give any of their contemporaneous parts ; for afluming
different values of the Correlate Quantity, we ma'!, thence have the
correfponding different values of the Relate, and then by fubtradion
we fhall obtain the contemporary differences of each. Thus if the

given Equation were y = x -{- - , where x is fuppos'd to be a quan-
tity equably increafmg or decreafing ; make x = o, i, 2, 3, 4, 5,
&c. fucceifively, then y = infinite, 2, 2|, 3.1, 4^, 5-^-, &c. refpec-
tively. And taking their differences, while x flows from o to i,
from i to 2, from 2 to 3, &c. y will flow from infinite to 2, from
2 to 2-i-, from 2| to 3.1, &cc. that is, their contemporaneous parts
will be i, i, i, i, &c. and infinite, i, , -{.I, &c. refpeclively.
Likewife, if we go backwards, or if we make x negative, we mall
have x = o, i, 2, &c. which will make _y= infinite, 2,
2-i-, &c. fo that the contemporaneous differences will be as be-
fore.

Perhaps it may make a ftronger impreffion upon the Imagina-
tion, to reprefent this by a Figure. To the rectangular Afymptotes
GOH and KOL let ABC and DEF
be oppofite Hyperbola's ; bifed the An-
gle GOK by the indefinite right Line
yOR, perpendicular to which draw the
Diameter BOE, meeting the Hyperbola's
in B and E, from whence draw BQP
and EST, as alfo CLR and DKU pa-
rallel to GOH. Now if OL is made
to reprefent the indefinite and equable

quantity x in the Equation y = x -f- -'

then CR may reprefent y. For CL = ^ = l - , (fuppofing

= OL = x , therefore CR =^ LR -4-

R r or

306 *The Method of FLUXIONS,

or y = x -f- ^ Now the Origin of OL, or x, being in O j if

x = o, then CR, or y, will coincide with the Afymptote OG, and
therefore will be infinite. If x= i = OQ^ then _y = BP=2.
If x = 2 = OL, then y = CR = 2 i. And fo of the reft. Alfo
proceeding the contrary way, if x = o, then y may be fuppofed
to coincide with the Afymptote OH, and therefore will be negative
and infinite. If x = OS = i, then y = ET = 2. If x
= OK = 2, then _y = Dv = 2~, &c. And thus we may
purfue, at leaft by Imagination, the correfpondent values of the flow-
ing quantities x and_y, as alfo their contemporary differences, through
all their poiTible varieties ; according to their relation to each other,

as exhibited by the Equation y = x +- - . .

The Transition from hence to Fluxions is fo very eafy, that it
may be worth while to proceed a little farther. As the Equation
expreffing the relation of the Fluents will give (as now obferved)
any of their contemporary parts or differences ; fo if thefe differences
are taken very fmall, they will be nearly as the Velocities of the
moving Bodies, or points, by which they are defcribed. For Mo-
tions continually accelerated or retarded, when perform'd in very
fmall fpaces, become nearly equable Motions. But if thofe diffe-
rences are conceived to be dirninifhed in infiriitum, fo as from finite
differences to become Moments, or vanifhing Quantities, the Mo-
tions in them will be perfectly equable, and therefore the Velocities
of their Defcription, or the Fluxions of the Fluents, will be accu-
rately as thofe Moments. Suppofe then x, y, z, &c. to reprefent
Fluents in any Equation, or Equations, and their Fluxions, or Ve-
locities of increafe or decreafe, to be reprefented by x, y, z, &c.
and their refpedlive contemporary Moments to be op, oq, or, &c.
where p, q, r, &c. will be the Exponents of the Proportions of
the Moments, and o denotes a vanifhing quantity, as the nature of
Moments requires. Then x, y, z, Sec. will be as op, oq, or, &c.
that is, as p, g, r, &c. So that ,v, y, z, &c. may be ufed inflead
of/>, ?> r -> ^ c - ni th e designation of the Moments. That is, the fyn-
chronous Moments of x, y, z, &c. may be reprefented by ox, oy,
oz, &c. Therefore in any Equation the Fluent x may be fuppofed
to be increafed by its Moment ox, and the Fluent y by its Moment
oy, &cc. or x -+- ox, y -{- oy, &c. may be fubftitnted in the Equation
inflead of x, y, &c. and yet the Equation will flill be true, becaufe
the Moments are fuppofed to be fynchronous. From which Ope-
ration

and INFINITE SERIES, 307

ration an Equation will be form'd, which, by due Redu&ion, muft
neceflarily exhibit the relation of the Fluxions.

Thus, for example, if the Equation y = x -+- z be given, by
Subftitution we fliall have y -f- oy =. x -f- ox -+- z -+- oz, which, be-
caufe y = x -+- z, will become oy = ox -f- oz, or y = x -f- z, winch

is the relation of the Fluxions. Here again, if we afllime z = - >
or zx = i, by increafing the Fluents by their contemporary Mo-
ments, we fliall have z -+- oz x A- -f- ox = i, or zx + ozx -f- oxz
-f- oozx = i. Here becaufe zx = i, 'tis ozx -f- o.\z -+- oozx = o,
or ~x-f- A- -+- 02X = o. But becaufe ozx is a vanifliing Term in

refpect of the others, 'tis zx -f- A z = o, or z === -f =

Now as the Fluxion of z conies out negative, 'tis an indication that
as A- increafes z will decreafe, and the contrary. Therefore in the

Equation y = x -+- z, if z = - , or if the relation of the Fluents
be y = x -+- - , then the relation of the Fluxions will be y = x

And as before, from the Equation y =: x -+- - we derived the
contemporaneous parts, or differences of the Fluents ; fo from the
Fluxional Equation y = x ^ now found, we may obferve the

rate of flowing, or the proportion of the Fluxions at different values
of the Fluents.

For becaufe it is x : y : : I : i \ : : x 1 : x 1 i ; when

# = o, or when the Fluent is but beginning to flow, (confequently
when y is infinite,) it will be x : y :: o : i. That is, the Ve-
locity wherewith x is defcribed is infinitely little in comparifon of the
velocity wherewith^ is defcribed; and moreover it is infinuated, (becaufe
of i,) that while x increafes by any finite quantity, tho' never fo
little, y will decreafe by an infinite quantity at the fame time. This
will appear from the infpeclion of the foregoing Figure. When
x= i, (and confequently _)'= 2,) then x : y : : i : o. That is,
x will then flow infinitely faflrer than y. The reafon of which is,
that y is then at its Limit, or the leaft that it can poflibly be, and
therefore in that place it is ftationary for a moment, or its Fluxion
is nothing in comparifon of that of x. So in the foregoing Figure,
BP is the lea ft of all fuch Lines as are reprcfented by CR. When
x=2, (and therefore y = 27,) it will be A- : y :: 4 : 3. Or

R r 2 the

Method of FLUXIONS,

the Velocity of x is there greater than that of^, in the ratio of 4
to 3. When x=3, then x : y :: 9 : 8. And fo on. So that
the Velocities or Fluxions conftantly tend towards equality, which

they do not attain till - (or CL) finally vanishing, x and y become

equal. And the like may be obferved of the negative values of
x and y.

SECT. V. 77je Refolution of Equations^ whether Algebrai-
cal or Fluxionalt by the ajfiftance of fuperior orders
of Fluxions.

ALL the foregoing Extractions (according to a hint of our Au-
thor's,) may be perform'd fomething more expeditioufly, and
without the help of fubfidiary Operations, if we have recourfe to
fuperior orders of Fluxions. To (hew this firft by an eafy Inftance.

Let it be required to extradl the Cube-root of the Binomial
a> -f- x 3 , or to find the Root y of this Equation y 1 = a 3 -\- x"' ;
or rather, for fimplicity-fake, let it be_) 3 =a* -+- z. Then y=.a,
&c. or the initial Term of y will be a. Taking the Fluxions of
this Equation, we fhall have -\$yy- = z = i, or y = -^y~ l . But
as it is y = a, &c. by fubftitution it will be y = jtf" 1 , 6cc. and
taking the Fluents, 'tis y= * -f- \a~ 1 z^ &c. Here a vacancy is
left for the firft Term of y, which we already know to be a. For
another Operation take the Fluxions of the Equation jj/=: j/~* ;
whence y = %yy~* = jy~ 5 - Then becaufe y = a, &c.
'tis y = ?&"*, &c. and taking the Fluents, 'tis y = *
\$a~ 3 z, &c. and taking the Fluents again, 'tis jy- = * * ^a-^z l ,
6cc. Here two vacancies are to be left for the two firfl Terms of
y, which are already known. For the next Operation take the

Fluxions of the Equation y=. ^y~ s , that is, y = -f- - 9 -yy~ 6 =.

_l_^.ji- 8 . Or becaufe _7=c,&c. 'tis _y=if(S- 8 , &c. Then taking
the Fluents, 'tis y= * L^. a -*z, &c. y = * * -^a-^z 1 , 6cc. and
y = * * * -v s T a~*z*, &c. Again, for another Operation take the

Fluxions of the Equation y =: ~~y^ ; whence y = -*f^~ 9
= Iry" 11 - Or becaufe y = a, &c. 'tis "y = IT^"", &c.
Then taking the Fluents, y = * ^a- 1 ^, &c. jy = * *

and INFINITE SERIES. 309

H-a-^z 1 , &c. y = * * * _+._ _.rt "z 5 , &c. and _)' = * * * *
'^-?-a-"z+, &c. And fo we may go on as far as we pleafe. We

have therefore found at laft, that v = a -\ -. , -f- ~g

y. 1 9 8i B

^ , &c. or for z writing x*. 'tis fa + x- = -+- ~ ^ (

C* 1* -

I - - n " ~ l\f\^

' XI ,3 ^1 I '

bi

Or univerfally, if we would refolve a -f- x \ m into an equivalent
infinite Series, make y = a -f- x \ ra , and we fhall have a m for the

firfl Term of the Series y, or it will be y = a m , &c. Then be-
j_ i j

caufe y = a +- x, taking the Fluxions we fhall have ~yy m =

x = i, or y = my m . But becaufe it is y = a m , &c. it will be
y = t/m m ~ 1 , &c. and now taking the Fluents, 'tis y = * ma m ~ t x y

6cc. Again, becaufe it is y = my m , taking the Fluxions it will

be y = m lyv *= m x ly m ; and becaufe y = a m , Sec.
'tis/ = wxw ia m ~' i , &c. And taking the Fluents, 'tis y = *
? x / i~% 6cc. and therefore y = * * m x ^ = - I d m ~ 2 x% &c.

_ j 2

Again, becaufe it is y = m x //; 17 "', taking the Fluxions it
will be y = m - - i x ;// 2j/y m = m x / i x

and becaufe 7 := a m , &c. 'tis _y = m x
And taking the Fluents, 'tis /== * m x w i x w/
y =**/;/ x '^^- x m 2a m -ix*, &c. and v = *
x CT ~y-3 X 3, 6cc. And fo we might proceed as far as we pleafe,
nifefl. So that we fhall have here a -+- x I = a* -f- ma m ~ l x -+-

ftt __ r m ^ 2 ni * * m ~~" ^

m x !LU fl x l 4- w x - - x - a m ~^ -+- m x - x -

2 2 3

?^-V-4;c<, &c.

Vhis is a famous Theorem of our Author's, tho' difcover'd by

him after a very different manner of Investigation, or rather by .,- lldufr

Indudion. It is commonly known by the name of his Binomial > '\$ vntrrrU*t> *> v
Theorem, becaufe by its amftance any Binomial, as a + x; may JujJf&rcL f
be railed to any Power at pleafure, or any Root of it may be ex- -^fajfifa &

Online LibraryIsaac NewtonThe method of fluxions and infinite series : with its application to the geometry of curve-lines → online text (page 27 of 30)