Isaac Newton.

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

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fake, will happen in the Equation y* -}-axy-+-a l y x> 20 3 = o,

D by


1 8 The Method of FLUXIONS,

by fubftituting -f-rf, or a, or za, or 2 } | T , &c. inftead
of .v. And thus you may conveniently fuppofe the Quantity x to
differ little from -j-tf, or a, or 2a, or za*l^, and thence
you may extract the Root of the Equation propofed after fo
many ways. And perhaps alfo after fo many other ways, by fup-
poling thofe differences to be indefinitely great. Befides, if you take
for the indefinite Quantity this or that of the Species which exprefs
the Root, you may perhaps obtain your defire after other ways.
And farther ftill., by fubftituting any fictitious Values for the inde-
finite Species, fuch as az + bz 1 , -> ~n^> &c. and then proceeding
as before in the Equations that will refult.

52. But now that the truth of thefe Conclufions may be mani-
feft ; that is, that the Quotes thus extracted, and produced ad libi-*
turn, approach fb near to the Root of the Equation, as at laft to
differ from it by lefs than any afilgnable Quantity, and therefore
when infinitely continued, do not at all differ from it : You are to
confider, that the Quantities in the left-hand Column of the right-
hand fide of the Diagrams, are the laft Terms of the Equations
whofe Roots are p, y, r, s, &c. and that as they vanifh, the Roots
p, q, r, s, &c. that is, the differences between the Quote and the
Root fought, vanifh at the fame time. So that the Quote will not
then differ from the true Root. Wherefore at the beginning of the
Work, if you fee that the Terms in the faid Column will all de-
ftroy one' another, you may conclude^ that the Quote fo far ex-
tracted is the perfect Root of the Equation. But if it be other-
wife, you will fee however, that the Terms in which the indefi-
nitely fhiall Species is of few Dimenfions, that is, the greate ft Terms,
are continually taken out of that Column, and that at laft none
will remain there, unlefs fuch as are lefs than any given Quantity,
and therefore not greater than nothing when the Work is continued
ad infinitum. So that the Quote, when infinitely extracted, will at
laft be the true Root.

53. Laftly, altho' the Species, which for the fake of perfpieuity I
have hitherto fuppos'd to be indefinitely little, fhould however be
fuppos'd to be as great as you pleafe, yet the Quotes will ftill be
true, though they may not converge fo faft to the true Root. This
is manifeft from the Anal'ogy of the thing. But here the Limits
of the Roots, or the greateft and leaft Quantities, come to be
confider'd. For thefe Properties are in common both to finite and
infinite Equations. The Root in thefe is then greateft or leaft,.



when there Is the greateft or leaft difference between the Sums of
the affirmative Terms, and of the negative Terms ; and is limited
when the indefinite Quantity, (which therefore not improperly I
fuppos'd to be fmall,) cannot be taken greater, but that the Mag-
nitude of the Root will immediately become infinite, that is, will
become impoffible.

54. To illuftrate this, let AC D be a Semicircle defcribed on the
Diameter AD, and BC be an Ordinate.
MakeAB = ^,BC=7,AD = ^. Then


as before.

Therefore BC, or y, then becomes greateft
when iax moft exceeds all the Terms

Sax -f- f- S^x 4- S ax > &c - that is > when * = ** i but

la " ga* V i6a> V

it will be terminated when x a. For if we take x greater than

a t the Sum of all the Terms ^ S ax s7 V ax TbTs *S ax >
&c. will be infinite. There is another Limit alfo, when x = o,
by reafon of the impoffibility of the Radical S ax ; to which
Terms or Limits, the Limits of the Semicircle A, B, and D, are cor^

Tranfttion to the METHOD OF FLUXIONS.

55. And thus much for the Methods of Computation, of which
I mall make frequent ufe in what follows. Now it remains, that ,
for an Illuftration of the Analytick Art, I mould give fome Speci-
mens of Problems, efpecially fuch as the nature of Curves will fup-
ply. But firft it may be obferved, that all the difficulties of thefe x
may be reduced to thefe two Problems only, which I mall propofe
concerning a Space defcribed by local Motion, any how accelerated '
or retarded. ~

56. I. The Length of the Space defcribed being continually ( that -*"*
?V, at fill Times) given; to find the Velocity of the Motion at any ffo^

Tune propofed. / SJLJ tt

57. II. The Velocity of the Motion being continually given ; to find JbotA.*** if*
the Length of the Space defcribed at any Time propofed.

58. Thus in the Equation xx=y, if y reprefents the Length of
the Space t any time defcribed, which (time) another Space x,

by increafing with an uniform Celerity #, mea/ures and exhibits as

D 2 defcribed :

20 ?%e Method of FLUXIONS,

defcribed : Then zxx will reprefent the Celerity by which the Space
y, at the fame moment of Time, proceeds to be defcribed ; and
contrary-wife. And hence it is, that in what follows, I confider
Quantities as if they were generated by continual Increafe, after the
manner of a Space, which a Body or Thing in Motion defcribes.

59. But whereas we need not confider the Time here, any
farther than as it is expounded and meafured by an equable local
Motion ; and befides, whereas only Quantities of the fame kind
can be compared together, and alfo their Velocities of Increafe and
Decreafe : Therefore in what follows I fhall have no regard to Time
formally conficter'd,, but I fhall fiippofe fome one of the Quantities
propofed, being of the fame kind, to be increafed by an equable
Fluxion, to which the reft may be referr'd, as it were to Time j
and therefore, by way of Analogy, it may not improperly receive
the name of Time. Whenever therefore the word Time occurs in
what follows, (which for the fake of perfpicuity and diftindlion I
have fometimes ufed,) by that Word I would not have it under-
ftood as if I meant Time in its formal Acceptation, but only that
other Quantity, by the equable Increafe or Fluxion whereof, Time
is expounded and meafured.

'60. Now thofe Quantities which I confider as gradually and
2 indefinitely increafing, I fhall hereafter call Fluents, or Flowing

Quantities, and fhall reprefent them by the final Letters of the
f Alphabet v, x, y, and z ; that I may diftinguifh them from other

Quantities, which in Equations are to be confider'd as known and.
T H > %f& f'df** determinate, and which therefore are reprefented by the initial
U _' .i V*> i*i~- Letters a, b, c, &c. And the Velocities by which every Fluent

is increafed by its generating Motion, (which I may call Fluxions,

( oi V* ffm***4t*'Qr fimply Velocities or Celerities,) I fhall reprefent by the fame

Letters pointed thus -y, x, y., and z. That is, for the Celerity of
K t4 JO the Quantity v I fhall put v, and fo for the Celerities of the other
id tti Quantities x, y, and z, I fhall put x, y, and z refpeftively.

J '(/ 6 1. Thefe things being premifed, I mall now forthwith proceed

to the matter in hand } and firft I fhall give the Solution of the:
two Problems juft now propofed.




P R O B. I.

The Relation of the Flowing Quantities to one another being
given, to determine the Relation of their Fluxions.


1. Difpofe the Equation, by which the given Relation is ex-
prefs'd, according to the Dimenftons of fome one of its flowing
Quantities, fuppofe x, and multiply its Terms by any Arithmetical

Progreflion, and then by - . And perform this Operation feparately

for every one of the flowing Quantities. Then make the Sum of
all the Products equal to nothing,, aad you will have the Equation

2. EXAMPLE i. If the Relation of the flowing Quantities A; and
y be X' ax* - {- axy ^ 3 =o; firft difpofe the Terms according
to x, and then according to y, and multiply them in the follow-
ing, manner.



makes %xx* zaxx -{- axy * zyy* -f- ayx *


The Sum of the Produdls is -jx** zaxx -k- axy W*-f- ayx=zo,

i .

which Equation gives the Relation between the Fluxions x and y.

For if you take x at pleafure, the Equation .v 3 ax 1 -{-axy yt
= o will give y. Which being determined, it will be x : y ::
7v* ax : yx^zax -{- ay.

3.. Ex. 2. If the Relation of the Quantities x, y,. and z r be ex-
preis'd by the Equation 2j 3 -f- x*y zcyz +- yz* z'' = QJ



+ ffxy-



JT axy.

ax 1




i y .





~- *







Mult. 2j 3 -i-xxxy z*

yx* -+- zy*

z* -fc- 3_>-2 1 zcyz + x'y

~f"~ 32;*

' zcyz

-h zy 3

ay y


; 2 ~

DV *"* O . "

. o .

. . - o.

' y y


z z z

makes 4^-* % 4-'~

zxxy %

- 2 zz*+6zzy-zc Z y .


22 *The Method of FLUXIONS,

Wherefore the Relation of the Celerities of Flowing, or of the
Fluxions ,v, v, and z, is tyy* -\- + 2xxy $zz l -f- 6zzy zczy


4. But fince there are here three flowing Quantities, .v, y, and
z, another Equation ought alfo to be given, by which the Relation
among them, as alfo among their Fluxions, may be intirely deter-
mined. As if it were fuppofed that x -\-y 2 = 0. From whence
another Relation among the Fluxions AT-HV z = o would be
found by this Rule. Now compare thefe with the foregoing Equa-
tions, by expunging any one of the three Quantities, and alfo any
one of the Fluxions, and then you will obtain an Equation which
will intirely determine the Relation of the reft.

5. In the Equation propos'd, whenever there are complex Frac-
tions, or furd Quantities, I put fo many Letters for each, and fup-
pofing them to reprefent flowing Quantities, I work as before. Af-
terwards I fupprefs and exterminate the afTumed Letters, as you fee
done here.

6. Ex. 3. If the Relation of the Quantities .v and y be yy aa

x\/aa ## = o; for x</aa xx I write z, and thence I
have the two Equations^' aa %,=.&., and a 3 -* 1 x 4 2*
i . o, of which the firfl will give zyy z = o, as before, for the
Relation of the Celerities y and z, and the latter will give 2<j*xx

o, or a * xx ~ **** = z, for the Relation of the

Celerities x and z. Now z being expunged, it will be zyy -

= o, and then reftoring x^aa xx for z, we fhall have zyy

-./** 4- g*.>* __ 0> for the Relation between x and y, as was re-

^ aa XX


7. Ex. 4. If .v 3 ay* 4- j4r XX \fay -+- xx = o, expreffes

the Relation that is between AT and v : I make ^^ = 5;, and

^x \/~ay-+-xx=v, from whence I fhall Lave the three Equations x-
a y* + & -u = o, az-\-yz ^ 3 =o, and ax*y + x 6 1^=0.
The firft gives 3**' zayy + z -0=0, the fecond gives az +
Z y^-yz 3^ & = o, and the third gives 4.axx>y-+-6xx'-i-a}>x*
2W= o, for the Relations of the Velocities -y, .v, y, and . But


and INF i NIT E SERIES. 23

the Values of & and i', found by the fecond and third Equations,
iSj ? for z and

/. /. v-. . 11 . ,. 7n vz

nrft Equation, and there anies %xx* 2a )y-^-~^T.

= o. Then inflead of z and v refloring their Values f and

a >

XX \/ ay -+- xx, there will arife the Equation fought ^xx*-2ayy

6*- A- 3 awMf ... . _ . . r ,

= o. by which the Relation or the


aa -f- 2^ + yy 2

Velocities x and y will be exprefs'd.

8. After what manner the Operation is to be performed in other
Cafes, I believe is manifefl from hence j as when in the Equation
propos'd there are found furd Denominators, Cubick Radicals, Ra-

dicals within Radicals, as v ax -+- \/ 'aa xx } or any other com-
plicate Terms of the like kind.

9. Furthermore, altho' in the Equation propofed there fhould
be Quantities involved, which cannot be determined or exprefs'd
by any Geometrical Method, fuch as Curvilinear Areas or the Lengths
of Curve-lines ; yet the Relations of their Fluxions may be found,
as will appear from the following Example.

Preparation for EXAMPLE 5*

10. Suppofe BD to be an Ordinate at right Angles to AB, ancL
that ADH be any Curve, which is defined by
the Relation between AB and BD exhibited
by an Equation. Let AB be called A;, and
the Area of the Curve ADB, apply 'd to Unity,
be call'd z. Then erect the Perpendicular AC
equal to Unity, and thro' C draw CE parallel
to AB, and meeting BD in E. Then conceiving
thefe two Superficies ADB and ACEB to be generated by the
Motion of the right Line BED ; it is manifeft that their Fluxions,
(that is,, the Fluxions of the Quantities i x z t . and i x v, or of the
Quantities s and x,) are to each other as the generating Lines BD
and BE. Therefore : x :: BD : BE or i, and therefore
z = * x BD.

1 1. And hence it is, that z may be involved in any Equation,
expre fling the Relation between .v and any other flowing'Quantityjv ;
and yet the Relation of the Fluxions x and y may however be dif-
cover'd, 12 .

24 < fhe Method <J/" FLUXION s,

12. Ex. 5. As if the Equation zz -\-axz _y*=r=o were pro-
pos'd to exprefs the Relation between x and; 1 , as alfo \/axxx
= BD, for determining a Curve, which therefore will be a Circle.
The Equation zz-^-axz j^=o, as before, will give 2zz-i-
azX -f- axz 4_y_y = o, for the Relation of the Celerities x,y,
and z. And therefore fince it is z = x x BD or -x \/ax xx t
iubftitute this Value inftead of it, and there will arife the Equation

2xz -t- axx \/ax-r xx 4- axz qyy* = o, which determines the
Relation of the Celerities x and y.

DEMONSTRATION of the Solution.

13. The Moments of flowing Quantities, (that is, their indefi-
nitely fmall Parts, by the acceffjon of which, in indefinitely fmall
portions of Time, they are continually increafed,) are as the Ve-
locities of their Flowing or Increafing.

14. Wherefore if the Moment of any one, as x, be reprefented
t>y the Product of its Celerity x into an indefinitely fmall Quantity
o (that is, by xo,} the Moments of the others <y, y, z, will be
reprefented by vo t yo, zo ; becaufe vo y xo, yo, and zo, are to each
other as v, x, y, and x.

,. p. , 15. Now fince the Moments, as xo and yo, are the indefinitely

/fc, // natti** cttA u tt i e ^cceflions of the flowing Quantities .v and y, by which thofe


And therefore the Equation, which at all times indifferently exprefles
the Relation of the flowing Quantities, will as well exprefs the
Relation between x -3- xo and y-+-yo, as between x and y: So
that x -+- xo and y -f- yo may be fubftituted in the fame Equation
for thofe Quantities, inftead of x and y.

1 6. Therefore let any Equation #' ax* -+- axy ^' = be
given, and fubftitute x~\-xo for x } and y -j- yo for y, and there
will arife

+ $x*oox -f- x*o''

ax 1 2axox ax*oo

axy +- axoy -h ayox -h axyoo

y: lyoy- ~ yfooy


17. Now by Suppofition x 3 ax - 3 raxy _}' 3 =o, which there-
fore being expunged, and the remaining Terms being divided by o,
there will remain ^xx* -f- ^ox -+- x>oo zaxx ax 1 o -f- axy -f-
ayx _f_ axyo 3_vy* 3y*oy y*oo = o. But whereas o is fuppofed
to be infinitely little, that it may reprefent the Moments of Qiian-
tities ; the Terms that are multiply'd by it will be nothing in relbedl
of the reft. Therefore I reject them, and there remains $xx*
zaxx -f- axy -+- ayx 3_yj*= o, as above in Examp. i.

1 8. Here we may obferve, that the Terms that are not multiply'd
by o will always vaniih, as alfo thole Terms that are multiply'd by o
of more than one Dimenfion. And that the reft of the Terms
being divided by o, will always acquire the form that they ought
to have by the foregoing Rule : Which was the thing to be proved.

19. And this being now fhewn, the other things included in the
Rule will eafily follow. As that in the propos'd Equation feveral
flowing Quantities may be involved ; and that the Terms may be
multiply'd, not only by the Number of the Dimenlions of the flow-
ing Quantities, but alfo by any other Arithmetical Progreilions ; fo
that in the Operation there may be the lame difference of the Terms
according to any of the flowing Quantities, and the ProgrefTion be
difpos'd according to the fame order of the Dimenlions of each of
them. And thele things being allow'd, what is taught belides in
Examp. 3, 4, and 5, will be plain enough of itfelf.

P R O B. II.

An Equation being propofed, including the Fluxions of
O^uantitieS) to find the Relations of tbofe Quantities to
one another.


i. As this Problem is the Converfe of the foregoing, it muft be
folved by proceeding in a contrary manner. That is, the Terms
multiply'd by x being difpofed according to the Dimenfions of x ;

they muft be divided by * x , and then by the number of their Di-
menfions, or perhaps by fome other Arithmetical Progreffion. Then
the fame work muft be repeated with the Terms multiply'd by v, y,

E or

26 The Method of FLUXIONS,

or z, and the Sum refulting muft be made equal to nothing, re-
jeding the Terms that are redundant.

2. EXAMPLE. Let the Equation propofed be ^xx* 2axx 4- axy
4- ayx = o. The Operation will be after this manner :

Divide 3 ATA?* 2axx-i-axy

by - Quot. 3A: 5 2ax* -\-ayx

Divide by 3 . 2 i.

Quote A; 5 ax 1 -{-ayx


by ^. Quot. 3

Divide by 3

Quote _y 5

* -f- ayx

* 4- axy

2 . i.

* 4- axy

Therefore the Sum # 3 ax* -f- axy y* = o, will be the required
Relation of the Quantities x and y. Where it is to be obferved,
that tho' the Term axy occurs twice, yet I do not put it twice in
the Sum x'> ax* -+- axy y* =. o, but I rejed the redundant
Term. And fo whenever any Term recurs twice, (or oftener when
there are feveral flowing Quantities concern'd,) it muft be wrote
only once in the Sum of the Terms.

3. There are other Circumftances to be obferved, which I mall/
leave to the Sagacity of the Artift -, for it would be needlefs to dwell
too long upon this matter, becaufe the Problem cannot always be
folved by this Artifice. I mail add however, that after the Rela-
tion of the Fluents is obtain'd by this Method, if we can return,
by Prob. i. to the propofed Equation involving the Fluxions, then
the work is right, otherwife not. Thus in the Example propofed,

after I have found the Equation x> ax 1 - -{- axy y* = o, if from

thence I feek the Relation of the Fluxions x and y by the firft
Problem, I mall arrive at the propofed Equation ^xx* 2axx 4-
axy i,yy* -f- ayx= o. Whence it is plain, that the Equation
AT 3 -ax*-+-axy _y 3 = o is rightly found. But if the Equation
xx xy -\- ay = o were propofed, by the prefcribed Method I
fhould obtain this ^x* xy + ay = o, for the Relation between
x and y ; which Conclufion would be erroneous: Since by Prob. i.
the Equation xx xy yx -+- ay = o would be produced, which
is different from the former Equation.

4. .Having therefore premiled this in a perfundory manner, I
lhall now undertake the general Solution.




5. Firft it mufl be obferved, that in the propofed Equation
the Symbols of the Fluxions, (fince they are Quantities of a diffe-
rent kind from the Quantities of which they are the Fluxions,)
ought to afcend in every Term to the fame number of Dimenfions :
And when it happens otherwife, another Fluxion of fome flowing
Quantity mufl be underflood to be Unity, by which the lower
Terms are fo often to be multiply'd, till the Symbols of the Fluxions
arife to the fame number of Dimenfions in all the Terms. As if
the Equation x -+ x'yx axx = o were propofed, the Fluxion z
of fome third flowing Quantity z mufl be underilood to be Unity,
by which the firfl Term x mufl be multiply'd once, and the lafl
axx twice, that the Fluxions in them may afcend to as many Di-
menfions as in the fecond Term xyx : As if the propofed Equation
had been derived from this xz -{-xyx- azzx*- = o, by putting
z = i. And thus in the Equation yx =}')'-, you ought to ima-
gine x to be Unity, by which the Term yy is multiply'd.

6. Now Equations, in which there are only two flowing Quan-
tities, which every where arife to the fame number of Dimenfions,
may always be reduced to fuch a form, as that on one fide may be

had the Ratio of the Fluxions, (as 4 , or - , or ~ ,&c.) and on the

\ x . y x

other fide the Value of that Ratio, exprefs'd by fimple Algebraic


Terms ; as you may fee here, 4- = 2 -h 2X y. And when the

foregoing particular Solution will not take place, it is required that
you fhould bring the Equations to this form.

7. Wherefore when in the Value of that Ratio any Term is de-
nominated -by a Compound quantity, or is Radical, or if that Ratio
be the Root of an affected Equation ; the Reduction mufl be per-
form'd either by Divifion, or by Extraction of Roots, or by the
Refolution of an affected Equation, as has been before fhewn.

8. As if the Equation ya yx xa -+- xx xy = o were pro-
pofed j firfl by Reduction this becomes T-=i-f - ^-, or -==

x ax y

av+y' And in the firfl Cafe, if I reduce the Term ^^., deno-
minated by the compound Quantity a x, to an infinite Series of

E 2 fimple

28 The Method of FLUXIONS,

fimple Terms j -f- - -f- ~ -+- ^ &c. by dividing the Numerator
y by the Denominator a x, I mall have - i +- - -f- ^ -f.

^ -f- 7; &c. by the help of which the Relation between x and
y is to be determined.

9. So the Equation _y_y = xy -j- .XVY.V A: being given, or ^- = 4,

A-* x

i- xx, and by a farther Reduction 4=4 +V/T -+- A-* : I extract


the fquare Root out of the Terms -J -f- xr, and obtain the infinite
Series f -{-x* x* -f- 2X 6 5*" -f- 14*', &c. which if I fubfti-

tute for \/t H- xx, I (hall have - = i -f- x* x* -f- 2x 6


&c. or. ~ = x^-ir-x* 2X 6 -+- 5* 8 , &c. according as

is either added to -I, or fubtracled from it.

10. And thus if the Equation y* -j- axx*y -f- a'-x^y x*x"> ~

2x*a>=o were propofed, or ' -f- ax -f- a 1 - >v 3 2rf 3 = o

A:5 A: x

I extract the Root of the affected Cubick Equation, and there.

/- V X XX 111*5 COQi'4

anfes ~ =a ^-_|_ ^_ _ 4. ^ & c . as may be feen

x 4 640 5i2 a 16384^3 ^


11. But here it may be obferved, that I look upon thofc Terms
only as compounded, which are compounded in refpect of flowing
Quantities. For I efteem thofe as fimple Quantities which are com-
pounded only in refpect of given Quantities. For they may be re-
duced to fimple Quantities by luppofing them equal to other givea

Quantities. Thus I eonfider the Quantities " -> "-TT, rr-

^ - ^^' c a*4- b' ax-\~bx >

1 4

~^, L , x i > v/tfA- H- bx, &c. as fimple Quantities, becaufe they may
may all be reduced to the fimple Quantities ^ i, -^-, , \/ex (or

x*} &cc. by fuppofing a -f- b =r= e.

12. Moreover, that the flowing Quantities may the more eafily
be diflinguifh'd from one another, the Fluxion that is put in the
Numerator of the Ratio, or the Antecedent of the Ratio, may not
improperly be call'd the Relate Quantify, and the other in the De-
nominator, to which it is compared, the Correlate : Alfo the



flowing Quantities may be diftinguifli'd by the fame Names refpec-
tively. And for the better understanding of what follows, you may
conceive, that the Correlate Quantity is Time, or rather any other
Quantity that flows equably, by which Time is expounded and
meafured. And that the other, or the Relate Quantity, is Space,
which the moving Thing, or Point, any how accelerated or retarded,
defcribes in that Time. And that it is the Intention of the Problem,
that from the Velocity of the Motion, being given at every Inftant
of Time, the Space defcribed in the whole Time may be deter-

13. But in refpedt of this Problem Equations may be diftinguifli'd

into three Orders.

14. Firft: In which two Fluxions of Quantities, and only one
of their flowing Quantities are involved.

15. Second: In which the two flowing Quantities are involved,
together with their Fluxions.

1 6. Third: In which the Fluxions of more than two Quantities
are involved.

17. With thefe Premifes I {hall attempt the Solution of the
Problem, according to thefe three Cafes.


1 8. Suppofe the flowing Quantity, which alone is contain 'd in
the Equation, to be the Correlate, and the Equation being accord-
ingly difpos'd, (that is, by making on one fide to be only the
Ratio of the Fluxion of the other to the Fluxion of this, and on
the other fide to be the Value of this Ratio in fimple Terms,) mul-
tiply the Value of the Ratio of the Fluxions by the Correlate Quan-
tity, then divide each of its Terms by the number of Dimenfions
with which that Quantity is there afTeded, and what arifes will be
equivalent to the other flowing Quantity.

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