Isaac Newton.

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

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- = t : The Conflruclion of which is in this manner.

c ex

96. With the Center A, principal Vertex P, and Parameter aAP,
defcnbe the Hyperbola PK. Then from the point C draw the right
Line CK, that may touch the Parabola in K : And it will be, as
AP to 2AG, fo is the Area CKPC to the Area required DPED.

97. EXAMPLE 5. Let the Norma GFE fo revolve about the Pole
G, as that its angular point F may continually flide upon the right
Line AF given in pofition ; then conceive the Curve PE to be de-
fcribed by any Point E in the

other Leg EF. Now that the
Area of this Curve may be
found, let fall GA and EH per-
pendicular to the right Line
AF, and compleating the Pa-
rallelogram AHEC, call AC
= 2, CE=j, AG = , and
EF=; and becaufe of the
Proportionals HF : EH : : AG :
AF, we mall have AF =

, bz . Therefore CE or y

V a zz

But whereas </cc zz is the Ordinate

of a Circle defcribed with the Semidiameter c ; about the Center A


*fhe Method of FLUXIONS,
let fuch a Circle PDQ_be defcribed, which CE produced meets ia

D ; then it will be DE = ^=rS : B ? the hel P of which EqUa ~
tion there remains the Area PDEP or DERQ^to be determin'd.

Suppofe therefore =:2, and G=^ 3 and it will be DE= i^~ >

V ft ^

sn Equation of the ift Species of the 4th Order of Table i. And
the Terms being compared, it will be b-= d, cc =e, and j ==/;

fo that bV cc zz = l>R=f.

98. Now as the value of t is negative, and therefore the Area
reprefented by / lies beyond the Line DE ; that its initial Limit
may be found, feek for that length of z, at which t becomes no-
thing, and you will find it to be c. Therefore continue AC to Q^>
that it may be AQ==c, and erect the Ordinate QR.; and DQRED
will be the Area whofe value now found is b\/cc zz.

99. If you fhould define to know the quantity of the Area
PDE, pofited at the Abfcifs AC, and co-extended with it, without
knowing the Limit QR, you may thus determine it.

100. From the Value which / obtains at the length of the Ab-
fcifs AC, fubtract its value at the beginning of the Abfcifs ; that is,
from b\/ cc zz fubtract &, and there will arife the defired
quantity A: b\/ LC zz. Therefore compleat the Parallelogram
PAGK, and let fall DM perpendicular to AP, which meets GK
in M ; and the Parallelogram PKML will be equal to the Area

101. Whenever the Equation defining the nature of the Curve
cannot be found in the Tables, nor can be reduced to limpler terms
by divifion, nor by any other means ; it muft be transform'd into
other Equations of Curves related to it, in the manner fhewn in
Prob. 8. till at laft one is produced, whofe Area may be known by
the Tables. And when all endeavours are ufed, and yet no fuch
can be found, it may be certainly concluded, that the Curve pro-
pofed cannot be compared, either with rectilinear Figures, or with
the Conic Sedions.

102. In the fame manner when mechanical Curves are concern'd,
they muft fir ft be transform'd into equal Geometrical Figures, as is
fhewn in the fame Prob. 8. and then the Areas of fuch Geometri-
cal Curves are to be found from the Tables. Of this matter take
the following Example.



103. EXAMPLE 6. Let it be propofed to determine the Area of
the Figure of the Arches of any Conic Section, when they aie
made Ordinates on their Right Sines. As let A be the Center of
the Conic Section,

AQ_and AR the ^ " V .' ^\

Semiaxes, CD the
Ordinate to the Axis
AR, and PD a Per-
pendicular at the
point D. Alfo let
AE be the fa id
mechanical Curve
meeting CD in E;
and from its nature
before defined, CE
will be equal to the
Arch QD. There-
fore the Area A EC
is fought, or com-
pleating the parallelogram ACEF, the excefs AEF is required. To
which purpole let a be the Latus rectum of the Conic Section, and
b its Latus tranfverfum, or 2AQ^_ Alfo let AC=z, and CD=_>';

then it will be V ^bb -f- -zz =y, an Equation to a Conic Section,

as is known. Alfo PC= -z, and thence PD = v/^H ~- zz.

104. Now fince the fluxion of the Arch QD is to the fluxion of
the Abfcifs AC, as PD to CD ; if the fluxion of the Abfcifs be fup-
pos'd i, the Fluxion of the Arch QD, or of the Ordinate CE,

will be i/ 4 . Draw this into FE, or z, and there

for the fluxion of the Area AEF.

will arife z /

If therefore in the Ordinate CD you take CG -




, the Area AGC, which is defcribed by CG

moving upon AC, will be equal to the Area AEF, and the Curve




n6 77je Method of FLUXIONS,

AG will be a Geometrical Curve. Therefore the Area AGC is
fought. To this purpofe let z* be fubflituted for z* in the laft

Equation, and it becomes &*-* \/^-j-, j-^ = CG, an Equa.-


tion of the ad Species of the i ith Order of Table 2. And from a
comparifon of terms it is d = i, e-=.i-bb =,/= - ~ , and

$= : fo that \/ ^bb ~] zz=x. \/ -f- xx r, i>. and

a * * a ' Afl a /

~s = t. That is, CD = x, DP = v, and Jj = /. And this is
the Conftruction of what is now found.

105. At Q^ erect QK perpendicular and equal to QA, and thro*
the point D draw HI parallel to it, but equal to DP. And the
Line KI, at which HI is terminated, will be a Conic Section, and
the comprehended Area HIKQ^will be to the Area fought AEF,
as b to a, or as PC to AC.

106. Here obferve, that if you change the fign of b, the Conic
Section, to whofe Arch the right Line CE is equal, will become an
Ellipfis; and befides, if you make b = , the Ellipfis becomes-
a Circle. And in this cafe the line KI becomes a right line parallel

107. After the Area of any Curve has been thus found and con-
ftrucled, we fhould confider about the demonftration of the con-
ftruction ; that laying afide all Algebraical calculation, as much as
may be, the Theorem may be adorn'd, and made elegant,, fo as to
become fit for publick view. And there is a general method of de -
monftrating, which I mail endeavour to iiluftrate by the follow-
ing Examples.

Demonftration of the Conjlruflion in Example 5.

1 08. In the Arch PQ^take a point d indefinitely near to D,
(Figure p. 113.) and draw de and dm parallel to DE and DM,
meeting DM and AP in p and /. Then will DE^/ be the mo-
ment of the Area PDEP, and LM// will be the moment of the
Area LMKP. Draw the femidiameter AD, and conceive the inde-
finitely fmall arch ~Dd to be as it were a right line, and the tri-
angles -D/^/ and ALD will be like, and therefore D/> : pd:: AL : LD.
But it is HF : EH :: AG : AF ; that is, AL : LD :: ML : DE; and
therefore Dj> : pd : : ML : DE. Wherefore Dp x DE = pd x ML




That is, the moment DEed is equal to the moment LM;;//. And
fince this is demonflrated indeterminately of any contemporaneous
moments whatever, it is plain, that all the moments of the Area
PDEP are equal to all the contemporaneous moments of the Area
PLMK, and therefore the whole Areas compofed of thofe moments
are equal to each other. C^JE. D.

Demonftration of the ConftruSfion in Example 3.

109. Let DEed be the momentum of the fuperficies AHDE, and
A</DA be the contemporary
moment of the Segment ADH.
Draw the femidiameter DK,
and let de meet AK in c -, and
it is Cc : Dd :: CD : DK.
Befides it is DC : QA (aDK) : :
AC : DE. And therefore
Cc : 2Dd :: DC : aDK ::
AC : DE, and Cc x DE =
zDd-x. AC. Now to the mo-
ment of the periphery Dd
produced, that is, to the tan-
gent of the Circle, let fall the
perpendicular AI, and AI will
be equal to AC. So that
zDd x AC = zDd x AI = 4

Triangles AD</. So that 4 Triangles AD^/=C^xDE= moment
DE^/. Therefore every moment of the fpace AHDE is quadruple
of the contemporary moment of the Segment ADH, and therefore
that whole fpace is quadruple of the whole Segment. Q^E. D.



"The Method of FLUXIONS,

Demonftration of the ConftruRion in Example 4.

no. Draw ce parallel to CE, and at an indefinitely fmall diflance
from it, and the tangent of the
Hyperbola ck t and let fall KM
perpendicular to AP. Now
from the nature of the Hyper-
bola it will be AC : A? ::
AP : AM, and therefore AC? :
GLq :: AC?: LE? (or AP V ') ::
AP? : AM? ; and divlfim* AG/ :
AL? (DE?) ::.AP?: AM?
AP?(MK?) ; And invent, AG:
AP :: DE : MK. But the
little Area DEed is to the Tri-
angle CKr, as the altitude DE is to half the altitude KM ; that is,
as AG to -LAP. Wherefore all the moments of the Space PDE
are to all the contemporaneous moments of the Space PKC, as AG
to 4-AP. And therefore thofe whole Spaces are in the fame ratio.

Demonjlration of the Conjlruftion in Example 6.

in. Draw c*/ parallel and infinitely near to CD, (Fig. in p. 115-)
meeting the Curve AE in e, and draw hi and fe meeting DCJ in p
and q. Then by the Hypothefis ~Dd= Eg, and from the fimi-
litude of the Triangles Ddp and DCP, it will be D/> : (Dd)
Eq :: ( P : (PD) HI, fo that Dp x HI = Eg xCPj and thence
Dp x HI (the moment HI/'/.)): Eg x AC (the moment EF/e) ::
E ? xCP : EyxAC :: CP : AC. Wherefore fince PC and AC
are in the given ratio of the latus tranlverfum to the Jatus rectum
of the Conic Section QD, and fince the moments HI//) and EFfe
of the Areas HIKQ^and AEF are in that ratio, the Areas them-
felves will be in the fame ratio. Q-^E. D.

112. In this kind of demonilrations it is to be obferved, that I
affume fuch quantities for equal, whofe ratio is that of equality :
And that is to be efteem'd a ratio of equality, which differs lefs
from equality than by any unequal ratio that can be affign'd. Thus
in the laft demon ftration I fuppos'd the rectangle E^xAC, or FE?/,
to be equal to the fpace FEt/j becaufe (by realon of the difference
Eqe infinitely lefs than them, or nothing in comparifon of them,)




they have not a ratio of inequality. And for the fame reafon I
made DP x HI = HI//6 ; and fo in others.

1 13. I have here made ufe of this method of proving the Areas
of Curves to be equal, or to have a given ratio, by the equality, or
by the given ratio, of their moments ; becaufe it has an affinity to
the ufual methods in thefe matters. But that feems more natural
which depends upon the generation of Superficies, by Motion or
Fluxion. Thus if the Confbuclion in Example 2. was to be de-
monftrated : From the nature of the Circle, the fluxion of the right
line ID (Fig. p.i 1 1.) is to the fluxion of the right line IP, as AI to
ID ; and it is AI : ID : : ID : CE, from the nature of the Curve

AGE ; and therefore CE x ID = ID x IP. But CE x ID = to
the fluxion of the Area PDI. And therefore thofe Areas, being ge-
nerated by equal fluxion, muft be equal. Q^E. D.

1 14. For the fake of farther illustration, I fliall add the demon-
flration of the Confrruc~r.ion, by which the Area of the Ciffoid is
determin'd, in Example 3. Let the lines mark'd with points in the
fcheme be expunged; draw the Chord DQ^ and the Afymptote
QR of the Ciffoid. Then, from the nature of the Circle, it Is
DQj- = AQ_x CQ^, and
thence (by Prob. i.)

Fluxion of DQj= AQjcCQ.

And therefore AQ_:

2DQj CX^ Alfo from the
nature of the Ciffoid it is ED :
AD :: AQ^: DQ^ There-

fore ED : AD : :

and EDxCC^=ADx2DQ^,

or 4xiADxDQ^ Nowfmce
DQ __ is perpendicular at the
end of AD, revolving about

A ; and i AD x QD = to the fluxion generating the Area

its quadruple alfo ED x CQ^== fluxion generating the Ciffoidal Area
QREDO. Wherefore that Area QREDO infinitely long, is gene-
rated quadruple of the other ADOQ^ Q^E. D.


120 The Method of FLUXIONS,


115. By the foregoing Tables not only the Areas of Curves, but
quantities of any other kind, that are generated by an analogous
way of flowing, may be derived from their Fluxions, and that by
the affiftance of this Theorem : That a quantity of any kind is to an
unit of the lame kind, as the Area of a Curve is to a fuperficial
unity ; if fo be that the fluxion generating that quantity be to an
unit of its kind, as the fluxion generating the Area is to an unit of
its kind alfo ; that is, as the right Line moving perpendicularly upon
the Abfcifs (or the Ordinate) by which the Area is defcribed, to a
linear Unit. Wherefore if any fluxion whatever is expounded by
fuch a moving Ordinate, the quantity generated by that fluxion will
be expounded by the Area defcribed by fuch Ordinate ; or if the
Fluxion be expounded by the fame Algebraic terms as the Ordinate,
the generated quantity will be expounded by the fame as the de-
fcribed Area. Therefore the Equation, which exhibits a Fluxion of
any kind, is to be fought for in the firft Column of the Tables, and
the value of t in the laft Column will mow the generated Quan-
tity. _

1 1 6. As if \/ 1 -h exhibited a Fluxion of any kind, make it
equal to y, and that it may be reduced to the form of the Equations
in the Tables, fubftitute z* for z, and it will be z~ ' </ 1 -+- z


7y, an Equation of the firft Species of the 3d Order of Table i.
And comparing the terms, it will be </= i, e=i,f=2. >

8a + i8z ,~ gz -id -p. _, _

and thence - \S i +- - a == R> =/. Therefore it is the
quantity Z ^~ 1/1 -4- which is generated by the Fluxion


3 17, And thus if v'l -f- J^l- reprefents a Fluxion, by a due re-

9 7

duftion, (or by extracting & out of the radical, and writing _
for 2~^) there will be had -or, */s& - ! =7, an Equation of

z ga*

the ad Species of the 5th Order of Table 2. Then comparing the



terms, it is d=. i, e = , and/= i. So that x 7 = - = **,


_j_ ' ^ = -u, and 4 J = - * = A Which being found, the


quantity generated by the fluxion v/ j + L^Z w ill be known, by

making it to be to an Unit of its own kind, as the Area j* is to
fuperficial unity ; or which comes to the fame, by fuppofing the
quantity t no longer to reprefent a Superficies, but a quantity of an-
other kind, which is to an unit of its own kind, as that fuperficies
k to fuperficial unity. _

1 1 8. Thus fuppofing \/i 4- l ~ to reprefent a linear Fluxion, I

9 T

imagine t no longer to fignify a Superficies, but a Line ; that Line,
for inftance, which is to a linear unit, as the Area: which (accord-
ing to the Tables) is reprefented by t, is to a fuperficial unit, or
that which is produced by applying that Area to a linear unit. On
which account, if that linear unit be made e, the length generated
by the foregoing fluxion will be ~ . And upon this foundation

thofe Tables may be apply'd to the determining the Lengths of
Curve-lines, the Contents of their Solids, and any other quantities
whatever, as well as the Areas of Curves.

Of ^uejlions that are related hereto.

I. To approximate to the Areas of Curves mechanically,

119. The method is this, that the values of two or more right-
lined Figures may be fo compounded together, that they may very
nearly conftitute the value of the Curvilinear Area required.

120. Thus for the Circle AFD which is denoted by the Equa-
tion .v xx = r zz } having found the value of

the Area AFDB, viz. ** #* /,**
J-x*, &c. the values of fome Rectangles are to
be fought, fuch is the value x\/x xx, or x*
z* T#* T V#% & c - of the rectangle
BD x AB, and x^/x, or #', the value of AD x
AB. Then thefe values are to be multiply'd by
any different letters, that ftand for numbers indefinitely, and then

R to

122 2^2 Method of FLUXIONS,

to be added together, and the terms of the fum are to be compared
with the correfponding terms of the value of the Area AFDB, that
as far as is poffible they may become equal. As if thofe Parallelo-
grams were multiply'd by e and f, the fum would be ex* \ex^

{$$, &c. the terms of which being compared with thefe terms
^ x * ,^ x * T V*% &c. there arifes +/=-!, and i^= 4.,
or e = , and /= % e = T * r So that ^-BD x AB -f- T 4 T AD x
AB = Area AFDB very nearly. For ^-BD x AB -f. T * T AD x AB is
equivalent to .!#* 4.** _^.v* - _L.,v*, &c. which being fub-
tracted from the Area AFDB, leaves the error only T '-# -j- T V#*,

121. Thus if AB were bifected in E, the value of the rectangle
AB x DE will be x\/x %xx, or x* -^x*

-2-#* - x*, &c. And this compared with

128 1024 r

the rectangle AD x AB, gives 8DE + zAD into
AB = Area AFDB, the error being only

J-x* -\ - x* &c. which is always lefs than
560 5760

TJ^JTJ. part of the whole Area, even tho' AFDB
were a quadrant of a Circle. But this Theorem may be thus pro-
pounded. As 3 to 2, fo is the rectangle AB into DE, added to a
fifth part of the difference between AD and DE, to the Area AFDB,
very nearly.

122. And thus by compounding two rectangles ABxED and
AB x BD, or all the three rectangles together, or by taking in ftill
more rectangles, other Rules may be invented, which will be fo
much the more exacT:, as there are more Rectangles made ufe of.
And the fame is to be understood of the Area of the Hyperbola, or
of any other Curves. Nay, by one only rectangle the Area may
often be very commodioufly exhibited, as in the foregoing Circle,
by taking BE to AB as v/io to 5, the rectangle AB x ED will be
to the Area AFDB, as 3 to 2, the error being only T f T AT* -fr-

II. The Area being g hen, to determine the Abfcifs and Ordinate.
123. When the Area is exprefs'd by a finite Equation, there can
be no difficulty : But when it is exprefs'd by an infinite Series, the
affected root is to be extracted, which denotes the Abfcifs. So for


W^^ ** w


the Hyperbola, defined by the Equation ^ = z, after we have
found * = bx -^ -+- - * , &c. that from the given Area
the Abfcifs x may be known, extract the affedled Root, and there

will arife x = + ^ + - 4- -JjjL , &c. And

moreover, if the Ordinate .5 were required, divide ab by /z 4- AT,
that is, by a -f- } -+ -^ -f- ~ s , &c. and there will arife z=l>>

124. Thus as to the Ellipfis which is exprefs'd by the Equation
ax -xx = zz, after the Area is found z = ^a?x* a%x *

1 i I , x ,

^!^ Hf_, &c. write i;' for , and / for x*, and it becomes
= t* ^ -i-j, &c. and extracting the root /=

&c. is equal to x. And this value being fubflituted inftead of x in
the Equation ax a -xx = zz, and the root being extracted, there
arifes * = **. ^L 3 38*' __ 4 Q7^ 7 5c c> So that from

5<: '7Sf* 225018

z, the given Area, and thence v or ./"I, the Abfcifs # will be

f za*

given, and the Ordinate z. All which things may be accommo-
dated to the Hyperbola, if only the flgn of the quantity c be changed,
wherever it is found of odd dimenfions.

R O B.


*The Method of FLUXIONS,

P R O B. X.

1o find as many Curves as we pleafe, vohofe Lengths
may be exprcfsd by finite Equations.

1. The following pofitions prepare the way for the foltirion of
this Problem.

2. I. If the right Line DC, ftanding perpendicularly upon any.
Curve AD, be conceived thus to move,

all its points G, g, r, &c. will defcribe
other Curves, which are equidiftant, and
perpendicular to that line : As GK, gk,
rs, &c.

3. II. If that right Line is continued
indefinitely each way, its extremities will
move contrary ways, and therefore there
will be a Point between, which will have
no motion, but may therefore be call'd
the Center of Motion. This Point will
be the fame as the Center of Curvature,
which the Curve AD hath at the point D,
as is mention'd before. Let that point

4. III. If we fuppofe the line AD not
to be circular, but unequably curved, fup-
pofe more curved towards <T, and lefs toward A; that Center will
continually change its place, approaching nearer to the parts more
curved, as in K, and going farther off at the parts lefs curved, as in.
k t and by that means will defcribe fome line, as KG.

5. IV. The right Line DC will continually touch the line de-
fcribed by the Center of Curvature. For if the Point D of this
line moves towards ^, its point G, which in the mean time pafTes
to K, and is fituate on the fame fide of the Center C, will move
the fame way, by pofition 2. Again, if the fame point D moves
towards A, the point g, which in the mean time paffes to k, and
k fituate on the contrary fide of the Center C, will move the con-
trary way, that is, the fame way that G moved in the former cafe,
while it pafs'd to K. Wherefore K and k lie on the fame fide of
the right Line DC. But as K and k are taken indefinitely f :>r any



points, it is plain that the whole Curve lies on the fame fide of the
right line DC, and therefore is not cut, but only touch'd by it.

6. Here it is fuppos'd, that the line <rDA is continually more
curved towards <T, and lefs towards A ; for if its greateft or leaft
Curvature is in D, then the right line DC will cut the Curve KC ;
but yet in an angle that is lefs than any right-lined angle, which is
the fame thing as if it were faid to touch it. Nay, the point C in
this cafe is the Limit, or Cufpid, at which the two parts of the
Curve, finishing in the moft oblique concourfe, touch each other ;
and therefore may more juftly be faid to be touch'd, than to be cut,
by the right line DC, which divides the Angle of contact.

7. V. The right Line CG is equal to the Curve CK. For con-
ceive all the points r, 2r, 3;-, ^.r, &c. of that right Line to defcribe
the arches of Curves rs, 2r2s, 3^3;, &c. in the mean time that they
approach to the Curve CK, by the motion of that right line ; and
fmce thofe arches, (by polition i.) are perpendicular to the right
lines that touch the Curve CK, (by pofition 4.) it follows that they
will be alfo perpendicular to that Curve. Wherefore the parts of
the line CK, intercepted between thofe arches, which by reafon of
their infinite fmallnefs may be confider'd as right lines, are equal to
the intervals of the fame arches ; that is, (by polition i.) are equal
to fo many parts of the right line CG. And equals being added
to equals, the whole Line CK will be equal to the whole Line

8. The fame thing would appear by conceiving, that every part
of the right Line CG, as it moves along, will apply itfelf fuccef-
fively to every part of the Curve CK, and thereby will meafure
them ; juft as the Circumference of a wheel, as it moves forward by
revolving upon a Plain, will meafure the diflance that the point of
ContacT; continually defcribes.

9. And hence it appears, that the Problem may be refolved, by
afiuming any Curve at pleaflue A/'DA, and thence by determining
the other Curve KC, in which the Center of Curvature of the
aftumed Curve is always found. Therefore letting fall the perpen-
diculars DB and CL, to a right Line AB given in pofition, and in
AB taking any point A, and calling AB = .v and BD = v ; to
define the Curve AD let any relation be affumed between x and v,
and then by Prob 5. the point C may be found, by which may be
determined both the Curve KC, and its Length GC.


Method of FLUXIONS,


10. EXAMPLE. Let ax =yy be the Equation to the Curve,
which therefore will be the Apollonian Parabola. And, by Prob. 5.
will be found AL=|
^ , and DC = 2if

* a

-+. ax.

Which being obtain'd, the Curve KC
is determin'd by AL and LC, and its
Length by DC. For as we are at
liberty to aflume the points K and C
anf where in the Curve KC, let us
fuppofe K to be the Center of Cur-
vature of the Parabola at its Vertex ;
and putting therefore AB and BD, or
x and y, to be nothing, it will be
DC = -irf. And this is the Length
AK, or DG, which being fubtracted
from the former indefinite value of

DC, leaves GC or KC = -^- V aa +. a x \a.

11. Now if you defire to know what Curve this is, and what is
its Length, without any relation to the Parabola ; call KL = z t
and LC = v, and it will be &==. AL \a = 3 x, or ^z = AT, and
- = ax =yy. Therefore 4v /- = S! = CL = v, or ' ==

2 ''' 27 t aa 2 7 #

u* j which fhews the Curve KC to be a Parabola of the fecond kind.
And for its Length there arifes llil ^/^aa -f- az a, by

writing ~z for >r in the value of CG.

12. The Problem alfo may be refolved by taking an Equation,
which fhall exprefs the relation be-
tween AP and PD, fuppofing P to

be the interfeclion of the Abfcifs and
Perpendicular. For calling AP=,v,
and PD =/, conceive CPD to move
an infinitely fmall fpace, fuppofe to
the place Cpd } and in CD and Cd ta-
king CA and CeT both of the fame
given length, fuppofe = r, and to
CL let fall the perpendiculars A^ and
fy y of which Ag, (which call =z)
may meet Cd inf. Then compleat
the Parallelogram gyfe, and making
x,y, and z the fluxions of the quantities ,v, y, and x, as before

Online LibraryIsaac NewtonThe method of fluxions and infinite series : with its application to the geometry of curve-lines → online text (page 11 of 30)