Isaac Newton.

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

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Font size A its Center, and B a given Point in

47. Let BK be
its Circumference.

a Circle,
be a Spiral, DC its Perpen-
dicular, and C the Center of
Curvature at the Point D.
Then drawing the right Line
equal to AK, as alfo the Per-
pendicular GF meeting CD
inF: Make AB or AK =
and GF = z. Then con-

.

ceive the Point D to move in the Spiral for an infinitely little Spree
Drf', and then through rfdraw the Semidiameter A/, and Cg parallel
and equal to it, draw gf perpendicular to gC, fo that G/ cuts gf
in/ and GF in P; produce GF to <p, fo that G p= < /, and draw
de perpendicular to AK, and produce it till it meets CD at I. Then
the contemporaneous Moments of BK, AD, and G<p, will be Kk, De
and Fa, which therefore may be call'd xo, yo, and zo.

48. Now it is AK : Ae (AD) :: kK : Je=yo, where I aflurne
x=i, as above. Alfo CG : GF :: de : eD = oyz, and there-
fore yz y f Befides CG : CF : : de : dD = oy x CF : : dD :
d\ = oy x CF?. Moreover, becaufe Z_PC<p (=Z-GG?) = LDAd,
and /.CPp (= LCdl = - eSQ -f- Red.) = L. ADJ, the Triangles
CP<p and AD</ are fimilar, and thence AD : Dd :: CP (CF) :
P<p = o x CFq. From whence take F<p t and there will remain PF
= oxCF^ ex z. Laftly, letting fall CH perpendicular to AD }
'tis PF : dl :: CG : eH or DH = LlHf . Or fubftituting i+zz

CFyx

for CFa, 'twill be DH =

y - ya!g

Here it may be obferved,

that

and IN FINITE SERIES. 69

that in this kind of Computations, I take thofe Quantities (AD and
Ae) for equal, the Ratio of which differs but infinitely little from
the Ratio of Equality.

49. Now from hence arifes the following Rule. The Relation
of x and y being exhibited by any Equation, find the Relation of
the Fluxions x and y, (by Prob. i.) and fubftitute i for x, and yz
for y. Then from the refulting Equation find again, (by Prob. i.)
the Relation between x, y, and z, and again fubftitute i for x.
The firft refult by due reduction will give y and z, and the latter
will eive z ; which being known, make = = DH, and raife

1 -f- Z.X.Z.

the Perpendicular HC, meeting the Perpendicular to the Spiral DC
before drawn in C, and C will be the Center of Curvature. Or
which comes to the fame thing, take CH : HD :: z : i, and
draw CD.

50. Ex. i. If the Equation be ax=y, (which will belong to

the Spiral at Archimedes,) then (by Prob. i.) ax=y y or (writing i
for x, and yz for_y,) 7 ^ =yz. And hence again (by Prob i.) o =
yz+y'z. Wherefore any Point D of the Spiral being given,, and
thence the length AD or y, there will be given z = - , and z=

( 3 - or) . Which being known, make i-t-zz-z :
H-iz :: DA (y) : DH. And i : z :: DH : CH.

And hence you will eafily deduce the following Conftrucftion.
Produce AB to Q, fo that AB : Arch BK :: Arch BK : BC^,
and make AB -+- AQ^: AQj: DA : DH :: a : HC.

51. Ex. 2. If ax 1 =_)" be the Equation that determines the Re-
lation between BK and AD; (by Prob. i.) you will have 2axx=.
3Jy,-*, or 2ax= 3y. Thence again 2a'x= ^zy s -+- gsiyy*. 'Tis
therefore z = ^7 , and z = ' a ~ 9 ~ z '- . Thefe being known, make

i-\-zz K : i-t-zz DA : DH. Or, the work being reduced
to a better form, make gxx 1 -f- 10 : gxx -f- 4 :: DA : DH.

52. Ex. 3. After the lame manner, if ax* bxy=yi determines

the Relation of BK to AD ; there will arife I"* ~ ' = z,, and

bxy -f- \$)*.

. g *~ ; *7~^;~ 9 *'- 8 = g. From which DH/ and thence the.
Point C, is determined as before.

5q
i-

yo I'he Method of FLUXIONS,

53. And thus you will eafily determine the Curvature of any-
other Spirals ; or invent Rules for any other kinds of Curves, in

4. And now I have finim'd the Problem ; but having made ufe
of a Method which is pretty different from the common ways of
operation, and as the Problem itfelf is of the number of thofe
which are not very frequent among Geometricians : For the illuflra-
tion and confirmation of the Solution here given, I mall not think
much to give a hint of another, which is more obvious, and has a
nearer relation to the ufual Methods of drawing Tangents. Thus if
from any Center, and with any Radius, a Circle be conceived to
be defcribed, which may cut any Curve in feveral Points ; if that
Circle be fuppos'd to be contracted, or enlarged, till two of the
Points of interfeclion coincide, it will there touch the Curve. And
befides, if its Center be fuppos'd to approach towards, or recede
from, the Point of Contadt, till the third Point of interfedtion fhall
meet with the former in the Point of Contadt ; then will that Circle
be cequicurved with the Curve in that Point of Contadt : In like man-
ner as I infmuated before, in the laft of the five Properties of the
Center of Curvature, by the help of each of which I affirm'd the
Problem might be folved in a different manner.

55. Therefore with Center C, and Radius CD, let a Circle be
defcribed, that cuts the Curve
in the Points d, D, and <f ;
and letting fall the Perpendi-
culars DB, db, <T/3, and CF,
to the Abfcifs AB ; call AB
= x, BD = y, AF = v,
FC=/,andDC=J. Then
BF=vx, and DB-f-FC
=_>>-{-/. The fum of the
Squares of thefe is equal to the
Square of DC ; that is, -D 1

2VX -+- X* -f- )" -h 2yt -+- /

=ss. If you would abbrevi-
ate this, make v* -f-/ 1 s 1 =f, (any Symbol at pleafure,) and it
becomes x 1 2vx -f-jy 1 -f- zfy -+- q 1 = o. After you have found

/, y, and q*, you will have s-=\/ r v 1 -+- 1* q*.

56. Now let any Equation be propofed for defining the Curve,
the quantity of whofe Curvature is to be found. By the help of
this Equation you may exterminate either of the Quantities x or y,

and

and INFINITE SERIES. 71

and there will arife an Equation, the Roots of which, (db, DB, <f/g,
&c. if y u exterminate x ; or A/>, AB, A/3, &c. if you exterminate
_y,) are "at the Points of interfedtion d, D, J\ &c. Wherefore fince
"three of them become equal, the Circle both touches the Curve,
and will alfo be of the fame degree of Curvature as the Curve, in
the point of Contact But they will become equal by comparing
the Equation with another fictitious Equation of the fame number
of Dimenfions, which has three equal Roots ; as Des Cartes has
fhew'd. Or more expeditioufly by multiplying its Terms twice by
an Arithmetical Progreflion.

57. EXAMPLE. Let the Equation be ax =yy, (which is an
Equation to the Parabola,) and exterminating x, (that is, fubftitu-

ting its Value - in the forego-
ing Equation,) there will arife * ^~y*_ -+ zty -f- ? a = o.
Three of whofe Roots ^ are to be _j_ yi
made equal. And for this purpofe 4*2 I o

I multiply the Terms twice by an * i o i

Arithmetical Progrellion, as you

fee done here j and there arifes -J 1 + 2 J X = -

Or u = + \a. Whence it is eafily infer'd, that BF = 2x -{-

\a, as before.

58. Wherefore any Point D of the Parabola being given, draw the
Perpendicular DP to the Curve, and in the Axis take PF = 2AB,
and erect FC Perpendicular to FA, meeting DP in C; then will C
be the Center of Curvity defired.

59. The fame may be perform'd in the Ellipfis and Hyperbola,
but the Calculation will be troublefome enough, and in other Curves
generally very tedious.

Of ^uefiions that have fome Affinity to the preceding

Problem.

60. From the Refolution of the preceding Problem fome others
may be perform'd ; fuch are,

I. To find the Point where the Curve has a given degree of Cur-
vature.

6 1. Thus in the Parabola, ax=yy, if the Point be required
whofe Radius of Curvature is of a given length f: From the Cen-
ter of Curvature, found as before, you will determine die Radius

72 7%e Method of FLUXIONS,

to be -~^ \/aa -+- ^.ax, which muft be made equal to f. Then
by reduction there arifes x = ^a -f- 1/^aff.
II. To find the Point of ReElitude.

62. I call that the Point of ReEiitude, in which the Radius of
Flexure becomes infinite, or its Center at an infinite diftance : Such
it is at the Vertex of the Parabola a*x=y*. And this fame Point
is commonly the Limit of contrary Flexure, whole Determination
I have exhibited before. But another Determination, and that not
inelegant, may be derived from this Problem. Which is, the
longer the Radius of Flexure is, fo much the lefs the Angle DCJ
(Fig.pag.6i.) becomes, and alfo the Moment <F/j fo that the
Fluxion of the Quantity z is diminim'd along with it, and by the
Infinitude of that Radius, altogether vanimes. Therefore find the
Fluxion z, and fuppofe it to become nothing.

63. As if we would determine the Limit of contrary Flexure in
the Parabola of the fecond kind, by the help of which Cartefius con-
ftructed Equations of fix Dimenfions ; the Equation to that Curve
is AT 3 bx* cdx -+- bed 4- dxy = o. And hence (by Prob. i .) arifes
3*** 2bxx - cdx -4- dxy -f- dxy = o. Now writing i for x t
and z for y, it becomes 3-v a zbx cd-{- dy -f- dxz=.o ; whence
again (by Prob. i,) 6xx zbx -+ dy + dxz +- dxz = o. Here again
writing i for x, & for y, and o for z, it becomes (>x zb -+- zdz
= o. And exterminating z, by putting b 3* for dz in the
Equation 3^,v zbx cd -+- dy -f- dxz = o, there will arife bx
cd-\$-dy = o ) ory=c-{-^; this being fubftituted in the room

of y in the Equation of the Curve, we fhall have x* +- bcd-=z. Q }
which will determine the Confine of contrary Flexure.

64. By a like Method you may determine
the Points of Rectitude, which do not come
between parts of contrary Flexure. As if the
Equation x* 4<w 3 -}- ba^x* b>y = o ex-
prefs'd the nature of a Curve ; you have firfl,
(byProb. i.)4^3 i2ax*-+- i2a*x faz=o,
and hence again 12X* 24^7^ -f- 12^' b*z
=o. Here fuppofe z = o, and by Reduc-
tion there will arife x = a. Wherefore take

ABi=fl, and erect the perpendicular BDj this will meet
Curve in the Point of Re&itude D, as was required.

III.

and IN FINITE SERIES. 73

III. To find the Point of infinite Flexure.

65. Find the Radius of Curvature, and fuppofe it to be nothing.
Thus to the Parabola of the fecond kind, whole Equation is A;* =

<7y a , that Radius will be CD = 4 " 6a q * \/q.ax-\- gxx , which be-
comes nothing when x = o.

IV. To determine the Point of the greatefl or leaft Flexure.

66. At thefe Points the Radius of Curvature becomes either the
greateft or leaft. Wherefore the Center of Curvature, at that mo-
ment of Time, neither moves towards the point of Contact, nor
the contrary way, but is intirely at reft. Therefore let the Fluxion
of the Radius CD be found; or more ex-

peditioufly, let the Fluxion of either of the
Lines BH or AK be found, and let it be

67. As if the Queftion were propofed con-
cerning the Parabola of the fecond kind
x l = o*y ; firft to determine the Center of

Curvature you will find DH = aa , 9X ->

ox

and therefore BH = 6 ^' ?AV ; make BH

Hence (by Prob. i.) "- - _j_ ^y == t}. But now fuppofe -y, or the
Fluxion of BH, to be nothing ; and belides, lince by Hypothecs
A- "' = rf 1 .y, and thence (by Prob. i.) yxx 1 =<?*.}', putting x= i, fub-
ftitute ^ for v, and there will arife 4.5x4=0+. Take therefore

^

AB ==a y'^j- =<7 x45| , and raifrng the perpendicular BD, it will"
meet the Curve in the Point of the greateft Curvature. Or, which
is the fame thing, make AB : BD : : 3^/5 : I.

68. After the fame manner the Hyperbola of the lecond kind
reprefented by the Equation xy l = 3 , will be
moft inflected in the points D and d, which you
may determine by taking in the Abfcifs AQ== r,
and erecting the Perpendicular QP_=z= v /5, and
Q^/> equal to it on the other fide. Then draw-
ing AP and A/>, they will meet the Curve in the
points D and d required.

V,

74 The Method of FLUXIONS,

V. To determine the Locus of the Center of Curvature, or to de-
fcribe the Curve, in which tbaf* Center is always found,

69. We have already {hewn, that the Center of Curvature of the
Trochoid is always found in another Trochoid. And thus the Cen-
ter of Curvature of the Parabola is found in another Parabola of
the fecond kind, reprefented by the Equation axx=y*, as will
eafily appear from Calculation.

VI. Light falling upon any Curve, to find its Focus, or the Con-
courje of the Rays that are ref rafted at any of its Points.

70. Find the Curvature at that Point of the Curve, and defcribe
a Circle from the Center, and with-the Radius of Curvature. Then
find the Concourfe of the Rays, when they are refracted by a Cir-
cle about that Point : For the fame is the Concourfe of the refrac-
ted Rays in the propofed Curve.

71. To thefe may be added a particular Invention of the Curva-
ture at the Vertices of Curves, where they cut their Abfcifles at right
Angles. For the Point in which the Perpendicular to the Curve,
meeting with the Abfcifs, cuts it ultimately, is the Center of its
Curvature. So that having the relation between the Abfcifs x,
and the rectangular Ordinate y, and thence (by Prob. i.) the rela-
tion between the Fluxions x and y ; the Value yy, if you fubftitute
r for x into it, and make y = o, will be the Radius of Curva-
ture.

72. Thus in the Ellipfis ax xX=yy, it is -* " = yy ;

which Value of yy, if we fuppofe^=o, and confequently x = />,
^writing i for x, becomes a for the Radius of Curvature. And fo
at the Vertices of the Hyperbola and Parabola, the Radius of Cur-
vature will be always half of the Latus rectum.

73-

and INFINITE SERIES. 75

73. And in like manner for the Conchoid, defined by the Equation
zbx xx = yy, the Value of yy t (found by

zicc + cc
~T bb

Prob. i.) will be ^ ""* IT ^ "~~ * Now fuppofing y = o,

and thence # = c or f, we mail have zb c, or

2(5 -f- f, for the Radius of Curvature. Therefore make AE : EG ::
EG : EC, and he : eG :: eG : ec, and you will have the Centers
of Curvature C and c, at the Vertices E and e of the Conjugate
Conchoids.

PROB. VI.

To determine the Quality of the Curvature, at a given

Point of any Curve.

I. By the Quality of Curvature I mean its Form, as it is more
or lefs inequable, or as it is varied more or lefs, in its progrefs thro'
different parts of the Curve. So if it were demanded, what is the
Quality of the Curvature of the Circle ? it might be anfwer'd, that
it is uniform, or invariable.
And thus if it were demand-
ed, what is the Quality of the
Curvature of the Spiral, which
is described by the motion of
the point D, proceeding from
A in AD with an accelerated
velocity, while the right
Line AK moves with an uni-
ter A ; the acceleration of

L 2 which

76 7&? Method of FLUXIONS,

which Velocity is fuch, that the right Line AD has the fame ratio
to the Arch BK, defcribed from a given point B, as a Number has
to its Logarithm : I fay, if it be afk'd, What is the Quality of the
Curvature of this Spiral 1 It may be anfwer'd, that it is uniformly
varied, or that it is equably inequable. And thus other Curves, in
their feveral Points, may be denominated inequably inequable, ac-
cording to the variation of their Curvature.

2. Therefore the Inequability or Variation of Curvature is re-
quired at any Point of a Curve. Concerning which it may be ob-
ferved,

3. I. That at Points placed alike in like Curves, there is a like
Inequability or Variation of Curvature.

4. II. And that the Moments of the Radii of Curvature, at thofe
Points, are proportional to the contemporaneous Moments of the
Curves, and the Fluxions to the Fluxions.

5. III. And therefore, that where thofe Fluxions are not propor-
tional, the Inequability of the Curvature will be unlike. For
there will be a greater Inequability, where the Ratio of the Fluxion
of the Radius of Curvature to the Fluxion of the Curve is
greater. And therefore that ratio of the Fluxions may not impro-
perly be call'd the Index of the Inequability or of the Variation
of Curvature.

6. At the points D and d, infinitely near to each other, in the
Curve AD^, let there be drawn the

Radii of Curvature DC and dc , and D</
being the Moment of the Curve, Cc
will be the contemporaneous Moment

of the Radius of Curvature, and -^
will be the Index of the Inequability of
Curvature. For the Inequability may
be call'd fuch and fo great, as the quan-
tity of that ratio 7^ mews it to be :

j Ja

Or the Curvature may be faid to be fo
much the more unlike to the uniform
Curvature of a Circle.

7. Now letting fall the perpendicular Ordinates DB and db t
to any line AB meeting DC in P j make AB = #, BD = y\

and thence B& = xo, it will be Cc = vo; and
-1 T^ = , making x = i.

Wherefore

>

II

and IN FINITE SERIES. 77

Wherefore the relation between x and y being exhibited by any
Equation, and thence, (according to Prob. 4. and 5.) the Perpendicu-
lar DP or /, being found, and the Radius of Curvature i 1 , and the

Fluxion <y of that Radius, (by Prob. i.) the Index '^ of the Inequabi-
lity of Curvature will be given alfo.

8. Ex. i. Let the Equation to the Parabola tax = vy be given ;
then (by Prob. 4.) BP = a, and therefore DP= ^ a -\-\y=^t.
Alfo (by Prob. 5.) BF = a -+- 2X, and BP : DP : : BF : "i)C =

- =1;. Now the Equations 2ax =}'}', aa-\-yy=tt, -and

t-~ =v, (by Prob. i.) give 2ax = 2jvy, and zyy = ztt, and
a t + Z fx + 2fx __ ^ Which being reduced to order, and putting

.v = i, there will arife y = -, / = r^ = ) - f > an< ^ v= -

And thus y, t, and v being found, there will be had ^ v the Index

of the Inequability of Curvature.

9. As if in Numbers it were determin'd, that^=j a or 2#==n > ,

and x= 4 ; then y (==

+ 7 + "= 3v /2. So that

j^= 3, which therefore is the Index of Inequability.

10. But if it were determin'd, that A: =2, then y = 2, ^'=T>
/ = v/5, f = </, and -17 = 3^/5. So that ^-=) 6 will be here

the Index of Inequability.

11. Wherefore the Inequability of Curvature at the Point of the
Curve, from whence an Ordinate, equal to the Latus reftum of the
Parabola, being drawn perpendicular to the Axis, will-be double to the
Inequability at that Point, from whence the Ordinate fo drawn is half
the Latus rectum ; that is, the Curvature at the firft Point is as unlike a-
gain to the Curvature of the Circle, as the Curvature at the fecond Point.

12. Ex. 2. Let the Equation be zax bxx-=.yy, and (by Prob. 4.)
it will be a &v=BP, and thence tf=(aa 2a6x-lrb

=) na byy - yy. Alfo (by Prob. 5.) it is DH =}' -{
where, if for yy byy you fubftitute // aa, there ariies DH =
Tis alfo BD : DP :: DH : DC= - =v. Now (hv Prob.i.)

f.ll U 1

the Equations zaxbxx^yy, aa byy-\-y\-=^t!, and

give

7 8 77je Method of FLUXIONS,

give a bx =}')', and yy byy = /'/, and ~ = v. And thus v

being found, the Index ^ of the Inequability of Curvature, will
aJib be known.

13. Thus in the Ellipfis 2X 3 ATA:
=}'}', where it is a = r, and b=-.^ ;
if we make x=-, then r- L v

* * S " a 3 x ~~ ~"

A

b P

Jl

o

V

and therefore ; v =|, which is the In-

dex of the Inequability of Curvature.

Hence it appears, that the Curvature of

this Ellipfis, at the Point D here af-

fign'd, is by two times left inequable,

(or 'by two times more like to the Cur-

vature of the Circle,) than the Curva-

ture of the Parabola, at that Point of

its Curve, from whence an Ordinate let fall upon the Axis is equaj

to half the Latus rectum.

14. If we have a mind to compare the Conclufions derived in

thefe Examples, in the Parabola 2ax=yy arifes (~ = > )^ v for the

V ' s a

Index of Inequability j and in the Ellipfis zax bxx=yy, arifes
(^7- =J - - x BP j and fo in the Hyperbola 2ax -+- bxx =yy,
the analogy being obferved, there arifes the Index ("2- ^ y+3b

\. t J &&

x BP. Whence it is evident, that at the different Points of any
Conic Section conn'der'd apart, the Inequability of Curvature is as
the Rectangle BD x BP. And that, at the feveral Points of the Pa-
raboh, it is as the Ordinate BD.

15. Now as the Parabola is the moft fimple Figure of thofe that
are curved with inequable Curvature, and as the Inequability of its
Curvature is fo eafily determined, (for its Index is 6x^ll^i,) t h er e-

.. .

fore the Curvatures of other Curves may not improperly be compared
to the Curvature of this.

1 6. As if it were inquired, what may be the Curvature of the
Ellipfis 2X \$xx=yy, at that Point of the Perimeter which is
determined by affuming x = : Becaufe its Index is 4., as before,
it might be anfwer'd, that it is like the Curvature of the Parabola

6.v

and IN FINITE SERIES.

79

6.v =)')', at that Point of the Curve, between which and the Axis
the perpendicular Odinate is equal to |.

17. Thus, as the Fluxion of the Spiral ADE is to the Fluxion
of the Subtenfe AD, in a certain given Ratio,
fuppofe as d to e; on its concave fide erect

y dd ee

and P will be the Center of Curvature, and

A P t

or r=? will be the Index of Inequa-

1J y a.i ee

bility. So that this Spiral has every where
its Curvature alike inequable, as the Parabola
6x = yy has in that Point of its Curve, from
whence to its Abfcifs a perpendicular Ordi-
nate is let fall, which is equal to the

1 8. And thus the Index of Inequability at any Point D of the

AB

Trochoid, (fee Fig. in Art. 29. pag. 64.) is found to be . Where-

fore its Curvature at the fame Point D is as inequable, or as unlike
to that of a Circle, as the Curvature of any Parabola ax - yy is at

AB

the Point where the Ordinate is ^a x -^

19. And from thefe Confiderations the Senfe of the Problem, as
I conceive, mufl be plain enough; which being well underftood, it
will not be difficult for any one, who obferves the Series of the
things above deliver'd, to furnifh himfelf with more Examples, and
to contrive many other Methods of operation, as occafion may re-
quire. So that he will be able to manage Problems of a like nature,
(where he is not difcouraged by tedious and perplex Calculations,)
with little or no difficulty. Such are thefe following ;

I. To find the Point of any Curve, where there is either no Inequabi-
lity of Curvature, or infinite, or tie grcatej?, or the leajl.

20. Thus at the Vertices of the Conic Sections, there is no In-
equability of Curvature; at the Cuf] id of the 1 rcchoid it is infi-
nite ; and it is greatefl at thofe Points of the Ellif.fis, where the
Rectangle BD x BP is greatefl, that is, where the Diagor.al-Lines
of the circumfcribed Parallelogram cut the Elliriis, whofe Sides
touch it in their principal Vertices.

II. 1o determine a Curve of fame definite Species, l'nfp r je a C r .n:c
Section, liioje Curvature at any Point may be cqiu:l and Jiitiilar to the
Curvature of any other Curve, at a given P./:./ of it.

8 o "The Method of FLUXIONS,

III. To iL-termine a Conk Sctfion, at any Point of which, the Cur-
ri?//i7V and Pojition of the tangent, (in refpeSt of the AxisJ) may be like
to the Curvature and Pofition of the Tangent, at a Point ajfigrid of
any other Curir.

21. The ufe of which Problem is this, that inftead of Ellipfes of
the fecond kind, whofe Properties of refradling Light are explain'd
by Des Cartes in his Geometry, Conic Sections may be fubftituted,
which mall perform the fame thing, very nearly, as to their Re-
fractions. And the fame may be underfhood of other Curves.

P R O B. VII.

To find as many Curves as you pleafe y ivbofe Areas may
be exhibited by finite Equations.

I. Let AB be the Abfcifs of a Curve, at whofe Vertex A let the
perpendicular AC = i be raifed, and let CE be D

drawn parallel to AB. Let alfo DB be a rectan-
gular Ordinate, meeting the right Line CE in E,
and the Curve AD in D. And conceive thefe
Areas ACEB and ADB to be generated by the
right Lines BE and BD, as they move along the
Line AB, Then their Increments or Fluxions will

be always as the defcribing Lines BE and BD. Wherefore make
the Parallelogram ACEB, or AB x i, =.v, and the Area of the
Curve ADB call z. And the Fluxions x and z will be as BE and

BD; fo that making x = i = BE, then z = BD.

2. Now if any Equation be a/Turned at pleafure, for determining
the relation of z and x, from thence, (by Prob. i.) may z be de-
rived. And thus there will be two Equations, the 'latter of which
will determine the Curve, and the former its Area.

EXAMPLES.

3. Aflume ##:=:, and thence (by Prob. i.) 2xx=s } or 2x=c:,
becaufe x=, i.

4. Aflame ^=z, and thence will arife =;s ? an Equation
to the Parabola.

5. A flume ax* =zz, or a' f x*=z, and there will arife \a^x' =^^,
or ^(?x = zz, an Equation again to the Parabola.

i 6.

and INFINITE SERIES. 81

6. Affume a 6 x~ 1 =zz,or a*x-' =z, and there arifes a*x f * = z,
or a'' -j-2xx = o. Here the negative Value of z only infinuates,