Isaac Newton.

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

. (page 7 of 30)
Font size about the Center A, let the Point D be con-
ceived to move any how, fo as to defcribe the
Spiral ADE. And fuppofe ~Dd to be an in-
finitely little part of the Curve thro' which
cD and Gg will be contemporaneous Moments
of the right Line AD and of the Periphery
BG. Therefore draw Af parallel to cd, that
is, perpendicular to AD, and let the Tangent
DT meet it in T ; then it will be cD : cd : :
AD : AT. Alfo let Gt be parallel to the Tangent DT, and it
will be cd : Gg :: (Ad or AD : AG ::) AT : At.

50. Therefore any Equation being propofed, by which the Re-
lation is exprefs'd between BG and AD ; find the Relation of their
Fluxions by Prob. i. and takeAi? in the fame Ratio to AD: And then
Gt will be parallel to the Tangent.

51. Ex. i. Calling EG = x, and AD=^, let their Relation be
A: 3 ax 1 -f- axy jy 5 = o, and by Prob. i. 3^* zax-\- ay : 3^*
ax : : (y : x : :) AD : At. The Point / being thus found, draw
Gt, and DT parallel to it, which will touch the Curve.

52. Ex. 2. If 'tis y =y> (which is the Equation to the Spiral

of Archimedes,} 'twill be j = y, and therefore a : b : : (y : x : :)

AD : At. Wherefore by the way, if TA be produced to P,
that it may be AP : AB :: a : b y PD will be perpendicular to
the Curve.

53. Ex. 3. If xx = by, then 2XX = by, and 2x : b :: AD :
A. And thus Tangents may be eafily drawn to any Spirals what-
ever.

Eighth

and INFINITE SERIES. 57

Eighth Manner : For Quad ratr ices.

CA. Now if the Curve be fuch, that any Line AGD, being drawn
from the Center A, may meet the Circular Arch inG, and the Curve in
D; and if the Relation between the
Arch BG, and the right Line DH,
which is an Ordinate to the Bafe
or Abfcifs AH in a given Angle,
be determin'd by any Equation
whatever : Conceive the Point D to
move in the Curve for an infinite-
ly {mail Interval to d, and the Pa-
rallelogram dhHk being compleat- Jf
ed, produce Ad to c, fo that

Ac = AD ; then Gg and D/' will be contemporaneous Moments of
the Arch BG and of the Ordinate DH. Now produce Dd ftrait
on to T, where it may meet with AB, and from thence let fall
the Perpendicular TF on DcF. Then the Trapezia Dkdc and DHTF
will be fimilar; and therefore D/fc : DC :: DH : DF. And befides
if Gf be raifed perpendicular to AG, and meets AF in f; becaufe
of the Parallels DF and Gf, it will be DC : Gg :: DF : Gf. There-
fore ex aquo, 'tis D : G^ : : DH : Gf, that is, as the Moments or
Fluxions of the Lines DH and BG.

55. Therefore by the Equation which exprefies the Relation of
BG to DH, find the Relation of the Fluxions (by Prob. i.) and in-
that Ratio take Gf, the Tangent of the Circle BG, to DH. Draw
DF parallel to Gf, which may meet A/* produced in F. And at
F creel the perpendicular FT, meeting AB in T; and the right
Line DT being drawn, will touch the Quadratrix.

56. Ex. i. Making EG = x, and DH=;', let it be xx = fy;
then (by Prob. i.)2xx = by. Therefore 2.x : b :: (y : x ::) DH :
GJ; and the Pointy being found, the reft will be determin'd as above.

But perhaps this Rule may be thus made fomething neater :
Make x :y :: AB : AL. Then AL : AD :: AD : AT, and then
DT will touch the Curve. For becaufe of equal Triangles AFD and
ATD, 'tis AD x DF= AT x DH, and therefore AT : AD : : (DF or

JB x Gf : DH or 1 G/::) AD : f- AG or) AL.

57. Ex.2. Let x=y, (which is the Equation to the Quadratrix
of the Ancients,) then #=v. Therefore AB : AD :: AD : AT.

I 8.

58 *fhe Method ^FLUXIONS,

58. Ex. 3. Let axx=y*, then zaxx=sMy*. Therefore make
3;-* : zax : : (x : y : :) AB : AL. Then AL : AD : : AD : AT. And

thus you may determine expeditioufly the Tangents of any other

Ninth Manner.

59. Laftly, if ABF be any given Curve, which is touch'd by the
right Line Bt ; and a part BD of

the right Line BC, (being an Or-
dinate in any given Angle to the
Abfcifs AC,) intercepted between
this and another Curve DE, has a
Relation to the portion of the
Curve AB, which is exprefs'd by
any Equation: You may draw a
Tangent DT to the other Curve,

by taking (in the Tangent of this ^ ^ <f-

Curve,) BT in the fame Ratio to

BD, as the Fluxion of the Curve AB hath to the Fluxion of the

right Line BD.

60. Ex. i. Calling AB ==x, and BD =y- t let it be ax==yy, and
therefore ax = zyy. Then a : zy : : (y : x : :) BD : BT.

6j. Ex.2. Let ^#==7, (the Equation to the Trochoid, if ABF
be a Circle,) then fX=y t and a : b :: BD : BT.

62. And with the fame eafe may Tangents be drawn, when the
Relation of BD to AC, or toBC, is exprefs'd by any Equation; or
when the Curves are refer 'd to right Lines, or to any other Curves,
after any other manner whatever.

63. There are alfo many other Problems, whofe Solutions are
to be derived from the fame Principles ; fuch as thefe following.

I. To find a Point of a Curve, where the Tangent is parallel to the
Abfcife, or to any other right Line given in pofition ; or is perpendicular
to it, or inclined to it in any given Angle.

II. To find the Point where the Tangent is moft or leajl inclined to
the Abfcifs, or to any other right Line given in 'pofition. That is, to find
the confine of contrary Flexure. Of this I have already given a Spe-
cimen, in the Conchoid.

III. From any given Point without the Perimeter of a Curve, to
draw a right Line, which with the Perimeter may make an Angle of

Contact.

and IN FINITE SERIES. 59

Contaft, or a right Angle, or any other given Angle, that is, from
a given Point, to draw 'Tangents, or Perpendiculars^ or right Lines
that Jhall have any other Inclination to a Curve-line.

IV. From any given Point within a Parabola, to draw a right
Line, which may make with the Perimeter the greateji or leaft Angle
poj/ible. And Jb of all Curves whatever.

V. To draw a right Line which may touch two Curves given in
pojition, or the fame Curve in two Points, when that can be done.

VI. To draw any Curve with given Conditions, which may touch
another Curve given in pojition, in a given Point.

VII. To determine the RefraSlion of any Ray of Light, that falls
upon any Curve Superficies.

The Refolution of thefe, or of any other the like Problems, will
not be fo difficult, abating the tedioufnefs of Computation, as that
there is any occalion to dwell upon them here : And I imagine if
may be more agreeable to Geometricians barely to have mention 'd
them.

; :
P R O B. V.

At any given Point of a given Curve^ to find the
Quantity of Curvature.

1. There are few Problems concerning Curves more elegant than
this, or that give a greater Infight into their nature. In order to c its
Refolution, I mufl: premife thefe following general Confederations.

2. L The fame Circle has every where trie fame Curvature, and
in different Circles it is reciprocally proportional to their Diameters.
If the Diameter of any Circle is as little again as the Diameter of
another, the Curvature of its Periphery will be as great again. If
the Diameter be one-third of the other, the Curvature will be thrice
as much, &c.

3. II. If a Circle touches any Curve on its concave fide, in any
given Point, and if it be of fuch magnitude, that no other tangent
Circle can be interleribed in the Angles of Contact near that Point ;
that Circle will be of the lame Curvature as the Curve is of, in that
Point of Contact. For the Circle that conies between the Curve
and another Circle at the Point of Contact, varies lefs from the
Curve, and makes a nearer approach to its Curvature, than that
other Circle does. And therefore that Circle approaches nea'-eil to its

I 2 Curvature,

60 *fbe Method of FLUXIONS,

Curvature, between which and the Curve no other Circle can in-
tervene.

4. III. Therefore the Center of Curvature to any Point of a
Curve, is the Center of a Circle equally curved. And thus the Ra-
dius or Semidiameter of Curvature is part of the Perpendicular
to the Curve, which is terminated at that Center.

5. IV. And the proportion of Curvature at different Points will
be known from the proportion of Curvature of aequi-curve Circles,
or from the reciprocal proportion of the Radii of Curvature.

6. Therefore the Problem is reduced to this, that the Radius, or
Center of Curvature may be found.

7. Imagine therefore that at three Points of the Curve <f , D, and d,
Peipendkulars are drawn, of which thofe that are

at D and ^ meet in H, and thofe that are at D
and d meet in h : And the Point D being in the /
middle, if there is a greater Curyity at the part Dj^
than at DJ, then DH will be lefs than db. But
by how much the Perpendiculars /H and dh are
nearer the intermediate Perpendicular, fo much the
lefs will the diftance be of the Points H and h :
And at laft when the Perpendiculars meet, thofe
Points will coincide. Let them coincide in the Point
C, then will C be the Center of Curvature, at the
Point D of the Curve, on which the Perpendicu-
lars ftand ; which is manifeft of itfelf.

8. But there are feveral Symptoms or Properties of this Point C',
which may be of ufe to its determination.

9. I. That it is the Concourfe of Perpendiculars that are on each
lide at an infinitely little diftance from DC.

10. II. That the Interfeftions of Perpendiculars, at any little finite
diftance on each fide, are feparated and divided by it ; fo that thofe
which are on the more curved fide D,f fooner meet at H, and thofe
which are on the other iefs curved fide -Dd meet more remotely
at h.

11. III. If DC be conceived to move, while it infifts perpendi-
cularly on the Curve, that point of it C, (if you except the motion
of approaching to or receding from the Point of Influence C,) will
be leaft moved, but will be as it were the Center of Motion.

12. IV. If a Circle be defcribed with the Center C, and the di-
ftance DC, no other Circle can be defcribed, that can lie between
at the Contact.

and INFINITE SERIES.

61

n. V. Laftly, if the Center II or b of any other touching Circle
approaches by degrees to C the Center of this, till at la it it co-
incides with 'it ; then any of the points in which that Circle mall
cut the Curve, will coincide with the point of Contact D.

14. And each of thefe Properties may fupply the means of folving
the Problem different ways : But we fliall here make choice of the
firlt, as being the moit fimple.

15. At any Point D of the Curve let DT be a Tangent, DC a
Perpendicular, and C the Center of Curvature, as before. And let
AB be the Abfcifs, to which let DB be apply 'd at right Angles,
and which DC meets in P. Draw

DG parallel to AB, and CG per-
pendicular to it, in which take
Cg of any given Magnitude, and
draw gb perpendicular to it, which
meets DC in <T. Then it will be
Cg : gf : : (TB : BD : :) the Fluxion
of the Ablcifs, to the Fluxion of
the Ordinate. Likewife imagine
the Point D to move in the Curve
an infinitely little diftance Dd, and
drawing de perpendicular to DG, and Cd perpendicular to the Curve,
let Cd meet DG in F, and \$g in/ Then will De be the Momen-
tum of the Abfcifs, de the Momentum of the Ordinate, and J/ the
contemporaneous Momentum of the right Line g. Therefore DF
-De^.^t . Having therefore the Ratio's of thefe Moments, or,

LJC ' *

which is the fame thing, of their generating Fluxions, you will have
the Ratio of CG to the given Line C^, (which is the fame as that of
DF to Sf,) and thence the Point C will be determined.

16. Therefore let AB = x, BD =y, Cg- = i, and g = z ;

then it will be i : z : : x : y, or z = r- . Now let the Mo-

X

mentum S-f of z be zxo, (that is, the Product of the Velocity

and of an infinitely fmall Quantity o,} and therefore the Momenta

Dt'==xxo, de=yx.o, and thence DF = .\o -f- . Therefore

X

'tisQ-(r) : CG :: (Jf : DF ::) zo : xo + ^ . That is, CG=

xx \y

J 7-

62 7%e Method of FLUXIONS,

17. And whereas we are at liberty to afcribe whatever Velocity
we pleafe to the Fluxion of the Abfcifs x, (to which, as to an
equable Fluxion, the reft may be referr'd j) make x = i, and
then y = z, and CG = '-^ . And thence DG = z -^. } and

J ' '

18. Therefore any Equation being propofed, in which the Rela-
tion of BD to AB is exprefs'd for denning the Curve ; firft find
the Relation betwixt x and y t by Prob. r. and at the fame time fub-
ftitute i for ,v, and z for y. Then from the Equation that arifes,
by the fame Prob. i. find the Relation between #, y, and z, and at
the fame time fubftitute i for x, and z for y, as before. And thus
by the former operation you will obtain the Value of z, and by
the latter you will have the Value of z ; which being obtain'd, pro-
duce DB to H, towards the concave part of the Curve, that it

may be DH = - - , and draw HC parallel to AB, and meet-

ing the Perpendicular DC in C j then will C be the Center of Cur-
vature at the Point D of the Curve. Or fince it is i -|- r.y. - 7

PT TM-T PT Tk/-> DP

make DH== ' or

z

19. Ex. i. Thus the Equation ax^-hx* y 1 =;o being pro-
pofed, (which is an Equation to the Hyperbola whofe Latus redtum

is a, and Tranfverfum 2 ; ) there will arife (by Prob. i.) a +. zbx
2zy o, (writing l for x, and z for y in the refulting Equation,
which otherwife would have been ax -+ 2&xx zyy = o ;) and
hence again there arifes zb 2zz 2zy = o, (i and z being again

wrote for ,v and y.) By the firft we have z = C L^L } an( j b y tne

i ^^
latter z = Therefore any Point D of the Curve being given,

and confequently xand y, from thence z and z will be given, which
being known, make 7 = GC or DH, and draw HC.

Z

20. As if definitely you make = 3, and b=i, fo that 3#-f-
xx=yy may be the condition of the Hyperbola. And if you
aliume x=i, ^11^ = 2, z=, z= T 9 T , and DH= gL.
li being found, raife the Perpendicular HC meeting the Perpendi-

cular

and IN FINITE SERIES. 63

cular DC before drawn ; or, which is the fame thing, make HD :
HC :: (i : z ::) i : . Then draw DC the Radius of Curva-
ture.

21. When you think the Computation will not be too perplex, you

may fabfHtute the indefinite Values of z and z into - , the

Value of CG. Thus in the prefent Example, by a due Reduction
you will have DH =y -j- 4 ' S ^ r * . Yet the Value of DH by

Calculation conies out negative, as may be feen in the numeral Ex-
ample. But this only fhews, that DH mufl be taken towards B ;
for if it had come out affirmative, it ought to have been drawn the
contrary way.

22. COROL. Hence let the Sign prefixt to the Symbol -\-b be
changed, that it may be ax -bxx yy=zo, (an Equation to the

Ellipfis,) then DH=; - f- ilLll^: .

23. But fuppofing b=. o, that the Equation may become ax
yy - o, (an Equation to the Parabola,) then DH = y -f- ~ ; and

thence DG = \a -f- 2X.

24. From thefe feveral Exprefilons it may eafily be concluded,
that the Radius of Curvature of any Conick Seftion is always

aa

25. Ex. 2. If x*=ay* xy- be propofed, (which is the Equa-
tion to the CiiToid of Diodes,") by Prob. i. it will be firft T > x l =.2azy

zxzy y-t and then 6x = 2azy-+-2azz -2zy zxzy 2xzz

1 3*x -4- yy , T.X a%z -4- 2cv+ *~~ n-.!

2Z\ : So that z= - 3-^. and z= - - ^ . There-

J zay 2.vy' ay xj

fore any Point of the Ciflbid being given, and thence .v and y,
there will be given alfo & and z, ; which being known, make -

K

= CG. _ _

26. Ex. 3. If b-jf-y^/cc yy =.vy were given, (which is the
Equation to the Conchoid, inpag.48;) make \/cc y\=zv, and
there will arife hi) -+- yv = xy. Now the firft of thele, (cc _vv
= vv,) will give (by Prob. i.) 2yz = 2vv, (writing z for v ;)
and the latter will give l>v -+-yv + zv =y -{- xz. And from thefe
Equations rightly difpofed v and z will be determined. But that z
may alfo be found; out of the laft Equation exterminate the Fluxion

i>, by fubilituting ^ , and there will arife 7 -I- ~"^

Method of FLUXIONS,

= y -f- xz, an Equation that comprehends the flowing Quantities,
without any of their Fluxions, as the Refolution of the firft Pro-
blem requires. Hence therefore by Prob. i. we mall have

^2* byz Ijzv 2)zs )? \vzv

" +- ZV = 2Z +- XZ.

This Equation being reduced, and difpofed in order, will give z.
But when z and z are known, make ' + zz = CG.

27. If we had divided the laft Equation but one by z, then
by Prob. i . we mould have had - -f- ^ -f- - - -f. -i; =

2 ^, ; which would have been a more fimple Equation than the

former, for determining z.

28. I have given this Example, that it may appear, how the ope-
ration is to be perform'd in furd Equations: But the Curvature of
the Conchoid may be thus found a fhorter way. The parts of the
Equation b -\-y ^/cc v\' = xy being fquared, and divided by yy,
there arifes ~ -f. *" ^ 2by y* = x*, and thence by Prob. i.

or

x

...

And hence again by Prob. i. ^^ -f- ~ z 1 m By

*^ J y4 y/9 z, zz

the firft refult z is determined, and z by the latter.

29. Ex. 4. Let ADF be a Trochoid [or Cycloid] belonging to
the Circle ALE, whofe Diameter is AE j and making the Ordinate
BD to cut the
Circle in L,

AB=x, BD

and the Arch
AL=/, and
the Fluxion of
the fame Arch
= /. And
firfl (drawing
the Semidia-
meterPL,)the
Fluxion of the

Bafe or Abfcifs AB will be to the Fluxion of the Arch AL, as BL

to

and INFINITE SERIES. 65

to PL ; that is, A* or I : / : : v : ~a. And therefore ^ = /. Then
from the nature of the Circle ax xx = -y-y, and therefore by
Prob. i. a 2X = 2-yy, or -~~* = v.

30. Moreover from the nature of the Trochoid, 'tis LD= Arch
AL, and therefore -y -M =y. And thence (by Prob. i ) v -h / =z.
Laftly, inftead of the Fluxions v and / let their Values be lubfti-
tuted, and there will arife a -^ =z. Whence (by Prob. i.) is de-
rived - -f- - = z. And thefe being found, make

*ut/ w *v z,

== DH, and raife the perpendicular HC.

31. COR. i. Now it follows from hence, that DH = 2BL, and
CH 2BE, or that EF bifeds the radius of Curvature CO in N.
And this will appear by fubftituting the values of z and z now
found, in the Equation ' . **= DH, and by a proper reduction of

the refult.

32. COR, 2. Hence the Curve FCK, defcribed indefinitely by the
Center of Curvature of ADF, is another Trochoid equal to this,
whofe Vertices at I and F adjoin to the Cufpids of this. For let
the Circle FA, equal and alike pofited to ALE, be defcribed, and
let C/3 be drawn parallel to EF, meeting the Circle in A : Then
will Arch FA = (Arch EL= NF =) CA.

33. COR. 3. The right Line CD, which is at right Angles to the
Trochoid IAF, will touch the Trochoid IKF in the point C.

34. COR. 4. Hence (in the in verted Trochoids,) if at theCufpid K
of the upper Trochoid, a Weight be hung by a Thread at the di-
ilance KA or 2EA, and while the Weight vibrates, the Thread be
fuppos'd to apply itfelf to the parts of the Trcchoid KF and KI,
which refift it on each fide, that it may not be extended into a
right Line, but compel it (as it departs from the Perpendicular) to
be by degrees inflected above, into the Figure of the Trochoid,
while the lower part CD, from the loweft Point of Contact, ftill
remains a right Line : The Weight will move in the Perimeter of
the lower Trochoid, becaufe the Thread CD will always be perpen-
dicular to it.

35. COR. 5. Therefore the whole Length of the Thread KA is
equal to the Perimeter of the Trochoid KCF, and its part CD is
equal to the part of the Perimeter CF.

K 36.

66 The Method of FLUXIONS,

36. COR. 6. Since the Thread by its ofcillating Motion revolves
about the moveable Point C, as a Center ; the Superficies through
which the whole Line CD continually pafles, will be to the Super-
ficies through whichjthe part CN above the right Line IF pafles at
the fame time, as CD* to CN*, that is, as 4 to i. Therefore the
Area CFN is a fourth part of the Area CFD ; and the Area KCNE
is a fourth part of the Area AKCD.

37. COR. 7. Alfo fince the fubtenfe EL is equal and parallel to
CN, and is converted about the immoveable Center E, juft as CN
moves about the moveable Center C ; the Superficies will be equal
through which they pafs in the fame time, that is, the Area CFN,
and the Segment of the Circle EL. And thence the Area NFD
will be the triple of that Segment, and the whole area EADF will
be the triple of the Semicircle.

38. COR. 8. When the Weight D arrives at the point F, the
KCF, and the Radius of Curvature will there be nothing. Where-
fore the Trochoid IAF is more curved, at its Cufpid F, than any
Circle ; and makes an Angle of Contact, with the Tangent /3F produ-
ced, infinitely greater than a Circle can make with a right Line.

39. But there are Angles of Contact that are infinitely greater
than Trochoidal ones, and others infinitely greater than thefe, and
fo on in infinitum ; and yet the greateft of them all are infinitely
lefs than right-lined Angles. Thus xx = ay, x 3 = y, x* ==ry 5 ,
x* = dy+, &cc. denote a Series of Curves, of which every fucceeding
one makes an Angle of Contact with its Abfciis, which is infinitely
greater than the preceding can make with the fame Abfcifs. And the
Angle of Contact which the firft xx=ay makes, is of the fame kind
with Circular ones; and that which the fecond x*-=by z makes, is of
the fame kind with Trochoidals. And tho' the Angles of the fucceed-
in Curves do always infinitely exceed the Angles of the preceding, yet
they can never arrive at the magnitude of a right-lined Angle.

40. After the fame manner x ==y, xx=ay, x*=l> 1 y, x4 = c*y,
&c. denote a Series of Lines, of which the Angles of the fubfequents,
made with their Abfcifs's at the Vertices, are always infinitely lefs
than the Angles of the preceding. Moreover, between the Angles
of Contact of any two of thefe kinds, other Angles of Contact may
be found ad infwitum, that mall infinitely exceed each other.

41. Now it appears, that Angles of Contact of one kind are in-
finitely greater than thofe of another kind ; fince a Curve of one
kind, however great it may be, cannot, at the Point of Contact,

I he

and INFINITE SERIES. 67

lie between the Tangent and a Curve of another kind, however fmall
that Curve may be. Or an Angle of Contacl of one kind cannot
necefTarily contain an Angle of Contact of another kind, as the whole
contains a part. Thus the Angle of Contaft of the Curve x* = cy*,
or the Angle which it makes with its Abfcifs, neceflarfly includes the
Angle of Contacl of the Curve x~' =^y i , and can never be contain'd
by it. For Angles that can mutually exceed each other are of the
fame kind, as it happens with the aforefaid Angles of the Trochoid,
and of this Curve x> = by*.

42. And hence it appears, that Curves, in fome Points, may be
infinitely more ftraight, or infinitely more curved, than any Circle, and
yet not, on that account, lofe the form of Curve-lines. But all
this by the way only.

43. Ex. 5. Let ED be the Quadratrix to the Circle, defcribed
from Center A; and letting fall DB

perpendicular to AE, make AB = x,
BD =y, and AE = i. Then 'twill

be yx yy* yx* =xy, as before.

Then writing i for x, and z for y, the

Equation becomes zx zy l zx*

= y ; and thence, by Prob. i. zx

zy* zx* -f- zx zzxx zzyy = y m Then reducing, and

again writing i for x and z for y, there arifes z

xxxjy

J,

But z and & being found, make ' T ** = DH, and draw HC as

above.

44. If you defire a Conftrudtion of the Problem, you will find it
very mort. Thus draw DP perpendicular to DT, meeting AT in P,

and make aAP : AE :: PT : CH. For * =r

and zy = g. = -BP; and;ey + x = AP, and -_^_..
into zy-\-x-=. z - into AP=2. Moreover it is i-4-zz =

AE x BTy

"PT* T> P\ TAT 1 . I nrfr T3T

r 1 /i f. BlJq U I a \ j i r 1 -j- ** r 1

:= i-{- rrTT =-T-:T I ,) and tnereiore : =

Bl? BI? " 2-

BT

= DH. Laftly, it is BT : BD :: DH : CH==^^. Here

the negative Value only mews, that CH mufl be taken the fame
way as AB from DH.

45. In the fame manner the Curvature of Spirals, or of any other
Curves whatever, may be determined by a very mort Calculation.

K 2 46.

68 7&e Method of FLUXIONS,

46. Furthermore, to determine the Curvature without any pre-
vious reduction, when the Curves are refer'd to right Lines in any
other manner, this Method might have been apply'd, as has beer*
done already for drawing Tangents. But as all Geometrical Curves,
as alfo Mechanical, (efpecially when the defining conditions are re-
duced to infinite Equations, as I mail mew hereafter,) may be re-
fer'd to rectangular Ordinates, I think I have done enough in this
matter. He that defires more, may eafily fupply it by his own in-
duftry ; efpecially if for a farther illuflration I mall add the Method
for Spirals.