Isaac Newton.

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

. (page 6 of 30)
Font size

that is, the Tranfmutation of a flowing Quantity by its connexion
with a given Quantity. Let AE and ae be two Lines indefinitely
extended each way, along which two moving Things or Points may
pafs from afar, and at the fame time

may reach the places A and a, B and A E c p E

b, C and c, D and d, &c. and let B '

be the Point, by its diftance from which, -4 : i ^ ?

the Motion of the moving thing or

point in AE is eftimated ; fo that BA, BC, BD, BE, fucceffively,
may be the flowing Quantities, when the moving thing is in the
places A, C, D, E. Likewife let b be a like point in the other Line.
Then will BA and ba be contemporaneous Fluents, as alfo
BC and be, BD andZv/, BE and be, 6cc. Now if inftead of the points
B and b, be fubftituted A and c, to which, as at reft, the Motions
are refer'd ; then o and ca, AB and cb, AC and o, AD and
cd, AE and ce, will be contemporaneous flowing Quantities. There-
fore the flowing Quantities are changed by the Addition and Sub-
traclion of the given Quantities AB and ac ; but they are not changed
as to the Celerity of their Motions, and the mutual refpect of their
Fluxion. For the contemporaneous parts AB and ab, BC and be,
CD and cd, DE and de, are of the fame length in both cafes. And
thus in Equations in which thefe Quantities are reprefented, the
contemporaneous parts of Quantities are not therefore changed, not-
withftanding their ablblute magnitude maybe increafed or diminimed
by fome given Quantity. Hence the thing propofed is manifeft :
For the only Scope of this Problem is, to determine the contempo-
raneous Parts, or the contemporary Differences of the abfolute Quan-
tities f, x, _>', or z, defcribed with a given Rate of Flowing. And
it is all one of what abfolute magnitude thofe Quantities are, fo that
their contemporary or correfpondent Differences may agree with the
prcpofed Relation of the Fluxions.

59. The reaibn of this matter may alfo be thus explain'd Al-
gebraically. Let the Equation y=xxy be propofed, and fup-

G 2 pole

44. 77je Method of FLUXIONS,

pofe x= i -+-Z- Then by Prob. i. x = z. So that for y =-. xxy ,
may be wrote y =. xy -h xzy. Now fince ,v=s, it is plain,, that
though the Quantities x and z be not of the fame length, yet that
they flow alike in refpecl: of y, and that they have equal contem-
poraneous parts. Why therefore may I not reprefent by the fame
Symbols Quantities that agree in their Rate of Flowing,; and to de-
termine, their contemporaneous Differences, why may not I uie

v === xy + xxy initead of y = xxy ?

60.. Lartly it appears plainly in what manner the contemporary
parts may be found, from an Equation involving flowing Quantities.

Thus if y = ~ -+- x be the Equation, when # = 2, then _y = 24.
But when x = 3, then y =. 3.1. Therefore while x flows from
2 to 3, y will flow from 2-i to 3.1. So that the parts defcribed in
this time are 3 2 = i, and 3-^ 2-i = f .

6 1. This Foundation being thus laid for what follows, I fhall
now proceed to more particular Problems.

PROB. m.

A ltijt'1 ^ determine the Maxima and Minima of H^

1. When a Quantity is the greateft or the leaft that it can be,
at that moment it neither flows backwards or forwards. For if it
flows forwards, or increafes, that proves it was lefs, and will pre-
fently be greater than it is. And the contrary if it flows backwards,
or decreafes. Wherefore find its Fluxion, by Prob. i. and fuppofe
it to be nothing.

2. Ex AMP. i. If in the Equation x> ax 1 + axy jy 3 = o the
greatefl Value of, x be required ; find the Relation of the Fluxions
of x and y, and you will have 3X.v a 2axx -f- axy %yy l -i-ayx
= o. Then making x = o, there will remain yyy 1 -\- ayx=o,
or 3j* = ax. By the help of this you may exterminate either x
or y out of the primary Equation, and by the refulting Equation you
may determine the other, and then both of them by 3^* -f-
ax = o.

3. This Operation is the fame, as if you had multiply 'd the
Terms of the propofed Equation by the number of the Dimenfions
of the other flowing Quantity.^. From whence we may .derive the

famous
2.

and INFINITE SERIES. 45

famous Rule of Huddenius, that, in order to obtain the greateft or
leaft Relate Quantity, the Equation mufl be difpofed according to
the Dimenfions of the Correlate Quantity, and then the Terms are
to be multiply 'd by any Arithmetical ProgrelTion. But fince neither
this Rule, nor any other that I know yet publiihed, extends to Equa-
tions affected with iiird Quantities, without a previous Reduction j
I fhall give the following Example for that purpofe.

4. EXAMP. 2. If the greatest Quantity y in the Equation x*

a y~ + 7+ - xx ^ a y ~+" xx = be to be determin'd, feek the
.Fluxions of xand^y, and there will arife the Equation 3^^* zayy-{-

^^v) 1 + 2^n5 Aaxxy-\-6x\* + atx 2 A j r \ r r

I __ - _ - = 0. And fince by fuppofition y = o, ,

a 1 -\- zay +j* 2 ^ ay -\- xx

omit the Terms multiply'd by y, (which, to fhorten the labour,
might have been done before, in the Operation,) and divide the reft

by xx, and there will remain %x ^- "*"-'** = o. When the Re-

a"xx

duction is made, there will arife ^ay-\- %xx = o, by help of which
you may exterminate either of the quantities x or y out of the pro-
pos'd Equation, and then from the refulting Equation, which will,
be Cubical, you may extract the Value of the other.

5. From this Problem may be had the Solution of thefe fol-
lowing.

I. In a given .Triangle, or in a Segment of any given Curve, ft>
ir.fcribe the greatejl Reft angle.

II. To draw the greatejl or the leafl right Line, 'which can lie:
between a given Point, and a Curve given in pofition. Or, to draw.
a Perpendicular to a Curve from a given Point.

III. To draw the greatejl or the leajl right Lines, which pajjin?.-
through a given Point, can lie bet-ween two others, either right Lines
or Curves.

IV. From a given Point within a Parabola, to draw a rivbt
Line, which Jhall cut the Parabola more obliquely than any other.
And to do the fame in other Curves.

V. To determine the Vertices of Curves, their greatejl or lealT
Breadths, the Points in which revolving parts cut each other, 6cc.

VI. To find the Points in Curves, where they hcrce the great ejT
or leajl Curvature.

VII. To find the Icaft Angle in a given EHi/is, in which the.
Ordinates can cut their Diameters.

VIII..

4.6 The Method of FLUXIONS,

VIII. Of EHipfes that pafs through four given Points, to deter-
mine the greateft, or that which approaches neareft to a Circle.

IX. 70 determine fuch a part of a Spherical Superficies, which
can be illuminated, in its farther part, by Light coming from a
great dijlance, and which is refracted by the nearer Hemijphere.

And many other Problems of a like nature may more eafily be
propofed than refolved, becaufe of the labour of Computation.

P R O B. IV.

To draw Tangents to Curves.

Firft Manner.

1. Tangents may be varioufly drawn, according to the various
Relations of Curves to right Lines. And firft let BD be a right
Line, or Ordinate, in a given Angle to

another right Line AB, as a Bafe or Ab-
fcifs, and terminated at the Curve ED.
Let this Ordinate move through an inde-
finitely finall Space to the place bd, fo
that it may be increafed by the Moment
cd, while AB is increafed by the Moment ^ A
Bb, to which DC is equal and parallel.
Let Da 1 be produced till it meets with AB in T, and this Line will
touch the Curve in D or d ; and the Triangles dcD, DBT will be
fimilar. So that it is TB : BD : : DC (or B) : cd.

2. Since therefore the Relation of BD to AB is exhibited by the
Equation, by which the nature of the Curve is determined ; feek for
the Relation of the Fluxions, by Prob. i. Then take TB to BD
in the Ratio of the Fluxion of AB to the Fluxion of BD, and TD
will touch the Curve in the Point D.

3. Ex. i. Calling AB = x, and BD =jy, let their Relation be
x -, ax* -h axy _y 3 = o. And the Relation of the Fluxions will
be 3xx -i 2axx-i-axy ^yy* -+- ayx-=. o. So that y : x :: ^xx
2ax -4- ay : ^ ax :: BD (;-) : BT. Therefore BT =
... w* ~~ f!X ~ Therefore the Point D being given, and thence DB
and AB, or v and x, the length BT will be given, by which the Tan-
gent TD is determined.

4-

and INFINITE SERIES. 47

4. But this Method of Operation may be thusconcinnated. Make
the Terms of the propofed Equation equal to nothing : multiply by
the proper number of the Dimenfions of the Ordinate, and put the
Refult in the Numerator : Then multiply the Terms of the fame
Equation by the proper number of the Dimenfions of the Abfcifs, and
put the Produdl divided by the Abfcifs, in the Denominator of the
Value of BT. Then take BT towards A, if its Value be affirmative,
but the contrary way if that Value be negative.

o o 13

5. Thus the Equation* 3 ax* -f- axy y*=o, being multi-

3 z 10

ply'd by the upper Numbers, gives axy 3_y 3 for the Numerator j
and multiply 'd by the lower Numbers, and then divided by x, gives
3-x- 1 zax -+- ay for the Denominator of the Value of BT.

6. Thus the Equation jy 3 by* cdy -f- bed -\-dxy = o, (which
denotes a Parabola of the fecond kind, by help of which Des Cartes
confirufted Equations of fix Dimenfions ; fee his Geometry, p. 42.
Amfterd. Ed. An. 1659.) by Infpeftion gives ^ - "fr+'^v ^ Qr

7. And thus a 1 r -x* y 1 = o, (which denotes an Ellipfis
whofe Center is A,) gives ^ , or ^ = BT. And fo in others.

- X
1

1

8. And you may take notice, that it matters not of what quantity
the Angle of Ordination ABD may be.

9. But as this Rule does not extend to Equations afFefted by furd
Quantities, or to mechanical Curves ; in thefe Cafes we mufl have
recourfe to the fundamental Method.

10. Ex. 2. Let A; S ay 1 -+- j- xx \/'ay -+- xx = o be the

Equation exprefling the Relation between AB and BD ; and by Prob. i.
the Relation of the Fluxions will be 3*** zayy -f. *"*"* + 2 V

=0. Therefore it will be <ixx

*/,.,,,

4 v ~

T^T- :: (y : x ::) BD : BT.

fay p ^^

II.

TJoe Method of FLUXIONS,

48

ii. Ex. 3. Let ED be the Conchoid of Nicomedes, defcribed with
the Pole G, the Afymptote AT, and the Diftance LD ; and let

'GA = , LD = c, AB=.v, andBD=;>. And becaufe of fimi-
lar Triangles DEL and DMG, it will be LB : BD : : DM : MG ;
that is, v/ 'cc yy : y : : x : b -+- y, and therefore b-\-y ^/cc yy
=yx. Having got this Equation, I fuppofe V cc yy = z, and
thus I fliall have two Equations bz ~\-yz =yx, andzz = cc yy.
By the help of thefe I find the Fluxions of the Quantities x, y, and
z, by Prob. i. From the firft arifes bz -+-yz -\- yz =y'x -+- xy,
and from the fecond 2zz = 2yy, or zz -j- yy = o. Out of

thefe if we exterminate z, there will arife -^ -i-yz =yx

-+ xy, which being refolved it will be y : z - x : :

(y : x ::) BD : BT. But as BD is y, therefore BT= .3-
That is, BT = AL -f- - ~ -; where the Sign

BL

iff !- J_l (_J

prefixt to BT denotes, that the Point T mufl be taken contrary to
the Point A.

12. SCHOLIUM. And hence it appears by the bye, how that
point of the Conchoid may be found, which Separates the concave
from the convex part. For when AT is the lea ft poffible, D will
be that point. Therefore make AT = v ; and fmce BT - z

x

then v = z -+- 2K -+-

by -\- yv

Here to morten

the work, for x fubftitute - ^l!5 > w hich Value is derived from what
is before, and it will be - ? -f. z -+- - - = v. Whence the
Fluxions v, y, and z being found by Prob. i. and fuppofing ^=0

and INFINITE SERIES. 49

.,, ... iy, )K ' iy-l-zyy Azy-4-zvy

bvProb. -3. there will anfe - - ~-t-z + - - =i; = o.

J J y jy z za

Laflly, fubftituting in this : - for z, and cc yy for zz, (which

values of z and zz are had from what goes before,) and making a
due Reduction, you will have y'- -+- ^by* -2.be* = o. By the Con-
ftrudlion of which Equation y or AM, will be given. Then thro'
M drawing MD parallel to AB, it will fall upon the Point D of
contrary Flexure.

13. Now if the Curve be Mechanical whofe Tangent is to be
drawn, the Fluxions of the Quantities are to be found, as in Examp.5.
of Prob. i. and then the reft is to be perform'd as before.

14. Ex. 4. Let AC and AD be two Curves, which are cut in
the Points C and D by the right Line

BCD, apply 'd to the Abfcifs AB in a
given Angle. Let AB = x, BD = y,

and - = z. Then (by Prob. i.

Preparat. to Examp. 5.) it will be z = x ~T> ^ ^ B~
xBC.

15. Now let AC be a Circle, or any known Curve ; and to deter-
mine the other Curve AD, let any Equation be propofed, in which
z is involved, as zz +- axz =_y 4 . Then by Prob. i. 2zz +- axz
-+- axz = 4X7*. And writing x x BC for z, it will be zxz x BC
-+- axx x BC H- axz = 4)7'. Therefore 2z x BC -+- ax x BC -{-
az : 4jy J :: (y : x ::) BD : BT. So that if the nature of the
Curve AC be given, the Ordinate BC, and the Area ACB or z ;
the Point T will be given, through which the Tangent DT will
pafs.

1 6. After the fame manner, if 32 = zy be the Equation to the

Curve AD ; 'twill be (3.3) 3^ x BC = zy. So that 3BC : 2 ::
(y : x ::) BD : BT. And fo in others.

17. Ex. 5. Let AB=,v, BD =y, as before, and let the length
of any Curve AC be z. And drawing a Tangent to it, as Cl, 'twill

x x C/

be Bt : Ct :: x : z, or z = ^-

18. Now for determining the other Curve AD, whofe Tangent
is to be drawn, let there be given any Equation in which z is in-
volved, fuppofe z ==)'. Then it will be z=y, fo that Ct : Bf ''
(y : x : :} : BD : BT. But the Point T being found, the Tan-
gent DT may be drawn.

H 19-

The Method of FLUXIONS,

19. Thus fuppofmg xzsssyy, 'twill be KZ + zx = zyj >, and
for z writing ^ there will arife xz -f- ^-^ = ayy. There-

y-> O/ ''

fore * -I- f~-' : 27 : : BD : DT.

20. E x . 6. Let AC be a Circle, or any other known Curve,
whofe Tangent is Ct, and let AD be any

other Curve whofe Tangent DT is to be
drawn, and let it be defin'd by afTuming
AB = to the Arch AC ; and (CE, BD
being Ordinates to AB in a given Angle,)
let the Relation of BD to CE or AE be
exprels'd by any Equation.

21. Therefore call AB or AC = x, BD =y, AE=z, and
CE = v . And it is plain that v, x, and z, the Fluxions of CE,
AC, and AE, are^to each other as CE, Ct, and Et. Therefore *x
C7 = i>, and .v x ^ = z.

22. Now let any Equation be given to define the Curve AD,

as y = . Then y = z ; and therefore Et : Ct :: (v x )
BD : BT. K "'

23. Or let the Equation be yz+vx, and it will be

. r~>T? I TT- . y-.

And therefore CE -4- Et

t. T

Ct : Ct :: (y : x ::) BD : BT.

24. Or finally, let the Equation be ayy = v* y and it will be
zayy = (3^ =) 3*1;' x . So that 31;* x CE : 2 ay x Ct ::
BD : BT.

25. Ex. 7. Let FC be a Circle, which is touched by CS in C;
and let FD be a Curve, which is de-
fined by affuming any Relation of the

Ordinate DB to the Arch FC, which is
intercepted by DA drawn to the Center.
Then letting fall CE, the Ordinate in
the Circle, call AC or AF=i, AB

CF = /; and it will be tz=(t^=)

K B

T ,S

. . ^..

v, and tv = (/x -^ =) z. Here I put z negatively, becaufe
AE is dirninifh'd while EC is increafed. And befides AE : EC ::

AB :

and INFINITE SERIES. 51

AB : BD, fo that zy = vx, and thence by Prob. i. zy -f- yx

vx -f- xv. Then exterminating v, z, and v, 'tis yx ty*

tx* = xy.

26. Now let the Curve DF be defined by any Equation, from

which the Value of t may be derived, to be fubftituted here. Sup-
pofe let ^=_y, (an Equation to the firft Quadratrix,) and by Prob. i.
it will be / = y, fo that yx yy* yx* = xy. Whence y : xx
x :: ( y : _ x : :) BD(;') : BT. Therefore BT = x*

- x; and AT = xx+yy = ^/.

27. After the fame manner, if it is // = ly, there will arife
= 6r, and thence AT= - x~ . And fo of others.

z/ / r

28. Ex. 8. Now if AD be taken equal to the Arch FC, the
Curve ADH being then the Spiral of Archimedes ; the fame names
of the Lines ftill remaining as were put

afore : Becaufe of the right Angle ABD
'tis xx -{-yy=tf ) and therefore (by Prob. i.)
xx +yy = //. Tis alfo AD : AC : :
DB : CE, fo that tv=y t znd thence (by
Prob. i.) tv -4- vf =y. Laftly, the Fluxion
of the Arch FC is to the Fluxion of the
right Line CE, as AC to AE, or as AD
to AB, that is, t : v : : t : x, and thence
ix = vf. Compare the Equations now found, and you will fee

'tis tv -+-ix=y, and thence xx -\-yy = (tt =) ^^ . And there-
fore compleating the Parallelogram ABDQ^_, if you make QD :
QP_ :: (BD : BT :: y : x ::) X : y ^ ; that is, if you

take AP = ; ! > PD will be perpendicular to the Spiral.

29. And from hence (I imagine) it will be fufficiently manifeft,
by what methods the Tangents of all fcrts of Curves are to be
drawn. However it may not be foreign from the purpofe, if I alfo
fliew how the Problem may be perform'd, when the Curves are re-
fer'd to right Lines, after any other manner whatever : So that hav-
ing the choice of feveral Methods, the eafieft and moil fimple may
always be ufed.

H 2 Second

\$2 The Method of FLUXIONS,

Second Manner.

30. Let D be a point in the Curve, from which the Subtenfe
DG is drawn to a given Point G, and let DB be anOrdinate in any given
Angle to the Abfcifs AB. Now let the

Point D flow for an infinitely fmall fpace

D^/ in the Curve, and in GD let Gk be

taken equal to Gd, and let the Parallelo-

gram dcBl> be compleated. Then Dk

and DC will be the contemporary Mo- - -

ments of GD and BD, by which they

are diminifh'd while D is transfer'd to d. Now let the right Line

~Dd be produced, till it meets with AB in T, and from the Point T to

the Subtenfe GD let fall the perpendicular TF, and then the Trapezia

Dcdk and DBTF will be like; and therefore DB : DF :: DC : Dk.

31. Since then the Relation of BD to GD is exhibited by the
Equation for determining the Curve ; find the Relation of the Fluxions,
and take FD to DB in the Ratio of the Fluxion of GD to the
Fluxion of BD. Then from F raife the perpendicular FT, which
may meet with AB in T, and DT being drawn will touch the
Curve in D. But DT muft be taken towards G, if it be affirmative,
and the contrary way if negative.

32. Ex. i. Call GD = x, and BD =_>', and let their Relation
be x ~, ax 1 -f- axy y"= = o. Then the Relation of the Fluxions
will be ^xx 1 2axx +- axy -f- ayx ^yy- = o. Therefore ^xx
zax -h ay : ^yy ax :: (y : x : :) DB (y) : DF. So that

.' V axy , . Then any Point D in the Curve being given,

~ 1

and thence BD and GD or y and x, the Point F will be given
alfo. From whence if the Perpendicular FT be raifed, from its
concourfe T with the Abfcifs AB, the Tangent DT may be
drawn.

3 3 . And hence it appears, that a Rule might be derived here, as well
as in the former Cafe. For having difpofed all the Terms of the given
Equation on one fide, multiply by the Dimensions of the Ordinatejy,
and place the refult in the Numerator of a Fraction. Then multiply
its Terms feverally by the Dimenfions of the Subtenfe x, and dividing
the refult by that Subtenfe x, place the Quotient in the Deno-
minator of the Value of DF. And take the fame Line DF to-
wards G if it be affirmative, otherwile the contrary way.. Where

you

and IN FINITE SERIES,

53

you may obferve, that it is no matter how far diftant the Point G
is from the Abfcifs AB, or if it be at all diftant, nor what is the
Angle of Ordination ABD.

34. Let the Equation be as before x* ax* -f- axy J 3 = o ;
it gives immediately axy 3>' 3 for the Numerator, and 3** 2ax
-+- ay for the Denominator of the Value of DF.

35. Let alfo a -+- -x~y=o, (which Equation is to a Conick
Sedtion,) it gives y for the Numerator, and for the Denomi-

fly

nator of the Value of DF, which therefore will be 7

36. And thus in the Conchoid, (wherein thefe things will be
perform'd more expeditioufly than before,) putting GA = b,

= c, GD=x, and BD=^, it will be BD (;) : DL (c) ::
G A (5) : GL (x <:). Therefore xy cy = cb, or xy cy
cb = o. This Equation according to the Rule gives ^-^ - , that

is, x <r=DF. Therefore prolong GD to F, fo that DF =
LG, and at F raife the perpendicular FT meeting the Alymptote
AB in T, and DT being drawn will touch the Conchoid.

37. But when compound or furd Quantities are found in the
Equation, you mufl have recourfe to the general Method, except you
fliould chufe rather to reduce the Equation.

38. Ex. 2. If the Equation

xv/cr yy =zyx, were gven

for the Relation between GD and BD ; (fee the foregoing Figure,
p. 52.) find the Relation of the Fluxions by Prob. i. As fuppoiing
v/ff )')' = z ) you will have the Equations bz -+- yz = yx, and
cc yy=.zz, and thence the Relation of the Fluxions bz-\-yx

= yx -f- yx, and 2yy=2Z,z. And now z, and z being
i exter-

T&e Method of FLUXIONS,
exterminated, there will arife v \/ cc yy 'Jjl v U \x = xy.

Therefore y : ^/cc yy J2^ .v :: (y : ,v ::) BD (ji 1 ) : DF.

Third Manner.

39. Moreover, if the Curve be refer'd to two Subtenfes AD and
BD, which being drawn from two given Points A and B, may
meet at the Curve: Conceive that Point
D to flow on through an infinitely little
Space Del in the Curve ; and in AD and
BD take Ak = Ad, and Bc = Bc/; and
then kD and cD will be contempora-
neous Moments of the Lines AD and -
BD. Take therefore DF to BD in

the Ratio of the Moment D& to the /r

Moment DC, (that is, in the Ratio of the Fluxion of the Line
AD to the Fluxion of the LineBD,) and draw BT, FT perpendicu-
lar to BD, AD, meeting in T. Then the Trapezia DFTB and DM:
will be fimilar, and therefore the Diagonal DT will touch the
Curve.

40. Therefore from the Equation, by which the Relation is
defined between AD and BD, find the Relation of the Fluxions by
Prob. i. and take FD to BD in the fame Ratio.

41. Ex AMP. Suppofing AD = x, andBD=;', let their Rela-
tion be a -f- e j y = o. This Equation is to the Ellipfes of

the fecond Order, whofe Properties for Refracting of Light are fhewn
by Des Cartes, in the fecond Book of his Geometry. Then the

Relation of the Fluxions will be e - y ==o. 'Tis therefore e :
d ::(>:# ::) BD : DF.

42. And for the fame reafon if a ^ y = o, 'twill be

e : _ d : : BD : DF. In the firft Cafe take DF towards A, and
contrary-wife in the other cafe.

43. COROL. i. Hence if d-=.e, (in which cafe the Curve be-
comes a Conick Section,) 'twill be

DF = DB. And therefore the Tri-
angles DFT and DBT being equal,
the Angle FDB will be bifected by
the Tangent. v -K A

44.

and INFINITE SERIES. 55

44. COROL. 2. And hence alfo thofe things will be manifeft of
themfelves, which are demonstrated, in a very prolix manner, by
Des Cartes concerning the Refraction of thcfe Curves. For as much
as DF and DB, (which are in the given Ratio of d to e,) in refpect
of the Radius DT, are the Sines of the Angles DTF and DTB,
that is, of the Ray of Incidence AD upon the Surface of the Curve,
and of its Reflexion or Refraction DB. And there is a like reafon-
ing concerning the Refractions of the Conick Sections, fuppofing
that either of the Points A or B be conceived to be at an infinite
diftance.

45. It would be eafy to modify this Rule in the manner of the
foregoing, and to give more Examples of it : As alfo when Curves
are refer'd to Right lines after any other manner, and cannot com-
modioufly be reduced to the foregoing, it will be very eafy to find
out other Methods in imitation of thefe, as occafion mall require.

Fourth Manner.

46. As if the right Line BCD mould revolve about a given Point
B, and one of its Points D mould defcribe a Curve, and another
Point C fhould be the

interfection of the right
Line BCD, with another
right Line AC given in
pofition. Then the Re-
lation of BC and BD be-
ing exprefs'd by any E-
quation ; draw BF pa-
rallel to AC, fo as to meet DF, perpendicular to BD, in F. Alfo
erect FT perpendicular to DF; and take FT in the fame Ratio to
BC, that the Fluxion of BD has to the Fluxion of BC. Then DT
being drawn will touch the Curve.

Fifth Manner.

47. But if the Point A being given, the Equation ihould exprefs
the Relation between AC and BD } draw CG parallel to DF, and
take FT in the fame Ratio to BG, that the Fluxion of BD has to
the Fluxion of AC.

Sixth Manner.

48. Or again, if the Equation exprefles the Relation between AC
and CD; let AC and FT meet in H ; and take HT in the fune
Ratio to BG, that the Fluxion of CD has to the Fluxion of AC. A. id
the like in others. Seventh

*fhe Method of FLUXION

Seventh Manner : For Spirals.

49. The Problem is not otherwise perform'd, when the Curves
are refer'd, not to right Lines, but to other Curve-lines, as is ufiial
in Mechanick Curves. Let BG be the Circumference of a Circle,
in whole Semidiameter AG, while it revolves