A History of Mathematics From Mesopotamia to Modernity

184 A History ofMathematics

T ABb

E

D

d

c

Fig. 8Newton’s picture of the tangent to a curve.

T

F

D

BLA

M

E

G

e

C

Fig. 9Newton’s ‘cissoid’.

by the momentBb, equal toDc[Fig. 8]. Now letDdbe extended till it meetsABinT: this will then

touch the curve inDordand the trianglesdcD,DBTwill be similar, so that

TB:BD=Dc:cd

When therefore the relationship ofBDtoABis exhibited in any equation by which the curve is to

be determined, seek the relation of the fluxions by Problem 1 and takeTBtoBDin the ratio of the

fluxion ofABto that ofBD; then willTDtouch the curve atD.

EXAMPLE 3. LetEDbe a conchoid of Nicomedes described with poleG, asymptoteATand

distanceLD, and letGA=b,LD=c,AB=x, andBD=y. [Fig. 9]. Because of the similar triangles

DBLandDMGthere will be

LB(

√

cc−yy):BD(y)::DM(x):MG(b+y)

and sob+ytimes

√

cc−yy=yx. Having gained this equation I suppose

√

cc−yy=zand thus

I have two equationsbz+yz=yxandzz=cc−yy. With their help I seek the fluxions of the

quantitiesx,y, andz(by Problem 1), [these are calledm,n,r, respectively] and from the first there

comesbr+yr+nz=nx+my, and from the second 2rz=− 2 ny,orrz+ny=0. On eliminating

r, there arises−bny/z−nyy/z+nz=nx+my. By resolving these there comes

y:z−

by

z

−

yy

z

−x(::n:m)::BD:BT