TheCalculus 185
In consequence, sinceBDis equal toy,BTwill bez−x−(by+yy)/z. This is−BT=AL+(BD×
GM/BL). Here the sign−prefixed toBTindicates that the pointTmust be taken on the sideaway
fromA.Note.Newton’s exposition of how to find tangents is very clear. Of his various examples,
I think that the ‘conchoid’, whose equation and picture he gives, is the clearest; as Whiteside
says in his note (1967–81) it is taken from an example of Descartes, who did not bother to write
the equation. It is given here more as an example of style than as an encouragement to follow
through the calculation One notational point: in Newton’s ms, the fluxions ofx,y,zare called
m,n,ras above. This makes it unclear which fluxion belongs to which variable, and once Leibniz’s
d-notation became common, Newtonians adopted the clearer practice of writing ‘pricked’ (dotted)
lettersx ̇,y ̇, ̇z. Whiteside, perhaps again for clarity, uses these in his translation—it is one of the few
points where he changes Newton’s text—and I have changed them back, since this is how they
appeared in the original.Appendix B. Leibniz
From ‘A new method for maxima and minima as well as tangents, which is neither impeded by fractional nor irrational
quantities, and a remarkable type of calculus for them’ (1684), reproduced from Fauvel and Gray 13.A.3.- Let an axisAX[Fig. 10]^11 and several curves such asVV,WW,YY,ZZbe given, of which the
ordinatesVX,WX,YX,ZX, perpendicular to the axis are calledv,w,y,zrespectively. The segment
Fig. 10Leibniz’s illustration for his 1684 paper.- Dupont and Roero point out (1991) that the original picture given by Leibniz in the paper is usually wrongly reproduced in
copies (in particular in Fauvel and Gray). Figure 10 is from Leibniz’s works, and hopefully correct.