50 PHILOSOPHICAL TRANSACTIONS. [aNNO 1735. 



Other parts of the treatise concerning the description of curve lines in this 

 supplement. 



That treatise requires these additions and illustrations the more, that though 

 the whole almost was new, it was published in a hurry, when the author was 

 very young, before he had time to consider sufficiently which were the best 

 ways of demonstrating the theorems, or resolving the problems, for which this 

 supplement he hopes will make some apology. 



The following paper, dated at Nancy, Nov. 27, 1722, is that which the 

 author mentions in his letter. 



Section I.— Pkop. 1. Which respects the Description of Lines. — About 

 the poles c, b, d, (fig. 15) let the lines cd, Bm, nr be moved; and let the 

 concourse of the legs sm, nr be drawn along the given line pg, and the con- 

 course of the legs cd, nr along the given line pa; then the concourse of the 

 legs cd, sd will describe a conic section. 



Draw rt parallel to the line bd given in position, meeting Bd in t; joint 

 pt, producing it to meet bd in f ; and it will give the point f. For as the 

 ratio of ru to rt is given, being the same as that of dg to db, because of the 

 similar triangles omBG and rmta ; and since ru is to rt, as og to qf, the 

 ratio of qf to ug will be also given ; so that, because of the given line qg, 

 there will be given qp, and hence the point f and the line pf. Since there- 

 fore Btand cr cut off the parts pt, pr, from the ITnes pf, pa, given in position, 

 their intersection d will always be in a given ratio in a conic section, by 

 Lem. 20, lib. \, Newton's Principia. 



If the point d be taken anywhere in the right line bf; and if dg be always 

 to aG, as bd to qf ; the conic section will be the same as d describes. 



The conic section passes through c, p, b and a, by completing the parallelo- 

 gram psav. It also passes through l, where the line bg produced meets pv, as 

 also through k, where cd cuts the given line pg. Hence the pentagon pkclb 

 is inscribed in the section. And if the 5 points ckpbl be given, through 

 which the conic section is to be drawn, or if the conic section is to be cir- 

 cumscribed about the given pentagon clbpk, let any 2 sides, ck, lb, be pro- 

 duced to their intersection d ; then join the rest pl, pk, and let the intersec- 

 tions of cd, Dr, and sd, dk, be silways drawn along those lines pl, pr ; then 

 the intersection d will describe the section. 



Prop. 2. About the given points f, c, g, s, (fig. l6) as poles, let the lines 

 FQ, cn, gq, sl be moved ; and let the intersections of the lines fq and cn, 

 FQ and Ga, Ga and sl, viz. the points m, a, l, always touch the lines ae, 

 BB, HL, given in position ; then the intersection of the lines cn, sl will describe 

 a conic section. 



