SEC. V.J OF NATURAL PHILOSOPHY. in H, and AH will be its diameter or latus transversum. to which the latus rectum will be as BG2 to AG X GH. If AG nowhere meets the locus, the line AH being infinite, the locus will be a par abola ; and its latus rectum corresponding to the diameter AG will be -.-7^ AC* But if it does meet it anywhere, the locus will be an hyperbola, when the points A and H are placed on the same side the point G ; and an ellipsis, if the point G falls between the points A and H ; unless, perhaps, the angle AGB is a right angle, and at the same time BG2 equal to the rectangle AGH, in which case the locus will be a circle. And so we have given in this Corollary a solution of that famous Prob lem of the ancients concerning four lines, begun by Euclid, and carried on by Apollonius ; and this not an analytical calculus, but a geometrical com position, such as the ancients required. LEMMA XX. If the two opposite angular points A and P of any parallelogram ASPQ touch any conic section in the points A and P ; and the sides AQ, AS of one of those angles, indefinitely produced, meet the same conic section in B and C ; and from the points of concourse, B and C to any fifth point D of the conic section, two right lines BD, CD are drawn meeting tlie two other sides PS, PQ of the parallelogram, indefinitely pro duced in T and R ; the parts PR and PT, cut off from the sides, will always be one to the other in a given ratio. And vice versa, if those parts cut off' are one to the other in a given ratio, the locus of the point D will be a conic section passing through the four points A, B, C, F CASE 1. Join BP, CP, and from the point D draw the two right lines DG, DE, of which the first DG shall be parallel to AB, and meet PB, PQ, CA in H, I, G ; and the other DE shall be parallel to AC, and meet PC, PS, AB, in F, K, E ; and (by Lem. XVII) the rectangle DE X DF will be to the rect angle DG X DH in a given ratio. But PQ is to DE (or IQ) as PB to HB, and con sequently as PT to DH ; and by permutation PQ, is to PT as DE to DH. Likewise PR is to DF as RC to DC, and therefore as (IG or) PS to DG ; and by permutation PR is to PS as DF to DG ; and, by com pounding those ratios, the rectangle PQ X PR will be to the rectangle PS X PT as the rectangle DE X DF is to the rectangle DG X DH. and consequently in "a given ratio. But PQ and PS are given, and there fore the ratio of PR to PT is given. Q.E.D.