# Lemma 19 ### LEMMA 19

Find a point P from which if four right lines PQ, PR, PS, PT an drawn to as many other right lines AB, CD, AC, BD, given by posi tion, each to each, at given angles, the rectangle PQ X PR, under any two of the lines drawn, shall be to the rectangle PS X PT, under the other tivo. in a given ratio. Suppose the lines AB, CD, to which the two PQ, PR, containing one of the rect drawn to meet two other lines, given are angles, by position, in the points A, B, C, D. From one right lines of those, as A, draw any right line AH, in which you would find the point P. Let this cut the H and I and, because opposite lines BD, CD, in the angles of the figure are given, the ratio of to PS, and therefore of to PA, and ; all PQ PA PQ to PS, will be also given. PQ X PR to PS X PT, Subducting this ratio from the given ratio oi PR to PT will be given and ad and PT to PH, the ratio of PI to PH. the ratio of ; ding the given ratios of PI to PR, and therefore the point P will be given. Q.E.I. COR. 1. Hence also a tangent may be drawn to any point D of the For the chord PD, where the points P and D locus of all the points P. AH is drawn through the point D, becomes a tangent. meet, that is, where will In which case the ultimate ratio of the evanescent lines IP and PH Therefore draw CF parallel to AD, meeting BD in be found as above. in the same ultimate ratio, then DE will be the tan in E F, and cut it evanescent IH are parallel, and similarly cut in and the CF gent because ; E and P. also the locus of all the points P may be determined. the of touching the locus, points A, B, C, D, as A, draw Through any and through any other point B parallel to the tangent, draw meeting in G, and find the point the locus in by this Lemma. Bisect

### Corollary 2

Hence AE F BF BF F ; will be the position of the dia

and, drawing the indefinite line AG, this

Let this meet the locus are ordinates. and meter to which BG FG AGOF NATURAL PHILOSOPHY. SEC. V.J AH H, and in will be its diameter or latus trans- BG AG X GH. If AG nowhere meets the locus, the line AH being infinite, the locus will be a par versum. to which the latus rectum will be as 2 to abola and ; diameter latus rectum corresponding to the its AG will be But -.-7^ AC* if it does meet it anywhere, the locus will be an hyperbola, when the points A and are placed on the same side the point G and an if the G falls between the points A and ellipsis, point unless, perhaps, 2 the angle is a right angle, and at the same time BG equal to the in which will a case the locus be circle. rectangle AGH, H ; H ; AGB 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.