Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/168

 the chord PD, where the points P and D meet, that is, where AH is drawn thro' the point D, becomes a tangent. In which cae the ultimate ratio of the evanecent lines IP and PH will be found aabove. Therefore draw CF parallel to AD, meeting BD in F, and cut it in E in the ame ultimate ratio, then DE will be the tangent; becaue CF and the evanecent IH are parallel, and imilarly cut in E and P.

Hence alo the locus of all the points P may be determined. Through any of the points A, B, C, D, as A. (Pl. 9. Fig. 1.) draw AE touching the locus, and through any other point B parallel to the tangent, draw BF meeting the locus in F: And find the point F by this lemma. Biect BF in G, and drawing the indefinite line AG, this will be the poition of the diameter to which BG, and FG are ordinates. Let this AG meet the locus in H, and AH will be its diameter or latus tranverum, to which the latus rectum will be as BG to AG x GH. If AG no where meets the locus, the line AH being infinite the locus will be a parabola; and its latus rectum correponding to the diameter AG will be $$\textstyle \frac {BG^2}{AG}$$. But if it does meet it any where, the locus will be an hyperbola, when the points A and H are placed on the ide the point G; and an ellipis, If the point G falls between the points A and H; unles perhaps the angle AGB is a right angle, and at the lime time $$\scriptstyle BG^2$$ equal to the rectangle AGH, in which cae the locus will be a circle.

And o we have given in this corollary a olution of that famous problem of the ancients concerning four lines, begun by Euclid, and carried on by Appolonius and this not an analytical calculus, but a geometrical compoition, uch as the ancients required.