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

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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 caſe the ultimate ratio of the evaneſcent lines IP and PH will be found aſabove. Therefore draw CF parallel to AD, meeting BD in F, and cut it in E in the ſame ultimate ratio, then DE will be the tangent; becauſe CF and the evaneſcent IH are parallel, and ſimilarly cut in E and P.

Plate 9, Figure 1
Plate 9, Figure 1

Cor. 2. Hence alſo 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. Biſect BF in G, and drawing the indefinite line AG, this will be the poſition 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 tranſverſum, 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 correſponding to the diameter AG will be . But if it does meet it any where, the locus will be an hyperbola, when the points A and H are placed on the 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 lime time 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 problem of the ancients concerning four lines, begun by Euclid, and carried on by Appolonius and this not an analytical calculus, but a geometrical compoſition, ſuch as the ancients required.