Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/188

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Proposition XXVI. Problem XVIII.

To deſcribe a trajectory that ſhall paſs through a given point, and touch four right lines given by poſition. Pl. 11. Fig. 1

Plate 11, Figure 1
Plate 11, Figure 1

From the common interſections of any two of the tangents to the common interſection of the other two draw an indefinite right line; and taking this line for the firſt ordinate radius tranſform the figure (by lem. 22.) into a new figure, and the two pairs of tangents each of which are concurred in the firſt ordinate radius will now become parallel. Let hi and kl, ik and hl, be thoſe points of parallels compleat, the parallelogram hikm. And let p be the point in this new figure correſponding to the given point in the firſt figure. Through O the centre of the figure draw pq; and Oq being equal to Op, q will be the other point through which the conic ſection mull paſs in this new figure. Let this point be tranſferred by the inverſe operation of lem. 22. into the firſt figure. and there we ſhall have the two points, through which the trajectory is to be deſcribed. But through thoſe points that trajectory may be deſcribed by prob. 17. Q. E. F.