Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/186

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to go off in infinitum, and parallel lines are ſuch tend to a point infinitely remote. And after the problem is ſolved in the new figure, if by the inverſe operations we tranſform the new into the firſt figure we ſhall have the ſolution required.

This lemma is alſo of uſe in the ſolution of ſolid problems. For as often as two conic ſections occur, by the interſection of which a problem may be ſolved; any one of them may be tranſformed, if it is an hyperbola or a parabola, into an ellipſis then this ellipſiss may be eaſily changed into a circle. So alſo a right line and a conic ſection, in the conſtruction of plane problems, may be traſformed into a right line and a circle.


Proposition XXV. Problem XVII.

To deſecribe a trajectory that ſhall paſs through two given points, and touch three right lines given by poſition. Pl. 10, Fig. 6.

Plate 10, Figure 6
Plate 10, Figure 6

Through the concourſe of any two of the tangents one with the other, and the concourſe of the third tangent with the right line which paſſes through two given points, draw an indefinite right line; and, taking this line for the firſt ordinate radius, tranſform the figure by the preceding lemma into a new figure. In this figure thoſe two tangents will become parallel to each other, and the third tangent will be parallel to the right line that paſſes through the two given points. Suppoſe hi, kl to be thoſe two parallel tangents. ik the third tangent, and bl a right line parallel thereto, paſſing through thoſ points a, b,