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

 to go off in infinitum, and parallel lines are uch tend to a point infinitely remote. And after the problem is olved in the new figure, if by the invere operations we tranform the new into the firt figure we hall have the olution required.

This lemma is alo of ue in the olution of olid problems. For as often as two conic ections occur, by the interection of which a problem may be olved; any one of them may be tranformed, if it is an hyperbola or a parabola, into an ellipis then this ellipiss may be eaily changed into a circle. So alo a right line and a conic ection, in the contruction of plane problems, may be traformed into a right line and a circle.

To deecribe a trajectory that hall pas through two given points, and touch three right lines given by poition. Pl. 10, Fig. 6.

Through the concoure of any two of the tangents one with the other, and the concoure of the third tangent with the right line which paes through two given points, draw an indefinite right line; and, taking this line for the firt ordinate radius, tranform the figure by the preceding lemma into a new figure. In this figure thoe two tangents will become parallel to each other, and the third tangent will be parallel to the right line that paes through the two given points. Suppoe hi, kl to be thoe two parallel tangents. ik the third tangent, and bl a right line parallel thereto, paing through tho points a, b,