Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/159

 a perpendicular ZS is let fall, this (ZS) hall be to AZ as the difference between AZ and CZ is to AC. Wherefore the ratio's of ZR and ZS to AZ are given, and conequently the ratio of ZR to ZS one to the other; and therefore if the right lines RP SQ meet in T, and TZ and TA are drawn, the figure TRZS will be given in pecie, and the right line TZ, in which the point Z is omewhere placed, will be given in poition. There will be given alo the right line TA, and the angle ATZ; and becaue the ratio's of AZ and TZ to ZS are given, their ratio to each other is given alo; and thence will be given likewie the triangle ATZ whoe vertex is the point Z. Q. E. I.

If two of the three lines, for example AZ and BZ, are equal, draw the right line TZ o as to biect the right line AB; then find the triangle ATZ as above. Q. E. I.

If all the three are equal, the point Z will be placed in the centre of a circle that paes thro' the points A, B, C. Q. E. I.

This problematic lemma is likewie olved in Apollonui's Book of Tactions retored by Victa.

About a give focus to decribe a trajectory, that hall pas through given points and touch right lines given by poition.

Let the focus S, (Pl. 8. Fig. 2.) the point P, and the tangent TR be given, uppoe that the other focus H is to be found. On the tangent let fall the perpendicular ST, which produce to T, o that TY