Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/206

 as the letters ACEA. Complete the egments into entire circles, and let P be the centre of the firt circle FSG, Q the centre of the econd FTH. Join and produce both ways the line PQ, and in it take QR in the ame ratio to PQ as BC has to AB. But QR is to be taken towards that ide of the point Q, that the order of the letters P, Q, R may be the ame, as of the letters A, B, C; and about the centre R with interval RF decribe a fourth circle FNc cutting third circle FVI in c. Join Fc cutting the firt circle in a and the econd in b. Draw aG, bH cI, and let the figure abcFGHI be made imilar to the figure abcFGHI; and the trpezium fghi will be taht which was requited to be decribed.

For let the two firt circles FSG, FTH cut one the other in K; join PK, QK, EK, aK, bK, cK, and and produce QP, to L. The angles FaK, FbK, FcK at the circumferences, are the halves of the angles LPK, LQK, LRK, the halves of thoe angles. Wherefore the figure PQRK is equiangular and imilar to the figure abcK, and conequently ab is to bc as PQ to QR, that is, as AB to BC. But by contruction, the angles fAg, fBh, fVi are equal to the angles FaG, FbH, FcI. And therefore the figure abcFGHI. Which done, a trapezium fghi will be contructed imilar to the trapezium FGHI, and which by its angles, f, g, h, i will touch the right lines ABC, AD, BD, CE. Q. E. F.

Hence a right line may be drawn whoe part intercepted in a given order, between four right lines given by poition, hall have a given