SEC. V.J OF NATURAL PHILOSOPHY. 15] may be the same as of the letters A, B, C ; and about the centre R with the interval RF describe a fourth circle FNc cutting (lie third circle FVI in c. Join Fc1 cut ting the first circle in a, and the second in / . Draw aG, &H, cl, and let the figure ABC/'4f/ii be made similar to the figure w^cFGHI; and the trapezium fghi will be that which was required to be de scribed. For let the two first circles FSG, FTH cut one the other in K ; join PK, Q,K, RK, "K, 6K, cK, and produce QP to L. The angles FaK, F6K, FcK at the circumferences are the halves of the angles FPK, FQK, FRK, at the centres, and therefore equal to LPK, LQ.K, LRK, the halves of those angles. Wherefore the figure PQRK is iquiangular and similar to the figure «6cK, and consequently ab is to be res PQ, to Q,R, that is, as AB to BC. But by construction, the angles ''Air, /B//,/C?', are equal to the angles F«G, F&H, Fcl. And therefore the figure ABCfghi may be completed similar to the figure abcFGHl. vVliich done a trapezium fghi will be constructed similar to the trapezium FGHI, and which by its angles/, g, h, i will touch the right lines ABC, AD, BD, CE. Q.E.F. COR. Hence a right line may be drawn whose parts intercepted in a given order, between four right lines given by position, shall have a given proportion among themselves. Let the angles FGH, GHI, be so far in creased that the right lines FG, GH, HI, may lie in directum ; and by constructing the Problem in this case, a right line fghi will be drawn, whose parts fg, gh, hi, intercepted between the four right lines given by position, AB and AD, AD and BD, BD and CE, will be one to another as the lines FG, GH, HI, and will observe the same order among them selves. But the same thing may be more readily done in this manner. Produce AB to K and BD to L, so as BK may be to AB as HI to GH ; and DL to BD as GI to FG; and join KL meeting the right line CE in i. Produce iL to M, so as LM may be to iL as GH to HI ; then draw MQ, parallel to LB, and meeting the right line AD in g, and join gi cutting AB, BD in f, h ; I M* say, the thing is done. For let MO- cut the right line AB in Q, and AD the right line KL iu II