BR it to BD as FH to FG. Wherefore fh is to fg as FH to FG. Since therefore, gi to hi likewiſe is as Mi to Li, that is, as GI to HI, it is manifeſt that the lines FL, fi, are ſimilary cut in G and H, g and h. Q. E. F.
In the conſtruction of this corollary, after the line LK is drawn cutting CE in i, we may produce iE to V, ſo as EV may be to Ei as FH to HI, and then draw Vf parallel to BD. It will come to the ſame, if about the centre i, with an interval IH, we deſcribe a circle cutting BD in X, and produce iX to Y ſo as iY may be equal to IF, and then draw Yf parallel to BD.
Sir Chriſtopher Wren, and Dr. Wallis have long ago given other ſolutions of this problem.
Proposition XXIX. Problem XXI.
To deſcribe a trajectory given in kind, that may be cut by four right lines given by poſition, into parts given in order, kind and proportion.
Suppoſe a trajectory is to be deſcribed that may be ſimilar to the curve line FGHI (Pl. 13 Fig. 4.) and whoſe parts. ſimilar and proportional to the parts FG, GH, HI of the other, may be intercepted between the right lines AB and AD, AD and BD, BD and CE given by poſition, viz. the firſt between the firſt pair of thoſe lines, the ſecond between the ſecond, and the third between the third. Draw the right lines FG, GH, HI, FI; and (by lem. 27.) deſcribe a trapezium
