Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/208

 140 BR it to BD as FH to FG. Wherefore fh is to fg as FH to FG. Since therefore, gi to hi likewie is as Mi to Li, that is, as GI to HI, it is manifet that the lines FL, fi, are imilary cut in G and H, g and h. Q. E. F.

In the contruction of this corollary, after the line LK is drawn cutting CE in i, we may produce iE to V, o as EV may be to Ei as FH to HI, and then draw Vf parallel to BD. It will come to the ame, if about the centre i, with an interval IH, we decribe a circle cutting BD in X, and produce iX to Y o as iY may be equal to IF, and then draw Yf parallel to BD.

Sir Chritopher Wren, and Dr. Wallis have long ago given other olutions of this problem.

To decribe a trajectory given in kind, that may be cut by four right lines given by poition, into parts given in order, kind and proportion.

Suppoe a trajectory is to be decribed that may be imilar to the curve line FGHI (Pl. 13 Fig. 4.) and whoe parts. imilar 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 poition, viz. the firt between the firt pair of thoe lines, the econd between the econd, and the third between the third. Draw the right lines FG, GH, HI, FI; and (by lem. 27.) decribe a trapezium fghi