Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/207

 proportion among themelves. Let the angles FGH, GHI, be o far increaed that the right lines FG, GH, HI, may lie in direction, and by contructing the problem in this cae, a right line fghi will be drawn, whoe parts fg, gh, hi, intercepted between the four right lines given by poition, AB and AD, AD and BD, BD and CE, will be one to another as the lines FG, GH, HI, and will oberve the ame order among themelves. But the ame thing may be more readily done in this manner.

Produce AB to K (Pl. 13. Fig. 2.) and BD to L, o as BK may be to B, 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, o 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 ay the thing is done.

For let Mg cut the right line AB in Q and AD the right line KL in S, and draw, AP parallel to BD, and meeting iL in P, and gM to Lb (gi to bi, Mi to Li, GI to HI, AK to BK) and, AP to BL will be in the ame ratio. Cut DL in R, o as DL to RL may be in that ame ratio; and becaue gS to gM, AS to AP, and DS to DL are proportional; therefore (ex æquo) as gS to Lb, o will AS be to BL, and DS to RL; and mixtly BL - RL to Lh - BL, as AS - DS to gS - AS. That is, BR is to Bh, as AD is to Ag, and therefore as BD to gQ. And alternately BR is to BD, as Bh to gQ, or as fh to fg. But by contruction the line BL was cut in D and R, in the ame ratio as the line FI in G and H; and therefore