^52 THE MATHEMATICAL PRINCIPLES [BOOK I. S, arid draw AP parallel to BD, and meeting iL in P, and §-M to Lh (g\ to hi, Mi to Li, GI to HI, AK to BK) and AP to BL, will be in the same ratio. Cut DL in 11, so as DL to RL may be in that same ratio; and be cause ffS to g~M, AS to AP. and DS to DL are proportional; therefore (ex ceqit.o) as gS to LA, so 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 Eh as AD is to Ag, and therefore as BD to gQ. And alternately BR is to BD as 13/i to g-Q,, or as fh to fg. But by construction the line BL was cut in D and R in the same ratio as the line FI in G and H ; and therefore BR is 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 FI, fi, are similarly cut in G and H, g and //.. 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, arid then draw V/~ 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 BO. Sir Christopher Wren and Dr. Wallis have long ago given other solu tions 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, 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 po sition, 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. XXVII) describe a trapezium fghi that may be similar to the trapezium FGHI, and whose an gles/, g, h, i, may touch the right lines given by posi tion AB, AD, BD, CE, severally according to their order. And then about bins trapezium describe a trajectory, that trajectory will be similar to the curve line FGHI. SCHOLIUM. This problem may be likewise constructed in the following manner. Joining FG, GH, HI, FI, produce GF to Y, and join FH, IG, and make