134

Again, in order to continue the curvebeyond the points E and F, take KT, andLU each equal to any one of the divisionsalready made on CK, say equal to NK, andthrough the points T and U draw TI andUW, making the former a third propor-tional to NO and KH, and the latter a thirdproportional to MP and LG ; join IW cut-ting the axis in X, and through X drawYXZ parallel to FE, cutting TI in Z andUW produced in Y, so shall Y and Z betwo points in the curve, as is evident fromarticle 21. In like manner by takingmean and third proportionals the construc-tion may be carried to any extent, and toany degree of exactness.

25. Problem. Given, any two of the sixquantities c, x, t, ?, z, and y, to constructthe catenary.

CASE I.

Fig. 27 . When c and x are the two given quan-tities.