Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/191



If two right lines as AC, BD given by poition, and terminating in given points A, B, are in a given ratio one to the other, and the right line CD, by which the indetermined points CD are joined, is cut in K in a given ratio; I ay that the point K will be placed in a right line given by poition. Pl. 11. Fig. 2.

For let the right lines AC, BD meet in E, and in BE take BG to AE, as BD is to AC, and let FD be always equal to the given line EG; and by contruction, EC will be to GD, that is, to EF, as AC to BD, and therefore in a given ratio; and therefore the triangle EFC will be given in kind. Let CF be cut in L o as CL may be to CF in the ratio of CK to CD; and becaue that is a given ratio, the triangle EFL will be given in kind, and therefore the point L will be placed in the right line EL given by poition. Join LK and the triangles CLK, CFD will be imilar; and becaue FD is a given line, and LK is to FD in a given ratio, LK will be alo given. To this let EH be taken equal, and ELKH will be always a parallelogram. And therefore the point K is always placed in the ide HK (given by poition) of that parallelogram. Q. E. D.