43. Numerical relations. Since three points, given in order, are sufficient to determine a fourth, as explained above, it ought to be possible to reproduce the process numerically in view of the one-to-one correspondence which exists between points on a line and numbers; a correspondence which, to be sure, we have not established here, but which is discussed in any treatise on the theory of point sets. We proceed to discover what relation between four numbers corresponds to the harmonic relation between four points.
Fig. 8
44. Let A, B, C, D be four harmonic points (Fig. 8), and let SA, SB, SC, SD be four harmonic lines. Assume a line drawn through B parallel to SD, meeting SA in A' and SC in C'. Then A', B', C', and the infinitely distant point on A'C' are four harmonic points, and therefore B is the middle point of the segment A'C'. Then, since [pg 26] the triangle DAS is similar to the triangle BAA', we may write the proportion
AB : AD = BA' : SD.
Also, from the similar triangles DSC and BCC', we have
CD : CB = SD : B'C.
From these two proportions we have, remembering that BA' = BC',
