*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. [Figure 8] 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 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’_, [formula] the minus sign being given to the ratio on account of the fact that _A_ and _C_ are always separated from _B_ and _D_, so that one or three of the segments _AB_, _CD_, _AD_, _CB_ must be negative. *45.* Writing the last equation in the form _CB : AB = -CD : AD,_ and using the fundamental relation connecting three points on a line, _PR + RQ = PQ,_ which holds for all positions of the three points if account be taken of the sign of the segments, the last proportion may be written _(CB - BA) : AB = -(CA - DA) : AD,_ or _(AB - AC) : AB = (AC - AD) : AD;_ so that _AB_, _AC_, and _AD_ are three quantities in hamonic progression, since the difference between the first and second is to the first as the difference between the second and third is to the third. Also, from this last proportion comes the familiar relation [formula] which is convenient for the computation of the distance _AD_ when _AB_ and _AC_ are given numerically. *46. Anharmonic ratio.* The corresponding relations between the trigonometric functions of the angles determined by four harmonic lines are not difficult to obtain, but as we shall not need them in building up the theory of projective geometry, we will not discuss them here. Students who have a slight acquaintance with trigonometry may read in a later chapter (§ 161) a development of the theory of a more general relation, called the _anharmonic ratio_, or _cross ratio_, which connects any four points on a line. PROBLEMS