Now the fraction on the right would be unchanged if instead of the points _A_, _B_, _C_, _D_ we should take any other four points _A’_, _B’_, _C’_, _D’_ lying on any other line cutting across _SA_, _SB_, _SC_, _SD_. In other words, _the fraction on the left is unaltered in value if the points __A__, __B__, __C__, __D__ are replaced by any other four points perspective to them._ Again, the fraction on the left is unchanged if some other point were taken instead of _S_. In other words, _the fraction on the right is unaltered if we replace the four lines __SA__, __SB__, __SC__, __SD__ by any other four lines perspective to them._ The fraction on the left is called the _anharmonic ratio_ of the four points _A_, _B_, _C_, _D_; the fraction on the right is called the _anharmonic ratio_ of the four lines _SA_, _SB_, _SC_, _SD_. The anharmonic ratio of four points is sometimes written (_ABCD_), so that [formula] If we take the points in different order, the value of the anharmonic ratio will not necessarily remain the same. The twenty-four different ways of writing them will, however, give not more than six different values for the anharmonic ratio, for by writing out the fractions which define them we can find that _(ABCD) = (BADC) = (CDAB) = (DCBA)_. If we write _(ABCD) = a_, it is not difficult to show that the six values are [formula] The proof of this we leave to the student. If _A_, _B_, _C_, _D_ are four harmonic points (see Fig. 6, p. *22), and a quadrilateral _KLMN_ is constructed such that _KL_ and _MN_ pass through _A_, _KN_ and _LM_ through _C_, _LN_ through _B_, and _KM_ through _D_, then, projecting _A_, _B_, _C_, _D_ from _L_ upon _KM_, we have _(ABCD) = (KOMD)_, where _O_ is the intersection of _KM_ with _LN_. But, projecting again the points _K_, _O_, _M_, _D_ from _N_ back upon the line _AB_, we have _(KOMD) = (CBAD)_. From this we have _(ABCD) = (CBAD),_ or [formula] whence _a = 0_ or _a = 2_. But it is easy to see that _a = 0_ implies that two of the four points coincide. For four harmonic points, therefore, the six values of the anharmonic ratio reduce to three, namely, 2, [formula], and -1. Incidentally we see that if an interchange of any two points in an anharmonic ratio does not change its value, then the four points are harmonic. [Figure 49] FIG. 49