Footnotes
The more general notion of anharmonic ratio, which includes the harmonic ratio as a special case, was also known to the ancients. While we have not found it necessary to make use of the anharmonic ratio in building up our theory, it is so frequently met with in treatises on geometry that some account of it should be given.
Consider any four points, A, B, C, D, on a line, and join them to any point S not on that line. Then the triangles ASB, GSD, ASD, CSB, having all the same altitude, are to each other as their bases. Also, since the area of any triangle is one half the product of any two of its sides by the sine of the angle included between them, we have
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
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
