This result is of fundamental importance.
The expression AC/CB : AD/DB has been called by Chasles the “anharmonic ratio of the four points A, B, C, D.” Professor Clifford proposed the shorter name of “cross-ratio.” We shall adopt the latter. We have then the
Fundamental Theorem.—The cross-ratio of four points in a line is equal to the cross-ratio of their projections on any other line which lies in the same plane with it.
§ 14. Before we draw conclusions from this result, we must investigate the meaning of a cross-ratio somewhat more fully.
If four points A, B, C, D are given, and we wish to form their cross-ratio, we have first to divide them into two groups of two, the points in each group being taken in a definite order. Thus, let A, B be the first, C, D the second pair, A and C being the first points in each pair. The cross-ratio is then the ratio AC : CB divided by AD : DB. This will be denoted by (AB, CD), so that
| (AB, CD) = | AC | : | AD | . |
| CB | DB |
This is easily remembered. In order to write it out, make first the two lines for the fractions, and put above and below these the letters A and B in their places, thus, A/*B : A/*B; and then fill up, crosswise, the first by C and the other by D.
§ 15. If we take the points in a different order, the value of the cross-ratio will change. We can do this in twenty-four different ways by forming all permutations of the letters. But of these twenty-four cross-ratios groups of four are equal, so that there are really only six different ones, and these six are reciprocals in pairs.
We have the following rules:—
I. If in a cross-ratio the two groups be interchanged, its value remains unaltered, i.e.