I′ in p′, viz. into the point where the parallel to p through S cuts p′. Similarly one point J in p will be projected into the point J′ at infinity in p′. This point J is of course the point where the parallel to p′ through S cuts p. We thus see that every point in p is projected into a single point in p′.
Fig. 5 shows that a segment AB will be projected into a segment A′B′ which is not equal to it, at least not as a rule; and also that the ratio AC : CB is not equal to the ratio A′C′ : C′B′ formed by the projections. These ratios will become equal only if p and p′ are parallel, for in this case the triangle SAB is similar to the triangle SA′B′. Between three points in a line and their projections there exists therefore in general no relation. But between four points a relation does exist.
§ 13. Let A, B, C, D be four points in p, A′, B′, C, D′ their projections in p′, then the ratio of the two ratios AC : CB and AD : DB into which C and D divide the segment AB is equal to the corresponding expression between A′, B′, C′, D′. In symbols we have
| AC | : | AD | = | A′C′ | : | A′D′ | . |
| CB | DB | C′B′ | D′B′ |
This is easily proved by aid of similar triangles.
 |
| Fig. 6. |
Through the points A and B on p draw parallels to p′, which cut the projecting rays in C2, D2, B2 and A1, C1, D1, as indicated in fig. 6. The two triangles ACC2 and BCC1 will be similar, as will also be the triangles ADD2 and BDD1.
The proof is left to the reader.
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.
II. If in a cross-ratio the two points belonging to one of the two groups be interchanged, the cross-ratio changes into its reciprocal, i.e.
From I. and II. we see that eight cross-ratios are associated with (AB, CD).
III. If in a cross-ratio the two middle letters be interchanged, the cross-ratio α changes into its complement 1 − α, i.e. (AB, CD) = 1 − (AC, BD).
[§ 16. If λ = (AB, CD), μ = (AC, DB), ν = (AD, BC), then λ, μ, ν and their reciprocals 1/λ, 1/μ, 1/ν are the values of the total number of twenty-four cross-ratios. Moreover, λ, μ, ν are connected by the relations
− λμν = 1;
this proposition may be proved by substituting for λ, μ, ν and reducing to a common origin. There are therefore four equations between three unknowns; hence if one cross-ratio be given, the remaining twenty-three are determinate. Moreover, two of the quantities λ, μ, ν are positive, and the remaining one negative.
The following scheme shows the twenty-four cross-ratios expressed in terms of λ, μ, ν.]
| (AB, CD) (BA, DC) (CD, AB) (DC, BA) | λ | 1 − μ | 1/(1 − ν) | (AC, DB) (BD, CA) (CA, BD) (DB, AC) | 1/(1 − λ) | 1/μ | (ν − 1)/ν | ||
| (AB, DC) (BA, CD) (CD, BA) (DC, AB) | 1/λ | 1/(1 − μ) | 1 − ν | (AD, BC) (BC, AD) (CB, DA) (DA, CB) | (λ − 1)/λ | μ/(μ − 1) | ν | ||
| (AC, BD) (BD, AC) (CA, DB) (DB, CA) | 1 − λ | μ | ν/(ν − 1) | (AD, CB) (BC, DA) (CB, AD) (DA, BC) | λ/(λ − 1) | (μ − 1)/μ | 1/ν |
§ 17. If one of the points of which a cross-ratio is formed is the point at infinity in the line, the cross-ratio changes into a simple ratio. It is convenient to let the point at infinity occupy the last place in the symbolic expression for the cross-ratio. Thus if I is a point at infinity, we have (AB, CI) = −AC/CB, because AI : IB = −1.
Every common ratio of three points in a line may thus be expressed as a cross-ratio, by adding the point at infinity to the group of points.
Harmonic Ranges
§ 18. If the points have special positions, the cross-ratios may have such a value that, of the six different ones, two and two become equal. If the first two shall be equal, we get λ = 1/λ, or λ2 = 1, λ = ±1.
If we take λ = +1, we have (AB, CD) = 1, or AC/CB = AD/DB; that is, the points C and D coincide, provided that A and B are different.
If we take λ = −1, so that (AB, CD) = −1, we have AC/CB = −AD/DB. Hence C and D divide AB internally and externally in the same ratio.
The four points are in this case said to be harmonic points, and C and D are said to be harmonic conjugates with regard to A and B.
But we have also (CD, AB) = −1, so that A and B are harmonic conjugates with regard to C and D.
The principal property of harmonic points is that their cross-ratio remains unaltered if we interchange the two points belonging to one pair, viz.
For four harmonic points the six cross-ratios become equal two and two:
| λ = −1, 1 − λ = 2, | λ | = 12, | 1 | = −1, | 1 | = 12, | λ − 1 | = 2. |
| λ − 1 | λ | 1 − λ | λ |
Hence if we get four points whose cross-ratio is 2 or 12, then they are harmonic, but not arranged so that conjugates are paired. If this is the case the cross-ratio = −1.
§ 19. If we equate any two of the above six values of the cross-ratios, we get either λ = 1, 0, ∞, or λ = −1, 2, 12, or else λ becomes a root of the equation λ2 − λ + 1 = 0, that is, an imaginary cube root of −1. In this case the six values become three and three equal, so that only two different values remain. This case, though important in the theory of cubic curves, is for our purposes of no interest, whilst harmonic points are all-important.
§ 20. From the definition of harmonic points, and by aid of § 11, the following properties are easily deduced.
If C and D are harmonic conjugates with regard to A and B, then one of them lies in, the other without AB; it is impossible to move from A to B without passing either through C or through D; the one blocks the finite way, the other the way through infinity. This is expressed by saying A and B are “separated” by C and D.
For every position of C there will be one and only one point D which is its harmonic conjugate with regard to any point pair A, B.
If A and B are different points, and if C coincides with A or B, D does. But if A and B coincide, one of the points C or D, lying between them, coincides with them, and the other may be anywhere in the line. It follows that, “if of four harmonic conjugates two coincide, then a third coincides with them, and the fourth may be any point in the line.”
If C is the middle point between A and B, then D is the point at infinity; for AC : CB = +1, hence AD : DB must be equal to −1. The harmonic conjugate of the point at infinity in a line with regard to two points A, B is the middle point of AB.
This important property gives a first example how metric properties are connected with projective ones.
[§ 21. Harmonic properties of the complete quadrilateral and quadrangle.
