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)= CB:ACADDB 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, P'B : PIB; 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 . (AB, CD) = (CD, AB) re (BA, DC) = (DC, BA) . 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 . (AB, CD) = I/(AB, DC) = i/(BA, CD) = I/(CD, BA) = 'ADC, AB) . From I. and II. we see that eight cross-ratios are associated with (AB, CD) . [§ i6 . If X.= (AB, CD), A= (AC, DB), v= (AD, BC), then A, At, v and their reciprocals i/A, I/µ, qv are the values of the

total number of twenty-four cross-ratios . Moreover, A, ti, v are connected by the relations P' A-i-Iiµ=,s+I/v=v+IIA= —Aµv= 1; this proposition may be Droved by substituting for A, is, v andreducing 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 A, µ, v are positive, and the remaining one negative .

The following

scheme shows the twenty-four cross-ratios expressed in terms of A, is, v.] (AB, CD) = (AB, DC) = (BA, CD) . For four harmonic points the six cross-ratios become equal two and two: A=—I,I—A=2,AI=2, =—I,IIA,Ar1=2 . Hence if we get four points whose cross-ratio is 2 or , then they are harmonic, but not arranged so that conjugates are paired . If this is the case the cross-ratio = — I . § 19 . If we equate any two of the above six values of the cross-ratios, we get either X= I, o, oo, or X= -1, 2, 1, or else A becomes a root of the equation A2—A+I =o, that is, an imaginary cube root of —I . 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 § I I, 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 —i . 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 . (AB, CD) ' (AC, DB)' (CD DC) - A t—µ I/(I—v) (CB CA I( —X — 1)/v A,' BD) (DC, BA) , (DB, AC)_ (AB, DC) ' (AD, BC)1 (BA, CD) . IjA IJ(I —ti) C, AB) Bv (~ —I)/A t+ l(µ—I) A) I(BC,AD) , D- (CB, DA) (DA, CB) J L (AC, BD) ' (AD, CB) (BD , AC) 1 —A µ v/(v—1) (BC, DD)~ ) Iµ--I)/i I/ V ( DB) (DA, BC) A/(A— ' (DB, .. § 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 = —i . Every common ratio of three points in a line may thus be ex-pressed as a cross-ratio, by adding the point at infinity to the group of points .