PROJECTION AND  CROSS-RATIOS § 12 . If we join a point A to a point S, then the point where the line SA cuts a fixed plane ,r is called the projection of A on the plane a from S as centre of projection . If we have two planes it and Zr and a point S, we may project every point A in 7 to the other plane, If A' is the projection of A, then A is also the projection of A', so that the relations are reciprocal . To every figure in ,r we get as its projection a corresponding figure in . We shall determine such properties of figures as remain true for the projection, and which are called projective properties . For this purpose it will be sufficient to consider at first only constructions in one plane . Let us suppose we have given in a plane two lines p and p' and a centre S (fig . 4); we may then project the points in p from S to p', Let A', B' .. be the projections of A, B . . ., the point at infinity in p which we shall denote by I will be projected into a finite point single point in the line AB . [Relations between segments of lines are interesting as showing an application of algebra to geometry . The genesis of such relations • C I' in p', viz. into the point where the parallel to p through S cuts Similarly one point J in p will be projected into the point JJ' 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 pro- tctions there exists therefore in general no relation . But between four points a relation does exist . § i . 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 . Through the points A and B on p draw parallels to p', which cut the projecting rays in s C2, D2, B2 and Al, Cl, DI, as indicated in fig . 6 . The two triangles ©s ACC2 and BCC' will be similar, as will also be the triangles ADD2 and BDDI .

The

proof is left to A the reader . This result of fundamental importance . The expression A' c• d _s' AC/CB:AD/DB has been called by Chasles the " anharmonic ratio of the four points A, B, C, D." Professor Clifford pro- posed the shorter name of " cross-ratio." We shall adopt the latter .