This association[35] is most easily effected by considering a class of projective relations of the line with itself, called by F. Schur (loc. cit.) prospectivities.
| Fig. 69. |
| Fig. 70. |
| Fig. 71. |
| Fig. 72. |
Let l (fig. 69) be the given line, m and n any two lines intersecting at U on l, S and S′ two points on n. Then a projective relation between l and itself is formed by projecting l from S on to m, and then by projecting m from S′ back on to l. All such projective relations, however m, n, S and S′ be varied, are called “prospectivities,” and U is the double point of the prospectivity. If a point O on l is related to A by a prospectivity, then all prospectivities, which (1) have the same double point U, and (2) relate O to A, give the same correspondent (Q, in figure) to any point P on the line l; in fact they are all the same prospectivity, however m, n, S, and S′ may have been varied subject to these conditions. Such a prospectivity will be denoted by (OAU²).
The sum of two prospectivities, written (OAU²) + (OBU²), is defined to be that transformation of the line l into itself which is obtained by first applying the prospectivity (OAU²) and then applying the prospectivity (OBU²). Such a transformation, when the two summands have the same double point, is itself a prospectivity with that double point.
With this definition of addition it can be proved that prospectivities with the same double point satisfy all the axioms of magnitude. Accordingly they can be associated in a one-one correspondence with the positive and negative real numbers. Let E (fig. 70) be any point on l, distinct from O and U. Then the prospectivity (OEU²) is associated with unity, the prospectivity (OOU²) is associated with zero, and (OUU²) with ∞. The prospectivities of the type (OPU²), where P is any point on the segment OEU, correspond to the positive numbers; also if P′ is the harmonic conjugate of P with respect to O and U, the prospectivity (OP′U²) is associated with the corresponding negative number. (The subjoined figure explains this relation of the positive and negative prospectivities.) Then any point P on l is associated with the same number as is the prospectivity (OPU²).
It can be proved that the order of the numbers in algebraic order of magnitude agrees with the order on the line of the associated points. Let the numbers, assigned according to the preceding specification, be said to be associated with the points according to the “numeration-system (OEU).” The introduction of a coordinate system for a plane is now managed as follows: Take any triangle OUV in the plane, and on the lines OU and OV establish the numeration systems (OE1U) and (OE2V), where E1 and E2 are arbitrarily chosen. Then (cf. fig. 71) if M and N are associated with the numbers x and y according to these systems, the coordinates of P are x and y. It then follows that the equation of a straight line is of the form ax + by + c = 0. Both coordinates of any point on the line UV are infinite. This can be avoided by introducing homogeneous coordinates X, Y, Z, where x = X/Z, and y = Y/Z, and Z = 0 is the equation of UV.
The procedure for three dimensions is similar. Let OUVW (fig. 72) be any tetrahedron, and associate points on OU, OV, OW with numbers according to the numeration systems (OE1U), (OE2V), and (OE3W). Let the planes VWP, WUP, UVP cut OU, OV, OW in L, M, N respectively; and let x, y, z be the numbers associated with L, M, N respectively. Then P is the point (x, y, z). Also homogeneous coordinates can be introduced as before, thus avoiding the infinities on the plane UVW.
The cross ratio of a range of four collinear points can now be defined as a number characteristic of that range. Let the coordinates of any point Pr of the range P1 P2 P3 P4 be
| λra + μr + a′ | , | λrb + μrb′ | , | λrc + μrc′ | , (r = 1, 2, 3, 4) |
| λr + μr | λr + μr | λr + μr |
and let (λrμs) be written for λrμs -λsμr. Then the cross ratio {P1 P2 P3 P4} is defined to be the number (λ1μ2)(λ3μ4) / (λ1μ4)(λ3μ2). The equality of the cross ratios of the ranges (P1 P2 P3 P4) and (Q1 Q2 Q3 Q4) is proved to be the necessary and sufficient condition for their mutual projectivity. The cross ratios of all harmonic ranges are then easily seen to be all equal to -1, by comparing with the range (OE1UE′1) on the axis of x.