33. Four harmonic lines. We are now able to extend the notion of harmonic elements to pencils of rays, and indeed to axial pencils. For if we define four harmonic rays as four rays which pass through a point and which pass one through each of four harmonic points, we have the theorem
Four harmonic lines are cut by any transversal in four harmonic points.
34. Four harmonic planes. We also define four harmonic planes as four planes through a line which pass one through each of four harmonic points, and we may show that
Four harmonic planes are cut by any plane not passing through their common line in four harmonic lines, and also by any line in four harmonic points.
For let the planes α, β, γ, δ, which all pass through the line g, pass also through the four harmonic points A, B, C, D, so that α passes through A, etc. Then it is clear that any plane π through A, B, C, D will cut out four harmonic lines from the four planes, for they are [pg 21] lines through the intersection P of g with the plane π, and they pass through the given harmonic points A, B, C, D. Any other plane σ cuts g in a point S and cuts α, β, γ, δ in four lines that meet π in four points A', B', C', D' lying on PA, PB, PC, and PD respectively, and are thus four harmonic hues. Further, any ray cuts α, β, γ, δ in four harmonic points, since any plane through the ray gives four harmonic lines of intersection.
35. These results may be put together as follows:
Given any two assemblages of points, rays, or planes, perspectively related to each other, four harmonic elements of one must correspond to four elements of the other which are likewise harmonic.
If, now, two forms are perspectively related to a third, any four harmonic elements of one must correspond to four harmonic elements in the other. We take this as our definition of projective correspondence, and say: