To prove this statement we construct a quadrangle K, L, M, N such that KL and MN pass through A, KN and LM through C, LN through B, and KM through D.  Take now any point S not in the plane of the quadrangle and construct the planes determined by S and all the seven lines of the figure.  Cut across this set of planes by another plane not passing through S.  This plane cuts out on the set of seven planes another quadrangle which determines four new harmonic points, A’, B’, C’, D’, on the lines joining S to A, B, C, D.  But S may be taken as any point, since the original quadrangle may be taken in any plane through A, B, C, D; and, further, the points A’, B’, C’, D’ are the intersection of SA, SB, SC, SD by any line.  We have, then, the remarkable theorem: 

32. If any point is joined to four harmonic points, and the four lines thus obtained are cut by any fifth, the four points of intersection are again harmonic.

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 {~GREEK SMALL LETTER ALPHA~}, {~GREEK SMALL LETTER BETA~}, {~GREEK SMALL LETTER GAMMA~}, {~GREEK SMALL LETTER DELTA~}, which all pass through the line g, pass also through the four harmonic points A, B, C, D, so that {~GREEK SMALL LETTER ALPHA~} passes through A, etc.  Then it is clear that any plane {~GREEK SMALL LETTER PI~} through A, B, C, D will cut out four harmonic lines from the four planes, for they are lines through the intersection P of g with the plane {~GREEK SMALL LETTER PI~}, and they pass through the given harmonic points A, B, C, D.  Any other plane {~GREEK SMALL LETTER SIGMA~} cuts g in a point S and cuts {~GREEK SMALL LETTER ALPHA~}, {~GREEK SMALL LETTER BETA~}, {~GREEK SMALL LETTER GAMMA~}, {~GREEK SMALL LETTER DELTA~} in four lines that meet {~GREEK SMALL LETTER PI~} 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 {~GREEK SMALL LETTER ALPHA~}, {~GREEK SMALL LETTER BETA~}, {~GREEK SMALL LETTER GAMMA~}, {~GREEK SMALL LETTER DELTA~} in four harmonic points, since any plane through the ray gives four harmonic lines of intersection.