*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
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,