two through _D_, and the remaining pair through _A_ and _C_.
_A_ and _C_ are thus harmonic conjugates with respect to _B_ and _D_. We
may sum up the discussion, therefore, as follows:

*30.* If _A_ and _C_ are harmonic conjugates with respect to _B_ and _D_,
then _B_ and _D_ are harmonic conjugates with respect to _A_ and _C_.

*31. Importance of the notion.* The importance of the notion of four
harmonic points lies in the fact that it is a relation which is carried
over from four points in a point-row _u_ to the four points that
correspond to them in any point-row _u'_ perspective to _u_.

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 DEL