*29. Four harmonic points.* Four points, _A_, _B_, _C_, _D_, related as in the preceding theorem are called _four harmonic points_. The point _D_ is called the _fourth harmonic of __B__ with respect to __A__ and __C_. Since _B_ and _D_ play exactly the same rôle in the above construction, _B__ is also the fourth harmonic of __D__ with respect to __A__ and __C_. _B_ and _D_ are called _harmonic conjugates with respect to __A__ and __C_. We proceed to show that _A_ and _C_ are also harmonic conjugates with respect to _B_ and _D_—that is, that it is possible to find a quadrangle of which two opposite sides shall pass through _B_, two through _D_, and of the remaining pair, one through _A_ and the other through _C_. [Figure 5] FIG. 5 Let _O_ be the intersection of _KM_ and _LN_ (Fig. 5). Join _O_ to _A_ and _C_. The joining lines cut out on the sides of the quadrangle four points, _P_, _Q_, _R_, _S_. Consider the quadrangle _P_, _K_, _Q_, _O_. One pair of opposite sides passes through _A_, one through _C_, and one remaining side through _D_; therefore the other remaining side must pass through _B_. Similarly, _RS_ passes through _B_ and _PS_ and _QR_ pass through _D_. The quadrangle _P_, _Q_, _R_, _S_ therefore has two opposite sides through _B_, 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._