102.  Self-polar triangle. In Fig. 26 it is not difficult to see that AOC is a self-polar triangle, that is, each vertex is the pole of the opposite side.  For B, M, O, K are four harmonic points, and they project to C in four harmonic rays.  The line CO, therefore, meets the line AMN in a point on the polar of A, being separated from A harmonically by the points M and N.  Similarly, the line CO meets KL in a point on the polar of A, and therefore CO is the polar of A.  Similarly, OA is the polar of C, and therefore O is the pole of AC.

103.  Pole and polar projectively related. Another very important theorem comes directly from Fig. 26.

As a point _A__ moves along a straight line its polar with respect to a conic revolves about a fixed point and describes a pencil projective to the point-row described by __A__._

For, fix the points L and N and let the point A move along the line AQ; then the point-row A is projective to the pencil LK, and since K moves along the conic, the pencil LK is projective to the pencil NK, which in turn is projective to the point-row C, which, finally, is projective to the pencil OC, which is the polar of A.

104.  Duality. We have, then, in the pole and polar relation a device for setting up a one-to-one correspondence between the points and lines of the plane—­a correspondence which may be called projective, because to four harmonic points or lines correspond always four harmonic lines or points.  To every figure made up of points and lines will correspond a figure made up of lines and points.  To a point-row of the second order, which is a conic considered as a point-locus, corresponds a pencil of rays of the second order, which is a conic considered as a line-locus.  The name ‘duality’ is used to describe this sort of correspondence.  It is important to note that the dual relation is subject to the same exceptions as the one-to-one correspondence is, and must not be appealed to in cases where the one-to-one correspondence breaks down.  We have seen that there is in Euclidean geometry one and only one ray in a pencil which has no point in a point-row perspective to it for a corresponding point; namely, the line parallel to the line of the point-row.  Any theorem, therefore, that involves explicitly the point at infinity is not to be translated into a theorem concerning lines.  Further, in the pencil the angle between two lines has nothing to correspond to it in a point-row perspective to the pencil.  Any theorem, therefore, that mentions angles is not translatable into another theorem by means of the law of duality.  Now we have seen that the notion of the infinitely distant point on a line involves the notion of dividing a segment into any number of equal parts—­in other words, of measuring.  If, therefore, we call any theorem that has to do with the line at infinity or with the measurement of angles a metrical theorem, and any other kind a projective theorem, we may put the case as follows: