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 [pg 60] the measurement of angles a metrical theorem, and any other kind a projective theorem, we may put the case as follows:
Any projective theorem involves another theorem, dual to it, obtainable by interchanging everywhere the words 'point' and 'line.'
105. Self-dual theorems. The theorems of this chapter will be found, upon examination, to be self-dual; that is, no new theorem results from applying the process indicated in the preceding paragraph. It is therefore useless to look for new results from the theorem on the circumscribed quadrilateral derived from Brianchon's, which is itself clearly the dual of Pascal's theorem, and in fact was first discovered by dualization of Pascal's.
106. It should not be inferred from the above discussion that one-to-one correspondences may not be devised that will control certain of the so-called metrical relations. A very important one may be easily found that leaves angles unaltered. The relation called similarity leaves ratios between corresponding segments unaltered. The above statements apply only to the particular one-to-one correspondence considered.