Many theorems of projective geometry are succinctly stated in terms of anharmonic ratios. Thus, the anharmonic ratio of any four elements of a form is equal to the anharmonic ratio of the corresponding four elements in any form projectively related to it. The anharmonic ratio of the lines joining any four fixed points on a conic to a variable fifthpoint on the conic is constant. The locus of points from which four points in a plane are seen along four rays of constant anharmonic ratio is a conic through the four points. We leave these theorems for the student, who may also justify the following solution of the problem: Given three points and a certain anharmonic ratio, to find a fourth point which shall have with the given three the given anharmonic ratio. Let A, B, D be the three given points (Fig. 49). On any convenient line through A take two points B' and D' such that AB'/AD' is equal to the given anharmonic ratio. Join BB' and DD' and let the two lines meet in S. Draw through S a parallel to AB'. This line will meet AB in the required point C.
Pappus, Mathematicae Collectiones, vii, 129.
J. Verneri, Libellus super vigintiduobus elementis conicis, etc. 1522.
Kepler, Ad Vitellionem paralipomena quibus astronomiae pars optica traditur. 1604.
Desargues, Bruillon-project d'une atteinte aux événements des rencontres d'un cône avec un plan. 1639. Edited and analyzed by Poudra, 1864.
The term 'pole' was first introduced, in the sense in which we have used it, in 1810, by a French mathematician named Servois (Gergonne, Annales des Mathéématiques, I, 337), and the corresponding term 'polar' by the editor, Gergonne, of this same journal three years later.
Euler, Introductio in analysin infinitorum, Appendix, cap. V. 1748.
Œuvres de Desargues, t. II, 132.
Œuvres de Desargues, t. II, 370.
Œuvres de Descartes, t. II, 499.