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_. 2 Pappus, Mathematicae Collectiones, vii, 129. 3 J. Verneri, Libellus super vigintiduobus elementis conicis, etc. 1522. 4 Kepler, Ad Vitellionem paralipomena quibus astronomiae pars optica traditur. 1604. 5 Desargues, Bruillon-project d’une atteinte aux événements des rencontres d’un cône avec un plan. 1639. Edited and analyzed by Poudra, 1864. 6 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. 7 Euler, Introductio in analysin infinitorum, Appendix, cap. V. 1748. 8 Œuvres de Desargues, t. II, 132. 9 Œuvres de Desargues, t. II, 370. 10 Œuvres de Descartes, t. II, 499. 11 Œuvres de Pascal, par Brunsehvig et Boutroux, t. I, 252. 12 Chasles, Histoire de la Géométrie, 70. 13 Œuvres de Desargues, t. I, 231. 14 See Ball, History of Mathematics, French edition, t. II, 233. 15 Newton, Principia, lib. i, lemma XXI. 16 Maclaurin, Philosophical Transactions of the Royal Society of London, 1735. 17 Monge, Géométrie Descriptive. 1800. 18 Poncelet, Traité des Propriétés Projectives des Figures. 1822. (See p. 357, Vol. II, of the edition of 1866.) 19 Gergonne, _Annales de Mathématiques, XVI, 209. 1826._ 20 Steiner, Systematische Ehtwickelung der Abhängigkeit geometrischer Gestalten von einander. 1832. 21 Von Staudt, Geometrie der Lage. 1847. 22 Reye, Geometrie der Lage. Translated by Holgate, 1897. 23 Ball, loc. cit. p. 261.