38 Mr. T. S. Davies on Pascal's Mystic Hexagram. 



theory of the pole and polar ;" the description of the conic 

 sections by revolving line9 or the sides of revolving angles, 

 first suggested by Newton, and followed out in detail by Mac- 

 laurin and Braikenridge, also flow at once from this theorem. 

 In short, for generality and facility of employment there is 

 only one other principle that can compete with it; which is 

 that of the anharmonic ratio of M. Chasles, as developed in 

 the notes to his Apercu Historique des Methodes en Geometric. 



The demonstration of this theorem was not, however, pub- 

 lished by Pascal; nor, I think, has there ever been given 

 a strictly geometrical demonstration in the manner of the an- 

 cients. For the circle the demonstration is very simple and 

 elegant; of which four specimens may be seen in the Mathe- 

 matical Repository, vol. iv. New Series, one of which by Mr. 

 Ivory is inserted by Dr. Bland in his Geometrical Problems. 

 The method of projection is employed to extend it to the other 

 conic sections : but admitting the theory of transversals, the 

 property admits of a very short and direct demonstration for 

 the conic sections generally. The proposition itself in the 

 general form was proposed in the Ladies' Diary for the present 

 year, to be established without any direct or implied use of the 

 circle ; and in reply to that, the demonstration above alluded 

 to has been given, and will appear in the next year's Diary. 



Many attempts, with different degrees of success and ele- 

 gance, have been made by the continental geometers to solve 

 this by the method of coordinates. I believe, however, that 

 except by Sir John Lubbock* and an imperfect sketch of my 

 ownf (which is here followed out and completed), no one of 

 our countrymen has looked at the subject in this light. I am 

 led, therefore, to think that the following investigation will 

 be interesting to geometers ; it being, I believe, very different 

 from any process published by other writers. 



The o rem. If the three pa irs of opposite sides of a hexagon 

 inscribed in a conic section be produced to meeU the three 'points 

 of concourse will be in one straight line. 



Take the opposite sides 

 A D, B C, uniting in O, 

 as axes of coordinates ; and 

 denote the distances O A, 

 O B, O C, O D by «, ft 

 y, 8, and the two remain- 

 ing angular points F and 

 E of the hexagon by (x x y^) 

 and (a? 8 y a ). 



* Phil. Mag. Third Series, vol. xiii. p. 83. 

 t Solutions to Hutton's Questions, p. 505. 



