334 Mr. T. S. Davies's Properties 



were never printed, or have at all events been totally lost, as 

 the Editor of his Works was never able to procure them either 

 printed or in manuscript. Our only knowledge of their having 

 existed is a letter of Leibnitz to M. Perrier, the nephew of 

 Pascal, which is printed at the end of the fifth volume of the 

 edition of his Works, before mentioned : and it seems from 

 that letter, that he attached some importance to the inquiry, 

 and perhaps some occult qualities to the figure, by the name 

 which he conferred upon it, " U Hexagramme Mystique." But 

 what were the properties which he discovered, cannot now be 

 known; nor even whether he had examined all the cases to 

 which the varied relations between the component parts of the 

 hypothesis may give rise. The most probable conjecture we 

 can form is, that his discoveries were those which most readily 

 suggested themselves to us in an examination of the same 

 figure. Those which follow are a few of the most obvious 

 and interesting which occurred to me in studying the subject 

 some time ago; and which I ought to add was before I had 

 seen Pascal's Works. I had been more than once surprised 

 to find that properties of the conic sections which were new in 

 appearance were but particular cases of this singular theorem. 

 I was thus led to examine its different cases, and the following 

 Form (virtually the same as Pascal's) was that which offered 

 itself to my consideration. 



Prop. I. 



DE and FG are two chords in any conic section, which 

 intersect in B ; and F, I are two points taken at pleasure in 

 the circumference of the section. Let EF, HI intersect in A, 

 and DI, FG in C : then A, B, C are one straight line. 



As B may be either within or without the conic section, 

 two Classes of cases corresponding thereto will result. 



It is unnecessary to insert the figures to this Prop, as the 

 reader can sketch them himself. 



Class I. — When B is without the conic section; and 



When I is in the arc GH and F in the arc 



4 



When I is in the arc DE and F in the arc 



When I is in the arc DH and F in the arc 



EG } . 



. . 1 



GHf . 



. . 2 



HDf . 



. . 3 



DE 1 . 



. . 4- 



EG \ . 



. . 5 



GHf . 



. . 6 



HDf . 



. . 7 



DE ) . 



. . 8 



EG \ . 



. . 9 



GHf . 



. . 10 



HD( . 



. . 11 



DE ) . 



. . 12 





When 



