104 GEOMETRICAL RESEARCHES, 



15. It is evident, from the projective properties of figures, 

 that analogous theorems and porisms, to those established con- 

 cerning plane conies, can be established in respect to " spherical 

 conies" It is also obvious that analogous problems concerning 

 spherical conies can be solved by analogous processes. 



HISTORICAL NOTES : 



The problem solved in this paper is famous from having been 

 the chief instrument in unfolding the theory of poles and polars 

 amongst the matchless geometers of France. 



In 1776, Castillon gave a solution to the particular case in 

 which the conic is a circle and n = 3, which appeared in the 

 " Nemoires of the Academy of Berlin" 



In 1776, Lagrange indicated a method of arriving at a solu- 

 tion to the particular case considered by Castillon, by means of 

 rather complicated trigonometrical equations. 



In 1776, the porism pertaining to this case appeared in 

 the " Opera Reliqua " of Professor Simson, of the University 

 of Glasgow. Simson solved the problem in 1731. 



In 1784, Ottajano and Malfatti (two distinguished Italian 

 Geometers) gave excellent solutions to the more general case in 

 which the curve is a circle and n = any whole number whatever. 

 These solutions were published in the " Memorie della Societa 

 Italiana " of Naples. 



In 1796, Lhuilier gave a solution to this case, or rather he 

 showed how its solution might be made dependent on the solu- 

 tion of trigonometrical equations. 



In 1803, the illustrious Carnot (the republican statesman 

 chosen by Napoleon I. to rally the shattered power of the 

 empire against the combined feudalism of Europe) gave a similar 

 solution to this particular case in his work entitled " Geometric 

 de Position" 



In 1810, Brianchon solved the general problem, in which the 

 curve is any conic, and n any whole number. This solution 

 appeared in the " Journal de P Ecole Poll/technique." 



In 1817, Poncelet (the celebrated French Engineer) gave an 



