16 THE TANGENCIES OP CIRCLES AND OF SPHERES. 



If the last problem in the Tangencies of Spheres had been solved algebraically, 

 by referring the spheres to three co-ordinate planes, the resulting equation would be 

 of the sixteenth degree. It will, therefore, be seen that still greater complexity 

 would attend any attempt thus to solve this problem, than was encountered by 

 Descartes in his trial of the questions in the Tangencies of Circles. As the num- 

 ber of solutions, in the separate problems in the Tangencies of Spheres and of 

 Circles, is 16, 8, 4, 2, and 1, all even multiples of 2, I had conjectured that the 

 resulting equation, in either case, might be in the form of a quadratic, but I do not 

 think it has ever been obtained by mathematicians. 



But it will be readily admitted that a geometrical solution, generalizing the 

 entire series, such as we have presented, is much more satisfactory than any which 

 could be obtained by the aid of algebra: 



Besides the elaborate dissertations referred to on the third page of this paper, fre- 

 quent geometrical solutions of the problems in the Tangencies of Circles have been 

 given — as in the first book of Newton's Principle/,, in Leslies Geometry, and in the 

 Geometry of the " Library of Useful Knowledge" — but none of them reduce the 

 question to the elementary principle above announced. 



The author offers this memoir for publication, under the belief that it is one of 

 the highest aims of science to simplify and reduce a complex subject to its elements. 

 An ingenious solution of detached problems would be less valuable, but the eluci- 

 dation and classification of an entire system, and the development of a principle, 

 however simple, which underlies said system, may be considered worthy of record. 



PUBLISHED BY THE SMITHSONIAN INSTITUTION, 



WASHINGTON, D. C. 



JANUARY, 1856. 



