A GEOMETRICAL SOLUTION 



TEN PROBLEMS IN THE TANGENCIES OF CIRCLES ; AND, ALSO, OP THE FIFTEEN PROBLEMS 

 IN THE TANGENCIES OF SPHERES, BASED UPON THE PRINCIPLE THAT THE TANGENT 

 LINE, OR TANGENT CURVE, IS THE LIMIT OF ALL SECANT LINES OR CURVES. 



The solution herein given to the problems in the tangencies of circles and of 

 spheres, is strictly based upon the principle that the tangent line, or curve, is the 

 limit to all secant lines or curves. This principle is frequently employed in de- 

 scriptive geometry, and in the discussion of tangents in analytical geometry, but I 

 am not aware that it has ever been employed in the manner set forth in this 

 memoir. Solutions are given of most of these problems in the eleventh volume of 

 the Annales de Mathematique (par J. D. Gergonne, Paris, 1820), and in the first 

 volume of Crelle's Mathematical Journal, Berlin, 1S2G. But the methods employed 

 are cpuite different from that herein explained. It is believed that the following 

 is a more complete generalization of this mass of problems than any heretofore 

 published. The classification given of the problems in the tangencies of spheres, 

 and also the classification of all the cases under each problem, both in the tan- 

 gencies of circles and in the tangencies of spheres, are believed to be novel. 



THE TANGENCIES OF CIRCLES. 



The following table exhibits the ten problems in the tangencies of circles pro- 

 posed by a Greek geometer, Apollonius Pergaeus, who lived A. C. 200. A trans- 

 lation of his Geometria Tactionum (the Geometry of Tangencies), was made by 

 Francis Vieta. An edition of Vieta, in French, was published by M. Herigon, in 

 1644. Apollonius recorded some complex solutions, but nothing of a general 

 character. 



