CIRCLES AND OF SPHERES. 15 



smallest sphere), a sphere tangent to these three auxiliary spheres. Its centre will 

 be the centre of one of the required tangent spheres. Pass three other auxiliary 

 spheres, A E', B I', and C G', outside the same three given spheres, and distant 

 also the length of the radius of the sphere D. By combining in all possible ways 

 these six auxiliary spheres, all the solutions, numbering sixteen, will be found. The 

 following table classifies the different tangent spheres : — 



No. 1 convex to or inside of all the spheres. 



No. 2 concave to or enclosing all. 



No. 3 convex to A and concave to the other spheres. 



No. 4 " B 



, No. 5 " G 



other three |^ a « J) " " " 



No. 1 concave to A and convex to the other spheres. 



No. 8 " B 



No. 9 " G 



I No. 10 " D 

 "No. 11 convex to A and B and concave to the other spheres. 



No. 12 " ^ and C 



No. 13 " A and D 



No. 14 " OandD 



No. 15 " A and G 



No. 16 " B and D 



Four are convex to one 

 and concave to the 



Four are concave to one 

 and convex to the 

 other three 



Six are convex to two 

 and concave to the 

 other two spheres 



From the auxiliary spheres AE,B I, and G G, are obtained, by Prob. 1 3, Nos. 1 and 2. 



From A E', C G, and B I " 3 and 7. 



From A E, < ' G, and B l "4 and 8. 



From A E, B I, and C G' " 5 and 0. 



From A E' , G G' , and B T " 6 and 10. 



From A E',B I, and G G " H and 14. 



From AE,B T, and G G' "12 and 13. 



From A E' , B I, and C G' "15 and 10. 



If, in this or in any of the previous problems, some of the spheres are enclosed 

 in one of the others, if the problem remains possible, it can be solved on analogous 

 principles, by making use of the " similar points" described in Prob. 6, in Tan- 

 gencies of Circles, see Fig. 3, Plate IV. 



It is interesting to notice that, if these problems had been solved algebraically — 

 as, for instance, by referring the circles, in the last problem in the Tangencies of 

 Circles, to co-ordinate axes, the abscissa of the centre of the required circle being 

 the unknown quantity — the resulting equation would be one of the eighth degree, 

 the eight roots giving the eight solutions. It is not surprising, then, that Descartes 

 should acknowledge (see Montucla's Eistory of Mathematics, p. 264) that, in attempt- 

 ing to apply algebraic analysis to this problem, he "obtained, by one process, an 

 expression so complicated that he would not undertake to construct it in a month ; 

 and, by another process, one less embarrassee, but which was so difficult as to deter 

 him from touching it." If a full analytical expression could be obtained to embrace 

 all cases in Prob. 10, in Tangencies of Circles, the other nine problems could all be 

 solved from it by different suppositions ; as, that the radius of one of the circles 

 was zero and the other infinity, which would reduce it to Prob. 5. 



