and Loci of Apof/onim, ^c. 67 



from thence, he himself acknowledges that one fm*nished him 

 with so complicated an expression, that he would not under- 

 take to constnict it in a month; while the other, though 

 somewhat less complicated, was not so very simple as to 

 encourage him to set about the construction of it. Lastly, 

 the Princess Elizal)eth of Bohemia, who it is well known 

 honored Pcscartes Avitli her correspondence, deigned to com- 

 municate a solution to this philosopher; but, as it is deduced 

 from the algebraic calcidus, it labours under the same incon- 

 veniences as that of Descartes/^ 



Euler, Fuss, T. Simpson, and other eminent analysts have 

 given algebraic solutions, though not at all commensurate 

 with the requirements of the problem. T. Simpson has also 

 given a geometrical solution in the appendix to his Elements 

 of Geometry, which, in reality, does not differ in principle 

 from Newton^s in the Principia. 



These solutions, like those of Euler, are very imperfect, 

 though complete ones of a similar nature, and much more 

 simple, can be easily formed. 



The late John ^lulcahy. L.L.D., professor. Queen's College, 

 Gahvay, after giving Gaulthier's solution as improved by Ger- 

 gonne, in article 68 of his Principles of Modern Geometry, 

 again returns to the subject in article 95, and deduces Gaul- 

 thier's original method depending on the circle Avhicli cuts 

 the three given ones orthogonally: this of course labours 

 under the disadvantage of being inapplicable when the radical 

 centre is witliin the three given circles. 



Those who are acquainted vn.\\\ the Principles of Modern 

 Geometry, or the Avi-itings of the late Professor Davies, of the 

 Royal ]\lilitary Academy, will at once see that all my methods 

 are applicable when, instead of three circles in a plane, there is 

 given three circles on the surface of a sphere. The only differ- 

 ence being that straight lines, Avhether in data or solution, 

 will be represented by great circles of the sphere. 



My solutions have also analogous ones answering to the 

 following celebrated problem, Avhich was proposed hy Des- 

 cartes to Fermat: — "Suppose four things, A, B, C, D, to be 

 given in position, consisting of points, planes, and spheres, 

 which may be taken of any one of these kinds exclusively , or of 

 any two of the kinds, or of all of the three kinds ; it is required 

 to describe a sphere ivhich shall pass through each of the given 

 points, and touch the given planes or spheres." 



All we have to do is (as in considering the Apollonian 

 problem) to form the solution for the general case in which 



r 2 



