22' The Three Sections, Tangencies, 



ment of fhe precise limiting values for the ratio '-^ . Surely, 

 it is not evident that there are two limiting positions for F, 

 such as/ and/', and that according as the ratio ~ lies outside 

 the limits ^ and S^J [h and h' being the points in Avhich/A 

 and /'A cut NN), or is equal to one of them, or is com- 

 prehended between them, so will the corresponding points C 

 be real and distinct, real and coincident, or imaginary. 



And similar remarks apply to his solutions to the Section 

 of Space and to the Determinate Section ; for the homogra- 

 phic theory will not establish the limits, nor even hint as to 

 their nature or namber. 



My sohitions are equally general with those given by 

 Chasles, and — as v/ill be seen in the Generating Problem — 

 one wording applies to the three questions in their most 

 general forms. Besides, they possess the distinguishing 

 characteristic of being intelligibly applicable to all the parti- 

 cular cases ; and the simple considerations, by means of 

 which the limits are established, will be found applicable to 

 the determinations of limits in numerous other important 

 questions. 



The next in order of the works of ApoUonius, after the 

 Determinate Section, was the " Tangencies.-" 



The enunciation of the problem, and some of the ""^ Lemmas'^ 

 used in its solution, which were preserved in the Mathemati- 

 cal Collection of Pappus, enabled Dr. Robert Simson, of 

 Glasgow, to reproduce one case (that of two circles and a 

 point) though not under its original form, — as may be seen in 

 the Appendix to his Opera Reliqua ; but a more elegant solu- 

 tion to the same Avas previously given by A^ieta, in his ApoUo- 

 nius Gallus. And since Dr. Simson^s, an entirely different 

 solution has been given by Monsieur Auguste Cauchy, in the 

 " Correspondence de I'Ecole Polyteehnique.^^ 



However, neither Simson, Vieta, nor Cauchy succeeded 

 in giving a direct solution to the general question. 



Newton virtually solved the general question in his Prin- 

 cipise, where it entered into some astronomical determina- 

 tions ; and, indeed, it is the only direct geometrical solu- 

 tion by a British geometer which applies to the various 

 cases, when we suppose the circles to have any value from 

 zero to infinit3^ 



But the most complete and elegant solution hitherto given 

 to the "Tangencies," is that of M. Gergonne, in the Annales 

 de Mathematiqve, which (according to M. Chasles) is an 



