50 The Three Sections, Tangencies, 



through the point of contact M, on circle C^, draw PM to cut 

 the circle C again in F^ and to cut the circle A in D similarly 

 to the point M on circle C ; draw ON to cut circle B in E 

 similarly to the point N on circle I ; describe the circle DEF. 

 Then is DEF a required circle. 



NOTES. 



This method of solution holds intelligibly good in all cases 

 in which neither of the circles A or C is infinitely great. 

 When the circles A and B are infinitely smalls the centre of 

 similitude O may have any position whatever in the line AB^ 

 as the ratio of their radii may be of any magnitude ; and 

 similar remarks apply to the centre of similitude Q when the 

 circles B and C are infinitely smalls. By fixing the ratio of 

 these infinitely small circles, we fix the positions' of the centres 

 of similitude ; and it is evident we may suppose one of them 

 infinitely small in respect to the other, so as to have the centre 

 of similitude coincident with this other in respect to finite 

 distances. And similar remarks apply as to the ratios of 

 infinitely great radii. 



This solution furnishes three methods to the case in which 

 two of the given circles are finite and the third infinitely 

 small. And that one in which we have A the infinitel}'^ small 

 circle is in substance the same as what is given by Monsieur 

 Auguste Cauchy. 



We are furnished with two methods for the case in which 

 two of the circles are infinitely smalls, and the third finite : — • 

 one of which (when A and B are the infinitely smalls), is in 

 substance the same as what is given by Pappus as the solution 

 to this case from the Work of ApoUonius. 



However, here as elsewhere, when I speak of a general 

 solution being inapplicable to any case or cases, it is to be 

 considered inapplicable only in a graphical point of view, 

 for a general solution holds mentally good in all cases, even 

 when quantities may be infinitely great or small ; and the 

 mind^s conviction in such cases is established by its knowledge 

 concerning properties of finite quantities and its own power 

 of legitimately applying the principle of ' continuity ' derived, 

 in degree, from this knowledge. 



It is also to be observed that owing to our imperfect know- 

 ledge of infinitesimal geometry, or to the nature even of this 

 geometry, it m^ay often happen that we cannot intelligibly 

 arrive at some necessary theorera from one point of view, so 



