64 The Three Sections, Tangencies, 



It will be seen that Vieta's solution, in the most improved 

 form, is the same as that of ApoUonius. 



And here, before closing my notes on this celebrated 

 problem, I may observe that Dr. Robert Simson, like many 

 others, certainly misunderstood the object of propositions 

 116, 117 and 118 of Book VII. of Pappus' Mathematical 

 Collections; and that through this he was led to imagine 

 he restored or reproduced the proposition to which they 

 were intended as subsidiary. 



However, as Dr. Simson's remarks are interesting in a 

 historical point, I will give them as translated from the 

 Appendix to his Opera Reliqua, by Professor Davies. They 

 are as follows : — 



'' In the Seventh Book of the Mathematical Collections of 

 Pappus Alexandrinus (every admirer of the ancient geome- 

 trical analysis ought to rejoice that this work has been 

 preserved to our times), among the lemmas which that most 

 eminent AViiter has handed down, there exists a problem for 

 one of the tangencies of ApoUonius, namely, in Prob. 117, 

 B. VII ; in which it is required, when a circle being given by 

 position and three points in a straight line, to inflect from two 

 of the points two lines meeting in the circumference, so as to 

 make the two points in which they intersect the circle and 

 the third given point in the same straight line. It is not 

 difficult to investigate the rest of the lemmas which are sub- 

 sidiary to the problems on the tangencies ; and some of these 

 Vieta has used in his ApoUonius Gallus ; but to what problem 

 the aforesaid lemma could be subsidiary, neither Vieta nor 

 any other geometer has attempted to conjecture. 



" Often, indeed, have I resolved the subject in my mind, 

 but I have never succeeded in arriving at any satisfactory 

 conclusion, except that the lemma, by no uncertain marks, 

 appeared to be necessary for the following problem : — Two 

 circles and a point being given by position, it is required to 

 describe a third circle which shall touch the given circles and 

 pass through the given point. In what manner, however, 

 the lemma might be subsidiary to this problem I did by no 

 means perceive. I have directed my attention to the solu- 

 tions of Vieta and others, hoping that by chance I might hit 

 upon the analysis requiring this lemma, but in vain; until 

 this day, after various trials, I discovered the true analysis of 

 ApoUonius, to M^hich, indeed, both this Prob. 117 of Pappus, 

 as well as Props. 116 and 118 are manifestly subsidiary. — 

 February 9, 1734.'' 



