and Loci of ApoUonius, ^-c. 63 



infinitely great eircle^ nor infinitely great in respect to the 

 circle B. The reason of such restriction arises from the 

 peculiar nature of infinitesimal geometry causing the indica- 

 ted operations to be graphically impracticable though men- 

 tally possible. 



However as regards the five principal cases of the problem; 

 viz., Avhcn the circles A, B, C are finite — Avhen A and B are 

 finite and C infinitely small — when A is finite and B and C 

 infinitely small — when A and C are finite and B infinitely 

 great — and when A is finite, C infinitely small and B infin- 

 itely great, this solution is remarkably elegant, and depends 

 on very simple Avell known elementary truths. 



That it is in substance the same as tlie one given by ApoUo- 

 nius, may be easily gathered from Pappus' commentaries on 

 the writings of the celebrated Greek geometers. 



He observed that the Apollonian sohition to the Tangen- 

 cies Avas of such a nature as to indicate a method of inscribing 

 a triangle in a given circle, whose sides Avoidd pass through 

 three given points in a straight line. And then, evidently, 

 in order to prepare for a construction to the general problem 

 of inscribing, in a given circle, a polygon, Avhose sides should 

 pass through given points, he gives the indicated method of 

 solution to the particular case just mentioned, both when the 

 three fixed points are at finite distances from each other, and 

 when one of them is at an infinite distance. 



Now, in the solution just given in the text, nothing would 

 be more apt to suggest itself than the fact that GH, a parallel 

 to FEQ, cuts OPQ in a point S, such that OS has to OQ the 

 known ratio which OG has to OE, or ^ihich rad. A has to 

 rad. B; and that we could .*. solve the problem: — '^Being- 

 given three points O, P, S, in a straight line ; to inscribe a 

 triangle DGH in a given circle so that its sides will pass 

 through these points.^' 



And the method which the present solution indicates is 

 exactly the same as is given in Pappus' INIathematical Collec- 

 tions, as well when the three points are at finite distances as 

 when one of them is at an infinite distance. 



These coincidences in peculiarities are, I consider, sufficient 

 to justify me in believing that I have reproduced the solution 

 of the celebrated Greek geometer. And I feel the better 

 pleased at this as it clears up a long disputed point concerning 

 the claims of the rival ' restoraiions' given by Vieta and 

 Simsoii to the case of the problem in wliich one of the circles 

 as C is infiuitolv small. * 



