and Loci of Apollonius, S^'c. 65 



How such an able geometer coiild look so long in vain for 

 a solution to the tangencies Avhieh might implicate the 11 7th 

 propositio 1 of Pappus' 7tli Book, I am at a loss to understand ; 

 though it evidently accounts for his implied opinion that the 

 general problem of the three circles was originally referred to 

 the particular case of tico circles and a point. 



Indeed, I may mention that the solution Avhich I give 

 as that of Apollonius was the fii-st one which suggested itself 

 to me for the general question of the three circles ; though 

 not exactly in the form in which I now present it : for after 

 arriWng at the point in the analysis in which M is shown to 

 be found, I proceeded as folloAvs : — 



Since HI is parallel to PM, a circle through M, P and D 

 touches the circle xV in D ; and .•., since P and M are knoA^Ti, 

 this touching circle ]MDP is known ; and hence ODE, PDF, 

 and circle DEF arc known. 



I may further note, that we may give another method of 

 solution implicating Pappus' lemma, by supposing K the point 

 in which DK parallel to OPQ cuts circle DEF, and V that in 

 which FK cuts OPQ. 



For as the angle FE right to K = DE right to K, it is= 

 OE right to V and .-. QV.QO being = QE.QF, the point V 

 is known. 



And if T be the point in which the tangent at D cuts OPQ ; 

 then since the angle FK riglit to D ;= DK right to T, it is= 

 TP riglit to D, and .-., PT.PV being = PD.PF, the point T is 

 known. Hence the tangent TD to circle A is known, and .•. 

 ODE, PDF and circle DEF are known. 



It is evident this method holds graphically good only when 

 A is finite and C neither infinitely small, nor infinitely small 

 in respect to the circle B. It applies to the case in which A. 

 is finite and B and C infinitely greats ; but does not to that 

 in which A is finite and B and C infinitely smalls. 



Similar solutions to the two just indicated are obvious from 

 the "Involution'' Theory as unfolded in the Geometric 

 Superieure : — 



1. Let D, E, F be the points of contact with the given 

 circles, and OPQ the known centres of similitude, in a 

 straight line, through which DE, DF and EF pass. If G and 

 II be the other points in which Dl^] and DF cut the circle A, 

 then G H is parallel to EF, and OG : OE : : rad A : rad B ; 

 and hence S the point in which GH cuts OQ is known. 



NoAv, if we suppose T the point in which a tangent to the 

 circle A at D cuts OQ ; then, since we may regard GHDD as 



r 



