116 BULLETIN OF THE 



A B, draw a tangent to the circle L (which will also be tangent 

 to the circle K), and from the point of tangency I of the circle 

 L, with the side A C, draw a tangent to the circle M (which 

 will also be tangent to the circle K) ; in the quadrilateral thus 

 formed, A I m* (wliich is a circumscriptible one), inscribe a 

 circle, and it is one of the circles required; in a similar manner 

 the remaining circles are found. 



The proof of this construction, as also the extension of it 

 which follows, is omitted, and the expression "jedoch oline 

 beweis" leaves us in doubt whether he merely omitted the proof, 

 or whether he had not yet found a proof to omit. 



It is proper to add that Steiner does not stop with merely 

 giving a solution of the problem, but by means of the principles 

 of Radical Axes, Centres of Similitude, etc., which be had 

 developed in the preceding paragraphs, he goes further and 

 solves this more general problem. Three circles in a plane are 

 given in magnitude and position, describe three other circles 

 each touching the other two and two of the given circles. And 

 even this extension is still further extended to the case of three 

 circles lying not in a plane but upon the surface of a sphere. 



After giving the solution of Malfatti's Problem without the 

 extension, he adds that the problem by no means admits of 

 merely a single solution, but that at least thirty-two different 

 solutions of the problem without the extension seem to be pos- 

 sible, all similar to the foregoing. 



These statements are given without proof, and, so far as known 

 to the writer, no geometric proof of this solution with the exten- 

 sions of the problem has been given. It is true that one of the 

 thirty-two solutions has been given by several persons, but no- 

 where apparently by the methods used by Steiner, unless per- 

 chance that by Andrew S. Hart, in Yol. I of the Quarterly 

 Journal of Pure and Applied Mathematics, be such. 



In the following year (May 2, 1821) Prof. Paucker, of Mitau, 

 presented to the Imperial Academy of Sciences of St. Petersburg, 

 a memoir on a question relative to the tangencies of circles, and 

 the memoir, which covers 84 4to. pages, is almost exclusively 

 devoted to Malfatti's Problem. This memoir seems to have 

 been generally overlooked ; at least no reference to it has been 



* is the centre of the inscribed circle, not placed upon the figure. 



