PHILOSOPHICAL SOCIETY OF WASHINGTON. 119 



panied with a trigonometric proof, and subsequently the same 

 construction was extended to spherical triangles. 



This makes three entirely different constructions, viz., the one 

 which Prof Paucker says results immediately from a theorem of 

 Tedeuat's, and which perchance might be called Tedenat's con- 

 struction ; second, Steiner's; and third, Schellbach's. A con- 

 sultation of all the authors referred to would doubtless reveal 

 others. 



A paper by Mr. Andrew S. Hart has been already referred to. 

 It is important, though brief, as apparently coming nearer in its 

 mode of thought to the reasoning that led Steiner to the discovery 

 of the solution than any paper met with. 



Prof. Cayley has also written upon the problem. He has a note 

 on Schellbach's solution in the first volume of the Quarterly Jour- 

 nal, and in the Transactions of the Royal Society, Yol. LXX, he 

 has a memoir entitled "Analytic Researches connected with 

 Steiner's extension of Malfatti's Problem." 



Lastly, w^e come to the memoir of Mr. Talbot, mentioned at 

 the outset. He says of Malfatti's Problem that "although it is 

 a question of elementary geometry which can be solved by a 

 simple and elegant geometrical construction, yet no geometrical 

 proof has ever been given, as far as I am aware, of the truth of 

 this construction. * * * j now offer," he says, "to the 

 Royal Society a purely geometrical solution of the problem ; and, 

 for the sake of clearness, I have divided it into several parts, 

 which I have called Lemmas." Then follow the Lemmas, of 

 which there are 11, "in the first 9 of which," he says, "I chiefly 

 follow Pliicker. " * * * 3^^ Lemmas 10 and 11 are original; 

 at least he believed them to be so. 



Lemma 10 is the general case of the first Lemma of Prof. 

 Paucker, already cited, i. e., if in Fig. 2, A M is not the bisector 

 of the angle A then the chords 



T^ ^<7 ^ M A T 

 Tt' ~"i^iM AT' 

 and this general case is proved by Mr. Hart. 



Mr. Talbot's 11th Lemma is identical with the 13th paragraph 

 of Paucker's memoir above cited. 



In conclusion it may be remarked that in none of the works 

 consulted in preparing this sketch has there been found a treat- 

 ment of the problem in the general way in which Steiner handled 



