114 BULLETIN OP THE 



In the Transactions of the Royal Society of Edinburgh, Vol. 

 24, 1864-67, pp. 127-138, Mr. H. F. Talbot inserts a memoir 

 entitled "Recent Researches on Malfatti's Problem." After a 

 brief introduction, in which the problem is given with some rea- 

 Bons why it has proved so interesting, he gives a history of the 

 problem ; but there are some important papers upon the subject 

 which Mr. Talbot has overlooked, and in consequence his history 

 contains some errors. The object of tlie present paper is to call 

 attention to the papers overlooked by Mr. Talbot, and to rewrite 

 the history of the problem. 



The problem to inscribe in a triangle three circles, each touch- 

 ing the other two, and also two sides of the triangle, was first 

 solved by an Italian geometer, Mr. John Francis Malfatti, in 

 1803, and from his solution being the first that has been given, 

 the problem is now called Malfatti's Problem. His solution is 

 given in the tenth volume of the Memoirs of the Italian Society 

 of Sciences. The proof which Mr. Malfatti gave of the correct- 

 ness of his construction is, according to Mr. Talbot, the follow- 

 ing : He deduces from his construction a value of one of the 

 radii of the required circles, and also deduces a value of the same 

 radius resulting from the conditions of the problem ; and these 

 results being found to be identical, he concludes the correctness- 

 of his construction. 



The expressions found by Mr. Malfatti, are the following : — 



In a triangle ABC, whose sides are a, b and c, call the radius 

 of the inscribed circle r, the radii of the three circles tonching^ 

 each other r^, r^ and r<,; retouching b and c, r^ touching c and a, 

 and r^ touching a and b ; also call the distances from the centre 

 of the inscribed circle to the three vertices A, B and G, o, /3 and y 

 respectively. Then we have as the expressions found by Mal- 

 fatti 



2r„ = 



s 



2,, = ^ 



— JS— r+a— ^— y I 

 —a I j 



*S — r — a+j3 — y X. 



2rj = — ^ is— r— o— /3 + y I 

 s—c \ ) 



The next writer upon the problem appears to have been Ger- 

 gonne, who proposed the problem in the first volume of his 



