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XI. — Researches on Malfatti" s Problem. By H. F. Talbot, Esq. 



(Read 20th March 1865.) 



The problem which bears the name of the Italian geometer Malfatti, by whom 

 it was first proposed and solved, has long attracted the attention and exercised 

 the ingenuity of mathematicians, and has been made the subject of many careful 

 and elaborate researches. 



The great attention which has been bestowed upon this problem has arisen 

 partly from its intrinsic difficulty, but chiefly on account of the extreme simplicity 

 of the solution finally obtained by Malfatti, which seemed to open new views of 

 geometrical research, and gave reason to hope that simple solutions might in like 

 manner be found of many other geometrical problems usually accounted very 

 difficult or insoluble. 



The problem of Malfatti offers another singularity. 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. It has been established hitherto only by a very 

 elaborate use of algebraic analysis, in the course of which, however indisputable 

 the result may be, all geometrical perception of its truth is lost. And yet there 

 can be little doubt, it should seem, that a geometrical reason must exist for any 

 simple series of facts belonging to elementary geometry. 



The necessity of calling in the aid of analysis can only arise from the true 

 connection of the geometrical principles involved in the problem being imperfectly 

 understood. 



I now offer 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. Some of these are well deserving of attention for their own sake, 

 and irrespective of Malfatti's problem. When these theorems have been 

 established, their combination affords a lucid proof of the truth of the solutions 

 which mathematicians have hitherto only obtained by the help of analysis. 



History of the Problem. 



In the year 1803, a distinguished Italian geometer, Signor Malfatti, proposed 

 the following problem in the Memoirs of the Italian Society of Sciences, vol. x. 

 part 1 : — * 



* See Gergonne, vol. i. p 347. 

 VOL. XXIV. PART I. 2 N 



