128 MR TALBOT ON MALFATTIS PROBLEM. 



" In a given triangle to inscribe three circles touching each other, and each of 

 them touching two sides of the triangle." 



He gave at the same time a remarkably simple geometrical solution which he 

 had discovered, but unaccompanied by any geometrical demonstration of its truth. 

 He contented himself with showing that, if we calculate the algebraic values of 

 the three radii which result from the above-mentioned geometrical construction, 

 these three values, when substituted in the analytical equations deduced from 

 the original conditions of the problem, do in fact satisfy them, and are therefore 

 demonstrated to be true. But he gives no indication of any process of reasoning 

 by which he arrived at the knowledge of these values. 



In the year 1810, Gergonne proposed this problem for solution in the " Annales 

 des Mathematiques," vol. i. p. 196, without knowing that it had been previously 

 solved by Malfatti ; and no solution of it being sent to him by his correspon- 

 dents, he took up the inquiry himself.* He makes the following preliminary 

 statement : — 



" II y a plus de 10 ans que ce difficile probleme s'est offert pour la premiere 

 fois aux redacteurs de ce recueil, mais bien qu'ils 1'aient attaque un grand nombre 

 de fois ils n'ont pu pendant longtemps parvenir a le resoudre ni meme a s'assurer 

 s'il etait resoluble par la ligne droite et le cercle 



" Ils ont cru devoir faire encore de nouvelles tentatives, et plus heureux cette 

 fois que les precedentes ils sont parvenus sinon a trouver une construction du 

 probleme, du moins a l'abaisser au premier degre." 



Then follows an analytical investigation, which finally gives an algebraic value 

 for the radius of any one of the circles in terms of known quantities. But this 

 value does not lead to any simple geometrical construction, nor is it easy to show 

 that it agrees with that previously found by Malfatti, which, however, must 

 necessarily be the case. 



Having succeeded to this extent, M. Gergonne did not believe the problem to 

 be susceptible of much further simplification, when he first became acquainted 

 with the previous researches of Malfatti. Having procured and perused the 

 memoir of that author, he found that it threw no light upon the point of chief 

 interest, viz., the mode of investigation by which a result so unexpectedly simple 

 had been obtained.f 



Nothing further appears to have been done till the year 1820, when M. Lech- 

 mutz, of Berlin, published a memoir in Gergonne's Annales, vol. x. p. 289, in 

 which he succeeded for the first time in solving the problem, by a course of a p?*iori 

 reasoning. His investigation is algebraical, and his results coincide with those of 

 Malfatti. In the year 1826, Steiner, a distinguished geometer of Berlin, threw 

 an entirely new light upon the problem, by giving, in Crelle's Annals (vol. i. p. 



* See Gergonne, vol. i. p. 343. f Ibid. vol. ii. p. 60. 



