130 MR TALBOT ON MALFATTl'S PROBLEM. 



In the following year, PlUcker of Bonn resumed the subject in Crelle's 

 "Annals," xi. p. 121. His memoir, which bears date October 1831, throws a 

 great deal of new light upon the subject. His object was, like that of Zornow, 

 to demonstrate the truth of Steiner's construction, in which attempt he succeeds 

 up to a certain point, by a well-conducted train of geometrical reasoning ; but 

 beyond that point he cannot proceed without the help of analysis. In fact, he 

 shows geometrically that there exists a certain point within any given triangle 

 ABC (see former figure), which possesses the property, that if AO, BO, CO, are 

 joined, and three circles are inscribed in the three triangles AOB, BOC, COA, as 

 in that figure, then the line DG will touch both the circles E, F, and also two of 

 the required circles. It remained to discover what point of the triangle the point 

 was, and to verify Steiner's assertion that it was the centre of the inscribed 

 circle, or that the lines AO, BO, CO, respectively bisect the three angles of the 

 given triangle. But of this capital point Plucker was unable to find any 

 geometrical proof. He has recourse, therefore, to a very free and prolix use of 

 trigonometry and algebra, through which I doubt whether any of his readers 

 have had the courage to follow him, but which finally conducts him to the con- 

 clusion that Steiner's assertion is true. It will be observed that Plucker's 

 memoir, though published subsequently to that of Zornow, preceded it in point 

 of date ; that of Zornow being dated in October 1832. He was, therefore, the first 

 who succeeded in demonstrating Steiner's theorem. Plucker concludes his 

 memoir with the following remarks upon the mode in which Steiner has treated 

 the question : — * 



" The construction which I have given is essentially the same with that pro- 

 posed by Steiner in vol. i. of this journal, p. 178. There is, however, at that 

 place no indication of a demonstration. The introductory words of the author— 

 ' to shorn the fraitfulness of the theorems set forth in paragraphs 1 , 2, 3, by a suitable 

 example, we add the geometrical solution, and also a greater generalisation of 

 MalfattV s problem, omitting the proof ' — might cause a person who (as I must con- 

 fess to be my own case) has no idea how the construction of that problem can 

 depend upon the well-known theorems explained in the above quoted paragraphs 

 concerning points of similitude, &c, &c , to think that the given construction is 

 not proved."! 



I have said that Plucker's recourse to a difficult and very prolix analysis in 

 order to justify the assertion that is the centre of the inscribed circle, is the 

 weak point of his able investigation. He has admitted this himself, for he says 

 (p. 126), " So soon as this theorem is brought into its proper connection there 



* Vol. xi. p. 126. 



f Die einleitenden Worte des Verfassers; " Um die Fruchtbarkeit," &c. &c, konnten demjeni- 

 gen, der, wie ich von mir bekennen muss, keine Idee davon hat, wie die Construction, &c, &c 

 den G-edanken aufdrangen, dass die gegebene Construction nicht bewiesen sei. 





