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XVI. On the Use of Functional Equations in the Elemcntaiy 

 Investigations of Geometry. 



To the Editors of the Philosophical Magazine and Journal. 

 Gentlemen, 

 A FEW years ago, when the French edition of Legendre's 

 -'^ Geometry first fell into my hands, I was considerably at- 

 tracted by the famous second Note. I bestowed some atten- 

 tion upon the subject, and was led to a particular artifice, which 

 I then conceived sufficient to destroy the force of Mr. Leslie's 

 objection to the legitimacy of the investigations. Accident 

 brought before me, some weeks ago, a volume of your Journal, 

 which recalled my former ideas. It contained two papers upon 

 the subject, signed Dis-Iota*, which it is needless further to 

 characterize than to attribute to Mr. Ivory. He firmly esta- 

 blishes, I think, the objections which he formerly made (like- 

 wise anonymously) in a letter to Mr. Leslie. There cannot 

 remain a doubt that the supposition of the identity of the func- 

 tions (f, ip', $", &c. involved in Legendre's " mise en equation" 

 is fully equivalent to Euclid's axiom. Legendre's method 

 then falls to the ground, in so far as he intended it to overcome 

 this great difficulty of geometry : but it still retains all its 

 interest as a method of investigating the elementary relations 

 of space. We all know the splendour of analytical mechanics, 

 compared with the old investigations by means of diagrams ; 

 and how pretty would it be, even although useless, were our 

 geometries condensed into a few families of formulae, each 

 bearing as wide a meaning as the well-known syS/i = in 

 mechanics. In this light alone am I inclined to view Legen- 

 dre's investigations ; and I feel a deep interest in attempting to 

 elucidate them. In a few points Mr. ivory's (or Dis-Iota's) papers 

 ajipeared to me somewhat deficient. They were in my hands 

 only for an hour or two, so that I cannot now refer to the par- 

 ticular passage which left this impression. I therefore submit 

 my ideas to you in the form of a few general critical remarks. 

 Let us state in the first place the exact nature of Legendre's 

 reasoning respecting the constitution of his functionary equa- 

 tion. He discovers from superposition, that a triangle is de- 

 termined by the constancy of the three quantities A, 13, and c ; 

 or, in odier words, that these are all the variables u|)on which 

 the value of C can depend. But he asserts, if C required the 

 combination of them all — for its determination an analytical 

 absurdity would exist in the equation C = ip(A, B, c). The 

 only possible manner then in wliich C can receive a deter- 

 minate character from the constancy oi" these (]unntities is ex- 



* On the Theory of Parallel Lints. I'liil. Mai:, vol. Ixiii. p. 161 & i).246. 



pressed 



