|14 BEPORT— 1861. 



Commercium Epistolicum of Wallls, Ep. 8) as a challenge to the English 

 mathematicians, and especially to Wallis. The problem was at first misunder- 

 stood by Lord Brouncker and Wallis, who each gave a method for its solution 

 in fractional numbers ; not attending to the restriction to integral numbers 

 implied, though not expressed, in Fermat's enunciation, without which the 

 problem is of a very elementary character. Ultimately, however, they ob- 

 tained a complete solution by a method, which Wallis describes in the Comm. 

 Epist. Epp. 17 (postscript) and 19, and in his Algebra, capp. xcviii. and xcix,, 

 attributing it to Lord Brouncker, though he seems himself to have had some 

 share in its invention. This method is the same as that which is given by 

 Euler in his Algebra, and in the first of the memoirs above cited, and which 

 is attributed by him to Pell *. It differs, in form at least, from that now em- 

 ployed, and was evidently suggested by the artifices of substitution employed 

 in Diophantine problems. It is most easily explained by an example. If 

 T*— 13U^=1 be the equation proposed, the process would stand thus: — 



(1) 3U<T<4U; letT=3U-ft;, ; -4^W+6\Jv^+v^^=l, 



(2) Vi<U<2«, ; }et\J=v^ + v^; ^v^—2v^v^—^v^^=l, 



(3) ^2<^i<2f3; \et v^ = v^-irv^; —Zv^-i[-'^v^v^-irSv^='[, 



(4) «3<W2<2r3; let v^=V3 + v^^, '^v^—2v^v^—Sv^=l, 



(5) v^<.v^<2v^; \etv^=v^ + v^; — »/-f6»^Vg + 4?;/=l, 



(6) 6v,<.v^<lv,; \eiv^=ev^ + v^; ^v.^—6v^v^—v^^ = l, 



(7) v^<v,<2v^; \etv^=v^ + v^; —W^+2v\v^+'^v^''=\, 



(8) r^<Vg<:2z?-; \eiv\:=v^ + v^\ 3v,- — iv.v^ — 3i>g^=l, 



(9) v^^v.^'-Zv^; letv^ = Vg + v^; —4^vi + 2VgVg + 3v^^=l, 

 (10) v,<t;,<2Vg; \etv^=v^+v^^; v^''—6v^t\^—4:vj = l. 



In the last equation we may put Vg = l, v^g=0 ; whence T=649, U=180. 

 It will be seen that the success of the method depends on its leading at last 

 to an equation in which the coefficient of one of the indeterminates is +1. 

 Wallis does not prove that such an equation will always occur ; and the de- 

 monstration which he has given of the resolubilityof the equation T^ — DU"=1 

 is inconclusive. (See his Algebra, cap. xcix. ; the reader will find the 

 paralogism which vitiates his reasoning in the proof of the lemma, upon which 

 it depends ; see also Lagrange's criticism in the 8th paragraph of the Additions 

 to Euler's Algebra ; and Gauss, Disq. Arith. art. 202, note.) It is evident that 

 the method of solution employed by Wallis really consists in the successive 



T 



determination of the integral quotients in the development of — in a con- 



T 



tinued fraction ; in addition to this, Euler observed that —is itself necessa- 

 rily a convergent to the value of VD ; so that to obtain the numbers T and U 

 it suffices to develope VD in a continued fraction. It is singular, however, 

 that it never seems to have occurred to him that, to complete the theory of 

 the problem, it was necessary to demonstrate that the equation is always re- 

 soluble, and that all its solutions are given by the development of VD. His 

 memoir (Comment. Arith. vol. i. p. 316) contains all the elements necessary 



* There does not seem to be any ground for attributing either the problem or its solution 

 to PeU ; and it is possible that Euler may have been misled by a confused recollection of 

 the contents of Wallis's Algebra, in -which an account is given of the method employed by 

 Pell in solving Diophantine problems. Nevertheless the equation T^— DU"= 1 is often called 

 the Pellian equation after him, probably upon Euler's authority. 



