igj^ CmrrOUS numerical proposition DEMONsTRAfEl)* 



lish mathematicians of his time; he also introduces it in & 

 note at page 6l of his dditioti of Dibphantus in the follow- 

 ing terms* 



*' Cithum Anient in duos cuhos, aut quadrato quadtdiutii 



in duos qUadrato quadrates, et generaliter millam in infinity fh. 



uttta quadratum potestafem in duos ejusdem nominis fas est 



dividere, cujus rei demonstrationem mirabilem sane detexi, 



Hanc marginis exiguii.as non caperet" 



Vfho never From which it appears^ that he was himself in possession 



published this ^f ^|,g demonstration *, thoiieh it tvas never published, nor 



demonstration . ' • i . ,• , , , 



has any mathematician since his time been able to restore 



it, notwithstanding they have succeeded in demonstrating 



many other of his propositions. 



Euler demon- Euler was, I believe, the first who undertook this task, 



strateditintwo ^^^^ succeeded in demonstruting the impossibility in the two 



* tases n zz 3, and nzz 4; that is, that the equations j:' + 



^' — 2^ and j:* -f 2/* m ♦* are impossible* and the same 



twro cases have alse been demonstrated upon similar prin* 



bs did Waring ciples by Waring, in his Meditationes Algehraica, and by 



and Legendre. Lggendre, in his Mssai sur la Thiorie des Nombres, the 



latter author concluding his chapter by the following re* 



joark. 



" Nous avons dtmontre dans ce paragrapke, que V equation 

 3c^ ■\-y^ zz z^ est impossible, aifisi V equation x^ -}- y*zz s*, et 

 4 plus fort f arson a* + y* = 2;*» Fermat a assurS de phis 

 [Ed. de DiopL pageGl) que t equation x" -f j/" r: s", est 

 gen'tralement impossible, lorsque n suppasse 2; mats cet pro- 

 position, passe le cas w rz 4, est du nombre de cellos que 

 restent a dcmontrcr, et pour lesquelles les mithodcs que nous 



Mistake in a * I ought here to correct an errolir, that I fell into in one of the notes 



iiote to the to the second English edition of Eulev's Algebra, wliere in mentioning 



translation of ^Yi\s theorem 1 have sai«J, " and the truth of it stiJl iest3,on no othet 



-uers ge- fg^j^ji^jjop jjign t^e bare assertion of Fermat, who probably baU never 



demohslrated it himself." I was led into this expression by writing the 



tiote in question from memory only, not recollecting at the same time 



th»; concluding part of his sentence, where he so posiTn^ely asserts h.ii 



being in possession of the demonstration : and as it was by no means my 



intention to impeach the veracity of this (lisiingui<;hed jjeom-^tar, I ought 



in justi«« to his memory to corf**i the mistak*. 



