It is well known that no satisfactory demonstration has ever been given of Fermat's celebrated theorem, which asserts that the equation an = bn + cn is impossible, if a, b, c, are whole numbers, and n is any whole number greater than 2. In Legendre's Théorie des Nombres, he demonstrates the cases of n = 3, n = 4, and n = 5, the latter only in his Second Supplement. In Crelle's Mathematical Journal, ix. 390, M. Dirichlet, a mathematician of Berlin, has demonstrated the case of n = 14, but I am not aware whether his demonstration is considered successful. Legendre informs us (Second Supplement, p. 3) that the Academy of Sciences, with the view of doing honour to the memory of Fermat, proposed, as the subject of one of its mathematical prizes, the demonstration of this theorem; but the Concourse, though prolonged beyond the usual term, produced no result.