It is with genuine regret that the reviewer has to point out one or two historical errors in an address which is otherwise so charming. On page 18 he refers to Fermat's "notorious assertion concerning Mersenne's numbers"; a letter to the reviewer indicates that this error probably arose through referring to Fermat a statement which was in fact made by Mersenne (and stated by W W Rouse Ball to be "probably due to Fermat"). From page 18 of Hardy's lecture I quote as follows:
No very laborious computations would be necessary to lead Waring to a highly plausible speculation, which is all I take his contribution to the theory to be; and in the theory of numbers it is singularly easy to speculate, though often terribly difficult to prove; and it is only proof that counts.It is hard to see in what sense the author can say that "it is only proof that counts" when he has before him a conjecture like that of Waring which has certainly influenced for good the development of a very fascinating chapter in the modern theory of numbers. Probably the same feeling that induced this statement led to Hardy's calling by the name "theorem of Lagrange" the theorem that every integer is a sum of four non-negative squares, whereas Fermat had stated that he had a proof of the theorem (both Fermat and Bachet ascribing the theorem to Diophantus) and Euler had made repeated efforts for forty years to prove it before Lagrange through the aid of Euler's work succeeded in giving the first proof in 1772. [See Dickson's History, vol. II, pp. ix, x, 275-303.] It appears to me to be unfortunate to have this theorem called by the name of Lagrange; it certainly represents one extreme of judgment concerning the question of attaching names of mathematicians to specific theorems.
The opposite extreme of the same thing recently came to my attention in another connection; curiously enough, it is again a case of a "theorem of Lagrange." The theorem that the order of a subgroup is a factor of the order of the group containing it has been called the "theorem of Lagrange" by at least two authors of high repute [see Pascal's Repertorium (in German), vol. I, 2d edition, 1910, p. 194, and Miller, Blichfeldt and Dickson's Finite Groups, 1916, p. 23 (in the part written by G. A. Miller)]
Now the facts seem to be that Lagrange knew the theorem only for the case of the subgroups of the symmetric group and that even for this case he had no satisfactory proof. Abbati (in 1803) completed the proof for subgroups of the symmetric group and also proved the theorem for cyclic subgroups of any group; but it was apparently more than seventy-five years after the publication of Lagrange's memoir (in 1770-1771) before the completed theorem became current (though it had appeared earlier in a paper by Galois in 1832). In this case we have attributed to Lagrange a theorem which he probably never knew or conjectured, on the ground (it would seem) that he knew a certain special case of it. In Hardy's paper we have a theorem referred to Lagrange apparently on the ground that he first published a proof of it though it had been in the literature long before. Somewhere between these two extremes lies the golden mean of proper practice in attaching the names of mathematicians to specific theorems; and this mean, in the opinion of the reviewer, is rather far removed from each of the extremes indicated.