ESSAYS 



THE THEORY OP NUMBERS (L. J. Mordell, Birkbeck College, London) 



THE Theory of Numbers is a subject which has aroused an extraordinary 

 enthusiasm among many of the world's greatest mathematicians. Fermat, 

 Euler, Lagrange, Legendre, Gauss, Cauchy, Jacobi, Eisenstein, Dirichlet, 

 Kummer, Kronecker, H. J. S. Smith, Minkowski, Sylvester, Hermite, and 

 Poincare among others have been fascinated by this branch of mathematics. 

 There seems to be some reason, therefore, for the statement which we have 

 heard attributed to Prof. Klein, that "One cannot be a great mathematician 

 unless he is acquainted with the Theory of Numbers." 



English mathematicians, however, have shown but little interest in the Theory 

 of Numbers, and excluding Prof. Smith, who was undoubtedly an arithmetician of 

 the first rank, and Sylvester, they ' have taken practically no part in its great 

 progress during the last century. 



This is due to a variety of causes. Foremost, perhaps, as it is one of the most 

 abstruse of studies and as its conclusions do not have any practical importance, it 

 does not tend to appeal to the practical genius of the English mathematician. 

 This tendency is accentuated by the English educational system, which seems to 

 regard the examination as its final aim : so that naturally the importance of a 

 subject, and the attention paid to it, depend upon its utility from an examination 

 point of view. One result of this is, that while the Englishman with any pretence 

 to a mathematical training is thoroughly acquainted with numerous properties of 

 conies, very frequently he is entirely ignorant even of the existence of some of the 

 most interesting and wonderful developments of pure mathematics. Another 

 result, let us hope it is a feature of the past, was that the interest and energy 

 of many mathematicians were directed to research which was closely related to 

 questions of the examination type ; or, as the late Prof. H. J. Smith put it, to 

 finding new expressions for the foci of a conic. This also accounts to some extent 

 for the production in England of so many elementary mathematical text-books 

 which, no doubt, are useful for the purpose for which they were written, while 

 really advanced mathematical books are still so few in number that the very 

 advanced student has to rely to a considerable extent upon works published 

 abroad. 



A further reason for the neglect of the Higher Arithmetic in England is to be 

 found in the Fellowship system of the ancient English universities. Under this 

 system the young mathematician is often expected to produce important research 

 within a few years of taking his degree. This is certainly no incentive to the 

 pursuit of a study in which research is difficult, and in which there may be little to 

 show in print as a result of many years' work. It can scarcely be expected that 

 the young mathematician will jeopardise his chances of substantial recognition at 



1 This does not apply to the Theory of Partitions, which, however, is really a 

 branch of Combinatory Analysis. Cf. the introduction to Major MacMahon's 

 Combinatory Analysis^ vol. i. 



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