130 SCIENCE PROGRESS 



is sad to think that his devotion to his favourite study contributed to his early 

 death at the age of twenty-nine, and was, perhaps, the greatest loss the Higher 

 Arithmetic has ever sustained. Similarly we read that Kronecker in 1858 writes 

 to Dirichlet that he requires a long time for the development of his ideas upon the 

 wonderful and intimate relation connecting elliptic function and the Theory of 

 Numbers, and it is only in 1885 that his results are perfected. 



The humorous element is not lacking since the establishment in 1908 of the 

 Wolfskell prize of about ,£5,000 for the first demonstration of Fermat's Last 

 Theorem. The opportunity of gaining immortal renown (not to mention the 

 ^5,000) has attracted the attention of numerous aspirants, whose chief qualifica- 

 tion, in many cases, seems to be their ignorance of elementary mathematics. 



The Higher Arithmetic presents many pitfalls even for the experienced in- 

 vestigator, and, as might be expected, the amateurs miss few opportunities of 

 making errors of every conceivable kind. We read with amusement of one 

 author who has been convinced for some time that the application of elementary 

 algebra can be greatly extended and supports his view by giving a proof of 

 Fermat's Last Theorem, which he considers is quite original (no one can doubt 

 this) and very fascinating. The author of another of the so-called solutions quotes 

 innumerable learned references ; but he makes mistakes of such a kind that one 

 is firmly convinced that his references were inserted unread and at random. 



English mathematicians are apt to forget that many of the developments of 

 modern mathematics had their origin in investigations arising from the Theory 

 of Numbers. The Higher Arithmetic throws out its tentacles in all directions, 

 and, in making a systematic study of the subject, the reader notices at once that 

 all the resources of pure mathematics are being drawn upon. The characteristic 

 properties of an invariant and a contravariant were introduced by Gauss in his 

 arithmetical investigations on the binary and ternary quadratics ; and it is not 

 unlikely that he was led by other arithmetic investigations to his discovery of the 

 elliptic functions twenty-five years before their discovery by Abel and Jacobi. 

 The first covariant, the Hessian of the binary cubic, was discovered by Eisenstein 

 in his researches on the arithmetic theory of the binary cubic. And it was in 

 applying the elliptic functions to the Theory of Numbers that he discovered 

 the Weierstrassian elliptic functions many years before their introduction by 

 Weierstrass. 



The modern developments of doubly periodic functions of the second and 

 third kinds are due to Hermite's efforts in proving Kronecker's Class Relation 

 Formulae. These formulae also proved a powerful stimulus to the Study of 

 Groups, the Theory of Functions, Modular Functions, and Automorphic Func- 

 tions. As a further illustration we may call attention to Poincare's review of a 

 large number of his published articles on Differential Equations, Theory of 

 Functions, and the Theory of Numbers. He emphasises the fact that he did not 

 pursue his researches on these three subjects separately, but that the results 

 obtained in the different subjects threw light on each other ; and that his work in 

 each of them was greatly assisted by his work in the others. 



The Theory of Dirichlet s series, which is attracting some attention at the 

 present time, arose from Dirichlet's famous investigations many years ago, when 

 he proved that every arithmetical progression, whose first term has no common 

 factor with the common difference, contained an infinite number of primes, and 

 also found the number of classes of binary quadratics with a given determinant. 



Finally, the analytical Theory of Numbers, which is exercising a powerful 

 attraction for many mathematicians now- a- days, is simply the study of functions 



