128 SCIENCE PROGRESS 



a period of his life when it is most useful, unless somehow he has become fasci- 

 nated by the Higher Arithmetic, and is resolved that he shall not be deterred 

 by reason of material considerations. 



The difficulty of securing a suitable introduction to the subject is a further 

 hindrance to its study. In the first place, there are no lectures upon it at most 

 British universities. In the second place, while there is no lack of introductory 

 text-books, a number of them either pay too much attention to details which 

 cannot be appreciated by the beginner, or limit their contents to the barest 

 elements of the subjects, giving not the slightest indication of its great extent 

 or any mention of many of its most beautiful and interesting results, a large 

 number of which could easily be mentioned in connection with many topics. 



Many of these reasons may also explain statements made and questions fre- 

 quently asked by English mathematicians. Thus Prof. Burnside in his presidential 

 address to the London Mathematical Society in 1908, deals with the fact that the 

 Theory of Groups attracts far more interest abroad than in England. In a similar 

 address in 1912, Prof. Baker asks, " Why is it so often the case that the early 

 history in England of a department of Pure Mathematics is a history of im- 

 portation ? " 



Gauss, to whose supreme genius is due the modern aspect of the Higher 

 Arithmetic, said, "Mathematics is the Queen of the Sciences, and the Theory 

 of Numbers is the Queen of Mathematics." In the introduction to Eisenstein's 

 Mathematische Abhandlungen, to quote H. J. S. Smith, 1 Gauss wrote: "The 

 Higher Arithmetic presents us with an inexhaustible storehouse of interesting 

 truths — of truths, too, which are not isolated, but stand in the closest relation to 

 one another, and between which, with each successive advance of the science, we 

 continually discover new and sometimes wholly unexpected points of contact. A 

 great part of the theories of Arithmetic derive an additional charm from the 

 peculiarity that we easily arrive by induction at important propositions, which 

 have the stamp of simplicity upon them, but the demonstration of which lies so 

 deep as not to be discovered until after many fruitless efforts ; and even then it is 

 obtained by some tedious and artificial process, while the simpler methods of 

 proof long remain hidden from us." 



The Theory of Numbers is unrivalled for the number and variety of its results 

 and for the beauty and wealth of its demonstrations. What reader cannot but be 

 impressed by such gems as, for example, Eisenstein's uniform demonstrations of 

 the laws of quadratic, cubic and bi-quadratic reciprocity ; Jacobi's arithmetic 

 investigation, as simplified by Dirichlet, of the representations of a number as a 

 sum of four squares ; or Tchebicheff s proof of Bertrand's postulate that at least 

 one prime number exists between the limits x and 2x - 2 if x is greater than 7/2 ? 



The best illustrations that the proofs of many simple results involve most 

 abstruse and complicated considerations are, perhaps, Eisenstein's results on the 

 representations of a number as a sum of five squares ; and Kummer's results on 

 the general law of reciprocity and Fermat's Last Theorem, one of which may be 

 stated as follows : If n is an odd prime which is not a divisor of the numerator of 

 one of the first \{n - 3) Bernoulli's numbers, the equation x n +y" + z n = o does 

 not admit of rational values for x, y and z apart from trivial solutions for which 

 one of the unknowns is zero; though 250 years ago Fermat asserted that he had 

 discovered a truly wonderful proof for all integer values of n greater than 2, but 



1 Collected Works, vol. ii. p. 169. I have much pleasure in taking this oppor- 

 tunity of acknowledging my great indebtedness to Prof. Smith's writings. 



