xx PREFACE. 



in the text. While many of these papers are of minor importance, the aim 

 has been to give an exhaustive account of the literature on the subject 

 rather than a selective account reflecting the author's imperfect views as to 

 relative importance. This work is intended as a source book not merely 

 for the fastidious professional mathematician, but also for the larger number 

 of amateurs who find endless fascination for the "queen of the sciences," 

 whose rule began centuries ago and has continued without interruption 

 to the present. 



Unfortunately, following the practice of Diophantus, many writers on 

 this subject have been content with a special solution of their problem, 

 obtained by making various assumptions which simplify the analysis. A 

 report which would give merely the final formulas in such a paper, without 

 indicating also the restrictive assumptions, would be useless. Instead, there 

 is given here a summary of the essential steps in the proof, and this plan is 

 followed especially in the case of papers not to be found in the average 

 large library. These papers which give only special solutions of the problem 

 attacked have at least the value of showing that the problem is not im- 

 possible. Moreover, an examination of many such papers reveals the fact 

 that there are a few constantly recurring types of auxiliary Diophantine 

 problems (such as that of making a quartic function equal to a square), 

 whose complete solution would permit the complete treatment of a very 

 large number of problems, and hence suggest specially useful subjects for 

 thorough investigation. Since there already exist too many papers on 

 Diophantine analysis which give only special solutions, it is hoped that all 

 devotees of this subject will in future refrain from publication until they 

 obtain general theorems on the problem attacked if not a complete solution 

 of it. Only in this way will the subject be able to retain its proper position 

 by the side of other virile branches of mathematics. 



It was initially planned to give this work the title " topical history of the 

 theory of numbers"; but the word topical was omitted at the advice of a 

 prominent historian. It is inconceivable that any one would desire this 

 vast amount of material arranged other than by topics. Again, conven- 

 tional histories take for granted that each fact has been discovered by a 

 natural series of deductions from earlier facts and devote considerable 

 space in the attempt to trace the sequence. But men experienced in 

 research know that at least the germs of many important results are dis- 

 covered by a sudden and mysterious intuition, perhaps the result of sub- 

 conscious mental effort, even though such intuitions have to be subjected 

 later to the sorting processes of the critical faculties. What is generally 

 wanted is a full and correct statement of the facts, not an historian's per- 

 sonal explanation of those facts. The more completely the historian 

 remains in the background or the less conscious the reader is of the his- 

 torian's personality, the better the history. Before writing such a history, 

 he must have made a more thorough search for all the facts than is necessary 

 for the conventional history. With such a view of the ideal self-effacement 

 of the historian, what induced the author to interrupt his own investigations 



