Translation:Disquisitiones Arithmeticae/Preface
AUTHOR'S PREFACE
THE INQUIRIES which this volume will investigate pertain to that part of Mathematics which concerns itself with integers. I will rarely refer to fractions and never to surds. The Analysis which is called indeterminate or Diophantine and which discusses the manner of selecting from the infinitely many solutions for an indeterminate problem those that are integral or at least rational (and usually with the added condition that they be positive) is not the discipline to which I refer but rather a quite special part, related to it roughly as the art of reducing and solving equations (Algebra) is related to the whole of Analysis. Just as we include under the heading ANALYSIS all discussion that involves quantity, so integers (and fractions in so far as they are determined by integers) constitute the proper object of ARITHMETIC. However what is commonly called Arithmetic hardly extends beyond the art of enumerating and calculating (i.e. expressing numbers by suitable symbols, for example by a decimal representation, and carrying out arithmetic operations). It often includes some subjects which certainly do not pertain to Arithmetic (like the theory of logarithms) and others which are common to all quantities. As a result it seems proper to call this subject Elementary Arithmetic and to distinguish from it Higher Arithmetic which includes all general inquiries about properties special to integers. We consider only Higher Arithmetic in the present volume.
Included under the heading "Higher Arithmetic" are those topics which Euclid treated in Book VIIII. with the elegance and rigor customary among the ancients, but they are limited to the rudiments of the science. The celebrated work of Diophantus, dedicated to undetermined problems, contains many results which excite a more than ordinary regard for the ingenuity and proficiency of the author because of their difficulty and the subtle devices he uses, especially if we consider the few tools that he had at hand for his work. However, these problems demand a certain dexterity and skillful handling rather than profound principles and, because the questions are too specialized and rarely lead to more general conclusions, Diophantus' book seems to mark an epoch in the history of Mathematics more because it presents the first traces of the characteristic art and Algebra than because it enriched Higher Arithmetic with new discoveries. Far more is owed to modern authors, of whom those few men of immortal glory P. de Fermat, L. Euler, L. Lagrange, A. M. Legendre (and a few others) opened the entrance to the shrine of this divine science and revealed the abundant wealth within it. I will not recount here the individual discoveries of these geometers since they can be found in the Preface to the appendix which Lagrange added to Euler's Algebra and in the recent volume of Legendre (which i shall soon cite). I shall also cite many of them in the proper places in these pages.
The purpose of this volume, whose publication I promised five years ago, was to present my investigations into Higher Arithmetic, both those begun by that time and later ones. Lest anyone be surprised that I start almost at the very beginning and treat anew many results that had been actively studied by others, I must explain that when I first turned to this type of inquiry in the beginning of 1795 I was unaware of the modern discoveries in the field and was without the means of discovering them. What happened was this. Engaged in other work I chanced on an extraordinary arithmetic truth (if I am not mistaken, it was the theorem of art. 108). Since I considered it so beautiful in itself and since I suspected its connection with even more profound results, [ concentrated on it all my efforts in order to understand the principles on which it depended and to obtain a rigorous proof. When I succeeded in this I was so attracted by these questions that I could not let them be. Thus as one result led to another I had completed most of what is presented in the first four sections of this work before I came into contact with similar works of other geometers. Once I was able to study the writings of these men of genius, I recognized that the greater part of my meditations had been spent on subjects already well developed. But this only increased my interest,and walking in their footsteps I attempted to extend Arithmetic further. Some of these results are embodied in Sections V, VI, and VII. After a while I began to consider publishing the fruits of my investigations. And I allowed myself to be persuaded not to omit any of the early results, because at that time there was no book that brought together the works of other geometers, scattered as they were among Commentaries of learned Academies. Besides, many results were new, most were treated by new methods, and the later results were so bound up with the old ones that they could not be explained without repeating from the beginning.
Meanwhile there appeared an outstanding work by a man to whom Higher Arithmetic already owed much, Legendre's "Essai d'une theorie des nombres." Here he collected together and systematized not only all that had been discovered up to that time but also many new results of his own. Since this book came to my attention after the greater part of my work was already in the hands of the publishers, I was unable to refer to it in analogous sections of my book. I felt obliged, however,to add Additional Notes on a few passages and I trust that this understanding and illustrious man will not be offended. The publication of my work was hindered by many obstacles over a period of four years. During this time I continued investigations which I had already undertaken and deferred to a later date so that the book would not be too large, and I also undertook new investigations. Similarly, many questions which I touched on only lightly because a more detailed treatment seemed less necessary (e.g. the contents of art. 37, 82 ff., and others) have been further developed and have led to more general results that seem worthy of publication (cf. the Additional Note on art. 306). Finally, since the book came out much larger than I expected, owing to the size of Section V, I shortened much of what I first intended to do and, especially, I omitted the whole of Section Eight (even though I refer to it at times in the present volume; it was to contain a general treatment of algebraic congruences of arbitrary rank). All these things, which will easily fill a book the size of this one, will be published at the first opportunity.
In several difficult discussions 1 have used synthetic proofs and have suppressed the analysis which led to the results. This was necessitated by brevity, a consideration that had to be consulted as much as possible.
The theory of the division of a circle or of regular polygons treated in Section VII of itself does not pertain to Arithmetic but the principles involved depend solely on Higher Arithmetic. Geometers may be as surprised at this fact itself as (I hope) they will be pleased with the new results that derive from this treatment. These are the things I wanted to warn the reader about. It is not my place to judge the work itself. My greatest hope is that it pleases those who have at heart the development of science, either by supplying solutions that they have been looking for or by opening the way for new investigations.