Gauss: Disquisitiones Arithmeticae
TO THE MOST SERENE
PRINCE AND LORD
CHARLES WILLIAM FERDINAND
DUKE OF BRUNSWICK AND LUNEBURG
MOST SERENE PRINCE,
I consider it my greatest good fortune that You allow me to adorn this work of mine with YOUR most honourable name. I offer it to YOU as a sacred token of my filial devotion. Were it not for YOUR favour, Most Serene Prince, I would not have had my first introduction to the sciences. Were it not for YOUR unceasing benefits in support of my studies, I would not have been able to devote myself totally to my passionate love, the study of mathematics. It has been YOUR generosity alone which freed me from other cares, allowed me to give myself to so many years of fruitful contemplation and study, and finally provided me the opportunity to set down in this volume some partial results of my investigations. And when at length I was ready to present my work to the world, it was YOUR munificence alone which removed all the obstacles that threatened to delay its publication. Now that I am constrained to acknowledge YOUR remarkable bounty toward me and my work I find myself unable to pay a just and worthy tribute. I am capable only of a secret and ineffable admiration. Well do I recognize that I am not worthy of YOUR gift, and yet everyone knows YOUR extraordinary liberality to all who devote themselves to the higher disciplines. And everyone knows that You have never excluded from YOUR patronage those sciences which are commonly regarded as being too recondite and too removed from ordinary life. YOU YOURSELF in YOUR supreme wisdom are well aware of the intimate and necessary bond that unites all sciences among themselves and with whatever pertains to the prosperity of the human society. Therefore I present this book as a witness to my profound regard for You and to my dedication to the noblest of sciences. Most Serene Prince, if YOU judge it worthy of that extraordinary favour which You have always lavished on me, I will congratulate myself that my work was not in vain and that I have been graced with that honour which I prize above all others.
MOST SERENE PRINCE
Your Highness' most dedicated servant
C F GAUSS
Brunswick, July 1801
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 an infinite set of solutions for an indeterminate problem those that are integral or at least rational (and especially with the added condition that they be positive) is not the discipline to which I refer but rather a special part of it, just as the art of reducing and solving equations (Algebra) is a special part of universal Analysis. And 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 properly includes more general inquiries concerning integers. We consider only Higher Arithmetic in the present volume.
Included under the heading "Higher Arithmetic" are those topics which Euclid treated with elegance and rigor in Book VII ff., and they can be considered an introduction to this science. The celebrated work of Diophantus, dedicated to the problem of indeterminateness, 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 skilful handling rather than profound principles and, because the questions are too specialized and rarely lead to more general conclusions, Diophantus' book seems to fit into that epoch in the history of Mathematics when scientists were more concerned with creating a characteristic art and a formal Algebraic structure than with attempts to enrich Higher Arithmetic with new discoveries. The really profound discoveries are due to more recent authors like those men of immortal glory P de Fermat, L Euler, L Lagrange, A M Legendre (and a few others). They opened the door to what is penetrable in this divine science and enriched it with enormous wealth. 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 give them their due praise in the proper places in these pages.
The purpose of this volume whose publication I promised five years ago is to present my investigations into the field of Higher Arithmetic. Lest anyone be surprised that the contents here go back over many first principles and that many results had been given energetic attention by other authors, I must explain to the reader that when I first turned to this type of inquiry in the beginning of I795 I was unaware of the more recent 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 [-1 is a quadratic residue of all numbers of the form 4n + 1 and a non-residue of all numbers of the form 4n + 3]). Since I considered it so beautiful in itself and since I suspected its connection with even more profound results, I 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. Only while studying the writings of these men of genius did I recognize 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 embodies in Sections V, VI, and VII. After a while I began to consider publishing the fruits of my new awareness. 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 of these results are so bound up with one another and with subsequent investigations that new results could not be explained without repeating from the beginning.
Meanwhile there appeared the outstanding work of that man who was already an expert in Higher Arithmetic, Legendre's "Essai d'une théorie des nombres." Here he collected together and systematized not only all that had been discovered up to that time but added 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 some observations in an Appendix 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, but 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 been replaced by more general considerations (cf. what is said in the Appendix about art. 306). Finally, since the book came out much larger than I expected, owing to the size of Section V, 1 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 indeterminate rank). All the treatises which complement the present volume will be published at the first opportunity.
In several difficult discussions I have used synthetic proofs and have suppressed the analysis which led to the results. This was necessitated by brevity, a consideration that must be consulted as much as possible.
The theory of the division of a circle or of a regular polygon treated in Section VII of itself does not pertain to Arithmetic but the principles involved depend uniquely on Higher Arithmetic. This will perhaps prove unexpected to geometers, but I hope they will be equally 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 and that it proposes solutions that they have been looking for or at least opens the way for new investigations.
JOC/EFR August 2007
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