Dedicatio

-- Carolus Fridericus Gauss {കാര്‍ള്‍ ഫ്രീഡ്രിഹ് ഗൌസ്സ് / கார்ள் ஃப்ரீட்ரிஹ் கௌஸ்ஸ் / कार्ल् फ्रीड्रिह् गौस्स्}.


Right: Carl Wilhelm Ferdinand, Duke of Brunswick and Lüneburg, by Pompeo Batoni Cavallino, Oil on canvas, 1767. 137 cm x 99 cm. Herzog Anton Ulrich Museum, Brunswick, Lower Saxony, Germany.

SERENISSIMO
PRINCIPI AC DOMINO
CAROLO GUILIELMO FERDINANDO
BRUNOVICENSIUM AC LUNEBURGENSIUM DUCI.

PRINCEPS SERENISSIME,

Summae equidem felicitati mihi duco, quod Celsissimo nomini TUO hoc opus inscribere mihi permittis, quod ut TIBI offeram sancto pietatis officio obstringor. Nisi enim TUA gratia, Serenissime Princeps, introitum mihi ad scientias primum aperuisset, nisi perpetua TUA beneficia studia mea usque sustentavissent, scientiae mathematicae, ad quam vehementi semper amore delatus sum, totum me devovere non potuissem. Quin adeo eas ipsas meditationes, quarum partem hoc volumen exhibet, ut suscipere, per plures annos continuare literisque consignare liceret, TUA sola benignitas effecit, quae ut, ceterarum curarum expers, huic imprimis incumbere possem praestitit. Quas quum tandem in lucem emittere cuperem, TUA munificentia cuncta, quae editionem remorabantur, obstacula removit. Haec TUA tanta de me meisque conatibus merita gratissima potius mente tacitaque admiratione revolvere, quam iustis dignisque laudibus celebrare possum. Namque non solum tali me muneri haud parem sentio, sed et neminem ignorare puto, solennem TIBI esse tam insignem liberalitatem in omnes qui ad optimas disciplinas excolendas conferre videntur, neque eas scientias, quae vulgo abstrusiores et a vitae communis utilitate remotiores creduntur, a patrocinio TUO exclusas esse, quum TU ipse intimum scientiarum omnium inter se et necessarium vinculum mente illa sapientissima omniumque quae ad humanae societatis prosperitatem augendam pertinent peritissima, penitus perspexeris. Quodsi TU, Princeps Serenissime, hunc librum, et gratissimi in TE animi et laborum nobilissimae scientiae dicatorum testem, insigni illo favore, quo me tamdiu amplexus es, haud indignum iudicaveris, operam meam me non inutiliter collocasse, eiusque honoris, quem prae omnibus in votis habui, compotem me factum esse, mihi gratulabor

PRINCEPS SERENISSIME
Celsitudinis Tuae servus addictissimus
C. F. Gauss.

Brunovici mense Iulio 1801.

Præfatio

Disquisitiones in hoc opere contentae ad eam Matheseos partem pertinent, quae circa numeros integros versatur, fractis plerumque, surdis semper exclusis. Analysis indeterminata quam vocant seu Diophantea, quae ex infinitis solutionibus problemati indeterminato satisfacientibus eas seligere docet, quae per numeros integros aut saltem rationales absolvuntur plerumque ea quoque conditione adiecta ut sint positivi), non est illa disciplina ipsa, sed potius pars eius valde specialis, ad eamque ita fere se habet, ut ars aequationes reducendi et solvendi (Algebra) ad universam Analysin. Nimirum quemadmodum ad Analyseos ditionem referuntur omnes quae circa quantitatum affectiones generales institui possunt disquisitiones: ita numeri integri (fractique quatenus per integros determinantur) obiectum proprium ARITHMETICAE constituunt. Sed quum ea, quae Arithmetices nomine vulgo traduntur, vix ultra artem numerandi et calculandi (i. e. numeros per signa idonea e. g. secundum systema decadicum exhibendi, operationesque arithmeticas perficiendi) extendantur, adiectis nonnullis quae vel ad Arithmeticam omnino non pertinent (ut doctrina de logarithmis) vel saltem numeris integris non sunt propria sed ad omnes quantitates patent: e re esse videtur, duas Arithmeticae partes distinguere, illaque ad Arithmeticam elementarem referre, omnes autem disquisitiones generales de numerorum integrorum affectionibus propriis Arithmeticae Sublimiori, de qua sola hic sermo erit, vindicare.

Pertinent ad Arithmeticam Sublimiorem ea, quae Euclides in Elementis L. VII sqq. elegantia et rigore apud veteres consuetis tradidit: attamen ad prima initia huius scientiae limitantur. Diophanti opus celebre, quod totum problematis indeterminatis dicatum est, multas quaestiones continet, quae propter difficultatem suam artificiorumque subtilitatem de auctoris ingenio et acumine existimationem haud mediocrem suscitant, praesertim si subsidiorum quibus illi uti licuit tenuitatem consideres. At quum haec problemata dexteritatem quandam potius scitamque tractationem quam principia profundiora postulent, praetereaque nimis specialia sint raroque ad conclusiones generaliores deducant: hic liber ideo magis epocham in historia Matheseos constituere videtur, quod prima artis characteristicae et Algebrae vestigia sistit, quam quod Arithmeticam Sublimiorem inventis novis auxerit. Longe plurima recentioribus debentur, inter quos pauci quidem sed immortalis gloriae viri P. DE FERMAT, L. EULER, L. LA GRANGE, A. M. LE GENDRE (ut paucos alios praeteream) introitum ad penetralia huius divinae scientiae aperuerunt, quantisque divitiis abundent patefecerunt. Quaenam vero inventa a singulis his geometris profecta sint, hic enarrare supersedeo, quum e praefationibus Additamentorum quibus ill. La Grange Euleri Algebram ditavit operisque mox memorandi ab ill. Le Gendre nuper editi cognosci possint, insuperque pleraque locis suis in his disquisitionibus Arithmeticis laudentur.

Propositum huius operis, ad quod edendum iam annos abhinc quinque publice fidem dederam, id fuit, ut disquisitiones ex Arithmetica Sublimiori, quas partim ante id tempus partim postea institui, divulgarem. Ne quis vero miretur, scientiam hic a primis propemodum initiis repetitam, multasque disquisitiones hic denuo resumtas esse, quibus alii operam suam iam navarunt, monendum esse duxi, me, quum primum initio a. 1795 huic disquisitionum generi animum applicavi, omnium quae quidem a recentioribus in hac arena elaborata fuerint ignarum, omniumque subsidiorum per quae de his quidpiam comperire potuissem expertem fuisse. Scilicet in alio forte labore tunc occupatus, casu incidi in eximiam quandam veritatem arithmeticam (fuit autem ni fallor theorema art. 108), quam quum et per se pulcherrimam aestimarem et cum maioribus connexam esse suspicarer, summa qua potui contentione in id incubui, ut principia quibus inniteretur perspicerem, demonstrationemque rigorosam nanciscerer. Quod postquam tandem ex voto successisset, illecebris harum quaestionum ita fui implicatus, ut eas deserere non potuerim; quo pacto, dum alia semper ad alia viam sternebant, ea quae in quatuor primis Sectionibus huius operis traduntur, ad maximam partem absoluta erant, antequam de aliorum geometrarum laboribus similibus quidquam vidissem. Dein copia mihi facta, horum summorum ingeniorum scripta evolvendi, maiorem quidem partem meditationum mearum rebus dudum transactis impensam esse agnovi: sed eo alacrior, illorum vestigiis insistens, Arithmeticam ulterius excolere studui; ita variae disquisitiones institutae sunt, quarum partem Sectiones V, VI et VII tradunt. Postquam interiecto tempore consilium de fructibus vigiliarum in publicum edendis cepi: eo lubentius, quod plures optabant, mihi persuaderi passus sum, ne quid vel ex illis investigationibus prioribus supprimerem, quod tum temporis liber non habebatur, ex quo aliorum geometrarum labores de his rebus, in Academiarum Commentariis sparsi, edisci potuissent; quod multae ex illis omnino novae et pleraeque per methodos novas tractatae erant; denique quod omnes tum inter se tum cum disquisitionibus posterioribus tam arcto nexu cohaerebant, ut ne nova quidem satis commode explicari possent, nisi reliquis ab initio repetitis.

Prodiit interea opus egregium viri iam antea de Arithmetica Sublimiori magnopere meriti, Le Gendre Essai d'une théorie des nombres, Paris a. VI, in quo non modo omnia, quae hactenus in hac scientia elaborata sunt, diligenter collegit et in ordinem redegit, sed permulta insuper nova de suo adiecit. Quum hic liber serius ad manum mihi pervenerit, postquam maxima operis pars typis iam exscripta esset, nullibi, ubi rerum analogia occasionem dare potuisset, eius mentionem iniicere licuit; de paucis tantummodo locis quasdam observationes in Additamentis adiungere necessarium videbatur, quas vir humanissimus et candidissimus benigne ut spero interpretabitur.

Inter impressionem huius operis, quae pluries interrupta variisque impedimentis usque in quartum annum protracta est, non modo eas investigationes, quas quidem iam antea susceperam, sed quarum promulgationem in aliud tempus differre constitueram, ne liber nimis magnus evaderet, ulterius continuavi, sed plures etiam alias novas aggressus sum. Plures quoque, quas ex eadem ratione leviter tantum attigi, quum tractatio uberior minus necessaria videretur (e. g. eae quae in artt. 37, 82 sqq. aliisque locis traduntur), postea resumtae sunt, disquisitionibusque generalioribus quae luce perdignae videntur locum dederunt (Conf. etiam quae in Additamentis de art. 306 dicuntur). Denique quum liber praesertim propter amplitudinem Sect. V in longe maius quam exspectaveram volumen excresceret, plura quae ab initio ei destinata erant, interque ea totam Sectionem octavam (quae passim iam in hoc volumine commemoratur, atque tractationem generalem de congruentiis algebraicis cuiusvis gradus continet) resecare oportuit. Haec omnia, quae volumen huic aequale facile explebunt, publici iuris fient, quam primum occasio aderit.

Quod, in pluribus quaestionibus difficilibus, demonstrationibus syntheticis usus sum, analysinque per quam erutae sunt suppressi, imprimis brevitatis studio tribuendum est, cui quantum fieri poterat consulere oportebat.

Theoria divisionis circuli, sive polygonorum regularium, quae in Sect. VII tractatur, ipsa quidem per se ad Arithmeticam non pertinet, attamen eius principia unice ex Arithmetica Sublimiori petenda sunt: quod forsan geometris tam inexspectatum erit, quantum veritates novas, quas ex hoc fonte haurire licuit, ipsis gratas fore spero.

Haec sunt, de quibus lectorem praemonere volui. De rebus ipsis non meum est iudicare. Nihil equidem magis opto, quam ut iis, quibus scientiarum incrementa cordi sunt, placeant, quae vel hactenus desiderata explent, vel aditum ad nova aperiunt.


Portrait, by Christian Albrecht Jensen, Oil on canvas, Jul. 1840. Sternwarte, Göttingen.

Dedication

-- Carl Friedrich Gauss {കാര്‍ള്‍ ഫ്രീഡ്രിഷ് ഗൌസ്സ് / கார்ள் ஃப்ரீட்ரிஷ் கௌஸ்ஸ் / कार्ल् फ्रीड्रिष् गौस्स्}.

TO THE MOST SERENE
PRINCE AND LORD
CARL WILHELM 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 recognise 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's most dedicated servant
C. F. GAUSS.

Brunswick, July 1801.

Preface

The enquiries 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 enquiries 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 VII sqq. with the elegance and rigour 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 skilful handling rather than profound principles and, because the questions are too specialised and rarely lead to more general conclusions, Diophantus's 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. LA GRANGE, A. M. LE GENDRE (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 La Grange added to Euler's Algebra and in the recent volume of Le Gendre (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 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 enquiry 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 connexion 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 rigourous 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 recognised 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 them from the beginning.

Meanwhile there appeared an outstanding work by a man to whom Higher Arithmetic already owed much, Le Gendre's Essai d'une théorie des nombres, 1798. Here he collected together and systematised 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 feel 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 sqq. 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 I 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.

Translated to English by Arthur A. Clarke, S. J., 1966.


© 2001. David C. Kandathil. All rights reserved.