It was probably for Zimmermann’s sake that Gauss was so careful of his Latin, for as the latter’s protégé he must have been anxious that his first work should in no wise lack the stamp of completion. His preliminary reports had so aroused the Duke’s high expectations that he assumed the cost of publication and the book was gratefully dedicated to him when it finally appeared in 1801.
For Zimmermann as well as for Gauss it would have been painful to a high degree if the critics had found any well-grounded defects in it. Meyerhoff was the proper person to relieve them of these cares. Zimmermann had known him from the Carolineum, Meyerhoff had entered at the same time with Bartels, and through him may perhaps have come into touch also with Gauss. How much the classic form of the Disquisitiones owes to his polishing is hard to establish.
The Disquisitiones arithmeticae, which is perhaps Gauss’ principal work, contains many important researches, one of which, known as the celebrated Fundamental Theorem of Gauss, or the Law of Quadratic Reciprocity of Legendre, of itself alone, R. Tucker wrote, “would have placed Gauss in the first rank of mathematicians.” In March, 1795, Gauss discovered by induction, before he was eighteen years old, the proposition that the quadratic residue of prime numbers of the form 4n+1 is a quadratic nonresidue of primes of the form 4n+3 and worked out the first proof of it, which he published the following year. He was not satisfied with this, but published other demonstrations resting on different principles, until the number reached six. The proposition which he had discovered appeared to its author to be one of no ordinary beauty; and as he conjectured that it was connected with others of still greater value and generality, he applied (as he himself assures us) all the powers of his mind to find the principles upon which it rested and to establish its truth by a rigid demonstration. Having fully succeeded in this aim, he felt himself so completely fascinated by this class of researches that he found it impossible to abandon them. He was thus led from one truth to another, until he had finished the larger part of the first and most original, if not the greatest, of his works, before he had read the writings of any of his precursors in this department of science, notably those of Euler and Lagrange. He had, however, been anticipated in enunciating the theorem, although in a more complex form, by Euler, Legendre had unsuccessfully tried to prove it. R. Tucker remarked that the question of priority of enunciation or of demonstrating by induction is in this case a trifling one; any rigorous demonstration of it involved apparently insuperable difficulties.
The subsequent study of the arithmetical researches of these great masters of analysis could hardly fail to expose Gauss to the chagrin which young men of premature and creative genius have so often experienced, of finding that they have been anticipated in some of their finest speculations. The crowning result of his labors was, as is well known, the complete solution of binomial equations, and a most unexpected achievement in placing the imaginary on a firm basis. He was the first to use the symbol i for √-1, giving it the interpretation of a geometric mean between +1 and –1. This convention rapidly spread into general use. Of his theorem on binomial equations he made two other distinct demonstrations in December, 1815, and January, 1816. But these works, though he was the first in the field on the subject, gave him little fame. Lagrange seems not to have heard of the first one; and Cauchy, whose subsequent demonstrations have been preferred in textbooks, received in France all the praise due to a first discoverer.
The fruits of the Helmstedt sojourn were manifold. His doctoral dissertation embodied the complete solution of binomial equations, and it was upon this that the philosophical faculty awarded him the Ph.D. degree, in 1799. Doctoral dissertations, even of the greatest scholars, rarely exhibit anything of more than passing value. But not so with Gauss. The foundation of the whole theory of equations is formed by the proposition that every expression placed as a sum of powers of one and the same unknown with positive, integral exponents, in order, is factorable into factors of the first or second degree of that unknown. The existence of this fundamental theorem of algebra had long been known. Many writers had published supposedly rigorous proofs of it. Gauss proceeded to show, in his first section, a model of historical and critical presentation, that all those early “proofs” were only pseudo proofs, were only miscarried attempts, and in its second section his thesis presented an indisputable proof of the theorem, of faultless, dogmatic sharpness. The proofs of 1815 and 1816 were just as rigorous. In 1849, at the golden-anniversary celebration of his doctorate, almost the last paper which he ever published gives still another proof. This is essentially the elaboration of a thought indicated in the thesis and presents the same proof in altered form. Which of these proofs is to be regarded as the most characteristic, and the most beautiful, is a matter of taste. It speaks highly for the four proofs when first this one and then that one receives the highest praise from various writers.
In dedicating his Disquisitiones arithmeticae to the Duke of Brunswick, Gauss acknowledged in very touching terms the wise and liberal patronage which had not only provided for the expenses of his work but also enabled him to exchange the humble pursuits of trade for those of science. The work itself, its author assures us, assumed many changes of form in its progress to maturity, as new views presented themselves from time to time to his mind; but, as is well known, the course which is followed in the discovery of new truths is rarely that which is most favorable to their clear exposition, more especially when it has been pursued in solitude, with little communication with other minds. The peculiar terminology which Gauss employed in the classification of numbers and their relations, and which is so completely embodied in the enunciation and demonstration of nearly every proposition that it can never be absent from the mind of the reader, rendered the study of this work so laborious and embarrassing that few persons have ever mastered its contents. The book is one of the standard works of the nineteenth century and has never been really surpassed or supplanted. Not an error has ever been detected in it. But though the character of Gauss’ subjects tempts few readers, though his own severe brevity renders these subjects even more difficult than they need be, yet the young reader of Euclid may be brought into contact with Gauss so as to understand the tone of his genius in a manner which would be utterly impossible in the case of Newton, or Lagrange, or Euler.
Even Legendre, who had written so much and so successfully on the same subject, and who, in the second edition of his Théorie des nombres (1808) makes the great discovery of what this book contains the occasion not merely of special investigation but of the most emphatic praise, complains of the extreme difficulty of adapting its forms of exposition to his own; while the writers of the Biographie des contemporains in a notice of the author at a much later period, when he had established many other and almost equally unquestionable claims to immortality, quote an extract from a report of a commission of the Institute of France, to whom it was referred in 1810, in which it is said, “that it was impossible for them to give an idea of this work, inasmuch as everything in it is new, and surpasses our comprehension even in its language.” The biographers then proceed to stigmatize the book as full of puerilities, and refer to its success, including its translation into two languages, as ground for the assumption that “charlatanism sometimes extended even to the domain of mathematics.” It is rather amusing that twelve years later we read in this work: “Suffice it to say that his works are esteemed in general by the most distinguished mathematicians and that they recommend themselves, rather by their exactitude than by clarity, by their precision and elegance of style.”
Another section in the Disquisitiones involves the theory of the congruence of numbers, or the relation that exists between all numbers that give the same remainder when they are divided by the same number. As Gauss was working on the book a new problem arose; it was much longer than originally planned and hence the funds promised by the Duke would not cover the cost of publication. He decided to close with the seventh section, to lay aside the eighth, and also to sh
orten it in many places. Even so, the publication suffered a long interruption until one day the Duke inquired about it and learned the difficulty. That was enough. The Duke supplied the needed funds, and in the summer of 1801 the final sheets of the folio left the press of Gerhard Fleischer in Leipzig. The sale of the book was greater than had originally been hoped for, but the author received almost no profit, because the bookseller in Paris, to whom most of the copies had been consigned, went bankrupt. Gauss was not the man to dwell on the misfortune.
The dedication, dated July, runs thus:
I account it the highest joy, most gracious Prince, that you allow me to grace with your exalted name this work which I offer to you in fulfillment of the holy obligations of loyal love. For if your Grace had not opened up for me the access to the sciences, if your unremitting benefactions had not encouraged my studies up to this day, I would never have been able to dedicate myself completely to the mathematical sciences to which I am inclined by nature. Indeed, the observations, some of which are presented in this volume . . . the fact that I could undertake them, continue them for several years, and publish them, I owe only to your kindness which freed me from other cares, and permitted me to devote myself to this work. Your magnanimity has pushed aside all the obstacles which delayed publication.
He also refers to the Duke’s understanding of the coherence of all the sciences, and to his wise insight, which did not deny assistance to those sciences which appear of little use in practical life and are criticized as abstract. This was not a phrase of flattery, and no one was more deeply convinced of the truth of the expression than Gauss himself.
The Duke’s long delay in granting Gauss an annuity on his return to Brunswick is explained by the fact that his state treasury was close to bankruptcy, depleted as it was by his father’s too generous spending. Tongues wagged in criticism of such gratuities as impractical and unwarranted. But in the end the Duke’s own better judgment settled the matter, and young Gauss’ financial worries were at least temporarily removed.
One can easily see now how the publication of the work was delayed for four years by various circumstances. Some of the material in the first six sections relates back many centuries to Diophantus. The presence of some old theorems is accounted for by the fact that the publications of the Berlin and St. Petersburg Académies were not available to Gauss. Two things stand out: the doctrine of quadratic forms, and the name “determinants,” which Gauss first introduced. “Mathematics,” he said, “is the queen of the sciences and the theory of numbers the queen of mathematics.” If this is true, one may continue by calling the Disquisitiones the Magna Charta of the theory of numbers. Gauss published memoirs on biquadratic residues in 1817 and 1831. After his death, an eighth section of the Disquisitiones treating of congruences of higher degree was found and published.
Lagrange, who was called by Napoleon the high pyramid of the mathematical sciences, wrote to the young scholar shortly after the Disquisitiones were published: “Your Disquisitiones have with one stroke elevated you to the rank of the foremost mathematicians, and the contents of the last section [theory of the equations of circle division] I look on as the most beautiful analytical discovery which has been made for a long time.”
Laplace, who was the author of the famous Mécanique céleste and before whose impressive greatness everyone in the Paris academy bowed down, is said to have exclaimed: “The Duke of Brunswick has discovered more in his country than a planet: a superterrestrial spirit in a human body!” This was just after the young Brunswick mathematician had made his first important astronomical contribution. Several years later Laplace recommended to Napoleon the Conqueror especial consideration for the university city of Göttingen, because “the foremost mathematician of his time dwells there.”
Gauss’ own opinion of the Disquisitiones, uttered in his closing years, is not without interest: “The Disquisitiones belong to history, and in a new edition, to which I am not disinclined, but for which I now possess no leisure, I would change nothing, with the exception of the misprints; I would merely like to attach the eighth section, which indeed was essentially worked out, but did not appear at the time in order not to increase the cost of printing the book.” Gauss wrote to Bolyai that he hoped to have published as a second volume of the work many supplementary pieces. These investigations as a matter of fact were later published in bulletins of the Royal Society of Göttingen and the university’s periodicals. The complete list of Gauss’ writings, exclusive of larger works, contains 124 papers. It is singular that Gauss devoted so few memoirs to subjects of a purely algebraic character; except for a rather unimportant paper on “Descartes’ Rule of Signs” in Crelle’s Journal (1828), his only algebraic papers relate to the doctoral thesis.
Very soon the Disquisitiones went out of print; Eisenstein, the brilliant mathematician who died very young, could not get possession of a copy of the original. On account of the scarcity of copies, other pupils of Gauss had the courage to copy the work by hand from beginning to end. Just as certain clergymen go about with their prayer book, so many a great mathematician of the nineteenth century was always accompanied by a much-read copy of the Disquisitiones, taken from the original volume. Nevertheless, it remained a book with seven seals.
By a mischance the thoroughly rigorous and simple proof of Lagrange’s theorem, which Gauss found in 1797, was never published. Gauss communicated it to Pfaff, who forwarded it to Hindenburg; Hindenburg soon died, and the manuscript never appeared again. Gauss never published this proof, of which a transcript is extant, because he later found that Laplace had applied a similar method. The theorem ran thus: “If p and q are positive uneven prime numbers, p has the same quadratic character with regard to q that q has with regard to p; except when p and q are both of the form 4n+3, in which case the two characters are always opposite, instead of identical.” Legendre’s unsuccessful attempt to prove this appeared in the Memoirs of the academy of Paris for 1784, Gauss was not content with once vanquishing the difficulty; he returned to it again and again in the fifth section of the Disquisitiones and there obtained another demonstration based on entirely different but perhaps still less elementary principles. In 1875 Kronecker showed that Euler was the real discoverer of this law, in that he enunciated it before Legendre.
Gauss rightfully called this theorem the fundamental one in quadratic residues, because it not only forms the real nucleus in the theory but also has fundamental meaning in the later portions of the theory.22 He could not foresee that scientists would furnish forty proofs of it! He did say that proofs would have to be created more ex notionibus rather than ex notationibus. Dirichlet succeeded in reducing Gauss’ eight cases to two. Gauss stated that he struggled for a year with one little point.
It is no wonder that he should have felt a sort of personal attachment to a theorem which he had made so completely his own, and which he used to call the “gem” of the higher arithmetic. It would be impossible to exaggerate the importance of the influence which this theorem has had on the later development of arithmetic, and the discovery of its demonstration by Gauss must certainly be regarded (as it was by Gauss himself) as one of his greatest scientific achievements. The fifth section (“these marvelous pages”) abounds with subjects each of which has been the starting point of long series of important researches by later mathematicians. It is curious that wonderful researches only alluded to in this section first saw the light sixty-three years later in the second volume of the collected edition of his works. Until the time of Jacobi the profound researches of the fourth and fifth sections were almost completely ignored. The seventh section on the theory of circle division and the theory of equations was received with great enthusiasm.
Very noteworthy is a passage (Article 335) where he observes that the principles of his method are applicable to many other functions besides circular functions, and in particular to the transcendents dependent on the integral
and have equal importance with respect to the l
emniscate, just as the trigonometrical functions are related to the circle. We know from his diary that on March 19, 1797, he investigated the equation which is used in the division of the lemniscate and on March 21 established the fact that this curve may be geometrically divided into five equal parts by means of a straightedge and compasses. This almost casual remark (as Jacobi observed) shows that Gauss before 1801 had already examined the nature and properties of the elliptic functions and had discovered their fundamental property, that of double periodicity. From May 6 to June 3, 1800, he formulated the general concepts in this field. Nothing of this was published; Gauss was ahead of his time. Abel and Jacobi found the doubly periodic or elliptic functions in the twenties. The elliptic modular functions which Gauss mastered in 1800 were fully developed only in recent times.
It was in the course of an astronomical investigation that Gauss arrived at some elliptic integrals, the evaluation of which he was able to effect by means of a transformation included in Jacobi’s series of transformations. It is said that this distinguished analyst was induced by his knowledge of the fact to seek (after his own discoveries were completed) an interview with the great mathematician who had thus intruded, prematurely as it were, into one of the deepest recesses of his own province. Jacobi submitted his various theorems to Gauss’ inspection, and they were met, as they successively appeared, by others of corresponding character and import produced from Gauss’ manuscript stores. The interchange concluded with an intimation that Gauss had many more in reserve. Such an anticipation of discoveries which totally changed the aspect of this difficult department of analysis constituted no infringement of the rights which Jacobi undeniably secured by priority of publication. Wide circulation was given to this story. The story is confirmed by the papers on elliptic transcendents, published in the third volume of Gauss’ Collected Works.
Carl Friedrich Gauss, Titan of Science_A Study of His Life and Work Page 7