## Disquisitiones Arithmeticae.

Leipzig: Gerh. Fleischer, 1801.

First edition, rare, of Gauss’ masterpiece, “a book that begins a new epoch in mathematics … Gauss ranks, together with Archimedes and Newton, as one of the greatest geniuses in the history of mathematics” (PMM). “Published when Gauss was just twenty-four, *Disquisitiones arithmeticae* revolutionized number theory. In this book Gauss standardized the notation; he systemized the existing theory and extended it; and he classified the problems to be studied and the known methods of attack and introduced new methods … The *Disquisitiones* not only began the modern theory of numbers but determined the direction of work in the subject up to the present time. The typesetters of this work were unable to understand Gauss’ new and difficult mathematics, creating numerous elaborate mistakes which Gauss was unable to correct in proof. After the book was printed Gauss insisted that, in addition to an unusually lengthy four-page errata, the worst mistakes be corrected by cancel leaves to be inserted in copies before sale … Gauss’s highly technical work was printed in a small edition, and the difficulty of understanding it was hardly alleviated by the sloppy typesetting” (Norman). “In the late eighteenth century [number theory] consisted of a large collection of isolated results. In his *Disquisitiones* Gauss summarized previous work in a systematic way, solved some of the most difficult outstanding questions, and formulated concepts and questions that set the pattern of research for a century and still have significant today. He introduced congruence of integers with respect to a modulus (*a ≡ b* (mod *c*) if *c* divides *a - b*), the first significant algebraic example of the now ubiquitous concept of equivalence relation. He proved the law of quadratic reciprocity, developed the theory of composition of quadratic forms, and completely analyzed the cyclotomic equation. The *Disquisitiones* almost instantly won Gauss recognition by mathematicians as their prince” (DSB).

“The awe that [*Disquisitiones arithmeticae*] inspired in mathematicians was displayed to the cultured public of the *Moniteur universel ou Gazette nationale* as early as March 21, 1807, when Louis Poinsot, who would succeed Joseph-Louis Lagrange at the Academy of Sciences six years later, contributed a full page article about the French translation of the *Disquisitiones arithmeticae*: ‘The doctrine of numbers, in spite of [the works of previous mathematicians] has remained, so to speak, immobile, as if it were to stay for ever the touchstone of their powers and the measure of their intellectual penetration. This is why a treatise as profound and as novel as his *Arithmetical Investigations* heralds M. Gauss as one of the best mathematical minds in Europe.’

“A long string of declarations left by readers of the book, from Niels Henrik Abel to Hermann Minkowski, from Augustin-Louis Cauchy to Henry Smith, bears witness to the profit they derived from it. During the XIXth century, its fame grew to almost mythical dimensions. In 1891, Edouard Lucas referred to the *Disquisitiones Arithmeticae* as an ‘imperishable monument [which] unveils the vast expanse and stunning depth of the human mind,’ and in his Berlin lecture course on the concept of number, Leopold Kronecker called it ‘the Book of all Books’ … Gauss’s book is now seen as having created number theory as a systematic discipline in its own right, with the book, as well as the new discipline, represented as a landmark of German culture …

“Gauss began to investigate arithmetical questions, at least empirically, as early as 1792, and to prepare a number-theoretical treatise in 1796 (i.e., at age 19 and, if we understand his mathematical diary correctly, soon after he had proved both the constructibility of the 17-gon by ruler and compass and the quadratic reciprocity law). An early version of the treatise was completed a year later. In November 1797, Gauss started rewriting the early version into the more mature text which he would give to the printer bit by bit. Printing started in April 1798, but proceeded very slowly for technical reasons on the part of the printer. Gauss resented this very much, as his letters show; he was looking for a permanent position from 1798. But he did use the delays to add new text, in particular to sec. 5 on quadratic forms, which had roughly doubled in length by the time the book finally appeared in the summer of 1801.

“The 665 pages and 355 articles of the main text are divided unevenly into seven sections. The first and smallest one (7 pp., 12 arts.) establishes a new notion and notation which, despite its elementary nature, modified the practice of number theory:

‘If the number *a* measures the difference of the numbers *b*, *c*, then *b* and *c* are said to be congruent according to *a*; if not, incongruent; this *a* we call the modulus. Each of the numbers *b*, *c* are called a residue of the other in the first case, a nonresidue in the second.’ The corresponding notation *b* ≡ *c* (mod *a*) is introduced in art. 2. The remainder of sec. 1 contains basic observations on convenient sets of residues modulo *a* and on the compatibility of congruences with the arithmetic operations …

“Section 2 (33 pp., 32 arts.) opens with several theorems on integers including the unique prime factorization of integers (in art. 16), and then treats linear congruences in arts. 29–37, including the Euclidean algorithm and what we call the Chinese remainder theorem. At the end of sec. 2, Gauss added a few results for future reference which had not figured in the 1797 manuscript, among them: (i) properties of the number *φ*(*A*) of prime residues modulo *A* (arts. 38–39); (ii) in art. 42, a proof that the product of two polynomials with leading coefficient 1 and with rational coefficients that are not all integers cannot have all its coefficients integers; and (iii) in arts. 43 and 44, a proof of Lagrange’s result that a polynomial congruence modulo a prime cannot have more zeros than its degree.

“Section 3 (51 pp., 49 arts.) is entitled ‘On power residues.’ As Gauss put it, it treats ‘geometric progressions’ 1, *a*, *a*^{2}, *a*^{3}, ... modulo a prime number *p* (for a number *a* not divisible by *p*), discusses the ‘period’ of *a* modulo *p* and Fermat’s theorem, contains two proofs for the existence of ‘primitive roots’ modulo *p*, and promotes the use of the ‘indices’ of 1, ..., *p -* 1 modulo *p* with respect to a fixed primitive root, in analogy with logarithm tables. After a discussion, in arts. 61–68, of *n*th roots mod *p* from the point of view of effective computations, the text returns to calculations with respect to a fixed primitive root, and gives in particular in arts. 75–78 two proofs – and sketches a third one due to Lagrange – of Wilson’s theorem, 1·2 ··· (*p* - 1) ≡ -1 (mod *p*) …

“Section 4 (73 pp., 59 arts.), ‘On congruences of degree 2,’ develops a systematic theory of ‘quadratic residues’ (i.e., residues of perfect squares). It culminates in the ‘fundamental theorem’ of this theory, from which ‘can be deduced almost everything that can be said about quadratic residues,’ and which Gauss stated as: ‘If *p* is a prime number of the form 4*n* + 1, then +*p*, if *p* is of the form 4*n* + 3, then −*p*, will be a [quadratic] residue, resp. nonresidue, of any prime number which, taken positively, is a residue, resp. nonresidue of *p*.’ Gauss motivated this quadratic reciprocity law experimentally, gave the general statement and formalized it in tables of possible cases … He also gave here the first proof of the law, an elementary one by induction. A crucial nontrivial ingredient (used in art. 139) is a special case of a theorem stated in art. 125, to the effect that, for every integer which is not a perfect square, there are prime numbers modulo which it is a quadratic nonresidue.

“The focus changes in sec. 5 of the *Disquisitiones arithmeticae*, which treats ‘forms and indeterminate equations of the second degree,’ mostly binary forms, in part also ternary. With its 357 pp. and 156 arts., this section occupies more than half of the whole *Disquisitiones**Arithmeticae*. Leonhard Euler, Joseph-Louis Lagrange, and Adrien-Marie Legendre had forged tools to study the representation of integers by quadratic forms. Gauss, however, moved away from this Diophantine aspect towards a treatment of quadratic forms as objects in their own right, and, as he had done for congruences, explicitly pinpointed and named the key tools. This move is evident already in the opening of sec. 5: ‘The form *axx *+ 2*bxy* + *cyy*, when the indeterminates *x, y* are not at stake, we will write like this, (*a, b, c*).’ Gauss then immediately singled out the quantity *bb – ac* which he called the ‘determinant’ – ‘on the nature of which, as we will show in the sequel, the properties of the form chiefly depend’ – showing that it is a quadratic residue of any integer primitively represented by the form (art. 154).

The first part of sec. 5 (arts. 153–222, 146 pp.) is devoted to a vast enterprise of a finer classification of the forms of given determinant, to which the problem of representing numbers by forms is reduced. Gauss defined two quadratic forms (art. 158) to be equivalent if they are transformed into one another under substitutions of the indeterminates, … Two equivalent forms represent the same numbers … After generalities relating to these notions and to the representation of numbers by forms … the discussion then splits into two very different cases according to whether the determinant is negative or positive. In each case, Gauss showed that any given form is properly equivalent to a so-called ‘reduced’ form (art. 171 for negative, art. 183 for positive discriminants), not necessarily unique, characterized by inequalities imposed on the coefficients. The number of reduced forms – and thus also the number of equivalence classes of forms – of a given determinant is finite … Gauss settled the general problem of representing integers by quadratic forms (arts. 180–181, 205, 212), as well as the resolution in integers of quadratic equations with two unknowns and integral coefficients (art. 216) …

“The classification of forms also ushers the reader into the second half of sec. 5, entitled ‘further investigations on forms’ … In art. 226, certain classes are grouped together into an ‘order’ according to the divisibility properties of their coefficients. There follows (arts. 229–233) a finer grouping of the classes within a given order according to their ‘genus’ … This rich new structure gave Gauss a tremendous leverage: to answer new questions, for instance, on the distribution of the classes among the genera (arts. 251–253); to come back to his favourite theorem, the quadratic reciprocity law, and derive a second proof of it … (arts. 261–262); to solve a long-standing conjecture of Fermat’s (art. 293) to the effect that every positive integer is the sum of three triangular numbers. For this last application, as well as for deeper insight into the number of genera, Gauss quickly generalized (art. 266 ff.) the basic theory of reduced forms, classes etc., from binary to ternary quadratic forms. This gave him in particular explicit formulae for the number of representations of binary quadratic forms, and of integers, by ternary forms, implying especially that every integer ≡ 3 (mod 8) can be written as the sum of three squares, which is tantamount to Fermat’s claim …

“Explicit calculations had evidently been part and parcel of number theory for Gauss ever since he acquired a copy of [Lambert, *Zusätze zu den logarithmischen und trigonometrischen Tabellen zur Erleichterung und Abkürzung der bey Anwendung der Mathematik vorfallenden Berechnungen*. Berlin: Haude und Spener, 1770] at age 15, and launched into counting prime numbers in given intervals in order to guess their asymptotic distribution. In these tables, Johann Heinrich Lambert made the memorable comment: ‘What one has to note with respect to all factorization methods proposed so far, is that primes take longest, yet cannot be factored. This is because there is no way of knowing beforehand whether a given number has any divisors or not.’ The whole *Disquisitiones arithmeticae* is illustrated by many non-trivial examples and accompanied by numerical tables. Section 6 (52 pp., 27 arts.) is explicitly dedicated to computational applications. In the earlier part of sec. 6, Gauss discussed explicit methods for partial fraction decomposition, decimal expansion, and quadratic congruences. Its latter part (arts. 329–334) takes up Lambert’s problem and proposes two primality tests: one is based on the fact that a number which is a quadratic residue of a given integer *M *is also a quadratic residue of its divisors and relies on results of sec. 4; the second method uses the number of values of √-*D* mod *M*, for -*D* a quadratic residue of *M*, and the results on forms of determinant -*D* established in sec. 5.

“The final Section 7 on cyclotomy (74 pp., 31 arts.) is probably the most famous part of the *Disquisitiones Arithmeticae*, then and now, because it contains the conditions of constructibility of regular polygons with ruler and compass. After a few reminders on circular functions … Gauss focused on the prime case and the irreducible equation

*X = x ^{n-}*

^{1}

*+ x*

^{n-}^{2}

*+ … + x*+ 1 = 0,

*n >*2 prime,

which his aim is to ‘decompose gradually into an increasing number of factors in such a way that the coefficients of these factors can be determined by equations of as low a degree as possible, until one arrives at simple factors, i.e., at the roots of *X*.’ Art. 353 illustrates the procedure for *n* = 19, which requires solving two equations of degree three and one quadratic equation (because *n* – 1 = 3·3·2); art. 354 does the same for *n* = 17 which leads to four quadratic equations (*n *– 1 = 2·2·2·2) …

“Complementary results on the auxiliary equations, i.e., those satisfied by the sums over all the roots of unity in a given period, are given in art. 359, applications to the division of the circle in the final arts. 365 and 366. As a by product of his resolution of *X *= 0, Gauss also initiated a study of what are today called ‘Gauss sums,’ i.e., certain (weighted) sums of roots of unity, like the sum of a period, or of special values of circular functions …

“Despite the impressive theoretical display of sec. 5, one cannot fully grasp the systemic qualities of the *Disquisitiones arithmeticae* from the torso that Gauss published in 1801. At several places in the *Disquisitiones arithmeticae* and in his correspondence a forthcoming volume II is referred to. The only solid piece of evidence we have is what remains of Gauss’s 1796–1797 manuscript of the treatise. This differs from the structure of the published *Disquisitiones arithmeticae* in that it contains an (incomplete) 8^{th} chapter (*caput octavum*), devoted to higher congruences, i.e., polynomials with integer coefficients taken modulo a prime and modulo an irreducible polynomial. Thus, according to Gauss’s original plan, sec. 7 would not have been so conspicuously isolated, but would have been naturally integrated into a greater, systemic unity. The division of the circle would have provided a model for the topic of the *caput octavum*, the theory of higher congruences; it would have appeared as part of a theory which, among many other insights, yields two entirely new proofs of the quadratic reciprocity law” (Goldstein & Schappacher).

PMM 257; Evans 11; Horblit 38; Dibner 114. Goldstein & Schappacher, ‘A book in search of a discipline (1801-1860),’ pp. 3-66 in *The Shaping of Arithmetic after C. F. Gauss’s Disquisitiones Arithmeticae*, 2007.

8vo (203 x 118 mm), contemporary green half morocco, pp [i-vi] vii-xviii [1] 2-668 [3:tables as the Horblit copy] [4:errata] with B7, G4, K3, 2F7, and 2T6 cancels (as usual), first and final leaves with some spotting as is often seen with this work. An entirely unrestored copy.

Item #4572

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Price:
$38,000.00
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