Disquisitiones Arithmeticae.
Leipzig: Gerh. Fleischer, 1801. First edition of the work that created modern number theory and established its twenty-four-year-old author as the foremost mathematician in Europe. Before the Disquisitiones Arithmeticae the theory of numbers consisted of a collection of isolated results — brilliant particular theorems due to Fermat, Euler, Lagrange, and Legendre, without a unifying framework or a systematic method of attack. Gauss brought the entire subject into order, filled the gaps, corrected the unsound proofs, solved the outstanding problems that had defeated his predecessors, and introduced concepts and notation of such power that they determined the direction of number-theoretic research for the next century and remain in daily use today. Leopold Kronecker, in his Berlin lecture course on the concept of number, called it the “Book of all Books,” and the epithet has stuck. The present copy carries a provenance of unusual interest, passing through a French private collection of 1881 and thence into the library of St. Laurence's Abbey at Ampleforth — the senior house of the English Benedictine Congregation, in historic descent from Westminster Abbey through the last Marian monk — where it was received as a memorial gift for a member of the Witham family of recusant Yorkshire and Durham. Carl Friedrich Gauss (1777–1855), the son of a bricklayer and gardener in Brunswick, had shown extraordinary mathematical precocity from early childhood. The Duke of Brunswick, recognising the gift, funded his education at the Collegium Carolinum and then at the University of Göttingen, which Gauss entered in October 1795. On 30 March 1796, one month before his nineteenth birthday, he proved that the regular heptadecagon — the polygon of seventeen sides — could be constructed with ruler and compass alone. This was the first advance in the constructibility of regular polygons since antiquity: Euclid had shown how to construct the regular triangle, square, pentagon, hexagon, and fifteen-gon, and no one in two thousand years had added to the list. The discovery confirmed Gauss in his vocation as a mathematician. On the same day he began his mathematical diary, and within weeks he had proved the law of quadratic reciprocity, the theorem that Euler and Legendre had both attempted without success. An early version of the Disquisitiones was drafted the following year. In November 1797 Gauss began rewriting it into the mature text he would deliver to the printer piecemeal; composition of the book and the search for a permanent position occupied him simultaneously through 1798–1801. He was still only twenty when he submitted his doctoral dissertation at Helmstedt, containing the first rigorous proof of the fundamental theorem of algebra; when the Disquisitiones appeared two years later, it announced to Europe that the foremost living mathematician was a man of twenty-four. Printing began in April 1798 at the Leipzig press of Gerhard Fleischer, but proceeded with excruciating slowness. The typesetters could not understand the mathematics they were setting, and the resulting errors were so numerous and so elaborate that Gauss could not correct them in proof. After the book was printed he insisted on a four-page errata sheet and, in addition, that the worst mistakes be corrected by cancel leaves inserted into copies before sale — five cancels in total (B7, G4, K3, Ff7, Tt6), an unusually drastic measure reflecting a level of typographic corruption that must have been agonising for an author of Gauss's precision. The edition was small. The 668 pages of dense Latin, laden with new notation and addressed to a readership that scarcely existed yet, ensured that copies sold slowly; first editions are rare in commerce, and copies preserving all five cancels alongside the original uncancelled leaves are rarer still. Every cancel called for in Horblit and in Norman is present in this copy. The book comprises seven sections of markedly unequal length and ambition. Section 1, the shortest, introduces the concept and notation that would prove the most consequential single innovation in nineteenth-century mathematics: congruence of integers with respect to a modulus. Two integers b and c are congruent modulo a if a divides their difference; Gauss writes this as b ≡ c (mod a). The notation is at once a simplification — it compresses an entire class of divisibility statements into a single symbolic relation — and a conceptual revolution: it identifies a new algebraic structure, the ring of integers modulo a, that is the first significant example of what modern algebra calls an equivalence relation. Its compatibility with the four arithmetic operations, established in the opening pages, makes it a tool of extraordinary power, and the three-bar sign ≡ that Gauss introduces here is still taught to every student of number theory on earth. Sections 2 and 3 develop the arithmetic of congruences. Section 2 proves the unique prime factorisation of integers — the theorem we now call the fundamental theorem of arithmetic, first stated with Gauss's rigour here — and treats linear congruences and the Chinese remainder theorem. Section 3, on the residues of powers, proves Fermat's little theorem, establishes the existence of primitive roots modulo a prime, and presents Gauss's own solution to the problem of the index and his table of indices to the modulus 97. Much of this material was partly known to Euler and Lagrange, but it had never before been assembled into a coherent logical sequence with rigorous proofs from a common foundation, nor furnished with a notation adequate to its expression. In Section 3 Gauss also gives his first proof of Wilson's theorem — a theorem which Lagrange had proved by a different route but which Gauss establishes as a corollary of his theory of primitive roots, demonstrating already the systematic economy that would make the Disquisitiones a model for mathematical exposition. Section 4 is the proof of the law of quadratic reciprocity, the theorem Gauss called the theorema fundamentale and regarded as the crown of the theory of quadratic residues. The law governs the relationship between two odd primes p and q: it determines, from the quadratic character of p modulo q, the quadratic character of q modulo p. Euler had conjectured the law; Legendre had stated it explicitly and attempted a proof that contained a gap; Gauss, at nineteen, gave the first rigorous proof and would eventually give eight proofs in total, each illuminating a different aspect of the theorem. The proof in the Disquisitiones, by induction, is the first; it depends on a non-trivial lemma about the existence of primes with specified quadratic character which Gauss establishes earlier in the section. Quadratic reciprocity remained a central object of research throughout the nineteenth century and was generalised successively by Eisenstein, Kummer, Hilbert, and Artin into the reciprocity laws that are the backbone of modern algebraic number theory; the twentieth-century Langlands programme, which organises much of contemporary number theory and representation theory, may be understood as a still vaster generalisation of the single theorem that Gauss first proved here. Section 5, which occupies more than half the entire book — 357 pages and 156 articles — is the theory of binary and ternary quadratic forms. Where his predecessors had studied the representation of integers by particular quadratic expressions (x² + ny² and its variants), Gauss treated quadratic forms as algebraic objects in their own right, classifying them by equivalence under substitution and introducing a chain of increasingly refined groupings — classes, orders, genera — whose interaction constitutes the first algebra of classification in the history of mathematics. The key invariant is what Gauss calls the “determinant” of the form (later called the discriminant); two forms of the same determinant can be further distinguished by the genus to which they belong, and the number of genera is itself computable from the determinant by means of Gauss's theory. From this apparatus he derives a second proof of quadratic reciprocity (by an entirely different method from the first), a proof of Fermat's conjecture that every positive integer is the sum of three triangular numbers, and a battery of results on the representation of integers by ternary forms. The section introduced the concept of equivalence classes of algebraic objects classified by invariants — a mode of mathematical thinking so fundamental to the modern discipline that later mathematicians would recognise it as the prototype of the group-theoretic and ring-theoretic methods that organised nineteenth- and twentieth-century algebra. Dedekind's theory of ideals, which grew directly out of his meditation on Gauss's composition of forms, is the most direct descendant; the class group of an algebraic number field, whose finiteness remains one of the central theorems of the subject, is a direct generalisation of Gauss's class number for binary forms. Section 6 turns to computation — explicit methods for partial fractions, decimal expansions, and two primality tests that Gauss derives from the theory of quadratic residues developed in Section 4 and the theory of forms in Section 5. These are not appendages but applications: Gauss is demonstrating that a deep theoretical apparatus has practical consequences for the testing of large numbers, a concern that traces back to the tables of Johann Heinrich Lambert that Gauss had acquired at fifteen and that had launched him into the empirical counting of prime numbers from which his conjecture of the prime number theorem — that the number of primes below x is asymptotic to x / log x — first emerged. Hadamard and de la Vallée Poussin would prove the conjecture in 1896, nearly a century after Gauss wrote it privately in the margin of his copy of Schulze's logarithmic tables; the theorem remains the single deepest result about the distribution of primes, and its proof in its various forms constitutes the founding work of analytic number theory. Section 7, on cyclotomy, is the most famous and the most surprising part of the book. Its subject is the division of the circle into n equal parts — equivalently, the solution of the equation xⁿ – 1 = 0 — and its culminating result is the determination of which regular polygons can be constructed with ruler and compass alone. Gauss shows that the constructibility of the regular n-gon depends on the prime factorisation of n: it is constructible if and only if n is a product of a power of 2 and distinct Fermat primes (primes of the form 2^(2^k) + 1). The regular 17-gon, 257-gon, and 65537-gon are constructible; the regular 7-gon, 9-gon, 11-gon, and 13-gon are not. The proof requires the “Gaussian periods” — sums of roots of unity grouped by the arithmetic of their indices — which anticipate the Galois theory of the 1830s and which Gauss uses to factorise the cyclotomic polynomial into components whose roots can be expressed by successive extraction of square roots. The section concludes with detailed constructions for n = 17 (requiring four nested quadratic equations) and n = 19 (requiring two cubics and a quadratic), the first new contributions to the problem of polygon construction since Euclid. Gauss was so proud of the heptadecagon that he asked for it to be carved on his gravestone; the mason, judging that a seventeen-sided figure would be indistinguishable from a circle, refused, and the memorial column at Gauss's birthplace in Brunswick bears a seventeen-pointed star instead. The published book is a torso. A planned eighth section on higher congruences — polynomials with integer coefficients taken modulo a prime and modulo an irreducible polynomial — was begun but not completed, and survived only in Gauss's manuscript of 1796–97, published posthumously as the caput octavum in the collected works. The unpublished section would have placed the cyclotomy of Section 7 within a larger algebraic framework and yielded two further proofs of quadratic reciprocity by new methods. In Gauss's original conception, the constructibility of regular polygons was not the climax of the book but a station on the way to a theory of which it was one application among several. The work as published is therefore both a monument and a fragment — the most architecturally complete mathematical treatise since Euclid, and at the same time an outline of a programme that its author never finished. The Disquisitiones was written in Latin — one of the last major mathematical works to be so composed. A French translation by Poullet-Delisle appeared in 1807 and was the edition through which most of the French mathematical community, including Cauchy and Dirichlet, first encountered the book; a German translation by H. Maser followed in 1889; the first English translation, by Arthur A. Clarke, was not published until 1966. The book's influence across the nineteenth century was immense and well-attested. Abel, Dirichlet, Dedekind, Kronecker, Eisenstein, Kummer, Minkowski, and Hilbert all acknowledged the Disquisitiones as the foundation of their work. Dirichlet, who carried his copy with him on every journey and is said to have slept with it under his pillow, transformed its methods into the analytic number theory of the Vorlesungen über Zahlentheorie; Dedekind, editing the posthumous editions of Dirichlet's lectures, built from Gauss's theory of forms the algebraic number theory that gave the nineteenth century one of its central mathematical achievements. As late as 1891 Édouard Lucas hailed the book as an “imperishable monument” unveiling the vast reach and depth of the human mind; Kronecker's Berlin tribute came more than eighty years after publication. Provenance. The early history of the present copy is French. At the head of the front pastedown a contemporary-looking pencil inscription reads cote 92 f° alg. Lefèvre 1881 — a French shelfmark identifying the book as the ninety-second item, folio size, in the algebra section of a private or institutional collection associated with the name Lefèvre, and dated 1881. Below this is a later pencil shelfmark, AS/P5, of the receiving library. At some point between 1881 and 1907 the book passed into English hands and entered the library of St. Laurence's Monastery at Ampleforth, where a printed armorial-style bookplate was affixed to the pastedown: Ex Libris Monasterii Sti. Laurentii apud Amplefordiam — Ora pro Anima Caroli Witham Herbert. Obiit die XX Aprilis MCMVII. Ampleforth Abbey in North Yorkshire is the senior house of the English Benedictine Congregation, descended in unbroken succession from the pre-Reformation community of Westminster Abbey through Father Sigebert Buckley (c. 1520 – c. 1610), the last surviving Marian monk, who before his death aggregated the first members of the exiled English Benedictine revival into the legal continuity of the medieval house. The Ampleforth community, driven from its refuge at Dieulouard in Lorraine by the French Revolution, settled at Ampleforth Lodge in 1802; the school opened the same year and the priory was raised to the dignity of abbey in 1890. A memorial bookplate of this form, of a type common in Catholic monastic libraries of the period, records a donation made in memory of a deceased member of one of the recusant Catholic families of the north of England. The surname Witham is borne by one of the most distinguished Catholic recusant dynasties of Yorkshire and Durham — a family that held Cliffe Hall at Piercebridge, that gave to the revived English Church the Venerable Bishop George Witham (1655–1725, Vicar Apostolic of the Northern District), that included Monsignor Thomas Edward Witham of Lartington Hall (1800–1897), and that maintained a secret chapel at Cliffe through the penal years from which its priests emerged to take up their posts in the newly established hierarchy of 1850. The coupling of the Witham name with Herbert, and the dating of the memorial — 20 April 1907, little more than five years after the new Abbey Church at Ampleforth received its first stones — places the donation firmly within the milieu of late-Victorian and Edwardian English Catholic learning, in which mathematical and scientific texts in Latin were still read, and in which the great books of the European learned tradition were thought to belong naturally in the library of a Benedictine monastery. The book subsequently left the Ampleforth library and passed into the modern trade. References: Printing and the Mind of Man 232 (‘a book that begins a new epoch in mathematics’) — Dibner, Heralds of Science 114 — Evans, Epochal Achievements 11 — Grolier/Horblit 37a — Landmark Writings in Western Mathematics 1640–1940 (Grattan-Guinness, ed., 2005), no. 22 — Goldstein, Schappacher, and Schwermer, eds., The Shaping of Arithmetic after C. F. Gauss’s Disquisitiones Arithmeticae (Springer, 2007), especially Goldstein & Schappacher, ‘A book in search of a discipline (1801–1860)’, pp. 3–66. 8vo (201 × 120 mm), pp. [i-vi] vii-xviii, [1] 2-668, [6, tables], [4, errata], with cancel leaves B7, G4, K3, Ff7, and Tt6 all present (Uu3 with small closed tear to head of blank margin, very faint damp-staining to bottom corner of page edges in the second half of the book). Contemporary marbled boards, spine with black lettering piece, all edges stained green (joints and spine ends rubbed, corners bumped). An excellent copy with no restoration.
Item #6460
Price: $65,000.00
