Theoria residuorum biquadraticorum. Commentatio prima [- secunda].

Göttingen: Dieterich, 1828 [- 1832].

First edition, very rare separately-paginated offprints, and an exceptional copy in original state, of these two important papers, in which Gauss coined the term ‘complex number’ and introduced the complex plane now often referred to as the ‘Gaussian plane’. Gauss offprints are especially difficult to find on the market in contemporary bindings, most having been removed from sammelbands and rebound later. “The foundations of the theory of algebraic integers were laid by Gauss in his important work Theoria residuorum biquadraticorum, Commentatio II, which appeared in 1832, in which he considered the numbers a + bi (i = √-1)” (Klein, p. 320). “In the Disquisitiones [Arithmeticae, 1801], Gauss gave the first rigorous proof of ‘the gem of arithmetic’ — the law of quadratic reciprocity. In a series of papers published between 1808 and 1817 Gauss worked on reciprocity laws for congruences of higher degree, and in two papers published in 1828 and 1832 stated (but did not prove) the law of biquadratic [i.e., quartic] reciprocity” (Ewald, p. 306). “In the second part of his study of biquadratic residues (1832), [Gauss] argued that number theory is revealed in its “entire simplicity and natural beauty” (Sect. 30) when the field of arithmetic is extended to the imaginary numbers. He explained that this meant admitting numbers of the form a + bi. “Such numbers,” he said, “will be called complex integers”. More precisely, he went on in the next section, the domain of complex numbers a + bi contains the real numbers, for which b = 0 and the imaginary numbers, for which b is not zero. Then, in Sect. 32, he set out the arithmetical rules for dealing with complex numbers. We read this as a step away from the idea that i is to be understood or explained as some kind of a square root, and towards the idea that it is some kind of formal expression to be understood more algebraically” (Bottazzini & Gray, p. 71). These papers directly influenced 20th and 21st century mathematics: after reading them in 1947, the great French mathematician André Weil was inspired to formulate the ‘Weil Conjectures’, which had a profound effect on the subsequent development of number theory and algebraic geometry (see below). No copies listed on ABPC/RBH in the last 50 years.

Quadratic reciprocity is a result about ‘modular arithmetic.’ Two integers (whole numbers) m and n are said to be equal modulo a positive integer p if mn is an integer multiple of p (so there are only p distinct integers modulo p). “The first really great achievement in the study of modular arithmetic was Carl Friedrich Gauss’s proof in 1796 of his celebrated law of quadratic reciprocity … It states that ‘if p is a prime number, then the number of square roots of an integer n in arithmetic modulo p [i.e., the solutions modulo p of the equation x2 = n (mod p)] depends only on p modulo 4n’” (Taylor). The attempt to generalize quadratic reciprocity for powers higher than the second was one of the main goals that led nineteenth-century mathematicians, including Gauss himself, as well as Richard Dedekind, Gustav Lejeune Dirichlet, Gotthold Eisenstein, David Hilbert, Carl Jacobi and Eduard Kummer, to the study of general algebraic number fields (it was Eisenstein who gave the first proof of biquadratic reciprocity). The ninth in the list of 23 unsolved problems which David Hilbert proposed to the International Congress of Mathematicians in 1900 asked for the “Proof of the most general reciprocity law for an arbitrary number field.”

“In studying the congruence x4 = k (mod p) Gauss concentrated on the central case where the modulus p is a prime of the form 4n + 1, and found himself obliged to examine the complex factors into which prime numbers of such a form can be decomposed. He was thereby led to widen his investigations from the ordinary integers to the complex integers, i.e., to numbers of the form a + bi, where a and b are real integers. Gauss showed that many of the properties of real integers are shared by the complex integers — for example that each complex integer has a unique prime factorization; that the Euclidean algorithm for finding the greatest common divisor of two integers carries over to the complex case; that Fermat’s theorem has a complex analogue; and the like.

“The principal methodological innovation in these investigations was the widening of the number concept to embrace the complex integers. Mathematicians in the seventeenth and eighteenth centuries had been almost as distrustful of imaginary numbers as of infinitesimals, and even Leibniz could write: “The nature of things, the mother of eternal manifolds, or rather the divine spirit, is more jealous of its splendid multiplicity than to allow everything to be herded together under a common genus. Therefore, it found a sublime and wonderful refuge in that miracle of analysis, the monstrum of the ideal world, almost an amphibium between being and non-being, which we call the imaginary roots.”

“Gauss was not the first to conceive of representing complex numbers by points in the plane … the idea had already occurred to John Wallis (in the Treatise of algebra, 1685), to Caspar Wessel in 1792, and to Jean Robert Argand in 1806; and William Rowan Hamilton was later to popularize the conception of a complex number as an ordered couple of real numbers. But Gauss was the first to use complex integers in a systematic way – in his work on biquadratic residues. His remarks on the complex integers touch upon a central theme in nineteenth-century mathematics: the widening of the number concept, and the growth of abstract algebra” (Ewald, pp. 306-7).

“Two of the papers that helped shape the research in number theory during the second half of this century directly referred to Gauss’s work on biquadratic residues: first, there is Weil’s paper from 1949 on equations over finite fields in which he announced the Weil Conjectures and which was inspired directly by reading Gauss: “In 1947, in Chicago, I felt bored and depressed, and, not knowing what to do, I started reading Gauss’s two memoirs on biquadratic residues, which I had never read before. The Gaussian integers occur in the second paper. The first one deals essentially with the number of solutions of ax4 – by4 = 1 in the prime field modulo p, and with the connection between these and certain Gaussian sums; actually the method is exactly the same that is applied in the last section of the Disquisitiones to the Gaussian sums of order 3 and the equation ax3 – by3 = 1. Then I noticed that similar principles can be applied to all equations of the form axm + bym + czr + … = 0, and that this implies the truth of the “Riemann hypothesis” … for all curves axn + byn + czn = 0 over finite fields … This led me in turn to conjectures about varieties over finite fields,” namely the Weil Conjectures.

“The other central theme in number theory during the last few decades came into being in two papers by Birch & Swinnerton-Dyer: while studying the elliptic curves y2 = x3 – Dx they were led to an amazing conjecture that linked local and global data of elliptic curves … in these papers, the quartic reciprocity plays a central role in checking some instances of their conjectures” (Lemmermeyer, pp. xii-xiii).

The Weil conjectures were proved largely by Alexander Grothendieck, using his vast reformulation and generalization of modern algebraic geometry, and by his student Pierre Deligne. The Birch & Swinnerton-Dyer conjecture, chosen as one of the seven Millennium Prize Problems listed by the Clay Mathematics Institute, is still unsolved.

These papers were read on April 5, 1825 and April 15, 1831 and were published in Vols. 6 & 7 of the Commentationes Societatis Regiae Scientiarum Gottingensis, pp. 27-56 & 89-148.

NDB VI, 103; Poggendorff I, 855; NDB VI, 103. Ewald, From Kant to Hilbert, Vol. 1, 1996. Klein, Vorlesungen uber die Entwicklung der Mathematik im 19. Jahrhundert, Bd. I, 1926. Lemmermeyer, Reciprocity Laws. From Euler to Eisenstein, 2000. For an introduction to reciprocity laws intended for non-mathematicians, see R. Taylor, ‘Modular arithmetic: driven by inherent beauty and human curiosity,’ Institute Letter, Summer 2012, Institute for Advanced Study, Princeton (

Two vols., 4to (270 x 220 mm), pp. 32; [ii], 60, both parts uncut. The first part bound in contemporary mute wrappers; the second part in unbound sheets as issued. Preserved in a clamshell box.

Item #6172

Price: $12,500.00