## Ueber die invarianten Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen.

Königsberg: Leupold, 1885.

First edition, very rare, of Hilbert’s inaugural dissertation at the University of Königsberg, ‘On the invariant properties of special binary forms, in particular the spherical functions,’ in which he began his study of invariant theory. This subject was at the forefront of mathematics in the second half of the 19^{th} century, and Hilbert’s work in this area made his name in the mathematical world. Hilbert (1862-1943) “studied at the University of Königsberg from 1880 to 1884 … Königsberg, the university where Immanuel Kant had studied and taught, became a center of mathematical learning through Jacobi’s activity (1827–1842). When Hilbert began his studies there, the algebraist Heinrich Weber, Dedekind’s collaborator on the theory of algebraic functions, was a professor at Königsberg. In 1883 Weber left. His successor was Lindemann, a famous but muddle-headed mathematician who the year before had had the good luck to prove the transcendence of π … Under his influence Hilbert became interested in the theory of invariants, his first area of research” (DSB). Hilbert himself was the first to assess the historical significance of his own work on invariant theory. In a review article read in his name at the International Congress of Mathematicians held in Chicago in 1893, Hilbert mentioned three clearly separated stages, that in his view mathematical theories usually undergo in their development: the naive, the formal and the critical. In the case of invariant theory, Hilbert saw the works of Cayley and Sylvester as representing the naive stage and the work of Gordan and of Clebsch representing the formal stage. In Hilbert’s assessment, his own work was the only representative of the critical stage in the theory of invariants. Ultimately Emmy Noether and others enlarged on Hilbert’s ideas to found modern algebra. OCLC lists four copies in US. No copies in auction records.

Invariant theory dates back at least to Gauss’ study of binary quadratic forms in his *Disquisitiones Arithmeticae* (1801). A ‘binary quadratic form’ is an expression

*ax*^{2} + *bxy* + *cy*^{2}

in the ‘variables’ *x*, *y*, the ‘coefficients’ *a*, *b*, *c* being numbers. If we make a change of variables

*x* = *αX + βY*, *y = γX + δY*,

the original quadratic form becomes

*AX*^{2} + *BXY* + *CY*^{2},

where *A*, *B*, *C* can be expressed in terms of *a*, *b*, *c* and the coefficients of the change of variables *α*, *β*, *γ*, *δ*. One finds that

*B*^{2} – 4*AC* = *D*^{2}(*b*^{2} – 4*ac*),

where *D* = *αδ – βγ* is the ‘determinant’ of the change of variables. In general, an ‘invariant’ of the binary quadratic form is any expression in the coefficients of the form that is altered by a change of variables only by a power of the determinant *D*. This can be generalized to forms involving more than two variables, and to forms of higher degree (i.e., involving higher powers of the variables). There are also the ‘covariants’, which are similarly defined functions involving both the coefficients and the variables of the form.

Even before Gauss, Lagrange in his *Méchanique analitique* (1788) had studied the transformation properties of forms under changes of variables. Lagrange’s work was taken up by Boole in his paper ‘Researches on the theory of analytical transformations’ (1839), which must be considered the foundation work in invariant theory and the birth of the British/Irish school of invariant theory. “Formally, invariant theory began with Cayley and Sylvester in the late 1840s. Cayley used it to bring to light the deeper connections between metric and projective geometry. Although important connections with geometry were maintained throughout the nineteenth and early twentieth centuries, invariant theory soon became an area of investigation independent of its relations to geometry. In fact, it became an important branch of *algebra *in the second half of the 19th century … Many of the major mathematicians of the second half of the nineteenth century worked on the computation of invariants of specific forms. This led to the major problem of invariant theory, namely to determine a complete system of invariants—a basis—for a given form. That is, to find invariants of the form—it was conjectured that finitely many would do—such that every other invariant could be expressed as a combination of these. Cayley showed in 1856 that the finitely many invariants he had found earlier for binary quartic forms (i.e., forms of degree four in two variables) are a complete system. About ten years later Gordan proved that every binary form, of any degree, has a finite basis. Gordan’s proof of this important result was computational—he exhibited a complete system of invariants” (Kleiner, pp. 92-93).

This was the situation in invariant theory when Hilbert arrived in Königsberg. “Having completed the eight university semesters required for a doctor’s degree, Hilbert began to consider possible subjects for his dissertation. In this work he would be expected to make some sort of original contribution to mathematics. At first he thought he might like to investigate a generalization of continued fractions; and he went to Lindemann, who was his ‘Doctor-Father’, with this proposal. Lindemann informed him that unfortunately such a generalization had already been given by Jacobi. Why not, Lindemann suggested, take instead a problem in the theory of invariants …

“The problem which Lindemann suggested to Hilbert for his doctoral dissertation was the question of the invariant properties for certain algebraic forms. This was an appropriately difficult problem for a doctoral candidate, but not so difficult that he could not be expected to solve it. Hilbert showed his originality by following a different path from the one generally believed to lead to a solution. It was a very nice piece of work, and Lindemann was satisfied.

“A copy of the dissertation was dispatched to Minkowski, who after his father’s recent death had gone to Wiesbaden with his mother. ‘I studied your work with great interest,’ Minkowski wrote to Hilbert, ‘and rejoiced over all the processes which the poor invariants had to pass through before they managed to disappear. I would not have supposed that such a good mathematical theorem could have been obtained in Königsberg!’

“On December 11, 1884, Hilbert passed the oral examination. The next and final ordeal, on February 7, 1885, was the public promotion exercise in the Aula, the great hall of the University. At this time he had to defend two theses of his own choice against two fellow mathematics students officially appointed to be his ‘opponents.’ The contest was generally no more than a mock battle, its main function being to establish that the candidate could perceive and frame important questions” (Reid, pp. 15-16).

The subject of Hilbert’s dissertation was special binary forms determined by algebraic differential equations, and their application to spherical functions, a special family of polynomials in one variable that arise from solutions of Laplace’s equation in three dimensions. As Hilbert explains in the introduction to his thesis: “The present investigation deals with the invariant properties and criteria of special binary forms determined by given algebraic differential equations. The first part deals with the general methods that can be applied to the direct derivation of invariant criteria from the given differential equation. The results obtained help both in constructing a system of invariant relations whose validity determines whether a form with general coefficients can be transformed into the special form, and they also serve in actually setting up the transformation. The second part is devoted to working out an especially interesting case of the general deductions of the first, namely for the differential equation of a spherical function, an investigation that I have undertaken at the urging of Professor Lindemann. As a result one obtains for a general spherical function of any degree and any order a sequence of invariant and simultaneous invariant relations using which one can answer the two fundamental questions, of the conditions for the possibility of transforming a form with general coefficients into a given spherical function, and the means of carrying out this transformation. The conclusion returns to more general points of view: on the one hand it describes the place of the special case of spherical functions within a more comprehensive class of problems, and, on the other hand, it contemplates a successful application of the principles of the first part to those more general questions” (translation from Ackermann & Hermann).

Hilbert continued his work on invariant theory, making his great breakthrough three years later. “In 1888 Hilbert astonished the mathematical world by announcing a new, conceptual approach to the problem of invariants. The idea was to consider, instead of invariants, expressions in a finite number of variables, in short, the polynomial ring in those variables. Hilbert then proved what came to be known as the *Basis Theorem*, namely that every ideal in the ring of polynomials in finitely many variables with coefficients in a field has a finite basis. A corollary was that every form, of any degree, in any number of variables, has a finite complete system of invariants. Gordan’s reaction to Hilbert’s proof, which did not explicitly exhibit the complete system of invariants, was that ‘this is not mathematics; it is theology.’ When Hilbert later gave a constructive proof of this result (which he, however, did not consider significant), it elicited the following response from Gordan: ‘I have convinced myself that theology also has its advantages’” (Kleiner, p. 93). “Hilbert’s basis theorem has passed into the very foundations of algebra. It marked an advance of the first order, not only in algebra but also in modern arithmetic and algebraic geometry” (Bell, p. 429).

“In this new light, the old questions and research programs, which had been directed toward calculating complete systems of in- and covariants, were gradually supplanted by new questions such as the fourteenth problem Hilbert posed in his renowned lecture before the International Congress of Mathematicians in 1900. There Hilbert asked: ‘whether it is always possible to find a finite system of relatively integral functions of *X*_{1}, …, *X*_{m} by which every other relatively integral function of *X*_{1}, …, *X*_{m} may be expressed rationally and integrally.’ In the twentieth century, researches directed at answering this and other, related questions came to characterize what we now know as modern algebra” (Parshall).

Ackermann & Hermann, *Hilbert’s Invariant Theory Papers. Lie Groups*: *History, Frontiers and Applications*, Vol. VIII, 1978. Bell, *The Development of Mathematics*, 1945. Kleiner, *A History of Abstract Algebra*, 2007. Parshall, ‘Toward a history of nineteenth-century invariant theory,’ pp. 157-206 in: *The History of Modern Mathematics,**Vol*. 1: *Ideas and their Reception* (Rowe & McCleary, eds.), 1989. Reid, *Hilbert-Courant*, 1986.

4to, pp. [iv], 30, [2]. Original paper spine strip (worn). Previous owner’s name stamp on front wrapper.

4to, pp. [iv], 30, [2]. Original paper spine strip (worn). Previous owner’s name stamp on front wrapper.

Item #4944

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Price:
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