ALEKSANDROFF (or ALEKSANDROV), Pavel and Heinz HOPF. Topologie. *Berlin, J. Springer, 1935*.

**£1400**

First edition, published in the series *Grundlagen der Mathematischen Wissenschaften*, of this ‘extremely influential book and … a sort of bible for the study of algebraic topology’ (Hilton).

‘Alekxandrov formed lifelong friendships during his summers in Göttingen, the most important of which, with Heinz Hopf, began in 1926. Their friendship grew during the academic year 1927-1928, which they spent at Princeton University. When they returned to conduct a topological seminar at Göttingen during the summer of 1928, they were asked by Richard Courant to write a topology book as part of his Yellow Collection for Springer. This request resulted in a seven-year collaboration that culminated in 1935 with the publication *Topologie*, a landmark textbook on topology’ (DSB).

‘Alexandroff being in Woscow and Hopf in Zürich, their collaboration was carried out chiefly by letter. The International Congress in Zürich (5-12 September 1932), of which Hopf was one of the organizers, enabled the two authors to meet; and it was the first international conference on topology (4-10 September 1935), this time organized by Alexandroff in Moscow, that they finished the book.

‘Alexandroff and Hopf is not properly speaking a textbook, but rather a coherent and systematic exposition of a large part of the subject. It gains its generality and coherence largely from the set-theoretic language in which it is uncompromisingly written: a good number of its distinctions and developments are bound up with this choice’ (*Landmark Writings in Western Mathematics*).

DINI, Ulisse. Serie di Fourier e altre rappresentazioni analitiche delle funzioni di una variable reale. *Pisa, Tipografia T. Nistri & C., 1880*.

**£550**

First edition of Dini’s important work on Fourier series.

A Fourier series is an expansion of a periodic function *f *(*x*) in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of the sine and cosine functions. The computation and study of Fourier series is known as harmonic analysis and is extremely useful as a way to break up an *arbitrary* periodic function into a set of simple terms that can be plugged in, solved individually, and then recombined to obtain the solution to the original problem or an approximation to it to whatever accuracy is desired or practical.

In particular, since the superposition principle holds for solutions of a linear homogeneous ordinary differential equation, if such an equation can be solved in the case of a single sinusoid, the solution for an arbitrary function is immediately available by expressing the original function as a Fourier series and then plugging in the solution for each sinusoidal component. In some special cases where the Fourier series can be summed in closed form, this technique can even yield analytic solutions.

‘Fourier frames go back to Dini (1880) and his book on Fourier series. There he gives Fourier expansions in terms of the set {eλ} of harmonics, where each λ is a solution of the equation

x cos πx + a sin πx = 0. (3.1)

Equation (3.1) was chosen because of a problem in mathematical physics from Riemann’s (1826–1866) and later Riemann-Weber’s classical treatise. Dini (1845– 1918) returned to this topic in 1917, just before his death’ (John J. Benedetto, online).

FREGE, Gottlob. Grundgesetze der Arithmetik. Begriffsgeschichtlich abgeleitet. *Jena, Hermann Pohle, 1893-1903.*

**£5850**

First edition, rare, of Frege’s *magnum opus*, the aim of which was to provide rigorous, gapless proofs that arithmetic was entirely reducible to pure logic.

‘In an attempt to realize Leibniz’s ideas for a language of thought and a rational calculus, Frege developed a formal notation for regimenting thought and reasoning. Though this notation was first outlined in his *Begriffsschrift* (1879), the most mature statement of Frege’s system was in his 2-volume *Grundgesetze der Arithmetik *(1893/1903)’ (*Stanford Encyclopedia of Philosophy*).

The first volume of *Grundgesetze *focused on natural-number arithmetic, the second volume is ‘concerned especially with the theory of real numbers, but a good deal of the space is taken up by criticism of current views, and at the end Frege allows that there is more to be done before real numbers can be properly defined. In a *Nachwort, *or appendix, written after most of the work had been printed, he sadly admits the ruin of his work by Bertrand Russell’s discovery of a contradiction implicit in his premisses; but he still maintains that his general conception of the relation between arithmetic and logic is correct, and considers methods by which the damage might be repaired’ (Kneale & Kneale, p. 437).

HILBERT, David. Ueber die Theorie der algebraischen Formen. *Leipzig, Teubner, *[*1890*]*.*

**£1400**

First edition of this famous paper, the rare offprint from the *Mathematische Annalen*, well preserved in the original printed wrappers, the wrappers a little chipped.

‘In his great 1890 paper *Ueber die Theorie der algebraischen* Formen, Hilbert mentions explicitly both Noether and Kronecker and uses Kronecker’s terminology for ideals, although his way of thinking of ideals was closer to Dedekind’s than to Kronecker’s. The theorems proved by Hilbert in this paper (Hilbert Basis Theorem and Hilbert Syzygy Theorem) are cornerstones of commutative algebra and algebraic geometry. He also introduced free resolutions, Hilbert functions, and Hilbert polynomials, which are important tools in modern algebra’ (I.G. Bashmakova and G.S. Smirnova, *The Beginnings & Evolution of Algebra* p. 146).

‘David Hilbert (1862-1943, German) presents the Hilbert Basis Theorem, asserting that every polynomial ideal has a finite basis, and extends the result of Paul Gordon that every binary form has a finite complete system of invariants and covariants to algebraic forms in *n*variables’ (Parkinson).

KLEIN, Felix. Analytische Mechanik. Theil I. Winter 1890-91. Göttingen [and] Theil II. Sommersemester 1891. Ausgearbeitet von (Friedrich Karl Arnold) Schwassmann. *[Göttingen, winter semester 1890 – summer semester 1891].*

**£4850**

Carefully executed lecture course on mechanics held by Felix Klein at Göttingen University in winter 1890 and summer 1891, and specifically on kinetic theory, statics, the Hamilton-Jacobi theory, planetary problems, and precession and nutation.

Klein’s main objectives apart from pure mathematics were the reinforcement of the connections between mathematics, the natural sciences and technology and the restructuring of education in mathematics and science from the earliest grades to the university. Both of these goals shaped much of his time in Göttingen. Gathering together interested professors and industry leaders, he created the Göttingen Association for the Advancement of Applied Physics and Mathematics in 1898.

The lecture was written down by Klein’s then student Friedrich Karl Arnold Schwassmann (1870-1964), who later gained renown in the field of observational astronomy.

LANDAU, Edmund. Theorie der Modulfunktionen. Erster Teil der Vorlesung: Über den Picard’schen Satz … [*N.p., n.d., but ?Berlin, c. 1905*].

**£3000**

An annotated typescript of Landau’s important lecture on Picard’s theorem.

‘In 1879 Émile Picard proved that a meromorphic function which omits more than two values is a constant. This theorem was justly hailed as an outstanding achievement and for many years it stirred the imagination of many prominent mathematicians with results that were ultimately epoch making. Picard’s proof was based on the elliptic modular function, a theory which was in the center of analysis at that time. Toward the end of the nineteenth century, mathematicians tried to replace Picard’s method by more elementary considerations, and in 1896 Émile Borel succeeded in giving such a proof. In the able hands of Edmund Landau this proof gave in 1904 some astonishing results concerning the influence of the first two coefficients of a power series on the properties of a function defined by the series. These results were taken as a vindication of the elementary methods, but the triumph was brief, for in 1905 Constantin Carathéodory proved that the modular function played the role of extremal function in Landau’s problem’ (Einar Hille, *Analytic Function Theory*, p. 219).

Distributed to Landau’s students in typescript form, this early modern method of reproduction very much highlights its innate deficiencies: uncorrected typos, manuscript corrections, insertions, paste-ons and, above all, the great difficulty of expressing or reproducing the virtuosity of mathematical formulae required to correctly express the subject in question by this means, a task therefore very much delegated to, and to be fulfilled or filled in by the diligent student.

THE - APPARENTLY UNRECORDED - ORIGINAL POLISH VERSION IN SEPARATELY PAGINATED OFFPRINT FORM, TOGETHER WITH THE CONTEMPORARY FRENCH TRANSLATION OF SMOLUCHOWSKI’S GROUND-BREAKING PAPER ON FLUCTUATION PHENOMENA

THIS, TOGETHER WITH EINSTEIN’S INDEPENDENT STUDY,

MARKS THE START OF THE STUDY OF STOCHASTIC PROCESSES IN MATHEMATICS

SMOLUCHOWSKI, Marian von. Zarys teoryi kinetycznej ruchów Browna I rotztworów mętnych [Outline of the kinetic theory of Brownian motion of turbid media]. *Cracow, Nakładem Akademii Umiejętności Skład Główny w Księgarni Spółki Wydawniczej Polskiej 1906*.

[*offered with*:] SMOLUCHOWSKI, Marian von. Essai d’une théorie cinétique du movement Brownien et des milieu troubles. *Cracow, Imprimerie de l’Université, 1906*.

**£14,000**

The extremely rare separately paginated offprint of the original, and largely unknown, Polish version, here offered together with the contemporary French translation, of Smoluchowski’s famous publication on Brownian motion.

Published one year after Einstein’s famous paper, but independently conceived, both contain ‘experimental predictions for fluctuation phenomena … [which] provided striking empirical successes for statistical mechanics’ (J. Uffink, *Compendium of the Foundations of Classical Statistical Physics* (2006), p. 59).

WEIERSTRASS, Karl. Theorie der Hyperelliptischen Funktionen. Nach einer Vorlesung von Professor Weierstrass. S.S. 1887. Ausarbeitung des M(athematischen). V(ereins) zu Berlin. *[Berlin, summer semester 1887].*

**£6500**

Carefully written lecture notes on hyperelliptic functions from the private collection of the distinguished German mathematician Paul Gustav Samuel Stäckel of a mathematical lecture held privatim by Karl Weierstrass at Berlin University in the summer of 1887.

Often cited as ‘the father of analysis’ unpublished notes such as the present are of particular interest, as Weierstrass published relatively little, since he was continually revising and expanding his ideas, often incorporating new results spontaneously into his lectures.

The unpublished lecture was handwritten on the request of the Mathematische Verein Berlin during the lecture, possibly by Victor von Dantscher (for his own research within this topic Felix Klein used an earlier version, copied by Adolf Hurwitz). The lecture was intended to appear in print by the Berlin Academy of Sciences as volume eight of the collected papers of Weierstrass, but the volume was never finished. The mathematician Georg Feigl (1890–1945), since 1935 a full professor at the University of Breslau, had been asked to edit the lecture.

Weierstrass introduced rigor into mathematical analysis and considerably enlarged the understanding of functions. ‘Admired by Poincare for his “unity of thought”, Weierstrass was the most important nineteenth-century German mathematician after Gauss and Riemann … Over the years Weierstrass developed a great lecture cycle, “Introduction to the Theory of Analytic Functions”; “Theory of Elliptic Functions” [etc.]. Within this cycle Weierstrass erected the entire structure of his mathematics, using as building blocks only that which he himself had proven’ (DSB XIV, 219 ff.).