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In 1903, Epstein published his proof of meromorphic continuation and a functional equation for Dirichlet series associated with quadratic forms, now called Epstein zeta-functions. However, already in 1889 (or even earlier) Hurwitz was aware of these results as his mathematical diaries and some unpublished notes (in an almost final form) found in his estate at the ETH Zurich show. In this article we present and analyze Hurwitz's notes and compare his reasoning with Epstein's paper in detail.

The date of publication of Helmholtz's first paper on the foundations of geometry is discussed.

That Euler was quite aware of the subtleties of assigning a sum to a divergent series is amply demonstrated in his paper De seriebus divergentibus which appeared in Novi commentarii academiae scientiarum Petropolitanae 5 (1754/55), 205–237 (= Opera Omnia (1) 14, 585–617) in the year 1760. The first half of this paper contains a detailed exposition of Euler's views which should be more readily accessible to the mathematical community.The authors present here a translation from Latin of the summary and first twelve sections of Euler's paper with some explanatory comments. The remainder of the paper, treating Wallis' hypergeometric series and other technical matter, is described briefly. Appended is a short bibliography of works concerning Euler which are available to the English-speaking reader.

One of the prettiest results in the global theory of curves is a theorem of Jacobi (1842): The spherical image of the normal directions along a closed differentiable curve in space divides the unit sphere into regions of equal area. The statement of this theorem is an afterthought to a paper in which Jacobi responds to the published correction by Thomas Clausen (1842) of an earlier paper, Jacobi (1836). In this note the context for this theorem and its proof are presented as well as a discussion of the ‘error’ corrected by Clausen.

Only a few short papers on probability and error theory by Peter Gustav Lejeune Dirichlet are printed in his Werke . However, during his Berlin period, Dirichlet quite frequently gave courses on probability theory or the method of least squares. Unpublished lecture notes reveal that he presented original methods, especially when deriving probabilistic limit theorems; e.g., the use of his discontinuity-factor. The following article discusses some central ideas in Dirichlet's printed papers and unpublished lectures on probability and error theory. These include his deduction of the approximately normal distribution of medians connected with a criticism of least squares as well as his improvement of Laplace's method of approximations relating not only to Stirling's formula but also to the treatment of the central limit theorem. Moreover, the study attempts to place the methods Dirichlet used in probability and error calculus within the broader context of his work in analysis. Nur einige kurze Artikel von Peter Gustav Lejeune Dirichlet zur Wahrscheinlichkeitsund Fehlerrechnung sind in seinen Werken gedruckt. Dirichlet hat aber wahrend seiner Berliner Zeit recht haufig Vorlesungen fiber Wahrscheinlichkeitsrechnung oder Methode der kleinsten Quadrate gehalten. Aus unveroffentlichten Vorlesungsmitschriften geht hervor, dab er gerade bei der Herleitung von Grenzwertsatzen eigene and neue Methoden, z.B. die Verwendung seines Diskontinuitatsfaktors vorgestellt hat. In folgender Arbeit werden Schwerpunkte der gedruckten Arbeiten and der Vorlesungen von Dirichlet fiber Wahrscheinlichkeitsrechnung beschrieben: Grenzwertsatz fur Mediane and Kritik der Methode der kleinsten Quadrate, Fortentwicklung der Laplaceschen Methode fur Approximationen im Hinblick auf die Stirlingsche Formel and die Behandlung des zentralen Grenzwertsatzes. Auberdem wird versucht, die von Dirichlet in Wahrscheinlichkeits- and Fehlerrechnung verwendeten Methoden in den Zusammenhang seiner analytischen Arbeiten zu stellen. Les oeuvres publiees de Peter Gustav Lejeune Dirichlet ne contiennent que quelques rares et courts articles sur le calcul des probabilitds et les erreurs. Pourtant, alors qu'il etait a Berlin, cc mathdmaticien avait assez frequement donne des cours sur les probabilitds ou la methode des moindres carrels: des manuscrits inedits de ses cours en temoignent et permettent de voir qu'a l'occasion de theoremes sur les limites, il introduisit des methodes nouvelles et originales, telle, par exemple, celle qui repose sur l'utilisation du facteur de discontinuity qui porte maintenant son nom. Dans le present article, nous nous proposons de presenter certains aspects fondamentaux des travaux de Dirichlet publies ou non et relatifs au calcul des probabilites, notamment: -la question de la determination de la distribution approximative des medianes, associee a une critique de la methode des moindres carrels; -l'amelioration de la methode d'approximation de Laplace relativement a la formule de Stirling et au traitement du theoreme limite central. Nous nous efforcerons aussi de resituer les methodes probabilistes de Dirichlet dans le contexte de ses travaux analytiques.

This paper gives a detailed analysis of the scientific interaction between Cantor and Dedekind, which was a very important aspect in the history of set theory during the 19th century. A factor that hindered their relationship turns out to be the tension which arose in 1874, due to Cantor's publication of a paper based in part on letters from his colleague. In addition, we review their two most important meetings (1872, 1882) in order to establish the possible exchange of ideas connected with set theory. The one-week meeting in Harzburg (September 1882) was particularly rich in consequences, among other things Dedekind's proof of the Cantor-Bernstein equivalence theorem. But the analysis of this episode will corroborate the lack of collaboration between both mathematicians.

The Italian mathematician Antonio Bordoni is mainly known for his adherence to the Lagrangian approach to the foundations of calculus and for his role in creating an important school of mathematics. In this paper, I consider his less known work on the application of probability to design exams and analyze their outcomes. Within this framework, he obtained in 1837, as Mondesir and Poisson, the result that would lead Catalan to formulate his “new principle” of probability ( Jongmans and Seneta, 1994 ). Moreover, in 1843, Bordoni also gave an early complete proof of the finite rule of succession.

Leibniz published his Euclidean construction of a catenary in Acta Eruditorum of June 1691, but he was silent about the methods used to discover it. He explained how he used his differential calculus only in a private letter to Rudolph Christian von Bodenhausen and specified a number that was key to his construction, 2.7182818, with no clue about how he calculated it. Apparently, the calculations were never divulged to anyone but were discovered later among his personal papers. They may be the earliest record of an accurate approximation of the number we label e and a demonstration of its role as the base of the natural logarithm and exponential function. This, at that time, was a remarkably precise estimate for e, accomplished more than 22 years before Roger Cotes published e to 12 significant digits, and some 57 years before Euler's treatment of the logarithm in his Introductio in Analysin Infinitorum. The Leibniz construction reveals a hyperbolic cosine built on an exponential curve based on his estimated value, which implies that he understood the number as the base of his logarithmic curve. The sheets of arithmetic used by Leibniz preserved at the Gottfried Wilhelm Leibniz Bibliothek (GWLB) in Hannover, confirm this. Those sheets show how Leibniz calculated e and applied it to his catenary construction. The data actually yield e to 12 significant figures: 2.71828182845, missed by Leibniz because of a misplaced decimal point. We summarize the construction and examine the worksheets. The unpublished methods seem entirely modern to us and could serve as enrichening examples in modern calculus texts.

The paper describes Sylow's discovery of the theorems named after him. He was led to this discovery by his study of Galois' work, in particular of Galois' criterion for the solvability of equations of prime degree. It is explained how Sylow used methods from Galois theory in his proofs. The paper also discusses relevant correspondence between Sylow, Jordan, and Petersen.ZusammenfassungDiese Arbeit behandelt Sylows Entdeckung der Sätze, die nach ihm benannt sind. Er gelangte zu ihnen durch seine Beschäftigung mit Galois' Arbeiten über die Auflösbarkeit von Gleichungen vom Primzahlgrad. Es wird beschrieben, wie Sylow im Beweis seiner Sätze Methoden aus der Galois-Theorie benutzt. Es wird weiterhin auf die diesbezügliche Korrespondenz zwischen Sylow, Jordan und Petersen eingegangen.

This paper is concerned with the question, “Is what a stored-program digital computer does thinking -in the full human sense of the term?” Several current controversies are examined, including the meaning and usefulness of the Turing test to determine “intelligence.” The Lucas controversy of the early 1960s is taken up, dealing with the philosophical issues related to the man-versus-machine debate, and Dreyfus' ideas against Machine Intelligence are explored. Searle's ideas in opposition to the validity of the Turing test are described, as are various interpretations of the Chinese room thought-experiment and its relation to real “thought”. Weizenbaum's opposition to the “information-processing model of man” is also developed. The paper concludes with a comparison of the 19th-century debates over Darwinian Evolution and those in this century over Artificial Intelligence.

The collection of Riemann's mathematical papers preserved in Gottingen University Library since 1895 includes none of Riemann's scientific correspondence nor any of his more personal papers. The present report gives an account of the documents (correspondence, lecture notes, etc.) discovered outside Gottingen in the course of a larger research project on Riemann, and briefly describes the history of the Riemann Nachlass. At the same time, readers are kindly requested to inform the author of the whereabouts of any further material relating to Riemann, so that it can be included in the collection of texts and sources currently in preparation.

The paper aims to give an insight into the meaning of Kronecker's program of Arithmetisierung of the whole mathematics through a reappraisal of the realization of a small part of it in the reinterpretation of numerical notions such as that of integer, rational, and algebraic number in terms of the fundamental notion of natural number. It tries to convey the flavor of Kronecker's strong and deliberate opposition to both the abstract and the set-theoretic trend which were blossoming during the last three decades of the 19th century, showing at the same time how an attentive reading of Kronecker's papers on the foundations of mathematics can shed some light on how one of his most celebrated aphorisms should properly be understood.ResumenL'articolo intende dare un'idea del significato del programma di Kronecker (di ciò che egli chiamò la Arithmetisirung dell'intera matematica) esponendo e commentando la realizzazione di una porzione specifica di esso: la reinterpretazione nella “aritmetica generale” della nozione di numero intero, di numero razionale e di numero algebrico. Esso mette in luce l'opposizione decisa e consapevole di Kronecker sia alla tendenza astratta sia a quella insiemistica che cominciavano a entrare nella matematica durante gli ultimi trent'anni del secolo XIX, mostrando al tempo stesso come un'attenta lettura dei lavori di Kronecker sui fondamenti della matematica possa far meglio capire il senso in cui egli intendesse uno dei suoi più celebri aforismi.

In 1912 the Finnish mathematical astronomer Karl Sundman published a remarkable solution to the three-body problem, of a type that mathematicians such as Poincaré had believed impossible to achieve. Although lauded at the time, the result dimmed from view as the 20th century progressed and its significance was often overlooked. This article traces Sundman’s career and the path to his achievement, bringing to light the involvement of Ernst Lindelöf and Gösta Mittag-Leffler in Sundman’s research and professional development, and including an examination of the reception over time of Sundman’s result. A broader perspective on Sundman’s research is provided by short discussions of two of Sundman’s later papers: his contribution to Klein’s Encyklopädie and his design for a calculating machine for astronomy.

The Bohr atom was a solar system in miniature. Despite many deep foundational questions related to the origin of quantized motion, rapid progress was made in its mathematical development and its apparently successful application to spectral line series. In United States, where celestial mechanics flourished throughout the 19th and well into the 20th century, mathematicians and physicists were well prepared for just this sort of problem and made it their own far faster than many areas of the new physics. This paper examines the link between classical problems of perturbation theory, three-body and N-body orbital trajectories, the Hamilton–Jacobi equation, and the old quantum theory. I discuss why it was comparatively easy for American applied mathematicians, astronomers, and mathematical physicists to make significant contributions quickly to quantum theory and why further progress toward quantum mechanics by the same cohort was, in contrast, so slow.

Based on a recently discovered letter of John von Neumann to his friend Howard Percy Robertson, the present paper discusses in detail the priority of the discovery and publication of the ergodic theorems by George David Birkhoff and von Neumann in 1931/1932. © 2002 Elsevier Science (USA). A la suite de la decouverte recente d'une lettre de John von Neumann a son ami Howard Percy Robertson, cet article examine en detail les priorites de George David Birkhoff et von Neumann dans la decouverte et la publication des theoremes ergodiques, dans les annees 1931/1932. © 2002 Elsevier Science (USA). MSC subject classifications: 01A60, 01A70.

The method of steepest descent, also known as the saddle-point method, is a natural development of Laplace's method applied to the asymptotic estimate of integrals of analytic functions. Mathematicians have often attributed the method of steepest descent to the physicist Peter Debye, who in 1909 worked it out in an asymptotic study of Bessel functions. Debye himself remarked that he had borrowed the idea of the method from an 1863 paper of Bernhard Riemann. The present article offers a detailed historical analysis of the creation of the method of steepest descent. We show that the method dates back to Cauchy and that, 25 years before Debye, the Russian mathematician Pavel Alexeevich Nekrasov had already used this technique and extended it to more general cases. Copyright 1997 Academic Press. La methode de la descente la plus rapide, que l'on appelle actuellement methode du col, represente un developpement naturel de la methode de Laplace pour l'estimation asymptotique des integrales des fonctions analytiques. La methode du col est generalement liee au nom du physicien Peter Debye, qui a utilise en 1909 les idees que Riemann a presentees dans son article de 1863 afin de donner une analyse asymptotique des fonctions de Bessel. Le present travail donne une etude bien detaillee de la creation de la methode du col. Tout d'abord nous montrons que la methode du col remonte aux travaux de Cauchy et ensuite que le mathematicien russe Pavel Alexeevich Nekrasov a applique cette methode un quart de siecle avant Debye et l'a etendue aux cas generaux. Copyright 1997 Academic Press. [FORMULA]

We address the problem of priority for the Strong Law of Large Numbers (SLLN) with a view to portraying Cantelli's role more accurately. The controversy over priority in 1928 was initiated by Slutsky and came to a head at the Bologna Congress of Mathematicians. We portray it through the media of letters (the centerpiece of our paper) written by Slutsky during the Congress, and notes in the Congress Proceedings. The technical focus of Cantelli's proof of the SLLN is Boole's Inequality. The publicity it received at the Congress very likely led to considerations of optimality of such bounds and to the Bonferroni Inequalities.

In a footnote to a short early paper (1844) G. Eisenstein gave an “analytic solution” of the general quintic equation. We discuss this remark in relation to the well-known work of Hermite (1858) and Kronecker (1861). We also discuss Eisenstein's solution from the point of view of Riemannian function theory.

This paper is concerned with (a) a brief outline of Chebyshev's life; (b) certain background material in connection with Chebyshev's work on approximation and integration in finite terms; (c) the question of whether he already was in Paris in 1842, 10 years prior to his established and presumed first trip to Western Europe; d) his contacts with Catalan, Liouville, Hermite, Lucas, d'Ocagne, Laussedat, and Dwelshauvers-Dery; (e) his contacts with Dirichlet, Borchardt, Kronecker, and Weierstrass; and (f) Chebyshev's last two trips to the West. It is argued that the great Russian scientist did not work in isolation at St. Petersburg. Instead, he was in personal contact, at least until 1884, with many of the greatest European scientists of the time.

In this paper we trace the early development of a method for finding the approximate solutions —called the WKB solutions—for a class of ordinary differential equations of second order. We also analyze the attempts made by the various contributors to this method to substantiate their results. These approximating solutions were subsequently shown to be asymptotic in the sense of Poincare. Also presented and examined are the several methods used to deal with the “connection problem” which arises in the use of the WKB solutions.

Leibniz considered the “ars combinatoria” as a science of fundamental significance, much more extensive than the combinatorics of today. His only publications in the field were his youthful Dissertatio de Arte Combinatoria of 1666 and a short article on probability, but he left an extensive (hitherto unpublished and unstudied) Nachlass dealing with five related topics: the basic operations of combinatorics, symmetric functions in connection with theory of equations, partitions (additive theory of numbers), determinants, and theory of probability and related fields. This paper concentrates on the first and third topics as they appear in published sources and the Nachlass. It shows that Leibniz was in possession of many results not published by other mathematicians until many decades later. These include a recursion formula for partitions of n into k parts (first published by Euler in 1751), the Stirling numbers of the second kind (first published in 1730), and several special cases of the general formula for partitions that was published only in 1840 by Stern.

One of the significant contributions of Chinese mathematicians is the method of solving numerical equations of higher degree. A number of scholarly works have established similarities between ancient and medieval Chinese root-extraction procedures and Horner's method of solving a numerical equation of any degree. The conceptual development of the Chinese methods, which began with the procedures of extracting square and cube roots during the Han dynasty, culminated in the solution of higher numerical equations in the 13th and the beginning of the 14th centuries. This paper attempts to show the intimate role played by the triangular array of numbers (known in the West as the Pascal triangle) in the process of the development of the Chinese methods, especially when the original geometrical concept was being replaced by the more general algebraic method.

The paper describes an approximation formula for sine (x + h) that differs from the first four terms of the Taylor expansion only by having 4 in place of 6 in the denominator of the fourth term. It appears in Sanskrit stanzas quoted in a work of about the fifteenth century and given here with translation and explanation.