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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 However, already in 1889 (or even earlier) Hurwitz was aware of these results as his mathematical diaries and some unpublished notes (in an almost final form) found in his estate at the ETH Zurich show.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.

As early as the 16th century, Simon Jacob, a German reckoning master, noticed that the worst case in computing the greatest common divisor of two numbers by the Euclidean algorithm occurs if these numbers are equimultiples of two consecutive members of the Fibonacci sequence.

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.

Konig's paradox, that he presented for the first time in 1905, preserved the same structure in all his papers: there was a number that at the same time was and was not finitely definable. Still, he changed the way for forming it, and both its consequences and its solutions changed as well. In the present paper we are going to follow the story of Konig's paradox, that is an intriguing mix of labelling, solving, criticising an “object” from different viewpoints and for different aims.

This paper traces the influence of the Boolean school, and more specifically of Peirce and his students, on the development of modern logic. In the 1890s it was Schroder's Algebra derLogik that represented the state of the art. This work mentions Frege, but the quantifier notation it adopts (a variant of the modern notation) is credited to Peirce and his students O. H. Mitchell and Christine Ladd-Franklin. This notation was widely adopted; both Zermelo and Lowenheim wrote famous papers in Peirce-Schroder notation. Even Whitehead (in 1908, in his Universal Algebra) fails to mention Frege, but cites the “suggestive papers” by Mitchell and Ladd-Franklin. (Russell credits Frege, with many things, but nowhere credits him with the quantifer; if the quantifiers in Principia were devised by Whitehead, they probably come from Peirce). The aim of this paper is not to detract from our appreciation of Frege's great work, but to emphasize that its influence came largely after 1900 (after Russell pointed out its significance). Although Frege discovered the quantifier in 1879 and Peirce's student Mitchell independently discovered it only in 1883, it was Mitchell's discovery (as modified and disseminated by Peirce) that made the quantifier part of logic. And neither Lowenheim's theorem nor Zermelo set-theory depended on Frege's work at all, but only on the work of the Boole-Peirce school.

This paper describes the work of mathematicians during the seventeenth and eighteenth centuries on the pressure-height problem of determining the relationship between atmospheric pressure and altitude. Omitting minor contributions by many other mathematicians, the paper describes the work of Pascal (atmospheric pressure decreases with altitude), E. Mariotte (height increases in geometric progression as pressure decreases in arithmetic progression), E. Halley (the use of logarithms), John Wallace, G.W. Leibniz, Jacques Cassini, Daniel Bernoulli, Pierre Bouguer, J.H. Lambert, G. Fontana, J.A. DeLuc, S. Horsley, J. Playfair, and P.S. Laplace whose formula summarized previous results.

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.

This paper studies the background to and content of George Peacock's work on symbolical algebra. It argues that, in response to the problem of the negative numbers, Peacock, an inveterate reformer, elaborated a system of algebra which admitted essentially “arbitrary” symbols, signs, and laws. Although he recognized that the symbolical algebraist was free to assign somewhat arbitrarily the laws of symbolical algebra, Peacock himself did not exercise the freedom of algebra which he proclaimed. The paper ends with a discussion of Sir William Rowan Hamilton's criticism of symbolical algebra.

In his pioneering papers of 1845 and 1846, Arthur Cayley (1821–1895) introduced several approaches to invariant theory, the most prominent being the method of hyperdeterminant derivation. This article discusses these papers in the light of Cayley's unpublished correspondence with George Boole, who exercised considerable influence on Cayley at this formative stage of invariant theory. In the 1850s Cayley put forward a new synthesis for invariant theory framed in terms of partial differential equations. In this period he published his memoirs on quantics, the first seven of which appeared in quick succession. This article examines the background of these memoirs and makes use of unpublished correspondence with Cayley's lifelong friend, J. J. Sylvester.

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.

Felix Klein's Erlanger Programm (1872) has been extensively studied by historians. If the early geometrical works in Klein's career are now well-known, his links to the theory of algebraic equations before 1872 remain only evoked in the historiography. The aim of this paper is precisely to study this algebraic background, centered around particular equations arising from geometry, and participating on the elaboration of the Erlanger Programm. Another result of the investigation is to complete the historiography of algebraic equations, in which those “geometrical equations” do not appear.

This text studies the Sanskrit and Malayalam versions of Citrabhānu’s Twenty-One problems – a discussion of quadratic and cubic problems from 16th century Kerala. It reviews the differences in the approaches of the two texts, highlighting the division between the Sanskrit integer arithmetic techniques and the Malayali fixed point reiterations. The paper concludes with some speculations on the possible transmission of algebraic knowledge between Kerala and the west., Historia Mathematica, 42 (3), ISSN:0315-0860, ISSN:1090-249X

This paper discovers the roots of symbolical algebra in a three-quarters-of-a-century British discussion of sound reasoning, general terms, and signs—a discussion in which mathematical and philosophical elements were freely and perhaps inseparably intermingled. It establishes, in particular, a link between early-19th-century symbolical algebra and nominalism. Opening with a review of recent attacks on the rigid internal/external dichotomy, which underlies much of the modern history of science, the paper also serves as an example of the fertility of suspending the dichotomy in pursuit of earlier mathematical subcultures.

As documented in a recent paper by Helene Gispert (Archive for History of Exact Sciences 28 (1983), 37-106), Gaston Darboux's calls in the mid-1870s for a more rigorous approach to analysis were largely ignored by the French mathematical community for quite some time. Gabriel Koenigs's 1884-1885 papers on the iteration of complex functions mark perhaps the earliest instance where Darboux directly influenced a French mathematician to develop a rigorous approach to his subject.

Prior to the development of real analysis in the 19th century, J.L. Lagrange had provided an algebraic basis for the calculus. The most detailed statement of this program is the second edition of his Lecons sur le calcul des fonctions 1806. The paper discusses Lagrange's conception of algebraic analysis and critically examines his demonstration of Taylor's theorem, the foundation of his algebraic program. Lagrange's striking algebraic style is further explored in two specific subjects of the Lecons : the theory of singular solutions to ordinary differential equations and the calculus of variations. A central theme of the paper concerns Lagrange's treatment of exceptional values in his demonstration of analytical theorems. The paper concludes that Lagrange's algebraic program was a natural one, but that the conception of a functional relation given by a single analytical expression was too restrictive to provide an adequate basis for analysis.

This paper describes an example of mathematical growth from scholarly curiousity to application. The contributions of Liouville, Riemann, and Laurent to the field of fractional operators are discussed. Motivation for the writing of this paper is based on the statement by Harold T. Davis [1927]: “The great elegance that can be secured by the proper use of fractional operators and the power they have in simplifying the solution of complicated functional equations should more than justify a more general recognition and use.”

Riemann's Collected Works contains a seldom mentioned paper on heat conduction, written in Latin and unsuccessfully submitted for a prize to the Academie des Sciences in Paris in 1861. This paper has been presented by many, including Riemann's editors, as a contribution to the development of his geometrical ideas, first outlined in his famous Habilitationsvortrag of 1854, Uber die Hypothesen welche der Geometrie zu Grunde liegen. Through a discussion of the paper, we offer a new perspective on its importance: rather than a development of the mathematics underlying new conceptions of space, it should be seen as a contribution to the development of what later became known as tensor analysis. This interpretation allows a fresh perspective to be brought to the history of a particular field of mathematics. Indeed, both Riemann's geometry and tensor analysis (as developed later) combine in general relativity. However, until then they were developed independently of one another despite both being present in different aspects of Riemann's work. Since the argument proceeds from a detailed consideration of the paper by Riemann, we give the first translation into English of that paper in the Appendix.

This paper deals with a sub-class of recreational problems which are solved by a simple memorized rule resulting from an elementary arithmetical or algebraic solution, called proto-algebraic rules. Their recreational aspect is derived from a surprise or trick solution which is not immediately obvious to the subjects involved. Around 1560 many such problems wane from arithmetic and algebra textbooks to reappear in the eighteenth century. Several hypotheses are investigated why popular Renaissance recreational problems lost their appeal. We arrive at the conclusion that the emergence of algebra as a general problem solving method changed the scope of what is considered recreational in mathematics.

Detlefsen (1986) reads Hilbert's program as a sophisticated defense of instrumentalism, but Feferman (1998) has it that Hilbert's program leaves significant ontological questions unanswered. One such question is of the reference of individual number terms. Hilbert's use of admittedly “meaningless” signs for numbers and formulae appears to impair his ability to establish the reference of mathematical terms and the content of mathematical propositions (Weyl, 2009/1949 and Kitcher, 1976). The paper traces the history and context of Hilbert's reasoning about signs, which illuminates Hilbert's account of mathematical objectivity, axiomatics, idealization, and consistency. Resumi Detlefsen (1986) legge il programma di Hilbert come una sofisticata difesa dello strumentalismo, ma Feferman (1998) sostiene che il programma di Hilbert lascia senza risposta alcune significative questioni ontologiche. Una fra queste e il riferimento dei termini individuali numerici. L'impiego da parte di Hilbert di simboli per i numeri e formule per la matematica finistista esplicitamente “privi di senso,” sembra impedire la possibilita di stabilire un riferimento per i termini matematici e un contenuto per le proposizioni (Weyl, 2009/1949 and Kitcher, 1976). Questo articolo ripercorre la storia e il contesto del pensiero di Hilbert concernente i simboli; tale contesto getta luce sulla concezione Hilbertiana dell'oggettivita matematica.

During his whole life, Leibniz attempted to elaborate a new kind of geometry devoted to relations and not to magnitudes, based on space and situation, independent of shapes and quantities, and endowed with a symbolic calculus. Such a “geometric characteristic” shares some elements with the perspective geometry: they both are geometries of situational relations, founded in a transformation preserving some invariants, using infinity, and constituting a general method of knowledge. Hence, the aim of this paper is to determine the nature of the relation between Leibnizʼs new geometry and the works on perspective, namely Desarguesʼ ones.

The algorithms used in Arabic and medieval European mathematics for extracting cube roots are studied with respect to algebraic structure and use of external memory (dust board, table, paper). They can be separated into two distinct groups. One contains methods used in the eastern regions from the 11th century, closely connected to Chinese techniques, and very uniform in structure. The other group, showing much wider variation, contains early Indian methods and techniques developed in central and western parts of the Arabic areas and in Europe. This study supports hypotheses previously formulated by Luckey and Chemla on an early scientific connection between China and Persia.

In this paper Newton’s persistent attempts to construct a unitary view of mathematics are examined. To reconcile the calculus of fluxions with Euclid’s Elements or Apollonius’s Conics appears, with the benefit of hindsight, an enterprise that cannot be accomplished simply by a widening of Greek mathematical thought. It requires a deep modification of the epistemological ground. Although Newton’s attempts remained for the most part in manuscript form, it is hardly doubtful that Newton’s ideas paved the way for the deep modifications that mathematics underwent in the succeeding centuries.

The principal aim of this paper is to shed some light on the algebraic content of the Tratado da arte de arismetica by Bento Fernandes, which was published in Porto in 1555 and is the earliest treatise of a Portuguese author that has come down to us in which algebra is studied. Since it therefore constitutes an important testimony of the state of development of algebra in Portugal in the middle of the 16th century, it deserves special attention. At a time when Pacioli's Summa , the first printed text that includes algebraic methods, was already so diffuse, it is surprising that it turns out not to have been the source of the algebraic material of Bento Fernandes. The comparative study I have carried out between the Tratado da arte de arismetica and a number of abacus books from the 14th and the 15th centuries shows that Bento Fernandes's algebra had its origin in abacus manuscripts antedating the Summa .

Historians have repeatedly asserted that invariant theory was born in two papers of George Boole (1841 and 1842). Although several themes and techniques of 19th-century invariant theory are enunciated in this work, in reacting to it (and thereby founding the British school of invariant theory), Arthur Cayley shifted Boole's research program.

In the introduction of the Arithmetica Diophantus says that in order to solve arithmetical problems one has to “follow the way he (Diophantus) will show.” The present paper has a threefold objective. Firstly, the meaning of this sentence is discussed, the conclusion being that Diophantus had elaborated a program for handling various arithmetical problems. Secondly, it is claimed that what is analyzed in the introduction is definitions of several terms, the exhibition of their symbolism, the way one may operate with them, but, most significantly, the main stages of the program itself. And thirdly, it is argued that Diophantus' intention in the Arithmetica is to show the way the stages of his program should be practically applied in various arithmetical problems.

This paper discusses two concepts of “moment” (infinitesimal) used successively by Newton in his calculus and relates these two concepts to the two concepts of force that Newton presented in Law II and Def. VIII of the Principia, to which the approximations to the action of a centripetal force known as the polygonal and parabolic models are considered to be related. It is shown that in the context of the application of the calculus to mechanics, the transition in the use of these concepts of “moment” took place in 1684, between the writing of De Motu and its first revision.

An important step in 17th-century research on quadratures involved the use of algebraic procedures. Pietro Mengoli (1625–1686), probably the most original student of Bonaventura Cavalieri (1598–1647), was one of several scholars who developed such procedures. Algebra and geometry are closely related in his works, particularly in Geometriae Speciosae Elementa [Bologna, 1659]. Mengoli considered curves determined by equations that are now represented by y = K ⋅ x m ⋅ ( t − x ) n . This paper analyzes the interrelation between algebra and geometry in this work, showing the complementary nature of the two disciplines and how their combination allowed Mengoli to calculate quadratures in a new way.

This paper focuses on an anonymous Catalan manuscript of the early 16th century dealing with algebra and commercial arithmetic. More than half of it consists of a series of notes concerning parts of Luca Pacioli's Summa de Arithmetica, Geometria, Proportioni et Proportionalita , while most of the rest is related to Joan Ventallol's commercial arithmetic ( 1521 ) . The use of a new kind of diagrams to work with equations is especially remarkable. The article throws new light on the cultivation of algebra in the Iberian peninsula before 1552, when the treatise on algebra by Marco Aurel, Libro Primero de Arithmetica Algebratica , was first printed in Spanish.

The second half of the 19th century witnessed a sudden and sustained revival of Italian mathematical research, especially in the period following the political unification of the country. Up to the end of the 19th century and well into the 20th, Italian professors—in a variety of institutional settings and with a variety of research interests—trained a number of young scholars in algebraic areas, in particular. Giuseppe Battaglini (1826–1892), Francesco Gerbaldi (1858–1934), and Luigi Bianchi (1856–1928) defined three key venues for the promotion of algebraic research in Rome, Palermo, and Pisa, respectively. This paper will consider the notion of “research school” as an analytic tool and will explore the extent to which loci of algebraic studies in Italy from the second half of the 19th century through the opening decades of the 20th century can be considered as mathematical research schools.

In this paper we give an overview of the interactions between Italian and American algebraic geometers during the first decades of the 20th century. We focus on three mathematicians—Julian L. Coolidge, Solomon Lefschetz, and Oscar Zariski—whose relations with the Italian school were quite intense. More generally, we discuss the importance of this influence in the development of algebraic geometry in the first half of the 20th century.

This article contains the hitherto unpublished text of Arthur Cayley's inaugural professorial lecture given at Cambridge University on 3 November 1863. Cayley chose a historical treatment to explain the prevalent basic notions of analytical geometry, concentrating his attention in the period (1638–1750). Topics Cayley discussed include the geometric interpretation of complex numbers, the theory of pole and polar, points and lines at infinity, plane curves, the projective definition of distance, and Pascal's and Maclaurin's geometrical theorems. The paper provides a commentary on this lecture with reference to Cayley's work in geometry. The ambience of Cambridge mathematics as it existed after 1863 is briefly discussed.Copyright 1999 Academic Press. Cet article contient le texte jusqu'ici inedit de la lecon inaugurale de Arthur Cayley donnee a l'Universite de Cambridge le 3 novembre 1863. Cayley choisit une approche historique pour expliquer les notions fondamentales de la geometrie analytique, qui existaient alors, en concentrant son attention sur la periode 1638–1750. Les sujets discutes incluent l'interpretation geometrique des nombres complexes, la theorie des poles et des polaires, les points et les lignes a l'infini, les courbes planes, la definition projective de la distance, et les theoremes geometriques de Pascal et de Maclaurin. L'article contient aussi un commentaire reliant cette lecon a l'oeuvre de Cayley en geometrie. L'atmosphere des mathematiques a Cambridge apres 1863 est brievement discutee.Copyright 1999 Academic Press. MSC Classification: 01A55, 01A72, 01A73.

This paper describes the importance of Euclidean geometry for the educational system in medieval Islamic India and surveys the kinds of sources available for study of this branch of Euclidean scholarship. It examines several types of source documents important for the study of Euclidean thought in India and its ties to other branches of the medieval Euclidean tradition. The major types of sources described are: (1) Arabic and Persian translations, (2) Recensions of these translations, (3) Summaries of the Euclidean corpus, (4) Encyclopedic works that include descriptions of Euclidean geometry, and (5) Rhymed prose (manzūmāt).

The Euclidean algorithm for computing the greatest common divisor of two integers is, as D. E. Knuth has remarked, “the oldest nontrivial algorithm that has survived to the present day.” Credit for the first analysis of the running time of the algorithm is traditionally assigned to Gabriel Lamé, for his 1844 paper. This article explores the historical origins of the analysis of the Euclidean algorithm. A weak bound on the running time of this algorithm was given as early as 1811 by Antoine-André-Louis Reynaud. Furthermore, Lamé's basic result was known to Émile Léger in 1837, and a complete, valid proof along different lines was given by Pierre-Joseph-Étienne Finck in 1841.

This paper describes the way in which the concept of quotient group was discovered and developed during the 19th century, and examines possible reasons for this development. The contributions of seven mathematicians in particular are discussed: Galois, Betti, Jordan, Dedekind, Dyck, Frobenius, and Holder. The important link between the development of this concept and the abstraction of group theory is considered.

This paper concerns a critical investigation of Tschirnhaus's role as a mathematician during the early stages of calculus, with emphasis on Tschirnhaus's relationship with Leibniz and on the significance of the Tschirnhaus transformation for the development of mathematics. The concluding section describes five main factors that influnced Tschirnhaus's life and work.

In the second part of his Algebra, Sharaf al-Dīn al- T ūsī (12th-century) correctly determines the number of positive roots of cubic equations in terms of the coefficients. R. Rashed has recently published an edition of the Algebra [al- T ūsī 1985], and he has discussed al- T ūsī's work in connection with 17th century and more recent mathematical methods (see also [Rashed 1974]). In this paper we summarize and analyze the work of al- T ūsī using ancient and medieval mathematical methods. We show that al- T ūsī probably found his results by means of manipulations of squares and rectangles on the basis of Book II of Euclid's Elements. We also discuss al- T ūsī's geometrical proof of an algorithm for the numerical approximation of the smallest positive root of x3 + c = ax2. We argue that al- T ūsī discovered some of the fundamental ideas in his Algebra when he was searching for geometrical proofs of such algorithms.

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.

Western culture is but one of several with an interest in continuous figure-tracing. This paper elaborates evidence of that interest in Oceania with emphasis on the dieas of the Malekula. Included are the figures, with cultural context, and with associated geometric and topological ideas. For many Malekula figures there is a record of actual order and direction of the tracing of each edge. This enables us to analyze their procedures and to show that basic procedures were transformed and combined into larger systematic procedures.