Modular arithmetic before C.F. Gauss: Systematizations and discussions on remainder problems in 18th-century Germany

Remainder problems have a long tradition and were widely disseminated in books on calculation, algebra, and recreational mathematics from the 13th century until the 18th century. Many singular solution methods for particular cases were known, but Bachet de Méziriac was the first to see how these methods connected with the Euclidean algorithm and with Diophantine analysis (1624). His general solution method contributed to the theory of equations in France, but went largely unnoticed elsewhere. Later Euler independently rediscovered similar methods, while von Clausberg generalized and systematized methods that used the greatest common divisor procedure. These were followed by Euler's and Lagrange's continued fraction solution methods and Hindenburg's combinatorial solution. Shortly afterwards, Gauss, in the Disquisitiones Arithmeticae, proposed a new formalism based on his method of congruences and created the modular arithmetic framework in which these problems are posed today.     

Nineteenth century roots of quasideterminants

In this expository paper I provide a complete record of the nineteenth century publications that bear on the development of quasideterminants in the twentieth century. Two important recursive feasible algorithms, Sylvester’s from 1851 and Dodgson’s from 1866 are discussed, and the antecedents of both are traced back to work by Jacobi.     

A note on extending Euler's connection between continued fractions and power series

Euler's Connection describes an exact equivalence between certain continued fractions and power series. If the partial numerators and denominators of the continued fractions are perturbed slightly, the continued fractions equal power series plus easily computed error terms. These continued fractions may be integrated by the series with another easily computed error term.     

Mathematical tables in Ptolemy's Almagest

This paper is a discussion of Ptolemy's use of mathematical tables in the Almagest. By focusing on Ptolemy's mathematical practice and terminology, I argue that Ptolemy used tables as part of an organized group of units of text, which I call the table nexus. In the context of this deductive structure, tables function in the Almagest in much the same way as theorems in a canonical work, such as the Elements, both as means of presenting acquired knowledge and as tools for producing further knowledge.     

Decimal periods and their tables: A German research topic (1765–1801)

At the beginning of the 18th century, several mathematicians noted regularities in the decimal expansions of common fractions. Rules of thumb were set up, but it was only from 1760 onward that the first attempts to try to establish a coherent theory of periodic decimal fractions appeared. J.H. Lambert was the first to devote two essays to the topic, but his colleagues at the Berlin Academy, J. III Bernoulli and J.L. Lagrange, also spent time on the problem. Apart from the theoretical side of the question, the applications (factoring, irrationality proofs, and computational advantages), as well as the tabulation of decimal periods, aroused considerable interest, especially among Lambert's correspondents, C.F. Hindenburg and I. Wolfram. Finally, in 1797–1801, the young C.F. Gauss, informed of these developments, based the whole theory on firm number-theoretic foundations, thereby solving most of the open problems left by the mathematicians before him.     

Renaissance notions of number and magnitude

In the 16th and 17th centuries the classical Greek notions of (discrete) number and (continuous) magnitude (preserved in medieval Latin translations of Euclid's Elements) underwent a major transformation that turned them into continuous but measurable magnitudes. This article studies the changes introduced in the classical notions of number and magnitude by three influential Renaissance editions of Euclid's Elements. Besides providing evidence of earlier discussions preparing notions and arguments eventually introduced in Simon Stevin's Arithmétique of 1585, these editions document the role abacus algebra and Renaissance views on the history of mathematics played in bridging the gulf between discrete numbers and continuous magnitudes.     

The difficult birth of stochastics: Jacob Bernoulli's Ars Conjectandi (1713)

Jacob Bernoulli (1654–1705) did most of his research on the mathematics of uncertainty – or stochastics, as he came to call it – between 1684 and 1690. However, the Ars Conjectandi, in which he presented his insights (including the fundamental “Law of Large Numbers”), was printed only in 1713, eight years after his death. The paper studies the sources and the development of Bernoulli's ideas on probability, the reasons behind the delay in publishing and the circumstances under which his masterpiece eventually reached the public.     

Robert Leslie Ellisʼs work on philosophy of science and the foundations of probability theory

The goal of this paper is to provide an extensive account of Robert Leslie Ellis?s largely forgotten work on philosophy of science and probability theory. On the one hand, it is suggested that both his ‘idealist’ renovation of the Baconian theory of induction and a ‘realism’ vis-à-vis natural kinds were the result of a complex dialogue with the work of William Whewell. On the other hand, it is shown to what extent the combining of these two positions contributed to Ellis?s reformulation of the metaphysical foundations of traditional probability theory. This parallel is assessed with reference to the disagreement between Ellis and Whewell on the nature of (pure) mathematics and its relation to scientific knowledge.     

Cristoforo Borri and the epistemological status of mathematics in seventeenth-century Portugal

This paper argues that the epistemological promotion of mathematics by the Jesuit Cristoforo Borri, while he was teaching at the Coimbra Jesuit College in the late 1620s, played a decisive role in the updating of cosmological ideas in 17th-century Portugal. The paper focuses on Borri's position on the celebrated quaestio de certitudine mathematicarum and on his understanding of the classification of sciences. It argues that by conferring on mathematics the status of Aristotelian causal science, Borri made it possible to integrate mathematical data into the philosophical debate, particularly with regard to the new cosmology.     

A treatise on proportion in the tradition of Thomas Bradwardine: The De proportionibus libri duo (1528) of Jean Fernel

The famous French physician Jean Fernel published in 1528 in Paris the De proportionibus libri duo. This treatise belongs to the tradition of texts on proportion that follow Bradwardine?s Tractatus de proportionibus seu de proportionibus velocitatum in motibus (1328). In the first book, Fernel presented a theory of ratios that is traditional but contains some distinctive features, on denominating ratios, on fractions, on irrational ratios. The second book is devoted to a theory of ratio of ratios of which I give an account in this paper.     

Princess Elizabeth of Bohemia and Descartes’ letters (1650–1665)

After Descartes’ death in 1650, Princess Elizabeth generously shared with others several letters she had received from the philosopher, which contained philosophically as well as mathematically exciting material. In this article I place the transmission of these copies in context, revealing that Elizabeth steadily became an intellectually inspiring figure, attracting international attention. In the 1650s she stayed at Heidelberg where she discussed Cartesian philosophy with professors and students alike, including the professor of philosophy and mathematics Johann von Leuneschlos. In the mid-1660s, an initiative was taken from the English side of the Channel (Pell, More) to obtain Descartes’ mathematical letters to Elizabeth that had not yet been published. One letter of Elizabeth herself on this very subject has been preserved. The letter, addressed to Theodore Haak, will be published here for the first time. It is of special interest, because the princess supplies a general outline of her solution to the mathematical problem Descartes gave her to solve in 1643. It substantiates the hypothesis regarding Elizabeth’s solution earlier proposed by Henk Bos.     

Polylogarithms,functional equations and more: The elusive essays of William Spence (1777–1815)

The little-known Scottish mathematician William Spence was an able analyst, one of the first in Britain to be conversant with recent continental advances, and having original views. His major work on “logarithmic transcendents” gives the first detailed account of polylogarithms and related functions. A theory of algebraic equations was published just after his early death; and further essays, edited by John Herschel, were published posthumously. The most substantial of these concern an extension of his work on “logarithmic transcendents”, and the general solution of linear differential and difference equations. But awareness of Spence?s works was long delayed by their supposed unavailability. Spence?s life, the story of his “lost” publications, and a summary of all his essays are here described.     

Die Habilitation von John von Neumann an der Friedrich-Wilhelms-Universität in Berlin: Urteile über einen ungarisch-jüdischen Mathematiker in Deutschland im Jahr 1927

The mathematician John von Neumann was born in Hungary but principally received his scientific education and socialization in the German science system. He received his Habilitation from the Friedrich-Wilhelms–Universität in Berlin in 1927, where he lectured as a Privatdozent until his emigration to the USA. This article aims at making a contribution to this early part of Neumann’s scientific biography by analyzing in detail the procedure that led to his Habilitation as well as the beginnings of Neumann’s research on functional analysis. An analysis of the relevant sources shows that in Berlin in the year 1927 Neumann was not yet regarded as the outstanding mathematical genius of the 20th century. Furthermore it will be seen that Neumann had great difficulties in developing the fundamental concepts for his path breaking work in spectral theory and only managed to do so with the support of the Berlin mathematician Erhard Schmidt.     

Local Stability of Euler's and Kahan's Methods

Two discretization methods, the forward Euler's method and the Kahan's reflexive method, are compared by looking at the local stabilities of fixed points of a system of differential equations. We explain why forward Euler's method is not as good from the viewpoint of complex analysis. Conformal mappings are used to relate the eigenvalues of the Jacobian matrices of the differential equations system and the resulting difference equations system. The Euler's method will not preserve Hopf bifurcation. The Kahan's method preserves the local stability of the fixed points of the differential equations.     

The evolution of transformation media in spherical trigonometry in 17th- and 18th-century China,and its relation to “Western learning”

Problems of spherical trigonometry in 17th- and 18th-century China were often reduced to problems in plane trigonometry and then solved by means of the proportionality of corresponding sides of similar right triangles. Nevertheless, in the literature on the history of Chinese mathematics, there is not much discussion on the transformation and reduction of spherical problems to the plane, and how the techniques utilized for such transformations evolved over time. In this article, I investigate the evolution of the transformation media involved. I will show that in the trigonometric treatises by Mei Wending (1633–1721) and Dai Zhen (1724–1777), the authors’ views on Western learning shaped their choices of transformation media, and conversely their choices of transformation media offered support to their views on trigonometry in the debate of Chinese versus Western methods. Based on my analysis, I also propose a reassessment of Dai’s treatise of trigonometry, which was controversial ever since its publication in the 18th century.     

Hilbert's objectivity

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 ( and ). 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.     

Beyond Cartesian limits: Leibniz's passage from algebraic to “transcendental” mathematics

This article deals with Leibniz's reception of Descartes' “geometry.” Leibnizian mathematics was based on five fundamental notions: calculus, characteristic, art of invention, method, and freedom. On the basis of methodological considerations Leibniz criticized Descartes' restriction of geometry to objects that could be given in terms of algebraic (i.e., finite) equations: “Descartes's mind was the limit of science.” The failure of algebra to solve equations of higher degree led Leibniz to develop linear algebra, and the failure of algebra to deal with transcendental problems led him to conceive of a science of the infinite. Hence Leibniz reconstructed the mathematical corpus, created new (transcendental) notions, and redefined known notions (equality, exactness, construction), thus establishing “a veritable complement of algebra for the transcendentals”: infinite equations, i.e., infinite series, became inestimable tools of mathematical research.