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.     

A dialogue on the use of arithmetic in geometry: Van Ceulen’s and Snellius’s Fundamenta Arithmetica et Geometrica

Snellius’s Fundamenta Arithmetica et Geometrica (1615) is much more than a Latin translation of Ludolph van Ceulen’s Arithmetische en Geometrische Fondamenten. Willebrord Snellius both adapted and commented on the Dutch original in his Fundamenta, and thus his Latin version can be read as a dialogue between representatives of two different approaches to mathematics in the early modern period: Snellius’s humanist approach and Van Ceulen’s practitioner’s approach. This article considers the relationship between the Dutch and Latin versions of the text and, in particular, puts some of their statements on the use of numbers in geometry under the microscope. In addition, Snellius’s use of the Fundamenta as an instrument to further his career is explained.     

Leonhard Euler’s use and understanding of mathematical transcendence

Leonhard Euler primarily applied the term “transcendental” to quantities which could be variable or determined. Analyzing Euler’s use and understanding of mathematical transcendence as applied to operations, functions, progressions, and determined quantities as well as the eighteenth century practice of definition allows the author to evaluate claims that Euler provided the first modern definition of a transcendental number. The author argues that Euler’s informal and pragmatic use of mathematical transcendence highlights the general nature of eighteenth century mathematics and proposes an alternate perspective on the issue at hand: transcendental numbers inherited their transcendental classification from functions.     

God, king, and geometry: revisiting the introduction to Cauchy’s Cours d’analyse

This article offers a systematic reading of the introduction to Augustin-Louis Cauchy’s landmark 1821 mathematical textbook, the Cours d’analyse. Despite its emblematic status in the history of mathematical analysis and, indeed, of modern mathematics as a whole, Cauchy’s introduction has been more a source for suggestive quotations than an object of study in its own right. Cauchy’s short mathematical metatext offers a rich snapshot of a scholarly paradigm in transition. A close reading of Cauchy’s writing reveals the complex modalities of the author’s epistemic positioning, particularly with respect to the geometric study of quantities in space, as he struggles to refound the discipline on which he has staked his young career.     

Symbolism, combinations, and visual imagery in the mathematics of Thomas Harriot

The mathematical work of Thomas Harriot (c. 1560–1621) is distinguished by extensive use of symbolism and other forms of visual imagery and by systematic use of combinations. This paper argues that these characteristics of his mathematical writing were already observable in the mid-1580s, in the phonetic alphabet he devised to record the speech of American Indians. The paper presents several little-known examples of Harriot's mathematics, demonstrating his use of symbolism both as a means of expression and as an analytic tool, and assesses Harriot's work in relation to the broader 17th-century trend toward symbolization in mathematics.     

The Mittag-Leffler Theorem: The origin,evolution, and reception of a mathematical result, 1876–1884

The Swedish mathematician Gösta Mittag-Leffler (1846–1927) is well-known for founding Acta Mathematica, often touted as the first international journal of mathematics. A “post-doctoral” student in Paris and Berlin between 1873 and 1876, Mittag-Leffler built on Karl Weierstrass? work by proving the Mittag-Leffler Theorem, which states that a function of rational character (i.e. a meromorphic function) is specified by its poles, their multiplicities, and the coefficients in the principal part of its Laurent expansion.     

The Luigi Cremona Archive of the Mazzini Institute of Genoa

Luigi Cremona (1830-1903) is unanimously considered to be the man who laid the foundations of the prestigious Italian school of Algebraic Geometry. In this paper we draw attention to the “Legato Itala Cremona Cozzolino”, which was given to the library of the Mazzini Institute, Genoa, Italy, by Cremona’s daughter, Itala, probably in 1939. This legacy, which contains over 6000 documents, mainly consisting of Cremona’s correspondence with scientific and institutional Italian interlocutors, can help us to understand the connections between the development of Italian mathematics in the second half of the XIX century and the main political issues of Italian history.     

Sets versus trial sequences,Hausdorff versus von Mises: “Pure” mathematics prevails in the foundations of probability around 1920

The paper discusses the tension which occurred between the notions of set (with measure) and (trial-) sequence (or—to a certain degree—between nondenumerable and denumerable sets) when used in the foundations of probability theory around 1920. The main mathematical point was the logical need for measures in order to describe general nondiscrete distributions, which had been tentatively introduced before (1919) based on von Mises’s notion of the “Kollektiv.” In the background there was a tension between the standpoints of pure mathematics and “real world probability” (in the words of J.L. Doob) at the time. The discussion and publication in English translation (in Appendix) of two critical letters of November 1919 by the “pure” mathematician Felix Hausdorff to the engineer and applied mathematician Richard von Mises compose about one third of the paper. The article also investigates von Mises’s ill-conceived effort to adopt measures and his misinterpretation of an influential book of Constantin Carathéodory. A short and sketchy look at the subsequent development of the standpoints of the pure and the applied mathematician—here represented by Hausdorff and von Mises—in the probability theory of the 1920s and 1930s concludes the paper.     

What did Gauss read in the Appendix?

In a clear analogy with spherical geometry, Lambert states that in an “imaginary sphere” the sum of the angles of a triangle would be less than π$\pi$. In this paper we analyze the role played by this imaginary sphere in the development of non-Euclidean geometry, and how it served Gauss as a guide. More precisely, we analyze Gauss’s reading of Bolyai’s Appendix in 1832, five years after the publication of Disquisitiones generales circa superficies curvas, on the assumption that his investigations into the foundations of geometry were aimed at finding, among the surfaces in space, Lambert’s hypothetical imaginary sphere. We also wish to show that the close relation between differential geometry and non-Euclidean geometry is already present in János Bolyai’s Appendix, that is, well before its appearance in Beltrami’s Saggio. From this point of view, one is able to answer certain natural questions about the history of non-Euclidean geometry; for instance, why Gauss decided not to write further on the subject after reading the Appendix.     

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.     

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.     

Why was Wantzel overlooked for a century? The changing importance of an impossibility result

The duplication of a cube and the trisection of an angle are two of the most famous geometric construction problems formulated in ancient Greece. In 1837 Pierre Wantzel (1814–1848) proved that the problems cannot be constructed by ruler and compass. Today he is credited for this contribution in all general treatises of the history of mathematics. However, his proof was hardly noticed by his contemporaries and during the following century his name was almost completely forgotten. In this paper I shall analyze the reasons for this neglect and argue that it was primarily due to the lack of importance attributed to such impossibility results at the time.     

Meta-mathematical rhetoric: Hero and Ptolemy against the philosophers

Bringing the meta-mathematics of Hero of Alexandria and Claudius Ptolemy into conversation for the first time, I argue that they employ identical rhetorical strategies in the introductions to Hero's Belopoeica, Pneumatica, Metrica and Ptolemy's Almagest. They each adopt a paradigmatic argument, in which they criticize the discourses of philosophers and declare epistemological supremacy for mathematics by asserting that geometrical demonstration is indisputable. The rarity of this claim—in conjunction with the paradigmatic argument—indicates that Hero and Ptolemy participated in a single meta-mathematical tradition, which made available to them rhetoric designed to introduce, justify, and bolster the value of mathematics.     

The origins of Euler's early work on continued fractions

In this paper, I examine Euler's early work on the elementary properties of continued fractions in the 1730s, and investigate its possible links to previous writings on continued fractions by authors such as William Brouncker. By analysing the content of Euler's first paper on continued fractions, ‘De fractionibus continuis dissertatio’ (1737, published 1744) I conclude that, contrary to what one might expect, Euler's work on continued fractions initially arose not from earlier writings on continued fractions, but from a wish to solve the Riccati differential equation.     

The dramatic episode of Sundman

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.     

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.