What Descartes knew of mathematics in 1628

The aim of this paper is to give an account of Descartes’ mathematical achievements in 1628–1629 using, as far as is possible, only contemporary documents, and in particular Beeckman’s Journal for October 1628. In the first part of the paper, I study the content of these documents, bringing to light the mathematical weaknesses they display. In the second part, I argue for the significance of these documents by comparing them with other independent sources, such as Descartes’ Regulae ad directionem ingenii. Finally, I outline the main consequences of this study for understanding the mathematical development of Descartes before and after 1629.     

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.     

The ovals in the Excerpta Mathematica and the origins of Descartes’ method of normals

The only occurrence of Descartes’ method of normals before La Géométrie (1637) is to be found in the Excerpta Mathematica. These mathematical fragments, published posthumously among others works in 1701, and dated by Tannery before 1629, deal with curves used in dioptrics which Descartes called ovals. I study in detail two of the texts on ovals together with the related texts in La Géométrie in order to shed light on the geometrical origins of Descartes’ method of normals.     

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.     

The Kujang sulhae 九章術解: Nam Pyoˇng-Gil’s reinterpretation of the mathematical methods of the Jiuzhang suanshu

In this article, a discussion and analysis is presented of the Kujang sulhae by Nam Pyoˇng-Gil (1820-1869), a 19th-century Korean commentary on the Jiuzhang suanshu. Nam copied the problems and procedures from the ancient Chinese classic, but replaced Liu Hui’s and Li Chunfeng’s commentaries with his own. In his postface Nam expressed his dissatisfaction with the earlier commentaries, because the approaches of Liu and Li did not match those of his contemporary readers well. This can be seen from the most important features of Nam’s commentary: the use of a synthesis of European and Chinese mathematical methods, easy explanations appealing to intuition, and disuse of the methods of infinitesimals and limits in Liu’s and Li’s commentaries. Based on his own postface and these features of his commentary, I believe that Nam Pyoˇng-Gil treated the Jiuzhang suanshu as a very important historical document, which he intended to explain according to the new mathematical canon in both Qing China and Chosoˇn Korea, the Shuli jingyun. Thus the Kujang sulhae is an example of the endeavor of 19th-century Korean mathematicians to reinterpret ancient Chinese mathematical texts with their contemporary knowledge.     

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.     

John Wallis and the French: his quarrels with Fermat,Pascal, Dulaurens,and Descartes

John Wallis, Savilian professor of geometry at Oxford from 1649 to 1703, engaged in a number of disputes with French mathematicians: with Fermat (in 1657–1658), with Pascal (in 1658–1659), with Dulaurens (in 1667–1668), and against Descartes (in the early 1670s). This paper examines not only the mathematical content of the arguments but also Wallis’s various strategies of response. Wallis’s opinion of French mathematicians became increasingly bitter, but at the same time he was able to use the confrontations to promote his own reputation.     

The rectification of quadratures as a central foundational problem for the early Leibnizian calculus

Transcendental curves posed a foundational challenge for the early calculus, as they demanded an extension of traditional notions of geometrical rigour and method. One of the main early responses to this challenge was to strive for the reduction of quadratures to rectifications. I analyse the arguments given to justify this enterprise and propose a hypothesis as to their underlying rationale. I then go on to argue that these foundational concerns provided the true motivation for much ostensibly applied work in this period, using Leibniz’s envelope paper of 1694 as a case study.     

Shifting the foundations: Descartes's transformation of ancient geometry

The aim of this paper is to analyse how the bases of Descartes's geometry differed from those of ancient geometry. Particular attention is paid to modes of specifying curves of which two types are distinguished — “Specification by genesis” and “Specification by property”. For both Descartes and most of Greek geometry the former was fundamental, but Descartes diverged from ancient pure geometry by according an essential place to the imagination of mechanical instruments. As regards specification by property, Descartes's interpretation of the multiplication of (segments of) straight lines as giving rise to a straight line (segment), together with newer methods of articifical symbolism, led to more concise and suggestive modes of representation. Descartes's account of ancient procedures is historically very misleading, but it allowed him to introduce his own ideas more naturally.     

Robustness in operational research and decision aiding: A multi-faceted issue

In Introduction, I explain the meaning I give to the qualifier term ”robust” and justify my preference for the expression robustness concern rather than robustness analysis, which I feel is likely to be interpreted too narrowly. In Section 2, I discuss this concern in more details and I try to clarify the numerous raisons d’être of this concern. As a means of examining the multiple facets of robustness concern more comprehensively, I explore the existing research about robustness, attempting to highlight what I see as the three different territories covered by these studies (Section 3). In Section 4, I refer to these territories to illustrate how responses to robustness concern could be even more varied than they currently are. In this perspective, I propose in Section 5 three new measures of robustness. In the last section, I identify several aspects of the problem that should be examined more closely because they could lead to new avenues of research, which could in turn yield new and innovative responses.     

Comparative analysis in Greek geometry

This article is a contribution to our knowledge of ancient Greek geometric analysis. We investigate a type of theoretic analysis, not previously recognized by scholars, in which the mathematician uses the techniques of ancient analysis to determine whether an assumed relation is greater than, equal to, or less than. In the course of this investigation, we argue that theoretic analysis has a different logical structure than problematic analysis, and hence should not be divided into Hankel’s four-part structure. We then make clear how a comparative analysis is related to, and different from, a standard theoretic analysis. We conclude with some arguments that the theoretic analyses in our texts, both comparative and standard, should be regarded as evidence for a body of heuristic techniques.     

A “lacuna” in Proposition 9 of Archimedes’ On the Sphere and the Cylinder, Book I

The proof of Proposition 9 in Archimedes’ On the Sphere and the Cylinder, Book i, contains an unproved statement that has been referred to as a “lacuna.” Most editors and experts in Archimedean texts have agreed on the existence of this gap and have offered different proofs for the statement, some of them with incomplete or even incorrect arguments. In this paper, I offer arguments of a mathematical, historical, and textual nature that show that it is not necessary to assume the presence of any gap in the text.     

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.     

Commentary on Robert Riley’s article “A personal account of the discovery of hyperbolic structures on some knot complements”

We give some background and biographical commentary on the posthumous article  [4] that appears in this journal issue by Robert Riley on his part of the early history of hyperbolic structures on some compact 3-manifolds. A complete list of Riley’s publications appears at the end of this article.     

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.     

Perspective in Leibnizʼs invention of Characteristica Geometrica: The problem of Desarguesʼ influence

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 Leibniz?s new geometry and the works on perspective, namely Desargues? ones.     

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.