Force,deflection, and time: Proposition VI of Newton's Principia

In this extended study of Proposition VI, and its first corollary, in Book I of Newton's Principia, we clarify both the statements and the demonstrations of these fundamental results. We begin by tracing the evolution of this proposition and its corollary, to see how their texts may have changed from their initial versions. To prepare ourselves for some of the difficulties our study confronts, we then examine certain confusions which arise in two recent commentaries on Proposition VI. We go on to note other confusions, not in any particular commentary, but in Newton's demonstration and, especially, in his statement of the proposition. What, exactly, does Newton mean by a “body [that] revolves … about an immobile center”? By a “just-nascent arc”? By the “sagitta of the arc”? By the “centripetal force”? By “will be as”? We search for the mathematical meanings that Newton has in mind for these fragments of the Proposition VI statement, a search that takes us to earlier sections of the Principia and to discussions of the “method of first and last ratios,” centripetal force, and the second law of motion. The intended meaning of Proposition VI then emerges from the combined meanings of these fragments. Next we turn to the demonstration of Proposition VI, noting first that Newton's own argument could be more persuasive, before we construct a modern, more rigorous proof. This proof, however, is not as simple as one might expect, and the blame for this lies with the “sagitta of the arc,” Newton's measure of deflection in Proposition VI. Replacing the sagitta with a more natural measure of deflection, we obtain what we call Platonic Proposition VI, whose demonstration has a Platonic simplicity. Before ending our study, we examine the fundamental first corollary of Proposition VI. In his statement of this Corollary 1, Newton replaces the sagitta of Proposition VI by a not quite equal deflection from the tangent and the area swept out (which represents the time by Proposition I) by a not quite equal area of a triangle. These two approximations create small errors, but are these errors small enough? Do the errors introduced by these approximations tend to zero fast enough to justify these replacements? Newton must believe so, but he leaves this question unasked and unanswered, as have subsequent commentators on this crucial corollary. We end our study by asking and answering this basic question, which then allows us to give Corollary 1 a convincing demonstration.     

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 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 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.     

Varieties of wonder:: John Wilkins' mathematical magic and the perpetuity of invention

Akin to the mathematical recreations, John Wilkins' Mathematicall Magick ( 1648) elaborates the pleasant, useful and wondrous part of practical mathematics, dealing in particular with its material culture of machines and instruments. We contextualize the Mathematicall Magick by studying its institutional setting and its place within changing conceptions of art, nature, religion and mathematics. We devote special attention to the way Wilkins inscribes mechanical innovations within a discourse of wonder. Instead of treating ‘wonder’ as a monolithic category, we present a typology, showing that wonders were not only recreative, but were meant to inspire Wilkins' readers to new mathematical inventions.     

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.     

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 introduction of logarithms into Spain

During the first half of the 17th century, logarithms were taught by some professors in Spain, but knowledge of this subject remained scanty until the publication of Architectura civil by Juan Caramuel (1678) and especially of Trigonometria española by José Zaragoza (1672). Logarithms were considered only as an aid for computation up to the second half of the 18th century. Only when the infinitesimal calculus became more widely spread in Spanish mathematics, analytical interpretations of logarithms were also taken into account in books such as Elementos de matemáticas by Benito Bails (1776).     

Some remarks on the meaning of equality in Diophantos’s Arithmetica

Diophantos in Arithmetica, without having defined previously any concept of “equality” or “equation,” employs a concept of the unknown number as a tool for solving problems and finds its value from an equality ad hoc created. In this paper we analyze Diophantos’s practices in the creation and simplification of such equalities, aiming to adduce more evidence on certain issues arising in recent historical research on the meaning of the “equation” in Diophantos’s work.     

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.     

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.     

Algebra in Roth,Faulhaber, and Descartes

Descartes' “multiplicative” theory of equations in the Géométrie (1637) systematically treats equations as polynomials set equal to zero, bringing out relations between equations, roots, and polynomial factors. We here consider this theory as a response to Peter Roth's suggestions in Arithmetica Philosophica (1608), notably in his “seventh-degree” problem set. These specimens of arithmetic-masterly problem design develop skills with multiplicative and other degree-independent techniques. The challenges were fine-tuned by introducing errors disguised as printing errors. During Descartes' visit to Germany in 1619–1622, he probably worked with Johann Faulhaber (1580–1635) on these problems; they are discussed in Faulhaber's Miracula Arithmetica (1622), which also looks forward to fuller publication, probably by Descartes.     

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.     

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.     

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.     

False position in Leonardo of Pisa's Liber Abbaci

We examine the rhetorical methods of Leonardo of Pisa in his exposition of single false position in Liber Abbaci. For example, Leonardo makes extensive use of formulaic phrases in his solutions. Some of these formulas also seem to indicate whether a particular solution needs further justification. Although he prefers proofs in terms of the pseudo-Euclidean canon of al-Khwārizmī, sometimes such proof eludes Leonardo and he resorts instead to justification by experiment. We also look at the extent to which using symbolic representations might distort our view of Leonardo's thinking.     

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.