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.     

Mathematical recreations of Dénes König and his work on graph theory

Dénes König (1884–1944) is a Hungarian mathematician well known for his treatise on graph theory (König, 1936). When he was a student, he published two books on mathematical recreations ( and ). Does his work on mathematical recreations have any relation to his work on graph theory? If yes, how are they connected? To answer these questions, we will examine his books of 1902, 1905 and 1936, and compare them with each other. We will see that the books of 1905 and 1936 include many common topics, and that the treatment of these topics is different between 1905 and 1936.     

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.     

W.W. Rouse Ball and the mathematics of string figures

This article examines a chapter of the popular book Mathematical Recreations and Essays (5th to 9th editions) written by the Cambridge mathematician Walter William Rouse Ball (1850–1925). This chapter is devoted to “String Figures”, a procedural activity which consists in producing geometrical forms with a loop of string and which is carried out in many traditional societies throughout the world. By analyzing the way in which Ball selected some string figures within ethnographical publications and conceived the structure of this chapter, it appears that he implicitly brought to light the mathematical dimension of this practice.     

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.     

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.     

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.     

The way of Diophantus: Some clarifications on Diophantus' method of solution

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.     

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.     

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.     

From anomaly to fundament: Louis Poinsot's theories of the couple in mechanics

In 1803 Louis Poinsot published a textbook on statics, in which he made clear that the subject dealt not only with forces but also with ‘couples’ (his word), pairs of coplanar non-collinear forces equal in magnitude and direction but opposite in sense. His innovation was not understood or even welcomed by some contemporary mathematicians. Later he adapted his theory to put forward a new relationship between rectilinear and rotational motion in dynamics; its reception was more positive, although not always appreciative of the generality. After summarising the creation of these two theories and noting their respective receptions, this paper considers his advocacy of spatial and geometrical thinking in mechanics and the fact that, despite its importance, historians of statics who cover his period usually ignore his theory of couples.     

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.     

Algebra and geometry in Pietro Mengoli (1625–1686)

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⋅xm⋅n(t−x)$y=K\cdot x^m\cdot {(t-x)}^n$. 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.     

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.     

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.     

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.     

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.     

One of Berkeley’s arguments on compensating errors in the calculus

This paper addresses three questions related to George Berkeley’s theory of compensating errors in the calculus published in 1734. The first is how did Berkeley conceive of Leibnizian differentials? The second and most central question concerns Berkeley’s procedure which consisted in identifying two quantities as errors and proving that they are equal. The question is how was this possible? The answer is that this was not possible, because in his calculations Berkeley misguided himself by employing a result equivalent to what he wished to prove. In 1797 Lazare Carnot published the expression “a compensation of errors” in an attempt to explain why the calculus functions. The third question is: did Carnot by this expression mean the same as Berkeley?