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.     

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.     

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.     

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).     

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 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.     

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.     

Abū al-Wafā’ Latinus? A study of method

This article studies the legacy in the West of Abū al-Wafā’s Book on those geometric constructions which are necessary for craftsmen. Although two-thirds of the geometric constructions in the text also appear in Renaissance works, a joint analysis of original solutions, diagram lettering, and probability leads to a robust finding of independent discovery. The analysis shows that there is little chance that the similarities between the contents of Abū al-Wafā’s Book and the works of Tartaglia, Marolois, and Schwenter owe anything to historical transmission. The commentary written by Kamāl al-Dīn Ibn Yūnus seems to have had no Latin legacy, either.     

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.     

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.     

On mathematical problems as historically determined artifacts: Reflections inspired by sources from ancient China

Is a mathematical problem a cultural invariant, which would invariably give rise to the same practices, independent of the social groups considered? This paper discusses evidence found in the oldest Chinese mathematical text handed down by the written tradition, the canonical work The Nine Chapters on Mathematical Procedures and its commentaries, to answer this question in the negative. The Canon and its commentaries bear witness to the fact that, in the tradition for which they provide evidence, mathematical problems not only were questions to be solved, but also played a key part in conducting proofs of the correctness of algorithms.     

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.     

The function of diorism in ancient Greek analysis

This paper is a contribution to our knowledge of Greek geometric analysis. In particular, we investigate the aspect of analysis know as diorism, which treats the conditions, arrangement, and totality of solutions to a given geometric problem, and we claim that diorism must be understood in a broader sense than historians of mathematics have generally admitted. In particular, we show that diorism was a type of mathematical investigation, not only of the limitation of a geometric solution, but also of the total number of solutions and of their arrangement. Because of the logical assumptions made in the analysis, the diorism was necessarily a separate investigation which could only be carried out after the analysis was complete.     

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.     

Une ambition éditoriale “universelle et confraternelle” : le bulletin bibliographique de L'Enseignement mathématique (1899–1920)

In 1899 Henri Fehr and Charles Laisant founded L'Enseignement mathématique (EM) with the ambition to involve teachers in the then-growing internationalization movement of mathematics. To this purpose, their editorial project gave an important place to a bibliographical bulletin reviewing periodicals which could be of interest for the world of mathematical education. This article is dedicated to the study of this bulletin, from its creation to the 1920s, and to the initiatives and choices that Laisant and Fehr made to carry out this internationalist editorial ambition, as well as to the limits and constraints of their project. During that time, many bibliographical initiatives for periodicals developed in the mathematical press, which can be considered as a first form of interaction between journals. Our study will concern initially the year 1899 and this interaction in which EM took part, dealing at first with the bulletin of EM, then, secondly, with the confrontation between bibliographical sections of other journals. Lastly, considering the first thirty years of the 20th century, we will study the different dynamics at work in the world of mathematical periodicals which the EM serves to depict.     

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.     

Simplifying equations in Arabic algebra

Historians have always seen jabr (restoration) and muqābala (confrontation) as technical terms for specific operations in Arabic algebra. This assumption clashes with the fact that the words were used in a variety of contexts. By examining the different uses of jabr, muqābala, ikmāl (completion), and radd (returning) in the worked-out problems of several medieval mathematics texts, we show that they are really nontechnical words used to name the immediate goals of particular steps. We also find that the phrase al-jabr wa'l-muqābala was first used within the solutions of problems to mean al-jabr and/or al-muqābala, and from there it became the name of the art of algebra.     

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.