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.     

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.     

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.     

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.     

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.     

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.     

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.     

n-Dimensional algebras over a field with a cyclic extension of degree n

We give a geometric method of classifying algebras A _n,K , n-dimensional over a field K, with a cyclic extension of degree n. Algebras A _n,K without zero divisors satisfying some conditions are classified. In particular, we determine all n-dimensional division algebras over a finite field F _q when n is prime and q is large enough.This research was supported in part by a grant from the M U R S T (40 % funds).     

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.     

Abstract convex optimal antiderivatives

Having studied families of antiderivatives and their envelopes in the setting of classical convex analysis, we now extend and apply these notions and results in settings of abstract convex analysis. Given partial data regarding a c-subdifferential, we consider the set of all c-convex c-antiderivatives that comply with the given data. Under a certain assumption, this set is not empty and contains both its lower and upper envelopes. We represent these optimal antiderivatives by explicit formulae. Some well known functions are, in fact, optimal c-convex c-antiderivatives. In one application, we point out a natural minimality property of the Fitzpatrick function of a c-monotone mapping, namely that it is a minimal antiderivative. In another application, in metric spaces, a constrained Lipschitz extension problem fits naturally the convexity notions we discuss here. It turns out that the optimal Lipschitz extensions are precisely the optimal antiderivatives. This approach yields explicit formulae for these extensions, the most particular case of which recovers the well known extensions due to McShane and Whitney.     

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.     

Comparison of volumes of convex bodies in real, complex, and quaternionic spaces

The classical Busemann-Petty problem (1956) asks, whether origin-symmetric convex bodies in Rn with smaller hyperplane central sections necessarily have smaller volumes. It is known, that the answer is affirmative if n?4 and negative if n>4. The same question can be asked when volumes of hyperplane sections are replaced by other comparison functions having geometric meaning. We give unified analysis of this circle of problems in real, complex, and quaternionic n-dimensional spaces. All cases are treated simultaneously. In particular, we show that the Busemann-Petty problem in the quaternionic n-dimensional space has an affirmative answer if and only if n=2. The method relies on the properties of cosine transforms on the unit sphere. We discuss possible generalizations.     

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.     

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.     

The evolution of the concept of homeomorphism

Topology, or analysis situs, has often been regarded as the study of those properties of point sets (in Euclidean space or in abstract spaces) that are invariant under “homeomorphisms.” Besides the modern concept of homeomorphism, at least three other concepts were used in this context during the late 19th and early 20th centuries, and regarded (by various mathematicians) as characterizing topology: deformations, diffeomorphisms, and continuous bijections. Poincaré, in particular, characterized analysis situs in terms of deformations in 1892 but in terms of diffeomorphisms in 1895. Eventually Kuratowski showed in 1921 that in the plane there can be a continuous bijection of P onto Q, and of Q onto P, without P and Q being homeomorphic.     

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.     

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.     

Fully nonlinear singularly perturbed equations and asymptotic free boundaries

In this paper we study one-phase fully nonlinear singularly perturbed elliptic problems with high energy activation potentials, ζε(u) with ζε→δ₀⋅∫ζ. We establish uniform and optimal gradient estimates of solutions and prove that minimal solutions are non-degenerated. For problems governed by concave equations, we establish uniform weak geometric properties of approximating level surfaces. We also provide a thorough analysis of the free boundary problem obtained as a limit as the ε-parameter term goes to zero. We find the precise jumping condition of limiting solutions through the phase transition, which involves a subtle homogenization process of the governing fully nonlinear operator. In particular, for rotational invariant operators, F(D²u), we show the normal derivative of limiting function is constant along the interface. Smoothness properties of the free boundary are also addressed.