Practicing algebra in late antiquity: The problem-solving of Diophantus of Alexandria

Medieval algebra is distinguished from other arithmetical problem-solving techniques by its structure and technical vocabulary. In an algebraic solution one or several unknowns are named, and via operations on the unknowns the problem is transferred to the artificial setting of an equation expressed in terms of the named powers, which is then simplified and solved. In this article we examine Diophantus? Arithmetica from this perspective. We find that indeed Diophantus? method matches medieval algebra in both vocabulary and structure. Just as we see in medieval Arabic and Italian algebra, Diophantus worked out the operations expressed in the enunciation of a problem prior to setting up a polynomial equation. Further, his polynomials were regarded as aggregations with no operations present.     

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.     

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.     

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.     

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.     

Why Rob Archimedes of his Lemma?

The method of exhaustion is one of the greatest achievements of Greek mathematics, but the history of its development is not clear. First and foremost Archimedes’ role has been keenly debated, by and large undermined, so that even his name seems condemned to disappear in the name of the Eudoxus-Archimedes Lemma. In this paper we try to revaluate his role by a new interpretation (or, more precisely, by the refinement of an old one) of the historical development of the theory, underlining the theoretical relevance of the problem of addition/subtraction and comparison between curves. Dedicated to the memory of Professor Aldo Cossu     

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.     

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.     

Rethinking geometrical exactness

A crucial concern of early modern geometry was fixing appropriate norms for deciding whether some objects, procedures, or arguments should or should not be allowed into it. According to Bos, this is the exactness concern. I argue that Descartes’s way of responding to this concern was to suggest an appropriate conservative extension of Euclid’s plane geometry (EPG). In Section 2, I outline the exactness concern as, I think, it appeared to Descartes. In Section 3, I account for Descartes’s views on exactness and for his attitude towards the most common sorts of constructions in classical geometry. I also explain in which sense his geometry can be conceived as a conservative extension of EPG. I conclude by briefly discussing some structural similarities and differences between Descartes’s geometry and EPG.     

The Square of Opposition and the Four Fundamental Choices

Each predicate of the Aristotelian square of opposition includes the word “is”. Through a twofold interpretation of this word the square includes both classical logic and non-classical logic. All theses embodied by the square of opposition are preserved by the new interpretation, except for contradictories, which are substituted by incommensurabilities. Indeed, the new interpretation of the square of opposition concerns the relationships among entire theories, each represented by means of a characteristic predicate. A generalization of the square of opposition is achieved by not adjoining, according to two Leibniz’ suggestions about human mind, one more choice about the kind of infinity; i.e., a choice which was unknown by Greek’s culture, but which played a decisive role for the birth and then the development of modern science. This essential innovation of modern scientific culture explains why in modern times the Aristotelian square of opposition was disregarded. This work was completed with the support of our -pert.     

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.     

A cogenerator for preseparated superconvex spaces

We construct a cogenerator for the category of preseparated superconvex spaces, and we describe separated convex spaces, i.e. convex spaces for which the morphisms into the unit interval separates points.Presented at the European Colloquium of Category Theory, Tours, France, 25–31 July 1994.     

On computing the number of subgroups of a finite Abelian group

We consider the numberN _A (r) of subgroups of orderp ^r ofA, whereA is a finite Abelianp-group of type =₁,₂,..., _l ()), i.e. the direct sum of cyclic groups of order ⁱⁱ. Formulas for computingN _A (r) are well known. Here we derive a recurrence relation forN _A (r), which enables us to prove a conjecture of P. E. Dyubyuk about congruences betweenN _A (r) and the Gaussian binomial coefficient .     

Euler’s fixed point theorem: The axis of a rotation

We give an elementary proof of what is perhaps the earliest fixed point theorem; namely Leonhard Euler’s theorem of 1775 on the existence of an axis v for any three-dimensional rotation R. The proof is constructive and shows that no multiplications are required to compute v. Dedicated to the memory of Leonhard Euler, “The Master of us all”, on the occasion of the 300th anniversary of his birth     

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.     

On the space of B-convex compacta

We investigate topology of the space of B-convex compacta of finite-dimensional Banach space (the notion of B-convexity space was introduced by M. Lassak). An answer to the question of M. van de Vel about a characterization of continuity of the closed B-convex hull is given. We prove that the space of B-convex compacta is a Q-manifold iff the map of the closed B-convex hull is continuous.     

Straddling centuries: The struggles of a mathematician and his university to enter the ranks of research mathematics, 1870–1950

This paper weaves two interlocking histories together. One strand of the fabric traces the development of the American mathematician Joseph B. Reynolds from a peripheral player to an active contributor to mathematics, astronomy, and engineering and to the founding of a sectional association of mathematicians. The other piece describes the evolution of his institution, Lehigh University, from its founding in 1865 to a full-fledged research department that began producing doctorates in 1939. Both Reynolds and Lehigh straddled the line between the pre- and post-Chicago eras in American mathematics.     

Cube roots of integers. A conjecture about Heron's method in Metrika III. 20

Did Heron (or his teachers) use sequences of differences to find an approximate value of the cube root of an integer? I venture a conjecture of his heuristics and a couple of possible mathematical proofs of his method.     

Non-Classical Stems from Classical: N. A. Vasiliev’s Approach to Logic and his Reassessment of the Square of Opposition

In the XIXth century there was a persistent opposition to Aristotelian logic. Nicolai A. Vasiliev (1880–1940) noted this opposition and stressed that the way for the novel – non-Aristotelian – logic was already paved. He made an attempt to construct non-Aristotelian logic (1910) within, so to speak, the form (but not in the spirit) of the Aristotelian paradigm (mode of reasoning). What reasons forced him to reassess the status of particular propositions and to replace the square of opposition by the triangle of opposition? What arguments did Vasiliev use for the introduction of new classes of propositions and statement of existence of various levels in logic? What was the meaning and role of the “method of Lobachevsky” which was implemented in construction of imaginary logic? Why did psychologism in the case of Vasiliev happen to be an important factor in the composition of the new ‘imaginary’ logic, as he called it?      

Covering a Finite Abelian Group by Subset Sums

Let G be an abelian group of order n. The critical number c(G) of G is the smallest s such that the subset sums set (S) covers all G for eachs ubset SG\{0} of cardinality |S|s. It has been recently proved that, if p is the smallest prime dividing n and n/p is composite, then c(G)=|G|/p+p–2, thus establishing a conjecture of Diderrich.We characterize the critical sets with |S|=|G|/p+p–3 and (S)=G, where p3 is the smallest prime dividing n, n/p is composite and n7p²+3p.We also extend a result of Diderrichan d Mann by proving that, for n67, |S|n/3+2 and S=G imply (S)=G. Sets of cardinality for which (S) =G are also characterized when n183, the smallest prime p dividing n is odd and n/p is composite. Finally we obtain a necessary and sufficient condition for the equality (G)=G to hold when |S|n/(p+2)+p, where p5, n/p is composite and n15p².* Work partially supported by the Spanish Research Council under grant TIC2000-1017 Work partially supported by the Catalan Research Council under grant 2000SGR00079