Mathematics

Heron, in Metrica III.20-22, is concerned with the the division of solid figures - pyramids, cones and frustra of cones - to which end there is a need to extract cube roots. We report here on some of our findings on the conjecture by Taisbak in C.M.Taisbak, Cube roots of integers. A conjecture about Heron's method in Metrika III.20. Historia Mathematica, 41 (2014), 103-104.

Mathematicians tend to use the phrase "arbitrarily close" to mean something along the lines of "every neighborhood of a point intersects a set". Taking the latter phrase as a technical definition for arbitrarily close leads to an alternative, or at least parallel, development of classical concepts in analysis such as closure and limits in the context of metric spaces as well as continuity, differentiation, and integration in the setting of real valued functions on the real line. In particular, a definition of integration in terms of arbitrarily close is presented here. The corresponding integral is distinct from and yet equivalent to the classical integrals of Riemann and Darboux.

We use Taylor's formula with Lagrange remainder to prove that functions with bounded second derivative are rectifiable in the case when polygonal paths are defined by interval subdivisions which are equally spaced. We discuss potential benefits for such an approach in introductory calculus courses.

The usual modelling of the syllogisms of the Organon by a calculus of classes does not include relations. Aristotle may however have envisioned them in the first two books as the category of relatives, where he allowed them to compose with themselves. Composition is the main operation in combinatory logic, which therefore offers itself for a new kind of modelling. The resulting calculus includes also composition of predicates by logical connectives.

We give an overview of several of the mathematical works of Gilles Lachaud and provide a historical context. This is interspersed with some personal anecdotes highlighting many facets of his personality.

This is an exposition of facts about Arithmetic with an approach via mathematical logic. In Section 1 we present Peano Arithmetic, PA, and the complete theory of N , and we show that N is a prime model of the theory of N . In Section 2 we deal with the Incompleteness Theorems. In Section 3 we deal with non-standard models of arithmetic, in Section 4 we present the Paris-Harrington principle and in Section 5 its independence. The results presented here are quoted from the references listed at the end.

A brief review of the history of the conic sections would not be complete without an exhaustively tolerable account of all the things related to the subject that can be found in the extensive work of the wise Archimedes. There is no strong evidence that the Syracusan genius wrote a treatise on Conics separately.

We systematically investigate properties of various triangle centers (such as orthocenter or incenter) located on the four faces of a tetrahedron. For each of six types of tetrahedra, we examine over 100 centers located on the four faces of the tetrahedron. Using a computer, we determine when any of 16 conditions occur (such as the four centers being coplanar). A typical result is: The lines from each vertex of a circumscriptible tetrahedron to the Gergonne points of the opposite face are concurrent.

Balancing square and rectangular tables by rotation has been a interesting way to illustrate the intermediate value theorem. The aim of this note is to show that the balancing act but with non-rectangular tables can be a nice application of the ergodic theorem (or more generally, invariant measures).

Barry Mazur published an article some year ago, where he showed, among other things, that the result in the so-called mathematical passage of Plato s Theatetus and Euclid s proposition X.9 in the Elements are very different, while almost all historians and commentators have claimed the opposite. In this lecture, we consider this question to try to understand how and why could have happened such an error, this 'strange delusion' to quote Mazur.