An elegant geometrical argument concerning Dido’s problem, traditionally credited to Jakob Steiner in about 1840, appears to have been invented by Thomas Simpson, almost 100 years earlier. The argument itself uses only the idea of triangle area and the converse of Thales’ theorem about the angle in a semicircle. It appears in Simpson’s book Elements of Geometry, published in 1760, as part of a highly original section on ‘The Maxima and Minima of Geometrical Quantities’, but traces of the argument can even be found in an earlier edition dated 1747.

The ancient Mesopotamians were able to estimate as , in modern decimal notation. How they calculated this estimate is uncertain. We furnish Neugebauer and Sachs' original explanation with a new geometric description and reconstruct this estimate using contemporary reciprocal approximation.

This paper attempts a new interpretation of Euclid’s Elements Book X. This study of irrational lines has long been viewed as an anomaly within the Euclidean corpus: it includes a tedious and seemingly pointless classification of lines, known as ‘the cross of mathematicians’. Following Ken Saito’s toolbox conception, we do not try to reconstruct the book’s mathematical process of discovery, but, instead, the kind of applications for which it serves as a toolbox. Our claim is that the book provides tools for solving questions about proportional lines inspired by results in music theory and a context of Pythagorean-Platonic interest in proportions. We show that the entire content of Book X can indeed be accounted for as a set of tools for these questions, augmented by the general editorial norms that govern the Elements. We conclude by explaining why the purpose of Book X as reconstructed here has disappeared from mathematical memory.

In late 1683, the physician and F.R.S. Martin Lister displayed to the Royal Society a new way of recording barometric observations, which amounted in all but name to the construction of line graphs. The innovation was communicated to the Oxford Philosophical Society, where Robert Plot used the method to display a year’s observations, published in the Philosophical Transactions early in 1685. At the Dublin Philosophical Society, William Molyneux displayed a month’s worth of observations kept in the same way during May 1684. Lister’s own engraved forms—what amounted to graph paper—circulated among these groups but are not known to survive; Molyneux had forms of his own engraved. Finally, the London instrument maker John Warner engraved his own version of a similar form for recording weather observations and offered it to Plot. Two exemplars survive, but neither Warner’s offer nor this graphical method itself seem to have been more widely taken up in this period. This paper reviews the evidence for this early interest in and promotion of line graphs and graph paper, a century before the wider uptake of these technologies.

Fostering students' relational understanding of the equals sign is a challenge for math educators that begins in the primary levels and persists into tertiary education. In this paper, I develop an entry point, especially for students who only have an operational understanding of the equals sign, to the core idea of equivalence in linear algebra. My approach is informed by the history of mathematics: In the 17th and 18th centuries, mathematics research underwent an algebraicization, with mathematicians replacing their classical geometric questions with novel algebraic investigations. In this paper, I will offer geometric interpretations of two operations developed at the precipice of this monumental shift: algebraic substitution and Gaussian elimination. I will then utilize Lakoff & Johnson's theory of conceptual metaphor to compare and contrast this historically-grounded geometric re-interpretation of modern linear algebra to the direct algebraic interpretation taken in most modern textbooks.

During the Spanish eighteenth century, a process of modernization took place in scientific knowledge, partly driven by the circulation and appropriation of new scientific ideas. In this context, the Spanish mathematician Benito Bails (1731–1797) published his course Elementos de Matemática (Elements of Mathematics) consisting of ten volumes (1779–1799), in which, among other subjects, he presented one of the most complete mathematical developments of logarithmic calculation methods of his time, by using the infinity through infinite series. The aim of our article is to demonstrate how algebraic analytical reasoning enabled Bails to obtain new and more efficient infinite algorithms that converge more quickly in the computation of logarithms in any system. We show how Euler's number ‘e’ is calculated, probably for the first time in Spanish teaching, in an eighteenth century mathematical text. Our analysis concludes that Bails’ course constituted an innovation and provides evidence of its creativity, originality and ingenuity.

We present the research outcomes of a project in Mathematics Education about the design and implementation of an instrument to learn the integral for the arc length of a function by using differential elements as the strategy for its construction. The research was done via a didactic intervention in a regular Integral Calculus course. The instrument was designed based on historical-epistemological analyses of the works of van Heuraet and Fermat. The main result of this research was that students achieve a more robust conceptualization of the integral for the arc length, supported by its construction with differential elements and its geometric foundation.