New paper in the De Morgan Gazette:
Abstract: The nine-point circle is established by Euclidean means; the nine-point conic, Cartesian.Cartesian geometry is developed from Euclidean by means of Thales’ s Theorem. A theory of proportion is given, and Thales’s Theorem proved, on the basis of Book I of Euclid’s Elements, without the Archimedean assumption of Book V. Euclid’s theory of areas is used, although this is obviated by Hilbert’s theory of lengths. It is observed how Apollonius relies on Euclid’s theory of areas. The historical foundations of the name of Thales’s Theorem are considered. Thales is thought to have identified water as a universal substrate; his recognition of mathematical theorems as such represents a similar unification of things.