Two Remarks on Ancient Greek Geometry
Modern physical theories tell us that space is not euclidean. Yet, to most people, euclidean geometry seems most natural. Furthermore, most mathematical theories of non-euclidean geometry refer back to euclidean geometry in one way or another. For example, in differential geometry, the differential areas are almost always approximately Euclidean. Why is this? I want to suggest here a reason rooted in our biological programming. I base this on two technical results. One, mentioned by Herbert Busemann [2], is an idea of the physicist Helmholtz about the mobility of rigid objects. As Busemann puts it (on p. 51, his emphasis),