In the North Atlantic countries that have been the main stage of our story, the nineteenth century was a time of enormous production, not only of manufactured goods but also of art and literature, science, and philosophy. From the great wealth of new ideas in nineteenth-century mathematical physics we can consider only a very small part, chosen mainly for their impact on twentieth-century physics and philosophy. I begin with the new geometries (§4.1), whose emergence is often treated as a chapter in the history of mathematics and its philosophy but which in fact attracted some of the mathematicians who developed them – notably Riemann – chiefly for their potential significance for physics (which Einstein subsequently made good in a wholly unexpected way). The next two sections deal with the concept of field, especially in electrodynamics (§4.2), and with the introduction of chance into thermal physics (§4.3). Finally, we shall take a glance at some nineteenth-century philosophies, which set the tone of twentiethcentury debates (§4.4)

Geometries

Euclid's Fifth Postulate and Lobachevskian Geometry

I can still recall my frustration when, early in my first term of high school geometry, the teacher “proved” – such was his word – Euclid's Theorem 1.29 “by parallel transport”, in effect by sliding a wooden square along a steel ruler pressed on the blackboard.