The Theory of Relativity and Geometry

Abstract

A FUTURE historian of geometry, if pressed for space, might devote to the period between Euclid and Einstein a passage somewhat like the following four paragraphs: (I) The Greeks began the systematic study of objects such as points, lines, planes, polygons, conic sections, and spheres. They discovered how to draw, from a very few assumptions about these objects, an astonishing number of conclusions. Euclid’s assumptions (some of which he never stated explicitly) about points, lines, and planes involved a two-fold idealization of the relations between small dots, rigid rods, and flat boards. First, he neglected the extension of the dots as well as the thickness of the rods and boards. Secondly, he assumed the length of the rods to exceed any finite bound. The Greeks also included in their studies a few curves and surfaces more complicated than the conic sections and the sphere. These turned up in the course of their pursuit of certain geometric hobbies, such as the trisection of angles. Archimedes discovered methods for computing the areas bounded by some curves as well as the slopes of their tangents. But it was not until the eighties of the nineteenth century that the foundation of this postulational or “synthetic” geometry was completed.