Geometry: The Parallel Postulate

Abstract

The first half of the nineteenth century was a time of tremendous change and upheavals all over the world. First the American and then the French revolution had eroded old power structures and political and philosophical belief systems, making way for new paradigms of social organization. The Industrial Revolution drastically changed the lives of most people in Europe and the recently formed United States of America, with the newly perfected steam locomotive as its most visible symbol of progress. The modern era began to take shape during this time (Exercise 1.1). No wonder that mathematics experienced a major revolution of its own, which also laid the foundations for the modern mathematical era. For twenty centuries one distinguished mathematician after another attempted to prove that the geometry laid out by Euclid around 300 B.C.E. in his Elements was the “true” and only one, and provided a description of the physical universe we live in. Not until the end of the eighteenth century did it occur to somebody that the reason for two-thousand years’ worth of spectacular failure might be that it was simply not true. After the admission of this possibility, proof of its reality was not long in coming However, in the end this “negative” answer left mathematics a much richer subject. Instead of one geometry, there now was a rich variety of possible geometries, which found applications in many different areas and ultimately provided the mathematical language for Einstein’s relativity theory.