II . 2 Geometry

Themodern view of geometry was inspired by the novel geometrical theories of hilbert [VI.63] and Einstein in the early years of the twentieth century, which built in their turn on other radical reformulations of geometry in the nineteenth century. For thousands of years, the geometrical knowledge of the Greeks, as set out most notably in euclid’s [VI.2] Elements, was held up as a paradigm of perfect rigor, and indeed of human knowledge. The new theories amounted to the overthrow of an entire way of thinking. This essay will pursue the history of geometry, starting from the time of Euclid, continuing with the advent of non-Euclidean geometry, and ending with the work of riemann [VI.49], klein [VI.57], and poincaré [VI.61]. Along the way, we shall examine how and why the notions of geometry changed so remarkably. Modern geometry itself will be discussed in later parts of this book.