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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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II.2 Geometry

Jeremy Gray

1 Introduction

The modern 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], ...

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