Euclid's greatest accomplishment was the Elements , his 13-chapter book outlining everything he knew about geometry. He based all of his geometrical theorems on just five postulates, making the work very rigorous and complete, but for two millenia mathematicians wondered if the fifth postulate (the so-called "Parallel Postulate") could in fact be derived from the other four. This was finally answered (in the negative) by Lobachevsky, Bolyai, and Gauss, leading to the branch of mathematics we now call Non-Euclidean Geometry .
Euclid also developed what may have been the first nontrivial algorithm, for computing the greatest common divisor of two natural numbers, and he discovered a marvelous proof that the number of primes is infinite, which is still arguably the clearest proof we know.
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