Euclid of Alexandria

Euclid wrote the Elements, the most influential textbook ever produced. Almost nothing is known about him personally: he taught at Alexandria around 300 BC, probably under Ptolemy I, and the two anecdotes attached to him are that he told the king there is no royal road to geometry, and that he had a servant give a coin to a student who asked what he would gain by learning it. The Elements is not an original research work but a systematisation. It gathers the mathematics of Eudoxus, Theaetetus and others and arranges it into thirteen books proceeding from twenty-three definitions, five postulates and five common notions to 465 propositions, each derived only from what precedes it. That structure is the achievement. It is the first sustained demonstration of the axiomatic method — that a large body of certain knowledge can be built from a small number of stated assumptions by explicit steps — and it became the model of rigorous argument for every field that aspired to it. Newton wrote the Principia in that form; Spinoza wrote his Ethics in it. The content covers plane geometry, proportion, number theory — including the proof that there are infinitely many primes and the Euclidean algorithm — and the five regular solids. The fifth postulate, on parallel lines, is longer and less self-evident than the others. Mathematicians tried for two thousand years to derive it from the rest, and the discovery in the nineteenth century that it can be denied consistently produced non-Euclidean geometry and, through it, the mathematics of general relativity. It was used as a school text into the twentieth century.

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