Almost nothing reliable is known about Euclid's own life beyond that he worked at Alexandria during the reign of Ptolemy I. His work, however, has been clear for centuries: he gathered the achievements of Greek mathematics from the Pythagoreans through Eudoxus and Theaetetus and presented them in a new form — beginning from definitions, postulates, and common notions, and deriving every subsequent theorem only from what had already been proved.

The Elements consists of 13 books. The first six treat plane geometry (the Pythagorean theorem, parallel lines, triangles); books 7–9 cover number theory (the proof that there are infinitely many primes is here); book 10 deals with irrational magnitudes; and the last three develop solid geometry and the five Platonic solids. The method outlasted the content: a proposition is established not by authority, intuition, or observation, but only by an argument resting on earlier steps. This idea became the ground not just of mathematics but of nearly all later rational thought.

The text quickly became canonical in the ancient world. Greek copies were multiplied throughout the Roman period and, in the 8th–9th centuries, translated into Arabic at the House of Wisdom in Baghdad; al-Hajjaj, Thabit ibn Qurra, and later Nasir al-Din al-Tusi commented on it and tried to fill what they saw as its gaps (above all the 'parallel postulate'). In the 12th century Adelard of Bath translated it from Arabic into Latin, and the Elements became the basic mathematics text of the European universities. Even Newton's Principia (1687) was deliberately written in Euclidean axiomatic style.

In the 19th century Lobachevsky, Bolyai, and Riemann showed that the fifth postulate could be denied, opening non-Euclidean geometries — one of which underlies Einstein's general relativity. This did not refute Euclid; it enlarged the horizon he had defined. The real legacy of the Elements is not the truth of one particular geometry but the practice of thinking by proof itself.