Elements

A thirteen-book compilation of geometry and number theory, written in Ptolemaic Alexandria, that organizes earlier Greek mathematics into a single deductive system built from definitions, postulates, and common notions. Its method of rigorous proof from minimal axioms made it the standard mathematics textbook in the West for over two thousand years and the model for deductive reasoning far beyond mathematics.

Euclid opens Book One with twenty-three definitions, five postulates, and five common notions, self-evident claims meant to require no further justification, from which every subsequent proposition is derived through explicit, checkable logical steps; the fifth postulate, concerning parallel lines, proved so much less self-evident than the others that mathematicians spent over two thousand years trying and failing to derive it from the rest, a project that eventually gave rise, in the nineteenth century, to non-Euclidean geometries once the postulate's denial was shown not to be contradictory. The early books cover plane geometry, triangles, and areas, culminating in Book One's Proposition 47, the Pythagorean theorem, proved by Euclid's own method rather than by appeal to Pythagoras's name.

Later books extend the system in directions modern readers might not expect from a geometry textbook: Books Five and Six develop a rigorous general theory of proportion, attributed to the earlier mathematician Eudoxus of Cnidus; Books Seven through Nine treat number theory directly, including Euclid's proof that there are infinitely many prime numbers and the Euclidean algorithm for finding the greatest common divisor of two numbers, still taught today; and Book Ten works through an elaborate classification of irrational magnitudes. The final three books turn to solid geometry, building up to the construction and proof that there are exactly five regular (Platonic) solids, a climax some ancient commentators believed was the work's ultimate organizing goal from the start.

Euclid compiled and systematized rather than discovered most of the individual results, drawing on generations of earlier Greek geometers including Eudoxus, Theaetetus, and the Pythagorean tradition, but the achievement of organizing this material into one deductively airtight structure was entirely his own and became the archetype of rigorous argument for every field that later aspired to mathematical certainty, from Spinoza's Ethics, structured explicitly 'in geometrical order,' to the axiomatic method of modern mathematics and logic itself.