Composed by Euclid of Alexandria c. 300 BCE. Thirteen books of geometry, number theory, and solid geometry, all derived from just 5 postulates and 5 common notions. The most widely read mathematics text in history after the Bible — in continuous use for over 2,300 years. Contains the first proof that there are infinitely many primes and the algorithmic procedure now called the Euclidean algorithm.
Around 300 BCE, a mathematician called Euclid, working in Alexandria, Egypt, wrote a book called the Elements. It is probably the most successful mathematics textbook ever written — used in schools for over 2,300 years, right up to the 20th century.
What made it extraordinary was its method. Instead of just listing facts about shapes, Euclid started with just 10 simple statements that seemed obviously true, and then — step by step — proved everything else from those starting points. It was the first time anyone had systematically built all of geometry from first principles.
The Elements was the model for rigorous mathematical proof for 2,000 years. Newton wrote his Principia Mathematica in the same style. Lincoln memorised it to improve his reasoning. It was translated into Arabic, Latin, Chinese, Hebrew, and dozens of other languages. Even today, when mathematicians write proofs, they are following a tradition Euclid established.
Euclid begins with 10 starting assumptions: 5 postulates (geometric) and 5 common notions (logical). Everything in the 13 books follows from these by deductive argument.
| # | Postulate |
|---|---|
| 1 | A straight line can be drawn from any point to any other point |
| 2 | A finite straight line can be extended indefinitely |
| 3 | A circle can be drawn with any centre and radius |
| 4 | All right angles are equal |
| 5 | (Parallel postulate) Through a point not on a line, exactly one parallel line exists |
Suppose there are finitely many primes p₁, p₂, …, pₙ. Consider N = (p₁ × p₂ × … × pₙ) + 1. Either N is prime (a new prime not in the list), or N has a prime factor. But that factor cannot be any of p₁, …, pₙ (since dividing N by any of them leaves remainder 1). Either way, there is a prime not in the original list — contradicting the assumption. Therefore the list of primes is infinite.
Euclid's fifth postulate (the parallel postulate) seemed less obvious than the others. For 2,000 years, mathematicians tried to prove it from the first four. In the 1820s, Bolyai, Lobachevsky, and Gauss independently showed it was independent — you could replace it with a different assumption and get a consistent but different geometry. This was the birth of non-Euclidean geometry.
Remarkably little is known about Euclid himself. He almost certainly lived and worked in Alexandria c. 300 BCE, during the reign of Ptolemy I Soter. The statement that he told Ptolemy "there is no royal road to geometry" (recorded by Proclus, c. 450 CE) is probably apocryphal. No original manuscripts survive — the oldest surviving copies date to the 10th century CE Byzantine period. The standard edition used for 1,500 years was that of Theon of Alexandria (c. 390 CE), whose revisions are now separable via a Vatican manuscript (Vat. gr. 190) that predates Theon.
The standard modern scholarly edition is T.L. Heath (1908, Dover reprint 1956), which provides the Greek text, English translation, and an extensive commentary on the mathematical and historical context of each proposition. Heath's commentary identifies logical gaps in Euclid — places where the argument implicitly uses results not yet established. These gaps were filled by Hilbert's rigorous axiomatization in the Grundlagen der Geometrie (1899).
David Hilbert showed that Euclid's geometry requires additional axioms Euclid did not state explicitly: axioms of order (betweenness), axioms of congruence, the axiom of continuity, and the parallel axiom. Hilbert's Grundlagen der Geometrie gives a complete, rigorous axiomatization of Euclidean geometry — the modern replacement for Euclid as a foundation, though Euclid remains the best introduction.
The Elements was translated into Arabic by al-Ḥajjāj ibn Yūsuf (c. 800 CE) under Hārūn al-Rashīd, and again c. 820 CE for Al-Maʾmūn. The Arabic tradition produced important commentaries (al-Nayrīzī, c. 900 CE). The text returned to Europe via Latin translation by Adelard of Bath (c. 1120 CE) from the Arabic, and by Gerard of Cremona (c. 1175 CE). The first printed edition appeared in Venice in 1482 — the first mathematical book ever printed. It was one of the most frequently reprinted books of the 15th–19th centuries.