# Euclid's Elements

**Type:** Work  
**Domain:** Geometry  
**Composed:** c. 300 BCE  
**Author:** Euclid of Alexandria  
**Books:** 13  
**Codex URL:** /mathematics/geometry/euclids-elements/  
**Entry status:** Live — v1.0 (2026-05-27)

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## Summary

13-book axiomatic treatment of geometry and number theory built from 5 postulates and 5 common notions. Most widely used mathematics textbook in history — in continuous use for 2,300 years. Contains first proof of infinitely many primes (IX.20), the Euclidean algorithm (VII.2), proof of the Pythagorean theorem (I.47), and proof that there are exactly 5 regular polyhedra (XIII.18).

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## 5 Postulates

1. A line can be drawn between any two points
2. A line segment 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 exists

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## Key results

- **I.47** — Pythagorean theorem
- **VII.2** — Euclidean algorithm (GCD)
- **IX.20** — Infinitely many primes (Euclid's proof)
- **X.9** — √2 is irrational
- **XIII.18** — Exactly 5 regular polyhedra

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## Sources

### Tier 1
- Heath, T.L. (1956). *Euclid: The Thirteen Books of the Elements*. 3 vols. Dover.
- Hilbert, D. (1902). *The Foundations of Geometry*. Open Court.

### Tier 2
- Artmann, B. (1999). *Euclid: The Creation of Mathematics*. Springer.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*