---
tag: 83WHDAV68J
kind: object
name: Euclid's Elements
msc: [01A20, 51M04, 51-03]
uses: [0VP3A1290K, 0TRNT4X7J0, 0292EPPPT0, 017E5PRS1Q]
sources:
  - url: https://en.wikipedia.org/wiki/Euclid%27s_Elements
same_as:
  wikidata: Q172891
  wikipedia: Euclid's Elements
---

## Description

The *Elements* is a mathematical treatise in thirteen books, written in Greek by Euclid of Alexandria around 300 BC.
Books I to VI treat plane geometry, and Books VII to IX treat the arithmetic of whole numbers.
Book X treats incommensurable magnitudes, and Books XI to XIII treat solid geometry.
Book I starts from definitions, five postulates and a list of common notions, and derives every proposition from these and from earlier propositions.
The fifth postulate of Book I is the [[parallel postulate|2PC3XY0PYZ]].
Book I ends with the [[Pythagorean theorem|0VP3A1290K]] and its converse.
Book VII contains the [[Euclidean algorithm|0TRNT4X7J0]], Book IX contains [[Euclid's theorem|0292EPPPT0]] that there are infinitely many primes, and Book XIII constructs the five [[Platonic solids|017E5PRS1Q]].

## Remarks

The book is the model of the axiomatic method: a short list of first principles and a chain of proofs.
Its axioms are not complete by modern standards; for example, the proof of its first proposition assumes that two circles meet without an axiom that says so.
Modern axiom systems such as [[Hilbert's axioms|KM1FB8XZ2R]] fill these gaps.

## Explanations

### L1 human

The *Elements* is the oldest mathematics textbook that is still read.
It starts from a few simple rules, such as "a straight line can be drawn between any two points", and proves hundreds of facts about figures and numbers from them.
For two thousand years, students learned geometry and the idea of a proof from it.