# Pythagorean Theorem

**Type:** Theorem  
**Domain:** Named Results · Geometry  
**Codex URL:** /mathematics/named-results/pythagorean-theorem/  
**Entry status:** Live — v1.0 (2026-05-27)

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## Statement

**a² + b² = c²**

In any right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides.

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## Curious Level

Known to Babylonian scribes by c. 1800 BCE (Plimpton 322), to Indian mathematicians in the Śulbasūtras by c. 800 BCE, to Chinese mathematicians in the Zhou Bi Suan Jing by c. 300 BCE, and proved deductively by Euclid (Elements I.47) c. 300 BCE. The name "Pythagorean" reflects European mathematical historiography, not the actual history.

**Four traditions:**

- **Babylon (c. 1800 BCE):** Plimpton 322 contains 15 rows of Pythagorean triples, generated systematically.
- **India (c. 800 BCE):** Baudhāyana Śulbasūtra: "The diagonal of a rectangle produces an area equal to the sum of the areas produced by its two sides." Lists specific triples (3,4,5), (5,12,13), (8,15,17), (7,24,25).
- **China (c. 300 BCE):** Zhou Bi Suan Jing — Gōu-Gǔ theorem (勾股定理), proof for 3-4-5 triangle.
- **Greece (c. 300 BCE):** Euclid, Elements I.47 — first rigorous deductive proof in an axiomatic system.

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## Exploring Level

### Proofs

**Euclid's proof (Elements I.47):** Area argument using squares on each side and auxiliary lines. Constructs altitude from right angle to hypotenuse; shows each leg-square equals corresponding hypotenuse-rectangle.

**Rearrangement proof:** Arrange four copies of the triangle in a square of side c. Central square has area = c² = (a+b)² − 2ab = a² + b². This proof appears in Bhāskara II's *Bījaganita* (1150 CE) with commentary "Behold!" (Paśya), and in the Zhou Bi Suan Jing.

**Similar triangles proof:** Drop altitude from right angle C to hypotenuse AB at D. Triangles ACD ~ ABC (AA), giving a² + b² = c².

Loomis (1927) catalogued 367 proofs. Over 400 are now known.

### Plimpton 322

Old Babylonian tablet, c. 1800 BCE. Contains hypotenuse and one leg of 15 Pythagorean triples, including (119,120,169), (3367,3456,4825), (65,72,97). Standard analysis: Neugebauer-Sachs 1945, Robson 2001.

### Śulbasūtras

Baudhāyana Śulbasūtra (c. 800 BCE): theorem stated for rectangles, with specific triples. Constructions for squaring the sum of two squares. Used for Vedic fire altar construction.

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## Deep Dive

### Generalisations

- **Law of Cosines:** c² = a² + b² − 2ab cos C. Pythagorean theorem is C = 90° case.
- **Inner product spaces:** ‖u + v‖² = ‖u‖² + ‖v‖² when ⟨u,v⟩ = 0.
- **Non-Euclidean geometry:** Fails in curved spaces. The theorem characterises flat (Euclidean) geometry.
- **Fermat's Last Theorem:** No integer solutions for aⁿ + bⁿ = cⁿ with n ≥ 3 (Wiles, 1995). The n = 2 case has infinitely many solutions.

### Historiography

No writing from Pythagoras survives. Attribution comes from Proclus (c. 450 CE) — nearly a millennium later. Burkert (1972) argued the attribution is largely legendary. Consensus among historians of mathematics (Neugebauer, Robson, Plofker, Cullen): the relationship was known to Babylonian and Indian mathematicians independently and prior to Greek mathematics; Euclid's contribution was rigorous deductive proof within an axiomatic system.

### Plimpton 322 — current scholarship

Neugebauer-Sachs (1945): generating formulae p² − q² and 2pq for triples. Robson (2002, *Isis*): alternative interpretation as teacher's aid for reciprocal pairs. Mansfield-Wildberger (2017): rational trigonometry interpretation (contested). The mathematical content — Pythagorean triples — is not disputed.

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## Sources

### Tier 1
- Heath, T.L. (1956). *Euclid: The Thirteen Books of the Elements*. Vol. 1. Dover.
- Plofker, K. (2009). *Mathematics in India*. Princeton University Press. pp. 17–23.
- Robson, E. (2001). Neither Sherlock Holmes nor Babylon. *Historia Mathematica*, 28(3), 167–206.

### Tier 2
- Cullen, C. (1996). *Astronomy and Mathematics in Ancient China: The Zhou Bi Suan Jing*. Cambridge University Press.
- Loomis, E.S. (1927). *The Pythagorean Proposition*. NCTM (1968 reprint).

### Tier 3
- Joseph, G.G. (2011). *The Crest of the Peacock*. 3rd ed. Princeton University Press. pp. 237–248.

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