In any right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides: a² + b² = c². Known to Babylonian scribes by c. 1800 BCE, 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 Greek mathematicians. One of the most widely known results in mathematics — and one of the most frequently mis-attributed.
The theorem and the attribution problem
In a right-angled triangle with legs a and b and hypotenuse c:
a² + b² = c²
This result is named after Pythagoras of Samos (c. 570–495 BCE). But calling it "Pythagorean" obscures a remarkable history: the result was known — and used — long before Pythagoras, in at least three other civilisations.
The four traditions
Babylon (c. 1800 BCE): The clay tablet Plimpton 322, held at Columbia University, contains 15 rows of numbers that are Pythagorean triples (integers satisfying a² + b² = c²). These were generated systematically — far beyond accident. Babylonian scribes knew and used this relationship well over a millennium before Pythagoras was born.
India (c. 800–600 BCE): The Śulbasūtras — Sanskrit manuals for constructing fire altars — state the theorem explicitly for specific cases and give methods to construct right angles using ropes. The Baudhāyana Śulbasūtra states: "The cord stretched along the diagonal of a rectangle produces an area which the vertical and horizontal sides together produce." This is the Pythagorean theorem in words, for rectangles, c. 800 BCE.
China (c. 300 BCE or earlier): The Zhou Bi Suan Jing (Arithmetical Classic of the Gnomon and Circular Paths) contains a proof of the theorem for the 3-4-5 triangle, attributed to Duke of Zhou (c. 1000 BCE), though the text dates to c. 300 BCE. The Chinese theorem is called the Gōu-Gǔ theorem (勾股定理).
Greece (c. 300 BCE): Euclid gave a rigorous deductive proof in Elements Book I, Proposition 47. This is the proof you probably learned — but it was not the first.
The name "Pythagorean theorem" reflects European mathematical history-writing, not the actual history of the result.
Proofs across traditions
Euclid's proof (Elements I.47)
Euclid's proof uses area arguments involving squares on each side and auxiliary lines. It does not use algebra. The proof constructs the altitude from the right angle to the hypotenuse and shows that each square on a leg equals the corresponding rectangle formed on the hypotenuse. The proof is entirely within Euclid's geometric framework and requires no numerical calculation.
The "rearrangement" proof
This proof — of disputed origin, appearing in India and China independently — arranges four copies of the right triangle in a square of side c, leaving a central square of area c² = (a+b)² − 4 · ½ab = a² + 2ab + b² − 2ab = a² + b². The same proof appears in Bhāskara II's Bījaganita (1150 CE) with the single-word commentary "Behold!" (Paśya). It is also the basis of the Zhou Bi Suan Jing argument.
Algebraic proof (modern)
Drop a perpendicular from the right angle C to the hypotenuse AB at point D. Then triangles ACD and ABC are similar (AA), giving AC/AB = AD/AC, so AC² = AB · AD. Similarly BC² = AB · BD. Adding: AC² + BC² = AB(AD + BD) = AB² — so a² + b² = c².
Plimpton 322 and Babylonian Pythagorean triples
Plimpton 322 (Old Babylonian, c. 1800 BCE) is a table with 15 rows, each containing the hypotenuse and one leg of a Pythagorean triple, along with a ratio column. The triples include (119, 120, 169), (3367, 3456, 4825), (65, 72, 97) — far beyond the common (3,4,5) and (5,12,13) triples. The standard analysis (Neugebauer and Sachs, 1945) identifies a generating formula for these triples. Mansfield and Wildberger (2017) proposed an alternative interpretation via rational trigonometry, which remains contested.
The Śulbasūtras
The Baudhāyana Śulbasūtra (c. 800 BCE) gives the theorem as: "The diagonal of a rectangle produces an area equal to the sum of the areas produced by its two sides." It lists specific cases: (3,4,5), (5,12,13), (8,15,17), (7,24,25) — the same triples appearing in Plimpton 322. Baudhāyana also gives a construction for a square whose area equals the sum of two given squares, which is a geometric version of the theorem.
Count of known proofs
Elisha Loomis catalogued 367 distinct proofs in his 1927 book The Pythagorean Proposition. The number of proofs is now in the hundreds (some lists exceed 400). In 2023, Calcea Johnson and Ne'Kiya Jackson (high school students) published a trigonometric proof not relying on circular reasoning — which had been incorrectly claimed to be impossible.
Generalisations, history, and scholarly debate
Generalisations
Law of Cosines: For any triangle, c² = a² + b² − 2ab cos(C). The Pythagorean theorem is the special case C = 90° (cos 90° = 0).
Inner product spaces: In any inner product space, if vectors u and v are orthogonal (⟨u, v⟩ = 0), then ‖u + v‖² = ‖u‖² + ‖v‖². The Pythagorean theorem is the ℝ² case.
Non-Euclidean geometry: The theorem fails in curved spaces. In spherical geometry, the correct analogue is cos(c) = cos(a)cos(b); in hyperbolic geometry, cosh(c) = cosh(a)cosh(b). The Pythagorean theorem characterises flat (Euclidean) geometry: a space satisfies the theorem if and only if it is flat.
Fermat's Last Theorem: There are no integer solutions to aⁿ + bⁿ = cⁿ for n ≥ 3 (Wiles, 1995). The Pythagorean theorem is the n = 2 case, where infinitely many integer solutions (Pythagorean triples) exist.
Historiographic debate on attribution
Whether Pythagoras or his school actually proved the theorem is unknown. No writing from Pythagoras survives. Ancient testimony (Proclus, c. 450 CE) attributes the result to Pythagoras, but Proclus wrote nearly a millennium after the fact. Burkert (1972) argued, on philological grounds, that the attribution is largely legendary. Van der Waerden (1983) argued for independent discovery across multiple traditions. The consensus among historians of mathematics (Neugebauer, Robson, Plofker, Cullen) is that the relationship between Pythagorean triples was known to Babylonian and Indian mathematicians independently and prior to the Greek tradition, and that Euclid's contribution was a rigorous deductive proof within an axiomatic system — not the discovery of the relationship.
Plimpton 322 — current scholarship
The Neugebauer-Sachs (1945) interpretation — that Plimpton 322 is a table of Pythagorean triples generated from a parametric formula — is standard. The tablet uses the generating formulae p² − q² and 2pq for integer p > q with gcd(p,q) = 1. Robson (2002) in Isis proposed an alternative interpretation as a teacher's aid for reciprocal pairs, disputing that the author intended to generate Pythagorean triples. This remains a live scholarly discussion; the mathematical content — Pythagorean triples — is not in dispute.
Sources
Heath, T.L. (1956). Euclid: The Thirteen Books of the Elements. Vol. 1. Dover. — Critical edition and commentary on Elements I.47 and I.48. Tier 1
Plofker, K. (2009). Mathematics in India. Princeton University Press. pp. 17–23. — Śulbasūtra treatment with source analysis. Tier 1
Robson, E. (2001). Neither Sherlock Holmes nor Babylon: A Reassessment of Plimpton 322. Historia Mathematica, 28(3), 167–206. — Standard current analysis of Plimpton 322. Tier 1
Neugebauer, O. and Sachs, A. (1945). Mathematical Cuneiform Texts. American Oriental Society. — Original Plimpton 322 analysis. Tier 2
Cullen, C. (1996). Astronomy and Mathematics in Ancient China: The Zhou Bi Suan Jing. Cambridge University Press. — Chinese tradition analysis. Tier 2
Loomis, E.S. (1927). The Pythagorean Proposition. National Council of Teachers of Mathematics (1968 reprint). — 367 proofs. Tier 2
Joseph, G.G. (2011). The Crest of the Peacock. 3rd ed. Princeton University Press. pp. 237–248. — Multi-civilisational treatment. Tier 3
Relationships
Generalised by Law of Cosines — Pythagorean theorem is the right-angle special case
Generalised by Pythagorean theorem in inner product spaces — orthogonal vectors
Fails in Non-Euclidean geometry — characterises flat space
Related theorem Fermat's Last Theorem — no integer solutions for aⁿ + bⁿ = cⁿ, n ≥ 3
Early evidence Plimpton 322 (Babylonian, c. 1800 BCE)
Early evidence Baudhāyana Śulbasūtra (Indian, c. 800 BCE)
Early evidence Zhou Bi Suan Jing (Chinese, c. 300 BCE)