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Every student meets the Pythagorean theorem in middle school, and most assume Pythagoras discovered and proved it. The historical record says something different, and the real story is more useful: it shows the same idea turning up in three civilisations for three different reasons. This post sets out what the evidence supports and gives you three checks you can do with a pencil.
What do we know about Pythagoras himself?
Less than you might think. The Stanford Encyclopedia of Philosophy entry on Pythagoras says he lived from about 570 to about 490 BCE, grew up on the island of Samos and later moved to Croton in southern Italy. He was famous in his own lifetime, but mainly as a teacher of a way of life and of ideas about the soul.
On the theorem, the SEP is blunt. None of the early sources, including Plato and Aristotle, link Pythagoras to it. The link appears almost a thousand years later, when Proclus, in the fifth century CE, reports that "those who wish to investigate ancient history" refer the theorem to Pythagoras and say he sacrificed an ox on discovering it. Even Proclus credits Pythagoras only with "knowing the truth of the theorem", and contrasts him with Euclid, who gave the proof. In the SEP's words, "there is not a jot of evidence for a proof by Pythagoras".
How old is the rule? The Babylonian evidence
Much older than Pythagoras. The SEP entry notes that the Babylonians "were aware of the basic Pythagorean rule and could generate Pythagorean triples" long before him, though they never stated or proved it as a general theorem.
The most famous evidence is a clay tablet called Plimpton 322, now at Columbia University. In Mansfield and Wildberger (2017), published in Historia Mathematica, the authors note that it has been dated by its script to between 1822 and 1762 BCE, roughly the time of Hammurabi. It has fifteen rows of numbers written in base 60. Their paper argues it is an exact table of ratios, a kind of trigonometric table, which other historians dispute. What nobody disputes is that the numbers fit the rule.
Check 1: the first row of Plimpton 322. The tablet's first row gives a short side and a diagonal, written in base 60 as 1,59 and 2,49.
Convert: 1,59 = 1 × 60 + 59 = 119. And 2,49 = 2 × 60 + 49 = 169.
Square: 169² = 28,561 and 119² = 14,161.
Subtract: 28,561 − 14,161 = 14,400.
14,400 = 120². So the triangle has sides 119, 120 and 169, a right triangle with whole-number sides.
You can generate this triple with the modern formula m² − n², 2mn, m² + n². With m = 12 and n = 5 you get 144 − 25 = 119, 2 × 12 × 5 = 120, and 144 + 25 = 169. Whether the Babylonian scribe used anything like this is a separate, open question.
What do the Indian Sulbasutras say?
The Sulbasutras are Vedic texts with rules for building fire altars to exact shapes and areas. The MacTutor article on the Indian Sulbasutras dates the Baudhayana Sulbasutra to about 800 BC and the Katyayana Sulbasutra to about 200 BC, while noting that we know almost nothing about their authors.
The rule is stated with ropes, because ropes were the measuring tools. In MacTutor's translation, the Baudhayana text explicitly gives the special case of a square: "The rope which is stretched across the diagonal of a square produces an area double the size of the original square." The Katyayana text gives the general version for a rectangle: the rope along the diagonal "produces an area which the vertical and horizontal sides make together". MacTutor also lists triples used in these texts, including (5, 12, 13), (8, 15, 17) and (12, 35, 37).
Check 2: Baudhayana's square root of 2. The square case needs the diagonal of a unit square, √2. The MacTutor biography of Baudhayana reports his approximation √2 ≈ 1 + 1/3 + 1/(3 × 4) − 1/(3 × 4 × 34) = 577/408.
Square it: 577² = 332,929 and 408² = 166,464.
2 × 166,464 = 332,928, so 577²/408² = 2 + 1/166,464.
That is about 2.000006. The approximation is correct to five decimal places, as MacTutor states.
Who wrote down the first proof?
The first proof that survives is in Euclid's Elements: Book I, Proposition 47. Euclid builds squares on all three sides and shows that the two smaller squares exactly fill the large one, using congruent triangles. Our post on Euclid's Elements explains why that style of proof mattered.
Check 3: a proof you can draw. Here is a short algebraic proof, not Euclid's. Arrange four copies of a right triangle with legs a and b inside a square of side a + b, so that their hypotenuses form a tilted square of side c in the middle.
Area of big square: (a + b)² = a² + 2ab + b².
The same area as pieces: four triangles, 4 × ½ab = 2ab, plus the inner square c².
Set them equal: a² + 2ab + b² = 2ab + c², so a² + b² = c².
A common mistake is to use the theorem on a triangle that is not right-angled. If you only know three sides, test first: in a triangle with sides 7, 8 and 11, 7² + 8² = 113, which is less than 11² = 121, so the angle opposite 11 is obtuse, not right.
What can a student take from this history?
Ask for the evidence behind a name. Many results are named after someone who did not discover them. Knowing this makes you a careful reader of any claim.
Notice that need drives maths. Babylonian scribes, Vedic altar builders and Greek geometers reached the same rule for accounting, ritual and proof.
Check numbers yourself. You just verified a 3,800-year-old tablet with subtraction.
For practice, generate five triples with m² − n², 2mn, m² + n², then prove one with the four-triangle picture. If a geometry problem uses the theorem in a hidden way, the Fermi Problem Library has a step-by-step solver that explains the reasoning, which helps you see where the right angle was.
Next step: check whether (65, 72, 97) is a Pythagorean triple, then find the m and n that produce it.
