History of the Pythagorean Theorem

Nobody knows who first discovered the Pythagorean theorem. Babylonian scribes were using the rule a² + b² = c² by about 1800 BCE, more than a thousand years before Pythagoras was born, and it was known in ancient India and China too. The oldest proof that survives is Euclid's, from about 300 BCE. Pythagoras' name stuck because later Greek writers gave him the credit.

This page tells the whole story in one place and links to a detailed page for each chapter. If you want the mathematics first, start with what the Pythagorean theorem says.

The Pythagorean theorem timeline

Dates before about 500 BCE are approximate, and several are debated by centuries. A range such as "c. 800–200 BCE" for the Śulba Sūtras reflects uncertainty about when the texts were composed, not a period of activity.

Who discovered the Pythagorean theorem?

The honest answer is that we don't know, and the question bundles together three different achievements. Historians separate them like this:

1. Using the rule. Finding the diagonal of a rectangle, or a missing side of a right triangle, from the other two sides. Old Babylonian scribes in what is now Iraq did this routinely by about 1800 BCE, and school tablets such as Plimpton 322 and YBC 7289 still survive.
2. Stating it as a general rule. Saying that it works for every right triangle, not just a few examples. The Indian Śulba Sūtras state it for the diagonal of any rectangle, and the Chinese Zhoubi Suanjing explains it through the 3-4-5 triangle.
3. Proving it. Showing, step by logical step, why it must be true. The oldest proof we can still read is Proposition 47 of Book I of Euclid's Elements, written about 300 BCE in Alexandria. Earlier Greek proofs may have existed, but none survives.

Some historians argue that the Babylonian scribes must have had a geometric reason to trust the rule, because they applied it confidently to very large numbers. That may be true, but no Babylonian text that proves it has been found. Egypt is often added to the list of discoverers, usually with a story about rope stretchers and 3-4-5 triangles. The evidence for that story, examined on the Egyptian rope stretchers page, is much thinner than its popularity suggests.

Why is it named after Pythagoras?

Pythagoras of Samos (about 570–490 BCE) founded a philosophical and religious community at Croton in southern Italy. He wrote nothing, and no surviving source from his lifetime or the two centuries after it connects him with the theorem. Plato and Aristotle, who both discussed the Pythagoreans, never mention it.

The connection first appears in two lines of verse by a writer known as Apollodorus the Calculator, whose date is uncertain: when Pythagoras found "that famous diagram", the lines say, he celebrated with a sacrifice of oxen. Cicero, Plutarch and Diogenes Laërtius later repeat the story. In the 5th century CE the philosopher Proclus, commenting on Euclid's proof, reported that writers on ancient history refer the theorem back to Pythagoras. Through Proclus and many centuries of commentary on Euclid, the name passed into modern textbooks.

So the name is a label of tradition, not a verdict of history. In China the same result is called the gougu theorem, after the traditional names of the two legs, and in India it is sometimes called the Baudhāyana theorem.

Before Pythagoras: Babylon, Egypt and India

Babylon, about 1800 BCE

The strongest early evidence comes from clay tablets of the Old Babylonian period. Plimpton 322 lists 15 rows of numbers that match Pythagorean triples, some as large as 12,709, 13,500 and 18,541. YBC 7289, a small round school tablet, shows a square whose diagonal is labeled with a base-60 value of √2 accurate to about six decimal places. Other problem texts use the rule to find the diagonal of a rectangle. Historians of Mesopotamian mathematics agree that the rule was in wide use by this time.

Egypt: less than the legend

Egyptian scribes were skilled with areas, volumes and slopes, and one papyrus from about 1800 BCE asks for two squares whose areas add up to a square of area 100 (the answer is sides 6 and 8). But no surviving Egyptian text states the theorem or uses it on a triangle. The famous claim that Egyptian surveyors made right angles with a 12-knot rope bent into a 3-4-5 triangle was proposed in the early 1880s by the German historian Moritz Cantor, as a guess.

India, first millennium BCE

The Śulba Sūtras are manuals for building Vedic fire altars to exact shapes and sizes. The oldest, attributed to Baudhāyana, is usually dated somewhere between about 800 and 500 BCE, though scholars disagree. It states that the square on the diagonal of a rectangle equals the squares on its two sides together, gives the special case of a square's diagonal, and lists whole-number triples used in altar designs.

Greece: the first surviving proof

Whatever Pythagoras himself did, later Pythagoreans such as Philolaus and Archytas worked seriously on number and proportion. By about 400 BCE Greek mathematicians had also discovered that the diagonal of a square cannot be measured exactly in units of its side, the fact we now express by saying that √2 is irrational.

Around 300 BCE Euclid collected and organized Greek geometry in the Elements. Book I ends with the theorem (Proposition 47) and its converse (Proposition 48). Euclid's proof splits the square on the hypotenuse into two rectangles and shows that each equals the square on one leg; the figure is often called the windmill. Proclus, writing some 750 years later, said that he admired Euclid's clear proof even more than the original discovery, and admired still more Book VI, Proposition 31, which extends the theorem to any similar shapes drawn on the three sides.

China, Baghdad and medieval India

The Chinese Zhoubi Suanjing, a text on astronomy and the gnomon (a vertical shadow stick) compiled during the Han dynasty, contains a conversation in which Shang Gao explains the 3-4-5 right triangle to the Duke of Zhou. In Chinese the shorter leg is the gou, the longer leg the gu and the hypotenuse the xian. A 3rd-century commentary by Zhao Shuang adds a "hypotenuse diagram" that many historians read as a proof, and in 263 CE Liu Hui gave a dissection proof in his commentary on The Nine Chapters on the Mathematical Art. The Chinese proof page animates it.

In 9th-century Baghdad, Thābit ibn Qurra (about 836–901) translated Greek mathematics into Arabic, gave new proofs of Euclid's theorem, and found a version that works for any triangle, not only right triangles. It is a cousin of the law of cosines; see Thābit ibn Qurra's theorem.

In 12th-century India, Bhāskara II (1114–1185) presented a famous dissection of the square on the hypotenuse into four copies of the triangle and a small square in the middle. Popular retellings say his only explanation was the single word "Behold"; the figure and its algebra are on the Bhāskara proof page.

Modern chapters: from Garfield to 2023

• 1876, James A. Garfield. Five years before he became president of the United States, Congressman Garfield published a proof in the New England Journal of Education. It uses a trapezoid built from three right triangles: see Garfield's proof.
• 1927 and 1940, Elisha Scott Loomis. His book The Pythagorean Proposition classified hundreds of proofs; the 1940 second edition holds about 370 (367 is the figure usually quoted). Loomis also asserted that no trigonometric proof is possible, because trigonometry itself rests on the theorem. Browse the proofs of the Pythagorean theorem.
• Albert Einstein, as a boy. In his autobiographical notes Einstein recalled proving the theorem, at around the age of twelve, from the similarity of triangles: see Einstein's proof.
• 1994–95, Andrew Wiles. Around 1637 Pierre de Fermat claimed that aⁿ + bⁿ = cⁿ has no solutions in positive whole numbers when n > 2, unlike the Pythagorean equation with its endless supply of triples. Wiles proved it: see Fermat's Last Theorem.
• 2023, Ne'Kiya Jackson and Calcea Johnson. As high-school students in New Orleans, they presented a trigonometric proof at a meeting of the American Mathematical Society, against the long-standing belief that such proofs were impossible (Jason Zimba had published one in 2009). Their paper "Five or Ten New Proofs of the Pythagorean Theorem" appeared in The American Mathematical Monthly in 2024: see trigonometric proofs.

Explore the history section

Each chapter of the story has its own page:

• HistoryWho Was Pythagoras?Who was Pythagoras of Samos? What we really know about his life, his community at Croton, the music ratios, the ox legend, and whether he proved the theorem.
• HistoryPlimpton 322 and Babylonian Pythagorean TriplesPlimpton 322 is a Babylonian clay tablet from about 1800 BCE listing Pythagorean triples in base 60. Learn to read it, see all 15 rows and weigh the theories.
• HistoryEgyptian Rope Stretchers and the 12-Knot RopeDid Egyptian rope stretchers use a 12-knot rope to make 3-4-5 right angles? What the Rhind and Berlin papyri really show, and how to make a 12-knot rope today.
• HistoryBaudhāyana and the Śulba Sūtras: the Pythagorean Theorem in Ancient IndiaThe Baudhayana theorem in the Sulba Sutras: the Sanskrit rule for a rectangle's diagonal, Vedic altar geometry, triples, the √2 rule 577/408 and who was first.
• HistoryThe Gougu Theorem: the Pythagorean Theorem in Ancient ChinaThe gougu theorem of ancient China: the Zhoubi Suanjing dialogue, what gou, gu and xian mean, Nine Chapters problems solved step by step, and Liu Hui's work.
• HistoryThābit ibn Qurra's Generalization of the Pythagorean TheoremThabit ibn Qurra's theorem extends the Pythagorean theorem to every triangle: AB² + AC² = BC·(BB′ + CC′). Statement, animation, proof and 9th-century Baghdad.