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Portrait of Pythagoras' theorem

Science · Mathematics

Pythagoras' theorem

c. 1800 BC – c. 530 BC

Mathematics — Euclidean geometry

The statement that in a right-angled triangle the square on the hypotenuse equals the sum of the squares on the other two sides. Babylonian tablets such as Plimpton 322 list triples that satisfy it more than a thousand years before Pythagoras of Samos, and Chinese and Indian texts state it independently, but it carries his name and appears as Proposition 47 of Book I of Euclid's Elements. Its most famous consequence was disastrous for the Pythagorean brotherhood: the diagonal of a unit square is √2, which cannot be written as a fraction.

How to spot them

  • ▸The square on the hypotenuse equals the sum of the squares on the other two sides
  • ▸Proposition 47 of Book I of Euclid's Elements, sometimes called the 'bride's chair'
  • ▸Known to Babylonian scribes on Plimpton 322 over a thousand years before its namesake
  • ▸Its converse is a test for a right angle, and the 3-4-5 triangle is the builder's version
  • ▸Generalised by the cosine rule, and its failure for cubes and higher powers is Fermat's Last Theorem

Key facts

Proposition 47 of Book I of Euclid's Elements, the 'bride's chair' (c. 300 BC)

Euclid's proof by dissecting the square on the hypotenuse into two rectangles

Plimpton 322, a Babylonian clay tablet listing triples more than a millennium early (c. 1800 BC)

Now at Columbia University; evidence the relation was known long before the Greeks

The discovery that a unit square's diagonal cannot be written as a fraction

The irrationality of √2, traditionally credited to Hippasus and said to have scandalised the brotherhood

The 3-4-5 triangle knotted into a rope to set out a right angle on a building site

The smallest and most useful of the integer triples

The gougu rule of the Chinese Zhoubi Suanjing, with its hsuan-thu diagram

An independent Chinese statement and proof by dissection

US President James Garfield's 1876 proof using the area of a trapezium (1876)

Published while he was a congressman, five years before his assassination

The case n = 2 of the equation that Fermat claimed had no solutions for higher powers

The link to Fermat's Last Theorem: infinitely many solutions for squares, none for cubes and above

Key ideas

  • The converse also holds: if the squares on two sides sum to the square on the third, the triangle is right-angled. That converse is how the theorem is used in practice, to check for squareness.
  • The squares need not be squares — any similar figures erected on the three sides obey the same area relation.
  • Extended to non-right triangles it becomes the cosine rule, with a correction term of −2ab cos C.

Quick-fire facts

  • •It appears as Proposition 47 in Book I of Euclid's Elements, with the converse as Proposition 48
  • •Plimpton 322, a Babylonian tablet of about 1800 BC, lists numbers related to what we now call Pythagorean triples
  • •Pythagoras of Samos lived roughly 570–495 BC and founded a brotherhood at Croton in southern Italy
  • •Hippasus of Metapontum is traditionally said to have been drowned at sea for revealing that √2 is irrational
  • •The irrationality proof supposes √2 = p/q in lowest terms and derives that both p and q must be even — the classic reductio ad absurdum
  • •The smallest integer triples are (3,4,5), (5,12,13), (8,15,17) and (7,24,25)
  • •Euclid gave a formula generating all primitive triples in Book X of the Elements
  • •James A. Garfield published an original proof in 1876, before becoming the twentieth US President
  • •Elisha Scott Loomis's The Pythagorean Proposition (1927) collected 367 distinct proofs
  • •In L. Frank Baum's book and the 1939 film The Wizard of Oz, the Scarecrow garbles the theorem, saying 'isosceles' and 'square roots'
  • •The medieval nickname for the diagram was the 'bride's chair' or, for the difficulty of the proof, the 'pons asinorum' — though that name properly belongs to Proposition 5

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