Babylonians used Pythagoras’s rule a millennium early
The Pythagorean theorem says that in a right triangle the square on the hypotenuse equals the sum of squares on the other two sides—a² + b² = c². Named for Pythagoras, born around 570 BC, it may own more distinct proofs than any other theorem. Old Babylonian tablets already used the rule more than a thousand years earlier.
Plutarch, writing centuries later, claimed Egyptians linked the sides of the 3:4:5 triangle with Osiris, Isis and Horus. Plimpton 322, from near Larsa around 1800 BC, lists what read as the sides and diagonals of 15 Pythagorean triples, and the tablet YBC 7289 from about the same time computes the diagonal of a square. India’s Baudhayana Shulba Sutra, dated between the eighth and fifth centuries BC, states the theorem for the isosceles and the general right triangle, as does the Apastamba Shulba Sutra of about 600 BC.
China knows it as the Gougu theorem: the Zhoubi Suanjing, surviving in texts from about the first century BC, reasons it out for the 3-4-5 triangle, and triples appear in the Han-era Nine Chapters. Proclus, in the fifth century AD, credited Pythagoras with a rule that starts from an odd number and yields a triple whose leg and hypotenuse differ by one, and Plato with another.
The book The Pythagorean Proposition collects 370 proofs, a total rivalled perhaps only by quadratic reciprocity. One rearrangement fits four copies of the triangle inside a square of side a + b, so the leftover area equals c squared or, rearranged, a squared plus b squared. Thomas Heath, commenting on Euclid’s Proposition I.47, notes that Bretschneider and Hankel thought Pythagoras might have known that argument. Another proof drops the altitude onto the hypotenuse to make two smaller triangles similar to the original; Euclid may have avoided it because his theory of proportions came later in the Elements.
A dissection sometimes credited to Albert Einstein uses that same altitude without moving any pieces: the two halves share the original triangle’s shape, their hypotenuses are its legs, and their areas add up to the whole. Euclid’s own proof instead splits the big square into two rectangles and matches each to a smaller square through congruent triangles.
Source: Pythagorean theorem
