Plimpton 322, the gougu and the many proofs

The relation $a^2+b^2=c^2$ between the sides of a right-angled triangle appears, in various forms, in all the great mathematical civilisations of antiquity, long before the Pythagorean school gave — most likely — its first general proof.

The Babylonian tablet Plimpton 322

The oldest and most surprising piece of evidence is the Babylonian clay tablet known as Plimpton 322, written in cuneiform around 1800 BC. Its columns of numbers, recorded in the sexagesimal (base-60) system, contain fifteen rows that read as Pythagorean triples: alongside simple cases appear huge numbers, such as the triple $(119,\,120,\,169)$ or even $(3367,\,3456,\,4825)$, impossible to find by trial and error. The Babylonian scribes therefore possessed a systematic method for generating right triangles with whole-number sides, more than a thousand years before Pythagoras (Katz, Boyer).

The Chinese gougu

In China the same theorem is called gougu (勾股), from the names of the two legs: gou the short side, gu the long side, while the hypotenuse is called xian. In the classic Zhoubi Suanjing (around the 1st century BC) the commentator Zhao Shuang attaches the celebrated hypotenuse diagram: four equal right triangles arranged around a small central square make up a large square of side $c$. A simple computation of areas, $c^2 = 4\cdot\frac{ab}{2} + (b-a)^2 = 2ab + b^2 - 2ab + a^2 = a^2+b^2,$ proves the theorem by pure dissection, with no need for proportions (Katz).

Euclid and the hundreds of proofs

In the Greek world the proof enters Euclid’s Elements as Euclid’s Proposition I.47, its figure nicknamed in the Middle Ages the pons asinorum or “bride’s chair”: the altitude to the hypotenuse cuts the large square into two rectangles, each equivalent to the square built on one leg — exactly the proof by equivalence we still use today. It is perhaps the most-proved theorem in history: in the book The Pythagorean Proposition Elisha Loomis collected 371 distinct proofs, among them the one found in 1876 by James Garfield, a future president of the United States, based on the area of a trapezoid (Maor, Dunham).

Links

Topics: Equivalence and Pythagoras
Concepts: Pythagoras’ theorem · Pythagorean triples · Equivalence · Equidecomposability
People: Euclid · Zhao Shuang · Elisha Loomis · James Garfield