Pythagorean theorem

The theorem that bears Pythagoras's name was in use more than a thousand years before he was born. Babylonian clay tablets from the Old Babylonian period (1900-1600 BCE) demonstrate not merely awareness of the relationship between the sides of a right triangle, but systematic methods for generating Pythagorean triples—sets of three whole numbers where the square of one equals the sum of the squares of the other two.

The tablet Plimpton 322, dating to approximately 1800 BCE and discovered near Larsa in southern Iraq, lists 15 different Pythagorean triples. This was no accident or isolated observation. The tablet represents a method, a systematic approach to finding integer solutions to what we now write as a² + b² = c². The Babylonians had discovered not just the theorem but the underlying mathematics that generates examples of it.

More recently discovered, the tablet Si.427 reveals why Babylonians cared. Ancient land surveyors used Pythagorean triples to create perpendicular boundaries—drawing accurate right angles on the ground. During this period, land was becoming privatized. As one scholar noted, "people started thinking about land in terms of 'my land and your land,' wanting to establish a proper boundary to have positive neighborly relationships." The Pythagorean theorem was a surveying tool before it was philosophy.

The practicality extended beyond boundaries. The same triples could determine layouts for canals, temples, and public buildings. Any construction requiring right angles—and nearly all construction does—benefited from integer solutions that simplified calculation. Using 3:4:5 or 5:12:13 ratios, workers could verify perpendicularity with rope and stakes, no abstract geometry required.

Egypt pursued parallel knowledge. The Berlin Papyrus 6619 (circa 1800 BCE) contains a problem involving squares whose areas sum to a third square, yielding the triple 6:8:10. According to Plutarch, Egyptians associated the 3:4:5 triangle with their gods Osiris, Isis, and Horus—giving mathematical relationships religious significance. The knowledge transmission chain ran from Babylon to Egypt to Greece.

Yet no surviving Greek text from the five centuries after Pythagoras attributes the theorem specifically to him. Modern scholarship increasingly questions whether Pythagoras himself contributed any mathematics at all, suggesting the Pythagorean school may have absorbed existing knowledge and attached their founder's name to it. The theorem's true origin lies not with a single genius but with practical problem-solvers across multiple civilizations who needed to measure land, build structures, and verify right angles.

What makes the theorem universal is its inevitability. Any civilization working with squares and right triangles would eventually notice the relationship. The Babylonians expressed it numerically through tables of triples. The Egyptians encoded it in religious symbolism. The Greeks formalized it as deductive proof. Each approach reflects cultural priorities, but the underlying mathematical truth existed independent of its discoverers—waiting in the geometry of flat space for anyone willing to measure carefully.