Pythagorean theorem

• 1. Introduction

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  The Pythagorean theorem states something almost absurdly simple: in any right triangle, the square drawn on the hypotenuse has the same area as the two squares drawn on the other two sides combined. That is the whole claim. And yet this relationship has been proved more times, by more different methods, than possibly any other theorem in all of mathematics. A book called The Pythagorean Proposition catalogues 370 separate proofs. One of those proofs was devised by a sitting member of the United States Congress named James A. Garfield, who had not yet become president. Another is sometimes attributed to Albert Einstein. The theorem carries the name of Pythagoras, a Greek philosopher born around 570 BC, even though the mathematical knowledge behind it was in circulation in Egypt and Mesopotamia well over a thousand years before he was born. How did one man's name come to dominate a discovery that belongs to the whole ancient world? And why does a statement about triangles turn out to underlie Euclidean distance, complex numbers, the least squares method in statistics, and geometry on the surface of a sphere?

• 2. Before Pythagoras

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  Written around 1800 BC, the Egyptian Middle Kingdom document known as the Berlin Papyrus 6619 poses a problem involving two squares whose areas sum to a third square. The solution uses the triple 6:8:10. The papyrus does not mention a triangle at all, yet the relationship is unmistakably there. Plutarch, writing many centuries after the fact, reported that the ancient Egyptians knew the 3:4:5 right triangle well enough to assign its three sides to the gods Osiris, Isis, and Horus. In Mesopotamia, historians of mathematics have concluded that the Pythagorean rule was in widespread use throughout the Old Babylonian period, which ran from the 20th to the 16th centuries BC. The tablet Plimpton 322, also written near the city of Larsa around 1800 BC, contains entries that can be read as the sides and diagonals of 15 different Pythagorean triples. A companion tablet, YBC 7289, works out the diagonal of a square directly. In India, the Baudhayana Shulba Sutra, dated variously between the 8th and 5th century BC, lists Pythagorean triples and states the theorem both for the special case of an isosceles right triangle and in the fully general form. The Apastamba Shulba Sutra, from around 600 BC, does the same. In China, the text known as the Zhoubi Suanjing works through a geometric reasoning for the 3:4:5 triangle, and some scholars believe the theorem was known there as early as the 11th century BC, attributed to an astronomer and mathematician named Shang Gao. The Chinese name for it today is the Gougu theorem.

• 3. The Name and the Man

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  Thomas L. Heath, who lived from 1861 to 1940 and produced the definitive English commentary on Euclid, observed that no surviving Greek text from the five centuries after Pythagoras explicitly credits him with the theorem. What exists instead is inference: when later writers like Plutarch and Cicero mentioned Pythagoras in connection with the result, they did so as though the attribution was already so well established it needed no argument. The classicist Kurt von Fritz addressed the attribution directly. He wrote that whether the formula is rightly credited to Pythagoras personally, one can safely assume it belongs to the very oldest period of Pythagorean mathematics. That is a careful formulation. It grants the Pythagorean school priority without granting any individual certainty. The philosopher Proclus, a Byzantine Neoplatonist writing in the fifth century AD, described two arithmetic rules for generating Pythagorean triples: one attributed to Pythagoras and one to Plato, who lived from 428/427 or 424/423 BC until 348/347 BC. The rule traced to Pythagoras starts from an odd number and produces a triple in which the leg and hypotenuse differ by exactly one unit. Recent scholarship has pushed further, casting increasing doubt on any direct creative role for Pythagoras in mathematics at all, though that debate has not been settled. What is settled is that around 300 BC, in Euclid's Elements, the oldest surviving axiomatic proof of the theorem was written down, along with Euclid's formula for generating all primitive Pythagorean triples.

• 4. An Embarrassment of Proofs

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  Euclid's proof, recorded as Proposition 47 of Book I of the Elements, proceeds by dividing the large square on the hypotenuse into two rectangles and then showing, through a careful chain of congruent triangles and shared altitudes, that each rectangle has exactly the same area as one of the two smaller squares on the legs. It relies on four elementary results: the side-angle-side criterion for congruent triangles, the fact that a triangle's area is half that of any parallelogram sharing its base and height, and two results about the areas of rectangles and squares. Heath, in his commentary on this very proposition, speculated that Pythagoras may have known a different proof based on the rearrangement of squares. That rearrangement approach is the one most people encounter in school: two identical large squares, each filled with four copies of the same right triangle arranged differently, leaving behind either one square of area c-squared or two squares of areas a-squared and b-squared. Because both leftovers come from the same large square, they must be equal. A dissection proof sometimes attributed to Einstein drops a perpendicular from the right-angle vertex to the hypotenuse, splitting the original triangle into two smaller ones. Each smaller triangle is similar to the original, and because similar figures scale by the square of their linear dimensions, the areas add up in exactly the right proportion. The proof using similar triangles was considered so elegant that historians have wondered why Euclid did not use it. One answer is that the argument requires a developed theory of proportions, a topic Euclid does not introduce until later in the Elements. James Garfield's proof, published while he was a U.S. representative and before his election to the presidency, used a trapezoid rather than a square, halving the area calculation partway through by bisecting along a diagonal. A proof using differential calculus treats the hypotenuse as a function of one leg and integrates the resulting differential equation directly.

• 5. Triples, Irrationals, and a Drowning

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  A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the theorem exactly, meaning all three sides of the right triangle are whole numbers. The most familiar examples are (3, 4, 5) and (5, 12, 13). A primitive triple is one where the three numbers share no common divisor. Euclid's formula, the most well-known method for generating them, takes any two positive integers m and n and produces a triple directly. The Pythagorean school treated numbers as fundamentally whole, understanding proportions through the comparison of integer multiples of a shared unit. The theorem broke that assumption. By the same reasoning that gives integer triples, it guarantees the existence of lengths that cannot be expressed as a ratio of two whole numbers at all. The hypotenuse of a right triangle whose two legs each measure one unit has length equal to the square root of two, which is irrational. One legend says that a member of the Pythagorean circle named Hippasus of Metapontum, dated to around 470 BC, was drowned at sea for revealing this fact publicly. Whether the story is literally true is uncertain, but the scholar Kurt von Fritz wrote a careful study of Hippasus's contributions. The practical implication of irrational lengths is that a straightedge and compass can construct them without anyone needing to name them as a fraction: the theorem's square-root operation builds incommensurable lengths geometrically even when arithmetic cannot pin them down.

• 6. Where the Theorem Lives Today

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  Euclidean distance in any number of dimensions is a direct extension of the theorem. In a flat plane with Cartesian coordinates, the distance between two points is the square root of the sum of the squares of the differences in each coordinate, which is the theorem stated one step removed. In n-dimensional Euclidean space the formula is identical in structure. The squared version of this distance, called the squared Euclidean distance, avoids the square root entirely and is a smooth, convex function of both points; it forms the basis of the least squares method used across optimization theory and statistics. Complex numbers use the theorem to define the absolute value or modulus of any complex number: the three quantities modulus, real part, and imaginary part satisfy the Pythagorean equation. The theorem also connects to the cross product and dot product of vectors through the Pythagorean trigonometric identity, and this connection can be used to define the cross product in seven dimensions. In Riemannian geometry the theorem appears inside the metric tensor, the object that measures distances in curved space and which may vary from point to point. The computer scientist Edsger W. Dijkstra restated the theorem's converse, classifying any triangle as right, acute, or obtuse by the sign of the difference between the square of its longest side and the sum of the squares of the other two. Outside flat Euclidean space the theorem fails: on the surface of a sphere, a right triangle whose two legs each have length pi-divided-by-2 also has a hypotenuse of that same length, which no Euclidean triangle could produce. De Gua's theorem, named for Jean Paul de Gua de Malves, extends the logic to three dimensions: for a tetrahedron with a right-angle corner, the square of the area of the face opposite that corner equals the sum of the squares of the areas of the other three faces.