Hippocrates of Chios was a Greek mathematician of the late 5th century BC. The biographical sources — Aristotle’s brief mentions, Eudemus of Rhodes’s (lost) History of Geometry as preserved through the 6th-century commentator Simplicius, and Iamblichus’s 4th-century AD Pythagorean Life — are sparse and partly contradictory. The standing reconstruction is that he was born on the Aegean island of Chios around 470 BC, was originally a merchant who was substantially defrauded of his cargo at Byzantium (or, in another tradition, at Athens by customs officers), and turned to mathematics in middle age while trying to recover his finances through teaching at Athens.
He was not the famous physician Hippocrates of Cos (c. 460–375 BC), to whom the Hippocratic Oath is attributed. The two men were contemporaries and were occasionally confused by later classical sources, but were substantively unrelated.
The first geometry textbook
Hippocrates is credited by Eudemus as the author of the first written Elements of Geometry — substantively the first systematic textbook of Greek geometry. The text is lost; the standing assumption is that approximately a century later Euclid’s substantially more famous Elements (c. 300 BC) drew heavily on Hippocrates’s original framework, in the same way that Euclid drew on Eudoxus’s later proportion theory.
The Hippocratean Elements is the earliest documented attempt at the systematic deductive presentation of Greek geometry — the methodological framework that Euclid’s Elements would substantively complete and that would substantively define European mathematical proof through the next two thousand years.
The lune
Hippocrates’s most lasting individual contribution was the quadrature of a specific curved figure called a lune — a crescent-shaped region bounded by two circular arcs. The classical Greek mathematical problem of quadrature asked: can you construct, with straightedge and compass alone, a square whose area equals the area of a given figure? For straight-sided figures (triangles, polygons) the construction is routine. For circular regions the problem becomes severe; the most famous instance is the quadrature of the circle, which had been the unresolved mathematical challenge of Hippocrates’s generation.
Hippocrates did not solve the quadrature of the circle (the problem was eventually shown to be impossible in 1882 when Ferdinand von Lindemann proved that π is transcendental). What he did do was solve the quadrature of a specific lune: he showed that the lune bounded by the upper semicircle of a triangle’s hypotenuse and the larger arc of the circumscribed circle has exactly the area of the right-angled isosceles triangle inside it. The proof uses the Pythagorean theorem and substantively the proposition that the areas of similar circular segments stand in the same ratio as the squares of their chords.
The achievement was substantively the first computation of the exact area of a region bounded by curves in European mathematical history. It opened the path that Eudoxus’s method of exhaustion and Archimedes’s geometric integration methods would substantively complete two centuries later.
What was lost
Almost all of Hippocrates’s original work was lost. The surviving fragments amount to perhaps fifteen pages of substantively reconstructed mathematical content, transmitted through later commentators (most importantly Simplicius’s 6th-century AD commentary on Aristotle’s Physics, which preserves the most substantively detailed surviving account of the lune proof). The original Elements of Geometry substantively did not survive to medieval Byzantium; the substantive Greek mathematical tradition of his successors had substantively absorbed and surpassed his work and the original treatise had substantively dropped out of the regular copying-and-teaching tradition by approximately the 3rd century AD.
The date of his death is unknown. He had stopped publishing by approximately 410 BC.