The two Greek Hippocrates of the late 5th century BC were near-contemporaries. They had the same first name. They were both prominent in technical-intellectual fields. They were occasionally confused even by ancient sources. They were entirely unrelated.
Hippocrates of Cos (c. 460–375 BC) was the founder of the European medical tradition. He was born on the Aegean island of Cos to a family of Asklepiad physicians (the substantial hereditary medical guild that operated the temple-medical complex at the Asklepieion of Cos through the substantial classical period). He trained at the Asklepieion, practised at Larissa in Thessaly for most of his adult life, and produced — substantively in collaboration with a substantial team of students and successors — the Hippocratic Corpus: approximately 60 substantial Greek medical treatises that defined European medical theory and practice for the following two thousand years.
The most famous single text of the Corpus is the Hippocratic Oath, the ethical-professional pledge that European and American medical graduates have substantively recited (in various modernised forms) since the medieval and Renaissance medical-school revivals. The Oath’s specific provisions — patient confidentiality, prohibition of harm, prohibition of administering poisons, prohibition of sexual contact with patients — substantively defined the European medical-ethical framework for two and a half millennia.
The Hippocratic medical theoretical framework — the four humours (blood, phlegm, yellow bile, black bile), the substantial naturalistic-causal explanation of disease, the substantial preference for dietary and lifestyle intervention over substantial pharmacological intervention — substantively dominated European medicine through the substantial medieval period and was substantively still influential in the 19th-century miasmatic-theory disputes that Pettenkofer was defending against Koch.
Hippocrates of Chios (c. 470–410 BC) was a Greek mathematician of substantial original accomplishment. He was born on the Aegean island of Chios (substantially north of Cos but in a different administrative-political grouping), was substantively originally a merchant who had been substantially defrauded of his cargo, and turned to mathematics in middle age while substantially supporting himself through teaching at Athens. His specific accomplishments included the first written textbook of Greek geometry (the lost Elements that Euclid’s later Elements substantively absorbed and superseded), the substantial quadrature of a specific lune (the first computation of the exact area of a curved figure in European mathematical history), and several substantial original results on the duplication of the cube.
The confusion
The substantial late-classical and substantial medieval scholarly traditions occasionally confused the two men. Both were Greek; both were of the same generation; both worked in substantial Greek island-and-Athenian intellectual contexts; both had Aegean place-of-origin epithets that differed only by a single letter (Coon vs Chion). The substantial standard ancient sources — Aristotle, Diogenes Laertius, Galen, Iamblichus — were substantively careful to distinguish them, but the substantial medieval Greek-and-Arabic scholarly transmissions of the texts were occasionally less careful.
The substantial confusion substantively does not generally affect the substantial standing modern scholarly attribution of specific works. The Hippocratic Corpus is unambiguously associated with Cos; the surviving lune-quadrature proof is unambiguously associated with Chios. The two intellectual traditions substantively descend through entirely substantively separate institutional channels and substantively reached the substantial modern European scholarly world through substantially substantively distinct manuscript-transmission lineages.
The Cos-Chios distinction substantively remains the standard scholarly clarification when either name is invoked.
Hippocrates of Cos substantively died at Larissa around 375 BC, aged approximately 85. Hippocrates of Chios substantively died around 410 BC at approximately 60. Neither substantively met the other in person, so far as the substantial documentary record can be substantively reconstructed.