# Hippocrates of Chios

**Hippocrates of Chios** (Greek: Ἱπποκράτης ὁ Χῖος) was an ancient Greek mathematician, geometer, and astronomer, who lived c. 470 – c. 410 BCE.

He was born on the isle of Chios, where he originally was a merchant. After some misadventures (he was robbed by either pirates or fraudulent customs officials) he went to Athens, possibly for litigation. There he grew into a leading mathematician.

On Chios, Hippocrates may have been a pupil of the mathematician and astronomer Oenopides of Chios. In his mathematical work there probably was some Pythagorean influence too, perhaps via contacts between Chios and the neighbouring island of Samos, a center of Pythagorean thinking: Hippocrates has been described as a 'para-Pythagorean', a philosophical 'fellow traveler'. The *reductio ad absurdum* argument (or proof by contradiction) has been traced to him.

## Mathematics

The major accomplishment of Hippocrates is that he was the first to write a systematically organized geometry textbook, called *Elements* (Στοιχεῖα, *Stoicheia*), that is, basic theorems, or building blocks of mathematical theory. From then on, mathematicians from all over the ancient world could, at least in principle, build on a common framework of basic concepts, methods, and theorems, which stimulated the scientific progress of mathematics.

Only a single, and famous, fragment of Hippocrates' *Elements* is existent, embedded in the work of Simplicius. In this fragment the area is calculated of some so-called *Hippocratic lunes* — see Lune of Hippocrates. This was part of a research programme to achieve the "quadrature of the circle", that is, to calculate the area of the circle, or, equivalently, to construct a square with the same area as a circle. The strategy apparently was to divide a circle into a number of crescent-shaped parts. If it were possible to calculate the area of each of those parts, then the area of the circle as a whole would be known too. Only much later was it proven (by Ferdinand von Lindemann, in 1882) that this approach had no chance of success, because the factor pi (π) is transcendental. The number π is the ratio of the circumference to the diameter of a circle, and also the ratio of the area to the square of the radius.

In the century after Hippocrates at least four other mathematicians wrote their own *Elements*, steadily improving terminology and logical structure. In this way Hippocrates' pioneering work laid the foundation for Euclid's *Elements* (c. 325 BC) that was to remain the standard geometry textbook for many centuries. Hippocrates is believed to have originated the use of letters to refer to the geometric points and figures in a proposition, e.g., "triangle ABC" for a triangle with vertices at points A, B, and C.

Two other contributions by Hippocrates in the field of mathematics are noteworthy. He found a way to tackle the problem of 'duplication of the cube', that is, the problem of how to construct a cube root. Like the quadrature of the circle this was another of the so-called three great mathematical problems of Antiquity. Hippocrates also invented the technique of 'reduction', that is, to transform specific mathematical problems into a more general problem that is more easy to solve. The solution to the more general problem then automatically gives a solution to the original problem.

## Astronomy

In the field of astronomy Hippocrates tried to explain the phenomena of comets and the Milky Way. His ideas have not been handed down very clearly, but probably he thought both were optical illusions, the result of refraction of solar light by moisture that was exhaled by, respectively, a putative planet near the sun, and the stars. The fact that Hippocrates thought that light rays originated in our eyes instead of in the object that is seen, adds to the unfamiliar character of his ideas, which belong to the realm of speculative philosophy.

## References

- Ivor Bulmer-Thomas, 'Hippocrates of Chios', in:
*Dictionary of Scientific Biography*, Charles Coulston Gillispie, ed. (18 Volumes, New York 1970–1990) pp. 410–418. - [Axel Anthon] Björnbo, 'Hippokrates', in: Paulys Realencyclopädie der Classischen Altertumswissenschaft, G. Wissowa, ed. (51 Volumes; 1894–1980) Vol. 8 (1913) col. 1780–1801.

## External links

- O'Connor, John J.; Robertson, Edmund F., "Hippocrates of Chios",
*MacTutor History of Mathematics archive*, University of St Andrews. - The Quadrature of the Circle and Hippocrates' Lunes at Convergence