Hippocrates of Chios (; c. 470 – c. 410 BC) was an ancient Greek mathematician, geometer, and astronomer. He was born on the isle of Chios, where he was originally a merchant. After some misadventures (he was robbed by either pirates or fraudulent customs officials) he went to Athens, possibly for litigation, where he became 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 neighboring island of Samos, a center of Pythagorean thinking: Hippocrates has been described as a 'para-Pythagorean', a philosophical 'fellow traveler'. "Reduction" arguments such as reductio ad absurdum argument (or proof by contradiction) have been traced to him, as has the use of power to denote the square of a line. 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, 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. This was part of a research program to square the circle, that is, 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), which was to remain the standard geometry textbook for many