Hippocrates of Chios - Mathematics

Mathematics

The major accomplishment of Hippocrates is that he was the first to write a systematically organized geometry textbook, called Stoicheia Elements, 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.