Hippocrates of Chios taught in Athens and worked
on the classical problems of squaring the circle and duplicating
the cube. Little is known of his life but he is reported to have been an excellent geometer who, in other respects, was stupid and lacking in sense. Some claim that he was defrauded of a large sum of money because of his naiveté.
Iamblichus  writes:-
One of the Pythagorean [Hippocrates] lost his property, and when this misfortune befell him he was allowed to make money by teaching geometry.
Heath  recounts two versions of this story:-
One version of the story is that [Hippocrates] was a merchant, but lost all his property through being captured by a pirate vessel. He then came to Athens to persecute the offenders and, during a long stay, attended lectures, finally attaining such proficiency in geometry that he tried to square the circle.
Heath also recounts a different version of the story as told by Aristotle:-
... he allowed himself to be defrauded of a large sum by custom-house officers at Byzantium, thereby proving, in Aristotle's opinion, that, though a good geometer, he was stupid and incompetent in the business of ordinary life.
The suggestion is that this 'long stay' in Athens was between about 450 BC and 430 BC.
In his attempts to square the circle, Hippocrates
was able to find the areas of lunes, certain crescent-shaped figures,
using his theorem that the ratio of the areas of two circles is
the same as the ratio of the squares of their radii. We describe
impressive achievement more fully below.
Hippocrates also showed that a cube can be
doubled if two mean proportional can be determined between a number
and its double. This had a major influence on attempts to duplicate
the cube, all efforts after this being directed towards the
mean proportional problem.
He was the first to write an Elements of Geometry and although his work is now lost it must have contained much of what Euclid later included in Books 1 and 2 of the Elements. Proclus, the last major Greek philosopher, who lived around 450 AD wrote:-
Hippocrates of Chios, the discoverer of the quadrature of the lune, ... was the first of whom it is recorded that he actually compiled "Elements".
Hippocrates' book also included geometrical solutions to quadratic equations and included early methods of integration.
The Greeks did not use numbers to measure the area of a figure. Equality of plane figures is verified by cutting in pieces and rearranging. Quadrature of a figure means finding a side of a square of the same area as the figure. Well--known quadrature date back to earliest Greek mathematics -- Thales (c. -600), Pythagoras (c. -540). The Greeks had accomplished the quadrature of polygons, but they were less successful in knowing the properties of circles and other curvilinear forms. It was known that all types of polygons could be equated, in measure, to the square, and the square seemed an ideal unit of areal measure. The next three links below show, in turn, the quadrature of the rectangle, triangle and any polygon which illustrate how the Greeks squared the figures. And finally the quadrature of the lune which was the outstanding problem of Greek mathematics.
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