**Hippocrates** was a Greek mathematician who worked on the classical problems of squaring the circle and duplicating the cube.

- 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.
- Hippocrates also showed that a cube can be doubled if two mean proportionals can be determined between a number and its double.
- Hippocrates' book also included geometrical solutions to quadratic equations and included early methods of integration.
- Eudemus of Rhodes, who was a pupil of Aristotle, wrote History of Geometry in which he described the contribution of Hippocrates on lunes.
- This work has not survived but Simplicius of Cilicia, writing in around 530, had access to Eudemus's work and he quoted the passage about the lunes of Hippocrates 'word for word except for a few additions' taken from Euclid's Elements to make the description clearer.
- We will first quote part of the passage of Eudemus about the lunes of Hippocrates, following the historians of mathematics who have disentangled the additions from Euclid's Elements which Simplicius added.
- Before continuing with the quote we should note that Hippocrates is trying to 'square a lune' by which he means to construct a square equal in area to the lune.
- To follow Hippocrates' argument here, look at the diagram.
- Then Hippocrates argues that the semicircle ABCABCABC with the two segments 1 removed is the triangle ABCABCABC which can be squared (it was well known how to construct a square equal to a triangle).
- Thus Hippocrates has proved that the lune can be squared.
- However, Hippocrates went further than this in studying lunes.
- There is one further remarkable achievement which historians of mathematics believe that Hippocrates achieved, although we do not have a direct proof since his works have not survived.
- In Hippocrates' study of lunes, as described by Eudemus, he uses the theorem that circles are to one another as the squares on their diameters.
- However, Eudoxus was born within a few years of the death of Hippocrates, and so there follows the intriguing question of how Hippocrates proved this theorem.
- Since Eudemus seems entirely satisfied that Hippocrates does indeed have a correct proof, it seems almost certain from this circumstantial evidence that we can deduce that Hippocrates himself developed at least a variant of the method of exhaustion.

Born about 470 BC, Chios, Greece. Died about 410 BC.

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Ancient Greek, Geometry, Origin Greece, Puzzles And Problems

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**Oâ€™Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive