One of the Pythagoreans [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 this impressive achievement more fully below.
Hippocrates also showed that a cube can be doubled if two mean proportionals 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 proportionals 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.
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. See  both for the translation which we give and for a discussion of which parts are due to Eudemus:-
The quadratures of lunes, which were considered to belong to an uncommon class of propositions on account of the close relation of lunes to the circle, were first investigated by Hippocrates, and his exposition was thought to be correct; we will therefore deal with them at length and describe them. He started with, and laid down as the first of the theorems useful for the purpose, the proposition that similar segments of circles have the same ratio to one another as the squares on their bases. And this he proved by first showing that the squares on the diameters have the same ratio as the circles.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. This is precisely what the problem of 'squaring the circle' means, namely to construct a square whose area is equal to the area of the circle. Again following Heath's translation in :-
After proving this, he proceeded to show in what way it was possible to square a lune the outer circumference of which is that of a semicircle. This he affected by circumscribing a semicircle about an isosceles right-angled triangle and a segment of a circle similar to those cut off by the sides. Then, since the segment about the base is equal to the sum of those about the sides, it follows that, when the part of the triangle above the segment about the base is added to both alike, the lune will be equal to the triangle. Therefore the lune, having been proved equal to the triangle, can be squared.
To follow Hippocrates' argument here, look at the diagram.
ABCD is a square and O is its centre. The two circles in the diagram are the circle with centre O through A, B, C and D, and the circle with centre D through A and C.
Notice first that the segment marked 1 on AB subtends a right angle at the centre of the circle (the angle AOB) while the segment 2 on AC also subtends a right angle at the centre (the angle ADC).
Therefore the segment 1 on AB and the segment 2 on AC are similar. Now
segment 1/segment 2 = AB2/AC2 = 1/2 since AB2+ BC2= AC2 by Pythagoras's theorem, and AB = BC so AC2= 2AB2.
Now since segment 2 is twice segment 1, the segment 2 is equal to the sum of the two segments marked 1.
Then Hippocrates argues that the semicircle ABC with the two segments 1 removed is the triangle ABC which can be squared (it was well known how to construct a square equal to a triangle).
However, if we subtract the segment 2 from the semicircle ABC we get the lune shown in the second diagram. Thus Hippocrates has proved that the lune can be squared.
However, Hippocrates went further than this in studying lunes. The proof we have examined in detail is one where the outer circumference of the lune is the arc of a semicircle. He also studied the cases where the outer arc was less than that of a semicircle and also the case where the outer arc was greater than a semicircle, showing in each case that the lune could be squared. This was a remarkable achievement and a major step in attempts to square the circle. As Heath writes in :-
... he wished to show that, if circles could not be squared by these methods, they could be employed to find the area of some figures bounded by arcs of circles, namely certain lunes, and even of the sum of a certain circle and a certain lune.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. This theorem is proved by Euclid in the Elements and it is proved there by the method of exhaustion due to Eudoxus. 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.
Article by: J J O'Connor and E F Robertson
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