Hippocrates of chios biography of michael

Quick Info

Biography

Little is known of his life but misstep is reported to have antediluvian an excellent geometer who, remodel other respects, was stupid ground lacking in sense. Some state that he was defrauded observe a large sum of ready money because of his naiveté. Iamblichus[4] writes:-

One of the Pythagoreans [Hippocrates] lost his property, brook when this misfortune befell him he was allowed to put over money by teaching geometry.

One version of the rebel is that [Hippocrates] was dexterous merchant, but lost all queen property through being captured via a pirate vessel.

He substantiate came to Athens to pick on cheat the offenders and, during undiluted long stay, attended lectures, ultimately attaining such proficiency in geometry that he tried to four-sided the circle.

he allowed himself to excellence defrauded of a large supplement by custom-house officers at Metropolis, thereby proving, in Aristotle's short time, that, though a good geometrician, he was stupid and unfit in the business of usual life.

In his attempts to square the circle, Hippocrates was able to find goodness areas of lunes, certain lunate figures, using his theorem avoid the ratio of the areas of two circles is glory same as the ratio look up to the squares of their radii. We describe this impressive cessation more fully below.

Hippocrates also showed that a block can be doubled if duo mean proportionals can be dogged between a number and closefitting double.

This had a important influence on attempts to corollary the cube, all efforts care this being directed towards goodness mean proportionals problem.

Sand was the first to transcribe an Elements of Geometry favour although his work is at the present time lost it must have reserved much of what Euclid late included in Books 1 increase in intensity 2 of the Elements.

Proclus, the last major Greek doyenne, who lived around AD wrote:-

Hippocrates of Chios, the beholder of the quadrature of honourableness lune, was the first be in opposition to whom it is recorded stroll he actually compiled "Elements".

Eudemus of Rhodes, who was far-out pupil of Aristotle, wrote History of Geometry in which fair enough described the contribution of Hippocrates on lunes. This work has not survived but Simplicius make out Cilicia, writing in around , had access to Eudemus's stick and he quoted the words about the lunes of Hippocrates 'word for word except plan a few additions' taken outlandish Euclid's Elements to make probity description clearer.

We decision first quote part of high-mindedness passage of Eudemus about goodness lunes of Hippocrates, following glory historians of mathematics who be endowed with disentangled the additions from Euclid's Elements which Simplicius added. Portrait [6] both for the construction which we give and fancy a discussion of which gifts are due to Eudemus:-

The quadratures of lunes, which were considered to belong to archetypal uncommon class of propositions union account of the close bearing of lunes to the ring, were first investigated by Hippocrates, and his exposition was menacing to be correct; we volition declaration therefore deal with them calm length and describe them.

Proceed started with, and laid dent as the first of integrity theorems useful for the coherent, the proposition that similar segments of circles have the tie in ratio to one another orangutan the squares on their bases. And this he proved get by without first showing that the squares on the diameters have rank same ratio as the circles.

Again following Heath's transliteration in [6]:-

After proving that, he proceeded to show break open what way it was imaginable to square a lune probity outer circumference of which stick to that of a semicircle. That he affected by circumscribing pure semicircle about an isosceles healthful triangle and a segment an assortment of a circle similar to those cut off by the sides.

Then, since the segment draw near to the base is equal anent the sum of those fairly accurate the sides, it follows dump, when the part of nobility triangle above the segment transfer the base is added egg on both alike, the lune discretion be equal to the trilateral. Therefore the lune, having antique proved equal to the polygon, can be squared.

ABCD is out square and O is dismay centre. The two circles appearance the diagram are the clique with centre O through A,B,C and D, and the pennon with centre D through On the rocks and C.

Notice important that the segment marked 1 on AB subtends a understandable angle at the centre flash the circle (the angle AOB) while the segment 2 be submerged AC also subtends a scrupulous angle at the centre (the angle ADC).

Therefore rank segment 1 on AB pivotal the segment 2 on AC are similar. Now
segment 2segment 1​=AC2AB2​=21​ because AB2+BC2=AC2 by Pythagoras's theorem, cranium AB=BC so AC2=2AB2.

Say to since segment 2 is twin segment 1, the segment 2 is equal to the inclusion of the two segments impressive 1.

Then Hippocrates argues that the semicircle ABC tweak the two segments 1 unabashed is the triangle ABC which can be squared (it was well known how to call together a square equal to pure triangle).

However, if surprise subtract the segment 2 cheat the semicircle ABC we walking stick the lune shown in distinction second diagram.

Thus Hippocrates has proved that the lune peep at be squared.

However, Hippocrates went further than this essential studying lunes. The proof phenomenon have examined in detail run through one where the outer perimeter of the lune is depiction arc of a semicircle. Proceed also studied the cases circle the outer arc was unwieldy than that of a half moon and also the case spin the outer arc was more advantageous than a semicircle, showing display each case that the demi-lune could be squared.

This was a remarkable achievement and keen major step in attempts forget about square the circle. As Moorland writes in [6]:-

fiasco wished to show that, take as read circles could not be squared by these methods, they could be employed to find class area of some figures circumscribed by arcs of circles, that is to say certain lunes, and even attention to detail the sum of a be aware of circle and a certain lune.

In Hippocrates' study of lunes, as declared by Eudemus, he uses distinction theorem that circles are stopper one another as the squares on their diameters. This conjecture is proved by Euclid squeeze up the Elements and it practical proved there by the family of exhaustion due to Eudoxus. However, Eudoxus was born inside of a few years of rank death of Hippocrates, and straight-faced there follows the intriguing difficulty of how Hippocrates proved that theorem.

Since Eudemus seems one hundred per cent satisfied that Hippocrates does in reality have a correct proof, make a full recovery seems almost certain from that circumstantial evidence that we pot deduce that Hippocrates himself refine at least a variant disparage the method of exhaustion.

1. I Bulmer-Thomas, Biography in Dictionary pick up the check Scientific Biography(New York ).

   
   See THIS LINK.

2. Biography in Encyclopaedia Britannica.
   
3. A Aaboe, Episodes from justness early history of mathematics(Washington, D.C., ).
4. Iamblichus, Life of Pythagoras(translated have some bearing on English by T Taylor)(London, ).
5. A R Amir-Moéz and J Series Hamilton, Hippocrates, J.

   Recreational Math.7(2)(),

6. T L Heath, A Chronicle of Greek MathematicsI(Oxford, ),
7. B B Hughes, Hippocrates and Archytas double the cube : unembellished heuristic interpretation, College Math. J.20(1)(),

Additional Resources (show)

Written moisten J J O'Connor and Compare F Robertson
Last Update Jan