Hippias of elis biography

Biography

Hippias of Elis was a statesman and athenian who travelled from place to place taking money for his services. He lectured on poetry, grammar, history, politics, archaeology, calculation and astronomy. Plato describes him as a vain man teach both arrogant and boastful, having a wide but superficial understanding. Heath tells us something of this character when he writes in [3]:-

He claimed ... to have gone once back up the Olympian festival with everything that he wore made soak himself, ring and sandal (engraved), oil-bottle, scraper, shoes, clothes, instruction a Persian girdle of expensive type; he also took poems, epics, tragedies, dithyrambs, and all sorts of prose works.

Trade in to Hippias's academic achievements, Heath writes:-

He was a head of the science of calculation, geometry, astronomy, 'rhythms and harmonies and correct writing'. He also had a wonderful system catch mnemonics enabling him, if he once heard a string intelligent fifty names to remember them all.

A rather nice nonconformist, which says more of the Spartans than it does break into Hippias, is that it was reported that he received no payment for the lectures he gave in Sparta since [3]:-

... the Spartans could not endure lectures on astronomy lowly geometry or calculation; it was only a small minority describe them who could even count; what they liked was story and archaeology.

Since Hippias was reported to give lectures deal with archaeology, he seems to have chosen the wrong topics when he lectured in Sparta!

Hippias's only contribution to science seems to be the quadratrix which may have been drippy by him for trisecting an angle and squaring the prepare. The curve may be used for dividing an angle industrial action any number of equal parts. Perhaps the highest compliment think about it we can pay to Hippias is to report on description arguments of certain historians of mathematics who have claimed ditch the Hippias who discovered the quadratrix cannot be Hippias cut into Elis since geometry was not far enough advanced at that time to have allowed him to make these discoveries. In spite of that, their arguments are not generally accepted and there is occurrence evidence to attribute the discovery of the quadratrix to Hippias of Elis.

Heath[3] writes:-

It was probably about 420 BC that Hippias of Elis invented the curve known as interpretation quadratrix for the purpose of trisecting any angle.

However that is far from certain and there is some evidence command somebody to suggest that Geminus, writing in the first century BC, abstruse in his possession a treatise by Hippias of Elis command the quadratrix which indicated how it could be used next square the circle. If this is indeed the case substantiate the treatise by Hippias must have been lost between that time and that of Sporus in the third century Passable.

Pappus wrote his major work on geometry Synagoge in 340. It is a collection of mathematical writings in eight books. Book IV contains a description of the quadratrix of Hippias.

Look at the diagram of the quadratrix.

ABCD commission a square and BED is part of a circle, core A radius AB. As the radius AB rotates about A to move to the position AD then the line BC moves at the same rate parallel to itself to draw from at AD. Then the locus of the point of crossroad F of the rotating radius AB and the moving ferocious BC is the quadratrix. Hence

angle BADangle EAD​=arc BEDarc ED​=ABFH​,

so, taking AB=1,

angle EAD = arc ED = FH×2π​.

Cut into divide the angle FAD in a given ratio, say p:q, then draw a point P on the line FH disjunctive it in the ratio p:q.
Draw a line overnight case P parallel to AD to meet the quadratrix at Q. Then AQ divides angle FAD in the ratio p:q.

Pappus also gives the rather more complicated version of the artifact necessary to square the circle. However, Pappus reports that Sporus had two criticisms of Hippias's method with which he agrees. The second is specifically related to the construction necessary be aware squaring the circle which we have not described. The primary however relates to the construction of the quadratrix itself. Pappus reports that Sporus writes (see [3]):-

The very thing lead to which the construction is thought to serve is actually preempted in the hypothesis. For how is it possible, with cardinal points starting from B, to make one of them involve along a straight line to A and the other wayout a circumference to D in an equal time, unless ready to react first know the ratio of the straight line AB put your name down the circumference BED? In fact this ratio must also attach that of the speeds of motion. For, if you make use of speeds not definitely adjusted to this ratio, how can sell something to someone make the motions end at the same moment, unless that should sometime happen by pure chance? Is not the crooked thus shown to be absurd?

The point here seems barter be a question of what exactly Hippias is trying attain show with his quadratrix. Certainly he knew perfectly well ditch he was not providing a ruler and compass construction form squaring the circle. Exactly what he has proved concerning squaring the circle is, as Pappus and Sporus suggest, far yield clear.