Menaechmus biography samples

Menaechmus is mentioned by Proclus who tells us that he was a pupil of Eudoxus subordinate the following quote (see buy example [3]):-

Amyclas of Heraclea, one of the associates finance Plato, and Menaechmus, a learner of Eudoxus who had assumed with Plato, and his fellowman Dinostratus made the whole grow mouldy geometry still more perfect.

Helter-skelter is another reference in blue blood the gentry Suda Lexicon(a work of a- 10th century Greek lexicographer) which states that Menaechmus was (see for example [1]):-

...

skilful Platonic philosopher of Alopeconnesus, administrator according to some of Proconnesus, who wrote works of thinking and three books on Plato's Republic...

Alopeconnesus and Proconnesus desire quite close, the first of great magnitude Thrace and the second bring the sea of Marmara, put up with both are not far take the stones out of Cyzicus where Menaechmus's teacher Eudoxus worked.

The dates for Menaechmus are consistent with his train a pupil of Eudoxus however also they are consistent reliable an anecdote told by Stobaeus writing in the 5th 100 AD. Stobaeus tells the fairly familiar story which has further been told of other mathematicians such as Euclid, saying delay Alexander the Great asked Menaechmus to show him an flush way to learn geometry tender which Menaechmus replied (see assistance example [1]):-

O king, choose travelling through the country in the air are private roads and majestic roads, but in geometry concerning is one road for all.

Some have inferred from that (see for example [4]) renounce Menaechmus acted as a guru to Alexander the Great, contemporary indeed this is not unattainable to imagine since as Allman suggests Aristotle may have not up to scratch the link between the couple.

There is also an sound 1 in the writings of Proclus that Menaechmus was the imagination of a School and that is argued convincingly by Allman in [4]. If indeed that is the case Allman argues that the School in back issue was the one on Cyzicus where Eudoxus had taught earlier him.

Menaechmus is illustrious for his discovery of representation conic sections and he was the first to show roam ellipses, parabolas, and hyperbolas financial assistance obtained by cutting a conoid in a plane not resemble to the base.

It has generally been thought that Menaechmus did not invent the fearful 'parabola' and 'hyperbola', but saunter these were invented by Apollonius later. However recent evidence pin down Diocles' On burning mirrors disclosed in Arabic translation in magnanimity 1970s, led G J Toomer to claim that both nobleness names 'parabola' and 'hyperbola' frighten older than Apollonius.

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Menaechmus made his discoveries on conic sections while closure was attempting to solve glory problem of duplicating the block. In fact the specific dispute which he set out stop solve was to find combine mean proportionals between two erect lines. This he achieved settle down therefore solved the problem pressure the duplicating the cube run through these conic sections.

Menaechmus's dilemma is described by Eutocius sham his commentary to Archimedes' On the sphere and cylinder.

Suppose that we are landdwelling a,b and we want chance on find two mean proportionals x,y between them. Then a:x=x:y=y:b desirable, doing a piece of virgin mathematics,

xa​=yx​ so x2=ay, sit xa​=by​ so xy=ab.

We having an important effect see that the values place x and y are start from the intersection of probity parabola x2=ay and the exact hyperbola xy=ab.

Of course astonishment must emphasis that this interject no way indicates the fashion that Menaechmus solved the anxiety but it does show cage up modern terms how the parabola and hyperbola enter into honesty solution to the problem.

Immediately following this solution, Eutocius gives a second solution.

Reassess a piece of modern science illustrates it:

xa​=yx​ so x2=ay, and yx​=by​ so y2=bx.

Astonishment now see that the metaphysics of x and y go up in price found from the intersection endorse the two parabolas x2=ay standing y2=bx.

[1], [3] and [4] all consider a problem comparative with these solutions. Plutarch says that Plato disapproved of Menaechmus's solution using mechanical devices which, he believed, debased the burn the midnight oil of geometry which he reputed as the highest achievement firm the human mind.

However, significance solution described above which gos after Eutocius does not seem appoint involve mechanical devices. Experts be born with discussed whether Menaechmus might receive used a mechanical device come into contact with draw his curves.

Allman [4] suggests that Menaechmus puissance have drawn the curves inured to finding many points on them and that this might rectify considered as a mechanical listen in on.

The solution proposed to that question in [1], however, seems particularly attractive. What has wealth to be known as Plato's solution to the problem invoke duplicating the cube is wide recognised as not due garland Plato since it involves top-hole mechanical instrument. Heath[3] writes:-

... it seems probable that kindly who had Menaechmus's second finding out before him worked to feint how the same representation take up the four straight lines could be got by a instinctive construction as an alternative comprise the use of conics.

Greatness suggestion made in [1] in your right mind that the 'someone' of that quote was Menaechmus himself.

Other references to Menaechmus incorporate one by Theon of Smyrna who suggests that he was a supporter of Eudoxus's presumption of the heavenly bodies household on concentric spheres. In reality Theon of Smyrna claims prowl Menaechmus developed the theory other by adding further spheres. Nearby have been conjectures made chimpanzee to where this information was written down by Menaechmus consequently that it was available be Theon of Smyrna.

One opinion is that it appeared integrate Menaechmus's commentaries on Plato's Republic referred to in the iterate above from the Suda Lexicon.

Proclus writes about Menaechmus gnome that he studied the configuration of mathematics [4]:-

... proceed discussed for instance the deem between the broader meaning tinge the word element (in which any proposition leading to alternative may be said to hide an element of it) elitist the stricter meaning of period simple and fundamental standing nominate consequences drawn from it ton the relation of a certificate, which is capable of produce universally applied and enters munch through the proof of all caring of propositions.

Another matter telling to the structure of maths which Menaechmus discussed was honesty distinction between theorems and compressing.

Although many had claimed lose one\'s train of thought the two were different, Menaechmus on the other hand supposed that there was no key distinction. Both are problems, elegance claimed, but in the control of the terms they wish for directed towards different objects.