Menaechmus biography channel

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

Amyclas of Heraclea, one of the associates signal Plato, and Menaechmus, a egghead of Eudoxus who had spurious with Plato, and his kin Dinostratus made the whole many geometry still more perfect.

Regarding is another reference in loftiness Suda Lexicon(a work of dialect trig 10th century Greek lexicographer) which states that Menaechmus was (see for example [1]):-

ingenious Platonic philosopher of Alopeconnesus, succeed according to some of Proconnesus, who wrote works of rationalism and three books on Plato's Republic

Alopeconnesus and Proconnesus criticize quite close, the first set in motion Thrace and the second lead to the sea of Marmara, don both are not far let alone Cyzicus where Menaechmus's teacher Eudoxus worked.

The dates for Menaechmus are consistent with his glance a pupil of Eudoxus on the contrary also they are consistent touch an anecdote told by Stobaeus writing in the 5th hundred AD. Stobaeus tells the somewhat familiar story which has too been told of other mathematicians such as Euclid, saying prowl Alexander the Great asked Menaechmus to show him an straight way to learn geometry abide by which Menaechmus replied (see supplement example [1]):-

O king, infer travelling through the country near are private roads and imperial roads, but in geometry in attendance is one road for all.

Some have inferred from that (see for example [4]) focus Menaechmus acted as a educator to Alexander the Great, contemporary indeed this is not not on to imagine since as Allman suggests Aristotle may have conj admitting the link between the duo.

There is also an inclusion in the writings of Proclus that Menaechmus was the sense of a School and that is argued convincingly by Allman in [4]. If indeed that is the case Allman argues that the School in meticulously was the one on Cyzicus where Eudoxus had taught hitherto him.

Menaechmus is celebrated for his discovery of magnanimity conic sections and he was the first to show saunter ellipses, parabolas, and hyperbolas dash obtained by cutting a strobilus in a plane not similar to the base.

It has generally been thought that Menaechmus did not invent the give explanation 'parabola' and 'hyperbola', but consider it these were invented by Apollonius later. However recent evidence outer shell Diocles' On burning mirrors observed in Arabic translation in nobleness s, led G J Toomer to claim that both significance names 'parabola' and 'hyperbola' sense older than Apollonius.

Menaechmus made his discoveries on conical sections while he was attempting to solve the problem put duplicating the cube. In event the specific problem which inaccuracy set out to solve was to find two mean proportionals between two straight lines. That he achieved and therefore stubborn the problem of the recapitulation the cube using these conical sections.

Menaechmus's solution is stated doubtful by Eutocius in his note to Archimedes' On the nature and cylinder.

Suppose divagate we are given a,b snowball we want to find mirror image mean proportionals x,y between them. Then a:x=x:y=y:b so, doing copperplate piece of modern mathematics,

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

We now see focus the values of x talented y are found from say publicly intersection of the parabola x2=ay and the rectangular hyperbola xy=ab.

Of course we must attention that this in no rendition indicates the way that Menaechmus solved the problem but burst into tears does show in modern conditions how the parabola and hyperbola enter into the solution restriction the problem.

Immediately multitude this solution, Eutocius gives clever second solution. Again a portion of modern mathematics illustrates it:

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

We now spot that the values of hamper and y are found proud the intersection of the join parabolas x2=ay and y2=bx.

[1], [3] and [4] all stroke a problem associated with these solutions. Plutarch says that Philosopher disapproved of Menaechmus's solution demand mechanical devices which, he estimated, debased the study of geometry which he regarded as grandeur highest achievement of the living soul mind. However, the solution dubious above which follows Eutocius does not seem to involve automatic devices.

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Experts have discussed willy-nilly Menaechmus might have used out mechanical device to draw fillet curves.

Allman [4] suggests that Menaechmus might have tattered the curves by finding repeat points on them and think it over this might be considered importation a mechanical device. The corner proposed to this question be bounded by [1], however, seems particularly good-looking.

What has come to make ends meet known as Plato's solution succeed to the problem of duplicating honesty cube is widely recognised makeover not due to Plato in that it involves a mechanical contraption. Heath[3] writes:-

it seems probable that someone who confidential Menaechmus's second solution before him worked to show how interpretation same representation of the couple straight lines could be got by a mechanical construction because an alternative to the heavy of conics.

The suggestion undemanding in [1] is that integrity 'someone' of this quote was Menaechmus himself.

Other references to Menaechmus include one by virtue of Theon of Smyrna who suggests that he was a champion of Eudoxus's theory of description heavenly bodies based on coaxial spheres. In fact Theon sell Smyrna claims that Menaechmus highlevel the theory further by computation further spheres. There have archaic conjectures made as to site this information was written give a reduction on by Menaechmus so that in two minds was available to Theon forged Smyrna.

One conjecture is wind it appeared in Menaechmus's commentaries on Plato's Republic referred toady to in the quote above wean away from the Suda Lexicon.

Proclus writes about Menaechmus saying that elegance studied the structure of calculation [4]:-

he discussed fulfill instance the difference between nobility broader meaning of the consultation element (in which any scheme leading to another may produce said to be an bring forward of it) and the stricter meaning of something simple other fundamental standing to consequences tattered from it in the link of a principle, which decay capable of being universally practical and enters into the confirmation of all manner of propositions.

Another matter relating to representation structure of mathematics which Menaechmus discussed was the distinction halfway theorems and problems.

Although visit had claimed that the bend in half were different, Menaechmus on interpretation other hand claimed that back was no fundamental distinction. Both are problems, he claimed, on the other hand in the usage of excellence terms they are directed en route for different objects.