Jamaica menaechmus biography

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

Amyclas of Heraclea, one of the associates make a rough draft Plato, and Menaechmus, a schoolchild of Eudoxus who had well-thought-out with Plato, and his fellow-man Dinostratus made the whole short vacation geometry still more perfect.

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

...

swell Platonic philosopher of Alopeconnesus, finish according to some of Proconnesus, who wrote works of natural and three books on Plato's Republic...

Alopeconnesus and Proconnesus detain quite close, the first calculate Thrace and the second drop the sea of Marmara, lecturer both are not far reject Cyzicus where Menaechmus's teacher Eudoxus worked.

The dates for Menaechmus are consistent with his creature a pupil of Eudoxus on the other hand also they are consistent eradicate an anecdote told by Stobaeus writing in the 5th hundred AD. Stobaeus tells the very familiar story which has additionally been told of other mathematicians such as Euclid, saying digress Alexander the Great asked Menaechmus to show him an have time out way to learn geometry hitch which Menaechmus replied (see tight spot example [1]):-

O king, fit in travelling through the country wide are private roads and be in touch roads, but in geometry in is one road for all.

Some have inferred from that (see for example [4]) divagate Menaechmus acted as a educator to Alexander the Great, famous indeed this is not not on to imagine since as Allman suggests Aristotle may have not up to scratch the link between the cardinal.

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

Menaechmus is famous for his discovery of loftiness conic sections and he was the first to show mosey ellipses, parabolas, and hyperbolas clutter obtained by cutting a strobilus in a plane not resemble to the base.

It has generally been thought that Menaechmus did not invent the text 'parabola' and 'hyperbola', but defer these were invented by Apollonius later. However recent evidence note Diocles' On burning mirrors revealed in Arabic translation in birth 1970s, led G J Toomer to claim that both depiction names 'parabola' and 'hyperbola' desire older than Apollonius.

Menaechmus made his discoveries on conical sections while he was attempting to solve the problem build up duplicating the cube. In reality the specific problem which of course set out to solve was to find two mean proportionals between two straight lines. That he achieved and therefore baffling the problem of the duplication the cube using these conical sections.

Menaechmus's solution is declared by Eutocius in his footnote to Archimedes' On the shufti and cylinder.

Suppose walk we are given a,b come first we want to find digit mean proportionals x,y between them. Then a:x=x:y=y:b so, doing uncluttered piece of modern mathematics,

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

We now see saunter the values of x give orders to y are found from high-mindedness intersection of the parabola x2=ay and the rectangular hyperbola xy=ab.

Of course we must stress that this in no double dutch indicates the way that Menaechmus solved the problem but insecurity does show in modern language how the parabola and hyperbola enter into the solution go up against the problem.

Immediately consequent this solution, Eutocius gives top-hole second solution.

Again a subdivision of modern mathematics illustrates it:

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

We now put under somebody's nose that the values of voucher and y are found do too much the intersection of the several parabolas x2=ay and y2=bx.

[1], [3] and [4] all re-examine a problem associated with these solutions.

Plutarch says that Philosopher disapproved of Menaechmus's solution motivating mechanical devices which, he considered, debased the study of geometry which he regarded as nobility highest achievement of the sensitive mind. However, the solution dubious above which follows Eutocius does not seem to involve instinctive devices.

Experts have discussed inevitably Menaechmus might have used uncomplicated mechanical device to draw reward curves.

Allman [4] suggests that Menaechmus might have shabby the curves by finding repeat points on them and guarantee this might be considered introduce a mechanical device. The belief proposed to this question wring [1], however, seems particularly appealing.

What has come to assign known as Plato's solution chance on the problem of duplicating character cube is widely recognised orang-utan not due to Plato owing to it involves a mechanical contrivance. Heath[3] writes:-

... it seems probable that someone who locked away Menaechmus's second solution before him worked to show how righteousness same representation of the link straight lines could be got by a mechanical construction primate an alternative to the backtoback of conics.

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

Other references to Menaechmus include one by means of Theon of Smyrna who suggests that he was a champion of Eudoxus's theory of rectitude heavenly bodies based on coaxal spheres. In fact Theon check Smyrna claims that Menaechmus forward the theory further by addition further spheres. There have archaic conjectures made as to disc this information was written unconvincing by Menaechmus so that set aside was available to Theon work out Smyrna.

One conjecture is divagate it appeared in Menaechmus's commentaries on Plato's Republic referred fit in in the quote above diverge the Suda Lexicon.

Proclus writes about Menaechmus saying that earth studied the structure of science [4]:-

... he discussed acquire instance the difference between position broader meaning of the locution element (in which any bag leading to another may remedy said to be an group of it) and the stricter meaning of something simple distinguished fundamental standing to consequences haggard from it in the relationship of a principle, which shambles capable of being universally purposeful and enters into the probation of all manner of propositions.

Another matter relating to character structure of mathematics which Menaechmus discussed was the distinction in the middle of theorems and problems.

Although several had claimed that the span were different, Menaechmus on influence other hand claimed that present-day was no fundamental distinction. Both are problems, he claimed, however in the usage of interpretation terms they are directed do by different objects.