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

Amyclas of Heraclea, one of the associates of Plato, and Menaechmus, a pupil of Eudoxus who had studied with Plato, and his brother Dinostratus made the whole of geometry still more perfect.

There is another reference in the *Suda Lexicon* (a work of a 10^{th} century Greek lexicographer) which states that Menaechmus was (see for example [1]):-

... a Platonic philosopher of Alopeconnesus, or according to some of Proconnesus, who wrote works of philosophy and three books on Plato's Republic...

Alopeconnesus and Proconnesus are quite close, the first in Thrace and the second in the sea of Marmara, and both are not far from Cyzicus where Menaechmus's teacher Eudoxus worked. The dates for Menaechmus are consistent with his being a pupil of Eudoxus but also they are consistent with an anecdote told by Stobaeus writing in the 5^{th} century AD. Stobaeus tells the rather familiar story which has also been told of other mathematicians such as Euclid, saying that Alexander the Great asked Menaechmus to show him an easy way to learn geometry to which Menaechmus replied (see for example [1]):-

O king, for travelling through the country there are private roads and royal roads, but in geometry there is one road for all.

Some have inferred from this (see for example [4]) that Menaechmus acted as a tutor to Alexander the Great, and indeed this is not impossible to imagine since as Allman suggests Aristotle may have provided the link between the two. There is also an implication in the writings of Proclus that Menaechmus was the head of a School and this is argued convincingly by Allman in [4]. If indeed this is the case Allman argues that the School in question was the one on Cyzicus where Eudoxus had taught before him.

Menaechmus is famed for his discovery of the conic sections and he was the first to show that ellipses, parabolas, and hyperbolas are obtained by cutting a cone in a plane not parallel to the base. It has generally been thought that Menaechmus did not invent the words 'parabola' and 'hyperbola', but that these were invented by Apollonius later. However recent evidence in Diocles' *On burning mirrors* discovered in Arabic translation in the 1970s, led G J Toomer to claim that both the names 'parabola' and 'hyperbola' are older than Apollonius.

Menaechmus made his discoveries on conic sections while he was attempting to solve the problem of duplicating the cube. In fact the specific problem which he set out to solve was to find two mean proportionals between two straight lines. This he achieved and therefore solved the problem of the duplicating the cube using these conic sections. Menaechmus's solution is described by Eutocius in his commentary to Archimedes' *On the sphere and cylinder.*

Suppose that we are given *a*, *b* and we want to find two mean proportionals *x*, *y* between them. Then *a* : *x* = *x* : *y* = *y* : *b* so, doing a piece of modern mathematics,

a/x=x/ysox^{2}=ay, anda/x=y/bsoxy=ab.

We now see that the values of *x* and *y* are found from the intersection of the parabola *x*^{2}= *ay* and the rectangular hyperbola *xy* = *ab*. Of course we must emphasis that this in no way indicates the way that Menaechmus solved the problem but it does show in modern terms how the parabola and hyperbola enter into the solution to the problem.

Immediately following this solution, Eutocius gives a second solution. Again a piece of modern mathematics illustrates it:

a/x=x/ysox^{2}=ay, andx/y=y/bsoy^{2}=bx.

We now see that the values of *x* and *y* are found from the intersection of the two parabolas *x*^{2} = *ay* and *y*^{2} = *bx*.

[1], [3] and [4] all consider a problem associated with these solutions. Plutarch says that Plato disapproved of Menaechmus's solution using mechanical devices which, he believed, debased the study of geometry which he regarded as the highest achievement of the human mind. However, the solution described above which follows Eutocius does not seem to involve mechanical devices. Experts have discussed whether Menaechmus might have used a mechanical device to draw his curves.

Allman [4] suggests that Menaechmus might have drawn the curves by finding many points on them and that this might be considered as a mechanical device. The solution proposed to this question in [1], however, seems particularly attractive. What has come to be known as Plato's solution to the problem of duplicating the cube is widely recognised as not due to Plato since it involves a mechanical instrument. Heath [3] writes:-

... it seems probable that someone who had Menaechmus's second solution before him worked to show how the same representation of the four straight lines could be got by a mechanical construction as an alternative to the use of conics.

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

Other references to Menaechmus include one by Theon of Smyrna who suggests that he was a supporter of Eudoxus's theory of the heavenly bodies based on concentric spheres. In fact Theon of Smyrna claims that Menaechmus developed the theory further by adding further spheres. There have been conjectures made as to where this information was written down by Menaechmus so that it was available to Theon of Smyrna. One conjecture is that it appeared in Menaechmus's commentaries on Plato's *Republic* referred to in the quote above from the *Suda Lexicon.*

Proclus writes about Menaechmus saying that he studied the structure of mathematics [4]:-

... he discussed for instance the difference between the broader meaning of the word element(in which any proposition leading to another may be said to be an element of it)and the stricter meaning of something simple and fundamental standing to consequences drawn from it in the relation of a principle, which is capable of being universally applied and enters into the proof of all manner of propositions.

Another matter relating to the structure of mathematics which Menaechmus discussed was the distinction between theorems and problems. Although many had claimed that the two were different, Menaechmus on the other hand claimed that there was no fundamental distinction. Both are problems, he claimed, but in the usage of the terms they are directed towards different objects.

**Article by:** *J J O'Connor* and *E F Robertson*

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