[Math Lair] Menaechmus

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Menaechmus was an ancient Greek mathematician from around the middle of the fourth century B.C. (a rough estimate of his lifespan would be from 380 B.C. to 320 B.C.). He was a pupil of Eudoxus and the brother of Dinostratus, who was also a mathematician.

It is said that he was tutor to Alexander the Great. It is said that, when Alexander once asked him to show him a shortcut to geometry, he replied, "O King, for travelling over the country there are royal roads and roads for common citizens, but in geometry there is one road for all." (This quote is reminiscent of a similar one attributed to Euclid).

Menaechmus was the first mathematician to study conic sections, which he found useful for solving the problem of finding two mean proportionals between two straight lines, which could be applied to the problem of duplicating the cube. Menaechmus solved the problem geometrically, but were we to approach the problem today, we might approach the problem algebraically as follows:

Let a and b be the two straight lines, and x and y the required mean proportionals. So:

a : x = x : y = y : b

from which it follows that

x² = ay
y² = bx
xy = ab

We could then proceed to solve for x and y in terms of a and b. What is important to note here though is that the first two equations are equations of parabolas, while the third equation is the equation of a rectangular hyperbola. Menaechmus' geometric solution made use of these properties.