Menaechmus
c. 380-c. 320 B.C.
Greek Mathematician
In the course of trying to solve the problem of doubling the cube, Menaechmus discovered the conic sections, which would have an enormous impact on mathematics in modern times. He was also responsible for distinguishing between the two meanings of the word element, which up to that point had caused confusion among Greek geometers.
Whereas historians know almost nothing about the life of Dinostratus (c. 390-c. 320 B.C.), who applied the quadratix of Hippias (c. 460-c. 400 B.C.) in an attempt to square the circle, information about Menaechmus is much more readily available—a salient point, since he was the younger brother of Dinostratus. Nonetheless, his family life is otherwise a mystery, as are the facts of his birth.
Various writers have provided details on Menaechmus's career, allowing a historian to piece together a picture of sorts. There was a Menaechmus, from the region of either Alopeconnesus or Proconnesus, who wrote three commentaries on the Republic. The first of these is in Thrace, the second along the Sea of Marmara, and both are close to Cyzicus in Asia Minor, where Menaechmus studied under Eudoxus of Cnidus (c. 408-c. 355 B.C.) After completing his schooling, Menaechmus apparently worked as tutor for Alexander the Great (356-323 B.C.).
Like many mathematicians of his time and place, Menaechmus became involved in attempts to solve the Delian problem, or the challenge of doubling the cube. This would be quite simple today, using the formula x3 = 2a3, where a is the length of the known cube and x that of the doubled one. The Greeks, however, lacked not only this notation, which makes it possible to conceive of the problem algebraically; they also lacked the algebraic formula itself, and approached the challenge with the simple geometric tools of compass and straightedge. Hippocrates of Chios (c. 470-c. 410 B.C.) had discovered that the solution lay in finding the mean proportionals between two given lines; Archytas of Tarentum (c. 428-350? B.C.) improved on this method; and Menaechmus took the solution one step further with his work in what came to be known as conic sections.
Eventually Menaechmus offered two methods for finding a solution, both of which involved slicing cones with planes as a way of finding the mean proportionals between two numbers. The greatest importance of his work, however, lay not in its application to the Delian problem, but in his introduction of conic sections. Later, Apollonius of Perga (c. 262-c. 190 B.C.) would develop terminology for the shapes made by slicing cones—hyperbola, parabola, and so on—and in modern times these forms would find application in everything from calculus to rocket science.
Also useful was Menaechmus's distinction of the two meanings inherent in the term element. According to Proclus (410?-485), "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...." This "stricter meaning" is encompassed in the modern term elemental (in its everyday, nonmathematical sense), which fits Proclus's definition as "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." As for Menaechmus, he placed an emphasis on the first meaning, which more clearly has an application to the structure of mathematics.
In his recognition that at least some of the challenges facing thinkers could be explained in terms of language, Menaechmus calls to mind modern thinkers such as Ludwig Wittgenstein (1889-1951). He also rejected the prevailing distinction between "problems" and "theorems," stating that both terms describe problems, the only difference being their purpose and the nature of the challenge they posed.
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