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- There is also a Menaechmus in Plautus' play, The
Menaechmus (Greek: Μέναιχμος, 380–320 BC) was an ancient Greek mathematician and
born in Alopeconnesus (within modern-day Turkey), who was known for his friendship with
the renowned philosopher Plato
and for his apparent discovery of conic sections and his solution to the
then-long-standing problem of doubling the cube using the parabola and hyperbola.
Menaechmus is remembered by mathematicians for his discovery of
the conic sections and his solution to the
problem of doubling the cube.
Menaechmus likely discovered the conic sections, that is, the ellipse, the parabola, and the hyperbola, as a by-product
of his search for the solution to the Delian problem.
Menaechmus knew that in a parabola y² = lx, where
l is a constant called the latus rectum, although
he was not aware of the fact that any equation in two unknowns
determines a curve.
He apparently derived these properties of conic sections and others
as well. Using this information it was now possible to find a
solution to the problem of the duplication of
the cube by solving for the points at which two parabolas
intersect, a solution equivalent to solving a cubic equation.
There are few direct sources for Menaechmus' work; his work on
conic sections is known primarily from an epigram by Eratosthenes, and the accomplishment of
his brother (of devising a method to create a square equal in area
to a given circle using the quadratrix), Dinostratus, is known solely from the
writings of Proclus. Proclus
also mentions that Menaechmus was taught by Eudoxus. There is a curious statement by Plutarch to the effect that
Plato disapproved of Menaechmus achieving his doubled cube solution
with the use of mechanical devices; the proof currently known
appears to be solely algebraic.
Menaechmus was said to have been the tutor of Alexander
the Great; this belief derives from the following anecdote:
supposedly, once, when Alexander asked him for a shortcut to
understanding geometry, he replied "O King, for traveling over the
country, there are royal road and roads for common citizens, but in
geometry there is one road for all" (Beckmann 1989, p. 34).
However, this quote is first attributed to Stobaeus, about 500 AD, and so whether
Menaechmus really taught Alexander is uncertain.
Where precisely he died is uncertain as well, though modern
scholars believe that he eventually expired in Cyzicus.
Cooke, Roger (1997). "The Euclidean
Synthesis". p. 103. "Eutocius and Proclus both attribute the
discovery of the conic sections to Menaechmus, who lived in Athens
in the late fourth century B.C.E. Proclus, quoting Eratosthenes,
refers to "the conic section triads of Menaechmus." Since this
quotation comes just after a discussion of "the section of a
right-angled cone" and "the section of an acute-angled cone," it is
inferred that the conic sections were produced by cutting a cone
with a plane perpendicular to one of its elements. Then if the
vetex angle of the cone is acute, the resulting section
(calledoxytome) is an ellipse. If the angle is right, the
section (orthotome) is a parabola, and if the angle is
obtuse, the section (amblytome) is a hyperbola (see Fig.
- ^ Boyer (1991). "The age of Plato and
Aristotle". p. 93. "It was consequently a signal achievement
on the part of Menaechmus when he disclosed that curves having the
desired property were near at hand. In fact, there was a family of
appropriate curves obtained from a single source - the cutting of a
right circular cone by a plane perpendicular to an element of the
cone. That is, Menaechmus is reputed to have discovered the curves
that were later known as the ellipse, the parabola, and the
hyperbola. [...] Yet the first deisovery of the ellopse seems to
have been made by Menaechmus as a mere by-product in a search in
ehich it was the parabola and hyperbola that proffered the
properties needed in the solution of the Delian
- ^ a
Boyer (1991). "The age of Plato and
Aristotle". pp. 94–95. "If OP=y and OD = x are coordinates of
point P, we have y<sup2 = R).OV, or, on substituting
equals, y2 = R'D.OV = AR'.BC/AB.DO.BC/AB =
AR'.BC2/AB2.xInasmuch as segments AR', BC,
and AB are the same for all points P on the curve EQDPG, we can
write the equation of the curve, a "section of a right-angled
cone," as y2=lx, where l is a constant, later to be
known as the latus rectum of the curve. [...] Menaechmus apparently
derived these properties of the conic sections and others as well.
Since this material has a strong resemblance to the use of
coordinates, as illustrated above, it has sometimes been maintains
that Menaechmus had analytic geometry. Such a judgment is warranted
only in part, for certainly Menaechmus was unaware that any
equation in two unknown quantities determines a curve. In fact, the
general concept of an equation in unknown quantities was alien to
Greek thought. [...] He had hit upon the conics in a successful
search for curves with the properties appropriate to the
duplication of the cube. In terms of modern notation the solution
is easily achieved. By shifting the cutting plane (Fig. 6.2), we
can find a parabola with any latus rectum. If, then, we wish to
duplicate a cube of edge a, we locate on a right-angled cone two
parabolas, one with latus rectum a and another with latus
rectum 2a. [...] It is probable that Menaechmus knew that
the duplication could be achieved also by the use of a rectangular
hyperbola and a parabola."
- Boyer, Carl B. (1991). A
History of Mathematics (Second ed.). John Wiley & Sons,
- Cooke, Roger (1997). The
History of Mathematics: A Brief Course. Wiley-Interscience. ISBN