Menaechmus (, c. 380 – c. 320 BC) was an

ancient Greek Ancient Greek (, ; ) includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Greek ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

, geometer and philosopher born in Alopeconnesus or Prokonnesos in the Thracian Chersonese, who was known for his friendship with the renowned philosopher

Plato Plato ( ; Greek language, Greek: , ; born BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...

and for his apparent discovery of

conic section A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...

s and his solution to the then-long-standing problem of

doubling the cube Doubling the cube, also known as the Delian problem, is an ancient geometry, geometric problem. Given the Edge (geometry), edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first ...

using the

parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...

and

hyperbola In mathematics, a hyperbola is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected component ( ...

Life and work

Menaechmus is remembered by mathematicians for his discovery of the

conic sections A conic section, conic or a quadratic curve is a curve obtained from a Conical surface, cone's surface intersecting a plane (mathematics), plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is ...

and his solution to the problem of doubling the cube. Menaechmus likely discovered the conic sections, that is, the

ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

, the

, and the

, as a by-product of his search for the solution to the Delian problem. Menaechmus knew that in a parabola y² = ''L''x, where ''L'' is a constant called the ''

latus rectum A conic section, conic or a quadratic curve is a curve obtained from a Conical surface, cone's surface intersecting a plane (mathematics), plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is ...

'', 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 Doubling the cube, also known as the Delian problem, is an ancient geometry, geometric problem. Given the Edge (geometry), edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first ...

by solving for the points at which two parabolas intersect, a solution equivalent to solving a cubic equation. In modern notation, let

xy = 1

be a hyperbola, and

y = ax^2 + bx + c

be a parabola, then their intersections are the solutions to

ax^3 + bx^2 + cx = 1

. Now set

a = 1/2, b = 0, c = 0

. There are few direct sources for Menaechmus's work; his work on conic sections is known primarily from an

epigram An epigram is a brief, interesting, memorable, sometimes surprising or satirical statement. The word derives from the Greek (, "inscription", from [], "to write on, to inscribe"). This literary device has been practiced for over two millennia ...

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 Dinostratus (; c. 390 – c. 320 BCE) was a Greece, Greek mathematician and geometer, and the brother of Menaechmus. He is known for using the quadratrix to solve the problem of squaring the circle. Life and work Dinostratus' chief contribution ...

, is known solely from the writings of

Proclus Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor (, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers of late antiquity. He set forth one of th ...

. Proclus also mentions that Menaechmus was taught by Eudoxus. There is a curious statement by

Plutarch Plutarch (; , ''Ploútarchos'', ; – 120s) was a Greek Middle Platonist philosopher, historian, biographer, essayist, and priest at the Temple of Apollo (Delphi), Temple of Apollo in Delphi. He is known primarily for his ''Parallel Lives'', ...

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 purely algebraic. Menaechmus was said to have been the tutor of

Alexander the Great Alexander III of Macedon (; 20/21 July 356 BC – 10/11 June 323 BC), most commonly known as Alexander the Great, was a king of the Ancient Greece, ancient Greek kingdom of Macedonia (ancient kingdom), Macedon. He succeeded his father Philip ...

; 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."* However, this quotation is first attested by

Stobaeus Joannes Stobaeus (; ; 5th-century AD), from Stobi in Macedonia (Roman province), Macedonia, was the compiler of a valuable series of extracts from Greek authors. The work was originally divided into two volumes containing two books each. The tw ...

, 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 Cyzicus ( ; ; ) was an ancient Greek town in Mysia in Anatolia in the current Balıkesir Province of Turkey. It was located on the shoreward side of the present Kapıdağ Peninsula (the classical Arctonnesus), a tombolo which is said to have or ...

References

Sources

* *

External links

Menaechmus' Constructions (conics)
a
Convergence
*
Article
at

Encyclopædia Britannica The is a general knowledge, general-knowledge English-language encyclopaedia. It has been published by Encyclopædia Britannica, Inc. since 1768, although the company has changed ownership seven times. The 2010 version of the 15th edition, ...

Wolfram.com Biography
* Fuentes González, Pedro Pablo, �
Ménaichmos
��, in R. Goulet (ed.), ''Dictionnaire des Philosophes Antiques'', vol. IV, Paris, CNRS, 2005, p. 401-407. {{Authority control Ancient Greek geometers History of geometry 380 BC births 320 BC deaths Philosophers and tutors of Alexander the Great 4th-century BC Greek mathematicians