Menaechmus
Encyclopedia
Menaechmus was an ancient Greek
mathematician
and geometer born in Alopeconnesus in the Thracian Chersonese
, who was known for his friendship with the renowned philosopher Plato
and for his apparent discovery of conic section
s and his solution to the then-long-standing problem of doubling the cube
using the parabola
and hyperbola
.
, 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
.
Ancient Greek
Ancient Greek is the stage of the Greek language in the periods spanning the times c. 9th–6th centuries BC, , c. 5th–4th centuries BC , and the c. 3rd century BC – 6th century AD of ancient Greece and the ancient world; being predated in the 2nd millennium BC by Mycenaean Greek...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and geometer born in Alopeconnesus in the Thracian Chersonese
Thracian Chersonese
The Thracian Chersonese was the ancient name of the Gallipoli peninsula, in the part of historic Thrace that is now part of modern Turkey.The peninsula runs in a south-westerly direction into the Aegean Sea, between the Hellespont and the bay of Melas . Near Agora it was protected by a wall...
, who was known for his friendship with the renowned philosopher Plato
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
and for his apparent discovery of conic section
Conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
s and his solution to the then-long-standing problem of doubling the cube
Doubling the cube
Doubling the cube is one of the three most famous geometric problems unsolvable by compass and straightedge construction...
using the parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
and hyperbola
Hyperbola
In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
.
Life and work
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 ellipseEllipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
, the parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
, and the hyperbola
Hyperbola
In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
, 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
Epigram
An epigram is a brief, interesting, usually memorable and sometimes surprising statement. Derived from the epigramma "inscription" from ἐπιγράφειν epigraphein "to write on inscribe", this literary device has been employed for over two millennia....
by Eratosthenes
Eratosthenes
Eratosthenes of Cyrene was a Greek mathematician, poet, athlete, geographer, astronomer, and music theorist.He was the first person to use the word "geography" and invented the discipline of geography as we understand it...
, and the accomplishment of his brother (of devising a method to create a square equal in area to a given circle using the quadratrix
Quadratrix
In mathematics, a quadratrix is a curve having ordinates which are a measure of the area of another curve. The two most famous curves of this class are those of Dinostratus and E. W...
), Dinostratus
Dinostratus
Dinostratus was a 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 :...
, is known solely from the writings of Proclus
Proclus
Proclus Lycaeus , called "The Successor" or "Diadochos" , was a Greek Neoplatonist philosopher, one of the last major Classical philosophers . He set forth one of the most elaborate and fully developed systems of Neoplatonism...
. Proclus also mentions that Menaechmus was taught by Eudoxus
Eudoxus
Eudoxus or Eudoxos was the name of two ancient Greeks:* Eudoxus of Cnidus , Greek astronomer and mathematician.* Eudoxus of Cyzicus , Greek navigator....
. There is a curious statement by Plutarch
Plutarch
Plutarch then named, on his becoming a Roman citizen, Lucius Mestrius Plutarchus , c. 46 – 120 AD, was a Greek historian, biographer, essayist, and Middle Platonist known primarily for his Parallel Lives and Moralia...
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
Stobaeus
Joannes Stobaeus , from Stobi in 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...
, 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
Cyzicus was an ancient town of Mysia in Anatolia in the current Balıkesir Province of Turkey. It was located on the shoreward side of the present Kapıdağ Peninsula , a tombolo which is said to have originally been an island in the Sea of Marmara only to be connected to the mainland in historic...
.
External links
- Menaechmus' Constructions (conics) at Convergence
- Article at Encyclopedia BritannicaEncyclopædia BritannicaThe Encyclopædia Britannica , published by Encyclopædia Britannica, Inc., is a general knowledge English-language encyclopaedia that is available in print, as a DVD, and on the Internet. It is written and continuously updated by about 100 full-time editors and more than 4,000 expert...
- Wolfram.com Biography