Menaechmus

Menaechmus
Menaechmus
Born	c. 380 BC; Alopeconnesus
Died	c. 320 BC; Cyzicus[?]

Menaechmus (Greek: Μέναιχμος, c. 380 – c. 320 BC) was an ancient Greek mathematician, geometer and philosopher^[1] born in Alopeconnesus or Prokonnesos in the Thracian Chersonese, 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.

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.^[2] 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.^[3] 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.^[4] 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.^[4]

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 $a=1/2,b=0,c=0$ .^[5]

There are few direct sources for Menaechmus's 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 purely 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."^[6] However, this quotation is first attested by 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.

References

^ Suda, § mu.140
^ Cooke, Roger (1997). "The Euclidean Synthesis". The History of Mathematics : A Brief Course. New York: Wiley. p. 103. ISBN 9780471180821. 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 vertex angle of the cone is acute, the resulting section (called oxytome) 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. 5.7).
^ Boyer (1991). "The age of Plato and Aristotle". A History of Mathematics. Wiley. p. 93. ISBN 9780471543978. 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 discovery of the ellipse seems to have been made by Menaechmus as a mere by-product in a search in which it was the parabola and hyperbola that proffered the properties needed in the solution of the Delian problem.
^ ^a ^b Boyer (1991). "The age of Plato and Aristotle". A History of Mathematics. Wiley. pp. 104–105. ISBN 9780471543978. If OP=y and OD = x are coordinates of point P, we have y² = R).OV, or, on substituting equals, y² = R'D.OV = AR'.BC/AB.DO.BC/AB = AR'.BC²/AB².In as much 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.
^ Stillwell, John (2020), "Algebraic Geometry", Mathematics and Its History: A Concise Edition, Undergraduate Texts in Mathematics, Cham: Springer International Publishing, pp. 85–97, doi:10.1007/978-3-030-55193-3_6, ISBN 978-3-030-55193-3, retrieved 2024-04-26
^ *Beckmann, Petr (1989). A History of Pi (3rd ed.). Dorset Press. p. 34.

Sources

Boyer, Carl B. (1991). A History of Mathematics (Second ed.). John Wiley & Sons, Inc. ISBN 0-471-54397-7.
Cooke, Roger (1997). The History of Mathematics: A Brief Course. Wiley-Interscience. ISBN 0-471-18082-3.

External links

Menaechmus' Constructions (conics) at Convergence
O'Connor, John J.; Robertson, Edmund F., "Menaechmus", MacTutor History of Mathematics Archive, University of St Andrews
Article at Encyclopædia Britannica
Wolfram.com Biography
Fuentes González, Pedro Pablo, “Ménaichmos”, in R. Goulet (ed.), Dictionnaire des Philosophes Antiques, vol. IV, Paris, CNRS, 2005, p. 401-407.

[1] Suda, § mu.140

[2] Cooke, Roger (1997). "The Euclidean Synthesis". The History of Mathematics : A Brief Course. New York: Wiley. p. 103. ISBN 9780471180821. 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 vertex angle of the cone is acute, the resulting section (called oxytome) 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. 5.7).

[Boyer_Menaechmus-3] Boyer (1991). "The age of Plato and Aristotle". A History of Mathematics. Wiley. p. 93. ISBN 9780471543978. 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 discovery of the ellipse seems to have been made by Menaechmus as a mere by-product in a search in which it was the parabola and hyperbola that proffered the properties needed in the solution of the Delian problem.

[Boyer_Menaechmus_2-4] Boyer (1991). "The age of Plato and Aristotle". A History of Mathematics. Wiley. pp. 104–105. ISBN 9780471543978. If OP=y and OD = x are coordinates of point P, we have y² = R).OV, or, on substituting equals, y² = R'D.OV = AR'.BC/AB.DO.BC/AB = AR'.BC²/AB².In as much 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.

[5] Stillwell, John (2020), "Algebraic Geometry", Mathematics and Its History: A Concise Edition, Undergraduate Texts in Mathematics, Cham: Springer International Publishing, pp. 85–97, doi:10.1007/978-3-030-55193-3_6, ISBN 978-3-030-55193-3, retrieved 2024-04-26

[6] *Beckmann, Petr (1989). A History of Pi (3rd ed.). Dorset Press. p. 34.

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