メナイクモス

メナイクモス（ギリシア語: Μέναιχμος、前380年 - 前320年）はトラキアのケルソネソスのアロペコネソス（英語版）またはプロコネソスで生まれた古代ギリシアの数学者、幾何学者、哲学者である^[1]。彼は著名な哲学者プラトンとの友人関係、円錐断面の発見、そして放物線と双曲線を利用し、当時長い間未解決問題として著名だった立方体倍積問題を解決したことで知られている。

生涯と業績

数学者にとって、メナイクモスは円錐断面の発見及び立方体倍積問題を解決したことで記憶されている^[2]。メナイクモスはおそらくデロス島の問題の解法を探るうちに副産物として円錐断面（楕円、放物線、双曲線を含む）を発見したと思われる^[3]。メナイクモスは放物線が y² = Lx （L は通径 latus rectum と呼ばれる定数）という式で表されることを知っていたが、2つの未知数に関するすべての方程式が曲線を決めるという事実には気付いていなかった^[4]。彼は、これらの円錐断面の性質やその他の性質も導出したと思われる。その情報を利用して2放物線の交点について解くことで、立方体倍積問題を解決した。このことは三次方程式の解を求めることと等価である^[4]。

メナイクモスの業績に関する直接的な資料はわずかである。円錐断面に関する業績は主にエラトステネスのエピグラムによって知られている。彼の弟であるディノストラトス（英語版）の業績（円積曲線（英語版）を利用した円積問題の解法）はプロクロスの著作からのみ知られる。プロクロスは、メナイクモスがエウドクソスから教えを受けたことにも言及している。プルタルコスは、プラトンがメナイクモスによる機械的手法を用いた解法を認めなかったという興味深い記述を残している。今日的な証明は純粋な代数学的なものに限られていたようである。

メナイクモスはアレクサンドロス大王の家庭教師を務めたといわれている。このことは次の様な小噺によって知られている。アレクサンドロスが彼に幾何学を理解する近道がないか尋ねた時、彼は「王よ、国々を旅するには国王のための道と庶民の道がある。だが、幾何学においては道は一つしか存在しない」と返答したという（Beckmann, A History of Pi, 1989, p. 34）。しかしながら、この逸話はストバイオス（500年頃の人物）以前に遡ることができず、それゆえメナイクモスが実際にアレクサンドロスを教育したかどうかは明らかでない。

彼がどこで死去したかも明らかでないが、現在、学者は彼がキュジコスで亡くなったと考えている。

脚注

^ 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. 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. 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."

参考文献

Beckmann, Petr (1989). A History of Pi (3rd ed.). Dorset Press
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

外部リンク

Menaechmus' Constructions (conics) at Convergence
O'Connor, John J.; Robertson, Edmund F., “メナイクモス”, 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. 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. 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."

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