Menaechmus

Menaxmus (grekcha: Μέναιχμος , miloddan avvalgi 380-320-yillar) qadimgi yunon matematiki, geometriyachisi va faylasufi^[1] Frakiya chersonezidagi Alopekonnes yoki Prokonnesda tugʻilgan, taniqli faylasuf Platon bilan doʻstligi hamda konus kesmalarini aniq kashf etishi va uning yechimi bilan mashhur boʻlgan. Parabola va giperbola yordamida kubni ikki barobarga oshirish masalasini yechgan.

Hayoti va ish faoliyati

Menaxm konus kesimlarini kashf etgani va kubni ikki barobarga oshirish masalasini yechigani uchun matematiklar xotirasida qoladi^[2] .Menexmus, ehtimol, Delian muammosini hal qilish yoʻlini izlashning qoʻshimcha mahsuloti sifatida konusning kesmalarini, yaʼni ellips, parabola va giperbolani kashf qilgan^[3]. Menaxmus parabolada y ² = L x ekanligini bilar edi, bu yerda L doimiy latus rektum deb ataladi, ammo u ikkita nomaʼlumdagi har qanday tenglama egri chiziqni aniqlashini bilmagan^[4]. Koʻrinishidan, u konusning bu xususiyatlarini va boshqalarni ham yecha olgan. Ushbu maʼlumotlardan foydalanib, endi ikkita parabola kesishgan nuqtalar uchun echish yoʻli bilan kubni takrorlash masalasiga yechim topish mumkin edi, bu yechim kub tenglamani yechishga teng^[4] edi.

Menaxm ijodi uchun bevosita manbalar kam; Uning konus kesimlari boʻyicha ishi, birinchi navbatda, Eratosthenes epigrammasidan maʼlum va uning ukasi (kvadratrix yordamida berilgan doiraga teng kvadrat hosil qilish usulini ishlab chiqqan) Dinostratning yutugʻi faqat uning Proclus yozuvlaridan ham maʼlum .Proclus, shuningdek, Menaxmusga Evdoks tomonidan oʻrgatganini eslatib oʻtadi. Plutarxning hayratlanarli bayonoti bor, Platon Menaxmusning mexanik qurilmalar yordamida oʻzining ikki baravar kubli yechimiga erishganini maʼqullamagan; Hozirda maʼlum boʻlgan dalil faqat algebraik koʻrinadi.

Menaxmus Makedoniyalik Iskandarning tarbiyachisi boʻlgan; Bu eʼtiqod quyidagi latifadan kelib chiqadi: goʻyoki, bir kuni Iskandar undan geometriyani tushuntirishi uchun yordam soʻraganida, u shunday javob berdi: „Ey podshoh, mamlakat boʻylab sayohat qilish uchun qirollik yoʻli va oddiy fuqarolar uchun yoʻllar bor, lekin geometriyada hamma uchun bitta yoʻl bor“. (Bekmann, Pi tarixi, 1989, p. 34) Biroq, bu iqtibos birinchi boʻlib miloddan avvalgi 500-yillarda Stobaeus tomonidan tasdiqlangan va shuning uchun Menaxmus haqiqatan ham Iskandarga oʻrgatganmi yoki yoʻqmi, noaniq.

Uning qayerda vafot etgani ham nomaʼlum, ammo zamonaviy olimlar u oxir-oqibatCyzicus vafot etgan deb hisoblashadi.

Manbalar

↑ Suda, § mu.140
↑ Cooke, Roger „The Euclidean Synthesis“, . The History of Mathematics : A Brief Course. New York: Wiley, 1997 — 103-bet. 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 „The age of Plato and Aristotle“, . A History of Mathematics, 1991 — 93-bet. 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.“
↑ ^4,0 ^4,1 Boyer „The age of Plato and Aristotle“, . A History of Mathematics, 1991 — 104–105-bet. 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.“

Beckmann, Petr. A History of Pi, 3rd, Dorset Press, 1989.
Boyer, Carl B.. A History of Mathematics, Second, John Wiley & Sons, Inc., 1991. ISBN 0-471-54397-7.
Cooke, Roger. The History of Mathematics: A Brief Course. Wiley-Interscience, 1997. ISBN 0-471-18082-3.

[1] Suda, § mu.140

[2] Cooke, Roger „The Euclidean Synthesis“, . The History of Mathematics : A Brief Course. New York: Wiley, 1997 — 103-bet. 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 „The age of Plato and Aristotle“, . A History of Mathematics, 1991 — 93-bet. 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] 4,0 ^4,1 Boyer „The age of Plato and Aristotle“, . A History of Mathematics, 1991 — 104–105-bet. 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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