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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Roger Cotes ： ウィキペディア英語版 Roger Cotes Roger Cotes FRS (10 July 1682 – 5 June 1716) was an English mathematician, known for working closely with Isaac Newton by proofreading the second edition of his famous book, the ''Principia'', before publication. He also invented the quadrature formulas known as Newton–Cotes formulas and first introduced what is known today as Euler's formula.〔Cotes wrote: ''"Nam si quadrantis circuli quilibet arcus, radio ''CE'' descriptus, sinun habeat ''CX'' sinumque complementi ad quadrantem ''XE'' ; sumendo radium ''CE'' pro Modulo, arcus erit rationis inter $EX\; +\; XC\; \backslash sqrt$& ''CE'' mensura ducta in $\backslash sqrt$."'' (Thus if any arc of a quadrant of a circle, described by the radius ''CE'', has sinus ''CX'' and sinus of the complement to the quadrant ''XE'' ; taking the radius ''CE'' as modulus, the arc will be the measure of the ratio between $EX\; +\; XC\; \backslash sqrt$ & ''CE'' multiplied by $\backslash sqrt$.) That is, consider a circle having center ''E'' (at the origin of the (x,y) plane) and radius ''CE''. Consider an angle ''θ'' with its vertex at ''E'' having the positive x-axis as one side and a radius ''CE'' as the other side. The perpendicular from the point ''C'' on the circle to the x-axis is the "sinus" ''CX'' ; the line between the circle's center ''E'' and the point ''X'' at the foot of the perpendicular is ''XE'', which is the "sinus of the complement to the quadrant" or "cosinus". The ratio between $EX\; +\; XC\; \backslash sqrt$ and ''CE'' is thus $\backslash cos\; \backslash theta\; +\; \backslash sqrt\; \backslash sin\; \backslash theta\; \backslash$. In Cotes' terminology, the "measure" of a quantity is its natural logarithm, and the "modulus" is a conversion factor that transforms circular arc length into a measure of angle (here, the modulus is the radius (''CE'') of the circle). According to Cotes, the product of the modulus and the measure (logarithm) of the ratio, when multiplied by $\backslash sqrt$, equals the length of the circular arc subtended by ''θ'', which for any angle measured in radians is ''CE'' • ''θ''. Thus, $\backslash sqrt\; CE\; \backslash ln\; =\; (CE)\; \backslash theta$. This equation has the wrong sign: the factor of $\backslash sqrt$ should be on the right side of the equation, not the left side. If this change is made, then, after dividing both sides by ''CE'' and exponentiating both sides, the result is: $\backslash cos\; \backslash theta\; +\; \backslash sqrt\; \backslash sin\; \backslash theta\; =\; e^$, which is Euler's formula. See: *Roger Cotes (1714) "Logometria," ''Philosophical Transactions of the Royal Society of London'', 29 (338) : 5-45 ; see especially page 32. Available on-line at: (Hathi Trust ) *Roger Cotes with Robert Smith, ed., ''Harmonia mensurarum'' … (Cambridge, England: 1722), chapter: "Logometria", (page 28. )〕 He was the first Plumian Professor at Cambridge University from 1707 until his death, on June 5th. ==Early life== Cotes was born in Burbage, Leicestershire. His parents were Robert, the rector of Burbage, and his wife Grace ''née'' Farmer. Roger had an elder brother, Anthony (born 1681) and a younger sister, Susanna (born 1683). At first Roger attended Leicester School where his mathematical talent was recognised. His aunt Hannah had married Rev. John Smith, and Smith took on the role of tutor to encourage Roger's talent. The Smiths' son, Robert Smith, would become a close associate of Roger Cotes throughout his life. Cotes later studied at St Paul's School in London and entered Trinity College, Cambridge in 1699. He graduated BA in 1702 and MA in 1706.〔Meli (2004)〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Roger Cotes」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.