Roger Cotes

Roger Cotes
FRS
Roger Cotes; FRS
	Patung Roger Cotes yang pembuatannya disponsori oleh Robert Smith dan dipahat oleh Peter Scheemakers pada tahun 1758.
Lahir	10 Juli 1682; Burbage, Leicestershire, Inggris
Meninggal	5 Juni 1716 (umur 33); Cambridge, Cambridgeshire, Inggris
Tempat tinggal	Inggris
Almamater	Trinity College, Cambridge
Dikenal atas	Spiral logaritma; Persegi tersempit; Formula Newton–Cotes; Formula Euler; konsep radian
	Karier ilmiah
Bidang	Matematika
Institusi	Trinity College, Cambridge
Pembimbing akademik	Isaac Newton; Richard Bentley
Mahasiswa ternama	Robert Smith; James Jurin; Stephen Gray
Terinspirasi	John Smith (pamannya)

Roger Cotes FRS (10 Juli 1682 – 5 Juni 1716) adalah seorang matematikawan Inggris yang dikenal karena bekerja sama dengan Isaac Newton dengan melakukan koreksi terhadap edisi kedua bukunya yang terkenal, Principia, sebelum buku tersebut diterbitkan. Dia juga dikenang karena menemukan rumus kuadrat yang dikenal sebagai rumus Newton-Cotes, dan membuat argumen geometris yang dapat ditafsirkan sebagai versi logaritmik dari rumus Euler.^[5] Dia adalah Profesor Plumian pertama di Universitas Cambridge dari tahun 1707 sampai hari kematiannya.

Masa muda

Cotes lahir di Burbage, Leicestershire. Kedua orangtuanya adalah Robert, rektor Burbage, dan Grace Farmer. Roger memiliki kakak laki-laki, Anthony (lahir pada tahun 1681), dan seorang adik perempuan, Susanna (lahir pada tahun 1683), keduanya meninggal di usia muda. Awalnya Roger bersekolah di Leicester School, di mana bakat matematikanya diakui. Bibinya, Hannah, menikah dengan Pendeta John Smith, dan Smith berperan sebagai tutor yang mendorong bakat Roger. Putra keluarga Smith, Robert Smith, menjadi rekan dekat Roger Cotes sepanjang hidupnya. Cotes kemudian belajar di St Paul's School di London dan masuk Trinity College, Cambridge, pada tahun 1699. Ia lulus dengan gelar BA pada 1702 dan MA pada 1706.^[2]

Astronomi

Kontribusi Roger Cotes untuk metode komputasi modern terletak pada bidang astronomi dan matematika. Cotes memulai karir pendidikannya dengan fokus pada ilmu astronomi. Ia menjadi anggota Trinity College pada tahun 1707, dan di usia 26 ia menjadi Profesor Astronomi dan Filsafat Eksperimental Plumian pertama. Pada pengangkatannya menjadi profesor, ia membuka pendaftaran penggunaan observatorium Trinity. Sayangnya, observatorium itu masih belum selesai ketika Cotes meninggal, dan dihancurkan pada tahun 1797.^[2]

Dalam korespondensi dengan Isaac Newton, Cotes merancang teleskop heliostat dengan cermin yang berputar dengan jarum jam.^[6]^[7] Dia menghitung ulang tabel matahari dan planet buatan Giovanni Domenico Cassini dan John Flamsteed, dan dia berniat membuat tabel pergerakan bulan berdasarkan prinsip Newton. Akhirnya, pada tahun 1707 ia membentuk sekolah ilmu fisika di Trinity dalam kemitraan dengan William Whiston.^[2]

Principia

Dari tahun 1709 hingga 1713, Cotes menjadi sangat terlibat dengan edisi kedua Newton's Principia, sebuah buku yang menjelaskan teori gravitasi universal Newton. Edisi pertama Principia hanya dicetak beberapa eksemplar dan membutuhkan revisi untuk memasukkan karya-karya Newton dan prinsip-prinsip teori bulan dan planet.^[2] Newton pada awalnya memiliki pendekatan kasual untuk revisi, karena dia telah menyerah pada pekerjaan ilmiah.^{[butuh rujukan]}Namun, melalui semangat kuat yang ditunjukkan oleh Cotes, rasa lapar ilmiah Newton sekali lagi dihidupkan kembali.^{[butuh rujukan]}Keduanya menghabiskan hampir tiga setengah tahun berkolaborasi pada karya tersebut, di mana mereka sepenuhnya menyimpulkan, dari hukum gerak Newton, teori bulan, ekuinoks, dan orbit komet. Hanya 750 eksemplar edisi kedua yang dicetak.^[2] Namun, salinan bajakan dari Amsterdam memenuhi semua permintaan lainnya.^{[butuh rujukan]} Sebagai hadiah untuk Cotes, ia diberi bagian dari keuntungan dan 12 salinan miliknya sendiri.^{[butuh rujukan]} Kontribusi asli Cotes untuk karya itu adalah kata pengantar yang mendukung keunggulan ilmiah prinsip-prinsip Newton atas teori gravitasi pusaran populer yang didukung oleh René Descartes. Cotes menyimpulkan bahwa hukum gravitasi Newton dibuktikan oleh pengamatan fenomena langit yang tidak sesuai dengan fenomena pusaran yang dipaparkan oleh para kritikus Cartesian.^[2]

Matematika

Karya asli utama Cotes adalah dalam matematika, terutama di bidang kalkulus integral, logaritma, dan analisis numerik. Dia hanya menerbitkan satu makalah ilmiah dalam hidupnya, berjudul Logometria, di mana dia berhasil membangun spiral logaritmik.^[8]^[9] Setelah kematiannya, banyak makalah matematika Cotes yang diedit oleh sepupunya Robert Smith dan diterbitkan dalam sebuah buku, Harmonia mensurarum.^[2]^[10] Karya tambahan Cotes kemudian diterbitkan dalam The Doctrine and Application of Fluxions karya Thomas Simpson.^[8] Meskipun gaya penulisan dan penjelasan Cotes tidak terlalu jelas, pendekatan sistematisnya terhadap integrasi dan teori matematika sangat dihargai oleh rekan-rekannya.^{[butuh rujukan]} Cotes menemukan teorema penting pada akar persatuan ke-n,^[11] meramalkan metode kuadrat terkecil,^[12] dan menemukan metode untuk mengintegrasikan pecahan rasional dengan penyebut binomial ^[8]^[13] Dia juga dipuji atas usahanya dalam metode numerik, terutama dalam metode interpolasi dan teknik konstruksi tabelny.^[8] Dia dianggap sebagai salah satu dari sedikit matematikawan Inggris yang mampu mengikuti karya hebat Sir Isaac Newton.^{[butuh rujukan]}

Kematian

Cotes meninggal karena demam hebat di Cambridge pada tahun 1716, di usia 33 tahun. Isaac Newton berkomentar, "Jika dia hidup, kita akan mengetahui sesuatu."^[2]

Lihat juga

Referensi

^ Gowing 2002, p. 5.
^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ Meli (2004)
^ Rusnock (2004) "Jurin, James (bap. 1684, d. 1750)", Oxford Dictionary of National Biography, Oxford University Press, retrieved 6 September 2007 berlangganan atau keanggotan Perpustakaan Umum Britania Raya diperlukan
^ Gowing 2002, p. 6.
^ 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 $EX+XC{\sqrt {-1}}$ & CE mensura ducta in ${\sqrt {-1}}$ ." (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 $EX+XC{\sqrt {-1}}$ & CE multiplied by ${\sqrt {-1}}$ .) 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 $EX+XC{\sqrt {-1}}$ and CE is thus $\cos \theta +{\sqrt {-1}}\sin \theta \$ . In Cotes' terminology, the "measure" of a quantity is its natural logarithm, and the "modulus" is a conversion factor that transforms a measure of angle into circular arc length (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 ${\sqrt {-1}}$ , equals the length of the circular arc subtended by θ, which for any angle measured in radians is CE • θ. Thus, ${\sqrt {-1}}CE\ln {\left(\cos \theta +{\sqrt {-1}}\sin \theta \right)\ }=(CE)\theta$ . This equation has the wrong sign: the factor of ${\sqrt {-1}}$ should be on the right side of the equation, not the left. If this change is made, then, after dividing both sides by CE and exponentiating both sides, the result is: $\cos \theta +{\sqrt {-1}}\sin \theta =e^{{\sqrt {-1}}\theta }$ , which is Euler's formula.
See:
^ Edleston, J., ed. (1850) Correspondence of Sir Isaac Newton and Professor Cotes, … (London, England: John W. Parker), "Letter XCVIII. Cotes to John Smith." (1708 February 10), pp. 197–200.
^ Kaw, Autar (2003-01-01). "cotes - A Historical Anecdote". mathforcollege.com. Diakses tanggal 2017-12-12.
^ ^a ^b ^c ^d O'Connor & Robertson (2005)
^ In Logometria, Cotes evaluated e, the base of natural logarithms, to 12 decimal places. See: Roger Cotes (1714) "Logometria," Philosophical Transactions of the Royal Society of London, 29 (338) : 5-45; see especially the bottom of page 10. From page 10: "Porro eadem ratio est inter 2,718281828459 &c et 1, … " (Furthermore, the same ratio is between 2.718281828459… and 1, … )
^ Harmonia mensurarum contains a chapter of comments on Cotes' work by Robert Smith. On page 95, Smith gives the value of 1 radian for the first time. See: Roger Cotes with Robert Smith, ed., Harmonia mensurarum … (Cambridge, England: 1722), chapter: Editoris notæ ad Harmoniam mensurarum, top of page 95. From page 95: After stating that 180° corresponds to a length of π (3.14159…) along a unit circle (i.e., π radians), Smith writes: "Unde Modulus Canonis Trigonometrici prodibit 57.2957795130 &c. " (Whence the conversion factor of trigonometric measure, 57.2957795130… [degrees per radian], will appear.)
^ Roger Cotes with Robert Smith, ed., Harmonia mensurarum … (Cambridge, England: 1722), chapter: "Theoremata tum logometrica tum triogonometrica datarum fluxionum fluentes exhibentia, per methodum mensurarum ulterius extensam" (Theorems, some logorithmic, some trigonometric, which yield the fluents of given fluxions by the method of measures further developed), pages 113-114.
^ Roger Cotes with Robert Smith, ed., Harmonia mensurarum … (Cambridge, England: 1722), chapter: "Aestimatio errorum in mixta mathesis per variationes partium trianguli plani et sphaerici" Harmonia mensurarum ..., pages 1-22, see especially page 22. From page 22: "Sit p locus Objecti alicujus ex Observatione prima definitus, … ejus loco tutissime haberi potest." (Let p be the location of some object defined by observation, q, r, s, the locations of the same object from subsequent observations. Let there also be weights P, Q, R, S reciprocally proportional to the displacements that may arise from the errors in the single observations, and that are given from the given limits of error; and the weights P, Q, R, S are conceived as being placed at p, q, r, s, and their center of gravity Z is found: I say the point Z is the most probable location of the object, and may be most safely had for its true place. [Ronald Gowing, 1983, p. 107])
^ Cotes presented his method in a letter to William Jones, dated 5 May 1716. An excerpt from the letter which discusses the method was published in: [Anon.] (1722), Book review: "An account of a book, intitled, Harmonia Mensurarum, … ," Philosophical Transactions of the Royal Society of London, 32 : 139-150 ; see pages 146-148.

Sumber

Pranala luar

"Harmonia Mensurarum". MathPages. Diakses tanggal 2007-09-07. - A more complete account of Cotes's involvement with Principia, followed by an even more thorough discussion of his mathematical work.
Roger Cotes at the Mathematics Genealogy Project

[1] Gowing 2002, p. 5.

[ODNB-2] ^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ Meli (2004)

[3] Rusnock (2004) "Jurin, James (bap. 1684, d. 1750)", Oxford Dictionary of National Biography, Oxford University Press, retrieved 6 September 2007 berlangganan atau keanggotan Perpustakaan Umum Britania Raya diperlukan

[4] Gowing 2002, p. 6.

[5] 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 $EX+XC{\sqrt {-1}}$ & CE mensura ducta in ${\sqrt {-1}}$ ." (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 $EX+XC{\sqrt {-1}}$ & CE multiplied by ${\sqrt {-1}}$ .) 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 $EX+XC{\sqrt {-1}}$ and CE is thus $\cos \theta +{\sqrt {-1}}\sin \theta \$ . In Cotes' terminology, the "measure" of a quantity is its natural logarithm, and the "modulus" is a conversion factor that transforms a measure of angle into circular arc length (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 ${\sqrt {-1}}$ , equals the length of the circular arc subtended by θ, which for any angle measured in radians is CE • θ. Thus, ${\sqrt {-1}}CE\ln {\left(\cos \theta +{\sqrt {-1}}\sin \theta \right)\ }=(CE)\theta$ . This equation has the wrong sign: the factor of ${\sqrt {-1}}$ should be on the right side of the equation, not the left. If this change is made, then, after dividing both sides by CE and exponentiating both sides, the result is: $\cos \theta +{\sqrt {-1}}\sin \theta =e^{{\sqrt {-1}}\theta }$ , which is Euler's formula.
See:

[6] Edleston, J., ed. (1850) Correspondence of Sir Isaac Newton and Professor Cotes, … (London, England: John W. Parker), "Letter XCVIII. Cotes to John Smith." (1708 February 10), pp. 197–200.

[7] Kaw, Autar (2003-01-01). "cotes - A Historical Anecdote". mathforcollege.com. Diakses tanggal 2017-12-12.

[mactutor-8] O'Connor & Robertson (2005)

[9] In Logometria, Cotes evaluated e, the base of natural logarithms, to 12 decimal places. See: Roger Cotes (1714) "Logometria," Philosophical Transactions of the Royal Society of London, 29 (338) : 5-45; see especially the bottom of page 10. From page 10: "Porro eadem ratio est inter 2,718281828459 &c et 1, … " (Furthermore, the same ratio is between 2.718281828459… and 1, … )

[10] Harmonia mensurarum contains a chapter of comments on Cotes' work by Robert Smith. On page 95, Smith gives the value of 1 radian for the first time. See: Roger Cotes with Robert Smith, ed., Harmonia mensurarum … (Cambridge, England: 1722), chapter: Editoris notæ ad Harmoniam mensurarum, top of page 95. From page 95: After stating that 180° corresponds to a length of π (3.14159…) along a unit circle (i.e., π radians), Smith writes: "Unde Modulus Canonis Trigonometrici prodibit 57.2957795130 &c. " (Whence the conversion factor of trigonometric measure, 57.2957795130… [degrees per radian], will appear.)

[11] Roger Cotes with Robert Smith, ed., Harmonia mensurarum … (Cambridge, England: 1722), chapter: "Theoremata tum logometrica tum triogonometrica datarum fluxionum fluentes exhibentia, per methodum mensurarum ulterius extensam" (Theorems, some logorithmic, some trigonometric, which yield the fluents of given fluxions by the method of measures further developed), pages 113-114.

[12] Roger Cotes with Robert Smith, ed., Harmonia mensurarum … (Cambridge, England: 1722), chapter: "Aestimatio errorum in mixta mathesis per variationes partium trianguli plani et sphaerici" Harmonia mensurarum ..., pages 1-22, see especially page 22. From page 22: "Sit p locus Objecti alicujus ex Observatione prima definitus, … ejus loco tutissime haberi potest." (Let p be the location of some object defined by observation, q, r, s, the locations of the same object from subsequent observations. Let there also be weights P, Q, R, S reciprocally proportional to the displacements that may arise from the errors in the single observations, and that are given from the given limits of error; and the weights P, Q, R, S are conceived as being placed at p, q, r, s, and their center of gravity Z is found: I say the point Z is the most probable location of the object, and may be most safely had for its true place. [Ronald Gowing, 1983, p. 107])

[13] Cotes presented his method in a letter to William Jones, dated 5 May 1716. An excerpt from the letter which discusses the method was published in: [Anon.] (1722), Book review: "An account of a book, intitled, Harmonia Mensurarum, … ," Philosophical Transactions of the Royal Society of London, 32 : 139-150 ; see pages 146-148.

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