Roger Cotes

Roger Cotes
FRS
Roger Cotes FRS
	This bust was commissioned byRobert Smithand sculpted posthumously byPeter Scheemakersin 1758.
Born	10 July 1682; Burbage, Leicestershire,England
Died	5 June 1716(aged 33); Cambridge,Cambridgeshire,England
Alma mater	Trinity College, Cambridge
Known for	Logarithmic spiral; Least squares; Newton–Cotes formulas; Euler's formula proof; Concept of the radian
	Scientific career
Fields	Mathematician
Institutions	Trinity College, Cambridge
Academic advisors	Isaac Newton; Richard Bentley
Notable students	Robert Smith; James Jurin; Stephen Gray

Roger CotesFRS(10 July 1682 – 5 June 1716) was an Englishmathematician,known for working closely withIsaac Newtonby proofreading the second edition of his famous book, thePrincipia,before publication. He also invented thequadratureformulas known asNewton–Cotes formulas,and made a geometric argument that can be interpreted as a logarithmic version ofEuler's formula.^[4]He was the firstPlumian ProfessoratCambridge Universityfrom 1707 until his death.

Early life

Cotes was born inBurbage, Leicestershire.His parents were Robert, therectorof Burbage, and his wife, Grace,néeFarmer. Roger had an elder brother, Anthony (born 1681), and a younger sister, Susanna (born 1683), both of whom died young. 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,became a close associate of Roger Cotes throughout his life. Cotes later studied atSt Paul's SchoolinLondonand enteredTrinity College, Cambridge,in 1699.^[5]Hegraduated BAin 1702 andMAin 1706.^[2]

Astronomy

Roger Cotes's contributions to moderncomputational methodslie heavily in the fields ofastronomyand mathematics. Cotes began his educational career with a focus onastronomy.He became afellowof Trinity College in 1707, and at age 26 he became the first Plumian Professor of Astronomy and Experimental Philosophy. On his appointment to professor, he opened a subscription list in an effort to provide anobservatoryfor Trinity. Unfortunately, the observatory was still unfinished when Cotes died, and was demolished in 1797.^[2]

In correspondence with Isaac Newton, Cotes designed aheliostattelescope with a mirror revolving by clockwork.^[6]^[7]He recomputed the solar and planetary tables ofGiovanni Domenico CassiniandJohn Flamsteed,and he intended to create tables of themoon'smotion,based on Newtonian principles.^{[citation needed]}Finally, in 1707 he formed a school of physical sciences at Trinity in partnership withWilliam Whiston.^[2]

ThePrincipia

From 1709 to 1713, Cotes became heavily involved with the second edition of Newton'sPrincipia,a book that explained Newton's theory ofuniversal gravitation.The first edition ofPrincipiahad only a few copies printed and was in need of revision to include Newton's works and principles of lunar and planetary theory.^[2]Newton at first had a casual approach to the revision, since he had all but given up scientific work.^{[citation needed]}However, through the vigorous passion displayed by Cotes, Newton's scientific hunger was once again reignited.^{[citation needed]}The two spent nearly three and half years collaborating on the work, in which they fully deduce, fromNewton's laws of motion,the theory of themoon,theequinoxes,and theorbitsofcomets.Only 750 copies of the second edition were printed^[2]although pirated copies fromAmsterdamwere also distributed to meet the demand for the work.^{[citation needed]}As a reward to Cotes, he was given a share of the profits and 12 copies of his own.^{[citation needed]}Cotes's original contribution to the work was a preface which supported the scientific superiority of Newton's principles over the then popularvortex theory of gravityadvocated byRené Descartes.Cotes concluded that the Newton's law of gravitation was confirmed by observation of celestial phenomena that were inconsistent with the vortex theory.^[2]

Mathematics

Cotes's major original work was in mathematics, especially in the fields ofintegral calculus,logarithms,andnumerical analysis.He published only onescientific paperin his lifetime, titledLogometria,in which he successfully constructs thelogarithmic spiral.^[8]^[9]After his death, many of Cotes's mathematical papers were edited by his cousin Robert Smith and published in a book,Harmonia mensurarum.^[2]^[10]Cotes's additional works were later published inThomas Simpson'sThe Doctrine and Application of Fluxions.^[8]Although Cotes's style was somewhat obscure, his systematic approach tointegrationand mathematical theory was highly regarded by his peers.^{[citation needed]}Cotes discovered an important theorem on then-throots of unity,^[11]foresaw the method ofleast squares,^[12]and discovered a method for integratingrational fractionswithbinomial denominators.^[8]^[13]He was also praised for his efforts in numerical methods, especially ininterpolationmethods and his table construction techniques.^[8]He was regarded as one of the few British mathematicians capable of following the powerful work of Sir Isaac Newton.^{[citation needed]}

Death and assessment

Cotes died from a violent fever inCambridgein 1716 at the early age of 33. Isaac Newton remarked, "If he had lived we would have known something."^[2]

References

^Gowing 2002, p. 5.
^^a ^b ^c ^d ^e ^f ^g ^h ⁱMeli (2004)
^Rusnock (2004) "Jurin, James (bap. 1684, d. 1750)",Oxford Dictionary of National Biography,Oxford University Press, retrieved 6 September 2007(subscription orUK public library membershiprequired)
^
Cotes wrote:"Nam si quadrantis circuli quilibet arcus, radioCEdescriptus, sinun habeatCXsinumque complementi ad quadrantemXE;sumendo radiumCEpro Modulo, arcus erit rationis inter $EX+XC{\sqrt {-1}}$ &CEmensura ducta in ${\sqrt {-1}}$ ."(Thus if any arc of a quadrant of a circle, described by the radiusCE,has sinusCXand sinus of the complement to the quadrantXE;taking the radiusCEas modulus, the arc will be the measure of the ratio between $EX+XC{\sqrt {-1}}$ $EX+XC{\sqrt {-1}}$ &CEmultiplied by ${\sqrt {-1}}$ ${\sqrt {-1}}$ .) That is, consider a circle having centerE(at the origin of the (x, y) plane) and radiusCE.Consider an angleθwith its vertex atEhaving the positive x-axis as one side and a radiusCEas the other side. The perpendicular from the pointCon the circle to the x-axis is the "sinus"CX;the line between the circle's centerEand the pointXat the foot of the perpendicular isXE,which is the "sinus of the complement to the quadrant" or "cosinus". The ratio between $EX+XC{\sqrt {-1}}$ $EX+XC{\sqrt {-1}}$ andCEis thus $\cos \theta +{\sqrt {-1}}\sin \theta \$ $\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 ${\sqrt {-1}}$ ${\sqrt {-1}}$ ,equals the length of the circular arc subtended byθ,which for any angle measured in radians isCE•θ.Thus, ${\sqrt {-1}}CE\ln {\left(\cos \theta +{\sqrt {-1}}\sin \theta \right)\ }=(CE)\theta$ ${\sqrt {-1}}CE\ln {\left(\cos \theta +{\sqrt {-1}}\sin \theta \right)\ }=(CE)\theta$ .This equation has the wrong sign: the factor of ${\sqrt {-1}}$ ${\sqrt {-1}}$ should be on the right side of the equation, not the left. If this change is made, then, after dividing both sides byCEand exponentiating both sides, the result is: $\cos \theta +{\sqrt {-1}}\sin \theta =e^{{\sqrt {-1}}\theta }$ $\cos \theta +{\sqrt {-1}}\sin \theta =e^{{\sqrt {-1}}\theta }$ ,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",p. 28.
^"Cotes, Roger (CTS699R)".A Cambridge Alumni Database.University of Cambridge.
^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 (1 January 2003)."cotes - A Historical Anecdote".mathforcollege.Retrieved12 December2017.
^^a ^b ^c ^dO'Connor & Robertson (2005)
^InLogometria,Cotes evaluatede, 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 mensurarumcontains a chapter of comments on Cotes' work by Robert Smith. On page 95, Smith gives the value of 1radianfor 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 especiallypage 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; seepages 146-148.

Sources

[Anon.]"Cotes, Roger".Encyclopædia Britannica.Vol. 7 (11th ed.). 1911.
Cohen, I. B. (1971).Introduction to Newton's "Principia".Harvard: Harvard University Press.ISBN 0-674-46193-2.
Edleston, J., ed. (1850).Correspondence of Sir Isaac Newton and Professor Cotes.viaInternet Archive
Gowing, R. (2002).Roger Cotes: Natural Philosopher.London: Cambridge University Press.ISBN 0-521-52649-3.
Koyré, A.(1965).Newtonian Studies.London: Chapman & Hall. pp. 273–82.ISBN 0-412-42300-6.
Price, D. J. (1952). "The early observatory instruments of Trinity College, Cambridge".Annals of Science.8:1–12.doi:10.1080/00033795200200012.
Turnbull, H. W.; et al. (1975–1976).The Correspondence of Isaac Newton(7 vols ed.). London: Cambridge University Press. vols.5–6.
Whitman, A., ed. (1972).Isaac Newton's Philosophiae Naturalis Principia Mathematica: The Third Edition (1726) with Variant Readings.London: Cambridge University Press. pp. 817–26.ISBN 0-521-07960-8.

External links

"Harmonia Mensurarum".MathPages.Retrieved7 September2007.- A more complete account of Cotes's involvement withPrincipia,followed by an even more thorough discussion of his mathematical work.
Roger Cotesat theMathematics Genealogy Project
O'Connor, John J.;Robertson, Edmund F.,"Roger Cotes",MacTutor History of Mathematics Archive,University of St Andrews
Meli, D. B. (2004) "Cotes, Roger (1682–1716)",Oxford Dictionary of National Biography,Oxford University Press, retrieved 7 September 2007(subscription orUK public library membershiprequired)

[1] Gowing 2002, p. 5.

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

[3] Rusnock (2004) "Jurin, James (bap. 1684, d. 1750)",Oxford Dictionary of National Biography,Oxford University Press, retrieved 6 September 2007(subscription orUK public library membershiprequired)

[4] Cotes wrote:"Nam si quadrantis circuli quilibet arcus, radioCEdescriptus, sinun habeatCXsinumque complementi ad quadrantemXE;sumendo radiumCEpro Modulo, arcus erit rationis inter $EX+XC{\sqrt {-1}}$ &CEmensura ducta in ${\sqrt {-1}}$ ."(Thus if any arc of a quadrant of a circle, described by the radiusCE,has sinusCXand sinus of the complement to the quadrantXE;taking the radiusCEas modulus, the arc will be the measure of the ratio between $EX+XC{\sqrt {-1}}$ &CEmultiplied by ${\sqrt {-1}}$ .) That is, consider a circle having centerE(at the origin of the (x, y) plane) and radiusCE.Consider an angleθwith its vertex atEhaving the positive x-axis as one side and a radiusCEas the other side. The perpendicular from the pointCon the circle to the x-axis is the "sinus"CX;the line between the circle's centerEand the pointXat the foot of the perpendicular isXE,which is the "sinus of the complement to the quadrant" or "cosinus". The ratio between $EX+XC{\sqrt {-1}}$ andCEis 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 byθ,which for any angle measured in radians isCE•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 byCEand exponentiating both sides, the result is: $\cos \theta +{\sqrt {-1}}\sin \theta =e^{{\sqrt {-1}}\theta }$ ,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",p. 28.

[5] 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

[6] Roger Cotes with Robert Smith, ed.,Harmonia mensurarum…(Cambridge, England: 1722), chapter: "Logometria",p. 28.

[5] "Cotes, Roger (CTS699R)".A Cambridge Alumni Database.University of Cambridge.

[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 (1 January 2003)."cotes - A Historical Anecdote".mathforcollege.Retrieved12 December2017.

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

[9] InLogometria,Cotes evaluatede, 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 mensurarumcontains a chapter of comments on Cotes' work by Robert Smith. On page 95, Smith gives the value of 1radianfor 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 especiallypage 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; seepages 146-148.

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