Editing Roger Cotes

{{Short description|English mathematician (1682–1716)}}
{{Use dmy dates|date=July 2021}}
{{Use British English|date=June 2012}}
{{Infobox scientist
| honorific_suffix  = {{postnominals|country=GBR|size=100|FRS}}
| image             = Roger Cotes.png
| image_size        = 250px
| caption           = This bust was commissioned by [[Robert Smith (mathematician)|Robert Smith]] and sculpted posthumously by [[Peter Scheemakers]] in 1758.
| birth_date        = {{Birth date|1682|7|10|df=y}}
| birth_place       = [[Burbage, Leicestershire]], England
| death_date        = {{death date and age|1716|6|5|1682|7|10|df=y}}
| death_place       = [[Cambridge]], [[Cambridgeshire]], England
| field             = [[Mathematician]]
| work_institutions = [[Trinity College, Cambridge]]
| alma_mater        = [[Trinity College, Cambridge]]
| doctoral_advisor  = <!--There was not doctorate in Cambridge before 1912-->
| academic_advisors = [[Isaac Newton]]<br>[[Richard Bentley]]<ref>Gowing 2002, p. 5.</ref>
| doctoral_students = <!--There was not doctorate in Cambridge before 1912-->
| notable_students  = [[Robert Smith (mathematician)|Robert Smith]]<ref name="ODNB"/><br>[[James Jurin]]<ref>Rusnock (2004) "[http://www.oxforddnb.com/view/article/15173 Jurin, James (bap. 1684, d. 1750)]", ''[[Oxford Dictionary of National Biography]]'', Oxford University Press, retrieved 6 September 2007 {{ODNBsub}}</ref><br>[[Stephen Gray (scientist)|Stephen Gray]]
| known_for         = [[Logarithmic spiral]]<br>[[Least squares]]<br>[[Newton–Cotes formulas]]<br>[[Euler's formula|Euler's formula proof]]<br>[[Radian|Concept of the radian]]
}}

'''Roger Cotes''' {{postnominals|country=GBR|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 ''[[Philosophiae Naturalis Principia Mathematica|Principia]]'', before publication. He also devised the [[quadrature (mathematics)|quadrature]] formulas known as [[Newton–Cotes formulas]], which originated from Newton's research,<ref>{{Cite book |url= |title=The Cambridge Companion to Newton |date=2016 |publisher=[[Cambridge University Press]] |isbn=978-1-139-05856-8 |editor-last=Iliffe |editor-first=Rob |edition=2nd |pages=411 |doi=10.1017/cco9781139058568 |editor-last2=Smith |editor-first2=George E.}}</ref> and made a geometric argument that can be interpreted as a logarithmic version of [[Euler's formula]].<ref>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 <math>EX + XC \sqrt{-1}</math>& ''CE'' mensura ducta in <math>\sqrt{-1}</math>."'' (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 <math>EX + XC \sqrt{-1}</math> & ''CE'' multiplied by <math>\sqrt{-1}</math>.) 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 <math>EX + XC \sqrt{-1}</math> and ''CE'' is thus <math>\cos \theta + \sqrt{-1} \sin \theta \ </math>.  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 <math>\sqrt{-1}</math>, equals the length of the circular arc subtended by ''θ'', which for any angle measured in radians is ''CE'' • ''θ''. Thus, <math>\sqrt{-1} CE \ln{\left ( \cos \theta + \sqrt{-1} \sin \theta \right ) \ } = (CE) \theta </math>. This equation has the wrong sign:  the factor of <math>\sqrt{-1}</math> 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:  <math>\cos \theta + \sqrt{-1} \sin \theta = e^{\sqrt{-1} \theta}</math>, which is Euler's formula.<br>
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:  [http://babel.hathitrust.org/cgi/pt?id=ucm.5324351035;view=2up;seq=38 Hathi Trust]
*Roger Cotes with Robert Smith, ed., ''Harmonia mensurarum'' … (Cambridge, England:  1722), chapter:  "Logometria",  [https://books.google.com/books?id=J6BGAAAAcAAJ&pg=PA28 p. 28].</ref> He was the first [[Plumian Professor of Astronomy and Experimental Philosophy|Plumian Professor]] at [[Cambridge University]] from 1707 until his death.

==Early life==
Cotes was born in [[Burbage, Leicestershire]]. His parents were Robert, the [[Rector (ecclesiastical)|rector]] of Burbage, and his wife, Grace, ''née'' Farmer. 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 (mathematician)|Robert Smith]], became a close associate of Roger Cotes throughout his life. Cotes later studied at [[St Paul's School (London)|St Paul's School]] in [[London]] and entered [[Trinity College, Cambridge]], in 1699.<ref>{{acad|id=CTS699R|name=Cotes, Roger}}</ref> He [[graduation|graduated]] [[Bachelor of Arts|BA]] in 1702 and [[Master of Arts|MA]] in 1706.<ref name="ODNB">Meli (2004)</ref>

==Astronomy==
Roger Cotes's contributions to modern [[Computational mathematics|computational methods]] lie heavily in the fields of [[astronomy]] and mathematics. Cotes began his educational career with a focus on [[astronomy]]. He became a [[Fellow#Oxford, Cambridge and Dublin|fellow]] of 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 an [[observatory]] for Trinity. Unfortunately, the observatory was still unfinished when Cotes died, and was demolished in 1797.<ref name="ODNB"/>

In correspondence with Isaac Newton, Cotes designed a [[heliostat]] telescope with a mirror revolving by clockwork.<ref>Edleston, J., ed. (1850) ''Correspondence of Sir Isaac Newton and Professor Cotes, …'' (London, England:  John W. Parker), [https://archive.org/stream/correspondenceof00newtrich#page/197/mode/2up "Letter XCVIII.  Cotes to John Smith."] (1708 February 10), pp. 197–200.</ref><ref>{{Cite web|url=http://mathforcollege.com/nm/anecdotes/cotes.html|title=cotes - A Historical Anecdote|last=Kaw|first=Autar|date=2003-01-01|website=mathforcollege.com|access-date=2017-12-12}}</ref> He recomputed the solar and planetary tables of [[Giovanni Domenico Cassini]] and [[John Flamsteed]], and he intended to create tables of the [[moon]]'s [[planetary motion|motion]], based on Newtonian principles.{{Citation needed|date=September 2007}} Finally, in 1707 he formed a school of physical sciences at Trinity in partnership with [[William Whiston]].<ref name="ODNB"/>

==The ''Principia''==
From 1709 to 1713, Cotes became heavily involved with the second edition of Newton's ''Principia'', a book that explained Newton's theory of [[universal gravitation]]. The first edition of ''Principia'' had only a few copies printed and was in need of revision to include Newton's works and principles of lunar and planetary theory.<ref name="ODNB"/> Newton at first had a casual approach to the revision, since he had all but given up scientific work.{{Citation needed|date=September 2007}} However, through the vigorous passion displayed by Cotes, Newton's scientific hunger was once again reignited.{{Citation needed|date=September 2007}} The two spent nearly three and half years collaborating on the work, in which they fully deduce, from [[Newton's laws of motion]], the theory of the [[moon]], the [[equinox]]es, and the [[orbit]]s of [[comet]]s. Only 750 copies of the second edition were printed<ref name="ODNB"/> although pirated copies from [[Amsterdam]] were also distributed to meet the demand for the work.{{Citation needed|date=September 2007}} As a reward to Cotes, he was given a share of the profits and 12 copies of his own.{{Citation needed|date=September 2007}} Cotes's original contribution to the work was a preface which supported the scientific superiority of Newton's principles over the then popular [[Mechanical explanations of gravitation|vortex theory of gravity]] advocated by [[René Descartes]]. Cotes concluded that the Newton's law of gravitation was confirmed by observation of celestial phenomena that were inconsistent with the vortex theory.<ref name="ODNB"/>

==Mathematics==
Cotes's major original work was in mathematics, especially in the fields of [[integral calculus]], [[logarithm]]s, and [[numerical analysis]]. He published only one [[scientific paper]] in his lifetime, titled ''Logometria'', in which he successfully constructs the [[logarithmic spiral]].<ref name="mactutor">O'Connor & Robertson (2005)</ref><ref>In ''Logometria'', Cotes evaluated [[e (mathematical constant)|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; [http://babel.hathitrust.org/cgi/pt?id=ucm.5324351035;view=2up;seq=16 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, … )</ref> After his death, many of Cotes's mathematical papers were edited by his cousin Robert Smith and published in a book, ''Harmonia mensurarum''.<ref name="ODNB"/><ref>''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, [https://books.google.com/books?id=J6BGAAAAcAAJ&pg=RA1-PA95 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.)</ref> Cotes's additional works were later published in [[Thomas Simpson]]'s ''The Doctrine and Application of Fluxions''.<ref name="mactutor"/> Although Cotes's style was somewhat obscure, his systematic approach to [[integral|integration]] and mathematical theory was highly regarded by his peers.{{Citation needed|date=September 2007}} Cotes discovered an important theorem on the ''n''-th [[root of unity|roots of unity]],<ref>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), [https://books.google.com/books?id=J6BGAAAAcAAJ&pg=PA113 pages 113-114.]</ref> foresaw the method of [[least squares]],<ref>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 [https://books.google.com/books?id=J6BGAAAAcAAJ&pg=RA1-PA20 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])</ref> and discovered a method for integrating [[rational fraction]]s with [[binomial (polynomial)|binomial]] [[denominator]]s.<ref name="mactutor"/><ref>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 [http://babel.hathitrust.org/cgi/pt?id=ucm.5324350998;view=2up;seq=166 pages 146-148.]</ref> He was also praised for his efforts in numerical methods, especially in [[interpolation]] methods and his table construction techniques.<ref name="mactutor"/> He was regarded as one of the few British mathematicians capable of following the powerful work of Sir Isaac Newton.{{Citation needed|date=September 2007}}

==Death and assessment==
Cotes died from a violent fever in [[Cambridge]] in 1716 at the early age of 33. Isaac Newton remarked, "If he had lived we would have known something."<ref name="ODNB"/>

==See also==
*[[Cotes's spiral]]
*[[Extended Euclidean algorithm]]
*[[Newton–Cotes formulas]]
*[[Lituus (mathematics)]]

==References==
{{reflist}}

==Sources==
* [Anon.] {{cite EB1911 |wstitle=Cotes, Roger |volume=7 |short=1}}
* {{cite book | author=Cohen, I. B. | title=Introduction to Newton's "Principia" | year=1971 | isbn=0-674-46193-2 | publisher=Harvard University Press | location=Harvard }}
* {{cite book | title=Correspondence of Sir Isaac Newton and Professor Cotes | editor=Edleston, J. | year=1850 |url=https://archive.org/details/correspondenceof00newtrich/page/n7}} via [[Internet Archive]]
* {{cite book | author=Gowing, R. | title=Roger Cotes: Natural Philosopher | year=2002 | isbn=0-521-52649-3 | location=London | publisher=Cambridge University Press }}
* {{cite book | author=Koyré, A. | author-link=Alexandre Koyré | title=Newtonian Studies | year=1965 | pages=273–82 | isbn=0-412-42300-6 | location=London | publisher=Chapman & Hall }}
* {{cite journal | author=Price, D. J. | title=The early observatory instruments of Trinity College, Cambridge | journal=Annals of Science | volume=8 | year=1952 | pages=1–12  | doi=10.1080/00033795200200012}}
* {{cite book | title=The Correspondence of Isaac Newton | author=Turnbull, H. W.| edition=7 vols | pages=vols.5–6 | year=1975–1976 | location=London | publisher=Cambridge University Press | no-pp=true |display-authors=etal}}
* {{cite book | title=Isaac Newton's Philosophiae Naturalis Principia Mathematica: The Third Edition (1726) with Variant Readings | editor=Whitman, A. | year=1972 | pages=817–26 | isbn=0-521-07960-8 | location=London | publisher=Cambridge University Press }}

==External links==
* {{cite web | url=http://www.mathpages.com/home/kmath192/kmath192.htm | title=''Harmonia Mensurarum'' | work=MathPages | access-date=2007-09-07 }}- A more complete account of Cotes's involvement with ''Principia'', followed by an even more thorough discussion of his mathematical work.
* {{MathGenealogy|id=103067}}
* {{MacTutor|id=Cotes}}
* Meli, D. B. (2004) "[http://www.oxforddnb.com/view/article/6386 Cotes, Roger (1682–1716)]", ''[[Oxford Dictionary of National Biography]]'', Oxford University Press, retrieved 7 September 2007 {{ODNBsub}}

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[[Category:1682 births]]
[[Category:1716 deaths]]
[[Category:18th-century English mathematicians]]
[[Category:Mathematical analysts]]
[[Category:Alumni of Trinity College, Cambridge]]
[[Category:Fellows of Trinity College, Cambridge]]
[[Category:Fellows of the Royal Society]]
[[Category:People educated at St Paul's School, London]]
[[Category:English scientific instrument makers]]
[[Category:People from Burbage, Leicestershire]]
[[Category:Plumian Professors of Astronomy and Experimental Philosophy]]