Editing Roger Cotes (section)

==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}}