Editing Roger Cotes (section)

{{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.