Roger Cotes

Roger Cotes
FRS
Roger Cotes FRS
	This bust was commissioned by Robert Smith and sculpted posthumously by Peter Scheemakers in 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 Cotes 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 Principia, before publication. He also invented the quadrature formulas known as Newton–Cotes formulas, and made a geometric argument that can be interpreted as a logarithmic version of Euler's formula.^[4] He was the first Plumian Professor at Cambridge University from 1707 until his death.

^ Gowing 2002, p. 5.
^ Cite error: The named reference ODNB was invoked but never defined (see the help page).
^ Rusnock (2004) "Jurin, James (bap. 1684, d. 1750)", Oxford Dictionary of National Biography, Oxford University Press, retrieved 6 September 2007 (subscription or UK public library membership required)
^
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 $EX+XC{\sqrt {-1}}$ $EX+XC{\sqrt {-1}}$ & CE multiplied by ${\sqrt {-1}}$ ${\sqrt {-1}}$ .) 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 $EX+XC{\sqrt {-1}}$ $EX+XC{\sqrt {-1}}$ and CE is 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 by θ, which for any angle measured in radians is CE • θ. Thus, ${\sqrt {-1}}CE\ln {\left(\cos \theta +{\sqrt {-1}}\sin \theta \right)\ }=(CE)\theta$ ${\sqrt {-1}}CE\ln {\left(\cos \theta +{\sqrt {-1}}\sin \theta \right)\ }=(CE)\theta$ . 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 by CE and 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.

[1] Gowing 2002, p. 5.

[ODNB-2] Cite error: The named reference ODNB was invoked but never defined (see the help page).

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

[4] 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:
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.

[1]

[2]

[3]

[4]

Roger Cotes FRS
This bust was commissioned by Robert Smith and sculpted posthumously by Peter Scheemakers in 1758.
Born	(1682-07-10)10 July 1682 Burbage, Leicestershire, England
Died	5 June 1716(1716-06-05) (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^[1]
Notable students	Robert Smith^[2] James Jurin^[3] Stephen Gray