Roger Cotes
Roger Cotes | |
---|---|
Born | Burbage, Leicestershire, England | 10 July 1682
Died | 5 June 1716 Cambridge, Cambridgeshire, England | (aged 33)
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 |
Roger Cotes
Early life
Cotes was born in
Astronomy
Roger Cotes's contributions to modern 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 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.[2]
In correspondence with Isaac Newton, Cotes designed a
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
Mathematics
Cotes's major original work was in mathematics, especially in the fields of
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."[2]
See also
References
- ^ Gowing 2002, p. 5.
- ^ a b c d e f g h i Meli (2004)
- Oxford Dictionary of National Biography, Oxford University Press, retrieved 6 September 2007 (subscription or UK public library membershiprequired)
- ^ 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 & CE mensura ducta in ." (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 & CE multiplied by .) 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 and CE is thus . 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 , equals the length of the circular arc subtended by θ, which for any angle measured in radians is CE • θ. Thus, . This equation has the wrong sign: the factor of 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: , 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.com. Retrieved 12 December 2017.
- ^ a b c d O'Connor & Robertson (2005)
- ^ In Logometria, Cotes evaluated 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; 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 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, 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 especially 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])
- ^ 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 pages 146-148.
Sources
- [Anon.] 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. via Internet Archive
- Gowing, R. (2002). Roger Cotes: Natural Philosopher. London: Cambridge University Press. ISBN 0-521-52649-3.
- ISBN 0-412-42300-6.
- Price, D. J. (1952). "The early observatory instruments of Trinity College, Cambridge". Annals of Science. 8: 1–12. .
- 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. Retrieved 7 September 2007.- A more complete account of Cotes's involvement with Principia, followed by an even more thorough discussion of his mathematical work.
- Roger Cotes at the Mathematics 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 or UK public library membershiprequired)