Roger Cotes
Encyclopedia
Roger Cotes FRS 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 first introduced what is known today as Euler's formula
. He was the first Plumian Professor
at Cambridge University from 1707 until his death.
. His parents were Robert, the rector
of Burbage, and his wife Grace née Farmer. Roger had an elder brother, Anthony (born 1681) and a younger sister, Susanna (born 1683). 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
, would become a close associate of Roger Cotes throughout his life. Cotes later studied at St Paul's School in London
and entered Trinity College, Cambridge
in 1699. He graduated
BA
in 1702 and MA
in 1706.
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 still was unfinished when Cotes died, and was demolished in 1797.
In correspondence with Isaac Newton, Cotes designed a heliostat
telescope with a mirror revolving by clockwork. He recomputed the solar and planetary tables of Giovanni Domenico Cassini
and John Flamsteed
, and he intended to create tables of the moon
's motion, based on Newtonian principles. Finally, in 1707 he formed a school of physical sciences at Trinity in partnership with William Whiston
.
, the theory of the moon
, the equinox
es, and the orbit
s of comet
s. Only 750 copies of the second edition were printed. However, a pirate copy from Amsterdam
met all other demand. As reward to Cotes, he was given a share of the profits and 12 copies of his own. Cotes's original contribution to the work was a preface which supported the scientific superiority of Newton's principles over the then popular vortex theory of gravity
advocated by René Descartes
. Cotes concluded that the Newton's law of gravitation was confirmed by observation of celestial phenomenon that were inconsistent with the vortex phenomena that Cartesian critics alleged.
s, and numerical analysis
. He published only one scientific paper in his lifetime, entitled Logometrica, in which he successfully constructs the logarithmic spiral
. After his death, many of Cotes's mathematical papers were hastily edited by Robert Smith and published in a book, Harmonia mensurarum. Cotes's additional works were later published in Thomas Simpson
's The Doctrine and Application of Fluxions. Although Cotes's style was somewhat obscure, his systematic approach to integration
and mathematical theory was highly regarded by his peers. Cotes discovered an important theorem on the nth roots of unity
, foresaw the method of least squares
, and he discovered a method for integrating rational fractions with binomial
denominators. He was also praised for his efforts in numerical methods, especially in interpolation
methods and his table construction techniques. He was regarded as one of the few British mathematicians capable of following the powerful work of Sir Isaac Newton.
in 1716 at the early age of 33. Isaac Newton remarked, "If he had lived we would have known something."
England
England is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, known for working closely with Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
by proofreading the second edition of his famous book, the Principia
Philosophiae Naturalis Principia Mathematica
Philosophiæ Naturalis Principia Mathematica, Latin for "Mathematical Principles of Natural Philosophy", often referred to as simply the Principia, is a work in three books by Sir Isaac Newton, first published 5 July 1687. Newton also published two further editions, in 1713 and 1726...
, before publication. He also invented the quadrature
Quadrature (mathematics)
Quadrature — historical mathematical term which means calculating of the area. Quadrature problems have served as one of the main sources of mathematical analysis.- History :...
formulas known as Newton–Cotes formulas and first introduced what is known today as Euler's formula
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...
. He was the first Plumian Professor
Plumian Professor of Astronomy and Experimental Philosophy
The Plumian chair of Astronomy and Experimental Philosophy is one of the two major Professorships in Astronomy at Cambridge University, alongside the Lowndean Professorship. The chair is currently held at the Institute of Astronomy in the University...
at Cambridge University from 1707 until his death.
Early life
Cotes was born in Burbage, LeicestershireBurbage, Leicestershire
Burbage is a civil parish in Leicestershire, England. It is a southern suburb of the town of Hinckley. According to the United Kingdom Census 2001 the parish had a population of 14,324.-History:...
. His parents were Robert, the rector
Rector
The word rector has a number of different meanings; it is widely used to refer to an academic, religious or political administrator...
of Burbage, and his wife Grace née Farmer. Roger had an elder brother, Anthony (born 1681) and a younger sister, Susanna (born 1683). 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
Robert Smith (mathematician)
Robert Smith was an English mathematician and music theorist.-Life:Smith was probably born at Lea near Gainsborough, the son of the rector of Gate Burton, Lincolnshire...
, would become a close associate of Roger Cotes throughout his life. Cotes later studied at St Paul's School in London
London
London is the capital city of :England and the :United Kingdom, the largest metropolitan area in the United Kingdom, and the largest urban zone in the European Union by most measures. Located on the River Thames, London has been a major settlement for two millennia, its history going back to its...
and entered Trinity College, Cambridge
Trinity College, Cambridge
Trinity College is a constituent college of the University of Cambridge. Trinity has more members than any other college in Cambridge or Oxford, with around 700 undergraduates, 430 graduates, and over 170 Fellows...
in 1699. He graduated
Graduation
Graduation is the action of receiving or conferring an academic degree or the ceremony that is sometimes associated, where students become Graduates. Before the graduation, candidates are referred to as Graduands. The date of graduation is often called degree day. The graduation itself is also...
BA
Bachelor of Arts
A Bachelor of Arts , from the Latin artium baccalaureus, is a bachelor's degree awarded for an undergraduate course or program in either the liberal arts, the sciences, or both...
in 1702 and MA
Master of Arts (postgraduate)
A Master of Arts from the Latin Magister Artium, is a type of Master's degree awarded by universities in many countries. The M.A. is usually contrasted with the M.S. or M.Sc. degrees...
in 1706.
Astronomy
Roger Cotes's contributions to modern computational methodsComputational mathematics
Computational mathematics involves mathematical research in areas of science where computing plays a central and essential role, emphasizing algorithms, numerical methods, and symbolic methods. Computation in the research is prominent. Computational mathematics emerged as a distinct part of applied...
lie heavily in the fields of astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
and mathematics. Cotes began his educational career with a focus on astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
. 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
Observatory
An observatory is a location used for observing terrestrial or celestial events. Astronomy, climatology/meteorology, geology, oceanography and volcanology are examples of disciplines for which observatories have been constructed...
for Trinity. Unfortunately, the observatory still was unfinished when Cotes died, and was demolished in 1797.
In correspondence with Isaac Newton, Cotes designed a heliostat
Heliostat
A heliostat is a device that includes a mirror, usually a plane mirror, which turns so as to keep reflecting sunlight toward a predetermined target, compensating for the sun's apparent motions in the sky. The target may be a physical object, distant from the heliostat, or a direction in space...
telescope with a mirror revolving by clockwork. He recomputed the solar and planetary tables of Giovanni Domenico Cassini
Giovanni Domenico Cassini
This article is about the Italian-born astronomer. For his French-born great-grandson, see Jean-Dominique Cassini.Giovanni Domenico Cassini was an Italian/French mathematician, astronomer, engineer, and astrologer...
and John Flamsteed
John Flamsteed
Sir John Flamsteed FRS was an English astronomer and the first Astronomer Royal. He catalogued over 3000 stars.- Life :Flamsteed was born in Denby, Derbyshire, England, the only son of Stephen Flamsteed...
, and he intended to create tables of the moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
's motion, based on Newtonian principles. Finally, in 1707 he formed a school of physical sciences at Trinity in partnership with William Whiston
William Whiston
William Whiston was an English theologian, historian, and mathematician. He is probably best known for his translation of the Antiquities of the Jews and other works by Josephus, his A New Theory of the Earth, and his Arianism...
.
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. Newton at first had a casual approach to the revision, since he had all but given up scientific work. However, through the vigorous passion displayed by Cotes, Newton's scientific hunger was once again reignited. The two spent nearly three and half years collaborating on the work, in which they fully deduce, from Newton's laws of motionNewton's laws of motion
Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...
, the theory of the moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
, the equinox
Equinox
An equinox occurs twice a year, when the tilt of the Earth's axis is inclined neither away from nor towards the Sun, the center of the Sun being in the same plane as the Earth's equator...
es, and the orbit
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...
s of comet
Comet
A comet is an icy small Solar System body that, when close enough to the Sun, displays a visible coma and sometimes also a tail. These phenomena are both due to the effects of solar radiation and the solar wind upon the nucleus of the comet...
s. Only 750 copies of the second edition were printed. However, a pirate copy from Amsterdam
Amsterdam
Amsterdam is the largest city and the capital of the Netherlands. The current position of Amsterdam as capital city of the Kingdom of the Netherlands is governed by the constitution of August 24, 1815 and its successors. Amsterdam has a population of 783,364 within city limits, an urban population...
met all other demand. As reward to Cotes, he was given a share of the profits and 12 copies of his own. Cotes's original contribution to the work was a preface which supported the scientific superiority of Newton's principles over the then popular vortex theory of gravity
Mechanical explanations of gravitation
Mechanical explanations of gravitation are attempts to explain the action of gravity by aid of basic mechanical processes, such as pressure forces caused by pushes, and without the use of any action at a distance. These theories were developed from the 16th until the 19th century in connection...
advocated by René Descartes
René Descartes
René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...
. Cotes concluded that the Newton's law of gravitation was confirmed by observation of celestial phenomenon that were inconsistent with the vortex phenomena that Cartesian critics alleged.
Mathematics
Cotes's major original work was in mathematics, especially in the fields of integral calculus, logarithmLogarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
s, and numerical analysis
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
. He published only one scientific paper in his lifetime, entitled Logometrica, in which he successfully constructs the logarithmic spiral
Logarithmic spiral
A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, "the marvelous...
. After his death, many of Cotes's mathematical papers were hastily edited by Robert Smith and published in a book, Harmonia mensurarum. Cotes's additional works were later published in Thomas Simpson
Thomas Simpson
Thomas Simpson FRS was a British mathematician, inventor and eponym of Simpson's rule to approximate definite integrals...
's The Doctrine and Application of Fluxions. Although Cotes's style was somewhat obscure, his systematic approach to integration
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
and mathematical theory was highly regarded by his peers. Cotes discovered an important theorem on the nth roots of unity
Root of unity
In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...
, foresaw the method of least squares
Least squares
The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...
, and he discovered a method for integrating rational fractions with binomial
Binomial
In algebra, a binomial is a polynomial with two terms —the sum of two monomials—often bound by parenthesis or brackets when operated upon...
denominators. He was also praised for his efforts in numerical methods, especially in interpolation
Interpolation
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....
methods and his table construction techniques. He was regarded as one of the few British mathematicians capable of following the powerful work of Sir Isaac Newton.
Death and assessment
Cotes died from a violent fever in CambridgeCambridge
The city of Cambridge is a university town and the administrative centre of the county of Cambridgeshire, England. It lies in East Anglia about north of London. Cambridge is at the heart of the high-technology centre known as Silicon Fen – a play on Silicon Valley and the fens surrounding the...
in 1716 at the early age of 33. Isaac Newton remarked, "If he had lived we would have known something."