Roger Cotes FRS (July 10, 1682 – June 5, 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 first introduced what is known today as Euler's formula. He was the first Plumian Professor at Cambridge University from 1707 until his death.

## Early life

Cotes was born in

Burbage, Leicestershire. 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 then at

Trinity College, Cambridge. He

graduated BA in 1702 and

MA in 1706.

## 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 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 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 motion, the theory of the

moon, the

equinoxes, and the

orbits of

comets. 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.

## Mathematics

Cotes's major original work was in mathematics, especially in the fields of

integral calculus,

logarithms, 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.

## 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."

## References

## Bibliography

## External links

- - A more complete account of Cotes's involvement with Principia, followed by an even more thorough discussion of his mathematical work.