COTES (Roger)
, a very eminent mathematician, philosopher, and astronomer, was born July 10, 1682, at Burbach in Leicestershire, where his father Robert was rector. He was first placed at Leicester school; where, at 12 years of age, he discovered a strong inclination to the mathematics. This being observed by his uncle, the Rev. Mr. John Smith, he gave him all the encouragement he could; and prevailed on his father to send him for some time to his house in Lincolnshire, that he might assist him in those studies: and here he laid the foundation of that deep and extensive knowledge in that science, for which he was afterward so deservedly famous. He was hence removed to St. Paul's school, London, where he made a great progress in classical learning; and yet he found so much leisure as to support a constant correspondence with his uncle, not only in mathematics, but also in metaphysics, philosophy, and divinity. His next remove was to Trinity College Cambridge, where he took his degrees, and became fellow.
Jan. 1706, he was appointed professor of astronomy and experimental philosophy, upon the foundation of Dr Thomas Plume, archdeacon of Rochester; being the first that enjoyed that office, to which he was unanimously chosen, on account of his high reputation and merits. He entered into orders in 1713; and the same year, at the desire of Dr. Bentley, he published at Cambridge the second edition of Newton's Mathematica Principia; inserting all the improvements which the author had made to that time. To this edition he prefixed a most admirable preface, in which he pointed out the true method of philosophising, shewing the foundation on which the Newtonian philosophy was raised, and refuting the objections of the Cartesians and all other philosophers against it.
The publication of this edition of Newton's Principia added greatly to his reputation; nor was the high opinion the public now conceived of him in the least diminished, but rather much increased, by several productions of his own, which afterward appeared. He gave in the Philos. Transactions, two papers, viz, 1, Logometria, in vol. 29; and a Description of the great fiery meteor that was seen March 6, 1716, in vol. 31.
This extraordinary genius in the mathematics died, to the great regret of the university, and all the lovers of the sciences, June 5, 1716, in the very prime of his life, being not quite 34 years of age.
Mr. Cotes left behind him some very ingenious, and indeed admirable tracts, part of which, with the Lo- | gometria above mentioned, were published, in 1722, by Dr. Robert Smith, his cousin and successor in his professorship, afterward master of Trinity College, under the title of Harmonia Mensurarum, which contains a number of very ingenious and learned works: see the Introduction to my Logarithms. He wrote also a Compendium of Arithmetic; of the Resolution of Equations; of Dioptrics; and of the Nature of Curves. Beside these pieces, he drew up, in the time of his lectures, a course of Hydrostatical and Pneumatical Lectures, in English, which were published also by Dr. Smith in 8vo, 1737, and are held in great estimation.
So high an opinion had Sir Isaac Newton of our author's genius, that he used to say, “If Cotes had lived, we had known something.”
COTESIAN theorem, in Geometry, an appellation used for an elegant property of the circle discovered by Mr. Cotes. The theorem is this:
If the factors of the binomial ac<01> xc be required, the index c being an integer number. With the centre O, and radius AO = a, describe a circle, and di- vide its circumferance into as many equal parts as there are units in 2c, at the points A, B, C, D, &c; then in the radius, produced if necessary, take OP = x, and from the point P, to all the points of division in the circumference, draw the lines PA, PB, PC, &c; so shall these lines taken alternately be the factors sought; viz, , according as the point P is within or without the circle.
For instance, if c = 5, divide the circumference into 10 equal parts, and the point P being within the circle, then will .
In like manner, if c = 6, having divided the circumference into 12 equal parts, then will .
The demonstration of this theorem may be seen in Dr. Pemberton's Epist. de Cotesii inventis. See also Dr. Smith's Theoremata Logometrica and Trigonometrica, added to Cotes's Harm. Mens. pa. 114; De Moivre Miscel. Analyt. pa. 17; and Waring's Letter to Dr. Powell, pa. 39.
By means of this theorem, the acute and elegant author was enabled to make a farther progress in the inverse method of Fluxions, than had been done before. But in the application of his discovery there still remained a limitation, which was removed by Mr. De Moivre. Vide ut supra.
COVERT-Way, in fortification, a space of ground level with the adjoining country, on the outer edge of the ditch, ranging quite round all the works. This is otherwise called the corridor, and has a parapet with its banquette and glacis, which form the height of the parapet. It is sometimes also called the counterscarp, because it is on the edge of the scarp.
One of the greatest difficulties in a siege, is to make a lodgment on the covert-way; because it is usual for the besieged to palisade it along the middle, and undermine it on all sides.
