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Roger Cotes (1682--1716) --- Historical Sketch

Roger Cotes was born Leicestershire, England, in 1682, and he died from an unexplained violent fever at age 32. Of this sad event, Isaac Newton said,MATHeven though relations between the two had become strained by the time of Cotes' death

By the time Roger Cotes was 12, it was clear to his teachers that he had an abundance of talent for mathematics. He then lived with his uncle who tutored him in the subject.

Cotes graduated from Cambridge University with a bachelor's degree at age 19 and continued his studies there. At age 25. he was appointed the university's first Plumian Professor of Astronomy and Experimental Philosophy. While he did not distinguish himself as a very observant and dedicated astronomer, his mathematical abilities shone through; in this regard, he was considered in his generation in England to be second only to Isaac Newton. He was elected as a fellow of the Royal Society at age 29 and ordained as a priest at age 30.

The period 1709--1713 was central to Cotes' short life. During these years he assisted Newton in revising Newton's seminal book Philosophiae Naturalis Principia Mathematica (often referred to as Principia or sometimes Principia Mathematica). In that book, Newton described universal gravitation, giving laws that govern motion---both terrestrial and celestial---and he gave a theoretical explanation of Kepler's laws of planetary motion. These laws argue that planets and comets travel in orbits that are elliptical, parabolic, or hyperbolic (the conic sections). When they began on the revision, Newton and Cotes were friends, and Newton penned a paragraph of thanks to Cotes, to be included in the book's preface. But by the time the book was published, Newton had omitted the paragraph.

You have learned in this course about the Newton-Cotes method of approximating integrals. That is just one of Cotes' many interests in approximations. He is said to have invented radian measure of angles. He calculated the continued-fraction representation of $e$, and he made substantial advances in calculus, interpolation processes, and logarithmic calculations

One of the mysteries associated with Cotes is how he arrived at the approximationMATHThe many guesses as to how he did this seems oddly to exclude generalized Greek ladders, coupled with the Farey fraction inequality MATHthat you were asked to establish in the Historical Sketch for Nicolas Chuquet. The Greek ladder for the square root of $2$ begins with a row of two $1$'s:MATHThen each row MATHis followed by the row MATHSo the ladder proceeds asMATHand the ratios of the second terms over the first terms MATHetc., are increasingly closer approximations to $\sqrt{2}.$ If you play this game starting with a row of three $1$'s, you get closer and closer to $\root{3} \of{2}$, and if you start with a row of four $1$'s, you get closer and closer approximations to $\root{4} \of{2}$, etc. So now try this for getting approximations to the fourth root of $2.$ You will in the second row get $7/6 $ as an approximation to $\root{4} \of{2}$ and in the third row an approximation of $37/31$. Now do the Farey fraction inequality to obtain $44/37.$ Your task here is simply to carry out these instructions. Does that settle the mystery?


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Newton-Cotes formulas

Historical sketches


Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.

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