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,

    If Cotes had lived, we might have known something

even 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, and 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; and 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, then ordained as a deacon and a short time later a priest, at age 30.

The period 1709-1713 was central to Cotes' short life, for it was during these years that he assisted Newton in revising his seminal book Philosophiae Naturalis Principia Mathematica (often shortened to simply Principia or sometimes Principia Mathematica). In this book, Newton describes universal gravitation, giving laws that govern motion---both terrestrial and celestial---and he substantiates Kepler's laws of planetary motion. These laws argue that planets, comets, etc., travel in conic sectional orbits that are elliptical, parabolic, or hyperbolic (the conic sections). When their work began on the revision, they operated as friends, and Newton penned a paragraph of thanks to Cotes, to be included in the book's Preface. But by the time for publication rolled around, Newton had omitted the paragraph.

You have learned in calculus about the Newton-Cotes method of approximating integrals, and that is just one of the many interests of Cotes in approximations. He also is said to have "invented" radian measure of angles. He found 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 approximation

which he did not explain. The many guesses as to how he did this seems oddly to exclude generalized Greek ladders, coupled with the Farey fraction inequality

that you were asked to establish in the Historical Sketch forNicolas Chuquet. The Greek ladder for the square root of 2 begins with a row of two 1's: 1, 1. Then each row a,b is followed by the row a+b, 2a+b. So the ladder proceeds as 1,1; 2,3; 5,7; 12,17. The ratios of the second terms over the first terms

etc., are increasingly closer approximations to the square root of 2. If you play this game starting with a row of three 1's, you get closer and closer to the cube root of 2, and if you start with a row of four 1's, you get closer and closer approximations to the fourth root 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 the fourth root 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. Maybe that settles the mystery?