Tuesday, December 11, 2007

Roger Cotes

Roger Cotes was born on July 10, 1682 in Burbage, England. His father was the rector of the town and he grew up with an older brother and younger sister. At 12, he began to show great mathematical ability and from this point on, his uncle, the Reverend John Smith, took responsibility for his education.

At 16, he entered Trinity College at Cambridge. He graduated from Cambridge 3 years later and became a fellow of Cambridge in 1705. By 26, he was a full professor of Astronomy and Experimental Philosophy at Cambridge. By this point, he already had attracted the attention and support of Sir Isaac Newton. In 1711, he became a fellow of the Royal Society.

In 1709, he began his important work of proof-reading the second edition of Newton's Principia. In one famous episode, he corrected Newton's approximation of the fourth root of 2. Newton had used 13/11 (= 1.181818...). Cotes suggested 44/37 (= 1.189189189) which closer to the actual value (= 1.189207...). This correction, it turned out, came out of an important advance Cotes had made in approximation of irrationals.

Cotes would later publish this discovery in the only paper published in his lifetime which appeared in the Philosophical Transactions of the Royal Society in March, 1714. In this paper, Cotes showed his method of approximating irrationals using continued fractions. He also wrote the preface to the second edition of Newton's magnus opus that compared his method to the vortex theory of Descartes.

Cotes made many other important discoveries but the rest were not published in his lifetime. He did very important work in relation to the nth roots of unity, worked out the continued fraction expansion of e, proposed radians as a measure of angles, did important work on least squares, interpolation, integration of rational functions, and graphs of tangents and secants.

Roger Cotes died suddenly from a severe fever on June 5, 1716. He was 33. He was buried at Trinity College. Later his cousin, Robert Smith, the son of the Reverend John Smith, edited and published his work. Robert Smith had been Cotes's assistant and later became master of Trinity College. In this role, he arranged for a statue of Cotes to be made which is displayed above.

Sir Isaac Newton said the following about Roger Cotes:
"...if he had lived, we might have known something."

Monday, December 10, 2007

Leibniz's Mistake

In 1702, Gottfried Leibniz wrote an important paper on the integration of rational fractions. It was well known that the time that ∫ dx/x = log x (see Lemma 1, here for proof) and Leibniz had previously discovered in 1675 that ∫ dx/(x2 + 1) = tan-1x (see Theorem, here).

Leibniz knew that if a polynomial could be decomposed into irreducible parts, then a rational fraction could be restated as the sum of partial fractions (see here for details).

In particular, Leibniz tried to figure out if all polynomials were decomposable into dx/x and dx/(x2 + 1). In answer, Leibniz believed that he had found a polynomial form that resisted any such attempt at decomposition.

Leibniz noted that:

x4 + a4 = (x2 + a2-1)(x2 + a2-1) =

Leibniz then concludes that:

"Therefore, ∫ dx/(x4 + a4) cannot be reduced to the squaring of the circle [ dx/(x2 + 1)] or the hyperbola [ dx/x ] by our analysis ..., but founds a new kind of its own." [see Tignol, p 75]

In short, Leibniz did not understand roots of unity. He did not realize that:

His mistake would later be corrected by Roger Cotes whose response to Leibniz will be the next part in our story of the roots of unity.

In 1719, N. Bernoulli put forward the following response to Leibniz:

x4 + a4 = (x2 + a2)2 - 2a2x2 =

= (x2 + a2 + √2ax)(x2 + a2 - √2ax)

In retrospect, Leibniz's mistake led to the clarification of the roots of unity.