How Isaac Newton Discovered the Binomial Power Series

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Isaac Newton was not known for his generosity of spirit, and his disdain for his rivals was legendary. But in one letter to his competitor Gottfried Leibniz, now known as the Epistola Posterior, Newton comes off as nostalgic and almost friendly. In it, he tells a story from his student days, when he was just beginning to learn mathematics. He recounts how he made a major discovery equating areas under curves with infinite sums by a process of guessing and checking. His reasoning in the letter is so charming and accessible, it reminds me of the pattern-guessing games little kids like to play.

It all began when young Newton read John Wallis’ Arithmetica Infinitorum, a seminal work of 17th-century math. Wallis included a novel and inductive method of determining the value of pi, and Newton wanted to devise something similar. He started with the problem of finding the area of a “circular segment” of adjustable width x. This is the region under the unit circle, defined by y=1x2−−−−−√, that lies above the portion of the horizontal axis from 0 to x. Here x could be any number from 0 to 1, and 1 is the radius of the circle. The area of a unit circle is pi, as Newton well knew, so when x=1, the area under the curve is a quarter of the unit circle, π4. But for other values of x, nothing was known.


https://d2r55xnwy6nx47.cloudfront.net/uploads/2022/08/Newton_2560_Lede.jpgMaggie Chiang for Quanta Magazine



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