All he needed to do was omit the denominators and reduce the powers of $latex x$ by 1 in the power series displayed above. Returning to the problem of the circular segment, Newton realized that the equation for the circle itself (not merely the area underneath it) could also be represented by a power series. Eventually he calculated the first 15 digits of pi. It was an important finding, but it turns out there are better ways to approximate pi by means of an infinite sum, as Newton himself soon discovered after this initial foray into these kinds of infinite sums, now called power series. This is the region under the unit circle, defined by $latex y=\sqrt$. But the greatest of them all was undoubtedly Sir Isaac Newton. He started with the problem of finding the area of a “circular segment” of adjustable width $latex x$. Sir Isaac Newton (1643-1727) In the heady atmosphere of 17th Century England, with the expansion of the British empire in full swing, grand old universities like Oxford and Cambridge were producing many great scientists and mathematicians. Wallis included a novel and inductive method of determining the value of pi, and Newton wanted to devise something similar. It all began when young Newton read John Wallis’ Arithmetica Infinitorum, a seminal work of 17th-century math. His reasoning in the letter is so charming and accessible, it reminds me of the pattern-guessing games little kids like to play. There a devout young man, Isaac Newton, was finally to discover the way to a new synthesis in which truth was revealed and God was preserved. He recounts how he made a major discovery equating areas under curves with infinite sums by a process of guessing and checking. The 17th century was a time of intense religious feeling, and nowhere was that feeling more intense than in Great Britain. In it, he tells a story from his student days, when he was just beginning to learn mathematics. A young Isaac Newton is sitting beneath an apple tree contemplating the mysterious universe. But in one letter to his competitor Gottfried Leibniz, now known as the Epistola Posterior, Newton comes off as nostalgic and almost friendly. Isaac Newton was not known for his generosity of spirit, and his disdain for his rivals was legendary.
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