The Fundamental Theorem of Calculus

“There have been several occasions in the history of mathematics when an important discovery was made independently and almost simultaneously by two individuals … To the extent that we can rationally explain these remarkable events, it must be on the basis of ideas already “in the air,” of conditions becoming favorable for their crystallization” — Mathematics and its history | John Stillwell

James Gregory (1638–1675) was a Scottish mathematician who spent most of his life just outside the broader academic community thriving on the European continent. Although analytic geometry was taking its place among the academics of the time, many were still arguing the legitimacy of transcendental numbers and in some cases, whether or not negative numbers should even be admitted into the mathematical techniques of the day. Gregory used Archimedean techniques for finding areas and, regarding his interest in finding tangents, used techniques that had been available to mathematicians ever since the Greek philosophers. Like many of his day, Gregory’s work was heavily biased towards pure geometry. None the less, he was one of the first to develop a series expansion for arctangent, tangent, and secant. Hidden in his ingenious geometric work was the Taylor series fourty years before Taylor and, although he didn’t realize it, he would provide a first, geometrically based, proof of the fundamental…