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 theorem of calculus.

Isaac Barrow (1630–1677), working out of Trinity college, gave a series of lectures from 1664 to 1666 that linked the problem of quadratures with the problem of tangents. These lectures where geometrically based arguments but in them was a proof that derivation and integration where inverse relationships. Barrow had provided the first rigorous proof of the fundamental theorem. Barrow’s lectures would be published posthumously in 1683 however, at the time that he was giving them, he was aware of another method of proof — the method of infinitesimals. Barrow was critical of the method but would later be convinced of its legitimacy by one of his students and attendee to his lectures, Isaac Newton.

Gregory and Barrow had proven the fundamental theorem. Issac Newton, using infinite series and the theory of infinitesimals (or the theory of fluxions as Newton later described them), would formalize the surrounding mathematical…