As the renowned author of Principia (1687) as well as a host of equally esteemed published works, it appears that Newton not only went much further in exploring the applications of calculus than Leibniz did, but he also ventured down a different road. Leibniz and Newton had very different views of calculus in that Newton’s was based on limits and concrete reality, while Leibniz focused more on the infinite and the abstract. However, regardless of the divergent paths these two scholars chose to venture down, the question of who took the first step remained the primary issue of debate. Unaware that Newton was reported to have discovered similar methods, Leibniz discovered “his” calculus in Paris between 1673 and 1676. By 1676, Leibniz realized that he was onto something “big”; he just didn’t realize that Newton was on to the same big discovery because Newton was remaining somewhat tight lipped about his breakthroughs. In fact, it was actually the delayed publication of Newton’s findings that caused the entire controversy. Leibniz published the first account of differential calculus in 1684 and then published the explanation of integral calculus in 1686.
Newton did not publish his findings until 1687. Yet evidence shows that Newton discovered his theories of fluxional calculus in 1665 and 1666, after having studied the work of other mathematicians such as Barrows and Wallis. Evidence also shows that Newton was the first to establish the general method called the "theory of fluxions" was the first to state the fundamental theorem of calculus and was also the first to explore applications of both integration and differentiation in a single work. However, since Leibniz was the first to publish a dissertation on calculus, he was given the total credit for the discovery for a number of years. This later led, of course, to accusations of plagiarism being hurled relentlessly in the direction of Leibniz.
There was speculation that Leibniz may have gleaned some of his insights from two of Newton's manuscripts on fluxions, and that that is what sparked his understanding of calculus. Many believed that Leibniz used Newton's unpublished ideas, created a new notation and then published it as his own, which would obviously constitute plagiarism. The rumor that Leibniz may have seen some of Newton's manuscripts left little doubt in most people’s minds as to whether or not Leibniz arrived at his conclusions independently. The rumor was, after all, believable because Newton had admittedly bounced his ideas off a handful of colleagues, some of who were also in close contact with Leibniz.
It is also known that Leibniz and Newton corresponded by letter quite regularly, and they most often discussed the subject of mathematics. In fact, Newton first described his methods, formulas and concepts of calculus, including his binomial theorem, fluxions and tangents, in letters he wrote to Leibniz. However an examination of Leibniz' unpublished manuscripts provided evidence that despite his correspondence with Newton, he had come to his own conclusions about calculus already. The letters may then, have merely helped Leibniz to expand upon his own initial ideas.
The question of the date at which these extracts were made is therefore all-important. It is known that a copy of Newton's manuscript had been sent to Tschirnhausen in May, 1675, and as in that year he and Leibniz were engaged together on a piece of work, it is not impossible that these extracts were made…