Like most discoveries, calculus was the culmination of centuries of work rather than an instant epiphany. Mathematicians all over the world contributed to its development, but the two most recognized discoverers of calculus are Isaac Newton and Gottfried Wilhelm Leibniz. Although the credit is currently given to both men, there was a time when the debate over which of them truly deserved the recognition was both heated and widespread. 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 (Struik, 1948). 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.

It is also known that Leibniz and Newton corresponded by letter quite regularly, and they most often discussed the subject of mathematics (Boyer, 1968). 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 (Ball, 1908). However an examination of Leibniz' unpublished manuscripts provided evidence that despite his correspondence with Newton, he had come to his own conclusions about calculus already.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. In 1669, he wrote a short manuscript on the method entitled De Analysi per Aequationes Infinitas (On analysis by Infinite Series), which he showed to a few people, including Isaac Barrow, the Lucasian Professor, who urged him to publish it. But, he would not agree. Why? ... because of his almost pathological fear of criticism.

He wrote De Methodis Serierum et Fluxionum (On the methods of series and fluxions) in 1671 but again, he didn't publish it; indeed, it did not appear until an English translation was produced in 1736. So, at age 23 years, and still a student, Newton had surpassed all the leading mathematicians in Europe - and almost no-one knew it except a few close colleagues! He returned to Cambridge early in 1667, and in March 1668 he obtained a MA degree and become a major Fellow of Trinity College. By then, however, he had developed his version of the calculus and he had turned to other scientific endeavors, principally continuing his studies of light and optics. 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 (Boyer, 1968). It is also known that Leibniz and Newton corresponded by letter quite regularly, and they most often discussed the subject of mathematics (Boyer, 1968). 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 (Ball, 1908). 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 then. It is also possible that they may have been made in 1676, for Leibniz discussed the question of analysis by infinite series with Collins and Oldenburg in that year, and it is a priori probable that they would have then shown him the manuscript of Newton on that subject, a copy of which was possessed by one or both of them. On the other hand it may be supposed that Leibniz made the extracts from the printed copy in or after 1704.

Leibniz shortly before his death admitted in a letter to Conti that in 1676 Collins had shown him some Newtonian papers, but implied that they were of little or no value, - presumably he referred to Newton's letters of June 13 and Oct. 24, 1676, and to the letter of Dec. 10, 1672, on the method of tangents, extracts from which accompanied the letter of June 13, - but it is remarkable that, on the receipt of these letters, Leibniz should have made no further inquiries, unless he was already aware from other sources of the method followed by Newton (Ball, 1908). In 1715, just a year before Leibniz death, the Royal Society handed down their verdict crediting Sir Isaac Newton with the discovery of calculus. It was also stated that Leibniz was guilty of plagiarism because of certain letters he was supposed to have seen (Ball, 1908). It later became known that these accusations were false, and both men were then given credit, but not until after Leibniz had already died. In fact, the controversy over who really deserved the credit for discovering calculus continued to rage on long after Leibniz’ death in 1716 (Struik, 1948).

Newton and his associates even tried to get the ambassadors of the London diplomatic corps to review his old manuscripts and letters, in the hopes that they would endorse the finding of the Royal Society that Leibniz had plagiarized his findings regarding calculus. Another argument on the side promoting the idea of Leibniz as a plagiarist was the fact that he used an alternate set of symbols. Leibniz specifically set out to develop a more meticulous notation system than Newton’s, and he developed the integral sign ( I ) and the 'd' sign, which are still used today (O’Connor, 1996) However this action was argued by many to be merely a way for Leibniz to “cover his tracks” so as not to get accused of stealing Newton’s material (Boyer, 1968).

The fact that the method was more efficient was considered to be an ancillary benefit. The fact is that Leibniz sent letters to Newton outlining his own presentation of his own methods, and these letters focused quite stringently upon the subject of tangents and curves. Because Newton had been approaching calculus primarily in regards to its applications to physics, he purported curves to be the creation of the motion of points while perceiving velocity to be the primary derivative. Conversely, the calculus of Leibniz was applied more to discoveries in geometry made by scholars such as Descartes and Pascal. Since "Leibniz' approach was geometrical," the notation of the differential calculus and many of the general rules for calculating derivatives are still used today, while Newton's approach, which has in many aspects, fallen by the wayside, was "primarily cinematical" (Struik, 1948).

Despite the ruling of the Royal Society, mathematics throughout the eighteenth century was typified by an elaboration of the differential and integral calculus in which mathematicians generally discarded Newton's fluxional calculus in favor of the new methods presented by Leibniz. Nevertheless, in England, the controversy was viewed as an attempt to pilfer Newton's glory simply because of international egotism. Consequently, as a matter of “national pride”, England refused to teach anything but Newton’s discoveries of geometrical and fluxional methods for over a century. So while other countries were integrating various findings that occurred over time and were progressing in their discoveries, England remained essentially stagnant in the realm of mathematic discovery. In fact, it wasn’t until 1820 that England finally agreed to recognize the work of mathematicians from any other countries (Ball, 1908).