The great controversy as to who discovered the Calculus first, either Isaac Newton or Gottfried Leibniz, is indeed a sordid affair, which has sullied the arena of science. Boyer speaks the truth when he says that no invention in science or mathematics can be said to be the accomplishment of one or two persons (1959, p. 187). Newton himself admitted “If I have seen further it is by standing on the shoulders of giants” (qtd. in Rees 2006, p. 340). Such self-effacement is part of the magnanimity that we expect from a true genius. But did Leibniz conduct himself similarly?
This, I believe, is the crux of the debate. Scientists not only stand on the shoulders of the giants of the past, but they also collaborate with each other. The very greatness of science stems from the fact that it is practiced in broad daylight. There should be no place for pride and vanity here. And yet the great controversy involves nothing but vanity. In the first instant it involved the vanities of two personalities, and then embroiled the vanities of two nations. If the accolade of the “inventor of the Calculus” must go to one among the two, I believe it must go to him who has conducted himself with most honor. And in this duel Newton emerges the winner.
I first catalogue all that can be said in favor of Leibniz. He was truly a philosopher, in contrast to the scientific genius that Newton was. If we examine his philosophy we will find that it is in complete harmony with what the science of the calculus describes. He postulated a theory of “monads”, which are infinitesimal units of reality in which the microcosm contains the macrocosm. Calculus is the analysis of infinitesimals, and we are able to see in it a reflection of the Monadology.
Therefore it is very likely that he came to an independent discovery. Calculus was on the verge of being discovered in any case, which the works of Huygens, Barrow and Fermat attest to. It is recorded that Leibniz began work on the Calculus in 1674, independently of Newton (?), and was the first to publish in 1684 (Stillwell 2002, p. 159). His unique approach (the dy/dx notation) demonstrates clearly his originality. And because he starts from a philosophical point of view, his analysis is more intuitive and suitable to demonstration. This is why the Leibnizean notation and approach that has become the norm.
But the fact remains that Newton was the first to come a thorough formulation of the Calculus. In a note to a paper written in 1666 we find him deriving a tangent to a curve using his “method of fluxions”. In this note there is as aside that reads “This is only a special case of a general method whereby I can calculate curves and determine maxima, minima, and centers of gravity” (Boyer 1959, p. 207). This clearly indicates that Newton had come to a complete formulation.
But he has no regard for the vanity of publication, being the consummate scientist that he was. In the height of the controversy Newton is reported to have said, “I have never grasped at fame among foreign nations, but I am very desirous to preserve my character for honesty” (Brewster 2004, p. 72). Calculus to Newton was merely a tool that he required to come to his universal theory of gravitation and motion, and not something that should be flouted separately. He was even reluctant to publish the revolutionary Principia, and did so only after the prodding of Edmund Halley.
Leibniz, on the other hand, was eager to publish and propagate his findings. While we admit to his originality to a large extent, the conduct of Leibniz is highly suspicious in the proceedings. He makes no defense of his integrity, as Newton does, but instead seem entirely intent on pushing the evidence alone, as if defending himself in a court of law, and this makes us feel that he is hiding something. Subsequent scholarship does indeed reveal that he manipulated documents before being released. He is also found to have possessed crucial papers of Newton which he fails to admit of, which C J Gerhardt unearthed in 1849, even though he did make such an admission shortly before his death (Cajori 1898, p. 240).
We must judge by circumstantial evidence, because it is all that we have at this distance. When we focus on the conduct of the two disputants, Leibniz is certainly the suspect one. There is no doubt that they both collaborated with each other. But plagiarism must be construed when any one among them fails to be completely honest and forthcoming. From this point of view the accusation falls on Leibniz, who has surely acted suspiciously. Even by his own admission he was aided by Newton’s papers, yet he failed to acknowledge his debt in time. This amounts to plagiarism. And since it is Newton that he plagiarized from, it is fair to name Newton as the inventor of the Calculus.
Boyer C B. (1959). The History of the Calculus and Its Conceptual Development. Chelmsford, MA: Courier Dover Publications.
Brewster D. (2004). Memoirs of the Life, Writings and Discoveries of Sir Isaac Newton Part 2. Whitefish MT: Kessinger Publishing.
Cajori F. (1898). A History of Elementary Mathematics. London: Macmillan.
Rees N. (2006). Brewer’s Famous Quotations: 5000 Quotations and the Stories. New York: Sterling Publishing Company.
Stillwell J. (2002). Mathematics and Its History. New York: Springer Publishing Company.