The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time (2006)

Epilogue

In 1737, a few years after Newton died, his treatise Method of Fluxions finally appeared. This was the exposition of his method of calculus that he had written long before, and it was not printed as a bit of posthumous hero worship. The wording of the preface shows just how revered Newton had become only a decade after his death: “The following treatise containing the first principles of fluxions, though a posthumous work, yet being a genuine offspring (in an English dress) of the late Sir Isaac Newton, needs no other recommendation to the public than what that Great and Venerable Name will always carry with it.”

Newton’s discourse was at times hard to read. One striking example comes on page sixty, where he explains: “When a quantity is the greatest or the least that it can be at that moment it neither flows backwards nor forwards: for if it flows forwards or increases then it was less, and will presently be greater than it is; and on the contrary if it flows backwards or decreases, then it was greater and will be presently be less than it is. Wherefore to find its fluxion by [Newton’s methods] and suppose it to be equal to nothing.” The same meaning can be much more succinctly described today as: “set the derivative equal to zero and solve.”

Nor was Newton’s notation as useful as the superior notation that Leibniz had invented and the advanced calculus that Johann Bernoulli and the other European mathematicians developed throughout the century. Leibniz had correctly surmised that his symbols would make for the easy development of calculus, and these symbols, which he first penned in his notebooks in Paris in 1675, can still be found to this day in every calculus textbook.

In this sense, the high esteem in which Newton was held in Britain was not always a good thing, because, many of the mathematicians and scientists living there in the eighteenth century were behind the iron curtain of Newton’s fame and glory. Ironically, as much as Leibniz’s reputation suffered in Great Britain, the whole country may have suffered a self-inflicted wound by so underappreciating him. After the calculus wars, British mathematicians were prevented from learning calculus using Leibniz’s notations, which were largely in use elsewhere, and they were not finally accepted in that country until the early nineteenth century.

It took until the mid-nineteenth century for explosion of scholarship to begin to redeem Leibniz and return to him the general recognition for his role in the creation of calculus. Even though he would no longer be regarded as its sole inventor, historians at that time would at least establish the facts that led to his now-universal regard as its co-inventor. It was their firm establishment of the basic facts of the calculus wars that led to this renewed appreciation of Leibniz’s contributions. As one scholarly review of a new Leibniz biography in 1846 put it:

Most persons of the present day, who have investigated the subject, have pretty well made up their minds as to the following points: first, that the system of Fluxions is essentially the same with that of the Differential Calculus—differing only in notation; secondly, that Newton possessed the secret of Fluxions as early as 1665—nineteen years before Leibniz published his discovery, and eleven before he communicated it to Newton; thirdly, that both Leibniz and Newton discovered their methods independently of one another—and that, though the latter was the prior inventor the former was also truly an inventor. . . . Whether Leibniz was truly an independent inventor of this method—in principle identical with that of Fluxions—is the only question, in our judgment, that really affects his fair name; and that he was so, is now, we may say, all but universally regarded as indisputable.

Despite this writer’s enthusiasm that the case was settled, some scholars were still arguing even when he wrote these words. Some nineteenth-century writers accepted Newton’s stance that the sole inventor was whoever had first come up with calculus and written it down—thus giving himself full credit. After all, he did discover calculus first, twenty years before Leibniz published anything. To Newton, the discovery and subsequent dissemination of calculus were not two parts of a whole discovery, and neither would they be to his subsequent champions.

To others, Leibniz was the one who deserved full credit, since his methods and notation were the ones that progressed and survived. He invented calculus independently, was the first to publish his ideas, developed calculus more than had Newton, had far superior notation, and worked for years to move calculus forward into a mathematical framework that others could use as well. Besides, history is full of examples of second inventors taking full or partial credit for an invention, including others from the seventeenth century.

Even so, in the mid-eighteenth century, many writers, like the author of the review quoted above, began to take a more conciliatory tone. In the century and a half since, some of Newton and Leibniz’s biographers have gone even further and dismissed their fight as a ridiculous waste of time.

Actually, there is a long history of this sort of reasoning, dating all the way back to the middle of the calculus wars, as Varignon, a contemporary of the two mathematicians, first aired when he wrote a letter to Leibniz in 1713. Calculus was so great, Varignon said, that it should have been enough for both of them.

Another possibility is that neither one of them deserves all the credit that they were both seeking to claim from the other. In some ways, the development of calculus owes just as much to all those who came before Leibniz and Newton, and to the Bernoulli brothers and the others like them who came afterward, took what was published, and turned it into a much richer subject with numerous applications.

For me, what’s really interesting about the calculus wars is not who won or lost, but how they fought. The real story is not about how relevant or ridiculous the entire squabble was but how rich it was—and how much it reveals about both men.

Their stories were completely different. Leibniz went to Paris to avert a war and stayed to enrich his mind. He was embarrassed about his lack of knowledge in mathematics, but more than made up for it when he invented calculus, developed it, published it, and corresponded with others about it. While he was mired in his non-calculus-related obligations to the court at Hanover decades later, he was forced to defend his invention. Then, near the end of his life, he struggled in vain to beat down the accusations and insinuations that he was a plagiarist. His story was tragic.

Newton’s was triumphant. He invented calculus, wrote it down, shared it with a few people, forgot about it for a while, was asked about it, and again forgot about it for years. Then he began working on the Principia and, when he was finished, he found out that Leibniz had published his own writings on calculus. For years, Newton held to the belief that he had been first to discover the process, and a few of his supporters came out and said as much in print, but he never did anything to win the glory of the invention for himself. Then, after a mid-life crisis, a new job at the mint, and a few years at the helm of the Royal Society, with the help of friends he launched a full-court effort to win recognition for his invention. And he ultimately succeeded.

Perhaps their argument reveals these men in their worst light. After all, theirs are two of the original profiles from which the archetypal myth of the modern scientist has been drawn—the ambitious, detached, hard working, prolific, and very nearly godlike genius—and one never likes to think of gods mired in nasty disputes. But then, perhaps the calculus wars reveal something more interesting.

It is a cautionary tale in the importance of publishing scientific discoveries, to be sure. Perhaps because Newton and Leibniz, fought the calculus wars at a time when each was at the height of his fame, the fight will forever be clouded in infamy to some. But to me it is one of the most fascinating stories in the history of science because it combines the most glorious heights of discovery with one of the most grueling and personal intellectual fights. And it is possibly the only dispute in the history of science that was ever fought by two such great minds—perhaps the greatest of their day.