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

Chapter 10. The Burden of Proof

1708–1712

“Justice is a social virtue, or a virtue which preserves society.”

—Leibniz, On Natural Law

Leibniz was dumbfounded and incensed upon hearing of Keill’s accusations. He assumed that Keill had erred because of some rash conclusion that he made, and resented that a man whom he did not regard as one of his legitimate peers was making such accusations in the first place. Who did Keill think he was? To obtain satisfaction, Leibniz would turn to the venerable Royal Society, of which he was a longtime member. This was the same course of action he had followed when Fatio had made his unsupported attack. Leibniz had been vindicated then, therefore he expected to be so again—not just because it was exactly the same situation but because he knew that he was right. He had not stolen anything from Newton, and he was confident the intelligent members of the Royal Society would see things his way. Leibniz was a great believer in intellectual societies, after all.

Scientific societies were a big part of his life, as they were for many of the scientists in the seventeenth and eighteenth centuries. In his lifetime, Leibniz had seen how the academies played an important role both in the collection and communication of as well as in the conducting of experiments. The French Académie des Sciences, for instance, sponsored major projects, such as an accurate mapping of the French Empire through South America, Africa, the West Indies.

And Leibniz was especially fond of these scientific societies because he saw the greatest of possibilities for them. The existing societies in Paris and London, venerable institutions with an august membership, were but trivial gentlemen’s clubs compared to what Leibniz envisioned. He had an almost unquenchable enthusiasm for the possibilities of scientific societies because they fit into his grand vision of a more perfect world, and he even had attempted to found such a society of sciences in Berlin.

In 1697, Leibniz found out from diplomat Johann Jakob Chuno that Sophie Charlotte wanted to build an observatory in Berlin, and he immediately sent her a letter saying that she should expand her plans and turn it into a scientific academy. Leibniz’s designs for the Berlin Society of Sciences were complicated by the fact that there were cool relations between Berlin and Hanover. Besides that, from George Ludwig’s point of view, the writing of the history of the House of Brunswick was Leibniz’s main task, unacceptably overdue.

At first, George Ludwig forbade Leibniz even to go to Berlin, but eventually the duke relented, and finally, in 1700, George allowed Leibniz to make the trip—but only after the elector in Berlin had personally requested Leibniz’s presence. The society was successfully launched with the support of Sophie Charlotte and Frederick III, who liked the idea that he would be seen as a patron of intellectual pursuits; and Leibniz was to be appointed the society’s first president. Frederick the Great would later say that Leibniz was a society of sciences all by himself.

In a way, this was nothing new for Germany. Groups that met and discussed philosophy, physics, mathematics, astronomy, or any number of other subjects on a regular basis were probably quite common. Leibniz had belonged to one at the University of Jena, while for one semester he was working toward his doctorate in law. There, a group of professors and students met once a week to discuss new and old books. He had joined a similar one at the University of Leipzig.

But these groups were nothing compared to institutions like the Royal Society or the French Académie des Sciences. What Leibniz had envisioned for the Berlin society was even grander than its English and French counterparts. “The labors of such a society should not be directed merely to the gratification of a scientific curiosity and the performance of fruitless experiments, or simply to the discovery of useful truths, without any application of the same; but the uses of science should be pointed out, even at the outset, and such inventions be made as would redound to the honor of the originator and the benefit of the public,” he wrote. “The aim of the society, accordingly, should be to improve not only the arts and sciences, but also agriculture, manufacture, commerce, and, in a word, whatever is useful in the support of life.”

His vision for his scientific society was something akin to the modern think tank, but perhaps with a lot more power. Leibniz thought his society should not simply advise, study, and report on the issues of the day but that it also should establish policies, practices, and progressive approaches to improving life. He desired it to not solely to be focused on science either, but to expand its interests to include history, art, and commerce.

Leibniz had harbored this vision for years. His scheme for draining the mines in the Harz Mountains had been predicated on the notion that it could fund such a society. From an earlier experience, when he had still been in the service of the elector of Mainz and Boineburg, he had learned the value of not proposing too much, after the elector had rejected his far-reaching plans as being too ambitious and expensive. These plans, incidentally, had called for changing everything from standard units of measurement to the church’s role in education, and sought to deliver a lot of the decision-making power into his proposed academy’s hands.

In 1700, Leibniz had learned to rein in his plans a lot more, but his schemes were still grand, of course. His Berlin society was to have an observatory, laboratory space, hospitals, libraries, a press, and museums. He did not underestimate how much money it would take to achieve his goals. He was keenly aware of the financial needs of such an enterprise, and this forced him to come up with an inordinate number of schemes to finance it.

To fund the academy, Leibniz unleashed a torrent of creative ideas. He suggested asking for donations from the church, creating a lottery, and instituting a number of new taxes, including a tax on wine, a small income tax increase, and a tax on foreign travel and paper. He wanted to obtain monopolies on the production of new calendars and almanacs, on the production of fire engines, and on the production of mulberry trees, which were used in the cultivation of silkworms.

In fact, Leibniz was so keen on mulberry trees that he tried for years to get them to grow. However, this was a failure because the silkworms did not thrive in the Germanic climate. The mulberry plantations were eventually abandoned and fell into ruin.

Like his mulberry trees, Leibniz’s grand vision fell into ruins as well. The problem was that the academy was in Berlin and he was in Hanover, and although he now had a legitimate reason to travel, he nevertheless had to obtain permission from George Ludwig each time he wished to do so. The duke, of course, had no interest in allowing Leibniz to spend long periods of time away from Hanover—not while the history needed his attention.

Leibniz’s situation was further complicated by the fact that relations between the courts at Hanover and Berlin were strained; this even led to his being accused of being a spy when he was in Berlin. The effect his absences had was to reduce Leibniz’s influence at the academy. He may have held the official title of president, but for most of the time in those early years of the academy he was out of sight and out of mind.

The two academy members who really had the power were a pair of characters known as the Jablonski brothers. One was secretary and, the other, the acting president. They eventually stopped consulting Leibniz on the appointment of new members, and added the ultimate insult by electing a Baron von Printzen as director of the academy in 1710. When the academy was officially inaugurated on January 19, 1711, Leibniz was not there, and in April 1715, his salary was abruptly halved. The final insult was that, when Leibniz died a year and a half later, the academy did nothing to mourn the passing of its creator.

Nevertheless, even if his own Berlin society had not turned out the way he envisioned in 1711, when he prepared to respond to Keill’s attack, Leibniz was still a great believer in scientific societies in general, and he had a great deal of respect for the Royal Society and felt they would justly decide his case if he put it before them.

For Newton, the only scientific society that really mattered was the Royal Society of London. When he became its president on November 30, 1703, the society had changed somewhat since its glory days in the 1670s, when Newton had been elected among its hundreds of members and it oversaw many important experiments. These issues were a faint memory in 1703. New membership was stagnant and total membership had declined.

The types of discussions and experiments at the Royal Society had become the subject of ridicule. Jonathan Swift satirized the Royal Society in Gulliver’s Travels by describing scientists who wanted to extract sunshine from cucumbers. England’s king was reportedly amused by the attempt by one Royal Society member to weigh air, and some of the society’s genuine discussions of the medicinal properties of common or uncommon substances are equally comical. In 1699, one Royal Society fellow, a Mr. Van de Bemde, remarked how cow piss “drank to about a pint” will cause a person to either purge or vomit “with great ease.”

But Newton brought renewed vigor to the Royal Society, and for the next twenty years, he ran it as a CEO would manage a personally financed startup. Newton presided over almost every meeting the society had for the next couple of decades, including the smaller meetings of the society’s council. His tenure was unusual. Every president prior to Isaac Newton had only served a few years at the most, and some had administrations so short they could almost be called “acting” presidents. Samuel Pepys, for instance, was president for exactly two years in 1684–1686, and Christopher Wren was also in office for two years, starting in 1680.

It’s no exaggeration to say that when Leibniz made his appeal to the Royal Society, he was really making his appeal to Newton himself. Newton was the Royal Society in those days.

THE YEAR THAT the calculus wars exploded into a full-blown battle, 1711, was a time of increasing accomplishments for Newton, in terms of publishing. A few years before, in 1707, William Whiston had published in Latin Newton’s Cambridge lectures on algebra, Arithmetica universalis, which the mathematician gave in accordance with the requirements of the Lucasian chair he held. As far back as 1672, Newton had begun compiling notes for these lectures. In 1712, the text for this would be translated and published in London in English. Meanwhile, Newton’s De Analysi was published under the editorship of William Jones. This book basically demonstrated some of the results obtained with Newton’s calculus, but with no formal treatment or notation. Jones had purchased John Collins’s library several years after Collins had died, and there among the books and papers he found Newton’s text from so many years before. Jones sought out Newton for permission to publish the book, which was granted, and he brought out the edition in 1711.

Leibniz probably could not have cared less about these publications. They were taken from material that was badly out of date, having been written decades before. He was more interested in the outrageous accusation published by Keill in the 1708 Philosophical Transactions of the Royal Society, which in 1711 he had just read because it took a few years to reach him in Hanover.

In March 1711, Leibniz sent a letter to Hans Sloane, who was the secretary of the Royal Society, complaining of the way he had been treated. The letter was read before the Royal Society on May 24, 1711, and in it, Leibniz essentially said, here we go again: “I could wish that an examination of the work did not compel me to make a complaint against your countrymen for the second time. Some time ago Nicholas Fatio de Duillier attacked me in a published paper for having attributed to myself another’s discovery. I taught him to know better in the Acta Eruditorum of Leipzig, and you [English] yourselves disapproved of this [charge] as I learned from a letter written by the Secretary of your distinguished Society (that is, to the best of my recollection, by yourself),” Leibniz wrote to Sloane on February 21, 1711.

As he did before when Fatio had published accusations against him, Leibniz’s approach was to acknowledge Newton’s greatness in mathematics. Ask Newton, Leibniz essentially said—he backed me up before and he’ll do so again. “Nobody knew better than Newton that this charge is false,” Leibniz wrote. “For certainly I never heard of the name of the calculus of fluxions nor saw with these eyes the characters which Newton used.

“Newton himself, a truly excellent person, disapproved of this misplaced zeal of certain persons on behalf of your nation and himself, as I understand,” he continued. “And yet Mr. Keill in this very volume, in the [Transactionsfor] September and October 1708, page 185, has seen fit to renew this most impertinent accusation when he writes that I have published the arithmetic of fluxions invented by Newton, after altering the name and the style of notation.”

Again, as he had done in the fight with Fatio, Leibniz distinguished between Newton, whom he held in high esteem, and Keill, who was at best mistaken and at worst a liar. And, in any case, Keill had said things that needed redress. “Although I do not take Mr. Keill to be a slanderer (for I think he is to be blamed rather for hastiness of judgement than for malice),” Leibniz wrote, “yet I cannot but take that accusation which is injurious to myself as a slander. And because it is to be feared that it may be frequently repeated by imprudent or dishonest people I am driven to seek a remedy from your distinguished Royal Society.”

What Leibniz wanted was for Keill to give a public statement in front of the Royal Society, retracting his accusation. Leibniz told Sloane that he wanted Keill to say that he didn’t mean to say what he had said, the slander, “as though I had found out something invented by another person and claimed it as my own,” Leibniz explained. “In this way he may give satisfaction for his injury to me, and show that he had no intention of uttering a slander, and a curb will be put on other persons who might at some time give voice to other similar [charges].”

On March 22, 1711, Keill appeared at a meeting of the Royal Society that was presided over by Newton, and agreed to write a letter of reply to Leibniz’s demand for satisfaction. Keill prepared his response for several weeks, probably with the help of Newton, and appeared in front of the Royal Society on April 5, 1711, to present it.

Keill was unrepentant. At that second meeting, he vigorously defended himself against the libel claim in the only way possible—by prosecuting his case against Leibniz. He answered Leibniz’s charges, saying that his attack on Leibniz was not without provocation but was merely a response to the anonymous review of Newton’s work in 1705. He was not unfairly harsh in his criticism, he claimed, because it was a proper response to the unfair attack on Newton. Keill declared that he would produce a written account of the history of calculus and the dispute.

Keill’s response was carefully crafted so that it did not accuse Leibniz of plagiarism as such but, rather, simply stated that Newton invented his calculus first, that Leibniz saw some of what Newton did, and that these “clear and obvious hints,” claimed Keill, “gave him an entrance into the differential calculus.”

He formally submitted this opinion in a letter to Sloane in May 1711, saying, “I have been impelled to write these lines by the publisher of the Acta Eruditorum of Leipzig, who in the account they have given of Newton’s work on fluxions or quadrature expressly affirm that Mr. Leibniz was the discoverer of this method.” Newton was the injured one, said Keill. “Whence, if I seem to have spoken pretty freely about Leibniz, I did so not with the intention of snatching anything from him but rather in order to vindicate Newton’s authorship of what I take to be his own.”

Finally, in a sort of courteous insult, Keill expressed amazement that Leibniz would even need to claim the invention of calculus: “Since he possesses so many unchallengeable riches of his own certainly I fail to see why he wishes to load himself with spoils stolen from others.”

Keill’s letter was formally presented to the Royal Society on May 24 and sent to Leibniz thereafter. Leibniz was shocked when he read Keill’s response. Not only did Keill not accept his generous offer to retract his words and humiliate himself in front of the Royal Society, but he now reiterated his outrageous case even more strongly than before! This was the final straw for Leibniz. If Keill would not retract his words, Leibniz was going to shove them back down Keill’s throat—or at least ask the Royal Society to make him eat them.

Although Leibniz was peeved, he did not stoop to anger. He obviosly saw himself on a whole different level intellectually from Keill and was sure that he could achieve satisfaction by getting the Royal Society (a body to which he still belonged, after all) to silence and censure Keill for his vanæ et injustæ vociferationes (“vain and unjust clamors”), as Leibniz perceived them.

On December 29, 1711, Leibniz wrote to Sloane again demanding redress and accusing Keill of being an upstart who was little acquainted with the facts of the case. He still had no harsh words for Newton, of course, because he respected his across-the-channel contemporary and equal. But Keill was someone for whom Leibniz had little regard—someone who was certainly not his equal.

In his letter, Leibniz said, “No fair-minded or sensible person will think it right that I, at my age and with such a full testimony of my life, should state an apologetic case for it, appearing like a suitor before a court of law, against a man who is learned indeed, but an upstart with little deep knowledge of what has gone before and without any authority from the person chiefly concerned . . .” He appealed to the society (and Newton) for redemption: “I throw myself upon your sense of justice, [to determine] whether or not such empty and unjust braying should not be suppressed, of which I believe even Newton himself would disapprove, being a distinguished person who is thoroughly acquainted with past events.”

In retrospect, it seems a ludicrous approach for Leibniz to have been taking. But at the time it was entirely reasonable. During all his years of silence on the subject of calculus, Newton had never really made any aggressive public statements on the level of the ones that Keill was making. And a few years before, when Fatio had accused Leibniz of plagiarism in much the same tone, Newton had been completely silent and done nothing to defend his close friend when Leibniz had protested. Leibniz may have fully believed that Newton would back him up in his appeal to the Royal Society in his case against Keill.

Nothing could be further from the truth. In fact, Leibniz was living his last taint-free days as the widely recognized inventor of calculus. The bear trap was set, and he walked right into it. From this moment to the day he died, he would have to answer the charge that he borrowed from Newton.

What Leibniz didn’t know in 1711 was that Keill had been discussing his accusations with Newton—that he was in fact writing with Newton’s approval. In 1711 Keill had sent Newton a copy of an anonymous review of Newton’s De quadratura curvarum, from the 1705 issue of Acta eruditorum, that basically implied the Englishman’s original work was adapted from Leibniz’s calculus—an insult that Keill was careful to point out in his accompanying letter: “I have here sent you the [article] where there is an account given of your book, I desire you will read from page 39 . . . to the end,” wrote Keill.

The article was like a bucket of gasoline dumped on a campfire. Newton must have been enraged when he read the review, because he took a long time to cool down—never really cooling down until long after Leibniz died. Newton was not fooled for an instant as to the identity of the author, and he assumed from the beginning that it was Leibniz, since the Acta Eruditorum was the journal with which the German was so closely associated. Even though Leibniz would deny authorship of this review until the day he died, Newton guessed absolutely right: of course Leibniz had written it.

Newton drafted several responses to the review, even though he never published any of them; meanwhile, circulation of the Acta Eruditorum article set off a flurry of writings against him. For years, his private writings would be filled with the occasional asides and long diatribes ranting against Leibniz, who was to Newton the new Hooke, a surrogate Flamsteed, a Judas . . . Cain . . . Satan.

Newton wrote multiple drafts of a letter to Hans Sloane, commenting on the dispute between Keill and Leibniz and the now-infamous review: “I had not seen those passages before, but upon reading them I found that I have more reason to complain of the collectors of the mathematical papers in those Acta then Mr. Leibniz hath to complain of Mr. Keill.”

Newton had a valid point. Leibniz’s review was more than a little ungenerous in its assessment of his original work. But Keill’s response to it, on the other hand, went for the jugular with its overt assertion that Leibniz had borrowed his ideas from Newton.

Putting on a facade of objective independence, Newton wrote to Sloane that the dispute was between Leibniz and Keill, and did not involve him: “Mr. Leibniz thinks that one of his age & reputation . . . should not enter into a dispute with Mr. Keill & I am of the same opinion, I think that it is as improper for me to enter into a dispute with the author of those papers. For the controversy is between that author & Mr. Keill.” Instead of involving himself directly, Newton set the wheels of justice into motion in another way.

Leibniz would deny vehemently that he ever borrowed ideas from Newton. His appeal to the Royal Society to decide the issue turned out to be a cataclysmic mistake, because Newton was not only the most famous and most respected scientist in this august body—he was its president. He could influence the society’s disposition in the matter as perhaps no other individual could. Newton’s interest was solely with Newton.

In response to Leibniz’s December 29, 1711, letter and his demand for satisfaction, the Royal Society appointed a committee on March 6, 1712, to look into the matter. On paper, it was a dispute between two Royal Society members, and the society was acting in good faith and striving to fairly settle the dispute.

In actuality, there was little about the committee or its work that was truly objective. Its members were largely Newton’s friends and countrymen—people like Halley. But perhaps in anticipation of the appearance of partiality toward their own countryman, several more people were appointed to the committee, including foreigners like De Moivre and Bonet, the Prussian minister.

On the strength of these appointments, Newton would later claim that that the committee was numerous in membership and international in character. Three hundred years after the fact, the claim seems flimsy, and the committee appears as if it were little more than a thinly veiled vehicle for putting forward its president’s arguments. The commission did not sit down prepared to decide which was better, fluxions or calculus. They began from the premise that they were the same but for the symbols used. Hence the question of authorship became a simple matter of priority: Was Newton first?

With documents at hand (Newton’s hand) proving that Englishman was first, a decision was a simple matter for the committee. What can one say about their deliberations? Their greatest achievement was that they seem to have set something of a speed record for the work of a committee.

They studied the issue for a mere six weeks, and on April 24, 1712, gave their lengthy and detailed report—a publication known as the Commercium Epistolicum D. Johannis Collins et Aliorum de Analysi Promota (The correspondence of the learned John Collins and others relating to the progress of analysis.) Not surprisingly, the document found in Newton’s favor and condemned Leibniz. It thrust Newton into an elevated limelight, casting him as the one who should be rightly recognized as the best mathematician in the last fifty years. It could not have been more damaging to Leibniz’s reputation, painting him as a compulsive plagiarist.

“We have consulted . . . the papers of Mr. John Collins,” the report began earnestly. I examined an original version of the Commercium at the Royal Society library in London (a reissued version from 1727). It is basically a large folder of such documents as De analysi, and letters to and from Collins and others, starting with Barrow to Collins in 1669 and ending with Leibniz’s final 1677 letter to Oldenberg. The Commercium selectively abstracts pieces of these correspondences and other relevant writings with the purpose of proving that Newton was the true inventor of calculus.

The authors of the Commercium Epistolicum seem to have started with the premise that Leibniz was guilty, and had spent their time cobbling together fragments from letters and papers written for some forty years, to prove it. They called attention to the fact that Leibniz had a history of misrepresenting the work of others as his own—such as the affair of the eyebrow, when Leibniz had talked to the mathematician Pell and claimed as his own some of the previous discoveries of another mathematician. “He persisted in maintaining it to be his own invention by reason that he had found it himself,” the committee wrote.

They also established that Newton invented calculus before 1669—as evidenced by the fact that a copy of De Anaylsi was found among Collins’s papers.

The Commercium Epistolicum concluded that Leibniz had been privy to certain writings of Newton’s while he was in London in 1673 and 1676, that he had received letters from Newton, and that there was no evidence that he had invented calculus before receiving those letters. It further found that Leibniz’s calculus was the same as Newton’s, but for its notation, created later than the British mathematician’s method of fluxions. Their decision: Keill was not libelous and therefore need not apologize.

“We believe that those who have reported Mr. Leibniz the first inventor knew little or nothing of his correspondence with Mr. Collins and Mr. Oldenburg long before,” the report concluded. “For which reasons, we reckon Mr. Newton the first inventor and are of the opinion that Mr. Keill, in asserting the same, has been noways injurious to Mr. Leibniz.”

The Royal Society and its president, Newton, accepted the report as correct and fair, and decided to pay for its publication. While an officially bound edition did not go on sale in bookstores, copies became available on January 8, 1713, and the Royal Society paid for some to be sent to key mathematicians in Europe. Several copies of the Commercium Epistolicum went to Paris, and one made its way into the hands of Abbé Bignon, who gave it to Nikolaus Bernoulli, who carried it to Basel and showed it to his uncle Johann, who wrote about it to Leibniz in a letter dated June 7, 1713.

The report was a stunning success, from Newton’s point of view. To him, the case was now drawn up and easily understood. It established his priority in the invention of calculus some forty years after the fact, and did it so convincingly that, from the time the committee published its report all the way up to today, very few have mentioned calculus and Leibniz in the same breath without first mentioning Newton.

From Leibniz’s perspective, the report was a slap in the face with a bag full of marbles. Even if one were to accept that the members of the committee were completely objective, their conclusions are still worth questioning. But Leibniz never had a chance to question these conclusions because the committee extended no invitation to the German to present his own case.

The Commercium Epistolicum, as flawed a document as it is, had a profound effect on the calculus debate. It effectively subjugated Leibniz to a lesser status of second inventor at best, and as opportunistic plagiarist at worst in the eyes of many. It turned the tide of popular opinion against him, and if it failed to knock him out completely, it at least knocked him onto his heels. He would spend the rest of his life fighting back but was never fully able to beat down Newton’s accusations.

Leibniz’s friends urged him to reply. “Most people may deduce from silence that the English case is a good one,” one of them wrote. The problem for Leibniz was that Newton had cast the argument in historical terms—specifically the version of history that had him inventing fluxions long before Leibniz invented calculus. This happened to be true, and ample proof was supplied in the Commercium Epistolicum. But Keill had asserted that Leibniz had been given access to Newton’s unpublished work, and that it had been sufficiently intelligible for him to be able to copy it. Because the Commercium did not attempt to disprove Keill’s accusation, Leibniz was left having to prove his own innocence. In absence of a credible counterproof, the case for Newton was made all the more strong.

These were the last years of Leibniz’s life, and they should have been filled with the joy of seeing his accomplishments blossoming into maturity, not a fight to retain his honor over work long past. As he never married and never had any children to surround him with grandchildren, he had to take pride in offspring of a different sort—his intellectual creations and intelligent European protégés who were inspired by those ideas to develop them further. Now Newton had taken custody of calculus, one of Leibniz’s most brilliant offspring.

Leibniz was honored in 1711 with an invitation to a conference with Czar Peter the Great, who had come to Germany to see his son married to the princess Wolfenbüttel. Leibniz advised the czar at one point to open libraries and observatories in Russia, and to appoint teachers in the arts and sciences. Despite the furor in London, Peter met with Leibniz again in 1712, and he asked the German’s advice on establishing and promoting math and science in Russia. Without ever setting foot in the country, Leibniz was given the title of privy counsellor of justice and awarded a nice salary. A year later the czar visited Hanover; though Leibniz was not there, he heard that Peter had had nice things to say about him.

At this time, Leibniz was not a well man. He was sick, old, and partly crippled from gout so severe that he had suffered an open lesion on his leg for two years. He ignored his leg. His attention was on matters less close to home.