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

Chapter 11. The Flaws of Motion

1713–1716

To examine the last years of the calculus dispute does not increase one’s admiration for some of the greatest of mankind.

—A. R. Hall, Philosophers at War

In 1712, Leibniz set out to Vienna, a city that appealed to him much more than the lonely haunts of Hanover, about which he complained to his friend Thomas Burnet, “The narrow limitations, both physical and mental, within which I am confined, are owing to the circumstance that I do not live in a big city, like Paris or London, abounding with learned men from whom one can learn something, and derive some assistance.” It was the last extended trip he would take in his life. He was to stay in Vienna for two years.

The Austrian capital had a lot to offer Leibniz, and it was there that he composed the Monadologie, one of the best-known sketches of his philosophy. He also came up with another plan for a scientific society, with all the various accompanying ambitions he was given to attaching to such schemes—a laboratory, a library, an observatory, a botanical garden, a geological collection, and a medical school. He wrote letters and memoranda in support of his vision, and wrote directly to the nobles whose support he needed most. The plan was seriously considered by the court of Charles VI in Vienna, where Leibniz had strong supporters, but they would not advance the money to carry it out.

Despite this disappointment, Leibniz was happy in Vienna. He stonewalled repeated requests in 1713 to return to Hanover after he had been in Austria for many months. The exasperated duke had his salary frozen until he returned, and even then he delayed. The money didn’t really mean much to Leibniz at that point. He had additional sources of income at this late stage in life, and was a relatively wealthy man. Upon his death in 1716, he had amassed a nest egg of around 12,000 taler, which was quite a fortune considering that an average week’s wages for a common worker was about one taler.

In 1714, he was still waiting things out in Vienna, and that summer received a letter from Hanover asking whether he intended to return to the city at all. Leibniz wrote back defending himself by rolling out his record of service through four decades of the court. He might well have stayed for the remainder of his days in Vienna, however, had not fate intervened.

On June 8, 1714, Sophia was walking through the gardens of one of her homes when suddenly she was struck ill, collapsed, and died at the age of eighty-four. A few weeks after that, Queen Anne died, and suddenly it was inevitable that Sophia’s son, George Ludwig, would become king of England. He left for London on September 3.

Though he took his time to make his way to it, George showed no hesitation at accepting his throne. And why should he? He was trading up from being the ruler of a small state with its seat in a second-tier European city, to becoming the monarch of one of Europe’s great powers—with the fringe benefit of a new residence in one of the biggest booming metropolises in the world. At the same time, he was coming to a Britain that was rife with political infighting, social problems, and when the great South Sea bubble burst in a few years, economic ones as well.

In England, riots were commonplace, and the highways thick with robbers and cutthroats. City gates were still sometimes adorned with the rotting heads of rogues set upon pikes, and public executions—sometimes brutal stoning affairs—were considered a source of entertainment.

Everything could be found on the streets of London. Livestock—cows, sheep, chickens, and all their concomitant noises and smells—dogs barking and leaving messes everywhere; soldiers brawling and boozing at all hours; tradesmen shouting out their wares; servants rushing about; beggars and prostitutes sneering and cursing about in the foul air; elegant men and women of fortune picking their way through the cobblestone lanes with their entourages; and waste most foul draining down open sewers in the streets.

Perhaps George was the perfect king to rule over this mess, since he has been described as as coarse and crude just like so many of his subjects. According to some of the descriptions I have read, this description is perhaps even generous. He was said to be cynical, selfish, and even pathologically cruel. He may have been thrust upon the English throne, but he was to govern his own terms.

Leibniz, having heard the news of Anne’s demise and knowing what it meant, proceeded back to Hanover. Surely he couldn’t miss the opportunity to go to London. Even before George became king, Leibniz had hatched a plan to spend part of his time in London—to partake in conversation with the “excellent persons in whom England is so rich,” as he explained to a friend.

They missed each other by three days. George, mockingly, said of Leibniz, “He comes only when I have become king.” Leibniz sought to follow George, proposing to accompany Princess Caroline, but he was not well enough to travel when she left. Instead of going to London, he went to nearby Zeitz, where apparently he was introduced to a talking dog that could recite the alphabet and bark out words like “chocolate” and “coffee.” The loneliness of being left behind!

In December 1714, Leibniz received a letter from Prime Minister von Bernstorff telling him not to come to London, and, a month or so later, George Ludwig, now King George I, expressly forbade Leibniz to come, ordering him to stay in Hanover until the still-incomplete history of the family was complete. Leibniz was apparently a victim of George I’s advisors, who thought he would do little more than seek to interfere with their efforts in London. Leibniz continued to work for George from Hanover, producing, for instance, an anti-Jacobite pamphlet—anonymously, of course.

Leibniz’s response to all this was to petition his employer to have him made historiographer of England. George I was not impressed by this request. “He must first show me that he can write history,” said the king to his daughter-in-law.

Stuck in the scientific and cultural backwater of Hanover, Leibniz was now more isolated than Newton, and he remained in Hanover until he died—sick, busy, and distracted with the never-ending history project and with the calculus wars.

He should have stayed in Vienna. There, Leibniz had found the time to produce some of his best writings. Aside from the Monadology, he wrote an exposition on the state of philosophy and science in China, for example.

It was also in Vienna where Leibniz first heard of the Commercium in a letter from Johann Bernoulli, one of his biggest supporters. Bernoulli was outraged, writing to Leibniz on May 27, 1713, “This hardly civilized way of doing things displeases me particularly; you are at once accused before a tribunal consisting, as it seems, of the participants and witnesses themselves, as if charged with plagiary, then documents against you are produced, sentence is passed; you lose the case, you are condemned.” He saw the Commercium Epistolicum as yet another blatant attempt of the British to take credit for discoveries made by intellectuals on the European continent.

Bernoulli derided Keill as Newton’s ape and also wrote that he believed some of the documents in the Commercium to be either fabricated or altered. Worse than that, said Bernoulli, the English were accusing Leibniz of doing exactly what Newton had done: stealing the idea for calculus. Bernoulli wrote essentially that Newton had not grasped—had not even dreamt—what he claimed to have accomplished until after he had read Leibniz’s work.

“Indeed, you can find no least word or single mark of this kind even in the Principia Philosophiæ Naturalis, where he must have had so many occasions for using his calculus of fluxions, but almost everything is there done by lines of figures without any definite analysis in the way not used by him only but by Huygens too, indeed by Torricelli, Roberval, Fermat and Cavalieri long before,” wrote Bernoulli, who was absolutely right. Newton had developed the Principia in the old, formalized geometric style rather than with the sort of algebraic mathematics that someone using calculus would employ.

Bernoulli had made the same accusation to Leibniz years earlier, in a 1696 letter, but at that time Newton was recovering from a severe bout of depression and Leibniz was regarded in many corners as Europe’s greatest mathematician. At that time, Leibniz must have seen no need to go public with such nasty accusations. Plus, Newton had never made any claim of priority in the invention of calculus, so Leibniz was probably satisfied to leave well enough alone. But in 1713, his good name sullied, Leibniz heeded Bernoulli’s words.

Upon hearing the news of the Commercium Epistolicum, Leibniz became convinced that it must be filled with malicious falsehoods, and he wrote to Bernoulli asking him to look into it. Bernoulli responded, on June 7, with a letter containing several pages of his opinions on the matter.

Leibniz replied a few weeks later that, while he still hadn’t seen the Commercium, he was sure that the “idiotic arguments” contained therein were worthy of ridicule. He expressed regret to Bernoulli that, for all his years of saying kind words about Newton when his comments about his counterpart were so solicited, that was the price he paid for his kindness.

He would not be so kind again, and in fact Leibniz turned very ungenerous to Newton after the Commercium Epistolicum appeared. He began to question whether Newton actually had invented his own version of calculus: “He knows fluxions, but not the calculus of fluxions which (as you rightly judge) he put together at a later stage after our own was already published.” In fact, said Leibniz in his letter to Bernoulli, “For many years now the English have been so swollen with vanity, even the distinguished men among them that they have taken the opportunity of snatching German things and claiming them as their own.”

Bernoulli received Leibniz’s letter and dashed one off in response, saying that his friend should consider proving the inferiority of the Brits by posing more challenging problems to them that could only be solved using calculus. “If such things were proposed to the English by way of a trial, it would be in my opinion the quickest way of stopping their mouths, particularly if they should reveal their extraordinary feebleness and the inadequacy of their calculus of whose antiquity they boast so greatly.”

The problem that Bernoulli had posed years earlier had been a success in the sense that the only people who submitted correct answers on time were the ones who knew calculus. As Newton had proved himself equal to the challenge then, it’s hard to fathom why Bernoulli and Leibniz thought a new challenge would stump him now. Nevertheless, they did seem to, and many months after Bernoulli suggested the idea, Leibniz proposed a new challenge to prove that Newton was inferior when it came to mathematics, which he included in a letter he wrote to a Venetian nobleman, the Abbé Conti. The letter to Conti ends with Leibniz stating that the purpose of the problem was, “to test the pulse of our English analysts,” though the purpose of this challenge was obvious—it was clearly intended for Newton.

The challenge was to determine the curve that should cut, at right angles, an infinity of curves expressible by the same equation. Unfortunately for Leibniz, this effort failed to reveal the inferiority of the English mathematicians because there was a problem with the way the challenge was written; it was interpreted to be asking for a specific example of such a curve rather than for a general solution for finding such a curve—the much harder question that Leibniz had intended.

The general way was more difficult, requiring a mastery of calculus. But Leibniz had used some unfortunate wording that caused a number of mathematicians in England to misinterpret the challenge. Conti wrote back to Leibniz in March that “several geometers, both in London and in Oxford have given the solution.”

The challenge may have been a failure, but it was not the only line of attack that Leibniz was following. In the same letter that Johann Bernoulli had sent to Leibniz telling him about the Commercium Epistolicum, he wrote about a mistake he had discovered a few years earlier in the Principia. In fact, it had been brought to Newton’s attention by Johann’s nephew, Nikolaus Bernoulli, who had gone to London and there met Newton in 1712.

Newton had written to Nikolaus on October 1, 1712, thanking him: “I send you enclosed the solution of the Problem about the density of resisting Mediums, set right. I desire you to show it to your Uncle & return my thanks to him for sending me notice of the mistake.” He was no doubt very happy to have the correction made prior to the printing of the second edition of the Principia in 1713, as the revisions that went into the making of the second edition were already extensive and required years of exhaustive work.

But the fact that Newton had made this mistake in the first place may have given Johann Bernoulli cause to wonder whether the Englishman had not fully understood calculus even as late as the 1680s, when the first edition of the Principia had appeared. If that were the case, then Newton could not possibly be its inventor, and Bernoulli said as much to Leibniz in 1713. Bernoulli also published his criticism of the Principia in the Acta Eruditorum, the German journal closely allied with Leibniz, in this same year. However, he was reluctant to enter the spotlight and attack Newton publicly as Keill had done to Leibniz. Instead, Bernoulli published this opinion anonymously.

Nevertheless, Bernoulli’s doubts about Newton’s abilities inspired Leibniz to write a short meditation on Newton and the whole dispute that managed, if nothing else, to stoke the flames a little more.

The Charta Volans (Flying Sheet) was a short printed sheet that appeared on July 29, 1713, with no author listed, though few could not guess who the author was. Like the Commercium Epistolicum, the Charta Volans was a flawed document. Leibniz, referring to himself in the third person throughout, used this paper as a vehicle to attack and mock Newton. The heart of the Charta Volans was Bernoulli’s mistaken argument that Newton had stolen the idea of calculus from Leibniz: “After many years there was produced by Newton something that he calls the calculus of fluxions similar to the differential calculus but with other notations and terminology.” It was essentially the same argument and phrases made by Newton’s camp, but in reverse.

However, Leibniz made a good case for himself as the one who was duped by the other’s treachery . . . because of his own trusting nature. “Leibniz on the other hand, judging others according to his own honest nature,” he wrote, “readily believed the man [Newton] when he declared that such things had come to him from his own ingenuity, and so he wrote that it appeared that Newton possessed something similar to the differential calculus.”

The Charta Volans argued that the root reason behind the position of Newton’s camp in general and Keill’s attack in particular was that the English suffered from an “unnatural xenophobia” that caused them to want to steal the credit from the continent and apportion the invention of calculus wholly to Newton. This would be an argument that Leibniz and his supporters would resort to over and over. It was no surprise really, as many of the figures they had known from Britain (most notably Wallis and Collins) were known to be quite protective of British accomplishments. As Bernoulli put it, in a letter to Leibniz a few months before the latter died, “It is a characteristic of the English that they begrudge everything to other [nations] and attribute all things to themselves or to their nation . . . I doubt whether you can expect [even] this much from them, that they will acknowledge Newton to be capable of error, or at any rate to have been mistaken in any one particular.”

In the Charta Volans, Leibniz declared that once he—still in third person)—became aware of the treacherous and unfair way he was being treated, he “considered the question more carefully, which otherwise he would not have examined because he was prejudiced in Newton’s favour, and began to suspect from that very procedure [of the English] which was so remote from fair-dealing that the calculus of fluxions had been developed in imitation of the differential calculus.”

To support this claim that Newton had copied Leibniz, the Charta Volans included the “impartial” opinion of a leading mathematician who pointed out that Newton had been second to publish, and referred to the error that Bernoulli spotted three years earlier as proof that Newton’s methods had been developed in imitation of Leibniz’s, after the mid-1680s. Newton’s supporters would later seize upon this section of the Charta Volans because, in addition to referencing this “leading mathematician” (which was revealed to be Bernoulli), the document referred to a certain eminent mathematician, whom they took to mean Leibniz. So Leibniz would later be mocked for calling himself an eminent mathematician.

But that summer in 1713, when the Charta Volans was produced, Leibniz would get the first barbs in. Perhaps the most stinging passage is where Leibniz mocked Newton’s attempt to steal the credit for calculus, which the German put down to the Englishman’s greed and pride: “He was too much influenced by flatterers ignorant of the earlier course of events and by a desire for renown. Having undeservedly obtained a partial share in this, through the kindness of a stranger, he longed to have deserved the whole—a sign of a mind neither fair nor honest.” Moreover, the Charta Volans pointed to Newton’s earlier troubles with Hooke over the Principia, and to his falling out with the astronomer John Flamsteed over his theory of lunar motion: “Of [Newton’s tendency not to give others full credit], Hooke too has complained, in relation to the hypothesis of the planets, and Flamsteed because of the use of [his] observations.”

Leibniz got one of his friends, a man named Christian Wolf, to print and circulate the Charta Volans for him. By early 1714, copies were being spread around Europe, and Johann Bernoulli wrote to Leibniz that May to share the good news: “Mr. Wolf has sent me many copies of the sheets containing your reply (for Wolf has said it is yours, and the statement appears publicly in the German journal, Büchersaal, which is printed in Leipzig; and has asked me to distribute it amongst the mathematicians known to me; of course I have already done so, and I have especially sent quite a number into France; but I was reluctant to send any to England, lest the English suspect that I am author of that reply.”

From there, the controversy and the battle increased in intensity. Though Leibniz denied that he was the paper’s author, few (and least of all Newton) doubted where it came from. Newton was sent a copy by a man named John Chamberlayne, and, after incredulously reading the Charta Volans, he became almost obsessive in his pursuit of the case against Leibniz. Newton wrote a number of drafts of responses, several of which were found among his papers when he died, though he ultimately never published them nor sent them to others in the form of letters.

Meanwhile, in the summer of 1713, a new Dutch journal, Journal Literaire de la Haye, was launched that carried a translation of the Commercium Epistolicum (done by Leibniz’s man Wolf) in its first issue. Playing to both sides of the fence, the journal also published a paper called the “Letter from London” that was written by Keill, which included an extract of a letter Newton had penned to Collins more than forty years before, in which Newton described his method for finding tangents. Keill claimed that the same letter had been sent to Leibniz. To this, Leibniz responded with an article, “Remarks on the Dispute,” at the end of the year. In it, he again touted the evidence that the mistakes made in the Principia supposedly proved.

Continuing to pay special attention to the dispute later that same year, another issue of Journal Literaire de la Haye reprinted the Charta Volans plus an anonymous review of the Commercium Epistolicum by Leibniz, along with an anonymous response to Keill’s remarks that was also written by Leibniz.

The reason for all this anonymity was simple: For Leibniz, the fact that Keill was attacking him was not acceptable. He did not seem to want to engage in a head-to-head fight with someone who was not only much younger and much less accomplished than he was, but who was fundamentally a less intelligent mathematician. Leibniz seemed to feel no need to denigrate himself by replying directly to an underling like Keill—but rather aimed to take on Newton directly.

But Newton and Keill already had a nice working arrangement, and neither was about to disrupt it. Keill wrote to Newton on February 8, 1714, telling him of the review of the Commercium and asking him, “I would gladly have your opinion what you think is needful further to be done in answer to Mr. Leibniz . . . I am of opinion that Mr. Leibniz should be used a little smartly and all his Plagiary and Bluders showed at large.” Then Keill wrote another two letters to Newton on the subject, saying of Leibniz’s remarks that he “never saw any thing writ with so much impudence falsehood and slander,” and that they must be answered immediately.

Newton replied casually, almost two months later, “If you please when you have it, to consider of what answer you think proper, I will within a post or two send you my thought upon the subject, that you may compare them with your own sentiments & then draw up such an answer as you think proper.” Newton wrote no fewer than seven drafts of a reply to Leibniz’s anonymous “Remarks” but never published any of them.

Instead, it was up to Keill. He sent Newton a draft of his answer in May, and it eventually grew into a forty-two-page article, which he sent to the Journal Literaire de la Haye for publication in their July/August 1714 issue. There is good reason to believe that Newton had played a major role in this “Answer to the Author of the Remarks,” as the article was called, as it was written at a high enough level to probably have been over Keill’s head.

Now that the dispute was fully out in the open and there were numerous published accounts of it, many more people were becoming aware of it, and a number of contemporaries of both men couldn’t help but to get involved. Newton’s enemies among the English intellectual elite, for instance, would send Leibniz copies of such publications as the Commercium Epistolicum, as well as word on what Newton was up to. The astronomer John Flamsteed sent Leibniz a list of errors in Newton’s lunar theories.

To some of Leibniz’s supporters, the Charta Volans was not enough. If Leibniz could respond directly to Newton with his own Commercium Epistolicum, his case would be greatly bolstered. Bernoulli suggests that doing so would bring about a sound victory. “I think that Mr. Newton will some time smart for so easily lending his ear to flatterers,” Bernoulli wrote. “Meanwhile it will be wise for you to concentrate on your reply to the Commercium Epistolicum, finish it in good time and lay it before the public, lest they should have reason to rejoice in the delay.”

Indeed, Leibniz made noises that his own Commercium Epistolicum would be more fair because it would include all the relevant letters and documents, insinuating that Newton had hand-chosen certain documents while ignoring others. When Newton heard of this criticism he said that if Leibniz had letters to produce, then he should go ahead and produce them. He added that there were even more damning letters than the ones that were included in the Commercium Epistolicum, and these were not published.

Leibniz wrote to Johann Bernoulli, toward the end of 1714, “Many distinguished men there [in England] do not at all approve the boldness of Newton’s toadies . . . I am resolved to publish some correspondence of my own, from which it will appear how weak Newton once was in other respects.”

But this was not the easiest thing for Leibniz to do. First of all, he was in Vienna from 1712 to 1714, and far out of range of access to all the relevant letters. Second, it would not have been easy for him to go through his papers and come up with only the most relevant bits—he had massive piles of correspondence stretching back over decades. Going through a stack of these letters would not have been so simple as simply flipping through documents already gathered toward an express purpose, as the committee that had assembled the Commercium Epistolicum had done. Moreover, many of the German’s papers were a mad scramble of tiny writing, some so small as to be barely legible without a magnifying glass. Add to this marginal notes in the same hand and multiple corrections—additions, deletions, and word changes . . . even to Leibniz, familiar with his own words, the unlikeliness of a quick skim through must have felt hopeless. And there, all the time in the background, was the pressure his employer continued to exert on him to finish the historical work.

Meanwhile Newton must have recognized that the Commercium Epistolicum might not be enough to support his own case. He wrote a paper called “An account of the Book entitled Commercium Epistolicum,” in 1714, and published it anonymously in the January–February issue of the Philosophical Transactions of the Royal Society. It filled all but three pages of the issue. He further had it translated into French and published it in the Journal Literaire de la Haye, arranged for a review of it to appear in another journal called the Nouvelles Litteraires, and had it printed as a separate pamphlet and had it distributed through Europe. Then, for good measure, he had it translated into Latin. Finally, Newton was the prolific author his contemporaries had wanted him to be for so many years.

In this “Account,” Newton attacked and devalued one of Leibniz’s greatest contribution to mathematics: his invention of the symbols of calculus, which had greatly enhanced the ability of mathematicians to learn and apply the methods of calculus that are still in use today. Newton, wrote Newton hautily, does not confine himself to symbols.

Feelings were equally hostile in the Leibniz camp, and Leibniz’s supporters generated a great deal of ill feeling—much of it directed at Keill. Christian Wolf wrote a letter to Leibniz in the second half of 1714 that complained of the man and his childish reasoning: “I wonder at the impudence of the man, and also I wonder at his boasting . . . that he fight not with his own weapons, but with Newton’s.”

Leibniz replied to Wolf several months later: “I cannot bring myself to make a reply to that crude man Keill. I have held what he has put forward hardly worth reading.” In another letter, Leibniz showed even more of his true feelings: “Since Keill writes like a bumpkin, I wish to have no dealings with a man of that sort. It is pointless to write for those who respond only to his bold assertations and boasting, for they do not examine the substance . . . I think of knocking the man down, some time, with things rather than words.” While Keill was several years younger and not crippled by gout as Leibniz was, my money would have been on Leibniz—angry as he was.

Leibniz, at this point, was desperate to bring Bernoulli into the fray so that he could champion him the way that Keill was championing Newton. Bernoulli was the perfect man to play that role. He was a master of calculus and had been using it for decades. He was also a very distinguished mathematician, unlike Keill who was secondary in skills and accomplishments to Newton and Leibniz both. In fact, Bernoulli was one of the few people alive who was the mathematical equal to both parties in the dispute—and perhaps even more brilliant and pure a mathematician than either man.

Bernoulli would have made a much more formidable second than Keill, and his disposition was perfect for Leibniz. He firmly came down on Leibniz’s side in the matter, and he was already a “leading mathematician” whose anonymous criticism was contained in the Charta Volans. So why not bring him out in the open?

Bernoulli did not want to be on the front lines of the calculus wars, and he asked Leibniz to keep him out of the controversies. Bernoulli did not want to have his name associated with the dispute because he was torn. On the one hand, he was loyal to his friend and longtime collaborator—Bernoulli’s own career as a mathematician was advanced as a result of his picking up the threads that Leibniz had spun and weaving calculus into a set of mathematical tools that could be grasped and applied by many mathematicians. At the same time, Bernoulli wanted to be diplomatic in his direct dealings with Newton because he personally harbored no ill will toward England’s greatest scientist. In fact, he must have felt the opposite—Newton was the friendly colleague who had helped Bernoulli gain admission into the Royal Society, and also had been the gracious host who had entertained Bernoulli’s son when he was in London.

Still, Leibniz was not going to accept no for an answer that easily. He did little to conceal Bernoulli’s true allegiance, and once, writing a letter referring to the most recent mathematical challenge that had been proposed “to test the pulse of the English analysts,” he outed Bernoulli as the one who had conceived the problem. He also sought to draw Bernoulli out by telling him that Newton knew the letter referred to in the Charta Volans was his. “I wonder how Newton could know that I was the author of the letter,” Bernoulli wrote back, “since no mortal knew that I wrote it except [you and I].”

Finally, Leibniz let slip that Bernoulli was the author of the letter referred to in the Charta Volans, when he anonymously reviewed Newton’s “Account of the Commercium Epistolicum” in 1715. To draw Bernoulli out, Leibniz also began naming him in correspondences as one of Newton’s critics.

Once Newton found out that Bernoulli was the mysterious “eminent mathematician,” he wasted no time in insulting him, calling Bernoulli a “pretended” mathematician in 1716.

Bernoulli would deny his authorship of this letter for years, and after Leibniz died, sought to make amends, letting Newton know that he was not the author and that Leibniz had been misled in attributing it to him. He wrote to the French mathematician Pierre Rémond de Monmort, “I desire nothing so much as to live in good fellowship with him, and to find an opportunity of showing him how much I value his rare merits, indeed I never speak of him save with much praise.”

Newton, for his part, accepted the olive branch from Bernoulli, writing to Monmort in France, “I readily welcome and court his friendship.”

THOUGH BERNOULLI BALKED at getting between Newton and Leibniz, there were many others who were more than willing to do so—and not solely because they were advocating for one man or the other. Indeed, as tempers flared and hostilities became more and more open, many third parties on both sides of the English Channel were anxious to see the dispute reach an amicable conclusion.

The ambitious John Chamberlayne, who was in correspondence with both Newton and Leibniz, tried to single-handedly settle the dispute. He sent a letter to Leibniz, then in Vienna, on February 27, 1714, telling him, “I have been inform’d of the differences fatal to learning between two of the greatest philosophers & mathematicians of Europe, and I need not say I mean Sr. Isaac Newton and Mr. Leibniz, one of the glory of Germany the other of Great Britain, and both of them men that honor me with the friendship which I shall always cultivate to the best of my power, tho’ I can never deserve it . . . yet as it would be very glorious to me, as well as advantagious to the commonwealth of learning, if i could bring such an affair to a happy end.”

But Chamberlayne’s desire to make harmonious wine of the vinegary dispute would die on the vine. Really, all that his efforts did for Leibniz was to allow him yet another outlet via which to vent his anger. Leibniz wrote back, in April 1714, in a harshly-worded letter, that Newton’s purpose in bringing out the Commercium Epistolicum had been to unfairly discredit him, and that he doubted whether Newton had invented calculus at all before reading Leibniz’s work. Nor was Newton any more willing to let bygones be bygones. Chamberlayne sent news of Leibniz’s letter to him, and Newton replied that he would not retract things that were true and that, because the Commercium Epistolicum was a true document, it in no way did Leibniz an injustice.

Leibniz wrote another letter to Chamberlayne in which he laid out his dissatisfaction with the Commercium, asking the Englishman to submit this letter to the Royal Society. It reads in part: “I do not at all believe that the judgment which is given can be taken for a final judgment of the Society. Yet Mr. Newton has caused it to be published to the world by a book printed expressly for discrediting me, and sent it into Germany, into France, and into Italy as in the name of the society. This pretended judgment, and this affront done without cause to one of the most ancient members of the Society itself and who has done it no dishonor will find but few approvers in the world.”

Newton translated this letter himself and had it read before the Royal Society. The members snubbed this effort, however, passing a resolution to coldly ignore the letter without comment. The journal of the royal Society records on May 20, 1714, “The Translation of [Mr. Leibniz’s] Letter to Mr. Chamberlayne produced the last meeting was read. It was not judged proper [since this letter was not directed to them] for the Society to concern themselves therewith, nor were they desired to do so . . .”

Keill, on the other hand was more than willing to take on whatever Leibniz had to offer, writing to Chamberlayne a few months later, “If Mr. Leibniz makes any more noise I will still give the world a greater knowledge of his merits and candor.”

This created such animosity among Leibniz, his supporters, and Keill that they began to regard the latter in the cruelest manner. For example, Leibniz’s friend Wolf wrote him a letter in which he spread the most stinging gossip about Keill: “A few days ago I learnt from someone from England who visited me that Keill had behaved so unlike the occupant of a professiorial chair because of his disgraceful morals (for he has frequented drinkshops and bawdy-houses with the students entrusted to his care, spending heavily on wine and women) that he may become notorious for some infamous proceedings arising from his want of morals. . . .”

Even while such whispers were spread against Keill, further evidence was spread against Newton. Leibniz wondered aloud in his letters, some of which he expected Newton to be privy to, about a famous paragraph in the Principia that was in the first edition but that Newton had retracted from the second.

He also wrote to such people as the Abbé Conti and a Madame de Kilmansegg, saying that Newton had accorded to him the invention of calculus years earlier, in the second lemma’s ending scholium of the second book of the Principia. In this paragraph, Newton wrote: “In a correspondence which took place about ten years ago between that very skillful geometrician, G.W. Leibniz, and myself, I announced to him that I possessed a method of determining maxima and minima, of drawing tangents, and of performing similar operations, which was equally applicable to rational and irrational quantities, and concealed the same in transposed letters . . . This illustrious man replied that he also had fallen on a method of the same kind, and he communicated to me his method, which scarcely differed from mine except in the notation.”

Strangely, this scholium, as the passage is called, had different meanings to Leibniz and his supporters than it did for Newton and his. Leibniz seemed to take this to mean that Newton was admitting that Leibniz was in possession of a method like Newton’s own. Newton and his supporters looked at it as establishing his priority as the inventor.

This difference of opinion was reflected in the pages of another book, History of Fluxions, by British mathematician Joseph Raphson, which appeared in 1715 to further Newton’s cause.

Raphson, who had died before his book hit the streets, had reviewed a half dozen previous published documents that were available to him. Although Newton’s work was not yet published or available to the public, he had allowed Raphson to read some of his personal papers periodically, through the years. The book was clearly biased toward Newton, reiterating in its preface that Newton had the priority and the genius both. Raphson went even further than seeking to set the record straight, by establishing a chronology in favor of Newton, suggesting at the same time, perhaps unfairly, that Leibniz’s calculus was “less apt and more laborious” than Newton’s.

Newton wrote a densely typeset seven-page supplement to the book in which he defended his earlier words in the scholium and claimed that it was a matter of misinterpretation on Leibniz’s part rather than any admission on his part: “It was written not to give away that lemma to Mr. Leibniz but, on the contrary, to assert it to myself.”

When Leibniz became aware of the History of Fluxions, he was already writing his own version of the history, calling it the History and Origin of the Differential Calculus. This was no new idea. Twenty years earlier, he had written to Huygens with essentially the same intention, to write a book on calculus (albeit one that was a little more forward looking). “Your exhortation confirms me in the purpose I have of producing a treatise explaining the foundations and applications of the calculus of sums and differences and some related matters,” Leibniz wrote. “As an appendix I shall add the beautiful insights and discoveries of certain geometricians who have made good use of my method, if they will be so kind as to send them to me. I hope that the Marquis de l’Hospital will do me this favor if you judge it fitting to suggest it to him. The Bernoulli brothers could also do it. If I find something in the works of Mr. Newton which Mr. Wallis has inserted in his algebra which will help get us forward, I shall make use of it and give him credit.”

But, as the Brunswick history, Leibniz never finished it. He may have lacked the patience that was needed to carefully comb through his old notes and letters or he may have simply been too busy with his other things. Nevertheless, his fragmented History and Origin of the Differential Calculus is a document that’s both beautiful and jarring. The opening paragraph, which I quoted at the beginning of chapter 5 of this book, is an outstanding statement of the importance of recording a discovery of any sort—particularly one of the importance of calculus. “Among the most renowned discoveries of the times must be considered that of a new kind of mathematical analysis, known by the name of the differential calculus; and of this, even if the essentials are at the present time considered to be sufficiently demonstrated, nevertheless the origin and the method of the discovery are not yet known to the world at large . . . ,” Leibniz wrote.

Then, in the paragraphs that followed, the History became much more bitter and mired in the dispute at hand:

Now there never existed any uncertainty as to the name of the true inventor, until recently, in 1712, certain upstarts, either in ignorance of the literature of the times gone by, or through envy, or with some slight hope of gaining notoriety by the discussion, or lastly from obsequious flattery, have set up a rival to him; and by their praise of this rival, the author has suffered no small disparagement in the matter, for the former has been credited with having known far more than is to be found in the subject under discussion. Moreover, in this they acted with considerable shrewdness, in that they put off starting the dispute until those who knew the circumstances, Huygens, Wallis, Tschirnhaus, and others, on whose testimony the could have been refused, were all dead.

Leibniz was in the middle of this History when he received a letter from Newton himself. This letter was the fruit of another effort to broker peace—ultimately not successful except that it led to one final exchange of letters between the two. It started when Newton had Abbé Conti arrange for the ambassadors and foreign ministers who were in London, including Baron de Kilmansegg, the ambassador from Hanover, to assemble and decide the issue for themselves. It was a confident and bold move, but one that was doomed for failure. While the ambassadors were more than happy to gather to discuss the dispute, they were not able to come to a decision.

I’m not surprised, really. Newton had arranged for them to view the Commercium Epistolicum and related papers for themselves. But these were no easy documents for anyone to peruse, let alone an international group of nonmathematicians who would have nevertheless prided themselves on their intellectual abilities, or at least interests, such that their need to save face would have prevented their admitting they were not up to the task.

As a solution, the baron urged the English mathematician to write to Leibniz himself, which the Abbé Conti reported back to Newton. Since he had been the one to arrange for the ambassadors to decide the issue, Newton had to follow through with a letter. He did so on February 26, 1716, which the Abbé Conti forwarded to Hanover.

Newton apparently spent many hours drafting this letter, though there was nothing new in it. It was yet another bitter rehash of all the evidence. To him, the Commercium Epistolicum was a factual and fairly collected pile of evidence published “by a numerous committee of gentlemen of several nations.” He displayed no intention of retracting a word of it.

Newton probably had the sense that his argument, solid thus far, was worth sticking to. The letter contained some criticisms of Leibniz’s philosophy, and then ended by stating that it was up to the German to prove his accusations of plagiarism against Newton. “But as he has lately attacked me with an accusation which amounts to plagiary; if he goes on to accuse me, it lies upon him by the laws of all nations to prove his accusation . . . he is the aggressor & it lies upon him to prove his charge,” Newton wrote.

The Abbé Conti wrapped Newton’s letter with his own; in this cover letter, he asked Leibniz directly who invented calculus first. Leibniz wrote to Bernoulli soon after, gloating, “Newton himself, since he saw that I regarded Keill as unworthy of an answer, has entered the ring, having written a letter to the Abbé Conti, who has sent [it] to me.”

Bernoulli replied to Leibniz, “It is a good thing that Newton has at last entered the ring himself, in order to fight under his own name, and laid aside his mask . . . Whatever it may be, I hope now the historical truth will be more clearly discovered, if only Newton will, with that candor which I suppose and trust him to possess, tell faithfully the things which have happened, and will publicly acknowledge the truth of what you have put forward.”

But the exchange would not bear such hopeful fruit. Newton, after getting Leibniz’s letter, responded with an even longer letter containing more reiteration.

Then Leibniz, perhaps sensing that he was finally beginning to confront Newton head-to-head as he had long sought, did the eighteenth-century equivalent of posting his opinion on a public Web site. Seeking to bring as many people into the fray as he could, he sent copies of the correspondence to Paris to be shared and distributed. Leibniz sent his response through Rémond de Montmort—telling him that it was a letter that he wanted communicated to all the mathematicians in Paris in order that they could all be his witnesses. In the letter-proper, Leibniz denied the accusation that he was the aggressor who was accusing Newton of plagiarism, and again blamed the influence of those who would flatter him. “The wicked chicanery of his new friends has greatly embarrassed him,” Leibniz wrote of Newton.

Newton’s “Observations” on Leibniz’s letter, which he recorded shortly thereafter, show how bitter he had become. “Mr. Leibniz accuses them [the committee appointed by the Royal Society] for not printing the letters entire (including as well what did not relate to the matter referred to them, as what did relate to it,) as if it were not lawful to cite a paragraph out of a book, without citing the whole book. Thus he complains, that the Commercium Epistolicum should have been much bigger. But when he is to answer it, he complains that it is too big, and would require an answer as big as itself.”

Where this might have led is anybody’s guess. But the correspondence did not continue. Instead, Leibniz stepped back from any discussion of calculus to attack Newton’s worldview—that is, Newton’s understanding of gravity. Here, Leibniz was no doubt sure, his rival was weak, because the Englishman believed in the hard-to-grasp and impossible-to-justify notion of universal gravitation—action at a distance. Like many of his other contemporaries, Leibniz had difficulty accepting Newton’s theory.

Leibniz had prefaced this attack in a letter to Bernoulli. “Newton in no way demonstrates by means of his experiments that matter is everywhere heavy, or that any part whatever is attracted by any other part, or that a vacuum exists, in accordance with his own boasts,” he had said.

Leibniz clearly wanted to shift the entire debate onto more philosophical grounds. He was, after all, one of the most preeminent philosophers in Europe (a distinction Newton could not claim) and he perceived his advantage in this regard. “His philosophy seems rather strange to me,” Leibniz wrote of Newton to the Abbé Conti. “I do not think it can be established.”

This was not something that Leibniz did whimsically. He probably really thought Newton was wrong, and he must have been convinced that what he saw as Newton’s ill-founded natural philosophy would sink him.

NEWTON WAS CONVINCED that there existed what we would today call a Newtonian universe—that gravity obeying deterministic laws governs all matter. In his earlier days, he had worked out his theory of universal gravitation as a way of describing things like the tides and the motion of the planets around the sun. He didn’t attempt to explain what gravity was, but rather satisfied himself and his readers by describing how it worked.

Gravity, for Newton, can best be understood by the equation he created to describe it. The force due to gravity that is exerted by two objects on each other is a function of the masses of the two objects and the inverse square of distance between them. It was, for Newton, a force that stretched across empty space.

Across the English Channel, Leibniz had profound problems with Newton’s physics because he was at heart a rationalist. He was perfectly willing to accept the mathematical formulation that gravity was inversely proportional to the square of the distance between two objects, but this purely mathematical formulation of reality was not enough for Leibniz. He needed it to be rational.

To Leibniz, one of the principles upon which science was founded was that of sufficient reason: that nothing happens without a reason sufficient for it to happen. He once wrote, “The fundamental principle of reasoning is, nothing without cause.” He also wrote, “This axiom, however, that there is nothing without a reason, must be considered one of the greatest and most fruitful of all human knowledge, for upon it is built a great part of metaphysics, physics, and moral science.”

Leibniz probably did not like Newton’s theory of universal gravitation on the simple premise that action at a distance (as in gravity, exerting a force even through the separation of millions of miles) had to be impossible. He outright rejected the theory as absurd. Or as Leibniz expressed it coldly, “I believe that one must have recourse to a kind of perpetual miracle to explain this effect.”

The once prevailing theory to which universal gravitation was an alternative was the notion that the planets are carried around the sun in vortexes, and Leibniz was a firm subscriber to this theory because it made much more sense than some mysterious miracle force called . . . what was it? . . . gravity?!

For him, the reason for the motion of the planets was simply one of matter—that is, the matter surrounding the planets pushing on the matter that is the planets. Leibniz looked at the fact that all the planets are in the same plane as the sun and reasoned that it was because they were spinning around in a massive vortex of matter. This motion is like the movement of a leaf in a stream, carried along by the billions of water molecules, and, just as without the water the leaf could not float downstream, without the vortex matter “nothing would prevent the planets from going in every direction,” he wrote.

This theory was robust and was employed by Leibniz to explain other things, such as the round shape of the earth, in a very convincing imagining: “If a body is surrounded by another which is more fluid and more agitated, to which it does not permit a sufficiently free passage into its interior, it will be struck from without by an infinity of waves which will help to harden and to press its parts together. A spherical body is less exposed to the blows of this surrounding fluid, because its surface is the smallest possible and because the uniform diversity of its internal motion as well as the external motions contributes to this roundness.”

Newton was likely, of course, enraged by what he regarded as Leibniz’s attempt to change the subject. He probably had no desire to enter into an extensive argument with Leibniz over natural philosophy, and he was spared from having to do so. Instead, another one of Newton’s proxies took up the debate.

Leibniz wrote letters that were critical of Newton’s worldview to Caroline, the Princess of Wales, in November 1715. She was the daughter-in-law of George Ludwig, who by then was sitting on the throne as England’s George I, and was someone who was familiar to Leibniz and somewhat of a champion of his philosophy and person. She passed the letters on to a man named Samuel Clarke, who was uniquely positioned to argue Newton’s worldview with Leibniz. Clarke, the king’s chaplain, had translated the book Opticks into Latin in 1706 for a large fee, and a decade later, was asked by Princess Caroline to translate Leibniz’s Theodicy into English. This he refused to do, but he did respond to Leibniz in writing.

In a letter to Caroline, Leibniz criticized Newton for relying upon divine intervention to explain phenomena and to maintain the function of the universe. Newton’s universe, as he saw it, was a badly constructed clock in need of occasional repair. He objected to this sort of need because he professed belief in the uniform rationality and morality of the universe, and he expressed that God’s decisions were behind everything. Those decisions, he believed, were derived from the same principles of rational and moral human decisions. Clarke responded by arguing against Leibniz, and this began one of the most famous exchanges in the history of philosophy—the so-called Leibniz-Clarke correspondence. This exchange, though short-lived, was significant enough to be published almost immediately, in 1717, and continues to be published today.

Ultimately, though, Leibniz’s attempt to draw Newton into an argument on either metaphysical or philosophical grounds amounted probably to less than he had hoped. Newton never took the bait and there was never a direct discussion between them on the subject of matter. Furthermore, while this may have been a smart and obvious choice for Leibniz to attempt at the time, it was a poor decision historically because his attack on Newton’s theory of universal gravitation weakened his own argument.

Despite the fact that Leibniz clearly saw himself to be on much higher ground, Newton was right about gravity. The arguments Leibniz made against him are somewhat embarrassing historically, because it was one area in which this brilliant man was dead wrong. As the eighteenth century wore on and after both men died, the balance of opinion was to sway in favor of Newton, and the scientists and mathematicians who followed these men began to realize more and more the reality of gravity. The theory of vortexes, while it had its supporters even into the eighteenth century, was destined for the dustbins of science.

And as gravity emerged triumphant, so too did many writers emerge to champion Newton. Perhaps the most famous of these was Voltaire, who derided the theory of the vortex and celebrated Newton for gravity. “Sir Isaac Newton seems to have destroy’d all these great and little vortices,” he wrote. And he added, “This power of gravitation acts proportionally to the quantity of matter in bodies, a truth which Sir Isaac has demonstrated by experiments.”

A consensus was reached years after Leibniz and Newton died: Because Newton was right on gravity, perhaps too, many must have thought, he was right on the true origins of calculus as well. Thus, it was an unfortunate side skirmish in the calculus wars that Leibniz chose to support his case by attacking Newton on gravity.