Farewell and Think Kindly of Me - Newton, Leibniz, and the Greatest Mathematical Clash of All Time - The Calculus Wars

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

Chapter 5. Farewell and Think Kindly of Me

1673–1677

It is an extremely useful thing to have knowledge of the true origins of memorable discoveries, especially those that have been found not be accident but by dint of meditation . . . the art of making discoveries should be extended by considering noteworthy examples of it.

—Leibniz, History and Origin of the Differential Calculus, 1714

In Paris, Leibniz still had to worry about his career, which was suddenly uncertain, and he began making inquiries about other jobs. The death of the elector presented problems for Leibniz, in that he was owed two years of back pay from the old elector. He enlisted young Schönborn to ask the new elector’s permission to remain in Paris, become a political emissary, and report on the political, scientific, and cultural events that were taking place. The response, which eventually came, was that he could stay “for a while” and keep his position as counselor, but he would receive no salary and would not be promoted to emissary.

Things were far from desperate, however, because before he died, Boineburg had made arrangements to send his son, who was a few years younger than Leibniz, to Paris so that he could study under Leibniz. Thus, Leibniz continued to be employed by the estate for over a year, tutoring Boineburg’s son, Philip William, who arrived in Paris on November 5, 1672. But Boineburg’s son clashed with his tutor—playboy aristocrat versus the solitary genius. When young Philip William grew up, he would become a famous governor, be elevated to the noble rank of count, and became known as the “Great Boineburg.” But in the 1670s, Philip William had no inclination toward serious study, especially not of the sort that Leibniz envisioned—a program that was to last from 6:00 A.M. until 10:00 P.M. The seventeen-year-old, a noble from one of Europe’s boon docks, was in the prime of his youth and set loose on the decadent courts of Paris. He preferred to spend his time with his friends, and this caused friction between Leibniz and Philip William. As one nineteenth-century account put it, the young baron was smart and talented, but he was of an age in which he “manifested at that time a greater fondness for the sports which invigorated the body, than for the severe studies designed to develop the mind.”

Leibniz wrote a letter to the Boineburg family complaining about his charge and asking for money to cover his expenses for tutoring and for his previous work on behalf of the boy’s now-deceased father. In response, early in 1673, the boy’s mother ended the tutoring and reduced Leibniz’s pay. Leibniz was dismissed coldly from the employ of the Boineburg estate on September 13, 1674.

Leibniz now sought other employment. Through his friend Christian Habbeus von Lichtenstern, he was offered a position as secretary to the chief minister of the king of Denmark. This, he politely refused. Leibniz desperately wanted to stay in Paris and, from 1673 to 1676, sought continuously to secure a diplomatic or academic position that would allow him to stay there. Unfortunately, the fact that he was not of noble birth was a deal killer for his lofty diplomatic ambitions. Despite his brilliance, despite his charm, and despite his command of seventeenth-century law, he was of little use in diplomacy.

He also tried to obtain a salaried position at the Paris Académie des Sciences—similar to the position his mentor Huygens enjoyed. As a foreigner, such a position was not easily obtained. The fact that the Académie des Sciences paid salaries to its members meant that there was additional scrutiny over who should and who should not be a member. And, like almost everything else in seventeenth-century France, this question was clouded by nationalistic pride. French members of the academy, apparently, felt that there were already enough foreigners in the organization, and that the position and money should properly go to another Frenchman.

Huygens, the most prominent foreigner in the Académie des Sciences, could have helped Leibniz, but he was too busy and distracted at the time. Leibniz tried unsuccessfully to get an audience with the French minister Colbert for help, but failed.

So he tried other ways to gain entry. In typical fashion of the seventeenth-century French society, securing one of these coveted positions required currying favor with important individuals, and this meant having to make bribes. Leibniz was willing to try anything, and he befriended the Abbé Gallois, a man who made up for what he lacked in intelligence with his ability to climb the social ladder. Gallois could have helped arrange for a position for him but, unfortunately, these designs went bust after Leibniz snickered during a presentation Gallois made regarding the war in Holland. The Frenchman was greatly offended and immediately dropped support for Leibniz’s cause.

In the end, Leibniz was forced to accept what perhaps was not his first choice of occupations: working for Duke Johann Friedrich of Hanover, a position he was offered on April 25, 1673. Leibniz had come to the duke’s attention a few years earlier, and Johann Friedrich had invited him to Hanover, but Leibniz had declined at the time since things were going so well in Mainz. He did, however, continue to correspond with the duke for the next few years. In 1671, for instance, he sent him two original papers, “On the Utility and Necessity of Demonstrating the Immortality of the Soul” and “On the Resurrection of Bodies.” Leibniz also sent him a letter with an account of his research in multiple fields, including his idea for making the alphabet of human thought—giving himself an intellectual CV of sorts.

After the affair of the eyebrow, and the deaths of Boineburg and the elector of Mainz, Leibniz—now on the job market—wrote to Johann Friedrich almost as soon as he arrived back in Paris. To the duke, it was not a subtle hint, and Johann Friedrich jumped at the opportunity of bringing him to his court. He wrote back offering a position with a salary, and, to sweeten the deal, did not demand that Leibniz return from Paris immediately.

For Leibniz, this was a sweet deal because he had no desire to leave Paris. In fact, even after he accepted the duke’s offer, he strung Johann Friedrich along for years, establishing the terms of the office, asking for more time to finish his calculating machine, asking to finish his mathematical research, and negotiating other matters with him. To the duke, Leibniz was boastful to a fault about his calculating machine, saying that it was regarded in both Paris and London as one of the great inventions of the time. He wrote to Johann Friedrich on January 21, 1675, asking if he wanted one of the calculating machines constructed for him.

Leibniz had set about supervising the completion of his calculating machine as soon as he returned from London. Always the optimist, Leibniz told Henry Oldenburg that he expected to be done very soon. But he was ultimately not satisfied with the design and decided to make radical revisions. Then, when the design was done, and the machine all but built, the craftsmen working on the project for Leibniz lost interest. Leibniz delayed writing to Oldenburg for months and months, Back in England, Oldenburg was probably wondering what had happened to him. It had been over a year since their last correspondence. Finally, in the fall of 1674, Leibniz had a Danish nobleman, Christian Walter, who was going to England, hand-deliver a letter to Oldenburg. In it, he said that his calculating machine was finally finished, and claimed that it could multiply a ten-figure number by a four-figure number, with but four turns of the crank to get the answer.

Once the machine was finished, Leibniz invited scientists to his rooms in Paris and demonstrated it—to the apparent wonderment of those who witnessed it. One of the people who came was Étienne Périer, who was the nephew of Blaise Pascal, who had invented the precursor machine in Paris some twenty years before. Leibniz’s machine was a vast improvement of Pascal’s machine, which was only able to add and subtract, as it added the two other fundamental algebraic operations, multiplication and division.

Leibniz was a larger-than-life figure, gangly, with long fingers and limbs, and a huge wig and courtly clothes. It’s easy to imagine him, with sweeping gestures, describing the uses of the machine: a marvelous speaker, now he’s talking about how addition and subtraction only required a few turns of the wheel. Whole pages of numbers can be added and subtracted faster than it would take to even write them down. Now he’s on to multiplication and division. The French finance minister, Colbert, wanted three—one for the king, one for the Royal Observatory, and one for his own financial offices.

Leibniz’s calculating machine was only a small part of what its inventor was doing during this time. He also threw himself into mathematical studies, teaching himself much of seventeenth-century mathematics within a few years. In fact, when Leibniz wrote to Oldenburg in the fall of 1674, after more than a year of silence, it was not about his model of the calculating machine but rather about some mathematical work he had been doing. By 1674, after more than a year of exhaustive work, Leibniz had arrived at the same place Newton had independently reached just a few years before. Leibniz still knew very little of the work of Newton, but that was about to change, thanks to Oldenburg.

Oldenburg is practically the inventor of modern scientific discourse—not because he developed any fundamental technology or pilloried the scientific journals with his papers, but because he was behind the success of what was really the first successful scientific journal—the Philosophical Transactions of the Royal Society. Oldenburg was the founding editor of the Philosophical Transactions, which he launched on July 3, 1665, and supervised until issue number 136 in June 1677.

The story of how Oldenburg came to play such an important role in the Royal Society is an interesting one. He was born in Bremen and came to England in 1653 as Bremen’s London consul during the reign of Cromwell. He lost his job a few years later and became a private tutor for a British nobleman’s children in London; when they moved to Oxford in 1656, their tutor moved with them. This was fortuitous for Oldenburg because in Oxford he made the acquaintance of those philosophers who would come together and form the Royal Society.

He was one of the first members of the Royal Society, and he was the secretary of the Royal Society from 1663 until his death. During the nearly fifteen years that he held that position, he was one of the most important members of the society. A prolific letter writer, he kept a correspondence with more than seventy philosophers and mathematicians. Many of these letters were written to communicate discoveries between various philosophers, mathematicians, and scientists throughout Europe. In addition to serving as secretary and furthering the science of British mathematicians through the publication of the Philosophical Transactions, he welcomed the cream of contemporary continental scientists into the society—men like the French astronomer Giovanni Cassini, the Dutch physicist and mathematician Christian Huygens, the Italian doctor and anatomist Marcello Malpighi, the early microbiologist Antoni van Leeuwenhoek, and of course Leibniz.

He kept so much correspondence, in fact, that he drew suspicion of certain officials and was arrested “for dangerous designs and practices”; he was locked in the Tower of London in the summer of 1667 but released after two months. Oldenburg actually deserves a great deal more credit than such biased suspicion afforded him. For the last few years of his life, if there was a discovery being made in England or the continent, he was probably in the middle of communicating it.

He was also involved in disputes between various people, such as when Huygens became embroiled in his fight with Hooke after he invented his balance spring, a device that uses oscillations to regulate the movement of a clock. It was a significant technological improvement at the time, and Huygens sought and was granted a patent for it by Colbert, the French minister of finance. Huygens also registered his invention of the balance spring with the British, in a manner of speaking, by sending to the Royal Society a letter containing a coded anagram description of it. Later, he sent a full description, and when this description was read at a meeting of the Royal Society on February 18, 1675, Hooke lashed out at Oldenburg, claiming that he had inverted the balance spring himself, accusing Oldenburg of spilling the beans to Huygens, and suggesting the venerable secretary was a French spy. The Royal Society backed Oldenburg against Hooke’s claims, but these charges would unfortunately linger over his head long after he died—complicated no doubt by the central role that he played in the calculus wars by fostering communication between Newton and Leibniz.

Twenty years after he first came to England, Oldenburg was perhaps the only person alive who was in continuous contact with Newton and Leibniz for the entire time the latter was in Paris. He enabled their first correspondence—two letters each, which were written by one, passed to Oldenburg, and then forwarded to the other.

What led to this exchange of letters was the correspondence Oldenburg himself carried on with Leibniz after Leibniz returned to Paris from London. They had been occasional correspondents for a few years before the two finally met in 1673, while Leibniz was visiting London, and afterward the two were in close contact. Oldenburg had taken an interest in Leibniz as his fellow countryman and a brilliant thinker. Leibniz had mutual admiration for the older German, since he was a friend of Boineburg’s, and believed Oldenburg would be a good source of information for him on the state of mathematical discoveries in Britain. And Leibniz was right. Oldenburg did as much as he could to share with him information about the state of British mathematics.

This mutual exchange would lead some to believe the sort of accusation that Hooke made—that, in fact, Oldenburg was some sort of spy. In fact, one nineteenth-century account of the calculus wars makes quite a lot of the fact that Oldenburg and Leibniz were both from northern Germany. “The Royal Society in London had committed the oversight of employing as their secretary, not an Englishman, but a German, Heinrich Oldenburg,” the writer, a Dr. H. Sloman, said. “This imprudence could not but soon have its consequence, and this consequence in particular, that when once the right man came, the interest of England was more or less sacrificed to a German friendship.”

Sloman’s book claimed Oldenburg promoted a young and overly ambitious Leibniz, who took advantage of Oldenburg’s natural bias toward his fellow countryman and made Oldenburg his “agent.” Sloman had Oldenburg conspiring with Leibniz and shuffling the younger man through a side door of the Royal Society into a prestigious membership on nothing more than the older German’s assurance of Leibniz’s genius and not on the merits of his nominee’s work.

“Oldenburg here again contrives his defense,” Sloman wrote of the secretary’s reaction to the affair of the eyebrow. “And as Leibniz had now become his pet and favorite, he exerted himself for his fame more than for his own . . . and so we see with astonishment the endeavors of the two friends quickly crowned in the access of the young man in the honor of becoming a member of the Royal Society.”

This is ludicrous for a few reasons, not the least of which is the fact that there were contemporary members of the Royal Society who were far less accomplished than Leibniz—even at that early age. Nevertheless, there is no question that Oldenburg’s communications with Leibniz did more to fan the flames of the calculus wars years later, when it exploded after the turn of the eighteenth century, than nearly anything else that happened in the 1670s.

A critical exchange took place in April 1673, when Leibniz received a long letter from Oldenburg. In the early 1670s, Oldenburg was compiling a roundup of all the great accomplishments of British mathematicians based on information he was gathering from others, particularly John Collins, who has been described as a pygmy standing between two giants. He was a minor government servant, an accountant—a mathematical hobbyist really—who, by luck, chanced to be central to one of the only exchanges of letters between the two greatest mathematical geniuses alive in his day.

The son of a poor preacher outside Oxford, Collins was a bookseller’s apprentice who later spent seven years as a seaman in service against the Ottoman Empire. Later, he became a mathematics teacher, an accountant, and finally (owing to the fact that he was a likable chap) a well-connected mathematician. Although he never contributed great mathematical discoveries like Newton and Leibniz, nor was he a consummate enabler of correspondence like Oldenburg, he nevertheless knew enough to comment on the work of others, and he could recognize great work when he saw it. Because he understood algebra, Collins was involved in Oldenburg’s communications with Newton and Leibniz. Oldenburg was not himself a mathematician, and could do little with obscure mathematical discoveries entrusted to him without help.

Collins was in a perfect position to be that help. He was one of the few who were privy to Newton’s early results as a mathematician. Newton had written letters to Collins in the early 1670s describing a number of his results, and they had carried on a lively correspondence for several years. Collins was happy to communicate these results to Oldenburg because he was what one might call a mathematical anglophile—one who wasted no opportunity to assert British superiority in math or science.

In his position as mathematical intermediary, Collins helped Oldenburg to draft Leibniz a letter detailing the status of mathematics in Britain—including the work of Newton. For Leibniz, the most valuable part of the letter was probably references to the contemporary British publications that Collins had meticulously compiled. This report contained references to books and papers that revealed to Leibniz the existence of a whole literature of mathematics that he scarcely knew existed.

The mathematical details sent to Leibniz were purposely vague, though, because Collins was cautious about revealing too much about his countrymen’s proprietary discoveries. He regarded the French with particular suspicion, and though Leibniz was not French, he carried the stain of living in Paris. Plus, the young German was a protégé of Huygens, who was then seen as one of the main competitors of English mathematicians.

So as much as Collins revealed, he withheld. To Leibniz, he described results of work by Newton and the Scottish mathematician James Gregory on infinitesimals, for instance, listing problems that Newton and Gregory could solve—but not their methods. This vagueness was unfortunate because it later led Leibniz to believe that his growth in mathematical discovery was completely fresh. There were plenty of other mathematicians who had solved the sort of problems calculus could solve, by using methods other than calculus. Leibniz would think that he was making completely original strides while, in fact, much of what he was discovering had already largely been worked out by Newton; it just hadn’t been published—partly because of the Great Fire of London and partly because of Newton’s trouble with Hooke.

An example of the level of detail, or rather the lack thereof, can be appreciated in the following passage:

As to solid or curvilinear geometry, Mr. Newton hath invented (before Mercator publish’t his Logarithmotechnia) a general method of the same kind for the quadrature of all curvilinear figures, the straightening of curves, the finding of the centers of gravity and solidity of all round solids and of their second segments . . . which doctrine, I hope, Mr. Newton is publishing. . . .

After receiving this letter, Leibniz went more than a year without writing anything to Oldenburg at the Royal Society. Following up on the references Collins provided, Leibniz was astounded to find out that, besides the material he presented to Pell that caused the affair of the eyebrow, much more of the mathematical work he was doing had already been done by others. Astounded and excited at the same time, he now knew what he didn’t know. Leibniz withdrew into the cell of his mind and began to work and rework the mathematics that he had to understand.

When Leibniz wrote to Oldenburg in the summer of 1674, after his many months of silence, Oldenburg had no way of knowing what an expert mathematician Leibniz had become, but that is how Leibniz presented himself in his letter, “In geometry, I have made some discoveries by rare luck . . . theorems of greater importance [including] certain analytical methods, completely general and widely extended, which I value more highly than particular theorems however excellent.” And as if too excited to wait for a reply, Leibniz wrote another letter a few weeks later, reiterating that he had made “a notable discovery” in the branch of geometry involving the analysis of curves.

Oldenburg replied on December 8, 1674, that Newton and Gregory both had general methods for all geometrical curves by which they could determine surface areas and volumes and other functions related to curves, such as tangents. Leibniz wrote Oldenburg yet again on March 30, 1675, excited about Newton and his work. “You write that your distinguished Newton has a method of expressing all squarings, and the measures of all curves, surfaces and solids generated by revolution, as well as the finding of centres of gravity, by a method of approximations of course, for this is what I infer it to be. Such a method, if it is universal and convenient, deserves to be appraised, and I have no doubt that it will prove worthy of its most brilliant discoverer.”

Thus began the exchange of letters involving Leibniz, Oldenburg, Collins, and eventually Newton in the last two years Leibniz was in Paris. They corresponded more or less continuously, playing a sort of cat-and-mouse game, with Leibniz sharing information, holding some back, and Collins doing the same thing. Leibniz began asking a number of questions about a specific type of geometrical problem called a quadrature. Quadratures were one of the hot topics in the 1670s, and many different mathematicians were working on difficult solutions to them. Calculus makes solving quadrature problems trivial. He also began to boast about his own methods—if in the vaguest possible terms, taking his cue from the previous exchange with Collins.

He began asking specific questions about Newton and Gregory’s results—did they have methods for rectifying the hyperbola and the ellipse? He offered to trade his own “far reaching” methods for some of Newton and Gregory’s methods that he knew Collins possessed. Leibniz was now very interested in what Newton had to offer, for it seemed to him that Newton had already made a lot of progress in this area.

Meanwhile, Leibniz made superb progress on his own. He had gotten a good start in mathematics thanks to his study of the work that Pascal and others had already done, and soon he began to make important discoveries of his own. One was a technique he called the transmutation rule, which was a way of figuring the quadrature of a curve, an important step along his way to inventing calculus.

By October 1675, having absorbed everything he could from his contemporaries, pulling together their work in his self-imposed retreat, he came out of his intellectual gestation and forged ahead. In 1675, Leibniz moved beyond the body of available knowledge and into the uncharted territory of differential calculus. In October and November of that year, he was able to bring these ideas together in a number of notes and papers he wrote containing the essence of calculus.

Moreover, Leibniz invented the symbols of differential and integral calculus, as we know them today. On October 29, for instance, he came up with the integral sign. Leibniz saw integration as summation. In fact, that’s why he gave it his symbol, “∫,” which is a fancy S that he invented. The new symbolism provided a general way to treat infinitesimal problems of calculus and would prove most useful for its spread.

This was a notion that appealed completely to Leibniz, who always favored utilitarian ends. Even in his younger days, when he was a mathematical novice, he was keen on communications being easily understandable. For instance, he praised the work of a philosopher, Nizolius, not for his philosophy itself, which contained many errors, in his opinion, but for its clear literary style. Nizolius, in fact, had suggested that anything that could not be described using simple terms expressed in everyday language was useless. In response to Nizolius, Leibniz recommended that jargon be avoided. In fact, one of his first introductions to mathematics while he was still in college was by a Professor Erhard Weigel, who had a reputation for taking apart other academics by asking them to repeat their Latin arguments in plain German. Weigel instilled in Leibniz a love for simplicity in discourse.

It’s no surprise, then, that following his discoveries in calculus, Leibniz saw the need for a clear way of describing them. In creating a clear and compact language for his work, he became a master mathematician. Leibniz proved this soon after, when a French mathematician, Claude Milliet Deschales, asked him to determine what the volume of that part of a circular cone would be if you cut off the tip with a plane parallel to the base, and Leibniz was able to work this out in a single evening.

In the next couple of years, Leibniz developed his methods of calculus, but he wouldn’t publish his work for another decade, which is something that is worthy of a comment.

Of all the nuanced differences between the work of a seventeenth-century scientist and a more modern one, none seems more pronounced than publishing. Today, publishing plays a central role in science and is integral to the advancement of nearly every scientist’s career. In fact, research findings are not finished in a sense until they are published in a peer-reviewed journal, and scientists make their reputations based on the number and quality of such publications. Competition among scientists is fierce, and there is often a rush to publish discoveries almost as soon as they can be written and reviewed. In recent years, scientific journals have even taken to publishing papers online as soon as they are ready—and in some cases even before they are edited. Today the idea of not publishing and jealously guarding a work as profoundly important as calculus is foreign. Today’s successful scientist making an original discovery that is worthy of publication will likely move with great speed to publish the results.

Leibniz might have published his calculus work earlier than he did, but the problem was that he needed to deal with much more pressing matters. As towering an achievement in the history of mathematics as inventing calculus as a relative novice may have been, it did little at the time to advance his career. His formal appointment to the Court of Hanover took place at the beginning of 1676, and, from that moment, the clock was ticking—the forces pulling him away from Paris were growing. At the end of February of that year, he was told that his patron duke wanted him to come to Hanover, and he would soon have to do so.

His uncertain future aside, Leibniz continued to work, correspond, and study. He wrote to his acquaintances on subjects that included law, gravity, and the logical underpinnings of experimental physics, and he had the desire to correspond on mathematics. He wrote to Oldenburg at the end of 1675, promising to show the solution to an unsolved problem in geometry that he had solved using new methods he had invented—an allusion to calculus.

Now the stage was set. These discoveries that Leibniz made in the waning months of 1675 would bring him, within a year, in contact with Newton. Just before he left Paris, Leibniz and Newton exchanged a few letters in which they danced around the subject of calculus. Their exchange had all the outward dull politeness of academic courtesy, and there is little there that anticipates the polemics that they wrote about each other decades later, when the calculus wars were at their climax.

Newton knew vaguely of Leibniz before their exchange, since he was familiar with one of Leibniz’s fellow Germans, Ehrenfried Walther von Tschirnhaus, who arrived in Paris from Saxony in August 1675. Tschirnhaus soon became friends with Leibniz, and the two made a few joint studies in addition to having many mathematical discussions (in which Leibniz was clearly the master). But Newton was not impressed with Tschirnhaus, and by extension probably wasn’t impressed with his countryman, Leibniz.

At the time Leibniz was inventing calculus, Newton was still dealing with the fallout of publishing his theory of colors and was still having problems with his optical theories. He had been defending himself against Hooke and Huygens for more than three years, and the affront would not go away any time soon. Newton sent Oldenburg a long letter enclosed with a document, “An Hypothesis explaining the Properties of Light Discoursed of in my Several Papers,” on December 7, 1675. The “Hypothesis” was an extensive defense of his optical theories.

Also in 1675, Newton made a trip to the Royal Society to attend a meeting—his first, even though he had been a member for three years. But far from striding triumphant into the hallowed halls, he was ready to cut himself off from communication just about the time when he would engage in the most important correspondence of the calculus wars. In fact, in five months’ time, Hooke renewed his attack on Newton in early May 1676, by standing up and declaring at a meeting of the society that Newton’s work on light was lifted from his own work, Micrographia. On May 25, a battered, agitated, and distracted Newton was approached by Collins and Oldenburg, and asked to write a letter to Leibniz. Newton was so embroiled with his battles over his optical work that he had little taste for opening himself up to a potential attack by revealing his work to a rival mathematician.

Still, Collins cajoled him to write to Leibniz because he was afraid that Leibniz was catching up with Newton. Collins was right. Leibniz was fast becoming every bit as brilliant the mathematician that Newton had been for a decade. Collins wasn’t in the greatest of positions to carry on a correspondence during these days. He was at the end of his life and not in the best of health. And, in 1676, he lost his job. Nevertheless, that May, Collins heard from Oldenburg that Leibniz was interested in further communication, and he began putting together a large account of the discoveries of James Gregory, who had recently died. This fifty-page document was later called the Historiola, and was meant to be a summary of English achievements in mathematics over the previous several decades.

In France and elsewhere on the continent, Descartes was still revered for his mathematical work, and his supremacy in mathematics was often asserted. But Collins felt that the Brits had made significant progress beyond Descartes, and the “Historiola” was an attempt to document this. Collins wrote the “Historiola” as a means to inform rather than instruct. He was not so much interested in teaching the mathematics of the British mathematicians to Leibniz as in securing their rights as inventors, and so he simply expounded which mathematicians had solved which problems, without going into methods or proofs.

Oldenburg thought the paper was running too long at fifty pages, so he asked Collins to abridge it. He then translated the abridgment into Latin. This was unfortunate because, in transcribing this complicated document into Latin, certain errors were made.

In any case, Leibniz would soon be receiving his first, enticing letter from a paranoid, battered Newton in the summer of 1676. Newton finished his epistola prior, as he would later call this letter, on June 13, and he sent it on to Oldenburg, who received it on June 23 and read it at the Royal Society a few days later. Sensing that this letter was of some importance, Oldenburg took extra measures to ensure that it was preserved and that Leibniz would get his copy. He had the letter copied and sent it to Leibniz some six weeks later, along with extracts from the letters of Gregory.

Not trusting the regular post, Oldenburg gave the package to a man named Samuel König. König, a German mathematician, would be leaving London around the beginning of August and heading to Paris. The timing was perfect—always better to wait a few more days and have it hand-delivered, Oldenburg must have reasoned. Once he got to Paris, however, König couldn’t find Leibniz, so he left the package at a local store, thinking the owner would soon see the German to conclude the relay. As it happened, the letter languished until Leibniz wandered by weeks later on August 24, 1676, and found it . . . what’s this? A letter from England?!—a letter from Newton himself!

The first letter is eleven pages long and is a catalog of the Englishman’s mathematical results, detailing several problems that Newton was able to solve with his methods. The centerpiece of this letter was Newton’s binomial theorem, a highly original discovery of great improtance in probability, algebra, and mathematical series, whereby roots of an equation can be extracted and a calculation simplified. The letter hints at “certain further methods” that Newton did not then have the time to explain. There was nothing in the letter of the central problem—that calculus could be used to solve these same infinite series problems.

Newton was being cautious; he may have suspected Leibniz was playing a complicated ruse to get him to reveal his secrets—by pretending that he had secrets of his own. Thus there was nothing in the letter that was not already known to Leibniz in some form or another. Nothing. The only new item, in fact, was added by Oldenburg—another reminder to Leibniz that his promised calculating machine was long overdue. “I would really like you, a German and a member of the said society, to fulfill the promise you gave, and in that way relieve me as soon as possible of an anxiety on account of a fellow citizen which vexes me very much,” Oldenburg wrote, concluding his cover to Newton’s epistola prior. “Farewell again, and pardon this frankness of mine.”

The fact that Newton did not send his methods would be an important point when the dispute raged decades later, because Leibniz would legitimately claim that he got nothing from the English and Newton as far as the methods of calculus were concerned. For all Leibniz knew, Newton had one method for solving a problem and he had another. In fact, Newton seemed to say the same thing himself in the opening to his epistola prior and was perfectly willing to acknowledge that Leibniz had something mathematically. “I have no doubt that he has discovered [speedy methods] . . . perhaps like our own if not even better,” he wrote in the first letter.

Oldenburg warned Leibniz that, in getting the letter transcribed, mistakes may have crept into the version that he now held, but that they should not present a problem for its recipient. “Your shrewdness will correct any errors,” the older German wrote in his cover letter.

Leibniz was blown away by the epistola prior. He immediately dashed off a reply for Oldenburg to give to Newton, commenting that the letter had “more numerous and more remarkable ideas about analysis then many thick volumes printed on these matters.” He called Newton’s series work to be worthy of the man who came up with the theory of colors and who invented the reflecting telescope.

In his response, Leibniz described his own mathematics and described an original discovery of his own, called his transmutation theorem, but withholding descriptions of his methods just as Newton had withheld his. He also included his arithmetical quadrature of the circle, as promised but, again, as was characteristic of this entire exchange, sent only the basic details, withholding the critical secrets that allowed him to solve it, feeling that since Newton gave only his results he needed to do so as well. On the other hand, he asked many questions, clearly intending to maintain the correspondence. Knowing that he would shortly be leaving for Germany, Leibniz wrote this reply after just three days, sending it on August 27, 1676. He ended the portion of the letter that was written for Oldenburg with a polite salutation: “Farewell and think kindly of one who is devoted to you.”

Leibniz was so excited and rushed, that his scrawled letter contained several mistakes and was written in a thick chicken scratch that was hard for Collins to copy over for Newton, and Collins amplified the sloppiness with his own mistakes in transcribing the letter. Significantly, the date on the cover of the letter was miscopied, so years later, when Newton was re-creating the chronology of that summer, he assumed that Leibniz received the letter shortly after he sent it in June. When Newton was poring back over this material, he incorrectly assumed that Leibniz had taken six weeks to reply—ample time to consider the material contained therein at great length. Years later, some of Newton’s supporters would also seize upon the mistakes as proof that Leibniz did not know what he was doing as opposed to being symptomatic of Leibniz’s haste—as was his excited tone.

When Newton received Leibniz’s reply, many weeks had in fact passed, and because he assumed that Leibniz had taken this time to write his letter, Newton decided to do the same thing. He took his own time to reply back—a tragedy, as it turns out, because after spending six weeks crafting his second letter to Leibniz, which he later called the epistola posterior, Newton sent it on November 3, 1676, but it was by then too late to send it to Leibniz in Paris. It did not reach Leibniz for nearly a year, because by the time Newton mailed it, he had already left Paris for the last time. When it finally did reach him, he was in Hanover.

While Newton was mulling over making his response, Leibniz, having delayed returning to Germany as long as possible, could delay no longer. The duke, turning up the heat, wrote to him several times in the summer of 1676, again asking Leibniz to come to his new job as quickly as possible. Leibniz stalled for a few more months. He got another letter from Hanover in July, and the tone of this one was different. Its writer, a court official named Kahn, expressed genuine surprise that he had delayed so long, perhaps sensing that Leibniz was not going to come at all. But rather than admonishing him the letter sought to sweeten the deal, and Kahn offered that, in addition to his post as a counselor, Leibniz could also be in charge of Johann Friedrich’s library.

Ah, books! The duke and his men knew exactly what they were doing by offering this to Leibniz. It was like offering an addict his favorite drug. In July, Leibniz was given his travel expenses from the Hanoverian ambassador in Paris and finally, on September 13, 1676, the duke put his foot down, writing that Leibniz could either come to Hanover or forget it.

Leibniz now had no choice. By the end of September, he had delayed leaving Paris for as long as he could. Within days, he was forced to leave Paris for good—riding out of town with the mail coach on October 4, 1676. He had come to the city a young man primarily interested in law and matters of state, knowing very little mathematics, and left four years later one of the top two or three mathematicians in Europe. (Today there is a street in Paris, the rue Leibniz, named in his honor.)

But he was still not yet on his way to Hanover. On his way there, Leibniz made a few stops. First, Calais, where the autumn storms blew against the docked boats for nearly a week until he was able to board one and set sail for England, where he arrived on October 18. He stayed in London for a little more than a week—several days that would shake his world forty-five years later and become the cornerstone of the claim that Leibniz had benefited from seeing Newton’s early work.

In London, Leibniz met Oldenburg again and showed him, at long last, the calculating machine. This meeting was rather insignificant historically—the much more important meeting on this trip was when Leibniz finally met Collins. Collins was apparently much charmed by his young guest despite the fact that he spoke no German and only poor Latin, and Leibniz had only poor English. But Collins liked the young man and he allowed him to peruse his correspondences and papers and to have access to the books in his possession, including some unpublished works of Newton’s. Collins was the Royal Society librarian in those days, and the society was still on recess for that week, so there was really no harm at all, he thought.

Leibniz looked at Newton’s “De Analysi” and took notes from it. He also looked at the long “Historiola,” which would become the subject of accusations against him decades later. Newton was convinced that Leibniz had the “Historiola” with him in Paris because there was a note on the cover asking him to return it when he was done. The note was, of course, referring to when he was done with the book while in London, where he spent a few hours over a few days looking at it. Newton assumed that Leibniz spent months studying it, as opposed to taking the quick look at it and making the few notes that he did.

Nevertheless, this seventeenth-century equivalent of a Post-it note became evidence for Newton and his supporters later that Leibniz had read the “Historiola” and other documents while in London. The “Historiola” detailed a lot of information about Gregory, Pell, and Newton, and, in particular, under the auspices of Collins, Leibniz saw a letter Newton had written that contained a detailed explanation of his rule for finding tangents—the slope of a curve at any given point—which would be something that Newton would claim Leibniz stole from him.

Collins tried to get Newton to publish his calculus, but Newton was too burned by the experience of publishing his theory of colors that he would not even consider it. “I could wish I could retract what has been done,” he wrote to Collins on November 8, 1676, “but by that, I have learnt what’s to my convenience, which is to let what I write lie by till I am out of the way.”

Newton also reassured Collins that his methods were superior to Leibniz’s. “As for the apprehension that Mr. Leibniz’s method may be more general or more easy than mine, you will not find any such thing. . . . The advantage of the way I follow you may guess by the conclusions drawn from it which I have set down in my answer to Mr. Leibniz: though I have not said all there.”

A few months after Leibniz left, Collins wrote to Newton about the German’s visit, saying that they had discussed some things taken from letters written by Gregory. However, Collins did not mention that he had let Leibniz see Newton’s papers—perhaps feeling guilty about showing him so much.

A few years later, Collins died without Newton ever realizing what he had shown Leibniz. Only decades later, long after Leibniz published his calculus papers, would Newton piece together what had transpired during that late autumn week in London, but even then imperfectly, of course, because he would draw far too much significance from the fact that Leibniz had read Collins’s copy of “De Analysi.” “De Analysi” was a crucial document because it and other pieces of evidence proved that Newton had invented calculus before Leibniz. But they were not enough to prove that Leibniz had borrowed any ideas from Newton, so the evidence that Leibniz was in London and looked at these works was essential for establishing the possibility that the German had stolen his work from Newton.

In fact, Leibniz did take notes from “De Analysi,” but the notes themselves are not really on the formulation of calculus but on some of the other things that are contained in the book. Today, there is little argument over the fact that Newton and Leibniz did their work independently of one another, because the documentation exists in Leibniz’s notes from October 1675—many months before he saw anything of Newton’s.

But the conflict was still to come; Leibniz was on his way back to Germany where he would start a new life, and Newton, apparently, was losing his interest in mathematics, which he referred to as dry and barren. Instead, he was becoming interested in alchemy and other subjects.

Leibniz left London feeling good about having finished his obligation with Oldenburg and having opened up a new line of communication with Collins. He set out for Germany aboard the yacht of Prince Ruprecht von der Pfalz, whom he met in London. He sailed first to Rotterdam, writing a discourse on the subject of a universal language while he was waiting to set sail, and complaining in a letter to an acquaintance that he had nobody to talk to but sailors.

From there, he made his way to Amsterdam, where he met with a few notable people, including Johann Hudde, the mathematician who had independently discovered many of the precursor methods to calculus—such as finding tangents to curves and doing the quadrature of the hyperbola. Then he went on a short tour of the surrounding country, visiting Haarlem, Leiden, Delft, the Hague, and finally back to Amsterdam. He met Antoni van Leeuwenhoek, who was a fellow member of the Royal Society and is still famous today for his discovery of microorganisms. He had long conversations with Benedict Spinoza on philosophy and theology.

Finally, he left for Germany and arrived in Hanover at the very end of 1676. As Leibniz’s time in Paris had come to an end, so too was the war that Leibniz had gone to Paris to prevent coming to an end. It would finally be over, with the Treaty of Nijmegen, in 1678. The treaty allowed Holland to remain intact, and as a concession to France, Louis XIV was allowed to keep the Lorraine. The preparations for this treaty took much time, and, even a year earlier, Leibniz had been busy writing documents supporting what would eventually be the peace conference when he got Newton’s very old and well-traveled second letter with a note from Oldenburg in June 1677.

As noted, this letter would not reach him for nearly a year after Newton had dispatched it. Oldenburg wrote his cover letter to Leibniz on February 22, 1677, explaining that he “put off writing to you until now, because I did not want to endanger what I have at hand for transmission to you, including a letter from Newton as weighty in argument as it is copious in expression.”

In the second letter, nineteen pages long, Newton was even more superlative with his praise: “Leibniz’s method of obtaining convergent series is certainly extremely elegant and would sufficiently display the writer’s genius even if he should write nothing else.” Newton also now expressed an interest in seeing Leibniz’s results. He wrote, “The letter of the most excellent Leibniz fully deserved of course that I should give it this more extended reply. And this time I wanted to write in greater detail because I did not believe that your more engaging pursuits should often be interrupted by me with this rather austere kind of writing.”

If Newton’s letter was warm on the surface, it was frozen in the middle. He was not particularly enthusiastic to carry on the correspondence. He gave a rich though veiled description of some of his most important mathematics, writing again about his series methods and on his discovery of the binomial theorem, and touching on his methods of fluxions (calculus) by showing three examples, tantalizing Leibniz by stating that he had arrived at “certain general theorems.” Of course, he was not willing to part with anything of real substance, so he refrained from going into too much detail.

What detail he did divulge, he pined over. After he sent the letter to Oldenburg toward the end of 1676, Newton sent another letter just days afterward, asking him to make a few changes. “Two days since, I sent you an answer to M. Leibniz’s excellent Letter. After it was gone, running my eyes over a transcript that I had made to be taken of it, I found some things which I could wish altered, & since I cannot now do it myself, I desire you would do it for me, before you send it away.”

So careful was Newton that, when he did disclose an important statement of calculus, he did so in an unintelligible form. He sent it in the form of an anagram—a common device in those days for asserting priority while not revealing anything. “The foundation of these operations is evident enough,” Newton wrote. “But because I cannot proceed with the explanation now, I have preferred to conceal it thus: 6accdoe13eff 7i319n4o4qrr4s8t12ux. . . .”

These secrets were transposed encoded characters. Once it was transposed properly and translated into Latin (and then into English), the sentence read: “Given in an equation the fluents of any number of quantities, to find the fluxions and vice versa.”

How hard would it have been for Leibniz to read these lines? Impossible. To give a flavor of the difficulty, imagine reading a single word thus coded, “coffeepots,” and trying to decipher its meaning. A simple cipher would be to replace each letter in “coffeepots” with the proceeding letter of the alphabet; the word would become “dpggffqput”; then, transposing these letters randomly would give something like “fpgqpufdtg.” The word “fpgqpufdtg” bears little resemblance to “coffeepots,” and likewise the sentence that Newton wrote was unrecognizable.

Writing anagrams was not so unusual. Huygens wrote his own anagram one time to conceal his invention of the spring balance for his pocket watch. Likewise, Newton was using an anagram to evidence the fact that he was in possession of his method of fluxions yet clearly did not intend to share that method; he would have known that Leibniz had absolutely no way of decoding the anagram. Moreover, even if he had the key to decipher the code, Leibniz would not have been able to decode the anagrams because one of them wasn’t even transcribed correctly in the copy that was sent to him.

Indecipherable bits of the letter aside, Leibniz was thrilled to receive it. He had been in the intellectual backwater of Hanover for several months and must have been going through withdrawal when he received the epistola posterior. He immediately responded to Newton and Oldenburg just days later, on June 11, 1677, in a letter full of praise and inquiry. He communicated the essence of his differential calculus, and he implored Newton for further correspondence. “I am enormously pleased that he has described by what path he happened on some of his really very elegant theorems,” he wrote, and he wrote again a few months later, practically begging Newton to open the communication. Leibniz further asked Oldenburg to send him copies of the Philosophical Transactions and news of other discoveries in Britain.

Oldenburg replied to Leibniz on August 9, 1677, telling him that Newton was preoccupied and thus he shouldn’t expect a reply right away. Newton never did reply. Overtired from the dispute over his theory of colors, the Englishman had neither the time nor the inclination to write further. In fact, he wrote to Oldenburg in the cover letter to his second letter to Leibniz, “I hope this will so far satisfy Mr. Leibniz that it will not be necessary for me to write any more about this subject. For having other things in my head, it proves an unwelcome interruption to me to be at this time put upon considering these things.”

Indeed, two days after sending the second letter to Oldenburg, Newton wrote to him again, begging, “pray let none of my mathematical papers be printed without my special license.” For the next few years, Newton hardly wrote any letters at all, to anybody.

In August 1678, Oldenburg went to Kent for a summer holiday with his wife, and while there they both contracted a severe fever and died. When Oldenburg died the communication between Leibniz and Newton died with him. The correspondence, slow in starting and marked by difficult interruptions with Leibniz suddenly moving countries in the middle, now ended abruptly.

In the ten years that followed, Newton and Leibniz completely lost track of each other. Newton shrunk back into his office in Cambridge University, and Leibniz became mired in the dealings of the court of Hanover—a position he would hold for the rest of his life.