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

Chapter 6. The Beginning of the Sublime Geometry

1678–1687

If it takes two to make a quarrel, it takes two men of genius to make a famous quarrel.

—A. R. Hall, Philosophers at War

Hanover Germany is a new city today—literally. Destroyed by allied bombing raids during World War II, it was rebuilt from the streets up. It is now home to a large university and a population of about a half million.

A large sign at the airport greets travelers with the salutation WELCOME TO HANOVER, THE CITY OF INTERNATIONAL FAIRS. When asked what type of fairs these were, one local resident said that they were industrial in nature, featuring computers and machines that build machines. Apparently, the fairs were from Leipzig, which had traditionally hosted them until after World War II, when Leipzig wound up in East Germany.

Like the fairs, Leibniz came to Hanover as an industrious transplant from Leipzig. He spent forty years there, the better part of his life, in the service of the dukes of Hanover, engaged in such endeavors as establishing the court library, researching the genealogy of the family, and writing its history.

The house where Leibniz took up residence in 1698 and lived on and off for the last two decades of his life, was built in 1499. Like the town itself, the house was completely rebuilt post–World War II, after it was utterly destroyed in a bombing raid in 1943. Following the war, there were discussions about what to do about the town in general and Leibnizhaus in particular. By the time the decision was made to rebuild the house, a shopping mall and a parking garage had already been built on top of the original site. So instead, construction was undertaken at an alternative site, the present-day Leibnizhaus. The new building had another problem in that the newly chosen site butted up against another building such that, if they had built an exact replica of the house, it would have overlapped its neighbor. Finally, the decision was made to construct a modern building covered with a genuinely old facade.

The new Leibnizhaus opened in the 1980s as part of the University of Hanover, and has a guesthouse, a conference center, and a small museum on the ground floor. At the museum are some original pieces, as well as a painting and a bust of Leibniz, and a casting of his skull.

The building’s incredibly ornamental baroque facade dates from 1651, just a few years before Leibniz moved to Hanover. The right side of the facade stands out from the rest of the face of the building, and forms a sort of three-story bay window, accentuated on each level by four fancy columns and decorated up and down with lots of angelic figures. The top of the building has several stepped, orthogonal levels and a series of small windows. The front rooms of Leibnizhaus look out onto a sort of small square. There is a fancy wrought iron monument in front of the building, which is situated on a street crowded with high-end secondhand shops, clothing stores, small restaurants, and a coffee shop or two. The building is flanked by a Spanish restaurant and an antique shop.

All the windows in the front of the building are divided into smaller, crosshatched squares. From the outside of the building, they are barely noticeable, as the building itself is so remarkable in other ways. But from the inside, the square panes dominate the view, which is framed terrifically by the tall windows. The rectangular frames appear dark against the buildings that are visible outside, across the street, echoing the shapes of the building that are themselves very rectangular and Bavarian, with exposed timbers and lots of windows.

Though the facade maintains period accuracy, the inside of Leibnizhaus is very different from the interiors of the mathematician’s time. And that is not all that has changed.

Hanover is a college town, and outside is every sign that the university is nearby. The university was, when I visited, the largest in lower Saxony, with more than 24,000 students—not a bad place to visit or to live. But the town was much smaller when Leibniz resided there.

His move into the court of Hanover was not an uncommon path for a man in his position to take. In his day, many people who were smart and social climbers would seek patronage in the courts of Europe. The best way to do this, of course, was to provide ways for the princes and dukes to increase their revenue. Wars, famines, and the lavish courtly lifestyles of the times all took their toll on the noble pocketbook, and a creative thinker who could come up with new schemes for making money was very valuable indeed. Still, probably few were as creative at schemes as Leibniz was.

The irony of Leibniz’s life is that, while he might seem so very academic today, he chose not to follow an academic path. Once he finally, reluctantly, arrived in Hanover, he stayed there most of his life, also serving in various capacities in the nearby courts of Celle, Wofenbüttel, Berlin, and Vienna—a career Bertrand Russell once called a lamentable waste of time. But to Leibniz, this pathway really made sense. Despite the fact that he was criticized in the eighteenth century for believing that this was the best of all possible worlds, he spent a considerable amount of time in the seventeenth century hoping to improve it.

Knowing the reality of his day was that power was concentrated in the hands of the few, Leibniz also held the neo-utopian beliefs that those holding this power should be wise and pious men, benevolent leaders who would be best suited to raise mankind to its greatest potential. While it would have been too much to expect that all nobles and hereditary rulers could themselves be wise men, he thought that any change to society should happen within the context of existing political power structures, and wanted to work within these structures. He desired to enlighten the princes, dukes, and other rulers of his day so that they could make the right choices. He was attracted to the job at Hanover because the duke appeared to Leibniz to be wise as well as powerful.

Johann Friedrich’s grandfather had been the great Duke William of Lüneberg, also known as William the Pious, who governed with religious discipline and left fifteen children to sort out the spoils of his kingdom among themselves. William the Pious went mad and blind and, on his deathbed, his children drew lots to decide the fate of the ducal lands. William’s sixth son, George, won. But George was not fit for the sedate and pious country life his father had established, and he went on a grand tour of Europe, indulging his every whim. Just after he tired of this and returned home, the Thirty Years’ War broke out, and George fought with the Holy Roman Empire in Lower Saxony and Italy. When he was finished, he took an abbey at Heldesheim as his personal booty. There he rested, and there he died. His oldest son, Christian Louis, succeeded him and became the new duke, but Christian Louis died childless a few years later. The territory was divided among the three remaining brothers, one of whom was Johann Friedrich. Thus Johann Friedrich became duke of the territory that included Hanover.

At Hanover, Leibniz took command of a library that contained 3,310 books and dozens of manuscripts. However, he was not satisfied with it and proposed a plan to the duke to expand its holdings. Having just come from one of the most learned centers of Europe, Leibniz was in a good position to claim the breadth of knowledge to be able to do so. In the years to come, he would add thousands and thousands of works to the collection.

For instance, he went to Hamburg in 1678 to look over the library of Martin Fogel. The availability of the Fogel collection was a tremendous opportunity for a book lover like Leibniz and a library builder like the duke, since it had 3,600 rare tomes on natural science and other subjects, so Leibniz convinced the duke to buy it. While he was there, he met Heinrich Brand, who discovered a way to manufacture phosphorus by accident, apparently, when he was following the instructions in an ancient alchemy book for extracting a chemical from urine that could turn silver into gold.

Leibniz convinced the duke to pay Brand to come to Hanover and set up a laboratory to manufacture phosphorus. The key starting ingredient in this process was urine. To produce a substantial amount of phosphorus, Brand needed a grand supply of his starting material. So he had barrels brought into the camps of the region’s soldiers, and these fighting men of war supplied him with his precious liquid, which was then shipped to Brand’s laboratory. I get this picture when I think about it: German soldiers from a mostly forgotten time standing around, foul mouthed, cackling and filling up the barrel. Liquid gold.

More books was not all Leibniz requested. Within a few months of arriving, he asked for and was given the honor of promotion to a higher rank of counselor, with an increase in salary. In the beginning, Leibniz was happy enough with his new life to write to some of his acquaintances abroad that he was pleased to be working for the duke who, in addition to being smart and discerning, was wise enough to allow him the freedom to pursue his own endeavors throughout the ticking hours of the day; he gave Leibniz ample time to devote to his intellectual pursuits. Leibniz even wrote one man in Leipzig, a Martin Geier, that he would rather work for Duke Johann Friedrich than enjoy every kind of freedom.

Meanwhile, the duke appears to have been impressed by the words of the philosopher Antoine Arnaud, a renowned scholar his new privy counselor knew in Paris, who paid Leibniz the great compliment of saying that the only thing that was possibly holding Leibniz back was his Protestantism.

Still, Hanover was certainly not the throbbing heart of the scientific revolution. Even though it was a large city by German standards, its population was only around 10,000—as opposed to cities like Madrid or Amsterdam, which had well over 100,000, or London, which had somewhere close to half a million. And, despite the fact that the court at Hanover is described as one of the most elegant and cultivated in all of seventeenth-century Germany, Leibniz was no longer in Paris. In Hanover, there was no scientific society comparable to those in London or Paris, and no community of intellectual peers—except perhaps the duke.

Their tastes happily coincided, and Johann Friedrich is said to have often joined Leibniz in his physical and chemical studies. Leibniz was teeming with ideas, and in the duke he found a patron who seemed willing to sponsor his ideas and had sufficient intelligence himself to grasp the vision. Together, the two could have been the dream team of intelligent governance.

Leibniz’s grand vision was to bring about improvements to a universal Christian society through the application of science and technology. He wrote three memoranda to the duke in 1678, proposing ways to improve everything from agriculture to public administration. He called for an economic survey to gauge the state of the state in terms of the number of workers and the amount of natural resources that would serve as the raw data for an analysis for improving economic output; the establishment of an academy to teach young people commerce; and the creation of something resembling the modern department store, where common goods could be purchased cheaply in one central place. He recommended that the state archives be organized under one director—himself, of course—so that information could be more easily accessed. He called for the creation of a bureau of information that would produce a magazine and provide a valuable eBay-like source for people looking to acquire rare goods and services. And he recommended incentives for farmers who followed good farming practices.

The above proposals were followed shortly by one for writing a book to be called Demonstrationes Catholicae, which would justify the reconciliation of Catholics and Protestants. At that time, Christianity was fragmented, after more than one hundred uncomfortable years of Protestant reformation that had started with Martin Luther’s questioning of papal authority in 1517 and continued when the French preacher John Calvin moved to Geneva in 1536. By the mid-seventeenth century, the influence of Luther and Calvin had spread rapidly throughout Europe, opening up pockets in England, throughout Scotland, in France, the Netherlands, large parts of the Holy Roman Empire, a few parts of Poland and other Eastern lands, and even large settlements in the New World.

Leibniz was not, by any means, the only figure in those days to see the value of reunifying the Christian churches, nor was he filled with unreasonable expectations as to its prospect for success. Nevertheless, he proposed finding some common ground and agreement between the theological systems, mainstream elements of both traditions, and engaged an extensive correspondence with various Catholics and Protestants in this regard.

In the last quarter of the seventeenth century, Leibniz was a chief negotiator to reunify the Lutherans and Roman Catholics. The main obstacle to reunification was that it required the reconciliation of beliefs and practices no longer compatible with one another. These were not necessarily obscure matters of theological philosophy but contentions so basic as to seem absurd. The Catholics, for instance, had to accept that the Protestants should no longer officially be regarded as sinners, and the Protestants had to agree to no longer call the pope the antichrist. (One wonders whether “Your Holiness, the antichrist” would have sufficed.) Not surprisingly, Leibniz found the positions of some of the religious authorities unyielding, and these negotiations, which began in Hanover in 1683, ultimately failed.

Long before his grand unification plans petered out, Leibniz suffered a personal and professional tragedy when Johann Friedrich died in 1679. Leibniz was struck with such sadness that he wrote three different eulogies dedicated to the memory of greatness of his friend and boss—including one in Latin and one in French verse.

Leibniz was confirmed in his position as counselor by the new Duke of Hanover, Ernst August, Johann Friedrich’s brother, and immediately began pressing his innovations upon his new employer. He had to tailor these proposals carefully. The new duke was not the philosopher his brother had been. Ernst August was a warrior who was recognized for his bravery. The library languished under the new administration. Ernst August spent a fraction of the amount his brother had on new acquisitions, and most of the money he did spend went toward paying bills left over from purchases predating his accession. Less pious and more rowdy than his late sibling, Ernst August is said to have loved the bottle, his stomach, and women—not necessarily in that order. He was given to long drinking bouts and outlandishness, and, in his youth, he had indulged in all manner of vices in Italy and France.

Ernst August’s primary concern was to enhance the power of his position and enrich his already extravagant lifestyle. Money was the fuel that could drive this desire, and Leibniz, recognizing this, responded in the only appropriate fashion—by sending the duke proposals that would increase the revenue stream of the court. Thus, money was the motivation for an ambitious project to drain water from the silver mines in the nearby Harz Mountains.

These mountains had been mined for centuries, and the sites were deep and prone to filling with seeping water. Draining them was a necessary step for continuous mining operations as, during the dry months of the year, rivers and streams dried up and pumps that operated on water power couldn’t be powered effectively, severely curtailing production in these dry months. A Dutch mining engineer, Peter Hartzingk, had come up with the idea of draining the mines using a combination of water and wind to keep the pumps operating continuously. In his ingenious design, wind power would be used to raise the water to an underground reservoir that could be opened up and emptied into a lower underground reservoir when the wind was not strong enough to operate the pumps.

Leibniz scoffed at this idea and claimed that he could switch the entire operation of the pumps to wind power alone, and he set about designing and implementing improved and more efficient windmills. If he could employ the wind to pump the water out in a steady outward flow, then the mines could be worked even in the winter months, and the silver could continue unabated to the royal coffers in a steady inward flow.

The increased profit, he suggested, could be also used to fund another idea that he had—the granddaddy of all proposals. He wanted to form an imperial scientific academy so impressive that it would surpass even the Académie des Sciences in France and the Royal Society in London. The academy, which was to be made up of forty-nine other scholars and himself, would then become the greatest in the world. Together the scholars would construct an encyclopedia of all human knowledge, wherein concepts would be collected, analyzed, and reduced to their component pieces, and the ways in which they were combined noted, and finally these same pieces and combinations used to build more concepts. Just as words are made up of letters strung together in a written language—or of a string of sounds in a spoken one—so, too, could ideas be thought of as having been formed by letters of the universal characteristic, or so thought Leibniz. The letters he envisioned were something like the unbreakable atoms of the molecule, the pure ingredients of a sauce, the indivisible organs of the body.

Moreover, the letters were only the beginning. Just as a language has a grammar to the way words are gathered together into sentences, so the ideas constructed with the universal characters obey a grammar. Leibniz and his helpers had only to discover these ideal grammatical rules, and they would be able to resolve all questions, from the greatest to the least, by properly resolving the question into the appropriate symbolic characters and then combining the characters into the logical form their internal grammar dictated. It was to be an analysis of human thought worthy of being thought of as a tribute to human analysis.

The universal language was a bold and beautiful idea, but it would not be an easy feat. Nor would it be without great expense, as Leibniz believed that the learned men of the academy, who no doubt would have been from scattered lands throughout Germany, should be freed of financial concerns by supplying them with stipends and the tools and facilities to conduct their research. That kind of funding would be hard to raise since Hanover, like all the German courts, did not have the advantage of large centralized states with extensive tax bases like France. Despite how much the court at Hanover longed for the greatness of the palace at Versailles, how could they possibly compete? The solution, according to Leibniz, was to increase the production of the nearby mines and pour the windfall into his project.

But first, he needed to drain the mines. His memoranda to the duke were vague at first, merely mentioning that he could increase production without mentioning how, but eventually he disclosed that he would design new pumps to eliminate friction and make the conversion of power more efficient using compressed air. He promised to build new and improved windmills that would work better in a slight wind than the existing ones would in strong gale, by implementing folding sails on the windmills that would open and close to adjust to the adjusting strength of the wind. He also came up with a scheme for a horizontal windmill—something that looks like a waterwheel turned on its side.

When he first proposed these ideas shortly before the old duke died, Johann Friedrich had not been an enthusiastic supporter of them, but he was an enthusiastic supporter of Leibniz, so he had agreed to the Harz project in October 1679, and even had a contract drawn up. When Johann Friedrich died, the project had enough momentum that it continued under the new duke, who was all too happy to back the venture financially—at least at first. Even so, he made Leibniz assume some of the costs of building the windmill.

Leibniz was to continually face bad cost overruns and unanticipated expenses. His original estimate of 330 taler had ballooned by the middle of 1683 to a cost of 2,270 taler. And from its inception the project was plagued by infighting. The mining office opposed Leibniz every step of the way. Probably because of their opposition, he began to suspect that his efforts were being sabotaged. He complained to Ernst August that the officials were putting up roadblocks at every juncture and poisoning the workers against him by using lies and threats. The mining office, for their part, poured an equal amount of scorn on Leibniz in their reports to the duke.

Ernst August grew tired of the project after the costs had ballooned and the project had failed to produce results by 1683, and, at the end of that year, cut off his funding for the project. Thereafter, Leibniz had to continue on his own dime. Leibniz did a series of tests in 1683, 1684, and again in 1685 with only partial success. Machines constantly broke down and caused extensive delays, requiring costly repairs. The fickle wind blew and ceased and made even testing the system an ordeal. By the middle of 1684, the weekly report of the mining office was filled with nothing but complaints about the project, and Leibniz faced what he perceived to be a worker’s revolt. He blamed the failures on the workers and administrators at the mines, whom he suspected feared for their livelihoods and sabotaged the project and, with it, progress.

Finally, on April 14, 1685, the duke pulled the plug entirely and ordered Leibniz to end construction of his windmills immediately and forever.

Whatever the cause of the project’s failures—the heavy expenditures; the initial or eventual lack of support by everyone else concerned, not to mention the uncooperative weather—it also brought about some unanticipated successes. It inspired Leibniz to visit many mining operations on his extensive travels around Europe. He had thrown himself into the work, studying and composing a review of all aspects of mines—from their management to the chemistry of the processes to the geology of the lands. Whenever he went to a region, he tried to make time in his schedule to visit a mine, and he became an expert in mining operations. He even came up with a scheme for altering the bar composition. The silver from Hanover’s mines was superior, Leibniz asserted, and so it should be mixed with an appropriate amount of some other ore when it was melted and cast into bullion.

Moreover, in the course of his investigations, Leibniz became interested in the rocks and how they got to be there. It has been said that during his subsequent travels, he never missed an opportunity to study fossils and geological formations. Leibniz looked at the minerals for evidence of their origins, and his insights were at times astounding. When he found an enormous prehistoric tooth in 1692, for instance, he took it as proof not of some ancient monster, but rather as evidence suggesting that oceans once covered the earth. He also proposed the theory that the early earth was molten. In some ways, Leibniz was the father of geology, because he wrote one of the first physical descriptions of the earth, anticipating modern earth science.

Despite his expertise and enthusiasm, his windmill project was an abysmal failure—it failed in its primary goal of drawing water out, producing extra revenue, and enabling the funding of the forty-nine scholars. It was a bit of a financial bomb for Leibniz as well. He spent a small fortune on the Harz project.

MEANWHILE, NEWTON HAD crawled into a deep hole of his own. He was moving steadily away from science and mathematics and into theological and alchemical endeavors, which had consumed him for most of the late 1670s and early 1680s. As much as he was repelled by the controversies surrounding his optical experiments, he was drawn toward these other subjects, which he regarded as highly important and that would variously occupy and consume him for the rest of his life.

He spent a great deal of energy in his alchemical research—untold hours collecting and copying alchemical texts, and working on an extensive chemical index. Manually databasing hundreds of topics, each with references to more than one hundred alchemical texts, plus other commentary, this was an exercise in tediousness. Reading these texts today is nearly impossible. Some of the writings are bizarre, especially to a layperson—full of so many strange symbols and references to mythology that one might have thought Newton mad. In fact, these symbols were annotations to denote different elements or substances to be combined, such as lead, copper, or mercury.

Newton was equally drawn to matters of theology. He wrote interpretations of biblical revelations, and worked for years on such projects as elucidating the prophesies of Daniel and John. He was convinced, for instance, that the scriptures had become corrupted during the fourth and fifth centuries. He wrote a few treatises on the subject of the trinity, such as one that he wrote to a friend in 1690 in which he explained, “Since the discourses of some late writers have raised in you a curiosity of knowing the truth of that text of Scripture concerning the testimony of the three in heaven . . . I have here sent you an account of what the reading has been in all ages, and by what steps it has been changed, so far as I can hitherto determine by records. And I have done it the more freely because to you who understand the many abuses which they of the [Catholic] Church have put upon the world. . . .”

He was something of a historian, and set out to correct ancient chronology and to improve it by basing it on mathematical principles. Newton was driven to matching historical facts with biblical references and to elucidating the details of history in general. He concluded, for instance, that the date given for the fall of Troy (then determined to be 1184 BCE) was wrong. He dated it as 904 BCE. Newton is also said to have been perhaps the most knowledgeable authority ever on the barbarian invasions in the fifth and sixth centuries. He studied writings from a variety of traditions extensively in order to reproduce the plans of the temple in Jerusalem; concerned with determining its exact dimensions, he examined ancient texts in which the temple was described and translated the ancient measurements into modern lengths.

When he died, Newton’s chronology work was recognized to be some of his most important, so much so that an unauthorized version of this historical research was published in 1725 in France by Nicolas Fréret. The official edition of the chronology came out a few years later, in 1728, just after Newton died. It was a history of mankind from the time of Alexander the Great, including Greek, Assyrian, Egyptian, Babylonian, and Persian chronologies, which makes it sound deceptively interesting.

These alternative studies together help to round out the figure of Newton. Like many great historical figures, Newton is an enigma. Not because he kept his work private from his wife or worked secretly for some government’s war efforts. He never married, in fact, and his political world revolved around scientific intrigue more than it did around the wars and problems of his day. Newton was an enigma because he contributed so much to humanity through his science and yet spent so many years in endless contemplation of religious and alchemical pursuits. Even though these endeavors really fit naturally with the time he was alive, it seems strange that such a brilliant scientist would have wasted so much time on alchemy, theology, and his chronology of historical and biblical events!

During the 1680s, while Leibniz was seemingly consumed by his windmill project, Newton’s calculus work was gathering layer after layer of thick dust. But—Leibniz had not spent every moment in the mines. He was about to publish the first paper ever in the field of calculus and thus fire the first shot in the calculus wars.

MATHEMATICS, FOR LEIBNIZ, had the power of demonstration. In the early 1690s, Prince Gasto of Florence, whom Leibniz had met during his travels through Italy, had sent him a problem for constructing a certain geometrical shape that he needed to solve, and the German was able to come up with a solution in just a few hours. But Leibniz dreamed of a mathematics that reached much further than the subject we today think of as math (as a pure discipline on its own, or one that finds application mostly in scientific applications, the social sciences, and so on). Leibniz saw possibilities for mathematics in ways that can hardly be imagined.

He thought that it might be possible to create an aesthetic calculus that would allow artists to create great works of art the way that a person can solve an equation by plugging in numbers and calculating. He even thought the same general approach could be used for creating poetry and music, which he defined as “an arithmetic of the soul, which knows not that it reckons.” That he never went anywhere with any of these other calculi in no way detracts from what he did with calculus, introducing it to the world before anyone else.

The story of his publication started during the Harz mines project, when Leibniz played host to Otto Mencke, a professor he knew from Leipzig where he had grown up. Mencke had an idea to start a scholarly journal that would keep the intellectuals in Germany abreast of the latest discoveries in the German states and throughout Europe, and Leibniz was a big supporter of this idea. He became a co-founder of this journal with Mencke, and in 1682, the Acta Eruditorum Lipsienium, or “The Acts of the Scholars of Leipzig” or sometimes as “Transactions of the Learned” began publication as a monthly scholarly journal.

It was the first scientific journal in Germany, and Leibniz was closely associated with it, publishing in it all the way up until his death in 1716. This was an important thing for Leibniz, who had experienced some difficulties in publishing and had tried repeatedly over the course of three years, from 1677 to 1680, to have one of his mathematical treatises published in Paris or Amsterdam without success. But now he could publish freely in this new organ, and he often contributed papers to it—including many of the key documents in the calculus wars.

Even in the early 1680s—years in which Leibniz witnessed the unhappy unraveling of his mine shaft windmill idea—he was so prolific that he might publish an important paper in mathematics one month and a seminal paper on his philosophy the next. In October of 1684, right in the most troubling time of the mining project, he published a paper whose short title is Nova Methodus Pro Maximis et Minimis (New method for maxima and minima) in the Acta Eruditorum. This was the first calculus publication anywhere in the world, and in it, Leibniz gave the rules for differentiation.

In the cover letter for the paper to his friend Mencke, he wrote that his calculus “will be of the utmost use in the whole of mathematics.” One of Leibniz’s later admirers gushed over the publication, “[In] 1684 he proceeded to publish the results of his labors in the Acta Eruditorum; and thereby called forth the admiration of the whole scientific world at the richness and brilliancy of his discovery.”

In actuality, the paper was more complicated. It was modeled after a half-century old work by Descartes called Geometry, and stylistically it was difficult to read. Jacob Bernoulli called it an enigma rather than an explanation. Though it was a mere six pages long, its full title was worthy of a much longer piece: “A New Method for maxima and minima as well as Tangents, Which Is Neither Impeded by fractional nor irrational quantities, and a Remarkable Type of calculus for them,” as translated into English.

But it had treasures aplenty. In the paper, Leibniz performed feats of mathematics with ease, such as deriving Snell’s law of sines. “Other very learned men,” he wrote boldly, “have sought in many devious ways what someone versed in this calculus can accomplish in these lines as by magic.” He solved with ease a problem that Descartes was unable to solve in his lifetime. “And,” Leibniz continued in the paper, “this is only the beginning of much more sublime geometry, pertaining to even the most difficult and most beautiful problems of applied mathematics, which without our differential calculus or something similar no one could attack with any such ease.”

Significantly, Leibniz had no historical introduction to his paper. Had he had one, he might have mentioned the work that developed his methods and the communication he carried out with Newton nearly a decade earlier. In the paper, Leibniz made no reference at all to the correspondence, and nowhere does he give Newton credit in this or any subsequent publication, and this may have been a mistake. Had he acknowledged Newton in some way, Newton may not have come back at him years later. But he had no such language. Instead he just plunges into a terse explanation of his own methods without ever once mentioning Newton.

Though Leibniz did not mention Newton in the article, he did mention him in the cover letter he sent to his friend Mencke in July of 1684. “As far as Mr. Newton is concerned, I have his and the late Mr. Oldenburg’s letters, in which they do not dispute my quadrature with me, but concede it,” Leibniz wrote. “I do not believe, either, that Mr. Newton will claim it for himself, but only some inventions relating to infinite series which he has in part also applied to the circle.” These inventions, Leibniz tells his friend, were first discovered by Mercator, then developed by Newton, and then continued by Leibniz “by another way.”

In this cover letter, Leibniz anticipated the calculus wars and dismissed them at the same time, determining that he had come up with one method, and Newton another. “I acknowledge,” he wrote, “that Mr. Newton already had the principles from which he could well have derived the quadrature, but all the consequences are not come upon at once: one man makes one combination and another man another.”

Leibniz could not be blamed in a sense if he underestimated Newton, since for Leibniz, the second letter Newton sent in 1676 held little more than a “bare enunciation” of concepts, none of which were even new to him. Nevertheless he seems to have recognized that Newton was in possession of certain mathematical techniques parallel to his own calculus, even if he was never quite satisfied in his desire to find out what Newton’s method of fluxions was exactly. When the calculus wars were in full tilt, in what would be the most bitter irony for Leibniz, Newton would turn things around and claim that he was in fact so explicit in explaining his fluxions to Leibniz that it was essentially what allowed Leibniz to put together his calculus.

This would not be for years, of course. In 1684, when Leibniz published his calculus, Newton had more or less abandoned mathematics. But he was about to be pulled back into it in a big way. Meetings and exchanges were taking place that would lead Newton to publishing the book for which he is most famous, the “Mathematical Principles of Natural Philosophy,” or Principia.

The earliest of these exchanges took place in the late 1670s and were initiated by Hooke, who extended Newton an olive branch in a letter he wrote on November 24, 1679. “I hope therefore that you will please to continue your former favors to the Society by communicating what shall occur to you that is philosophical, and in return I shall be sure to acquaint you with what we shall Receive considerable from other parts or find out new here,” Hooke wrote to Newton.

In the same letter, he tried to make amends for their earlier troubles. “I am not ignorant that both heretofore and not long since also there have been some who have endeavored to misrepresent me to you and possibly they or others have not been wanting to do the like to me, but difference in opinion if such there be (especially in philosophical matters where interest hath little concern) me thinks should not be the occasion of Enmity—tis not with me I am sure,” Hooke wrote. “For my own part I shall take it as a great favor if you shall please to communicate by Letter your objections against any hypothesis or opinion of mine.” And then he added, “particularly if you will let me know your thoughts of that of compounding the celestial motions of the planets of a direct motion by the tangent & an attractive motion towards the central body.”

This last bit was really the reason why Hooke was interested in chatting Newton up. He knew that Newton was an outstanding mathematician and natural philosopher, and Hooke had become interested in a subject about which he suspected Newton had a great deal of insight—the gravitational nature of planetary motion. Hooke wrote again on January 17, 1680 reiterating his interest in the properties of the path a body would take under the influence of a central attractive power—essentially what path would something like a comet or the Earth follow in its course around the sun if it were attracted to the gravitational pull of the sun. “I doubt not but that by your excellent method you will easily find out what that Curve must be, and its properties, and suggest a physical reason of this proportion,” Hooke wrote. “If you have had any time to consider of this matter, a word or two of your thoughts of it will be very grateful to the society (where it has been debated).”

Hooke may have found it easier to suggest that the Royal Society at large was interested in Newton’s opinions than admitting that he was the primary one who was. As secretary of the Royal Society, he certainly had the authority to speak for the body as a whole. And the exchanges were, on the surface, very cordial, with Newton signing his letters, “Your very much obliged & Humble Servant Isaac Newton” and Hooke signing, “Your most affectionate humble Servant Robert Hooke.” But they never went anywhere—that is, until a few years later when Edmond Halley came into the picture.

Halley met Hooke and Christopher Wren in a coffee shop sometime in the spring of 1684. Coffeehouses flourished in London in the seventeenth century, and by the end of the century there were thousands. These provided a forum for meetings, and I imagine them to be like the best coffee shops today with strange men reeking of tobacco meeting each other and sitting leaning over broad thick tables stained black with coffee bean oil. Halley was curious about the comet that today carries his name, and he posed a simple question to these two other men: what sort of path would a celestial object like a comet take?

Hooke had a physical explanation, and it was the right one. Celestial objects, he said, would follow an inverse square law of attraction. Wren, perhaps unconvinced, asked Hooke to demonstrate how he knew it was an inverse square law, but Hooke demurred. Wren challenged Hooke to prove it, promising that he would reward him with a valuable book worth forty shillings if he could do it, but Hooke had no such mathematical proof, so this was not a bet that he could accept. Instead he declined. Meanwhile, Halley sat there unsatisfied. He thought he had the answer, but how could he be sure?

Wren told Halley that a certain mathematics professor he knew in Cambridge named Isaac Newton might be able to answer the question, and so several months later, in August, just as the printers in Germany were about to press Leibniz’s famous first calculus paper to ink, Halley took a trip to Cambridge.

The dusty, bumpy, meandering fifty-mile ride must have been hell for Halley inside a shaking deathtrap of a seventeenth-century horse-drawn carriage. Today, it’s a breeze. For a pocketful of small bills, anyone can buy a ticket for a train ride to Cambridge from London that leaves two to three times an hour, and takes about forty minutes non-stop. The scenery on the journey is probably in many ways still similar to how it was in Newton’s day. You pass through rolling fields and farmlands separated by old stone walls. Aluminum siding, diesel tractors, and the occasional satellite dish are the only objects that give the landscape a more modern appearance.

When Halley got to Cambridge and walked through the grand gate at Trinity College, he sought out Newton and asked him the same question he had posed to Wren and Hooke months before: What sort of path does a celestial body follow? Newton answered immediately: an ellipse. The orbit of the planets around the sun follow an inverse square law, and the path is elliptical. It was a simple answer that would change both men’s lives forever.

Halley was “struck with joy and amazement” at hearing Newton’s words. This was the sweetest music to Halley’s ears to hear Newton say what Hooke had already told him. But could Newton prove it? Halley asked him how he knew. Newton replied that he knew because he had done the math. He had calculated it.

At this, Halley immediately asked to see the publications. Newton had worked many of these things out years before and was not exactly sure where to find the calculations—certainly not while Halley was waiting anxiously in his rooms. So he bid Halley return to London and promised to send the calculations afterward. This was a promise that Newton kept, sending two proofs after Halley left. He also wrote a short book, De Motu Corporum (On the Movements of Bodies), which he sent to Halley as well. Halley, recognizing its importance, began cajoling Newton to write more.

Newton did. He started in 1685 and sent the first part to the Royal Society in time to be recorded in the April 28 minutes of the Royal Society. While some may think that Halley’s greatest contribution was predicting the return of the comet he ultimately gave his name to, one could argue that in fact his greatest accomplishment was to convince Newton to publish one of the greatest books ever written—the Principia.

In fact, Halley did not only cajole Newton into writing the Principia, he also oversaw the production of the book and personally underwrote the expense of publishing it in 1687, since the Royal Society could not scrape together the funds to do so. “I have at length brought your Book to an end,” Halley wrote to Newton on July 5, 1687, “and hope it will please you.” Halley gave Newton twenty-seven copies and provided forty for booksellers in Cambridge—which could be had, back then, for a few shillings.

And Halley wrote proudly to King James II, in July of 1687, “And I may be bold to say, that if ever Book was so worthy of the favorable acceptance of a Prince, this, wherein so many and so great discoveries concerning the constitution of the visible world are made out, and put past dispute, must needs be grateful to your majesty; being especially the labors of a worthy subject of your own, and a member of that Royal Society founded by your late royal brother for the advancement of natural knowledge, and which now flourishes under your majesty’s most gracious protection.”

While Newton was first starting to work on the Principia, Leibniz’s work was spreading in Europe and had reached across the English Channel. A Scotsman, John Craig, who lived in Cambridge and was a friend of Newton’s, published the first publication on calculus to appear in England in 1685, a year after Leibniz had published his paper. Craig wrote a book, The Method of Determining the Quadratures of Figures, which described Leibniz’s work on differentials and used Leibniz’s notation. This effectively introduced England to calculus—or at least most of England, since Newton had been sitting on his own methods for two decades.

Craig was a mathematical enthusiast and something of a forgotten player in the invention of calculus. He published more on the subject than perhaps anyone else alive in his time. In addition to his 1685 book, he wrote another in 1693, and he also contributed articles on calculus to the Philosophical Transactions of the Royal Society in 1701, 1703, 1704, and 1708. Perhaps because he was so heavily indebted to both Newton and Leibniz, his is not a name readily associated with calculus today.

Nor is he a central fighter in the calculus wars—probably because he was willing to seek out and acknowledge his sources of inspiration. Craig had spoken with Newton prior to publishing and had gotten the binomial theorem from him prior to the 1685 book. In his 1693 book, Craig wrote what can be regarded as the model of elegant acknowledgment of Leibniz. “In order not to seem to assign too much to myself or detract from others, Craig wrote, “I freely acknowledge that the differential calculus of Leibniz has given me so much assistance in discovering these things that without it I could hardly have pursued the subject with the facility I desired.”

Leibniz, aware of Craig’s 1685 book and of the efforts of mathematicians and natural philosophers elsewhere in Europe, was inspired to send his second paper on calculus in 1686 to the Acta with the title “On Recondite Geometry and the Analysis of Indivisibles and Infinities.” It was on what he thought of as the inverse of differentiation—integration. He began the paper by boasting that the methods he presented in his previous paper “won no slight approval from certain learned men and are gradually indeed being introduced into general use.” And in this second, longer paper, Leibniz promised to illuminate calculus further.

“Like powers and roots in ordinary calculations, so here sum and differences . . . are each other’s converse,” Leibniz wrote. Both his 1684 and his 1686 papers are noteworthy as the first published descriptions of calculus and for their introduction of notation for differentiation and integration (the twin tools of calculus-based analysis) that are still in use today—though the word “integral” and integral calculus, now commonly used, was not actually mentioned in the 1686 paper. In fact, Leibniz had never intended to call his “recondite geometry” integral calculus. The term integral was first used in a paper by one of the Bernoulli brothers in 1690 and “integral calculus” first appeared as a term in a paper written by Johann Bernoulli with Leibniz in 1698.

The year 1686 was one in which things really crystallized for Leibniz. He published his famous “Discourse on Metaphysics” that year, his first actual systematic description of his philosophy, and this enabled him to begin a correspondence with Antoine Arnaud—something that he had tried to do nearly twenty years before. Leibniz sent Arnald the title headings from his “Discourse on Metaphysics” as a way of opening up the conversation, and what transpired is one of the most famous discourses in the history of philosophy, the Arnaud–Leibniz correspondence, which is still in print today.

In a way it is strange to think about how this philosophical work inspired a lengthy and interesting discourse because the same thing could have happened with Leibniz’s mathematical papers. They might have had the same effect this philosophical paper did and have been the cause for a discourse on mathematics to begin between him and Newton. But they had no such effect.

Newton was busy writing the Principia, a mammoth and all-consuming project. In fact, it’s fair to say that in these years, Newton had a second, midlife anni mirabiles in which he wrote the work in a mere eighteen months.

On May 22, 1686, Halley wrote proudly to Newton, “Your Incomparable treatise . . . was by Dr. Vincent presented to the R. Society on the 28^th past, and they were so very sensible of the great honor you do them by your dedication, that they immediately ordered you their most hearty thanks, and that a council should be summoned to consider about the printing thereof.”

Around the same time, Halley was the bearer of bad news. When Hooke found out about the Principia, he was furious. He had sent Newton a letter some six years before, and he was not about to sit quietly with his suspicion that Newton was stealing his thunder yet again.

“There is one thing more that I ought to inform you of,” Halley wrote to Newton, “that Hooke has some pretensions upon the invention of the rule of the decrease of gravity, being reciprocally as the squares of the distances from the center. He says you had the notion from him, though he owns the demonstration of the curves generated thereby to be wholly your own.”

Hooke wanted Newton to give him his due credit, and Halley wrote to Newton and politely suggested that he do it. “Mr. Hooke seems to expect you should make some mention of him, in the preface,” wrote Halley.

Newton bristled at the notion. After Halley wrote to Newton sending him the first proof of the Principia, Newton wrote back on June 20, 1686 asking that his intelligence not be insulted. “I hope I shall not be urged to declare in print that I understood not the obvious mathematical conditions of my own Hypothesis. But grant I received it afterwards from Mr. Hooke,” Newton wrote. After pages of defense in the dispute with Hooke, Newton finally addresses Halley’s letter by saying, “The Proof you sent me I like very well.”

Then he adds a note consisting of several more pages: “Since my writing of this letter I am told by one who had it from another lately present at one of your meetings, how that Mr. Hooke should there make a great stir pretending I had all from him & desiring they would see that he had justice done him. This carriage towards me is very strange & undeserved.” Newton was so infuriated that he threatened to kill the third part of the Principia altogether. Eventually, he did calm down and capitulated to Halley’s suggestion that he mention Hooke, but only in the context of Christopher Wren and Halley.

In the Principia, Newton also mentions his earlier exchange of letters with Leibniz. “When, in letters exchanged between myself and that most skilled geometer G. W. Leibniz ten years ago, I indicated that I possessed a method for determining maxima and minima, of drawing tangents and performing similar operations . . . that famous person replied that he too had come across a method of this kind, and imparted his method to me, which hardly differed from mine, except in words and notation.”

These words would become bandied about by both sides in the calculus wars years later, but in 1687, they went almost without notice. That year, just as a decade before in the 1670s, was a lost moment. Just when Newton may have otherwise taken a hard look at what Leibniz was printing and what people were saying about calculus, he was distracted by more troubles with Hooke. Instead of starting a conversation that could have resulted in acknowledging their co-invention of calculus, they became aware of the publications of the other and began to form, on opposite sides of the English Channel, a quiet competition—quiet for now at least.