The Beautiful and the Damned - 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 7. The Beautiful and the Damned

1687–1691

The benefits which, in the course of almost half a century, would have accrued to science from the harmonious connection, thus unceremoniously dissolved, of these two great philosophers, can hardly be too highly estimated.

—John Milton Mackie, from Godfrey William von Leibniz, 1845

Samuel Pepys lived a charmed life. He was not one of the greatest men of his day and yet his name still rings out today because he was a witness, through his diaries, to one of the most interesting times in the history of England. And the short period of his life when he was the head of the Royal Society was no exception. It was during his brief presidency, slightly more than a year—one of the shortest tenures anyone ever spent in that position, in the 350-year history of the Royal Society—that Newton finished the Principia. And because Pepys oversaw the delivery of it, the “Julii 5, 1686” imprimatur on the title page carries the name S. Pepys Reg. Soc. Praeses. Along with some information of the printers and, finally, the date of publication MDCLXXXVII (1687).

Many commented swiftly on the importance of the book. David Gregory wrote to Newton on September 2, 1687, that “having seen and read your book I think my self obliged to give you my most hearty thanks for having been at the pains to teach the world that which I never expected any man should have known. For such is the mighty improvement made by you in the geometry, and so unexpectedly successful the application thereof to the physics that you justly deserve the admiration of the best Geometers and Naturalists, in this and all succeeding ages.”

What a book it was! Its more than five hundred pages of seventeenth-century scholarly Latin were filled with complicated diagrams, illustrations, tables of astronomical observations, geometrical drawings, and a parade of propositions, problems, corollaries, definitions, and scholia, the Philosophiae Naturalis Principia Mathematica (“Mathematical Principles of Natural Philosophy”), or simply Principia was a book that was destined to have a major impact on science. It would become one of the single greatest scientific books ever written, containing one of the greatest bodies of knowledge ever conceived: Newtonian or “classical” mechanics, an understanding, with mathematical descriptions, of the mechanics of motion, which is still the gateway subject to a physics degree today.

Anyone who has ever written a technical paper or an original work of science, no matter how broad in scope, can appreciate the sheer magnitude that was the Principia. In it, Newton put forward the laws of mechanics as applicable on the earth as in outer space. He laid out proofs of Kepler’s laws based on his original work of centers of mass and gravity. He also used gravity to explore the attraction between two massive objects. This allowed him to explain how Jupiter and its moons interact. Newton explained the flattening of the Earth at its poles and the Earth’s bulging at its equator. He described phenomena that became the basis for fluid mechanics, and considered such topics as resistance to motion, pendulum motion with and without resistance, and the motion of waves. He worked out the theory of the tides, explaining it in terms of gravity and the moon’s pull on the Earth.

Finally, looking to the rest of the solar system, he described the motions of the planets and explained the precession of the equinoxes. He wrestled with comets and showed that they are part of the solar system. He estimated the density of the Earth and calculated the masses of the planets, the sun, and the Earth, and he was actually close on the mass of the earth. He disposed of the vortex theory of planetary orbitals favored by Descartes and many others in the seventeenth century, and went on to reconcile the orbits of Jupiter’s and Saturn’s moons with his theory of universal gravitation.

Most significantly, he developed his theory of universal gravitation—that objects are attracted to one another by virtue of the force of gravity. This was by far the most radical idea presented in the Principia—the notion that there was some force by which every mass in the universe attracts every other mass. It has been called the most important scientific discovery of all time, and because Newton used the theory to conceive of the universe as an expanse of objects interacting with one another through this gravitational force, he has been called the father of modern astronomy—even though he was unlike other astronomers of his day, in the sense of their looking through a telescope and observing the motions of heavenly bodies.

Universal gravitation was truly a revolution in science, flying in the face of both conventional wisdom and logic. To the student of science, this discovery is an inspiring example of a brazen young scientist at his most successful, who through a combination of hard work and genius puts together something that can truly be considered a new paradigm. To the philosopher, universal gravitation is pregnant with profound consequences—every particle in the universe attracting every other particle. And to the historian, the gradual way that resistance gave way to acceptance and then finally to championing of Newton’s theory by others in his lifetime and after his death is a remarkable story.

Finally, to the historian of science, the Principia represents a turning point. During the seventeenth century, it wasn’t just that people began to change their views of the world. They were beginning to realize that they could also change how they arrived at those views in the first place—through empiricism, i.e., observation and data. But while the ability to comprehend the world through reason and to quantify it through observable measurement was not something unique to Newton, it came into full flower with the Principia because he kept to observations as a way to formulate the nature of something and describe it mathematically.

To avoid being embroiled in disputes the way that he was over his optics, Newton felt no need to justify gravity in the Principia. He later wrote to a man named Bently that he did not pretend to know the cause of gravity. In fact, Newton wrote in the Principia the famous phrase, Non fingo hypotheses, or “I do not invent hypotheses.” With that phrase in mind, he resisted guessing as to the nature of gravity and restricted himself, instead, to describing the behavior of gravity. He believed in the experimental over the hypothetical, even when the hypothetical made more sense.

It was not always so easy for his contemporaries to accept a theory that didn’t logically make sense. This was made obvious in an extremely favorable review of the book that appeared in a journal on the continent, which pointed out the good things about the Principia. “The work of Mr. Newton is the most perfect treatise on mathematics that can be imagined, it not being possible to provide a more precise or more exact demonstration than those he gives,” it read. Nevertheless, the review also put forward this criticism: “He was not considered the principles in question as a physicist but purely as a geometer.”

Others, most notably Leibniz, were even less enthusiastic about the idea of gravity and went on to reject the notion. Leibniz could never accept the basic premise of Newton’s worldview—that outer space is essentially a vacuum and that the Earth and the planets revolve around the sun by virtue of gravitational attraction. For Leibniz, theory based upon observation was not enough. He could not accept how gravity might act upon an object millions of miles away through apparently empty space.

Diagrams of optical phenomena from Opticks by Isaac Newton.

SOURCE: Library of Congress.

Tough problems that calculus solves with ease #1.

Isaac Newton.

Isaac Newton.

SOURCE: Royal Society.

Gottfried Wilhelm Leibniz.

Gottfried Wilhelm Leibniz.

SOURCE: Royal Society.

Christopher Wren’s plan...

Christopher Wren’s plan for rebuilding London after the fire of 1666 was impressive—but so was Isaac Newton’s plan for rebuilding the world based on universal gravitation.

SOURCE: Library of Congress.

Newton’s own drawing...

Newton’s own drawing of his reflecting telescope.

SOURCE: Royal Society.

A page from the Philosophical Transactions of the Royal Society showing the experiment that led Newton to conclude that white light is made up of rays of different colors.

SOURCE: Library of Congress.

Engraving of a fly...

Engraving of a fly, as seen through the microscope. From Hooke’s Micrographia.

SOURCE: Library of Congress.

Christian Huygens.

Christian Huygens.

SOURCE: Royal Society.

A model of Leibniz’s...

A model of Leibniz’s calculating machine.

SOURCE: Gottfried Wilhelm Leibniz Bibliothek, Niedersächsische Landesbibliothek.

Henry Oldenburg.

Henry Oldenburg.

SOURCE: Royal Society.

Tough problems that calculus solves with ease #2.

Leibnizhaus prior to its destruction in WWII, where Leibniz spent his final days.

SOURCE: Library of Congress.

Leibniz’s notes on his horizontal windmills.

SOURCE: Gottfried Wilhelm Leibniz Bibliothek, Niedersächsische Landesbibliothek.

Edmond Halley...

Edmond Halley.

SOURCE: Royal Society.

Nicholas Fatio de...

Nicholas Fatio de Duiller.

SOURCE: Library in Geneva.

John Wallis...

John Wallis.

SOURCE: Royal Society.

Brachistochrone problem

George Ludwig, who...

George Ludwig, who later became George I, King of England, governed Hanover during Leibniz’s final years.

SOURCE: Library of Congress.

When Newton took...

When Newton took over the British Mint, it was housed in this row of buildings in the Tower of London.

PHOTOGRAPHER: Jason S. Bardi.

Part of a letter in Leibniz’s own hand describing some of his calculus work.

SOURCE: Gottfried Wilhelm Leibniz Bibliothek, Niedersächsische Landesbibliothek.

A copy of the Charta Volans.

SOURCE: Gottfried Wilhelm Leibniz Bibliothek, Niedersächsische Landesbibliothek.

The front of Westminster...

The front of Westminster Abbey, where Newton was interred with great fanfare on March 28, 1726.

PHOTOGRAPHER: Jason S. Bardi.

The final resting...

The final resting place of Leibniz’s remains is at this church in Hanover, Germany.

PHOTOGRAPHER: Jason S. Bardi.

Love it or hate it, Newton’s work spread throughout the seventeenth-century intellectual world in a completely modern fashion. Reviews appeared in the literature summarizing, praising, and sometimes critiquing it, which was how Leibniz first came to find out about the book, in fact. He read a long review of it in the June 1688 issue of Acta Eruditorum, which praised Newton as “a distinguished mathematician of our time.” The review was twelve pages of dry summary.

In a letter he wrote to a friend, Otto Mencke, in 1688, Leibniz said that he had been traveling and had not gotten his publications of late. But, he wrote, he had received a letter from a friend with a review of Newton’s Principia. “I came across an account of the celebrated Isaac Newton’s Mathematical Principles of Nature,” he said. “That remarkable man is one of the few who have advanced the frontiers of the sciences.”

High praise notwithstanding, Leibniz was reluctant to acknowledge the merits of the theory of universal gravitation as presented in the Principia. His own general view was that, while such phenomena as planetary motion could be explained mechanically and mathematically, the laws that governed them must arise from higher reasons. These higher reasons, he believed, were intelligible and logical, and, for him, the vortex theory that Newton was overthrowing made more sense.

Leibniz did do some interesting work in dynamics. He had postulated what is essentially the conservation of potential and kinetic energy—that, for instance, a ball held from a few feet above the ground and dropped will strike the ground with a kinetic energy equal to the potential energy it had by virtue of its position a few feet above the ground. Because he had done a great deal of thinking about some of the same sorts of problems that Newton had tackled, Leibniz was inspired to write three papers of his own on physical subjects, a few years after the Principia appeared.

One of these was his defense of the vortex theory, in his “Essay concerning the causes of the motions of the Heavenly Bodies,” published in the Acta Eruditorum. In this work, Leibniz describes planetary motion in terms of harmonic vortices in which the sun is at the center of the world system, and the planets are carried around in the vortex. Because the planets were all revolving in the same plane around the sun, he couldn’t think of a logical reason why they would do this if not for the existence of something like a vortex medium in which planets were spinning while they were carried around the sun.

Another of the papers he wrote after hearing about Newton’s book was on a physical problem involving the resistance of a medium to motion. He used the publication concerning the problem of resistance of the medium to promote the ease with which such problems could be solved through his calculus. Leibniz had begun to feel enthusiastic about the possible applications of calculus, especially after a 1690 paper by Jacob Bernoulli appeared. Bernoulli’s was an important document because it was written more accessibly than Leibniz’s own papers, and it was the first in a long series that applied calculus to solving problems in mathematics.

One thing was sure, for Leibniz. He would be paying attention to the goings-on in England more closely for the rest of his life. In the fall of 1690, he sent a letter to a German ambassador in London, asking him to send news of discoveries and publications there. He had not gotten the Philosophical Transactions since 1678.

LIKE MOST OF the rest of Europe, Leibniz also was consumed with the dramatic events unfolding along France’s eastern border. Europe was in turmoil in the late 1680s just when he was publishing his calculus papers and Newton was preparing to publish the Principia. At the end of the Franco-Dutch war in 1678, which Leibniz had crafted his Egyptian plan to avert, Holland was left free of French dominion but Louis XIV retained the Lorraine region. The king kept the latter militarily occupied, so he was poised in position to invade Holland or Germany again, should he choose to do so. And Louis was not the type to leave his troops idle forever—something that prompted Leibniz in 1683 to write a political satire, Mars Christianissimus (Most Christian War God), in which he called the French king the most powerful person in the world aside from the devil.

But for many in France, the actions of the most Christian war god were anything but a laughing matter. In the years leading up to 1685, a series of laws were passed in France that shut Protestants out of certain careers and encouraged the children of Protestants to declare their allegiance to Catholicism and be brought up as wards of the king. In addition, for the previous two decades, a number of French Protestants—Huguenots—had accepted an official government offer to convert to Catholicism and be exempt from taxes. Many more publicly converted to Catholicism because Louis XIV had exerted military force to influence their decision.

Soon thereafter, things went from bad to worse for Protestants in France. On October 18, 1685, Louis signed an edict that basically suspended all civil rights for the Huguenots, and the ripples that emanated from this chilling decree were profound. The edict ordered the demolition of Protestant chapels, called for an end to Protestant practices, closed Protestant schools, forced the baptism of Protestant children and authorized them to become the wards of the local judges, and allowed for the exile of pastors though not their flocks. This led to an exodus of as many as 200,000 refugees who fled France to seek asylum in Protestant countries. French Huguenots immigrated to England by the tens of thousands.

Meanwhile, a parallel political situation in England was throwing that country in turmoil. After several years of exile in France, Charles II had returned and become king of England on May 8, 1660. He rode into London, wearing his courtly finery, on May 29, 1660—his thirtieth birthday. The locals lined the streets and cheered. He had left England in defeat and now he returned in triumph, proving that if you cannot count on your own abilities, you may very well be able to count on the ineptitude of others—in this case, those who had followed Oliver Cromwell.

Voltaire described Charles II as having “a French mistress, French manners, and above all French money.” Charles II was nicknamed the Merry Monarch because of his wit, charm, and love of good cheer. But he had a funny way of showing it at times. One of his first acts was to order the execution of ten people who had been involved in the trial of his father Charles I more than a decade before. Also, Oliver Cromwell was treated to the ironic insult of a posthumous execution. His cadaver was exhumed, and he was hung, drawn, and quartered, and dragged through the streets of London. Cromwell’s head was placed on a pole in front of Westminster Abbey for the next fifteen years, until Charles II himself died.

Despite such demonstrations of pique, over the long run Charles II proved himself an amazing politician. He accepted kickbacks from Louis XIV because that king hated the Whigs (Puritans) so much that he was glad to pay Charles a salary while the latter ruled as king of England. And Charles very effectively kept the whigs in check throughout his reign. He even was able to dissolve parliament and have many of his Whig opponents arrested, without a civil war breaking out.

But when Charles II’s son James—a Catholic—came to power in 1685, the stability vanished. For years, many Puritans in England had tried to get Charles II to disinherit his son—to bar him from ascension—on the basis of his faith, but the king never did. And when he came to power, James II was confident in living out the life he believed he was born to do—to rule his country. Upon becoming king, he met with his council and declared, “I have often heretofore ventured my life in defense of this nation; and I shall go as far as any man in preserving it in all its just rights and liberties.”

In three years’ time, James would flee England without fighting to keep his throne, and he may have become feebleminded in his later years because of syphilis. Whether or not he suffered from this communicable disease, one thing is certain regarding James: He was a disaster of a king. He had judges declare him the right to suspend laws at will, which he did—especially laws that curbed the power of Catholics. He also raised a substantial number of Catholic soldiers from Ireland and stationed many of them near London, which was an infuriating if not frightening deed for the largely non-Catholic capital.

In the midst of all this, the Principia appeared, and in the front matter of the book was a dedication to James II as king. A year after that, James II was forced to flee England for good. This wasn’t exactly James’s fault. All his military and civil leaders abandoned him. Even his escape was ill fated. He was captured on his way to France—not by the English navy, nor the English army, nor the forces of the Dutch, but by salty, stinking fisherman. He escaped again, his second escape apparently helped by the fact that he was allowed to escape.

Nevertheless James had nobody to blame but himself, ultimately, since he alienated his allies as effectively as he did his enemies; he managed to unite the Tories and the Whigs against him. This inspired a number of Whigs to send a letter to Prince William of Orange in the Netherlands, in 1688, inviting him to become king.

By 1688, Louis XIV had been creeping toward war for several years. Ironically, his justification for his declaration of war was that the Ottoman Empire, he claimed, was planning to attack France and that Europe’s eastern borders were not secure. So instead of attacking the Ottoman Empire to avert war in Europe, as Leibniz and Boineburg had proposed sixteen years before, Louis XIV attacked Europe to avert war with the Ottomans. The French king was very much opposed to William’s taking over the throne of England because the two were adversaries in more ways than one.

William was an active leader in European resistance to Louis XIV, and, in 1686, he urged the reorganization of the Grand Alliance he had created in 1672 of the Dutch, the Holy Roman Empire, Spain, and Brandenburg. He created the League of Augsburg, whose members were the Holy Roman Empire, Spain, Holland, Sweden, Saxony, Bavaria, Savoy, and eventually England. The League of Augsburg was formed as an alliance against France after French troops invaded a German state and declared war against the Holy Roman Empire.

Louis threatened to declare war on England if William went there. But William, calculating that Louis was too busy invading Palatinate in Germany to make good on his threat, sailed to England with 15,000 men, landed in November 1688, and took the throne a few months later. As the third William to rule England, he became William III. The first was perhaps the most famous—William the conqueror, the first Norman king who had ruled the realm centuries before.

Unlike his renowned namesake, William III did not arrive the victor of a hard-fought conquest. He led an army that landed in England and deposed the king without a shot being fired (James II fleeing to France rather than facing war). Nevertheless, William looked the part of the gallant conqueror. A portrait of him by Sir Peter Lely, which resides today in the National Portrait Gallery in London, shows him in a suit of shiny black armor. A portrait of his wife, Mary, hangs nearby. She has wild brown hair and a gorgeous orange and crimson dress. They must have been a striking couple.

On February 13, 1689, England had its first double coronation—of William and Mary. Because William and Mary’s coronation was unique in the sense that the two were crowned at the same time, they needed, for the first time, a second coronation chair. Apparently this chair had to be set lower than the other coronation chair because Mary was taller than her husband when seated.

It was a strange transition. William III was the deposed King James’s nephew, and his wife Mary was James’s daughter. William and Mary ruled England from 1689 to 1702, but the glorious revolution of 1688, as it was called, seriously curbed the power of the crown. William agreed to become king and his wife became queen, but they had to agree to a bill of rights that seriously curbed the power of the monarchy and established parliament as the rulers of the realm. Even though the parliament that emerged after 1688 was not representative of the people in the sense that it is today (then being controlled largely by the elite landowners, merchants, and nobles), it was nevertheless a stepping-stone to modern forms of government.

Moreover, England was now not only Protestant again but ruled by a king with a very personal interest in checking France’s aggression toward Europe. These were strange times. French Huguenots fought with the English against their native countrymen, and English Jacobites (supporters of James II) joined the French to oppose the English.

Britain won a number of military victories against the French in the coming years. The English navy defeated the French fleet in 1692, and the war dragged on for another half-decade on land. The 1690s were a terrible time of war in Europe, and things were not helped by poor harvests, famine, and all the social problems that these spurred on. Against this backdrop, Leibniz and Newton were moving invariably toward a war of their own.

FOR NEWTON, THE Principia was a turning point in his life. It gave him the confidence to write the text that would become Opticks later. Meanwhile, demand fueled work toward a second and then third edition of the Principia, and he carried on extensive (perhaps even neurotic) correspondence helping others correct, revise, expound, and improve it—work that occupied Newton part time for most of the rest of his life.

As the editions of the Principia grew, so did the legend of Newton—and the popularizations of his science both profound and profane. A good example of the latter appeared in 1739, when a book, Sir Isaac Newton’s Philosophy Explain’d for the Use of the Ladies, was published. An Italian named Francesco Algarotti was the author, and he lauded his own efforts for bringing a new kind of amusement to the ladies of the continent, whom he felt should be obliged to thank him. “If I have brought into Italy a new mode of cultivating the mind, rather than the present momentary fashion of adjusting their head dress and placing their curls.”

Fame for his science aside, the Principia really changed Newton’s life. Just after it was published, he was elected to parliament, a post that brought him to London. And this led him to meet Christian Huygens, Leibniz’s old mentor, who visited London for the first time in the late 1680s. This meeting was a significant one not only because it brought together these two stellar intellectuals, but also because it would introduce Newton to a young mathematician and astronomer, Nicolas Fatio de Duiller, a Swiss national who lived for several years in London and would play a crucial role in Newton’s life.

Fatio is a fascinating character, and is a key player in the calculus wars. He entered the lives of Leibniz and Newton separately (the latter, in a most peculiar way), and was really the first person to stir up trouble between them.

Born in Basel, Switzerland, on February 16, 1664, Fatio was the bright son of a wealthy Swiss family. He went to Paris in the early 1680s to be educated, with a generous allowance and leave to study anything he wished. His father had made several attempts to study divinity, but Fatio chose to pick up mathematics and astronomy instead, for which he showed a great propensity, though his real talent in his early years seemed to be having the ability to be in the right place at the right time.

After Paris, Fatio went to the Hague to study, and at that point, as a young man of twenty-one, he met a certain Count Fenil. Fenil had been working as a military officer in France, when he shot dead his commanding officer and subsequently had to flee the country. He stayed at Fatio’s home for a while, during which time he confided in Fatio a plot he was hatching.

As a way of making amends, Fenil had proposed to France’s minister of war, the Marquis de Louvois, that he would seize William of Orange, then still the Dutch prince, and deliver him to King Louis XIV and France. The marquis took the bait. He sent Fenil a letter approving of the plot, promising a full pardon if Fenil succeeded and offering to pay for the operation. It was to be an ambush kidnapping raid. Prince William liked to take walks at the beach at Scheveling, about three miles from the Hague, where he lived. Count Fenil proposed to steer a light ship through the surf, land it in the shallows, hit the beach with about a dozen men, grab up the prince, and sail off to Dunkirk with him.

Bold as it was, the plan failed. Fenil’s only mistake was to tell Fatio—and Fatio immediately told the plan to a doctor who was traveling to Holland, who passed word on to William.

This won Fatio the favor of the Dutch court, and he was rewarded for his disloyalty to Count Fenil with the promise of a professorship of mathematics at the university at The Hague, with a nice salary and the comfy job of instructing nobles and the gentry. However, while these arrangements were being made, Fatio went to England, where he eventually became a mathematics teacher in Spitalfields. He made a few trips home in the 1690s, but otherwise resided in England for most of the rest of his life.

When Fatio arrived London in 1687, he wasted no time establishing himself among the British scientific elite, managing to get himself elected to the Royal Society in just two weeks. Newton had just published the Principia, and it was the talk of town in the circles Fatio frequented.

Huygens came to London that summer, and Fatio, exploiting his relationship with William of Orange, won the right to escort the famous older scientist around town. Huygens and he hit it off, and they became friends. Huygens became something of a mathematical mentor for Fatio, much as he had been for Leibniz. Then Fatio was introduced to and charmed one of the most important men he would ever meet—Isaac Newton. Escorting Huygens brought Fatio into contact with the Englishman.

Fatio had become infatuated with Newton’s theory of gravitation from the moment he arrived in London, and he became friends with Newton after the two met at a meeting of the Royal Society that they both attended on June 12, 1689. Newton had come to the meeting to meet Huygens, and Fatio was there with Huygens, but the real connection was between Fatio and Newton.

Their intense friendship in the early 1690s is the source of some historical speculation. Newton’s letters to Fatio are unusually warm, and some have suggested that Fatio was the object of Newton’s latent homosexual affections. It is more than tempting—indeed fun—to read the Englishman’s close relationship with his young protégé as one that reveals the root cause of his affection and to try to read between the lines of their correspondence. For his part, Fatio wrote to Huygens that he was “frozen stiff” when he saw what Newton had accomplished. Likewise, one might raise an eyebrow at the reports that Newton liked to build doll furniture and preferred the company of girls—suggesting that his preference for girls (as opposed to adult women) might have belied a preference for boys.

However, there is scant historical evidence that Newton had any interest in either sex. According to Voltaire, Newton died a virgin after more than eighty years on Earth—a virtue, Voltaire adds. For Newton, sex may have been as enticing as a tray of pudding and tea left outside the door of his study by his servant—the pudding goes cold and uneaten on one side of the door while Newton works all night scribbling strange symbols in notebooks on the other.

Whatever their relationship, the two were unusually close and were great admirers of each other, as is evidenced from their correspondence, which makes it clear that Fatio was very fond of Newton. Within a few months of their first meeting, Fatio wrote a letter to his friend Jean-Robert Choet, calling Newton the most honest man he knew and the ablest mathematician who ever lived. Fatio offered to sit with Newton and help him read a new book that Huygens had just published (in French).

The passion of their friendship was mutual. The earliest letter between the two was written by Newton later that year, on October 10. Newton wrote to Fatio and asked if there would be any rooms at the Swiss’s residence in London. “I intend to be in London the next week,” Newton wrote. “And I should be very glad to be in the same lodgings with you.”

Over the next two years, Fatio and Newton became closer and closer. Even when Fatio left England for fifteen months in June 1690, Newton had him on his mind, writing to John Locke, for instance, on October 28, 1690, “I suppose Mr. Fatio is in Holland for I have heard nothing from him the half year.” When Fatio returned in September 1691, Newton rushed to London to meet him in private as soon as he was back and, after that, their relationship became all the more intimate. They were frequently seen together in the Royal Society in London, so much so that, when their presence was recorded in the attendance notes, they were often marked down as a single unit. Hooke, still Newton’s nemesis in the 1690s, began calling Fatio “Newton’s ape.”

Fatio fancied himself as more than Newton’s ape, and he offered to supervise the revisions of the Principia to make the second edition of the book. He envisioned his role as something approaching Newton’s collaborator, and he wrote to Huygens that this second edition would be much longer because of his additions.

If Newton got along famously with Fatio, Leibniz had a strange relationship with him—nothing like the mutual admiration society Fatio formed with Newton. Huygens tried to get Leibniz and Fatio to correspond, but the German didn’t see the need. Leibniz was already serving as a mentor for a growing cadre of European mathematical intellectuals—and Fatio would not be one of them.

Calculus was also already on the move. In 1691, Johann Bernoulli went to Paris and became the teacher of the Marquis de L’Hôpital. This was a fruitful connection because L’Hôpital would write a few years later, in 1696, one of the first ever calculus textbooks, Analyse des Infiniment Petits (Analysis of Infinitely Small Quantities), with a great deal of help from Bernoulli.

Leibniz was on the move as well.