The Affair of the Eyebrow - 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 4. The Affair of the Eyebrow

1666–1673

Many of his dreams have been realized and have been shown to be more than the fantastic imaginings that they seemed to all his successors until the present day. . . .

—Bertrand Russell in the 1937 preface to his critical exposition on Leibniz’s philosophy

For most of his life, Leibniz rarely worried about being over-shadowed by Newton or anybody else. He was one of the most prolific thinkers of his day, and his far-flung interests led him to contribute advances in fields as diverse as medicine, philosophy, geology, law, physics, and of course mathematics. It was exactly the sort of ambition that led Leibniz to plunge into mathematics in the early 1670s—not simply to understand everything that had been done by his contemporaries, but also to synthesize everything known at that time into one general system that could serve as a tool for future discoveries.

Mathematics was not his main interest in his early days. In fact, Leibniz did not plunge far enough into mathematics to invent calculus until he was nearly thirty. And even then, calculus seemed but one facet of his grand vision for knowledge in general. He saw all human ideas, concepts, reasonings, and discoveries to be a combination of a small number of simple, basic fundamental elements—like numbers, letters, sounds, colors, and so on. Leibniz hit upon the idea of creating a universal system that would provide a way of representing ideas and the relationships among them—an alphabet of human thought with which ideas, no matter how complicated, could be represented and analyzed by breaking them down into their component pieces, like the letters that make up w-o-r-d-s a-n-d s-e-n-t-e-n-c-e-s.

The characteristica universalis or alphabet of human thought was first attempted in his doctoral thesis, Dissertatio de Arte Combinatori (Dissertation on the combinatorial art). A little later in life, he described his idea in the most visionary and optimistic terms: “Once the characteristic numbers for most concepts have been set up, the human race will have a new kind of instrument which will increase the power of the mind much more than optical lenses strengthen the eyes and which will be as far superior to microscopes or telescopes as reason is superior to sight. The magnetic needle has brought no more help to sailors than this lodestar will bring to those who navigate the sea of experiments.”

Such a reduction of complex ideas may sound foolishly simple, but the attempt to come up with an alphabet of human thought is what led Leibniz to calculus. He knew little mathematics when he wrote his “Dissertation on the Combinatorial Art,” but in a way the dissertation prepared him to discover calculus because it allowed him to appreciate the need that calculus would fulfill. Calculus, after all, is a body of knowledge dealing with the analysis of geometry and numbers, and for Leibniz this was one example of a larger logical system for analyzing all his characteristica universalis.

Moreover, “Dissertation on the Combinatorial Art” had a very direct impact on the calculus wars because it set into motion a sequence of events that would lead Leibniz to Paris, where he would invent calculus, and to London.

However brilliant his work, he was denied his doctorate at the University of Leipzig in 1666. Why this occurred is not entirely clear. One of the stories is that the wife of the university’s dean convinced her husband not to award the doctorate to young Leibniz for some personal reason. But perhaps he simply fell victim to the academic politicking of the university. There were a limited number of spots available for graduation and, had Leibniz’s thesis been accepted, he would have prevented a more senior student from graduating.

Undeterred by this setback, Leibniz left Leipzig, matriculated to the nearby University of Altdorf in October 1666, and graduated from there a few months later, receiving his doctorate from the university in February 1667. His dissertation, De Casibus Perplexis (On difficult cases [in law]), held that the law had to answer a certain number of uncertain cases, which in his day were often decided by drawing lots and other arbitrary means. Leibniz argued that these difficult cases should instead be decided by reason and the principles of natural justice and international law.

He claims that his thesis dazzled the audience. “I received the degree of a doctor from the University of Altdorf, with great applause,” Leibniz once bragged. “In my public disputation, I expressed my thoughts so clearly and felicitously, that not only were the hearers astonished at this extraordinary and, especially in a jurist, unexpected degree of acuteness; but even my opponents publicly declared that they were extremely well satisfied.”

Following the awarding of his doctorate, the education minister at the university, a man by the name of Johann Michael Dilherr, told him that he could guarantee Leibniz a professorship if he was so inclined. Leibniz was not. “My thoughts were turned in an entirely different direction,” he said later in life. “I gave up all other pursuits and confined my attention exclusively to that occupation upon which I was to depend for a livelihood.”

What was this livelihood that caused Leibniz to reject the offer? It was an occupation through which he sought to do something more practical—work that would confer the greatest benefit to mankind. He decided to pursue law. The thought that a lawyer would have more opportunities to do good than would a university professor would no doubt make many modern university faculty members laugh or wince. Nevertheless, once Leibniz finished his doctorate in 1667, he left university life forever. He would face the world, an ambitious and brilliant young lawyer with a keen interest in politics and learning, but not much knowledge of mathematics.

He settled in nearby Nuremberg, and had no trouble fitting into the learned societies there. One of the groups he became acquainted with was an alchemical society. The story is that he wanted to gain access to their society and secrets but, since he was an outsider, he did not have a way in. So he devised a plan. He consulted the most difficult alchemical textbooks he could find and wrote down the most obscure words that they contained, and he came up with a paper that was both impressive and meaningless. He later admitted that it made no sense whatsoever, even to him. But he so impressed the alchemists at his ability to write profoundly that they gladly welcomed him into their society and made him their secretary. For months, he joined them in discussion and debate. Later, though, he was to denounce the cult of alchemy as the “gold-making fraternity.”

In 1667, Leibniz’s life took a dramatic turn. He met a wealthy and well-connected German statesman, Baron Johann Christian von Boineburg, a man of prestige and learning known in many of the German capitals. In the next five years, Leibniz became a close friend of Boineburg, serving as his secretary, assistant, advisor, librarian, and lawyer for several years. This relationship would prove crucial in Leibniz’s life because it would be Boineburg who would convince him to go to Paris a few years later.

The baron saw in Leibniz a great protégé, and from the beginning, his assistant’s intellect impressed him. Boineburg once wrote to an acquaintance introducing Leibniz in the grandest of terms. “He is a young man from Leipzig, of four and twenty,” Boineburg wrote. “Doctor of laws and learned beyond all credence.”

Boineburg helped Leibniz get into the good graces of the archbishop elector of Mainz, Johann Philipp von Schönborn, who was a regional political leader of some prominence. During this time, Germany was something of an amalgam of states, dozens of which were ruled by bishops and archbishops like Schönborn. Mainz was a German state but also like a small country, in that it was part of the Holy Roman Empire. (Voltaire once quipped that the Holy Roman Empire was neither holy nor Roman nor even an empire.) Boineburg had been close to the archbishop and was formerly a minister of the court at Mainz (he was fired in 1664, but shortly thereafter reconciled with the elector after his daughter married Schönborn’s nephew).

This meant that Boineburg was well positioned to introduce Leibniz to Schönborn. Leibniz wrote an impressive essay, “A New Method of Teaching and Learning Law,” which is said to be rich in ideas. Boineburg convinced him to dedicate it to Schönborn and arranged for Leibniz to have an audience with the archbishop, to present his essay to him in person. Schönborn’s response was to make Leibniz a judge in the High Court of Appeal at the age of twenty-four.

Leibniz was assigned to work with a man named Herman Andrew Lasser on a project revising the legal code. Together they wrote a large work, Leibniz writing two parts and Lasser also contributing two. Leibniz’s opened powerfully: “It is obvious that the happiness of mankind consists in two things—to have the power, as far as is permitted, to do what it wills and to know what, from the nature of things, ought to be willed.” Some modern, some antiquiated, Leibniz sought to find a systematic basis for this diverse set of laws.

Legal reform was a hot topic in those days because the Holy Roman Empire was complicated by an intricate system of laws that varied from state to state. One effect this had was to fractionate Germany, and because the various states acted autonomously, various rulers considered only themselves when deciding with whom to form alliances. Since Germany was centrally located in the middle of Europe with bordering states on the east, west, north, and south, these alliances were key.

Moreover, a number of uncomfortable divisions had arisen out of the reformation following Martin Luther’s introduction of Protestantism more than a century before. States were divided between the Protestants and the Catholics. The Peace of Augsburg in 1555 allowed local princes to determine the religion of the land, but it only further divided Germany and subjected states to the whims of their rulers. Perhaps the most dramatic example of this was in Germany in the state of Rhineland-Palatinate, which switched from Catholic to Lutheran in 1544, from Lutheran to Calvinist in 1559, from Calvinist back to Lutheran in 1576, and from Lutheran again to Calvinist in 1583.

During the five years that Leibniz was a close advisor to Boineburg, he had his first taste of ambassadorial politics. When John Casimir, the king of Poland, stepped down from the throne in 1668, a number of people aspired to take his place. One of these, the prince of Neuberg, was supported by Boineburg, and Boineburg asked Leibniz to help toward this end. What Leibniz did in response was to write a pamphlet in which he not only gave the merits of Neuburg’s cause but also investigated the nature of Poland in general—its government, its conditions, and so forth. Although Neuburg did not become king, Boineburg rewarded Leibniz by recommending him to be a member of the elector of Mainz’s council.

It was through his relationship with Boineburg that Leibniz was thrust into Paris, London, and eventual conflict with Newton. By the beginning of 1672, the drums of war were deafening as France, Europe’s main superpower, was once again turning an aggressive eye toward other European countries. Louis XIV was furious with the Dutch, who had been his allies, because in 1668, Holland joined with England to thwart France’s attempt to annex the Spanish Netherlands. This set off a commercial dispute, with France slapping heavy tariffs on Dutch goods. By 1671, the situation was dire and Europe was on the brink of what could be another major war.

This created a confusing scenario where many of the states in Germany had various alliances with or against France. Johann Friedrich, the Duke of Hanover, was a good example of this. His foreign policy was to support France in exchange for money. But all alliances would be truly tested after France began amassing troops along its eastern borders as it prepared to invade Holland.

Schönborn was forced to abandon his alliance with the Duke of Lorraine after the duke pushed the elector to form an alliance with England, Holland, and Sweden against France at a meeting that took place in July 1670. Boineburg and Leibniz were both at this meeting, and they both opposed the prospect of such an alliance.

Leibniz even wrote a pamphlet with the unwieldy title, Reflections upon the Manner in which, under Existing Circumstances, the Public Safety, Both Internal and External, May Be Preserved, and the Present State of the Empire Be Firmly Maintained. This pamphlet warned of the dangers of taking sides against France, and Schönborn heeded the advice, sitting idly by as tens of thousands of French troops poured into Lorraine, and the Duke, his erstwhile ally, was forced to flee.

Boineburg saw the stupidity of standing against the great military superpower of France. Besides that, he had too much to gain from keeping Mainz on France’s good side—he had property and a pension in France that were owed to him, and he believed he could recover this small fortune if he played his cards right. He hoped to be sent to France to collect his money and, at the end of 1671, just as Newton was preparing to present his new theory of light and colors, Boineburg was positioning himself to go there.

But things became unglued when the French foreign minister died. It was several months before a new minister, Simon Arnauld de Pomponne, was appointed in January 1672. By then, a French ambassador had arrived in Mainz on a mission to ask for free passage of his war ships on the Rhine River so that Louis XIV’s troops could attack Holland more easily. The presence of the French ambassador in Mainz made Boineburg’s trip to Paris irrelevant. So Boineburg decided he would send Leibniz to Paris instead.

Leibniz drafted a rather vague document and sent it to Louis XIV on January 20, 1672, mentioning how he and France could benefit from “a certain undertaking” that had advantages for France. He gave no details as to what this undertaking would be, and the document must have piqued a great deal of curiosity in France and in the new French foreign minister, Simon Arnauld, Marquis de Pomponne, because a reply arrived on February 12, 1672, asking Boineburg to come and present his proposal. Boineburg sent word on March 4, 1672, that he would send Leibniz in his place.

Leibniz’s plan was bold almost to the point of being far-fetched. He wanted to convince Louis XIV that France should not go to war with Holland, by making a case for how profitable it would be to instead turn his country’s aggression and ambitions toward Ottoman Empire–controlled Egypt. Egypt, with its command of important trading route points, was a much more lucrative target, Leibniz argued, and attacking the Ottomans in Egypt would also shore up the eastern portion of Europe, where such cities as Vienna were under threat of attack from the east.

It may seem strange to propose an invasion of Egypt as a plan for peace, but the idea of turning war within Europe toward the outside world was nothing new. In the 1300s an Italian, Marino Sanuto, wrote a book, Secreta Fidelium Crucis, which proposed essentially the same thing to the pope. In fact, Leibniz drew upon this centuries-old work when he came up with his up-to-date version of the plan. But in the initial letter, none of the specifics were presented. In fact, it was so lacking in detail that nowhere did it even mention the word “Egypt.”

Leibniz and one servant set out for Paris on March 19, 1672, to present his eleventh-hour appeal. He carried with him a letter of attorney from Boineburg, a letter of introduction, traveling expenses, and a sincere desire to convince the French king of the value of forgetting about war in Europe and instead turning to parts of the non-Christian Middle East. His mission was kept semisecret, and he traveled under the cover of representing the personal interests of Boineburg, arriving in Paris at the end of the month.

Leibniz must have been excited about the trip, like any young man on his way to the big city for the first time. Paris was one of the largest and grandest cities in Europe, and it was the playground of Europe’s rich and elite. Even though much of Germany was at war with France at some point or other throughout his lifetime, France was nevertheless a model of seventeenth-century courtly life. Its features were to be admired and its courtly manners to be imitated in as many lavish details as possible.

Moreover, Leibniz was going there to present a proposal to the highest levels of French government. This was very appealing to him because one of the things he liked to do was act ambassadorial. He might have entertained ambitions of actually being an ambassador, but he lacked the one crucial asset that would have allowed him to do so—the pedigree of a high birth. He may have been representing Boineburg, but he was himself no Boineburg. Nevertheless, there was a real possibility that he would be presenting his work to Louis XIV, who was a fabulously powerful monarch.

Louis XIV had been a boy king, inheriting the throne from his father when he was only four years old. Because he was a child, he was completely unprepared to rule, and a regency government was installed instead, with Louis’s mother as regent and her close advisor, the Cardinal Mazarin, in charge for the next dozen years. When Cardinal Mazarin died, Louis XIV took over and became the longest-ruling king in the history of France. He was the model of the absolute monarch: Though he governed France with the help of myriad advisors and confidants, he kept absolute power, and if any one person had the power to change the course of history at will and to stop a war upon hearing a petition, that was Louis XIV.

As a military strategist, Leibniz was more than a century ahead of his time. France would indeed eventually invade Egypt under Napoleon, who grasped the value of the peninsula exactly as Leibniz had suggested. In fact, when Napoleon invaded Germany and occupied Hanover in 1803, he was annoyed to learn that Leibniz had anticipated him by more than a century.

However flattering this might have been for Leibniz had he known, the proposal was an ill-timed flop in his lifetime—as it turned out, he never had an opportunity to present his proposal.

On April 6, 1672, Louis XIV and his subordinates published a short document, “Declaration of War against the Dutch.” Issuing it from Versailles, the king ordered it disseminated throughout France and its dominions where, as all his subjects were to read, he commanded them to “fall upon Hollanders.” With the French already in a position to invade, the Dutch were forced to open up dikes and flood the countryside to slow the French advance. The Franco-Dutch War, as it is called, had begun, and it would drag on for the next six years.

When Leibniz finally did arrive in Paris, his original proposal was now moot. Nevertheless, Leibniz and Boineburg, keeping in communication, did not abandon their plan, but modified it making invasion of Egypt an enticement to end the war as opposed to a proposal to prevent it. Once the battles within Europe were concluded, they proposed, an invasion of Egypt could begin. Leibniz wrote a paper to this effect, “Consilium Aegyptiacum,” which is said to argue the case with eloquence, learning, and mastery.

To bolster their case, Leibniz and Boineburg brought the elector of Mainz on board. Schönborn thought it was a great idea, and he immediately sent word to Louis XIV, who was encamped with his army at the time, offering to mediate peace so that the French could quickly set sail for North Africa. The answer was, in effect, an eloquent “no thank you, the crusades are over.” “As to the project of the holy war, I have nothing to say,” read the response to the German court. “You know that since the days of Louis the Pious, such expeditions have gone out of fashion.”

But for Leibniz, the crusades were just beginning. He decided to make the most of his time in the French capital anyway. What an opportunity this was for him! In Paris, he was alone and without major day-to-day duties. After spending several months learning French and setting himself up in this new urban setting, he buried himself in the libraries for days; also, because he arrived in Paris as the representative of Boineburg and carried with him letters of introduction, many doors were opened to him.

With these open doors came numerous opportunities, and, in the few years he spent in Paris, he was able to partially support himself with legal work. He was, after all, a lawyer who could bring skills to the elite society, drawing up documents, taking legal briefs, or providing other services and actions on behalf of the well-to-do. He secured the release of a foreign prince from jail, for instance, and he arranged for the divorce of the archduke of Mecklenburg—a man who had been hated by his subjects at home, which forced him to flee Mecklenburg in 1674 to Paris, to which his temperament was more suited. However, Mecklenburg had converted to Catholicism, which presented him with a problem. Before, when he was Protestant, he had no trouble divorcing his first wife. But now that he was married to a nice Catholic lady, divorce was not so easy. So Leibniz helped him out.

Leibniz was busy enough with this type of work and other social obligations. In what would become a theme throughout his life, he became distracted from what he saw as his more interesting intellectual work. “My mind is burdened by a great variety of labors, in part required of me by my friends, and in part by persons of rank,” he wrote to the secretary of the Royal Society in the summer of 1674. “Therefore I have much less time than I could wish to devote to the study of nature and to mathematical investigations. Nevertheless, I steal as much of it as I can. . . .”

Luckily, Paris was an intellectual capital of Europe and boasted some of the finest minds alive in Leibniz’s day. He met many of them there, and was inspired to come up with highly original, although sometimes impractical, ideas—such as a way to determine longitude; a pneumatic gun; a concept for how a boat might be able to dive, submarine-style, to escape from pirates; and an idea for improving watches. In Paris, Leibniz truly began his lifelong career of scholarship, acquiring a breadth of learning and acquaintance that covered the whole of the “republic of letters” as the philosopher Bertand Russell once described it. And he made a few discoveries in mathematics.

His journey on the road to discovering calculus began in the fall of 1672, when he met Christian Huygens. A Dutch physicist and mathematician, Huygens was the son of a famous literary and diplomatic figure in the Netherlands, and he had something of a gift for words himself, once declaring, “The world is my county,” and adding that promoting science was his religion. His father was a friend of Descartes, and Huygens was a strict Cartesian for his entire life, which had a strange influence on his work at times. For instance, after he discovered a moon of Saturn, he stopped looking for more moons in the sky because Cartesian symmetry held that, since these were six planets, there should be six moons.

Despite how silly that reasoning seems today, Huygens is still regarded as one of the greatest scientists in the seventeenth century. When Leibniz first went to visit him, Huygens was perhaps the foremost natural philosopher living in Paris and one of the best-connected intellectuals in Europe. As a measure of how great a mathematician and scientist Huygens was, even though he was Dutch and living under the highly xenophobic regime of France’s Louis XIV, Huygens was still the leading member of the Académie des Sciences, an organization that he had helped found.

Huygens’s status was well deserved. A gifted craftsman who developed methods for making lenses in the mid-seventeenth century, he made several important contributions to science in his lifetime. In 1655, using his improved lenses in his telescope, he observed the rings of Saturn. Master of the latest mathematics, Huygens studied the pendulum, analyzed it mathematically, and used it as an engine to drive clocks of his own invention.

Huygens and Leibniz hit it off right away, and in the next few years they became friends. More importantly, the older, wiser Huygens became Leibniz’s inspirational mentor, encouraging the German to look deeply into mathematics. “I began to find great pleasures in geometrical investigations,” Leibniz wrote years later, as he remembered that time in a letter to the Countess Kielmansegge near the end of his life.

Huygens must have gotten a great deal of pleasure out of this interaction as well, because his protégé was beginning to make rapid progress by the end of 1672. That fall, Huygens gave Leibniz a challenging problem involving the sum of a mathematical series, specifically the sum of an infinite number of fractions, each smaller than the last: , and on and on. Huygens asked Leibniz to calculate the sum of the infinite series; Leibniz sat down and was able to come up with a solution (the answer is 2). Huygens was impressed and urged Leibniz onto further studies, suggesting books that the younger man should study. One of these was Arithmetica Infinitorum, by the English mathematician John Wallis, which had so inspired Newton just a few years before.

Another book was by the Belgian Jesuit mathematician Gregory St. Vincent, which Leibniz borrowed from the Royal Library in Paris and began to study as soon as Huygens suggested he read it. St. Vincent thought of a geometrical area as being the sum of an infinite number of infinitely thin rectangles. This work anticipated integral calculus, the second side of the calculus coin that can be used to determine the area or volume of a geometrical shape by applying a set of algebraic tricks that essentially add up all these little rectangles.

Leibniz also read Bonaventura Cavalieri, a friend of Galileo’s and professor of mathematics at Bologna. Cavalieri had developed the idea of the indivisible—a small section of a geometrical shape which, when taken with all the other small sections, would constitute the initial shape itself. He considered a line as being made up of an infinity of points, an area an infinity of lines, and a solid an infinity of surfaces. Think of this as a stack of pancakes: The stack is made up of all the individual flat pancakes. Cavalieri’s 1635 book, Geometria, proved such facts as that the volume of a cone is one-third the volume of the cylinder that fits around it.

While studying these works, Leibniz started to go further and do some original mathematics, which he thought of publishing in a French journal until it unexpectedly folded. Aside from this minor setback, by the end of 1672 Leibniz was beginning the most outstandingly productive time in his life—certainly the greatest time he spent considering mathematics. In his four and a half years in Paris, he grew from a lawyer with little formal training in mathematics into a scholar who not only understood the furthest mathematical advances of his contemporaries but pushed them forward—for example, inventing calculus.

However, during this time, Leibniz would also feel the biting sting of personal defeats, the first of which came less than a year after he arrived in Paris, when Boineburg died on December 15. This was not just the loss of a patron. Boineburg, whom he later called one of the greatest men of the century, was someone for whom Leibniz had great respect and affection. And this was not the only death that Leibniz would have to deal with. A month after Boineburg’s demise, Leibniz’s sister died.

But perhaps the greatest personal defeat would come a few months later on a trip to London in the winter of 1673, where he headed on another diplomatic mission in early 1673 with Boineburg’s son-in-law, Melchior Friedrich von Schönborn, the nephew of the elector of Mainz. Young Schönborn showed up in Paris on another peace mission, as his uncle wanted him to have an audience with Louis XIV to plead the case for peace talks to take place in Cologne. If this didn’t work, Melchior was to go to London and appeal to Charles II.

Since Leibniz was already in Paris and had also worked for the elector of Mainz, he was enlisted to help Melchior. But when the day came to seek an audience with the king, Melchior alone was permitted to see Louis XIV and little came of the meeting.

At that time the French and English offensive in Holland had stalled. Leibniz and Melchior continued with their plan and sought to seize the opportunity to further peace efforts, by seeking an urgent consultation with the English court and presenting their proposal there. They set out in the middle of winter for London, and arrived in Dover on January 21, 1673. It was almost exactly one year after Hooke had attacked Newton for his optics work.

In London, Leibniz’s and Melchior’s efforts to plead the case to the British king fell flat. And why wouldn’t they? Charles II had agreed to join France in war and attack Holland. England and Holland had been at odds for years, and for his pains, Charles was rewarded with a yearly pension of £100,000 from Louis XIV.

However, this trip had a dual purpose for Leibniz. When in London, he also met with members of the Royal Society and made contact with some preeminent British scientists, particularly Robert Boyle, John Pell, and Robert Hooke, who discussed natural philosophy, mathematics, and chemistry with him. In this sense, London held as much excitement for Leibniz as Paris had. One figure Leibniz did not have the opportunity to meet, however, was Newton, who was in Cambridge at the time. Leibniz certainly would have been aware of him—as one of the brilliant young mathematicians who, like Leibniz, had just been elected to the Royal Society.

Leibniz had been aware of the Royal Society for a few years. In 1670, he had written a paper on the collision of bodies, called “A New Physical Hypothesis,” in response to essays by Christopher Wren in England and Christian Huygens in France. The first part was on “concrete” motion, and the second on “abstract” motion. He dedicated the former to the Royal Society in London and the latter to the Académie des Sciences in Paris.

Academic societies were nothing new. Leibniz belonged to more than one when he was in college—but those were more informal than what emerged in Paris and London in the seventeenth century. The Académie des Sciences, for instance, was granted a royal charter and a room in the royal library at the Versailles Palace in 1666, and the moment when the charter was signed was regarded as such an important event that it was the subject of a painting by the artist Henri Testelin. In the painting, Louis XIV is depicted presenting the charter to a group of Académie founders.

In England, a group of churchmen, mathematicians, natural philosophers, and other scholars founded what would eventually become the Royal Society when they began meeting once a week in 1645 to “discourse upon such subjects” as natural and experimental philosophy. A number of individuals, including the mathematician John Wallis, the astronomer Seth Ward, the chemist Robert Boyle, the statistical theorist William Petty, and the architect Christopher Wren attended these meetings, which were sometimes at a Dr. Jonathan Goddard’s home, and sometimes at the lodgings of John Wilkins. When Wallis moved to Oxford as a professor a few years later, the group continued to meet in London and also began meeting in Oxford. The “Invisible College,” as Boyle called it, was the home to lively discussions in math, physics, astronomy, architecture, magnetism, navigation, chemistry, and medicine—all the important subjects of the day.

The meetings continued on and off through the years when Newton and Leibniz were in school. When Oliver Cromwell died in 1658, the Invisible College stopped meeting due to the turmoil, but after the monarchy was restored and King Charles II came to the throne, the Invisible College was resurrected and reborn on July 15, 1662, as the Royal Society for London for Improving Natural Knowledge, with ninety-eight charter members. In the next twenty-five years, about three hundred new members were added, including Leibniz and Newton.

Part of the reason for the success of these societies was that science was becoming fashionable. There was great patronage of scientists among the wealthy and noble of Europe. Members of the Académie des Sciences received salaries from the government and funds for their experiments. High-society types attended chemistry lectures in Paris and London, and joined the Académie des Sciences and the Royal Society. King Charles II had his own chemical laboratory built, and aristocrats read scientific publications.

And what a sweet time of discovery the seventeenth century was. The diameter of the Earth was estimated to within a few yards, and a sophisticated modern view of the solar system evolved, with the orbits of heavenly bodies accurately tracked by telescopes and faithfully described by mathematics. The circulation of blood through the body was carefully charted, and microscopes led to the discovery of cells and a world of tiny organisms too small to be seen with the naked eye.

In 1673, when Leibniz was visiting the Royal Society, he was thinking of presenting an invention he had been working on for a while in Paris—a mechanical calculating machine, which Huygens called “a promising project” in a letter to Henry Oldenburg, the secretary of the still new Royal Society. As a friend of Boineburg’s and fellow countryman, Oldenburg not only knew who Leibniz was, but had been in correspondence with him for a several years. Oldenburg was committed to helping Leibniz, who expected to make a splash in London with his calculating machine.

The Royal Society extended an invitation to Leibniz to demonstrate the machine. This wooden and metal device used a mechanical wheel to manipulate numbers. The famous French mathematician, Blaise Pascal, had invented a similar machine that could add and subtract, but Leibniz’s could add, subtract, multiply, and divide. Or at least it was supposed to. In 1673, his calculating machine was an incomplete, nonfunctioning prototype when Leibniz hauled it across the English Channel. Leibniz’s machine was something of a flop because he chose to demonstrate it even though it was not finished. He could explain it all very well, but his demonstration must have been something like the traveling vacuum cleaner salesman trying to sell his goods door-to-door during a blackout. The machine is great, and it would be very useful if only the darned thing worked.

Particularly unimpressed was Robert Hooke, who was spoiling for a fight. In addition to being one of the most prolific minds in the seventeenth century, Hooke was great with his hands and had made many scientific instruments; produced works of importance in astronomy, physics, biology; proposed a wave theory of light; discovered a new star in the constellation Orion; proposed the kinetic theory of gasses; and is still famous today for the discovery of the law governing masses on springs that bears his name.

Hooke was the toast of the Royal Society when Leibniz came to demonstrate his unfinished calculator in 1673, and, as Newton had already discovered, Hooke was infamous for engaging in brutal disputes—not always within the boundaries of fair and open scientific debate—with his rivals. One example is Hooke’s reaction to the spring balance, which Huygens had discovered as a by-product of his work in the 1650s, while trying to make a pendulum clock. Hooke not only disputed Huygens’s discovery, he claimed it as his own, constructing a pocket watch and presenting it to England’s king in the summer of 1675. Hooke went so far as to accuse Oldenburg, the secretary of the Royal Society, of betraying his ideas to Huygens.

Hooke clashed equally fiercely with Leibniz over the calculating machine. After looking carefully at all sides of it, and examining it in detail on February 1, 1673, Hooke expressed a desire to take it apart completely to examine its insides. This is no surprise—the machine is a tempting object for the curious.

In Hanover, there is a replica of Leibniz’s machine on display. It’s a fascinating object. Eight dials across the top allow a user to dial in a number, and add or subtract the numbers, which would reset the dials. The machine would keep track of the accumulating sum or difference. A knob on the machine acts to multiply or divide. Crank the handle one way and it divides, turn it the other way and it multiplies. The machine has a row of pentagons to address the problem of incorrectly adding columns of numbers with a different number of significant digits. The curator who put the machine on display had the foresight to place it on a mirror so that, by peering over it and peeking around it, one can examine nearly every inch of it. It’s a fascinating machine and I can easily imagine how much Hooke wanted to take it apart.

A few days after Leibniz’s presentation, Hooke attacked him in public, making derogatory comments about the machine and promising to construct his own superior and working calculating machine, which he would present to the society. At the same meeting, Hooke attacked Newton, lambasting him in a letter he read in front of the entire assembled Royal Society. Neither Newton nor Leibniz were there to defend themselves, and Leibniz had to hear about the attack from Oldenburg, who assured him that Hooke was quarrelsome and cantankerous, and urged him that his best course of action would be to finish his machine as quickly as possible.

Hooke finished his machine, based upon the designs of his countryman Samuel Morland, and presented it on March 5, 1673, as promised. This must have made Leibniz’s machine seem that much more unsatisfactory. Hooke made his in a matter of a few days, after all, and it worked as he said it would. Leibniz had been working on his machine for untold months, and so far it couldn’t be shown to do anything.

Hooke’s attack notwithstanding, the Royal Society later elected Leibniz a fellow on April 19, 1673, with the backing of Oldenburg. Leibniz committed a social faux pas by not immediately sending the Royal Society a formal letter of acceptance, as was the fashion of the times. Instead he sent a short note of thanks a few weeks later, which caused some grumblings among the fellows at the Royal Society. Oldenburg had to inform Leibniz that he was expected to write the formal letter, which he finally did several weeks later.

But worse embarrassment was yet to come for Leibniz after he visited Robert Boyle on February 12—the occasion of an event I like to call “The affair of the eyebrow.”

Leibniz had been happy to meet Boyle, the long, gaunt older scientist, because he was interested in his experiments—and for good reason. Boyle, one of the founders of the Royal Society, was a brilliant experimentalist and was given to stunning audiences with his scientific demonstrations—such as when he proved that sound was carried by air by enclosing a bell within a glass jar from which he could remove all the air with a vacuum pump. When he rang the bell with the air removed from the jar, it made no noise whatsoever. Boyle also did carefully controlled experiments designed to demonstrate relationships between factors such as the pressure and volume of gasses or reactions between two compounds. He discovered the fact that certain vegetable abstracts change color when subjected to acids or bases—the technology behind litmus tests. And finally, he published his book, The Skeptical Chymist, in 1661. In this he abandons air, fire, water, and air as the elements and argues that the real elements are more primitive and simple.

The affair of the eyebrow began at Robert Boyle’s house, when Leibniz met John Pell. Pell is a somewhat obscure figure today, but at the time was considered one of the top two or three mathematicians in Britain, an accomplishment made all the more remarkable by the fact that his reputation seems to have surpassed his actual work. But then Pell survived on his reputation alone. He had been a diplomat under Cromwell, stationed in Switzerland, so it is no wonder that when Charles II returned and Cromwell’s head was placed on a spit above the streets of London, Pell’s political career ended. Leibniz met him after these events.

Still, Pell was an expert on the sort of mathematics that Leibniz had been working on in Paris, in fact the type that Leibniz was presenting that night. Leibniz had arranged to have some of the work he had done in Paris written out, and he brought it with him to London so that he could present it to anyone who was interested. At Boyle’s house, Leibniz tried to impress the company by telling them that he had an original mathematical method for performing a difficult algebraic trick—employing the subtractions of square roots.

After looking at some of these “original” discoveries, Pell informed Leibniz that, a few years earlier, another mathematician, Gabriel Mouton, had published the same results in a book about the diameter of the sun and moon: Observationes diametrorum solis et lunæ apparentium. Mouton had reported in his book the results of a French mathematician, François Regnauld, which mirrored Leibniz’s supposed original discoveries. The same night Pell told Leibniz about the book, Leibniz grabbed a copy of it from Oldenburg, who lived close by. He opened it up and discovered that Pell was absolutely right. What an embarrassment. The book was available in France, and even though Leibniz had never heard of the book, there was the possibility that he might have read it.

This caused some, no doubt, to raise an eyebrow. Had Leibniz borrowed his ideas? Was he a plagiarist? Oldenburg asked him to write an explanation and deposit it in the papers of the Royal Society, which he did in haste. The letter that he wrote explaining the whole event was to become one of the key documents in the calculus wars. Even though it seemed like a simple misunderstanding, the letter proved that there had been a controversy—the possibility that Leibniz had plagiarized before. And for this reason it was an important document. Newton had a copy of it, apparently, in his possession when he died.

The affair of the eyebrow was a painful episode, but it revealed to Leibniz exactly how much mathematics—or rather how little—he understood, and he was left somewhat shaken at this humbling realization. At the very end of his life, Leibniz reflected on his lack of knowledge when he was visiting London. “Mathematics were studied by me only incidentally,” he admitted. “I had not the least knowledge of the infinite series of Mercator; and as little of the advancement then made in the science of geometry, by the adoption of the new methods of investigation,” he wrote. “I was not even thoroughly versed in the analysis of Descartes.”

He would soon have the opportunity to know the works of Descartes and many others quite well. Though the affair of the eyebrow gave him a certain amount of grief, that grief gave him resolve to redouble his efforts to learn mathematics, and he would soon have ample opportunity to do so. The same night he visited Boyle, February 12, 1673, Johann Philipp von Schönborn, the elector of Mainz, died. Shortly thereafter, Leibniz and Melchior received the news and rushed back to Paris. Melchior went on to Germany to be close to the new prince, to whom he was related and who appointed him to the new court.

Leibniz left a letter for Oldenburg before he departed, requesting membership in the Royal Society, and he sent Oldenburg several letters from Paris in 1673—one in March, another in April, another in May, again in June and July, and another in October. Then he stopped writing for a time. Back in Paris, he redoubled his efforts to learn mathematics. The affair of the eyebrow showed him how much work he had yet to do. In this sense, Leibniz was not led to calculus so much as he was driven from a mixture of ambition and embarrassment.

For Newton, already in possession of publishable material on calculus, there was no running back to Paris or ignoring correspondence. He continued to correspond with Hooke and others on his theory of colors, and the effect this had was to make him crawl further and further away from any possibility that he would publish his mathematical work.

Oldenburg and Collins had given Leibniz a letter to deliver to Huygens upon his return to Paris, and when he delivered it, Huygens gave him many suggestions on what he could read. Leibniz was reading Barrow’s book, which he had purchased in London, at the time and followed the meeting by seeking out the works of all the important mathematicians of his day—buying copies where he could, borrowing others, and transcribing information by hand. He read, absorbed, and sought the common threads in everything, and he made tremendous strides in the coming months.

Leibniz read the works of René Descartes, who had been such a profoundly important mathematician a generation before, and was even privy to some of Descartes’ unpublished writings. Leibniz read Bonaventura Cavalieri’s 1635 book, Geometria, in which he had developed new ways of analyzing geometrical shapes—a method of finding areas and volumes of geometrical shapes, which could be considered precursor work to calculus. Leibniz read Evangelista Torricelli, who developed methods for finding areas under parabolic curves and rendered a clear explanation of them. He read Gilles Personne de Roberval and Blaise Pascal, whose work on indivisibles and infinitesimals anticipated integral calculus. Leibniz knew of Johann Hudde, who in 1659 had given his own rule for constructing tangents and for geometrically finding the maxima and minima of algebraic equations. And he read René François de Sluse, who had made a rule for constructing tangents to a point on a curve.

He had an incredible propensity toward mathematics, and his lack of formal training in the subject probably helped him in the long run, contributing to the originality of his work (though it may have hurt him in the long run as well, since his lack of training also predisposed him to making errors). Mistakes aside, by the end of 1673, Leibniz had developed a way to use a series of rational numbers to find the solution to a problem that had vexed his contemporaries for a few years—the squaring of a circle, or a square equal to the area of a circle. Huygens described Leibniz’s solution as being “very beautiful and very successful.”

That was not all. Leibniz realized that Pascal’s work could be combined with Sluse’s tangent rule and applied to any geometrical curve, not just a circle. That is what led him to calculus.