The Shortest Possible Descent - 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 8. The Shortest Possible Descent

1690–1696

Men act like beasts insofar as the succession of their perceptions is due to the principle of memory alone . . .

—Leibniz, Monadology, published in 1720

It’s nighttime somewhere along the coast of Italy in the last decade of the seventeenth century, and a small ship bobs upon the Adriatic Sea with a small crew and few passengers, including one foreigner with courtly German manners and a quiet demeanor. The crew is worried. A storm, a storm is blowing! The ship is tossed, and all aboard are shocked, discomforted, fearful. The ship’s crew is probably cursing in five different languages before one of them finally says in Italian to his fellow sailors that the cause of the storm is their German passenger—a Protestant!

That Lutheran Judas has brought the wrath of God upon us! Throw him overboard! Throw him overboard!

But they note that the stranger is sitting quietly, like the calm eye at the center of a storm, passing something through his hand. What is it? A rosary?! Look at him praying! He must be a true believer. Let him live . . . He is one of us.

Strange as it sounds, the basis of this tale is true. To Catholics three hundred years ago, a rosary in the hands of a Protestant traveler was like a Canadian Flag on an American traveler today. Leibniz saved himself from murder at the hands of the superstitious sailors ferrying him between towns in Italy because, unbeknownst to the crew, he could understand enough Italian to know he was in danger unless he feigned Catholicism double-quick.

The event is one of the most interesting from a long trip that he took through Germany and Italy from the autumn of 1687 until the summer of 1690, to do the research he needed to write a short history of Ernst August’s family, the House of Brunswick-Lüneburg, which Leibniz had proposed doing a few years earlier within a few weeks of the failure of the project to drain the mines at the Harz Mountains.

Such histories were common in those days because the fortunes of the state ultimately depended on the fortunes of the noble heads of state. Nobility was hereditary, pedigree of utmost importance, and so genealogies became an important way of justifying, if not furthering, the sociopolitical positions of Europe’s leaders. In the seventeenth century, many scholars hired themselves out to noble patrons to research such family histories, often tracing the family back through the centuries to the Middle Ages or even earlier.

Because so much was at stake, flattery would often supplant history, as noblemen and women were often mapped back in lineage to Charlemagne, who by the seventeenth century must have had more descendants on paper than Genghis Khan. Many of these works were downright ridiculous. A Venetian theologian even claimed he could trace the royal Habsburg family back to Noah’s ark. Ernst August himself once received a bit of this kind of flattery from a Dutch nobleman who traced his line through Augustus Caesar all the way back to Romulus and Remus.

The duke was not so foolish as to believe all this, but it sparked in him an interest to learn the real history of his family. Other historians had asserted that the family was related to the House of Este, one of the oldest and most noble in Europe. If this were true, then it would lend a great deal of credibility to Ernst August’s ambition of furthering the fortunes of his family. In those days, one of the best ways of doing this was to show a noble pedigree, which in his case would have meant “Estefication,” but nobody had ever been able to prove the noble Brunswicks were related to the Estes.

Leibniz had a very realistic goal: He wanted to trace the family back about one thousand years, to AD 600, and fill in all the gaps in between. But to do this he would need to travel widely in search of sources in state archives and monasteries strewn across Germany and Italy; there was no way he could accomplish this by staying within Hanover. As soon as the mine project ailed and he was ordered to cease, Leibniz began petitioning Ernst August to allow him to undertake the research. He was not just seeking permission to travel and write, but paid expenses and a secretary.

Ernst August was sufficiently impressed by the proposal to approve Leibniz’s plans, appoint him court historian, and authorize him to research and write the history. This was a major coup for Leibniz. Finally he could travel, study, write, meet, and correspond with other scholars without having to worry about where his money was coming from.

Leibniz departed in the fall of 1687, and for the next two and a half years went to cities all over Germany, Italy, and throughout southern Europe: Bologna, Dresden, Frankfort, Florence, Marburg, Modena, Munich, Naples, Padua, Parma, Prague, Rome, and Vienna. Indeed, Leibniz was to happily indulge in this kind of travel for most of the rest of his life. He did so much traveling, away from his home base for weeks, months, or years at a time, that he designed and commissioned a folding leather chair to accompany him so that he would have a place to work wherever he went. This ornately designed chair had a seam that ran down the middle, the bottom struts hinged so that it could fold easily. This invention was characteristic of Leibniz—he was constantly trying to adapt the world to fit his desires or needs, and was not solely interested in how things were but how they could be. He lived a life that was not fit for the world at times, so he made his world fit for him.

On his travels, he took many detours and took in many sights, for instance climbing to the top of Mount Vesuvius in Naples and exploring the catacombs in Rome. He also met many people and discussed many subjects that were unrelated to his purpose—something that he was wonderfully happy about, as he indicated a letter to Antoine Arnauld: “As this journey has served to free me in part from my ordinary occupations, and to furnish my mind with recreation, so have I had the satisfaction of engaging in conversation with many gifted persons respecting science and learning.”

When he was in Rome, Pope Innocent XI died, and Leibniz schmoozed with the cardinals who came from France for a conclave. When a new pope, Alexander XIII, was chosen, Leibniz conscientiously wrote a long poem hailing him.

Several of the people he met, he would later correspond with for years to come. For instance, he met a Jesuit priest, Claudius Philip Grimaldi, who was about to depart to go to China as a missionary. Leibniz was very interested in Chinese things, and he believed that the Chinese language was based on a profound philosophy that had been forgotten—even by the Chinese themselves. For the rest of his life Leibniz was to have a singular passion on matters related to China and a cultural exchange between east and west, and so he relished his correspondence with Grimaldi.

Leibniz met the celebrated Italian doctor Bernardino Ramazzini, who has been called the father of industrial medicine. They held each other in very high esteem. Leibniz was a strong advocate of health care, and he believed it was the moral duty of governments to provide it. He heavily promoted a cure for dysentery that was found in a root from South America. Leibniz also advocated preventative medicine, once writing a memorandum in 1681 prompting military health through such peacetime activities as sports. He proposed the idea of health councils and strongly advocated the isolation of cases during disease outbreaks to curtail epidemics. After being encouraged by Leibniz, Ramazzini produced a statistical record about health in 1690 that was championed by Leibniz in Vienna and to some of his acquaintances in France.

Upon a trip to Vienna, Leibniz was given his first audience with the Holy Roman emperor, and he took the opportunity to pitch a wild ride of ideas, following them up with several memoranda. He was exploding with ideas: a tax on fancy clothes; street lamps for the city of Vienna (which was eventually carried out); central archives and libraries; major economic reforms; and ways to improve manufacturing.

The history of the House of Brunswick was in a certain way a tremendous success, and Leibniz was able to deliver on his promise to research the origins of the duke’s family. There had been a hypothesis that a marriage between a family in northern Italy and one in Bavaria had taken place several centuries earlier, which had involved one of the duke’s ancestors in the Guelf House. Following up on this, Leibniz tracked down old Este monuments and, in 1689, found a tomb in Modena that was engraved with the names of the deceased. He also located papers supporting the families’ legal connection, and so the physical and ephemeral evidence together served as a fairly good verification that the marriage had indeed taken place. By the beginning of 1690, he had pored over enough documents to make Milton go blind a second time, and was proud to report to Ernst August that he had firmly established the relationship of the duke’s family with the House of Este.

This effectively increased the prestige of the House of Brunswick, ultimately enabling the elevation of the Hanoveran dukes to the electorate of the Holy Roman Empire—one of the handful of German nobles who could vote for the Holy Roman emperor. (The emperor had been chosen, since the year 1356, by certain German princes who were known as “electors” and who fancied themselves heirs to the glory of the Roman senate among the plebian mishmash of the 350 other, mostly smaller, political entities that composed the Holy Roman Empire.)

Elevating Ernst August to elector was not a straightforward matter, as several of the other German princes opposed the move for a variety of reasons. Leibniz wrote a number of papers to support the Brunswicks’ cause, based on historical analysis, legal precedent, and diplomatic arm-wrangling. In all, for eight years, beginning in 1684, he was to work hard behind the scenes on the negotiations for the new electorate. Finally, in 1692, Ernst August achieved his ambitions and was made an elector, an honor his heirs would thereafter inherit without any need other than birthright for qualification.

In 1696, Leibniz was promoted to privy counselor of justice—an office of high rank probably awarded, at least in part, for his involvement in raising the duke to an elector. This resulted in the addition of a bonus to his salary.

From this perspective, Leibniz’s trip was a smashing success. Had he been able to stop working on the history at the point he established the connection between the duke and the House of Este, the project would have been a total success. But this was something he could not do. He still had to actually write the history of the House of Brunswick. Substantiating the Este connection had been only part of the deal. In January 1691, a year after he wrote to Ernst August from Italy, telling him the good news that he had established the favorable family tree, Leibniz now prepared an outline and presented it to the duke, saying that he estimated the history might take two years to write.

He had no idea what he was in for.

The project was a major undertaking and, even with the help of assistants, Leibniz was never able to finish it. In fact, the history dogged him for the rest of his life, and there was little in his later years that was not clouded by this incomplete assignment. The assignment took time away from his other studies in mathematics, physics, and philosophy, and when he was on his deathbed, in the throes of the calculus wars with Newton, the history was still hanging over his head like a paper sickle.

He expressed his frustration well in 1695, in a letter to a man named Vincent Placcius: “I cannot describe to you how distracted a life I am leading. I search for different things in the archives and look over old papers and manuscripts never printed, hoping to get some light respecting the history of the House of Brunswick. Letters I receive and answer in great numbers. But I have so much that is new in the mathematics, so many thoughts in philosophy, so numerous literary observations of other kinds, which I do not wish to lose, that I am often at a loss what to do first, and feel the truth of Ovid’s exclamation, Inopem me copia fecit [plenty has made me poor].”

In 1696, a premature report of Leibniz’s death was circulating in England. Hearing of this, Leibniz wrote a letter to Thomas Burnet in England, complaining about how busy he was: “If death will only grant me the time requisite for the execution of the works already projected by me, I will promise to enter upon no new undertaking, and industriously to prosecute the old ones; and even such an agreement would defer the end of life no inconsiderable period.”

Unfortunately, there would be no respite—more than twenty-five years later, Leibniz died still working on the bloody thing. It became his opus tedium and, later in life, he wrote to mathematician Adam Kochanski that the history was his Sisyphean stone to which he was bound. When Leibniz died, he had only gotten as far as the year 1005, and it would not be until more than a century after he died, that the history was finally published in three volumes.

Perhaps the reason for Bertrand Russell’s lament that much of Leibniz’s time in service of the dukes was a waste of time stemmed from the untold hours the German mathematician had spent during some of his most productive years, working on what seems now to be not only a mammoth but also pointless exercise in genealogical research. It’s true that establishing the Este connection helped in the elevation of the duke to electoral status, but the rest of the genealogical research did not do much to contribute to the family’s ultimate improvement—Ernst August’s son, George Ludwig, being elevated to a regent, which would happen when Leibniz’s third boss, this son, became George I, king of England, in 1714.

The decision to make George king came not from the ancient pedigree of the family but because of more recent ancestry and a solid Protestant pedigree. He was the great-grandson of England’s King James I, but, more important, he was thoroughly Protestant. And when he assumed the throne, what should have been a happy time for Leibniz—as one who was nominally in the inner circle of the court at Hanover—was in fact bitter. The writing of the Brunswick family history kept Leibniz away from the new court because George Ludwig used it as an excuse to not allow Leibniz to accompany him to England.

Even so, at least one interesting thing did emerge from the project. The preface, the Protogaea, which he wrote in 1693, was a fascinating natural history of the Earth and the region where the duke and his ancestors lived. In it, Leibniz delved into prehistory, going back before human creation.

In his Protogaea, Leibniz proposed that the planet was originally hot, and that it had cooled, formed a crust, and then water had condensed on its surface. He explained the influence of volcanic activity on geological history and sedimentation, discussed fossils, and anticipated Darwin’s theory of evolution by proposing that the earliest animals were marine and that land animals came later. One nineteenth-century commentator notes that the Protogaeacontains “the germ of some of the most enlightened speculations of modern geology.”

Leibniz published an account of his Protogaea in the Acta Eruditorum in 1693, but the essay itself was not published until after his death. Some writers have suggested that the project was the perfect example of a Leibnizian endeavor. “The mode in which he prosecuted his task, the immense gyrations of thought in which he indulged, the number of subjects which were successively taken up, the eagerness with which he pursued each, the gigantic scale on which he framed his plan, and not least of all, the scanty fragments he left of the whole, are so remarkably characteristic of his genius and his habits.”

NEWTON WAS ALSO experiencing the utmost highs and lows that his life had to offer, during the last decade of the seventeenth century. The Principia had been well received, and the year after it was printed, he had been elected to Parliament as a representative of Cambridge University. This election, which brought him to London, gave him a taste for public service that he would never lose. Newton also began to lobby his friends and contacts for a permanent administrative job. He tried to get John Locke to get him a position at the Mint in 1691, and another friend tried to get him a position at King’s College, London. His friend Charles Montague, who came to Trinity in 1679 and knew Newton’s genius firsthand, was also enlisted to get a post for Newton. Though he was unsuccessful at first, Montague ultimately did help to secure a governmental post for him.

In 1693, some of Newton’s mathematical work finally made it into print for the first time. He did not publish this work himself, but rather allowed John Wallis to publish it within some volumes of Wallis’s own mathematics. Wallis was a charming man and a brilliant mathematician, though perhaps a little flawed as well since he was first and foremost a British mathematician and went out of his way to promote the supremacy of British accomplishments. In Wallis’s 1693 and 1695 books, he devoted pages to Newton’s contributions, and compared fluents and fluxions to the calculus that Leibniz had published a few years earlier. “Here is set out Newton’s method of fluxions, to give it his name, which is of a similar nature with the differential calculus of Leibniz, to use his name for it,” Wallis wrote, “as anyone comparing the methods will observe well enough, though they employ different notations. . . .”

Wallis also referred to Newton letters of 1676 and said that in them Newton explained his methods to the German mathematician. This is a significant moment in the calculus wars because, in going over these passages, many of Wallis’s readers were for the first time encountering the notion that Newton had developed methods that, lo and behold, had actually preceded and were identical to Leibniz’s calculus. And with this revelation came the first suggestion that one man’s work was better than the other’s, because the much-respected Wallis championed the ease of Newton’s fluxions and fluents over Leibniz’s calculus. In one passage, for instance, Wallis wrote, “And although at first glance fluents and their fluxions seem difficult to grasp, since it is usually a hard matter to understand new ideas; yet he thinks the notion of them quickly becomes more familiar than does the notion of moments or least parts or infinitely little differences.” This claim did not do much to turn opinions away from Leibniz, but it was really the first salvo.

Some of the book’s readers on the continent were astounded by its claims regarding calculus. After all, Leibniz’s papers on calculus had been read all over Europe, and since Leibniz never mentioned in them anything about Newton, many Europeans didn’t know what to make of the subsequent British publication of the Newton’s earlier work. They still had not seen anything of Newton’s methods that Wallis was touting. Nor could any of these methods be found in print anywhere. Leibniz’s calculus, on the other hand, had been in print for a decade, and it was really starting to bear fruit, with the Bernoulli brothers and others learning, developing, and beginning to apply the methods to solving complicated problems.

Johann Bernoulli was a little peeved at what he saw as a slight against Leibniz, after he read the relevant sections in Wallis’s books. He wrote to Leibniz saying as much. Leibniz took the higher ground at this point. “It must be admitted that the man is outstanding,” he wrote to Bernoulli. But Bernoulli regarded Newton’s work as derivative of Leibniz’s, and he was so blunt as to suggest the possibility of plagiarism—that Newton had borrowed his ideas from Leibniz: “I do not know whether or not Newton contrived his own method after having seen your calculus, especially as I see that you imparted your calculus to him, before he had published his method.”

Leibniz was not silent about Wallis’s books and their treatment of his mathematics. He wrote a letter to Thomas Burnet complaining, “I am very satisfied with Mr. Newton, but not for Mr. Wallis who treats me a little coldly in his last [volume of] works in Latin, through an amusing affectation of attributing everything to his own nation.” In this letter, he employed a device that he would use for the next two decades every time the issue of calculus came up. “Ask Newton,” Leibniz essentially said. Newton knows—he’ll tell you.” And Leibniz apparently took Newton’s apparent silence on the matter as an acknowledgment that Leibniz was within his rights to claim his own independent invention of calculus.

Interestingly, Wallis was a much better ally to Leibniz than to Newton. Wallis was not out to get Leibniz, but saw him as a legitimately esteemed mathematician who made an independent discovery of calculus, and, in the last few years of Wallis’s life, he and Leibniz exchanged some eighty pages of letters. In actuality, Newton may himself have ghostwritten the personally praiseful passages that appeared in Wallis’s book. At the very least, Wallis was writing what Newton wanted him to write and, two decades later, when the calculus wars came to a head, it would be Newton who would point to Wallis for support of his case: Ask Wallis. He knew.

At this point Leibniz, for his part, was perfectly willing to put on a good public face and give Newton his due credit. In the early 1690s, Leibniz and Huygens were in communication again, until the latter’s death in 1695 cut short their renewed correspondence. Huygens wrote a letter to Leibniz after he had seen a volume of Wallis’s Algebra. In it, he told Leibniz, he had encountered “differential equations very much like yours, apart from the symbols.” Leibniz obtained extracts of Wallis’s book from Huygens in 1694, and after reading what was written of Newton’s work, wrote to Huygens saying, “I see that his calculus agrees with mine,” but adding that his own methods were “more fitted for enlightening the mind.”

Even so, there is some evidence that Leibniz didn’t want things to stand at that. An anonymous review of Wallis’s work appeared in the scholarly journal Acta eruditorum that treated Newton’s work as if it were merely a celebration of Leibniz’s skills as a mathematician. This review was most likely written by Leibniz himself, who loved to write anonymous scientific letters in which he both attacked other mathematicians and praised himself. (He once wrote an anonymous review of some of his own work in which he referred to himself as “the illustrious Leibniz.”)

Meanwhile Leibniz probably didn’t really see Newton as a threat because he was seeing tremendous success in his own intellectual endeavors—he was at the top of his game. Finally, in 1694, Leibniz had found a skilled artisan to help him perfect a working model of his calculating machine, which could multiply numbers up to twelve digits.

Leibniz published the fullest account of his philosophy in 1695 in a French journal under the title “Systéme nouveau de la nature et de la communication des substances.” It was his account of his metaphysics, which went all the way back to his logical studies as a college student and on which he had been working more or less continuously in the three decades since. This put Leibniz on the intellectual map of Europe. Many had already known of him through his vast correspondences and his various mathematical and philosophical papers, but the article really made his name well known. He became even more of a public persona for his philosophy after Frenchman Pierre Bayle wrote a dictionary and included a critique of Leibniz’s work in it.

Mathematically, on the continent, Leibniz was the grandfather of calculus—its utmost authority. When L’Hôpital planned to write a calculus book in 1694, he first wrote to Leibniz about it, spelling out some of the problems he intended to solve.

Had Leibniz chosen to attack Newton during the last decade of the seventeenth cetury, he surely would have won the calculus wars. Newton was not yet in his position of maximum strength as the president of the Royal Society, and he may well have never recovered if Leibniz, then at the peak of his fame, had come after him. But Leibniz would not have done this, because he felt no malice toward Newton at this point. He even wrote to Newton, in 1693, a letter full of praise and veneration for his esteemed colleague.

This brief exchange of letters that Newton and Leibniz made directly to each other was both friendly and meaningless. “How great I think the debt owed to you, by our knowledge of mathematics and of all nature, I have acknowledged in public also when occasion offered,” Leibniz wrote, opening up the exchange. “You had given an astonishing development to geometry by your series; but when you published your work, the Principia, you showed that even what is not subject to the received analysis is an open book to you.”

Leibniz added that he wished to see Newton continue with his studies of the mathematical nature of the world. “In this field you have by yourself with very few companions gained an immense return for your labor,” he wrote.

In Newton’s reply to Leibniz’s letter some six months later, he paid his correspondent an incredible compliment as a person he regarded as one of the top mathematicians of the day: “I value your friendship very highly and have for many years back considered you as one of the leading geometers of this century, as I have also acknowledged on every occasion that offered.” Also in that letter, Newton translated the anagram from the letter he had sent Leibniz in Paris two decades earlier, which the German was happy to finally receive. Clearly, Leibniz saw no need to challenge a seemingly chummy Newton at this time. He did not view him as a threat.

Johann Bernoulli, as one of Leibniz’s loyal followers, was not so willing to let things go. Bernoilli hatched the idea that he would reveal Newton’s inability to compete with his friend when it came to mathematics. In 1696, he issued a challenge called the “brachistochrone problem,” and addressed it, with no small amount of gregariousness, “to the shrewdest mathematicians in the world.” Individual copies were posted to Wallis and Newton in England, and Leibniz published an article on the problem in the German journal Acta Eruditorum, as well as had it advertised in the French Journal des Sçauans. Solutions were to be accepted up until the following Easter.

This sort of competition was one that Leibniz had established a few years earlier, when he issued such a challenge to Abbé Catelan in 1687. The problem was to find the curve along which a body would descend without friction and at a constant speed. Huygens, Leibniz, and the Bernoullis had all participated in it.

These sorts of problems served to demonstrate the power of calculus. Jacob Bernoulli had proposed a similar problem in 1690, and when Leibniz worked out the solution, he sent it to the Journal des Sçavans for publication in 1692. In his article, he touted the power of infinitesimal calculus to solve this and other problems with ease and speed. He sent another letter to the journal a few months later, and another in 1694 where he reiterated the power of calculus over Descartes’ inferior analysis. He also praised Johann and Jacob Bernoulli for applying his calculus, mentioned L’Hôpital and his work, and, interestingly, wrote that Newton had a similar method but used inferior notation.

In 1696, Johann Bernoulli wanted to test just how powerful Newton’s “similar” method was when he came up with the brachistochrone or “shortest time” problem. Bernoulli’s challenge was to determine the curved line that connects two given points, one not directly beneath the other, along which a heavy body falling under the influence of gravity alone would descend in the shortest possible time. This is a classic example of the type of problem that calculus could solve—a problem for which a general solution can be found that expresses the curve without defining any specific parameters of the problem, such as the mass of the object or the distance between the two points. And it was the ultimate challenge to test Newton’s abilities, since only true masters of calculus could possibly solve it.

The problem was a painful one, as I recall from my encounter with it in a junior-level physics class that I took more than a decade ago. I remember spending most of a Saturday working toward a solution, but I couldn’t get it right. A few days later, I showed up early to class and confessed to my professor that I hadn’t been able to solve it after exhaustive efforts. “Don’t feel so bad,” said my professor; “three hundred years ago, there were only three or four mathematicians in the entire world who could solve it.”

Actually only five mathematicians were able to solve the problem—or at least sent the solution back in the agreed upon time frame. These were, not surprisingly, perhaps the only five people on the planet who had mastered calculus: Leibniz, Newton, L’Hôpital, and the Bernoulli brothers. Newton, of course, had no problem solving it, and he did so with apparent ease. He received it on January 29, 1697, when, after working a full day at the mint, he came home and solved the problem in a single night, and sent his answer back to Bernoulli anonymously.

This fact was not missed on Leibniz, who gloated, “they only solved the problem whom I had guessed would be capable of solving it, as being those alone who had penetrated sufficiently deeply into the mystery of our differential calculus.”

The gambit failed to ferret out Newton as one with less skill in mathematical analysis, but it proved the supremacy of calculus. For Leibniz, calculus was an elite club of which he was the founder. He was not threatened by the fact that there was another member—Newton—across the English channel who had apparently come up with his own independent methods and was able to apply them with great success. The Leibniz school of calculus was dominant and rising, and, to him, the Newton school was . . . a footnote, really.

If anything, Leibniz rather pitied the man. After all, his own rise to intellectual supremacy in the early 1690s Europe had coincided with Newton’s deteriorating mental state. The British mathematician was not well, and rumors had spread through Europe that he had the worst possible illness a genius could have.

In 1693, Newton is reported to have had an almost complete mental breakdown, the cause of which has been the source of a great deal of historical speculation through the years. His symptoms, in modern terms, were insomnia, loss of appetite, memory loss, melancholia, and paranoid delusions. The delusions were manifested in letters he sent to his associates, and his insomnia and other symptoms are gleaned from those same sources. In one of these letters, he wrote that he had slept only nine hours over the course of two weeks. And he had refused food during that time as well.

Various reasons have been suggested for his illness, the most obvious symptom of which was his almost complete lack of sleep. Of the sleep, it must be said that Newton had been spending much of his time in excessive study, as he was wont to do, but even for a workaholic his sleeplessness was extreme.

Some have suggested that the lack of sleep was really just a symptom of a much deeper cause—Newton may have been suffering from chronic mercury poisoning. He certainly showed the symptoms—sleeplessness, digestive problems, loss of memory, and paranoid delusions. There is also no question that he was exposing himself to perhaps dangerous amounts of chemicals in the course of his alchemical experiments. In the late 1680s and early ’90s, Newton made experiments on different alloys of iron, tin, antimony, bismuth, and lead. His notes indicate that he was finding ways to combine different amounts of metals into alloys. He found one alloy, for instance—two parts lead, three parts tin, and four parts bismuth—that melted in the summer sun.

However, the case for mercury poisoning was weakened by the lack of additional symptoms that one would expect to accompany mercury poisoning severe enough to cause insomnia, including hard-to-miss symptoms such as gastrointestinal problems, gingivitis, neurological deficits, and chronic fatigue.

At least one psychiatric professional has argued against the mercury poisoning and instead in favor of the idea that Newton’s mental state was not toxic in nature, but rather psychological—that he suffered from manic depression (or bipolar disorder, as it is now called). Strong support of this theory may be in the fact that Newton seems to have suffered from insomnia many times in his life, which is consistent with manic depression and its tendency to manifest episodically throughout a person’s life.

Other signs from Newton’s childhood include things like the facts that he was often unkempt, a loner, shy, and did not seem to engage in any sort of recreational activity. His college years were marked by isolation, and manic depression may have been the root cause of many of the problems he had in his life—such as his battles with Hooke and Leibniz.

This analysis, while perfectly plausible, is impossible to prove. And it is far from the only theory out there. Another theory is that the mental breakdown was caused by a severe professional trauma that Newton suffered in 1692. According to legend, tragedy was a candle left burning and a window left open. One day in February of that year, he went to church and left a candle burning on his table. It somehow blew over without extinguishing and set fire to a ream of papers, including the sole copies of some of his valuable notes on optical experiments, physical observations, and other subjects that he had been perfecting for decades. Newton arrived home to discover that the fruit of many years’ labor had burned to crumbling black flakes of ash.

It’s not clear how much of an impact the loss of these papers had, but the theory is that it may have been the cause of Newton’s crumbling to the edge of sanity.. The loss of such an irreplaceable collection of notes comprising about half of his life’s work would certainly have been a devastating blow to any scientist before our age of backup disks.

Another version of the same story has Newton’s dog Diamond knocking the candle over onto the papers, again reducing them into ashes. In this account, Newton appears at the door like a swooning Southern belle with a British accent and laments, “Oh Diamond! Diamond! Thou little knowst the mischief done!”

As amusing as this latter scenario is to imagine, there is no evidence that Newton ever even had a dog named Diamond. The story may be no more accurate than the one about the fallen apple giving Newton the idea for universal gravitation. And there is evidence that the rumor of the fire itself was just that—a rumor. There was apparently a fire years earlier, in 1678, which had indeed burned some of his papers after Newton left a candle burning in the empty house, and there may have been some confusion about this when the rumor spread in the 1690s of a fire destroying a substantial quantity of his writings. In fact, the rumor became so overblown that at least one person reported that Newton’s entire house had burned down.

Another theory is that, fire or no fire, his incapacity was more to do with his relationship with Fatio, which had been growing more and more intense in the months leading up to the breakdown. A dramatic turn in their relationship occurred when Fatio fell ill with pneumonia in 1692, after returning to London from Newton’s Cambridge residence on November 17 of that year. “My head is something out of order, and I suspect will grow worse and worse,” he wrote to Newton. Fatio went on to detail his symptoms—a congestion that felt bigger than his fist in his chest—and he said that he tried all the normal medicines and treatments to no avail. “I have Sir almost no hopes of seeing you again,” Fatio wrote. “Were I in a lesser fever, I should tell you sir many things.”

Newton wrote that he could not even express how much he was affected to hear of it. He offered Fatio money, and wanted to keep him in Cambridge and nurse him back to health. “For I believe this air will agree with you better,” Newton responded, signing his reply “Your most affectionate and faithful friend to serve you, Isaac Newton,” and he sent it special delivery to London to Fatio, who by that time had already almost recovered.

Nonetheless, a few months later, Fatio wrote that he would like to take Newton up on his offer and stay with him in Cambridge—especially if he could be able to stay in the rooms next to Newton: “I should be glad to know sir what prospect you had before you of a way for me to subsist at Cambridge.”

Unfortunately, “The chamber next me is disposed of,” answered Newton. Still, he again offered to give Fatio money, an allowance, whatever it would take to get him to stay near him in Cambridge and make his stay there easier. To this Fatio replied, “I could wish sir to live all my life, or the greatest part of it, with you, if it was possible.”

Yet, instead, Fatio decided to leave England and return home to Switzerland. After two final meetings in May and June of 1693, he dropped out of Newton’s life—nearly for good. We will never know what passed between these two men. What we do know is that, in 1693, all intimate contact between the two came to an abrupt and final end, and this was about the time that Newton fell into a severe depression.

Whatever the cause of his madness, it manifested in strange ways. He sent disturbing letters to Samuel Pepys and John Locke saying that he had not slept or eaten in months; he wanted to cut off all correspondence with Montague, convinced that he was false; he apologized at length for minor snubs to which he had subjected Locke; and so on.

There is one final possibility to consider regarding Newton’s condition: that he was not poisoned by a toxin, wracked with depression, or overwrought at the loss of a friendship at all, but was quite simply being Newton. His sleeplessness might not have been a symptom of some underlying neurological defect but rather an ordinary bout of restless energy, the likes of which fueled him at many times in his life. Likewise, paranoid anger, which is often listed as one of his primary symptoms, was something that characterized many of his relationships. Not exclusive to the 1690s, the famous Newton temper was to rear its ugly head throughout his life. He struck up a nasty fight with the astronomer John Flamsteed, for instance, convincing himself that Flamsteed was to blame for his not being able to come up with an adequate theory of the moon’s motions. Newton had not been satisfied with lunar theory as it was laid out in the Principia as he wrote it in the 1680s, and he worked on improving it in subsequent years. In 1694, he began to use Flamsteed’s observations to elucidate the moon’s orbit. He worked on this on and off for several years.

This was to have been one of the first examples of what would become a standard sort of scientific collaboration between the theorist with the experimenter, the perfect marriage of theory and experiment. Newton, though himself a skilled experimenter, would act the theorist and apply his penetrating geometrical skills to the data sets that Flamsteed, the astronomer, would provide him with. The experiment, like the collaboration, failed—in large part because Newton was so overbearing that he spoiled their relationship.

But even if Newton did not have some sort of nervous breakdown, the effect on Leibniz was the same. He heard that Newton had had . . . something, and he was sympathetic for the man, whose greatness he recognized. Leibniz had genuine concern for his British rival.

This concern cropped up again a few years later in 1695, when Burnet, then royal physician of Scotland, visited Hanover and befriended Leibniz. When Burnet returned to Britain, Leibniz kept in touch with him and used him to feel the pulse of life and events in London.

Actually, he relied on Burnet to keep tabs on Newton as well. After their brief exchange in 1693, Leibniz took an occasional interest in Newton and his affairs, and had at least one more occasion to get a note to him through Burnet in 1696. Burnet reported back that its recipient was gracious and thankful for the letter but busy because he had just become warden of the mint, a position Newton had been trying to get for some time.

Newton’s friend Montague had written to him on March 19, 1696, with the good news of his appointment: “I am very glad that at last I can give you a good proof of my friendship, and the esteem the king has of your merits . . . the King has promised me to make [you] Warden of the Mint, the office is the most proper for you ’tis the chief officer in the mint, ’tis worth five or six hundred pounds per annum, and has not too much business to require more attendance then you can spare.”

Newton swore out an oath to keep secret the mint’s technology for making new coins, and he signed it on May 2—with that, he became the mint’s new warden. In this capacity, he would oversee an annual budget of £7,500, or the equivalent of more than £700,000 today (nearly a million and a half dollars). Plus, this job would bring him to London, which was a much more interesting place to live than Cambridge. Cambridge was a small town, whereas London was a major metropolis with a population of a half million.

Leibniz failed to see the value in this move, however, and he expressed regret that the job had apparently pulled Newton away from his serious work in science and mathematics. This was true, in a way. Although Newton did not give up mathematics entirely, his creative years as a working mathematician were now long gone. In the legacy of notes, unpublished papers, and other pieces of writing that he was to pass to his heirs upon his death, are many papers and letters written after 1696 that had related to mathematics, but most of these concerned revisions of the Principia and were far from original, new works. He did do a considerable amount of work in lunar theory, theories related to atmospheric refraction, and the determination of a form of solid of least resistance, which were all applications that heavily depended on his mathematical ability, but these were dwarfed by the considerable literature that Newton produced on matters relating to the mint, which he threw himself into despite Montague’s assurance that it was a job he could do with little effort.

Perhaps, though, Leibniz’s worry about Newton’s new career had little to do with the demands of the mint itself. Leibniz was no stranger to mint operations—at least in theory. He had drafted a memorandum for Ernst August years before, in which he proposed a new way of coining to take into account the fact that the Hanoverian region had some of the best silver. Leibniz had suggested introducing equivalent—as opposed to actual—weights for the coins. So a lighter coin from Hanover would be worth the same as a heavier coin from another region. This way, the value of the superior silver could be accounted for.

Leibniz may have been subtly or subconsciously referring to his own situation, pulled as he was away from more serious matters by the dreaded history that he was constantly having to work on. Perhaps he was simply expressing for Newton what he wished he could have for himself—freedom from the tedium of the history he was writing and the day-to-day petty intrigues of the court he served. Had Leibniz his choice, he would probably have preferred to spend his days conversing and writing on important matters. As it was, the court intrigues in Hanover at the time were enough to make a soap-opera-loving housewife blush.

The best way to illustrate that morally unhealthy setting is to illustrate it with a story, and the most intriguing one involving George Ludwig, the son of Ernst August, his wife, and his wife’s lover in the 1690s. It started when George Ludwig married his cousin, Sophia Dorothea, the daughter of his uncle, George William. George Ludwig was cold and stern, in contrast to his gorgeous and affectionate wife, who was said to be attractive and well loved, the only child of her parents to survive childbirth.

Court life in Hanover in the second half of the seventeenth century was grand, despite the wanton destruction wreaked in the rest of Germany during the Thirty Years’ War. Ernst August was said to have stables with six hundred horses, twenty coachmen, and dozens of smiths, grooms, horse doctors, and other helpers. The halls were filled with chamberlains, ushers, pages, physicians, fencing masters, dancing masters, barbers, musicians, cooks, gentlemen of the bedchamber, and others. Entertainment was lavish, and Ernst August turned Hanover into a lavish playground celebrating his tastes. Its features included a new palace, an Italian-style opera house, and a months-long carnival.

Sophia Dorothea arrived in all innocence to marry George Ludwig on November 21, 1682. Little did she know what misery lay in store for her. Her father-in-law had a mistress in residence, the Countess Platen, who plotted against her. In the seventeenth and eighteenth centuries, fashionable men of Europe had mistresses, and not to have one was considered strange—even unmanly. However, in Hanover, the nobles’ rompings tipped into incest. The countess enticed her sister to have an affair with young George. When he tired of her, the countess encouraged her own daughter (Ernst August’s daughter, George’s own half-sister) to become George’s mistress.

Into this incestuous scene rode Count Philip Christopher von Königsmarck, a dashing young noble from a well-to-do Swedish family. He was a friend of George’s brother Charles, and at a masquerade ball he met Sophia Dorothea whom, as fate would have it, he had met years before and nurtured a boyhood crush on. In 1688, he became smitten with her, and he returned a year later to settle down in Hanover, becoming a colonel in the service of the duke and settling into the welcoming arms of Sophia Dorothea, who had by then learned a thing or two about court life.

Unfortunately, Königsmarck was not a one-mistress sort of lad, and he slept with the much older Countess of Platen on the side. To her, he boasted of his affair with Sophia Dorothea. This was a very dangerous game Königsmarck was playing. His boss, Ernst August, was not a gentle and forgiving man, and Königsmarck was taking a great risk by sleeping with both his daughter-in-law and his mistress.

The duke was dangerous. He had an accomplice of his son Charles, who was involved in a plot to wrestle some inheritance from George, killed in a most heinous way in 1691. The plotter, a von Moltke, was “broken on the wheel” as they called it: His arms and legs were smashed to bits by a heavy cart wheel and then von Moltke was strapped prostrated to the wheel, which was raised on a pole and left in the sun so that he died slowly, his butt up in the air, with all the blood rushing to and swelling his broken limbs.

Königsmarck grew increasingly jealous of having to share Sophia Dorothea with George Ludwig, and when she had to spend much time in official duties during a three-month festival to mark the duke and duchess’s becoming electors, he flew into a rage. In such a state, he rejected the matronly Countess Platen as a poor substitute for his young and beautiful lover. He began to blame Platen for all his troubles, including some of his financial ones, and swore he would pick a fight with the countess’s son—a duel that would have been deadly for the boy because the dashing Königsmarck was a master swordman.

Rejected as a lover and threatened as a mother, Countess Platen had spies watch the lovers’ every move. When Sophia Dorothea and Königsmarck decided to cut and run away together, the countess told all to Ernst August. The duke flew into a rage and had his men intercept the dashing and desperate Königsmarck before he could rendezvous with his beautiful daughter-in-law. In the melee that ensued, Königsmarck was cut down and, as he died, the Countess Platen, who had been waiting in the shadows watching throughout the ambush, stood over him. It is said that, with his last breath, he cursed and spit at her, and that she dug her heel into his mouth and twisted aside his curses.

Such were the pleasantries of the intrigues at Hanover during the 1690s.

The official account of the Swede’s disappearance simply stated that Königsmarck had wandered off on that night, never to be seen again. But the damage was done. The lovers’ letters had been found and the scandal took on a very public flavor throughout Europe. In a stab at damage control, Sophia Dorothea was put on trial, and a divorce was granted on December 28, 1694. George Ludwig was now free to remarry. Sophia Dorothea, on the other hand, became a prisoner at a nearby fortress, and her children were taken from her. She lived thirty-two more years alone and forgotten—abandoned by her unfaithful husband, bereft of her murdered lover, and missed by her children.

In the years to come, George Ludwig grew to dislike his children—especially his son, who greatly resembled Sophia Dorothea. George was indifferent when his grandchildren were born, and by then was so nasty toward his own offspring that, a few years after he became king of England, he ordered his grandchildren, ages five, seven, and nine years old, forcibly removed from the children’s parents. His orders even went so far that his newborn grandson was ripped from his mother’s arms and died a few weeks later—possibly as a result of the heavy-handed act.

In this interesting yet empty dramatic scene that was Hanover toward the end of the seventeenth century, Leibniz languished with little if any intellectual company. He confided to Thomas Burnet in 1695 that he simply had hardly anyone to talk to and that, if it were not for his discussions with the aging Queen Sophie, Ernst August’s wife, he would have almost none. He had to rely upon his correspondence with people like Burnet for intellectual company. From his point of view, Leibniz could not wish that anyone else, especially a mathematician as brilliant and seemingly fragile as Newton, be mired in such tedious governmental intrigues. In 1696, the same year that Newton began working at the mint, a curious thing happened to Leibniz—he nearly married. While he was in Frankfurt at one point in his travels, some of his friends suggested he pursue a rich young spinster, and apparently he did make certain overtures, but it was probably closer to a legal negotiation than an intimate wooing, and nothing came of it. In the end the lady took her time to consider his proposal, and he lost interest. He was fifty years old at the time, and a lifelong bachelor. For my part, I can’t help but wish I knew more about her.