Purged of Ambiguity - 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 12. Purged of Ambiguity

1716–1728

Death troubles himself neither with the execution of our projects, nor with the improvement of science.

—Leibniz, from a letter to Thomas Burnet, 1696

Toward the end of Leibniz’s life, as the battle with Newton was reaching full throttle, it had the potential to take on an increasingly political tone, as his boss was now King of England. But anyone who might have assumed that George I would have more reason to side with Leibniz was completely wrong. Newton was a Whig, and the Whigs were generally loyal to the House of Hanover, so Newton was surely okay as far as George I was concerned.

In fact, the attitude of George I toward the calculus dispute seemed to be one of indifference—not so much out of a lack of interest but more an indifference that comes from knowing that, regardless of whom was right in the dispute, he was lord of both participants. “I think myself happy in possessing two kingdoms, one in which I have the honor of reckoning a Leibniz, and in the other a Newton, among my subjects,” he once said.

Besides, George had a strange relationship with Leibniz, ever leaning on him to stop stalling and complete the history of his family. Because of this and other reasons, Leibniz spent his dying days in Hanover while George and most of his court were in England—an abandonment perhaps, or something that shows lack of favor at the very least. Perhaps more revealing of their relationship is an incident that occurred in 1711. When Leibniz injured himself in a fall that year, sick, old, and partly crippled man that he was, George is said to have been amused and even saw it as fitting that it happened. He fell well short of benevolence toward his family’s longtime employee.

The injury was just one in a long line of physical insults that Leibniz would endure in the final years of his life. Leibniz was suffering from gout, which is an extremely painful form of arthritis caused by the buildup of needlelike crystals of uric acid in the connective tissues and joints. These buildups cause inflammation and shooting pains in the joints, and such attacks can take days to subside. Toward the close of Leibniz’s life, his gout worsened. “I suffer from time to time in my feet; occasionally the disease passes into my hands; but head and stomach, thank god, still do their duty,” Leibniz wrote in 1715.

He also developed a nasty abscess in his right leg that made it difficult for him to walk, perhaps because of his tendency toward a lack of movement. He is said to have often been given to sitting for hours—sometimes days on end—working from his chair.

Nevertheless, he never let the pain get the best of him. He would deal with the attacks by lying perfectly still in bed and at times by tightening wooden vices around the affected joints. Unfortunately this, apparently, damaged his nerves so badly that he became permanently bedridden.

In November 1716 he lay in bed for eight days, finally agreeing to see a doctor, a Dr. Seip, on Friday the thirteenth. One history paints an interesting picture of the patient as a living encyclopedia, with an in-depth knowledge of the art and application of medicine, discussing alchemy and history with the doctor while he was wracked with pain and his pulse weakened. Leibniz had broken into a cold sweat across his forehead and was perspiring profusely. He was shaking uncontrollably, surrounded by books and notes and other work, and though he tried to work, he could not write anything.

The doctor gave a dire prognosis: Leibniz would surely have no chance of recovery. He gave him some medicine. Leibniz lasted through to the next day, and on November 14, 1716, this most famous son of Leipzig died in his reluctant longtime residence of Hanover.

His coffin had to be built, and Leibniz’s secretary, Eckhart, ordered an ornately designed and expensive one that was decorated with lines from Horace, symbols of mathematics and rebirth. The funeral was a couple of days later, after which Leibniz was transferred to the Neustädter church where he was to be buried. He was buried inside the church, which was rare for a commoner back then. There is a sandstone marker with the inscription “Ossa Leibnitii” over what are today believed to be his remains.

Leibniz’s star grew in brightness after his death. In the eighteenth century, he was regarded as a very important intellectual, and a monument was erected in his honor around 1780, which was again extremely rare for a non-noble. This is described as a circular temple with a white marble bust in the middle and the inscription “Genio Leibnitii.” A measure of his worth was that, years later, when the church was renovated, the bones of the people buried inside it were exhumed. Only Leibniz was reburied within the renovated structure.

Still, many historians have commented on the paltry attendance at his funeral. A man named John Ker, of Kersland, who happened to arrive in town the day that Leibniz died, was apparently struck by the lack of attention paid by the locals. He commented, apparently, that Leibniz was buried more like a common thief than one of the ornaments of his country.

Most of George’s court was in London but the king and his entourage were hunting nearby when word reached them of Leibniz’s death. History records that, despite the fact that the entire court had been invited, the members of the court, most notably George I himself, did not attend it.

Several obituaries appeared in honor of Leibniz. The Journal des Savants published an account of his death in 1717, and another publication in the Hague appeared in 1718 with an “Éloge historique de M. de Leibniz.” The Académie des Sciences in Paris took notice, and the secretary there wrote an eulogy to Leibniz that he read to the members in 1717.

The Royal Society gave no notice of Leibniz’s passing, however, even though he was still a member. But perhaps the greater insult was that the Society of Sciences in Berlin did nothing to mark the occasion, even though Leibniz had been its first president and founder.

Shortly after Leibniz died, the Abbé Conti wrote to Newton to inform him of the fact. “Mr. Leibniz is dead,” Conti wrote, “and the dispute is finished.” But it was not nearly over for Newton.

As soon as Newton heard that Leibniz was dead, he pushed a reissued edition of Raphson’s book into print, and into this he inserted his own words in response to the letter Leibniz had sent him. Newton’s feelings toward Leibniz did not seem to soften with the passing years, not even after the death of his archrival. Two years later, the Englishman wrote a long, gloating passage about how Leibniz had never been able to refute his arguments. He continued to write bitter letters and treatises for years after Leibniz’s death, though he kept many private and those were not discovered until after his own death a decade later.

The letters that were in his possession when he died reveal how deeply wronged he felt by the whole affair, that he had been unfairly treated by Leibniz. He maintained to the grave that Leibniz was the aggressor and he, Newton, was the one who was defending himself from accusations of plagiarism. There can be but one true inventor of anything, Newton insisted, regardless of who improves upon the invention.

He was quite successful in spreading his belief in his greatness to the detriment of Leibniz’s—as were his followers. Voltaire, of course, was Newton’s greatest champion in France. After spending a few years in England, he wrote a number of essays that extolled Newton and Newtonianism, including one of the first popularizations of the Englishman’s ideas. Voltaire was rather unsparing in his treatment of Leibniz and his philosophy, many years after the older man had died. Leibniz was spoofed and ridiculed by Voltaire as the silly Dr. Pangloss of the novel Candide. His very name, Pangloss (broad summary), is a reference to the philosophy that came to oversimplify Leibniz’s outlook after he died—the notion of the best of all possible worlds.

Leibniz theorized that the total exclusion of evil in the world was impossible but that humans did live in the best of all possible worlds in the sense that the least amount of evil was allowed. Leibniz was not saying by “the best of all possible worlds” that every aspect of the world was perfectly without flaws. He was a witness to too many wars and too much suffering to think anything that stupid. All he was really saying was that, of the infinite number of possible worlds, this was the best. Suffering and the horrors of the world were part of a larger order, in Leibniz’s view, that remained harmonious. Moreover, he argued that the universe must be imperfect, because otherwise it could not be distinct from a perfect creator.

Though Leibniz was ridiculed by Voltaire’s cursory mocking of his philosophy, Bertand Russell, who wrote one of the definitive expositions on Leibniz’s outlook, called it an unusually complete and coherent system. But, however admired by Russell Leibniz would become, and however simple and elegant a concept his best of all possible worlds was, its Hollywood-style simplicity came to represent Leibniz’s philosophy after he died, and the phrase “the best of all possible worlds” became a mantra that was to tar and feather many aspects of Leibniz’s work in the eighteenth century and beyond. In the years immediately following and for centuries, he suffered from the perception that he was overly optimistic—that he was, in the words of one historian, the best of all possible worlds.

Even in the twentieth century, the best of all possible worlds is still the subject of some amusement. In Woody Allen’s Love and Death, Diane Keaton’s character holds up two dried, perfect leaves, comments on their beauty, and says that their beauty demonstrates that this certainly is the best of all possible worlds.

“It’s certainly the most expensive,” replies Allen.

Being mocked by Voltaire was certainly not the only knock that Leibniz took. For a century after he died, he was something of a pariah in England for his dispute with Newton and for his earlier opposition to John Locke, both national heroes.

Newton was the last man standing in the calculus wars, and he lived for another decade after Leibniz died. As an old man, he became a scientist of celebrity status in England and his fame spread abroad. Newton spent his autumn years constantly sought after by intellectuals and the well-to-do from England and abroad, who were excited to meet one of their heroes and one of the great minds of all time. Some of the scholars who visited him moved back to Europe, where they continued to champion his work.

Thus, Newton became more and more appreciated for his books, Principia and Opticks, in the last decade of his life, and he oversaw the publications of new editions of them. In the 1720s, his physics works were translated and lauded throughout Europe, and, in the decade after the calculus wars were cut short by Leibniz’s death, his work in mathematics began to catch on outside of England.

It first happened in Holland. Even though England and Holland had fought more than one war in the seventeenth century, the rise of William of Orange to the throne of England had warmed relations dramatically. Besides, the Dutch were now free of the French and German bonds to Descartes and Leibniz, both of whom were threatened by Newton and his philosophy.

Hermann Boerhaave taught at Leiden, in Holland, and wholeheartedly embraced and disseminated Newton’s philosophy. He called Newton the “Prince of Philosophers.” Another advocate was Willem Jacob Gravesande, who has been called Newton’s great Dutch popularizer. Gravesande also taught at Leiden—thanks in no small part to Newton, who had helped him obtain the position in 1717.

Even in France, with its long history of warfare and animosity with Great Britain, Newton was making headway—despite the fact that Opticks and Principia both were major challenges to aspects of Cartesian philosophy, and anti-Newtonianism had naturally arisen to counter the threat. The cooling of these tensions began in 1715, when an eclipse that was not visible in Paris but was in England brought a group of prominent intellectuals to London. Newton, as their gracious host, arranged for them to witness his optical experiments. He also saw that they were duly elected to the Royal Society. So full of gratitude was one member of the group, Pierre Rémond de Monmort, that he sent Newton fifty bottles of French champagne.

France began to warm to Newton after he was ultimately proven correct in one of his theories—namely that the earth is not a perfect sphere but an oblate spheroid that is flattened at the poles. In 1736, Pierre-Louis Moreau de Maupertuis went to Lapland to measure a minute of arc along the meridian. His careful survey proved Newton was right, and Maupertuis became Newton’s champion in France—so much so that he was dubbed Sir Isaac Maupertuis.

By 1784, Newton’s fame in France had grown so much that several competitions were held to design a monument in his honor. One of these was won by a man named Étienne-Louis Boullée, who designed a cenotaph—a tomb in which Newton’s remains would not actually be held. It was a sphere several hundred feet high, with Newton’s sarcophagus in the middle surrounded by a massive space. Another competition, held by the French Academy of Architecture the following year, called for design proposals “dedicated to the glory of the great genius, ought not to be magnificent so much as imposing in its dignified grandeur and noble simplicity.”

After Newton died, he was the face of science, discovery, and other abstract notions of genius in the eighteenth century—much as Einstein was the face of genius in the twentieth—and his fame would continue to grow unabated. His image appeared in paintings, sculpture, and other art throughout the eighteenth century. Perhaps the most famous of all these statues is the one by Roubilliac that was erected on July 4, 1755, and now resides at Cambridge University. Newton is depicted standing on a pedestal in a loose gown, holding a prism and looking upward.

The well-to-do in Europe commissioned busts that they placed on their mantels or in other prominent places of display, and it became popular for people to have their portraits painted with such a bust in the background. Benjamin Franklin had one such portrait painted of himself.

The celebration of Newton appeared in literature as well as art. Joseph-Louis Lagrange, who is considered by some to be the greatest mathematician of the eighteenth century, called Newton the greatest and the luckiest of all mortals for what he accomplished. James Thomson wrote “A Poem Sacred to the Memory of Sir Isaac Newton” in which he referred to Newton as the all-piercing sage: “Shall the great soul of Newton quit this Earth/To mingle with the stars and every Muse/Astonish’d into silence, shun the weight/Of honours due to this illustrious name.” Voltaire put it simply, “Newton is the greatest man who has ever lived.”

Even in recent years, the accolades continue to accumulate. An “Address from the Masters, Fellows, and Scholars of Trinity College to a Conference in Jerusalem Commemorating the 300th anniversary of the Birth of Isaac Newton” in February 1943 stated that “Homage to Newton is homage to the spirit of pure science.” A few years ago, Time magazine named Newton the “man of the Seventeenth Century.” And on September 12, 1999, The Sunday Times (London) named Newton the “Man of the Millennium,” beating out other scientists such as Darwin and Einstein, as well as British politicians, poets, and patriots alike.

When Newton died, he left an estate valued at £32,000 that was willed to his closest living relatives, his half nephews and nieces from his mother’s second marriage. More valuable than this sizeable fortune, however, was his reputation. He had become a living legend and was a highly sought-after London personality. By the time he died in 1727, he was at the absolute height of his fame, and dying was the only thing left for him to accomplish.

Death came to Newton shortly after he went to London at the end of February, to preside over his last Royal Society meeting on March 2. He looked great, and apparently felt great as well. He told his nephew-in-law, John Conduitt, that he had slept nine hours straight through a few days earlier.

However, on Friday, March 3, Newton became ill and returned home to rest. Unfortunately, he waited a week before contacting a doctor. On March 11, Conduitt heard that his uncle was ill, and he sent for a Dr. Mead and a Mr. Cheselden. These medical professionals diagnosed a stone in Newton’s bladder, which probably caused Newton severe pain in his last few days. Despite the pain, he is said to have remained upbeat, and would smile while talking to visitors even as the beads of sweat rolled down his forehead. He seemed to recover slightly by the middle of the following week, and by Saturday, March 18, he was well enough to read the newspaper. Things were beginning to look as though he might survive the episode.

But by that night Newton was insensible, and he grew worse the next day, slowly succumbing over the course of many hours to his acute illness until he died at 1:00 A.M. on Monday, March 20, 1726. It was headline news in the British newspapers. One periodical declared Newton to be “the greatest philosophers and the glory of the English Nation.” James Thomson quickly composed and published his “Poem Sacred to the Memory of Sir Isaac Newton,” and, before the year was over, five separate editions of this poem had been published.

Compared to Leibniz’s, Newton’s funeral was an event for the ages. Newton had been larger than life, and he had a funeral worthy of such celebrity. He was interred in the nave at Westminster Abbey on March 28, 1726, where the kings and queens of England are crowned when they come to power and where they are buried when they die. Next to him lies the cream of the last several centuries of British society—architects, scientists, poets, generals, theologians, and politicians—and he is buried among the likes of Dryden, Chaucer, Charles Darwin, Henry VIII, and Cecil Rhodes, and Mary, Queen of Scots.

His pallbearers were England’s Lord Chancellor, the Dukes of Montrose and Roxburghe, and the Earls of Pembroke, Sussex, and Macclesfield. Along the procession, there were choirs and throngs of adoring masses paying their respects. The funeral mass itself was presided over by no less than a bishop.

Newton is buried beneath a marker on the floor of the nave—a big black stone that reads Hic Depositum Est Quod Mortale Fuit Isaaci Newtoni (The Mortal Remains of Isaac Newton). This stone is flanked by stones dedicated to the memories of Michael Faraday and James Clerk Maxwell—the highest company of British physicists.

An expensive monument was soon built in Westminster Abbey in Newton’s honor, and the dean of Westminster found a very conspicuous place for it in the nave. Fatio assisted Conduitt with the design and inscription for the monument, and it was erected in 1731. It is a grand affair—a full-size statue of Newton at rest, reclined on a stack of books that represent what his contemporaries viewed as his major contributions to human knowledge when he died—his still-famous books on physics and optics, and his now-almost-forgotten contributions to theology and the chronology of ancient kingdoms.

To Newton’s left are a couple of young angels displaying a diagram of the solar system. Above his head is a globe with a woman weeping atop it—Lady Astronomy, the queen of the sciences, in mourning. Beneath Newton rests a marble sarcophagus base with a relief work depicting either children or cherubs wielding the scientific tools of experiments that had made him famous: a reflecting telescope, a prism, a furnace, and a steelyard for weighing the planets, and money newly coined. One is decanting some liquid from one vial into another. Two youths stand before him with a scroll with a diagram of the solar system on it. Above that is a converging series.

The epitaph, translated, reads:

Here lies

Sir Isaac Newton, Knight,

Who, by a vigor of mind almost supernatural

First demonstrated

The motions and figures of the planets,

The paths of the comets and the tides of the ocean.

He diligently investigated

The different refrangibilities of the rays of light,

And the properties of the colors to which they rise.

An assiduous, sagacious, and faithful interpreter

Of nature, antiquity, and the holy scriptures,

He asserted in his philosophy the majesty of God

And exhibited in his conduct the simplicity of the gospel.

Let Mortals Rejoice

That there has existed such and so great

An ornament to the human race.

Born 25 December 1642 Died 20 March 1727

A 1726 portrait of a surprisingly young-looking Isaac Newton at the age of eighty-three portrays the distinguished scholar in his robes shortly before he died. He is depicted seated at a table with a copy of the newly printed third edition of his famous Principia open on his lap. The picture is inspiring—one of the greatest mathematicians of all time together with his greatest work. Newton is to mathematics and physics what Elvis Presley is to rock and roll—the icon who practically invented iconography. And Newton’s Principia, his opus magnum, is a classic that ranks with Darwin’s Origin of the Species as one of the most famous and most influential science books of all time. It continues to be translated from its original Latin even today.

The third edition of the Principia depicted in the painting is truly a handsome volume. I examined a copy at the Wren Library in Cambridge, and was impressed by its beauty. The frontispiece is a portrait print of Newton from 1725. This edition has more extensive tables of data than did previous volumes. It also includes a page with Newton’s name and an homage to the king—George II, George I’s son, the second Hanoverian to rule England.

Newton rewrote this book throughout his entire life, and, through it and his other writings, he opened up whole new worlds of studies with his contributions to physics and optics, as well as invented the mathematical underpinnings needed to advance those disciplines. He developed mathematics as a way of rigorously describing physical phenomena—something that modern science takes almost for granted. Students of physics today may not ever read the Principia, but whether they know its text or not, the book has an indelible impact on their studies. Any student studying physics at the college level today will likely start the semester with a few weeks’ worth of what is now called either classical mechanics or Newtonian mechanics.

And yet something is missing from this third edition. What is not in the picture hanging at the National Portrait Gallery is any indication of Newton’s great rival Leibniz. Nor does the book that is open in front of Newton mention Leibniz’s name. In the first edition of the Principia, Newton had acknowledged that Leibniz had invented his own form of calculus and that Leibniz’s calculus had differed from his own only in notation and in the words they chose to describe this new branch of mathematics. That was in the 1680s. But for the second edition, which appeared in 1713, and for the edition of 1724, Newton had Leibniz edited out.

On display at the Leibnizhaus museum in Hanover, Germany, is a portrait of Leibniz that was painted prior to Newton’s. Leibniz’s shows him with a serious gaze and slightly furrowed brow. He has a bulbous nose, a slight double chin, a large head, and an even larger wig—a big, black, curly affair. One eyebrow looks ever so slightly raised, almost as if he is slightly amused. Or is he annoyed?

Leibniz left many things unfinished in his life—some, like the history of George I’s family, were left to future generations to complete. When the books were finally released, it was not due to some overwhelming interest in the history itself but because of people’s interest in publishing Leibniz’s complete works. Other designs, ideas, and dreams of his will never be realized. He left a trail of these incomplete projects in his wake: the failed windmill project for the mines, advanced watches that he never built, his never-completed alphabet of human thought, new mechanical engines that never advanced beyond theory, and some swift carriages he dreamed up because the roads throughout Europe in his days were terrible

Ironically, despite all these unfinished projects, it was one of his most successful inventions, calculus, that would wind up defining failure for Leibniz. Had he been born at another time and accomplished the sorts of things he did without being under anyone else’s shadow, he would be remembered now as the greatest mathematical and scientific mind of his day.

Leibniz was a mathematical novice who became, of his own volition, a math wizard. He was revolutionary for creating binary mathematics and advocating its use. He developed the use of determinants—a standard tool in linear algebra—and was of course revolutionary for both his invention and his dissemination of calculus. Indeed, he may have had one of the greatest minds of all time. He once boasted that he could recite almost all of Virgil’s Aeneid by heart (one wonders if even Virgil could have done that). He was an accomplished lawyer and advisor whose skills were highly sought after. He was one of the most important philosophers of his day, a father of modern geology, and an expert on everything from biology and medicine to theology and statistics. A pen pal to scientists, diplomats, kings, queens, clergy, and medical doctors alike, he maintained lifelong correspondence with hundreds of his contemporaries on every imaginable subject.

He may have known as much about China as any European of his day—its history, technology, culture, religions, and even its flora, and fauna—and yet he never went there. All his information was obtained through books and by corresponding with Jesuit missionaries in China.

In short, aside from being an expert mathematician, he was a polymath—a man who not only had an interest in many different fields of knowledge, but who could contribute advances to these fields—and has been called a universal genius.

But in 1700, when he was generally regarded as the sole inventor of calculus and commanded the respect of most of the leading mathematicians in Europe, he took a mighty fall. Perhaps his fault was that he underestimated the threat that Newton’s camp represented. He must have thought that he had truly invented calculus and had not borrowed anything from Newton, and that Newton himself recognized this fact. But in the years after Leibniz’s death, there were probably few who would dispute that, at the very least, Newton was the first inventor of calculus, and many would buy Keill’s argument that Leibniz may have indeed borrowed some of his calculus from Newton.

Did Leibniz lose the calculus wars?

In one sense, he did.

His life and legacy were both indelibly marked by the dispute, and even though he still had his supporters among the cadre of mathematicians he influenced and those mathematicians who followed them, that facet of his star faded after he died. He was never really able to promote his point of view concerning calculus’s origins to the extent that popular opinion reverted back to where it had been two decades before he died—when, prior to any publication of Newton’s mathematical discovery, Leibniz had been the unquestionable inventor of calculus.