The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time (2006)
Chapter 2. The Children of the Wars
1642–1664
Be it known to all . . . that for many years past, discords and civil divisions being stirred up in the Roman Empire, which increased to such a degree, that not only all Germany, but also the neighboring kingdoms, and France particularly, have been involved in the disorders of a long and cruel war.
—from The Treaty of Westphalia, 1648
Leibniz knew the stink and pain of war from having grown up in a land that was poisoned by it. He was born during one of the most horrible chapters in the history of Europe—the desperate and desolate times during the three-decade-long horror that was the Thirty Years’ War. It was a complicated, drawn-out affair involving multiple European states—Danish, Spanish, French, Swedish—that were vying for political gain and German land. The war was long enough that, by the time it ended, it didn’t matter much what the causes had been (a complicated fusion of territorial desire and Protestant rebellion). What mattered was that Germany had been utterly shattered by it.
One problem was that the burden of paying for the war was shifted in part from the countries commanding the armies to the lands where the battles were fought. It was no small price. During the Thirty Years’ War, assaulting towns and strongholds became difficult, making large, well-organized armies necessary. As a result, European armies swelled to sizes not seen since the times of Julius Caesar—many of the ranks filled with mercenary soldiers. But these large armies meant that there were suddenly tens of thousands that had to be equipped, fed, and perhaps most important, paid.
During the Thirty Year’s War, looting was the rule rather than the exception, as poorly paid soldiers would seek their recompense by sacking occupied towns. Moreover, looting became an outright policy for some of the warring armies, implemented so successfully by Sweden’s army that, in 1633, the army’s expenses cost a fraction of what it had in 1630. And the Swedes were not alone. A Bavarian monk named Mauros Friesenegger quipped, “On 30 September [1633] another troop of one thousand Imperial Spanish cavalry passed through. Although as new recruits they understood no military discipline, they did understand blackmail and robbery.”
Nor was the behavior confined to the rank-and-file. For years in the occupied lands, some of the armies’ top military and social ranks were occupied by individuals out for personal gain. In Wallenstein’s 1632 contract to become general of the Spanish-led army, he held the right to confiscate lands and to grant pardons.
When Leibniz was born in 1646, the war was almost over. He was born in Leipzig, which had been in the heart of the war. In fact, just south of Leipzig was Lutzen, which on the morning of November 16, 1632, about a dozen years before Leibniz was born, had been the site of one of the bloodiest battles of the war. Five thousand men were killed, including Gustavus Adolphus, the king of Sweden, who was cut down leading a blind charge through the fog into the opposing forces.
Two years after Leibniz was born, the war would finally end, when the treaty of Westphalia was signed. The treaty called for a “Christian and universal peace” and a pardon of all war crimes. If peace was Christian, the war had been anything but. Tens of thousands of villages and towns were ruined, and by some estimates, a third of all the houses in Germany were destroyed. Humanity was hit even harder. Perhaps a quarter of the population was killed, and many people were subjected to some of the worst forms of torture and cruelty.
During the Seige of Breisach in 1638, for instance, people were trading furs and diamonds for a kilo of wheat. According to a printed account, “News concerning the Great Famine and emergency that arose during the siege of Breisach,” all manner of animals were consumed. Meats that were palatable were sold at an incredible markup and those that were not were still consumed and sometimes traded. “Many mice and rats were sold at high prices,” the account reads. “[And] nearly all the dogs and cats eaten.” Late into the siege, the residents turned to cannibalism.
Cannibalism is the perfect metaphor for the Thirty Years’ War—Europe devouring itself. A man named William Crowne, who was traveling through Germany in 1636, wrote, “From Cologne to Frankfurt all the towns, villages and castles are battered, pillaged and burnt.” Industry and commerce did not recover until the eighteenth century, and it has been said that German economic development was thrown back one hundred years.
Leibniz was born at 6:45 a.m. on July 1, 1646, in a home near the University of Leipzig. His parents were Friedrich Leibniz and Catharina Schmuck, both moral and well-educated individuals. Catharina was the daughter of a “celebrated” lawyer in Leipzig, and Friedrich an ethics professor and vice chairman of the faculty of philosophy at the university. Friedrich had been married three times, Catharina was his much younger third wife. She was devoted to her two children, Gottfried and his sister.
Legend has it that Leibniz opened his eyes upon the baptismal font, which his father took to be a prestigious sign of the goodness of his being. “I prophetically look upon this occurrence as a sign of faith, and a most sure token,” Frederick wrote, “that this my son will walk through life with eyes upturned to heaven . . . abounding in wonderful works.” Later in life, Leibniz claimed that he had shown such an aptitude in learning that, even by the age of five, his father was indulging the “brightest anticipations of my future progress.” Unfortunately, these anticipations were all that Leibniz’s father would ever have. He died in 1652 when his son was only six years old.
One of the things that Friedrich left behind was a library of books—though Leibniz was not given access to these until an incident with his headmaster at his grammar school. One day Leibniz found two books that had been misplaced by an older student, and he began to read them. Leibniz’s headmaster was shocked. Though the books were good texts for an older boy, no boy Leibniz’s age should have been allowed such adult books, the headmaster believed. He confronted Leibniz’s mother, demanding that these books be taken from Leibniz at once.
Leibniz may have even been flunked into a lower grade had it not been for a chance benefactor, “a certain erudite and well-traveled knight,” as Leibniz described him. “He disliking the envy of or stupidity of the [headmaster], who, he saw, wished to measure every stature by his own, began to show, on the contrary, that it was unjust and intolerable that a budding genius should be repressed by harshness and ignorance.”
As it happened, this nobleman took issue with the headmaster, arguing that the boy’s acute interests in the advanced books was a sure sign of his budding keen intellect, which should be encouraged rather than stifled. The nobleman convinced Leibniz’s relatives not only that he should not be punished for reading the inappropriate book, but that he should be allowed to read all the books in his father’s library at his leisure. “This announcement was a great source of delight to me, as if I had found a treasure,” Leibniz would write years later in his personal confessions. So, at eight years old, Leibniz was allowed to enter his father’s study. He found books by Cicero, Pliny, Seneca, Herodotus, Xenophon, Plato, and many others, and he was free to avail himself of all the Latin classics, metaphysical discourses, and theological manuscripts shelved there. “These works I seized upon with the greatest avidity,” Leibniz said.
Being alone in the study—alone with the books—also awakened in him a love for contemplative independent learning, the sort that he would employ throughout his entire life. He spent many an hour studying the treasures of this library, and he began to read more Latin than a busload of pre-law students at a debate camp. By the time he was twelve, he boasted years later, he “understood the Latin writers tolerably well, began to lisp Greek, and wrote verses with singular success.” His Latin was so good, apparently he was able, at the age of eleven, to tackle a difficult assignment in composition in the ancient language in a matter of just a few hours’ work. The assignment was to deliver a poetic discourse in the place of his schoolmate, who had fallen ill. “Shutting myself up in my room,” Leibniz said, he was able to compose straight through, in a single morning, “three hundred hexameters, of such a character as to gain the [praise] of my instructors.”
He was not exposed to mathematics to any significant degree in his early schooling and, as a young man, he would have to teach himself mathematics. He and Newton were similar in this respect.
Their lives paralleled each other’s in another way as well. Isaac Newton was also the son of a torn land. In the seventeenth century, England was something of a European oddball in that it was never sucked into the Thirty Years’ War—largely because of its geographic isolation from the continent. Britain was also different from many countries in Europe, which were becoming highly centralized states led by a supreme ruler. Instead, it was already highly centralized. If anything, the British monarchy was in danger of losing power rather than consolidating it.
When Newton was born, England’s King Charles I had a precarious hold on power. In fact, the country was slipping rather hastily from his grip. The king was warring with Parliament, quite literally, and he resented the check on his power this body represented. He believed in the divine right of kings and thought that he shouldn’t be second-guessed or embroiled in petty disputes with Parliamentary officials. During one period of his reign, in fact, he had dissolved parliament for more than a decade, beginning in 1629.
His clash with parliament presented the king with a major financial crisis, though, because the political body had one power the king did not—voting for taxes. He survived for a while by raising fees and fines, but in 1637 a Scottish revolt forced him into the position of needing to raise an army, so he called parliament again.
Five years later, a few months before Newton was born, a civil war between the royal and parliamentary forces erupted. Parliament assumed control of the British fleet, all the major cities including London, and the lands surrounding London. It retained the ability to impose and collect tariffs and otherwise raise funds to supply the war. Charles, on the other hand, financed his armies by pawning off lands, jewels, and other assets. He even took loans from Spain to buy off the Scots.
At the beginning of the war, Charles enjoyed the advantage that his royalist troops were professional soldiers, whereas the parliamentary troops were a rabble. On January 4, 1642, confident in this advantage, Charles stormed parliament: armored, accompanied by armed goons, and intent upon arresting those members of parliament who had earlier defied him. But these opposition leaders were well informed as to the king’s movements, and they were gone by the time Charles and his entourage arrived. This was more than just an embarrassing mishap—it was a fatal mistake for Charles and his monarchy. By nightfall, many in the city had gathered together and taken to armed protest, practically making Charles a prisoner in his own castle. Crowds of zealots jeered outside the palace, and the cacophony was impossible to escape anywhere within the palace walls. The situation worsened, and Charles was forced to leave London and escape to more hospitable parts of the country—never to return except for his own execution.
The royalist troops drove up the Great Northern Road, which passed close by the farm where Newton’s mother sat pregnant with Isaac. Later the parliamentary troops marched up the same road in pursuit of the king. Though Charles’s troops may have been better trained, the parliamentary army led by Oliver Cromwell was disciplined and highly motivated. In the end, Britain’s king was executed on January 30, 1649, in London, and his son Charles II fled England a few years later.
Although Newton was born the same year the civil war started, another coincidence is more often noted by his biographers—that Newton was born the same year Galileo died. This has been hailed as significant because in a sense, Galileo was Newton’s scientific godfather. Newton would follow in Galileo’s footsteps and ultimately describe, using mathematics of his own invention, the physical universe Galileo had observed with his telescope, although an inconvenient fact for anyone who embraces this romantic notion is that Newton was born on January 4, 1643, according to the Gregorian calendar—the year after Galileo’s death. England did not follow this calendar in the seventeenth century because Protestants there resisted what they perceived as catholic contamination.
What is perhaps more remarkable than the year of his birth is the way Newton came into the world in the middle of the night so tiny and premature that the women looking after his mother during the delivery thought it a foregone conclusion that he would die—after all, in those days more than a third of all children died before they reached their sixth birthday. Two of these ladies who were sent out to get some medicine for the infant didn’t expect Newton to live long enough for them to return. Little could they know that he would outlive them all, not dying until he was more than eighty years old.
Newton’s family was unremarkable and largely uneducated. His forefathers were yeoman farmers—not an uncomfortable but certainly a humble lifestyle. His father apparently could neither read nor write, and Isaac was the first Newton who could sign his own name. Newton’s father has been described as wild, extravagant, and weak, an interesting guy to know, perhaps, but by the time Newton was born, his father had been dead two months. Newton’s father, also named Isaac, died at thirty-seven, just a few months after marriage to Newton’s mother, Hannah Ayscough Newton. Hannah, the daughter of a slightly better family, was left as a pregnant widow with a small estate of 46 cows, 234 sheep, and a couple of barns full of corn, hay, malt, and oats near the English town of Westby, in the county of Lincolnshire.
When he was three, on January 27, 1645, Mrs. Newton remarried. Her new husband, Barnabas Smith, was an Oxford-educated clergyman who was the rector of a nearby village. Born in 1582, Smith was sixty-three when the marriage vows were exchanged. The reverend Mr. Smith had his own needs, and the new couple would soon have three children of their own—Newton’s half brother and sisters Mary, Benjamin, and Hannah Smith. Newton’s mother moved into the good reverend’s rectory in North Witham. For whatever reason, Newton did not fit neatly into this picture, so he was sent to be raised by his grandparents in nearby Woolsthorpe, a hamlet in the parish of Colsterworth, in Lincoln County, about six miles south of Grantham. Newton was apparently close to neither of his grandparents, and his attitude toward his stepfather Barnabas Smith was even more volatile. As a child he once threatened to burn his mother and stepfather alive “and the house over them.” Newton later regretted saying this, especially after the Reverend Mr. Smith passed away when his stepson was ten, leaving Newton a collection of a few hundred theology books.
At twelve, Newton went to grammar school in nearby Grantham. There, he studied Latin and a few other subjects and boarded in a house that was an apothecary shop owned by a Mr. Clark. It is here, no doubt, that Newton was first exposed to the mixing of chemicals—the spark that ignited his lifelong love for alchemy.
Years later, Newton confessed that he was extremely inattentive in studies and was a bad student. Nevertheless, his vast intelligence, which probably made him seem strange to the other boys, would have been obvious at his grammar school.
During this time he was not known to play much with other boys, but rather spent most of his free time by himself tinkering in his room drawing and constructing things. For instance, he was deeply impressed with a windmill being constructed nearby, and he determined to build his own, which he did and which which was said to be as good as the original. Not satisfied with the fickle blowing of the wind, he built an additional device that allowed a mouse to spin the wheel. He is said to have filled the room with hand-drawn pictures. He constructed a paper lantern that he could fold up and carry in his pocket when he wasn’t using it. He also attached the lantern to a kite and flew it at night. He made so many sundials and became so good at it that the neighbors apparently began to come over to see what time it was.
Newton also built doll furniture for a childhood friend—a Miss Storer, who was two or three years younger than Newton and the daughter of Mr. Clark, with whom he boarded. Miss Storer, whose first name has been lost to time, was later to become Mrs. Vincent and is perhaps most famous for describing the young Newton as a “sober, silent, thinking lad.” She later confessed to one of Newton’s earliest biographers that Newton had been in love with her, though Newton himself left no indication that he had any such feelings.
Building the doll furniture for Miss Storer may have been more interesting to Newton than was Miss Storer herself. This tinkering translated years later into science, and he spent a lifetime building contraptions and conducting experimental work at the same time he did the theoretical work for which he is now so famous.
But Newton had a lot of science and mathematics still to learn—learning that he was not about to do in his grammar school, where he was not exposed at all to any mathematics of consequence. Rather, his grammar school training included Latin and a little Greek. Newton did learn Latin well, which was to be important in his later career, since many of the books of his day were written in Latin.
Leibniz’s schooling was equally uninteresting. He later remarked that because his education in mathematics was so poor, his progress was retarded. The scholastic tradition of Germany at the time meant learning Aristotle and logic. Leibniz excelled at logic in school. He claimed that he not only mastered the rules of Aristotelian logic before any other students, but that he also saw some of the limitations of the system.
As a young man Leibniz was to rely upon the method of self-teaching that he cultivated during the years he spent holed up with all his father’s books in the old man’s library. He was the sort of scholar who threw himself into his work with abandon, gathering his knowledge from books. “I did not fill my head with empty and cumbersome teachings accepted on the authority of the teacher instead of sound arguments,” he said once. Another time he reflected that his greatest obligation to his early teachers was “that they interfered as little as possible with my studies.”
Leibniz followed in his father’s footsteps, studying academic philosophy and law at the University of Leipzig, and he defended his master’s thesis, De Principio Individui (On the principle of the individual), in February 1664, at the age of seventeen. Leibniz’s advisor Jacob Thomasius praised the seventeen-year-old’s thesis, declaring publicly that although he was only a teenager, he was capable of investigating anything, however complicated.
Newton was less gifted in practical matters. As a fifteen-year-old, he had to make weekly trips back to Grantham to conduct business. His manservant, necessary because of Newton’s young age, was supposed to offer young Isaac advice as he found his way into the world of commerce and adulthood. In fact, Newton was not interested in any such education, and he let the manservant conduct all the business while he busied himself in reading.
In 1659, when Newton was seventeen, he was pulled away from his studies to take over stewardship of his family farm. As the oldest boy, he was expected to become farmer and sheep rancher, and he would have to spend several angry months at home in miserable exile before attending college. But his complete inadequacy for this line of work soon became apparent. Newton’s scholarly disposition made him entirely unsuitable to be a farmer of anything but ideas. There is a painting described in a famous biography of Newton that was written in the nineteenth century that captures this time perfectly. It depicts the sheep wandering off, the cattle making feed of the growing crops, and Newton sitting distracted under a tree.
Finally his mother realized that he should be devoted to a life of the mind and sent him back to Grantham for nine months to prepare for university studies. His uncle, the Reverend W. Ayscough, was a Trinity College man and was determined to send Newton there as well. So it came that in June 1661, at the age of eighteen, Newton enrolled in Cambridge University’s Trinity College, which historian John Strype called “the famousest college in the university.” In these early days Isaac Newton and Gottfried Wilhelm Leibniz knew little of the mathematics that would eventually make them famous, they knew nothing of each other, and they were both inexorably heading toward a similar intellectual destiny.