Reading Descartes, Leibniz underwent a strange transformation. While the unpublished writings of Descartes brought Leibniz into the fold of Cartesianism, the mathematics of Descartes also held a special attraction for him. Leibniz saw that Descartes had developed mathematical ideas and methods similar to those that he himself had been entertaining. In particular, Leibniz recognized that Descartes was so gifted as a mathematician that he could “use” calculus without being in possession of a complete methodology. In other words, Descartes had been able to find the derivative of a mathematical function without, perhaps, recognizing that he was doing completely new mathematics.
Leibniz’s perusal of Descartes’s manuscripts convinced him that no one could reasonably claim that he had stolen the calculus idea from Descartes. But a question remained in the minds of many people: Did Leibniz meet John Collins, a friend of Newton’s, during his first visit to Britain in 1673 and, through Collins, learn about Newton’s work on calculus? Or did this meeting take place in 1676, after Leibniz’s own published formulation of calculus? As one historical testimony maintains, “It seems that he only met Collins on his second voyage.”9 But the phrase “it seems” reveals historical doubt. The British were quick to attack Leibniz as a plagiarist, and the Germans defended him. Before long, the controversy about who had invented calculus took the nationalistic tones of England versus Germany.
French author Bernard le Bovier de Fontenelle, who wrote Leibniz’s eulogy, concluded, “If it was a theft, then it was a theft that only Leibniz could carry out.”10 In fact, the calculus of Leibniz is more comprehensive and fertile than Newton’s calculus. Whereas Newton approached the idea of the variation in a function in terms of bodies in motion and the concepts of speed and acceleration, Leibniz used the idea of mathematical infinitesimals in his approach. In 1684 Leibniz published a work entitled Nova Methodus Pro Maximis et Minimis (A New Method for Finding Maxima and Minima), the main application of differential calculus. Newton’s work on calculus, which he termed “the method of fluxions and fluents,” appeared in print only three years later, in 1687.
Leibniz strove to apply his new mathematics to metaphysics and theology. The infinitesimals he invented for his work—or, rather, adapted from the works of the ancient Greeks—held mystical powers in his eyes, and he hoped to use them in metaphysical investigations. While Newton, too, was a religious man, his calculus was purely a response to the needs of physics rather than anything metaphysical. Ultimately, both Leibniz and Newton are equally credited with independently developing the modern theory of calculus based on the work of Eudoxus, Archimedes, Fermat, Descartes, and other mathematicians.
But this was not the understanding in the late seventeenth century, as the virulent controversy raged in Europe. When Newton, whom we will soon meet, heard that Leibniz had published a paper on calculus and had previously made contact with British mathematicians familiar with his own work, he became testy and deeply suspicious. He immediately wrote Leibniz a letter about his own work and dispatched it to him through Henry Oldenburg at the Royal Society. Then he anxiously waited for Leibniz’s answer and explanation as to how, exactly, he had derived his method and whether he had any knowledge of Newton’s own work. But as fate would have it, the letter took a long time to arrive—about six weeks. Thus, when Newton received Leibniz’s response, he assumed that Leibniz had taken a very long time to craft an answer and became even more suspicious.
In Hanover, Leibniz received a second letter from Newton. Because the letter was forwarded to him from his previous address, there was once again a long delay in his response. Newton’s letter was written on October 24, 1676, while Leibniz was traveling, and arrived in Hanover in June 1677—eight months later! Leibniz replied immediately, explaining that he had developed his methods all on his own with no input about Newton’s work, but this second delay further convinced Newton that Leibniz was stalling. The controversy raged on, fanned by Newton’s anger.
In 1711 the Scottish mathematician John Keill published a paper in the Transactions of the Royal Society of London that squarely accused Leibniz of plagiarizing Newton’s work. Upon reading it, Leibniz wrote the Royal Society saying that he had never heard about Newton’s version of calculus before the publication of his own calculus. Keill wrote him back saying that Newton’s letters gave ample proof of plagiarism. Leibniz then wrote the society, asking that his name be cleared. In response, a committee of inquiry was established. As we will soon see, the committee’s report was written by none other than Newton himself.
In the meantime, Leibniz had to give up his beloved Paris and return to Germany. By late 1676 he was out of money, and Huygens’s attempts to help him gain acceptance into the French Academy of Sciences were in vain. Ironically, Leibniz would become the force that led to the founding of other academies of science in Europe, including the Prussian and Russian academies. Although the prime minister of Denmark offered Leibniz a position as councilor to the Royal Danish Court, Leibniz was not interested in relocating to Denmark. Before returning to his homeland, he spent a week in London, where he met with Boyle, Oldenburg, and Collins. Then he spent a month in Holland, where he met with Spinoza. In December of that year, Leibniz finally returned to Germany to assume the post of librarian, diplomat, tutor, court philosopher, and counselor to the Duke of Hanover.
His employer, Johann Friedrich, was chiefly concerned with reorganizing his army. The duke had recently converted to Catholicism and now zealously oversaw the adoption of his new religion by all his subjects. Leibniz admired his sovereign for his faith but disagreed with his politics. In fact, the Duke of Hanover was the military ally of Louis XIV, which made Leibniz uncomfortable. Still, ever since France’s attack on Holland a few years earlier, the House of Hanover had remained nominally neutral.
Leibniz wrote a thesis about the sovereignty of the German principalities and argued for a German federalism. He continued to push for a union of the religions of Europe and for a federalism of the entire continent under the supreme leadership of the Holy Roman Emperor—controversial ideas that nevertheless contributed to his appointment to the chancellery of Hanover in 1678. Later on he became involved in manufacturing and heavy industry. He tried to negotiate for the export of metals mined in Germany’s Harz Mountains and proposed to the Austrians that streetlamps in Vienna be fueled by rapeseed oil, which would save money and provide efficient lighting.
Leibniz remained a bachelor all through his life. When he was fifty, he finally proposed to a woman, but she took so long to consider his proposal that he decided to withdraw it. He died in Hanover at age seventy. Unlike Newton, who was buried in a place of honor—Westminster Abbey—Leibniz was put to rest in an unmarked grave. Only his secretary and a few bystanders attended the funeral.
ISAAC NEWTON
Like his mortal rival Leibniz, Newton, too, remained a bachelor throughout his life. And like Leibniz, he was also a man of many great achievements, each of which was monumental.
Isaac Newton was born on Christmas Day in 1642, the year that Galileo died. He came from a farming family in Woolsthorpe, in the county of Lincoln, England. Newton was a premature baby, and his mother once described him as having been so small at birth that he could fit inside a quart mug. Two neighbors who visited his mother shortly after his birth and then went away for a short while said they had expected him to be dead when they returned; but the baby survived.11
Isaac Newton was born in this house, Woolsthorpe Manor, in Lincolnshire, England. Today, it is maintained by the National Trust and is open to the public.
Newton’s father, also named Isaac, died at age thirty-seven and never had the chance to get to know his new son. When Isaac was three years old, his mother, Hannah Ayscough, married a much older man named Barnabas Smith, the minister of the church in a neighboring village. Upon remarrying, she left baby Isaac to be raised by his grandmother, Margery Ayscough. Hannah then moved away and, with her new husband, had three children.
As a child living wit
h his grandmother, Newton invented and built many mechanical toys, including a small flour mill. In one prank he put lanterns on kites and flew them up in the night sky to scare the villagers. Newton was on his way to becoming a farm boy. Later, when he was ten, his mother returned to the village after Barnabas Smith had died, and Newton was expected to stay in the village to assist her with chores.
In 1659 Newton’s mother yanked him out of the free grammar school in Grantham, which he had been attending. He then lived at home, part of a large household that included his mother, his grandmother, and his half siblings. But Newton was rescued from the unhappy fate of a farm boy by his uncle, William Ayscough, who had graduated from Cambridge and who recognized the boy’s promise. He convinced his sister, Newton’s mother, to send her bright son to be educated at that great university. In order to prepare for Cambridge, Isaac was allowed to return to the Grantham School to finish his education.
While attending school, Newton lodged with the family of the village apothecary, a Mr. Clarke, and fell in love with Clarke’s stepdaughter. The two got engaged, but in June 1661 Newton left for Cambridge and was soon so immersed in his studies and reading that he never married her—or anyone else. Newton supported himself as a student by doing menial work at the university but, nevertheless, excelled in his studies. His mathematics professor, Isaac Barrow—the first Lucasian Professor at Cambridge—taught geometry and used his own methods for finding tangents to curves and areas of geometrical figures. Not only would Newton surpass him by inventing calculus, a methodology for performing these very tasks in a systematic way, but he would eventually replace Barrow as the Lucasian Professor.
Newton was equally occupied with theology and alchemy, the forerunner of our modern science of chemistry. He was a religious man and spent much time during those years trying to make sense of the prophesies of Daniel and the Apocalypse. He also tried to determine the date of the creation of the world based on biblical writings, which he interpreted literally. At the university, Newton studied hard, but he also found time to relax, going to taverns and playing cards.12 Because Newton was secretive about his work, we don’t know about any discoveries he might have made during his period as an undergraduate. In 1664 he earned his bachelor’s degree, and then began his most fecund period.
Newton spent nearly forty years at Trinity College, Cambridge, first as a student and then as a professor. The rooms in which he lodged are located on the first floor in the low center building just to the right of the entrance gate.
DESCRIBING WHAT HAPPENED NEXT, Newton famously said, “If I have seen a little farther than others it is because I have stood on the shoulders of giants.” Presumably, the giants on whose work the modest Newton had relied included Descartes, Kepler, and Galileo. Cartesian logic led him forward, as did Descartes’s mathematical work. Galileo’s investigations of falling bodies, the pendulum, and other physical phenomena inspired his own theory of gravity. And Kepler’s laws of planetary motion were later abstracted by Newton, making them corollaries of his laws of universal gravitation. Standing on the shoulders of his giants, Newton saw much farther than anyone. He was not a man of universal interests, as was his contemporary and calculus cofounder Leibniz, but in the realm of physics and mathematics, Newton’s intellect was supreme.
In 1664 Britain was ravaged by bubonic plague, and Cambridge University was closed down. Newton left for Woolsthorpe, where he spent two years alone, thinking about the universe and its laws. It was here, for the purpose of explaining the laws of gravitation he deduced from physical investigations, that Newton invented calculus. Newton viewed variables as flowing quantities, and to describe this flow—the rate of change of a quantity with time—he devised what he called the “method of fluxions.” Before he achieved this great breakthrough, he generalized the binomial theorem—the rule for operations such as squaring the sum of two quantities—by extending it to higher exponents and also to cases where the exponent is not a positive integer. In such cases the series is infinite, and proving the result is much harder.
Newton’s law of universal gravitation states that any two particles of matter attract one another gravitationally with a force that is proportional to the product of their two masses and inversely proportional to the square of the distance between them. The constant of proportionality in the equation is known as Newton’s constant, G. Newton also developed laws of motion, which include the following:
1. Every body will continue in its state of rest or inertia (unaccelerated motion) in a straight line unless acted upon by a force.
2. The rate of change of momentum (mass times velocity, in Newtonian physics) is directly proportional to the force acting on a body and inversely proportional to the mass of the body.
3. For every action, there is an equal and opposite reaction.
In his laws of motion, Newton neatly outlined the relationship between the acceleration, velocity, and position of an object (i.e., acceleration is the rate of change of velocity, and velocity is the rate of change of position). These rates of change became the derivatives of differential calculus, and in order to measure the distance traveled by an object whose velocity changes over time, Newton invented integral calculus. Further, understanding that these two operations—finding a rate of change (i.e., computing a derivative), and finding a distance or area or volume (i.e., computing an integral)—are opposite operations, he formulated the fundamental theorem of calculus.
In 1684, in order to settle a dispute with fellow scientist Christopher Wren, Edmond Halley of the Royal Society—of which Newton was a member—asked him which law of attraction would explain the elliptical orbits of the planets. Newton replied that it would be an inverse-square law—that is, a law by which the force of attraction is decreased according to the square of the distance between a planet and the sun. “How do you know that?” asked Halley, to which Newton answered, “Why, I have calculated it.”13 In fact, Newton had performed a calculation in 1666 demonstrating that his law of universal gravitation led directly to Kepler’s laws of planetary motion. But he did not publish it until eighteen years later.
Why did he wait so long between derivation and publication? Although Newton understood that every point in a solid body exerted the same gravitational force on every point in the second body, it was not clear to Newton how to perform the computation for a large number of points. Eventually, he figured out that if he assumed that a body’s mass was concentrated at its center, he could calculate the gravitational force on another body.
In 1667 Newton was elected fellow of Trinity College, and two years later he was made Lucasian Professor of Mathematics at Cambridge, succeeding Isaac Barrow. Newton gave lectures on optics that included his own discoveries in the field. In contrast to the theory of Christiaan Huygens and Robert Hooke, which held that light was a wave, he espoused a corpuscular theory of light. Today we know that light is both a particle and a wave, as quantum theory has taught us.
In 1668 Newton built a new kind of telescope that used a reflecting mirror. (Such telescopes are still called Newtonian.) He then used his reflecting telescope to look at the night sky and discern the Galilean satellites. That year, Dutch mathematician Nicholas Mercator had produced a calculation of an area under the hyperbola using an infinite series that was closely related to Newton’s calculus. Since Newton’s method had not yet been published, the work of Mercator provided Newton with the impetus to circulate his own work among mathematicians at Cambridge.
In 1672 Newton was elected to the Royal Society. There, he communicated to the world of science his work on telescopes. He also read papers in front of the Society about his “particle” theory of light, prompting an argument with Robert Hooke about the nature of light. Other members sided with Hooke’s wave theory, and Newton’s letters to the Royal Society about the issue became angrier and angrier as time went by. In a letter dated November 18, 1676, he wrote, “I see a man must either resolve to put out nothing new, or become a slave to defend it.”14
Newton’s friend Edmond Halley eventually persuaded him to put his ideas to paper, lest credit for them go to other scientists and mathematicians. Thus in 1684 Newton began writing his masterpiece, Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy). Writing the Principia was a major undertaking, and it took a toll on its author, who worked for hours on end, sleeping little, waking up to write while still in bed, and eating poorly. It was a vast amount of knowledge, acquired over decades, that now had to be put down on paper.
The title page of the first edition of Newton’s Principia Mathematica shows that it was published in London in 1687.
In the dispute about who first discovered calculus, Newton prevailed over Leibniz—at least during his lifetime. Today, however, we recognize the contributions of both men to its development. This engraving depicts Newton at his prime.
Newton wrote about astronomy, physics, and the mathematics he had invented. But the main focus of the book was the dynamic structure of the universe—i.e., the solar system, with planets revolving around the sun, according to Newton’s law of universal gravitation. Newton’s calculus methods underlie the dynamics, but he reduced the calculations to geometrical arguments that could be easily understood. Newton deduced Kepler’s laws of planetary motion from his own law of universal gravitation, and he showed how to calculate the sun’s mass.
Newton also managed to show that comets’ return to Earth’s vicinity could be predicted by his gravity law, and he explained that Earth’s flatness at the poles was due to its rotation about its axis. Other observations tied to what he called the System of the World, under the rubric of his all-powerful law of universal gravitation, include tides as a result of the moon’s gravitational pull, the precession of the equinoxes, and the variation in the weight of an object with latitude. Newton’s Principia became part of the university curriculum in England around the turn of the eighteenth century, being taught at both Cambridge and Oxford. Meanwhile, news of his groundbreaking scientific and mathematical discoveries spread throughout the world.
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