Johannes Kepler, depicted in this 1610 portrait, served as Tycho Brahe’s assistant and mathematically deduced his famous laws of planetary motion.
1. Each planet moves around the sun in an elliptical orbit that has the sun at one of the two foci of the ellipse.
2. The radius line that connects the planet to the sun sweeps equal areas in equal time.
3. The square of a period of a planet is proportional to the cube of its orbit’s semimajor axis.
Kepler viewed the areas swept by the radius as comprised of an infinite number of tiny triangles, each with an infinitesimally small area. By “summing up” these areas mathematically, he was using a forerunner of the idea of integral calculus, which would be further developed by Descartes and later formalized by Leibniz and Newton.
After Tycho Brahe’s unexpected death in 1601, Kepler succeeded him as Imperial Mathematician to Holy Roman Emperor Rudolph II (the former king), a prestigious position he held until the emperor’s abdication eleven years later. In 1612 his wife and son contracted illnesses and died. That year, Kepler found an effective way to estimate the volumes of casks of wine. In carrying out this estimating procedure, Kepler went far beyond the work of Archimedes on volumes and, as he had done with areas, built an argument about the summation of an infinite number of elements of volume, each infinitesimally small—another close forerunner of integral calculus.
Kepler is also famously known for incorporating Plato’s five solids into a model of the universe. There were only five known planets at the time, including Earth, so Kepler concluded that there were five “separators” among their orbits around the sun. Thinking that perhaps he had found a cosmic meaning for the five mathematical elements of Plato—as the Greeks had done two millennia earlier by ascribing the elements of nature (earth, wind, water, fire, and quintessence) to these solids—he placed each planetary orbit on a sphere that inscribed one solid and was inscribed by another. It was a valiant attempt to impose mathematical structure on heavenly bodies, but—of course—it was false. (And it contradicted Kepler’s own discovery of elliptical orbits.)
Kepler used the idea of Platonic solids (see Eudoxus of Knidus) to create this diagram of the solar system, which appeared in his 1596 book Mysterium Cosmographicum.
By the end of the seventeenth century, not only did Newton and Leibniz incorporate the methods used by Kepler to find areas and volumes into integral calculus, but Newton’s laws of gravitation evidenced the overarching principles from which Kepler’s laws could be mathemati-cally derived—using the same calculus.
In 1617 Kepler’s mother, Katharina, an herbalist who had encouraged his interest in celestial events, was accused of witchcraft. Ursula Reingold, who was involved in a financial dispute with Kepler’s brother, claimed that she had contracted an illness from an “evil brew” Katharina had given her. Johannes went to great lengths to defend his mother in court, and after fourteen months in jail, she was released on technical grounds relating to the use of torture to extract “evidence.” (Not every accused witch was so lucky; of the fifteen women accused of witchcraft under the reign of Lutherus Einhorn, an overseer in Leonberg, eight were executed.) Kepler himself wrote horoscopes and indulged in mysticism, which, in combination with his advocacy for the Copernican heliocentric system, frequently landed him in political trouble. In 1625 measures enacted under the Catholic Counter-Reformation placed almost his entire body of work under seal. He died on a visit to Regensburg in 1630, where he had traveled to collect money owed him for one of his books.
GALILEO GALILEI
A contemporary of Kepler’s, Galileo Galilei (1564–1642) was born in Pisa, Italy, to a famous lutenist and musical composer. His contributions to physics, astronomy, and cosmology are immense. In 1609–10 Galileo used the recently invented telescope to look at the night sky and discovered the moons of Jupiter—now referred to as the Galilean satellites. This discovery lent much support to the developing arguments against Aristotelian philosophy and the Ptolemaic cosmological system, which placed Earth at the center of creation. Galileo also performed many experiments with falling objects—perhaps even dropping them from the top of the Leaning Tower of Pisa, as has been conjectured—which led him to discover that an object’s time of descent is independent of its mass. Galileo made important mathematical discoveries, too. By studying trajectories in the air, he was able to show that the path of a projectile in the air is a parabola. He also studied the cycloid—the curve traced by a point on the rim of a wheel as the wheel moves along a horizontal path.
Although he never married, Galileo fathered three children out of wedlock with a young Venetian woman named Marina Gamba. His two daughters were thus considered unmarriageable and were sent to a convent in Arcetri. Galileo’s eldest daughter took the name Sister Maria Celeste and remained devoted to her father until her death in 1634.
In pure mathematics, Galileo made a key discovery about infinity, referred to as Galileo’s paradox. He found that even though only some whole numbers are squares, and hence the set of whole numbers should be larger than the set of squares, he could set up a one-to-one correspondence between the set of whole numbers and the set of squares of whole numbers. Simply by pairing all positive integers with their corresponding squares—e.g., 1→1, 2→4, 3→9, 4→16, and so on to infinity—both sets could be “counted” against each other. This led him to the conclusion that there are “as many squares as there are numbers.”3 Later on, we will see why this discovery was so important in the study of infinity.
THE INFINITE HOTEL
In the twentieth century the German mathematician David Hilbert gave an entertaining example of the fact that an infinite set can be “counted” against a subset of itself, and thus both infinite sets can be “equal” in size despite the fact that one set contains the other. The example is called Hilbert’s Hotel, or the Infinite Hotel, and it proceeds thus: A person arrives late at night at the Infinite Hotel but is turned away.
“Sorry, we are completely full.”
The visitor protests, “But the brochure says that this is an infinite hotel—you have infinitely many rooms.”
“Yes,” responds the receptionist. “But all our infinitely many rooms are full.”
“Okay,” says the visitor, “this is what you should do: move the guest from room number 1 to room number 2; the one in room number 2 to room number 3; and so on. Since you have infinitely many rooms, you can do this for all your guests. Then room number 1 becomes available for me.”
Here, we have the one-to-one correspondence 1→2, 2→3, 3→4 … so the set of all integers and the set of all integers save the number 1 are equivalent to each other, and hence include the same number of elements in each (though it is infinite). Indeed, this trick shows–as Galileo had understood in the 1600s–that an infinite set can still be put into a one-to-one correspondence with a proper subset of itself.
As portrayed in this 1857 painting by Cristiano Banti, the great Italian mathematician Galileo Galilei faced trial by the Roman Inquisition because of his view that Earth revolved around the sun.
Galileo’s contributions to philosophy, mathematics, physics, and astronomy are far too numerous to list here. Despite his immense influence, however, his scientific discoveries led him to question scripture, and his publications incurred the ire of the Inquisition. In 1633 he was tried for heresy in Rome and condemned to house arrest for the rest of his life. He died in 1642, a broken man.
As the story of mathematics continues, we move from the Father of Modern Science to the Father of Modern Philosophy. René Descartes and his contemporaries extended mathematical principles to philosophy, and the seventeenth century saw the rise of rationalism, which anchored reality in geometry and the powers of deduction.
PART IV
TO CALCULUS AND BEYOND
René Descartes—famous for his dictum “Cogito ergo sum”—was hugely influential in mathematics and philosophy. His system of numerical coordinates served as the first codified link bet
ween algebra and geometry.
EIGHT
THE GENTLEMAN
SOLDIER
In the century after Cardano’s life, the great French philosopher and mathematician René Descartes (1596–1650) read the Ars Magna but wasn’t very impressed with it. He was much more enchanted with Galileo’s work; he felt a kinship with the Italian thinker—so much so that he worried he might encounter a fate similar to that of Galileo. Like Galileo, Descartes was engaged in research that was leading him in the direction of the Copernican system so abhorred by the Church.
On his own, this exceptional genius had generalized a number of Greek and Arab discoveries in mathematics and was on his way to unifying geometry with algebra. In a moment of almost careless doodling, he was able to solve one particular kind of fourth-order equation (although it should be noted that the Italian algebraists of the previous century could solve some such equations as well). René Descartes was almost too great a genius to even worry about solutions to equations; he had grander ambitions.
Nothing about René Descartes—the great French mathematician, philosopher, physicist, and natural scientist—is what meets the eye. In one of the greatest ironies in the history of ideas, the man who gave us the strict, perfectly logical Cartesian rules seems to have defied every rule of reason in his personal life.
RENÉ DESCARTES
René Descartes was born to a wealthy aristocratic family on March 31, 1596, in the town of La Haye (now named Descartes, in his honor), in the region of Touraine, France. His father, Joachim Descartes, was a councilor to the king of France, working on legal and legislative matters in the Parlement of Brittany in the city of Rennes, roughly one hundred miles to the west. The Descartes family was a major landowner in the agriculturally rich area of Châtellerault, and its members could easily live off the rents paid by their tenants without having to worry much about money.
In those days people were born at home, but the house in which Descartes was born was not his family home. Just as she was about to give birth, Descartes’s mother, Jeanne Brochard, left the family mansion in Châtellerault, in the region of Poitou—some twelve miles to the south—crossed the Creuse River, and gave birth at her mother’s house in La Haye. She didn’t feel comfortable having the baby in her own house, given that her husband was away.
At the turn of the seventeenth century, the two neighboring regions, Touraine and Poitou, were very different. Whereas Poitou was mostly Protestant, Touraine was predominantly Catholic—as is most of France today. This accident of birth—being born in a heavily Catholic part of France to a family that, although Catholic, hailed from a Protestant region—affected Descartes’s feelings about religion and society throughout his life. One key example of this would be his excessive, almost irrational fear of the Inquisition and what it might do to him if he published scientific writings contrary to Church doctrine; at the same time, he remained almost naively unprepared for the attacks on his views and writings that would come from Protestant theologians after he developed his theories as an adult.
Marin Mersenne (1588–1648), a monk, music theorist, and mathematician, served as an intermediary between Descartes and other scholars throughout Europe, including Pascal and Fermat.
Descartes was a weak child with many minor health problems, so from early childhood he had the privilege of sleeping late while the other schoolchildren were in class, joining them only when he felt ready to face the day. Not having to deal with the nuisances of school routine, the young genius was able to derive mathematical results while lying in bed. As he grew stronger, Descartes learned to fence, and he practiced swordsmanship throughout his adolescence and early adulthood.
Once he recovered from whatever had ailed him as a young boy, Descartes’s family sent him to study at the Jesuit College of La Flèche, in the nearby region of Anjou, to the north. At La Flèche, Descartes continued to practice his swordsmanship. His studies were focused primarily on the classics and Greek mathematics—especially geometry. What he loved most were the ancient Greek mathematicians’ constructions with a straightedge and compass, at which he excelled.
During his studies he met Marin Mersenne, who was a few years ahead of him at school and equally interested in mathematics and science. Mersenne, who would later become a Minim monk in Paris, became a lifelong friend and confidant of Descartes. Often, while Descartes traveled throughout Europe, Mersenne would be the only person who knew his whereabouts and could reach him by mail. And when Descartes wrote a letter to a mathematician, he usually used Mersenne as an intermediary.
Upon graduation Descartes moved to the big city: Paris. There he lived the exciting and carefree life of a young, rich dilettante. He gambled and drank and caroused with beautiful women—especially young women with pretty eyes. He also pursued mathematics, but it was difficult to find the solitude he needed in order to study. Often he would hide in his room when working on a theorem or problem, but his many friends would come and beckon him to join them in cafés and bars or on the streets. Hiding out became a tough task indeed, and Descartes looked for ingenious ways of avoiding disruption. But his friends were good at finding him.
At the time, Saint-Germain-des-Prés was an area at the edge of Paris by the meadows (prés in French) where young men would go to duel in secret. Dueling, which was a common practice in Europe among the aristocracy, had been recently made illegal by royal decree, so young men were forced to seek out deserted areas in the fields in order to indulge in it. Although Saint-Germain-des-Prés is now one of the central locations in Paris, with its famous church and trendy cafés, in Descartes’s time it was a forgotten backwater just outside the city walls. Descartes told no one that he had taken a room in this remote area—except for his valet, who followed him on his travels throughout the continent. One of his friends discovered his whereabouts, however, when he followed Descartes’s valet through the streets when the latter was buying food for his master. Thus the valet inadvertently led the young man right to his master’s hideout.
DESCARTES WAS A RESTLESS SORT who hated to settle in one place. In 1618 the twenty-two-year-old decided he’d had enough of Paris and its attractions; he now wanted to become a soldier. Having had much practice with his swordsmanship both in Saint-Germain and in school, he wanted to put his skills to the test, so he traveled with his valet to Breda in Holland and joined the army of Maurice of Nassau, Prince of Orange. Maurice was a Protestant ruler who was planning a battle against Catholic armies during the Thirty Years’ War. Descartes accepted a gold coin as token compensation for his entire future service to the prince, and, ever accompanied by his valet, the dandy soldier traveled throughout the region, dressed in green taffeta and carrying a shiny sword. On November 10, 1618, at the central square of Breda, Descartes had an unexpected revelation that would change his life. Many people were crowded around a tree in the square, to which a piece of paper had been posted. Descartes could not understand Dutch, so he asked the person nearest to him in Latin—the lingua franca of intellectuals throughout Europe at the time—what the writing was. The man Descartes happened to ask was Isaac Beeckman, a physician and would-be mathematician who fancied himself much smarter than some random French soldier.
“It’s a mathematical problem,” he answered.
“I can see that!” snapped Descartes. “But what does it say? I don’t understand Dutch.”
Beeckman translated it for him and then added arrogantly, “And I suppose you’ll give me the answer once you’ve solved it?”
“Of course I will!” answered Descartes, looking him intently in the eye. He asked for his address, and Beeckman explained that he was from out of town and staying with his uncle to help him slaughter his pigs. He was also looking for a wife, he added. Descartes took Beeckman’s uncle’s address and departed.
The next morning, very early, Descartes knocked at Beeckman’s uncle’s door and gave the stunned Dutchman the solution to the puzzle from the poster, which had stumped not only all passersby but also the mathem
atics professors at the local university. We don’t know what the problem was, but thanks to Beeckman’s journal, which was unexpectedly discovered in a Dutch library in 1905, we know that it was a geometrical one, perhaps based on ancient Greek theorems.
From that triumphant moment, Descartes knew he was a gifted mathematician, but he was also determined to never again require translations from Dutch. During long stays in Prince Maurice’s camp, Descartes taught himself the language of his fellow soldiers. Meanwhile, Beeckman and Descartes enjoyed a long correspondence, mostly about mathematical problems and ideas.
Descartes had heard that in Bohemia a Protestant king, Frederick II—later to be called the Winter King, as his rule would last only one season—was besieged by Catholic forces. Prince Maurice’s army could not travel there, however, since the Protestant leader’s predecessor had signed a treaty with Spain in 1609 that prevented the Dutch army from waging war on anyone for twelve years. As a volunteer soldier, Descartes was eager to see action, so one day he gathered his valet and headed southeast toward Bohemia (the modern-day Czech Republic). On the way, he stopped in Frankfurt, arriving there just in time to witness the elaborate ceremony of the coronation of the Holy Roman Emperor. Then he continued south and arrived in the German city of Ulm, where he met a mystic mathematician.
This depiction of the Temple of the Rosy Cross was designed by alchemist Daniel Mögling (a.k.a. Theophilus Schweighardt) in 1618. The temple has wheels to signify that the abode of the Rosicrucians is “nowhere but everywhere”; the hand of God holds the rope that guides it from above.
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