Johann Faulhaber was a rumored member of the Order of the Rosy Cross. The Rosicrucians, as they were called, were a secret society of scholars concerned with promoting science—which included alchemy—and mathematics. They wanted to cure all the ill people in the world and, being opposed to institutionalized knowledge, they supposedly favored Protestant causes that challenged the authority of the Church. But no one had ever met a Rosicrucian: they were known as the Invisibles, since they were always in hiding. Because no one ever saw them, one could say anything about them, including that they had never existed. Descartes’s first biographer, Adrien Baillet, who wrote about the philosopher-mathematician’s life only forty years after his death, claimed that Descartes was involved with the Rosicrucians.
Faulhaber received Descartes warmly and took him into his library. He presented him with some algebraic problems that Descartes solved admirably well and very quickly. Faulhaber laughed in delight and gave the young soldier more problems—all of which he solved with ease and elegance. The two men became close friends who exchanged mathematical ideas throughout life, and Descartes went as far as to adopt Faulhaber’s mystical notation. For example, he used the sign of Jupiter in his algebraic manipulations. This sign originates in astrology and alchemy.
The symbol for Jupiter—which Descartes used in his algebraic calculations—is also the alchemical symbol for tin.
The Battle of White Mountain was fought on November 8, 1620, near the Bohemian capital of Prague, which was then in the grip of rebel forces. The battle, in which Descartes fought for Maximilian in the combined forces under Ferdinand II, is memorialized in this painting by the Flemish artist Pieter Snayers.
Descartes continued his travels and—in a reversal, having served in a Protestant army—now joined the Catholic army of Duke Maximilian of Bavaria in the battle for Prague, which unseated Frederick. The Winter King, his wife, and their young daughter Elisabeth fled Prague in the middle of the night as Descartes and his fellow soldiers stormed the city. Thereafter, only a Habsburg would rule over Bohemia. Descartes and Elisabeth, “brushing” against each other that night, as it were, would meet in Holland years later and become close.
During the long winter of encampment outside Prague in 1619–20, Descartes lived in a pôele (French for “oven”), as he described it. It was a small hut containing a woodstove for cooking and heat. On the night of November 10, 1619, which was coincidentally the anniversary of his encounter with Beeckman in Breda, Descartes, sleeping in his “oven” by the Danube, experienced three intense dreams that apparently had a very strong influence on his life and quest for truth. The next day, he wrote in his notebook about a key mathematical discovery he claimed to have made that night. He did not specify what the discovery was or what area of mathematics it applied to, and the nature of this early breakthrough by the young Descartes remains a mystery.
After the battle for Prague, Descartes continued on his travels. One day, returning to the region of Poitou to sell some of his land in order to finance more travel, Descartes stopped by a roadside inn at the crossroads of two major routes near Orleans, south of Paris. Just outside the inn, he recognized a young woman with whom he had had an infatuation some years earlier. She recognized him, too, and rushed to meet him—and it was as if the intervening years had done nothing to cool their passion for each other. Suddenly her current suitor leaped toward Descartes and challenged him to a duel. Apparently, the man had no idea whom he was dealing with: an expert swordsman.
Needless to say, Descartes defeated the suitor easily. With one quick parry-and-attack move, Descartes made the man’s sword fly up in the air and then quickly put the point of his sword to his opponent’s neck. He said, “The lady has beautiful eyes, and for that I will spare you your life.” And with that, he straightened up, gathered his valet and their horses, and galloped away, leaving his stunned former love and her suitor in the dust.1
In July 1621 Descartes traveled from Germany to Hungary, Moravia, and Silesia, where he observed the war’s devastation on the city of Breslaw. The following autumn, he continued on to Pomerania, by the Polish border, stopping in the city of Stettin. He went on to the Baltic coast, to Brandenburg, the Duchy of Mecklenburg, and then to Holstein. Descartes was determined to visit most of the continent of Europe.
In November 1621 Descartes decided to make one last trip before settling down in one place, choosing to see the Frisian Islands off the northern coast of Germany and Holland. He and his valet hired a Dutch boat with a crew to show them the islands in detail. They had barely reached the open sea when the disreputable crew, eyeing the French gentleman with what was clearly a bagful of money, spoke freely in Dutch and within earshot of Descartes about their plan to rob him and throw both men overboard. Having made a special effort to learn Dutch while serving Prince Maurice, Descartes was fully aware of the unfolding conspiracy, however. Once he had heard enough, he lunged toward them with his sword and cursed them in their own language. Apparently, they were so stunned by this surprise move that “they failed to consider the advantage of their number and surrendered and peacefully brought him and his valet to their destination,” as Adrien Baillet described it in 1691.
Meanwhile, Descartes’s good friend Marin Mersenne had taken up residence at the Minim monastery in what is now the Place des Vosges in the Marais section of Paris—a medieval district that was not razed to the ground by Baron Haussmann to make way for wide boulevards in the nineteenth century and, therefore, still looks as it did in the time of Descartes. Engaging in an extensive correspondence with the most prominent mathematicians and scientists throughout Europe, Mersenne participated in what would later be called the Republic of Letters. When he died, ten thousand letters were found in his quarters in the abbey. These letters revealed an extensive, if indirect, correspondence between Descartes and several prominent mathematicians of the time, including Fermat, Pascal, Roberval, Desargues, and other mathematicians, with Mersenne as intermediary.
Progressively, Descartes became obsessed with the Inquisition. He was painfully aware of the travails of Galileo and was determined not to encounter a similar fate. In letter after letter to Mersenne, he expressed his fear that if he published a work proving that the earth rotates around the sun, he would be hunted down and imprisoned—or worse. He had written a book called Le Monde (The World) but refused to publish it because it espoused the heliocentric view. To further protect himself, Descartes would usually post his letters to Mersenne from locales near his residence rather than from wherever he was residing at the time.
MERSENNE PRIMES
Mersenne was a good mathematician in his own right. He was especially interested in prime numbers and looked for ways to identify them more efficiently and accurately than was then possible through use of the ancient Sieve of Eratosthenes. In particular, Mersenne was able to identify a relationship between prime numbers in his formula Mp = 2p – 1, where Mp is potentially a prime number and p is a known prime number (e.g., 22 – 1 = 3, a prime number; 23 – 1 = 7, another prime number). This relationship holds true until p = 11, as 211 – 1 = 2,047, which is the product of 23 and 89, and thus not prime.
Even though it turns out that so-called Mersenne numbers (Mp) are not always prime, the largest prime number found to date is a Mersenne prime (243,112,609 – 1). The search for greater and greater prime numbers continues to this day, and thanks to Mersenne, a project called the Great Internet Mersenne Prime Search facilitates the process of identifying prime numbers by inserting the largest known prime number into Mersenne’s formula to see if the result is prime. If it is, then the new prime number is inserted into Mersenne’s formula as the search continues.
In October 1628 Descartes witnessed the horrific siege of La Rochelle by Louis XIII and his army. Here was a French king waging a religious battle against his own citizens, the Huguenots. Fleeing persecution, these French Protestants found refuge in the city of La Rochelle on the Atlantic coast of Brittany, where they were promptly surrounded by Fr
ench forces that blocked all attempts by land or sea to resupply the city’s inhabitants with food and other necessities. After fourteen harrowing months, more than twenty thousand Protestants had starved to death, and the remaining population of five thousand—which had been reduced to eating rats and leather belts—finally surrendered. According to Baillet, this was the last battle that Descartes ever observed.
ALTHOUGH THE KING of France had given him special privileges based on his status as a prominent mathematician, philosopher, and scientist, Descartes—who was then settled in Paris—saw a danger to himself in living in a predominately Catholic nation. By the end of 1628, he had left for Holland, seeking tranquillity. It did not last.
Upon receiving word of Galileo’s 1633 trial by the Inquisition, Descartes’s obsession with the Italian scientist’s fate intensified, and he became increasingly distraught. Between 1628 and 1649, he changed addresses more than a dozen times and kept withholding publication of his book Le Monde, which described the motion of planets around the sun. Also, his longtime friendship with Beeckman fell apart when the latter boasted that he had “taught Descartes all the mathematics he knows.”
This detail of a painting by an anonymous artist depicts the surrender of the Huguenots at the conclusion of the 1627–28 Siege of La Rochelle, the last battle at which Descartes was present.
While renting an apartment at present-day 6 Westermarkt Street, near the West Church in Amsterdam, Descartes fell into a romance with his landlord’s servant, Hélène Jans van der Strom. Unlike most servants, Hélène was literate, and the two continued to exchange letters for years after their affair ended. The couple kept their relationship secret, and on July 19, 1635, they welcomed a daughter into the world, whom Descartes named Francine, meaning “little France.” Descartes had intended to send her to France so that she could receive a good education, but in 1640, at the age of five, she succumbed to scarlet fever and died. Distraught by their loss, the couple eventually parted ways.
In 1637 Descartes finally published his masterpiece, The Discourse on the Method, which contained the essence of his philosophy, including the timeless statement “Cogito ergo sum” (I think, therefore I am). The book’s appendix, entitled La Géométrie, contained Descartes’s breakthrough work on uniting geometry with algebra, which laid the foundation for the ubiquitous Cartesian coordinate system. Discourse made Descartes immensely famous all over Europe.
Elisabeth, Princess of Bohemia, was living at that time in exile in The Hague, an administrative center in the west of the country. Two decades his junior, the beautiful princess—who had crossed his path as she fled Prague with her parents—became his student in philosophy and mathematics. They corresponded for many years on questions of philosophy, and he even sent her derivations of theorems in geometry, which apparently delighted her. Although Descartes had no shortage of friends in high places, enemies began to emerge out of the woodwork. In 1647, while living in the Dutch countryside, he became entangled in one of the most vicious academic disputes in history. Protestant theologians at the University of Utrecht took issue with Cartesian philosophy, which is based on doubt. Descartes defended his views in print and was subsequently accused of libel by Gisbert Voetius, a man who had also accused Descartes of atheism. A court decided against Descartes, and he was forced to write an official apology, taking back what the court believed to be libelous statements against Voetius. The philosopher reluctantly complied, but there was a growing feeling in his heart that there was no longer a place for him in Holland.
Descartes (far right) and Queen Christina of Sweden (far left) are depicted in this detail from a painting of the philosopher among members of the queen’s court. Descartes tutored the monarch daily.
The possibility of escape came from an unexpected source: Queen Christina of Sweden had heard about the famous French philosopher living in self-imposed exile in Holland and decided that she wanted him for herself—as a private tutor. She wrote him letters, which flattered him, but he was still not ready to leave the comforts of Holland. The queen—a strong-headed woman, even though she was only twenty at the time—refused to give up. At one point she simply sent Admiral Fleming of the Swedish Royal Fleet to pick up the reluctant philosopher. Descartes relented and came aboard the ship. After settling in Sweden, he gave the queen lessons in philosophy at 5:00 a.m. every day in the unheated library of her palace.
But Descartes had enemies in Sweden as well. Some court toadies didn’t like the newcomer’s influence on their queen and were determined to put a stop to it. One of them was the queen’s doctor.
In early 1650 Descartes fell ill. He had contracted the flu from the French ambassador, and his condition deteriorated. The queen’s doctor insisted on bleeding him, which was a standard medical practice in those days. Descartes knew enough about medicine to refuse the treatment, but eventually, when his condition took a sharp turn for the worse, he consented. On February 11, 1650, the great French philosopher and mathematician died—whether from the flu or from poisoning, we do not know. His remains, along with his mathematical and philosophical papers, were brought back to France, and his bones were buried inside the church of Saint-Germain-des-Prés in Paris—in the area in which he loved to roam. Descartes’s skull, however, is now part of a tasteless display of anthropological finds in the Musée de l’Homme (Museum of Man) near the Trocadéro in Paris.
THE JURIST FROM TOULOUSE
Pierre de Fermat (1601–65) could not have been more different from his rival Descartes. Whereas Descartes was a soldier and adventurer, Fermat always led an exceptionally quiet existence—in fact, some might have called him boring. Unlike Descartes, who was interested in metaphysics, philosophy, and the applications of mathematics in physics, Fermat was interested almost exclusively in pure mathematics—save for some applied work in optics. In the field of pure mathematics, however, Fermat is sometimes described as the best of his generation.
Pierre de Fermat was born in the French town of Beaumont-de-Lomagne to a family of leather merchants. He studied classics at the University of Toulouse. (Interestingly, there is no evidence that he studied much mathematics there.) At age thirty he was appointed commissioner of requests in Toulouse and married in the same year.
With his wife, Louise de Long—a cousin of his mother’s—Fermat had three sons as well as two daughters, who eventually became nuns. In 1631 he was appointed councilor in the Parlement of Toulouse, a position he held until he died in 1665. According to the historian Eric T. Bell, Fermat fulfilled his duties to the king “with dignity, integrity, and great ability for seventeen years,” but he spent all his free time studying mathematics.2
Fermat studied graphs of equations, developing the kind of connection between algebra and geometry that Descartes explored, and he looked for ways to study rates of change as well as the maxima and minima of functions. Thus Fermat was another mathematician whose work aided the development of calculus. According to the historian Michael Mahoney, almost everything that Fermat did in mathematics—for example, solving complicated equations—was inspired by the work of François Viète.3
After reading Diophantus’s Arithmetica, Fermat became very interested in Diophantine equations, which led to his preoccupation with number theory. It was, in fact, in the margin of a Latin copy of the Arithmetica that he wrote his famous Last Theorem.
Fermat’s Last Theorem is the statement that the equation xn + yn = zn has no solutions in integers for any power n that is greater than 2. Produced by someone who was technically an amateur mathematician (Fermat, after all, had a “day job”), the theorem has attracted thousands of amateur and professional mathematicians over three and a half centuries to the cause of proving it. Only in the mid-1990s did Andrew Wiles of Princeton University, with the help of another British mathematician, Richard Taylor, present papers proving that the conjecture is, indeed, correct. In 1637, when he proposed the theorem, Fermat wrote on the margin of the book, “I have a marvelous proof of this assertion, but the margin is
too small to contain it.” Did he possess a proof? Probably not. When the final proof was obtained by Wiles, he had to use more than two hundred pages of derivations based on mathematics that had been developed centuries after Fermat’s time.
Pierre de Fermat, portrayed in this undated engraving, was a “pure” mathematician, interested chiefly in mathematical theories and principles rather than their applications.
Descartes, when exchanging information with Fermat (through Mersenne) about pre-calculus ideas and the union of algebra and geometry, referred to Fermat not by name but as “the jurist from Toulouse,” which took the mild-mannered Fermat aback. Each of them vied to be the greatest mathematician of his time.
DESCARTES LEFT AN IMMENSE LEGACY in mathematics and philosophy. His “I think, therefore I am,” statement, as well as his work on metaphysics and the mind-body connection, are mainstays of modern philosophical thought. In mathematics he understood enough about rates of change of functions, and about areas, to be considered one of the key theorists who paved the way for the invention of calculus.
While on the road, in the countryside of various European nations, or hiding out in Paris, Descartes successfully managed one of the greatest feats in mathematical history: he wed geometry with algebra. In other words, Descartes was able to show how every geometrical object can be associated with an algebraic equation. For example, a parabola, as every high-school student learns, is associated with a quadratic equation. In setting up this correspondence between geometry and algebra, Descartes gave us the Cartesian coordinate system, which allows us to associate each point in space with a set of numbers that describes its position. A modern piece of technology directly enabled by Descartes’s discovery is the Global Positioning System (GPS), which uses the Cartesian coordinate system to identify every point on Earth by its longitude, latitude, and altitude (height above sea level). In addition to this crucial development in mathematics, Descartes made a large number of beautiful mathematical discoveries, but he did not publish his findings until many years later. Other discoveries, found in notebooks he had left by his deathbed in Stockholm, remained unpublished until after his death.
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