When he was a child, Rene had a crush on a young girl who had a lazy-eye problem. This childhood memory made him always focus his attention on women's eyes. As a young man in Paris, he found himself especially attracted to women who had beautiful eyes.
Among Rene's new friends in Paris was Claude Mydorge, who had been the treasurer of the city of Amiens, and who enjoyed a reputation as a first-rate mathematician. Mydorge was a dozen years older than Descartes, a man of the world, and he had an especially engaging and vivacious personality and a good sense of humor. Descartes was attached to him, and the two men spent many hours together enjoying themselves as well as discussing mathematics. Among Rene's acquaintances from La Fleche was Marin Mersenne. Mersenne had by then finished a course of study at the Sorbonne and had received the habit of the Minim order on July 17, 1611, in the monastery of Nigeon, near Paris. The Order of the Minim Brothers was founded in 1435 by Saint Francis of Paola in Calabria, Italy. To the Minims, humility is the primary virtue, and the name Minim derives from minimi, since they view themselves as “the least” of all the religious.
Six months after Descartes arrived in Paris, Mersenne was ordained a priest and became a friar of the Minim monastery at the Place Royale (today's Place des Vosges). Descartes would often visit Mersenne, who had a very wide range of scientific and mathematical interests, and talk with him and explore new ideas. Mersenne quickly became Descartes' closest friend. According to Baillet, the stimulating interactions Descartes enjoyed with Mersenne served as a counterweight to the general lack of purpose that characterized his early days in Paris.
Descartes also loved music. Some scholars have drawn a connection between his interest in music and his great ability in mathematics. At any rate, Descartes and his many friends pursued musical interests together, attending concerts and performances throughout the capital.
After about a year of play and enjoyments of all kinds, Rene Descartes felt the need to become more serious. In Paris he had come in contact with new ideas. French intellectuals were studying Greek geometry, trying to embellish the works of the ancient Greeks. Euclid's Elements had by then been extended to include three more volumes beyond the original thirteen. Physics, too, was developing, as scientists were studying the nature of falling objects and exploring the riddle of gravity. Descartes longed to study these new ideas, and his feverish social life became an impediment to achieving this goal. To concentrate on his work, the young man now felt he had to distance himself from his many friends—but they made it hard for him to do so. Whenever Descartes stayed home to read and work, his friends would come over to his apartment and beseech him to join them on the streets and at the nightspots of the city.
In desperation, Descartes took a drastic measure—he secretly moved and did not give his friends his new address. He needed to be in an area where people would not recognize him whenever he went out for a walk, or recognize his servants or valet when they shopped or ran errands for him. So Descartes moved outside the walls of the city, to the neighborhood around the ancient Church of Saint-Germain-des-Pres. He loved this part of Paris because it was peaceful and quiet, and more rural. There were open fields here, to which young men would occasionally come to duel, an activity discouraged by the authorities. Descartes began to view himself as a spectator rather than an actor in the drama of daily life around him. For over a year, no one saw Descartes. His friends worried, and they suspected that he might have left Paris to return to his father's house in Brittany. They complained to one another about his incivility in leaving without saying good-bye. Some of them made inquiries in Rennes, but learned that he was not in Brittany, nor in Touraine or Poitou. They kept looking for him in the capital, but found him at no ball or banquet or reception. His friends were close to giving him up as lost.
Hiding out in a new part of the city worked for Descartes for some time. But his friends were still searching for him everywhere. One day, Descartes' valet was recognized on the street by one of these acquaintances, who then secretly followed the valet out of the city walls and into the area of Saint-Germain. The man waited for the valet to disappear up a flight of stairs, and then followed up the same stairway. He looked through the keyhole into Descartes' bedroom.
The man observed Descartes lying in bed reading, then sitting up in bed to write in a notebook, then lying down again to read, and a little later sitting up again to write in the notebook. The friend realized that Descartes was immersed in work he seemed determined to keep from the rest of the world. He would not disturb him, now that he understood Descartes' need to withdraw from the excited life of the capital to pursue his private work, and he quietly left.
Scholars believe that Descartes likely wrote his enigmatic Preambles, which included the statement “I advance masked,” in his hideaway in Saint-Germain. Descartes' Preambles continued:
The sciences are now masked; the masks lifted, they appear in all their beauty. To someone who can see the entire chain of the sciences, it would seem no harder to discern them than to do so with the sequence of all the numbers. Strict limits are prescribed for all spirits, and these limits may not be trespassed. If some, by a flaw of spirit, are unable to follow the principles of invention, they may at least appreciate the real value of the sciences, and this should suffice to bring them true judgment on the evaluation of all things.
Descartes' ideas about the “chain of the sciences” and the sequence of the natural numbers bore a striking resemblance to mysterious writings of a mystical nature that had begun to appear in Europe in the early part of that same decade. The authors of these treatises on science and mathematics had remained anonymous.
Two years after he arrived in the French capital, Rene Descartes was ready to move on. He had always enjoyed a fast lifestyle, and he liked to fence and ride horses; now he longed for a life of action. Descartes had heard that up in Holland, Maurice of Nassau, the new prince of Orange and the Protestant champion in the religious wars, was gathering men from several countries—including two French regiments—and training them in his camps for war with the gathering Catholic forces of Spain and Austria.
Although he was a Catholic, Descartes was interested in joining Prince Maurice's army. He felt that he could learn much about the art of war from the prince and his generals, and religion did not weigh in his decision, perhaps because he would join as a volunteer and would not have to fight if he chose not to do so. Descartes sent his servants back to his father in Rennes and, taking only his valet with him, traveled to Breda in southern Holland to volunteer his services. He would learn the art of war, but would not be paid for his military service—other than a token single gold doubloon. Unpaid, he would retain many freedoms, and these rights would allow him to pursue his research in mathematics and science, and to try to uncover their hidden meanings. Descartes would use the army as a vehicle for travel and adventure. In the words of Baillet, Descartes would use the army as “a passport to the world.”
Chapter 3
The Dutch Puzzle
“QUID HOC SIGNIFICAT?” THE YOUNG French soldier asked the slightly older Dutchman standing next to him, addressing him in Latin, the lingua franca of the educated throughout Europe. They were both among a group that had crowded around a curious poster attached to a tree trunk in the main square of the city of Breda, on the morning of November 10, 1618. “What does this mean?”
The Dutchman was from Middleburg and had just finished his studies in medicine and mathematics, and was hoping to take up the position of assistant principal of the Latin School of Utrecht. He had come to Breda to help his uncle slaughter his pigs, and was also hoping to find a wife. He took a long look at the young French soldier facing him. Rene Descartes wore the distinctive uniform of a volunteer in the army of Maurice of Nassau, the prince of Orange.
The Dutchman also noticed that the soldier was wearing a fancy plumed green hat, and that a silver sword hung from his hip—not the usual musket other soldiers carried. He looked not much older than twenty-two or twenty-thr
ee, was of medium height, slightly less perhaps, with longish thick and wavy dark brown hair, and a mustache and goatee to match. And he had piercing, earnest brown eyes.
The soldier was looking at him expectantly. “It is a mathematical puzzle,” the Dutchman answered.
“I can see that,” responded Descartes, “but what exactly does it say? I do not understand Flemish.”
The man took out a piece of paper and a pencil and began to copy the geometrical designs on the poster, labeling each in Latin, rather than Flemish, and then translating into Latin the paragraph written below the diagram. He handed the paper to the young soldier and said, “They want you to prove this statement,” pointing to the last sentence. Descartes looked intently at the paper in his hand, and the man added: “And I suppose you will give me the solution, once you have solved this problem?”
Descartes turned quickly from the paper and gazed intently at the man.
“Yes, of course I will give you the solution,” he said with determination. “Would you please give me your address?”
The Dutchman offered him his hand and said, “Isaac Beeckman is my name.”
“Rene Descartes,” said the soldier. “Or Rene le Poitevin, as they call me, since my family comes from the French region of Poitou; although I was actually born in Touraine.”
They shook hands, and Beeckman told Descartes that he was from Middleburg but was staying in Breda to help his uncle with his pigs. He gave him his uncle's address and they bade each other good-bye. From his journal, which was discovered in a Dutch library in 1905, and from other sources, we know that Beeckman did not believe that the young soldier would be able to solve the puzzle.
The next morning, as Isaac Beeckman was about to have breakfast at his uncle's house, there was a persistent loud knock on the door. The servant opened the door and let in the young soldier. He was accompanied by his valet. Descartes showed Beeckman his solution to the Dutch puzzle. Beeckman, who was a competent mathematician, was amazed by the soldier's solution to a very difficult mathematical problem. He had not expected a random person to solve a problem that many trained mathematicians and professors could not. Descartes' brilliant solution cemented a friendship between the two men. It was also a watershed event in the life of the young Descartes, since this was the moment he first realized that he was a very gifted mathematician.
The puzzle Descartes solved and demonstrated to his new friend was not an isolated problem suddenly appearing on a poster in southern Holland. The seventeenth century saw a revival of the classical geometry of ancient Greece as educated people throughout Europe sought intellectual challenges and the hidden meanings of mathematics. Ancient Greek texts were being republished in Latin, foremost among them Euclid's classic volumes of the Elements, written in Alexandria about 300 B.C. Euclid's work was, in fact, the most important textbook published on the new printing presses invented less than a century and a half before Descartes' time. Other ancient texts, such as Diophan-tus's Arithmetica, written around A.D. 250, were also being printed in seventeenth-century Europe. It was on the margin of a copy of this book that Pierre de Fermat (1601-65) wrote his famous Last Theorem, which would haunt mathematicians and amateurs alike until its final, dramatic proof late in the twentieth century.
These newly republished mathematical texts enabled a revival of the study of geometry in the schools and universities of Europe, and with these new publications arose a whole class of intellectuals who avidly pursued new solutions to ancient problems, challenging one another to solve problems they proposed and publicized on posters placed in public places. The problem Descartes solved on November 10, 1618, was one such example of a challenge issued by a mathematician through a public posting. Similar challenges issued a century earlier by mathematicians living in northern Italy had led to great developments in the area of algebra, resulting in the solution of complicated equations that the ancient Greeks and the medieval Arab mathematicians who followed them had not been able to achieve.
We don't know exactly what the problem was that Descartes solved and showed Beeckman in November 1618. We do know that it involved angles in a geometrical drawing, and that it was a very difficult problem. But one unusual, and perhaps incidental, aspect of Descartes' geometrical thought apparently impressed Beeckman very much. Descartes may have raised this particular issue on meeting Beeckman, even before he presented him with his solution to the problem from the poster the next day. In his journal, Beeckman described it on the day after he met Descartes:
Angulum nullum esse maL probavit Des Cartes:
Yesterday, which was November 10, 1618, at Breda, a Frenchman from Poitou tried to prove the following: “In truth, there is no such thing as an angle.” This was his argument: “An angle is the meeting point of two lines at one point, so that the line ab and the line cb meet at point b. But if you intersect angle abc by the line de, you divide point b into two parts, so that half of it is added to line ab, the other half to line be. But this contradicts the definition of a point, since it has no size. Hence, there is no such thing as an angle.
This, of course, is unlikely to have been the complicated mathematical puzzle Descartes solved, but rather Descartes being clever with Beeckman, showing off his mastery of Greek geometry with its axiomatic definitions—to the point of making an absurd argument. The point in the angle does not, in fact, get cut into two halves—precisely because it has no size. Descartes was philosophizing, and Beeckman was impressed with his mastery of the intricacies of Greek geometry.
At the time Descartes showed Beeckman his solution to the Greek geometrical problem, the two areas, geometry and algebra, were considered two different parts of a wider, and somewhat nebulous, field called mathematics. Geometry was about straight lines and triangles and circles— idealized visual images of the elements of the physical world. Algebra, however, was the study of equations—symbols and numbers on two sides of an equal sign, which had to be solved to result in some meaningful quantity. No one had imagined that these two fields could be unified into a larger discipline. But within two decades, Rene Descartes would do just that.
Descartes told Beeckman that he was hoping to go to war. Beeckman worried about this prospect, and hoped that his new friend would stay in the area and that the two of them could meet often and work together on problems of mathematics and science. After he left Isaac Beeckman on November 11, 1618, and returned to his army camp, Descartes realized that he was not yet going to be sent to war. He stayed on in Prince Maurice's army for several months while the troops remained stationed outside Breda. Since he was, after all, a volunteer, Descartes did more or less what he wanted to do. He spent time learning Flemish, so that he would never again have to ask strangers for translations. Descartes had excellent facility with language—he had perfect mastery of French and Latin—and before long, he could understand Flemish well and even speak it with some fluency. This new language ability also gave him fluency in other, related, Germanic dialects. Descartes was proud of his new achievement, and on January 24,1619, he wrote to Beeckmanfrom his army camp: “I devote my time here to painting, military architecture, and the study of Flemish. You will soon see the progress I have made in this language, when I come to see you in Middleburg, God permit it, at the beginning of Lent.” Descartes could not have known it at the time, but his new mastery of Flemish and related Germanic dialects would literally save his life.
Descartes' solution of the Dutch puzzle excited him about mathematics. For it demonstrated to him that he had a unique gift. He began to believe that mathematics held the secret to understanding the universe. He stayed most mornings in bed at camp reading and writing about mathematics and exploring its applications. He worked out an cient Greek problems in geometry, but he soon concluded that the power of geometry transcended pure mathematics: geometry held the secret to all creation.
Three years before they met in Breda, Isaac Beeckman had penned an article—which was discovered in his journal—about the mathematics of music. In this
article, Beeckman tried to use Greek geometry to explain the harmonics of a vibrating string. While Beeckman's analysis was not very deep, Descartes did not show any lack of respect for his new friend's work. The two men worked together, trying to create a theory of music based on mathematics. They also worked on mechanics and on pure geometry. Beeckman would suggest problems, and Descartes would solve them using his brilliant mathematical abilities. By now Beeckman was back home in Middleburg, and Descartes would visit him whenever he could. When they were not together, the two friends exchanged ideas through letters.
On March 26, 1619, Descartes wrote his friend from his camp in Breda. He unfolded his plan to invent a method of solving a very wide variety of problems. He wrote: “I desire to give the public not an Ars brews of Lull, but a science based on new foundations.” Descartes alluded here to the work of Ramon Lull (c. 1235-1315), a medieval mystic who was born on the Spanish island of Majorca and wrote 260 books, among them the Ars brevis (“Brief Art”) Descartes mentioned. Lull's works were a mixture of Cabbala and mysticism whose elements were ways of combining letters and numbers in an attempt to extract new knowledge about the universe.
A month later, on April 29, 1619, Descartes again wrote to Beeckman about the work of the Majorcan mystic: “Three days ago I met at an inn in Dordrecht a learned man and discussed with him the Ars parva [Ars brews] of Lull. He said that he could use the Art so successfully that he could discuss any topic whatsoever for an hour; and if one then asked him to speak about the same topic for another hour, he could do so without repeating what he had already said, and so on for another twenty hours.”
Descartes's Secret Notebook Page 4