Descartes's Secret Notebook

by Amir D. Aczel

  The first person in history known to have succeeded in solving cubic equations was Scipione del Ferro (1465-1526), who was a professor of mathematics at the University of Bologna. Nobody knows how del Ferro made his amazing discovery. He did not publish it and did not divulge it to anyone—until he was on his deathbed. Then he passed the secret on to his mediocre student Antonio Maria Fior. Soon afterward, word was out that Fior could solve cubic equations, which was considered a great achievement since no one had been able to do so even though people had been trying for many centuries.

  In 1535, Fior challenged Tartaglia to a competition. Each person was to submit thirty problems for the other to solve. At that time, the person who won such a competition could expect money, prestige, and sometimes a professorship at a university. Fior was confident that his ability to solve cubic equations would be enough to defeat Tartaglia, but del Ferro had shown Fior how to solve only one type of equation: a cubic equation of the simple form: x'=ax+b (that is, the coefficient of x' is 1, and there is no x2 term). Tartaglia, on the other hand, submitted to Fior a variety of different problems. Fior's poor performance, since he had only the one recipe given him by del Ferro, exposed him as an inferior mathematician. Fior also gave Tartaglia thirty problems of a variety of kinds—being sure that he could not solve them since he himself didn't know how to do so. Unbeknownst to him, in the early hours of February 13, 1535, Tartaglia had a revelation: he discovered a general method of solving cubic equations, that is, equations of the very general type: ax1+bx2+cx+d=0. Tartaglia was now able to quickly solve all thirty of Fior's problems—he completed his task in less than two hours. It was clear to everyone present that Tartaglia was both the winner of the competition and the greatest mathematician.

  At this point we meet another Italian mathematician, Girolamo Cardano (1501-76), who was a medical doctor and mathematics lecturer at the Piatti Foundation in Milan. Cardano was well aware of the importance of solving cubic equations, and was intrigued when he heard about the results of the contest in Venice. He immediately set to work on trying to discover Tartaglia's secret, but he was unsuccessful. A few years later, in 1539, he contacted Tartaglia through an intermediary. Cardano told Trataglia that he wanted to include his method for solving cubic equations in a book he was planning to publish that year. Tartaglia declined the offer, saying that he wanted to publish his own book. Cardano then asked Tartaglia if he would mind showing him his method anyway, promising that he would keep it secret. Tartaglia again refused.

  Not giving up, Cardano now wrote Tartaglia a letter in which he hinted that he had been discussing Tartaglia's brilliance with the chief of the army in Milan, Alfonso d'Avalos, who was one of Cardano's powerful sponsors. Tartaglia took the bait. He was a poorly paid mathematics teacher, and the thought of meeting an influential and wealthy individual who might be able to help him appealed to him. He wrote back to Cardano, and Cardano then invited him to his house, promising that he would arrange a meeting between Tartaglia and d'Avalos.

  On March 25, 1539, Tartaglia left Venice for Milan. To his dismay, he found that d'Avalos was not there as Cardano had promised. Cardano, however, wined and dined him in his house, and tried every method he could to convince Tartaglia to reveal his secret. Late that night, after he had drunk much wine and Cardano had sworn to him that he would never reveal his secret, Tartaglia divulged to him his secret formula. He did it by way of a poem in Italian in which he embedded his formula.

  In 1545, Cardano published his now-famous book Ars magna (“The Great Art”), which contained solutions to the cubic equation based on Tartaglia's secret formula, as well as solutions to the quartic (fourth-order) equations, which had been obtained by Cardano's student Ludovico Ferrari (1522-65). Cardano thanked Tartaglia in his book. But he had broken his promise—the oath he swore to Tartaglia never to reveal his secret. Understandably, Tartaglia was furious and for years kept writing letters to everyone he knew, attacking Cardano. He even published their conversation in Milan and the broken promise, including the formula he had divulged. But Cardano's book, the Ars magna, had established him as a leading mathematician, and he was untouched by Tartaglia's attacks. To add to his misery, Tartaglia was never given the chance to meet the wealthy patron he had hoped would help him. After a short period of teaching at a university, he returned to his position as teacher in Venice, which he kept until his death.

  Today, Tartaglia is remembered together with Cardano for a formula for solving cubic equations. Tartaglia also wrote a popular arithmetic text, and was the first Italian translator and publisher of Euclid's Elements, in 1543. He also published Latin editions of Archimedes' works.

  Descartes was well aware of the genesis of algebra and the development of solutions to equations of third and fourth order. He spent time working on such problems, and early on derived a result in this area. Descartes had shown that if a quartic equation has a special form (has no cubic term) and can be factored into two quadratic equations—

  x4+px2+qx+r=(x2+ax+b)(x2+cx+d)

  —then the number a2 is the root of a cubic equation; and also, b, c, and d are then rational numbers (meaning fractions or integers) that depend on a. This is a useful result that sometimes helps solve equations. It was a good start, and a continuation by Descartes of the work done a century earlier by the Italians. It was also strikingly close to work done by Faulhaber.

  The Italian mathematicians, the early algebraists, were called cos-sists. The word “cossist” comes from the Italian cosa, meaning “thing.” The cosa was the mystery that algebra was designed to solve—it was the name given to the unknown quantity in an equation (our modern x). Descartes' missing notebook contained an original alchemical and astrological sign, the sign of Jupiter. But it also included early cossist notation:

  Descartes learned to use this sign from the Italian cossists, whose algebra he had studied. Descartes himself later invented our modern notation, used in algebra today. He taught us the use of x and y for variables to be solved, and a and b and c, and so forth, for known quantities. But tantalizingly, his hidden notebook used a different notation—the mystical notation inspired by alchemy, astrology, and Rosicrucianism, and the old cossist notation. The secret notebook contained yet a third sign, which no one has been able to trace to any previous source.

  Incidentally, Descartes never used the equal sign (=), even though it had been invented by Michel Recorde in 1557. Descartes persisted in his use of a backward-facing Greek alpha denote equality. Interestingly, it would be Leibniz who decades after Descartes' death would revive the use of the equal sign we use today.

  Chapter 11

  A Duel at Orléans, and the Siege of La Rochelle

  DESCARTES TRAVELED ONCE MORE to Touraine and Poitou, and then went back to the capital and settled there for a few months. But family issues in Touraine, Poitou, and in Rennes kept him traveling to these areas frequently. He spent time with his sister and his brother-in-law, and with his father. Rene traveled to Poitou to sell more of his lands and to liquidate other assets so he could take the money to Paris and live comfortably on the wealth he had inherited. He also traveled frequently to Touraine to visit his governess and family members who had remained there.

  When he was just becoming a young adult, some years earlier, Descartes was rumored to have had an intimate relationship with a mysterious woman of Touraine called La Menaudiere. His family, disapproving of his relationship with someone they considered unsuitable for their son, began to look to find him a wife. They thought that getting married might bring stability to the life of the young and restless adventurer, and that once he married, he might settle down in the area and establish a business. The search began in earnest to find Rene a wife “of good birth and much merit.”

  There was a very beautiful young woman, who later became known as Mme. de Rosay, whose family also lived in Touraine. She was indeed of good birth and suitable to the Descartes family. Rene and the young woman met several times, and they were attracted to each other. But Re
ne soon left on his travels, and their relationship never progressed.

  But now, in 1625, Rene was again traveling the roads of Touraine and the French countryside south of Paris to see his family and take care of his affairs. He hadn't seen her for a few years. On one of his trips between La Haye and Paris, Descartes stopped his horse-drawn carriage at a major intersection near the city of Orleans, which lies on the main road south from Paris. Descartes had to stop for a while to let his horses rest, eat, and drink. He entered a roadside inn, one that was popular with travelers on these routes. France had many such establishments along its highways. The inn had a closed courtyard in which the horses were sheltered and fed and watered. Inside the inn there was a large room with a ceiling supported by dark wooden beams. It had large, arched windows. There were several wooden tables surrounded by simple rustic chairs. People crowded around these tables drinking and eating.

  Descartes and his valet spent some time at the inn, enjoyed a meal and rested, and were ready to continue their journey north to Paris. They walked into the courtyard, collected their horses, and went into the sunshine. They harnessed the horses to the carriage and were just about ready to leave when Descartes suddenly looked up. And there she was: the woman we now know as Mme. de Rosay.

  The two people looked at each other, and it seemed that the altered? vening years had done nothing to cool the attraction between them. Descartes, dressed in green taffeta and looking very smashing with his plumed hat and sword, approached her. She looked right into his eyes. They stood there speechless for a moment, just staring at each other. And then her companion rushed over. He was a jealous man, and he unsheathed his sword and challenged Descartes to a duel.

  The man apparently didn't know whom he was dealing with— someone with much experience with swords and battles. The two men locked swords, and swung and parried for a few moments. Swiftly, Descartes brought his sword in one last time and delivered a final blow. His opponent's sword flew up into the air. Descartes put the point of his sword to his challenger's throat and, glancing at Mme. de Rosay, said to him: “The lady has beautiful eyes, and for that I will spare your life.” He let him go, and pulled back in disgust. The lady rushed over to Descartes' side. One last time, Descartes stared into those beautiful eyes, and turning away from her, he said: “Your beauty is unmatched, but I love truth the most.” He left the two stunned figures by the roadside and in a minute gathered his valet, and in a whirl of dust they were off to Paris.

  Years later, when she was married and had become Mme. de Rosay, and when Descartes had become a famous philosopher, the lady confessed the story to her priest, who in order to protect his identity when he repeated the tale—violating the secrecy of the confessional—remained known only as “Father P.”

  According to Mme. de Rosay, she saw Rene Descartes for the first time when he was a young man and was one day in the company of several other young men who were joyfully playing around and talking about women. He confessed to them that he had never yet met a woman he found irresistible. Then he said: “I find that a beautiful woman, a good book, and a perfect preacher are the three most difficult things to find in this world.”

  But then, according to the lady, he met her. And he found her to be beautiful and irresistible, and from then on he desired only her. “Rene Descartes was a young cavalier who was guided by a love for me,” she said. “It made him distinguish himself in great deeds on my behalf.”

  According to Mme. de Rosay, Descartes was accompanying her, along with other ladies, on a trip to Paris when they were approached at Orleans by Descartes' bitter rival for her affections. When Descartes had won the duel and put his sword to his rival's neck, he told him the following: “You owe your life to this beautiful woman, to whom I devote my own.” But alas, things stopped there, and Descartes never married the lady. Was her version of the story true? Or did Descartes really say that he loved truth more than her beauty? Most likely, Descartes' account was the correct one, since it is in line with his general behavior and his rather cool approach to relationships. And we know that Descartes loved beautiful eyes.

  His frequent trips to Touraine and Poitou over, Rene Descartes was living in Paris in 1628, hiding from his friends once again so he could work in peace. He was writing extensively, deriving important results—some of which would become public within nine years. This was a difficult period for Descartes because he was anxious about not being found by his many friends as well as strangers who were attracted to him because of his growing fame as a philosopher, scientist, and mathematician. “The displeasure he felt by having been chased out of his favorite quarter into hiding brought about in him a desire to go to see the siege of La Rochelle,” Baillet tells us.

  Of all the places in which the Huguenots were once safe, by the early seventeenth century there remained only the city of La Rochelle, on the Atlantic coast of France. La Rochelle is a medieval walled city built in the twelfth century. It has a fortified harbor with two imposing ancient crenellated towers guarding its entrance from the sea. There is a fifteenth-century tower called the Tour Lanterne, which served as a powerful lighthouse guiding merchant vessels from around the world to this prosperous city. La Rochelle came to prominence with an economy based on salt, wheat, and wine. By 1620, 85 percent of its population was Huguenot. But the French state, whose religion was Catholicism, was threatened by the power of the Huguenots. King Louis XIII and his minister, Cardinal Richelieu, made a decision to crush Huguenot power.

  In 1627, the people of La Rochelle requested and obtained the aid of the British fleet against the French. King Louis XIII then sent his forces to La Rochelle in response to this act, to confront the British fleet, which made its base on the nearby Isle of Re. The French army wanted to prevent the British forces from aiding the Huguenots. The king himself ordered and oversaw a siege of the city—in what became one of the most dramatically recorded conflicts in French history. Alexandre Dumas devoted part of his masterpiece The Three Musketeers to the siege, sending Athos, Pathos, Aramis, and d'Artagnan to the scene. Cardinal Richelieu had his headquarters outside the city, commanding half the French forces, while the king commanded the other half.

  The siege of La Rochelle lasted thirteen months, from September 1627 until the city fell in October 1628. As the siege began, French forces encircled the city from all sides, preventing food and supplies from the rich lands it owned in the countryside from reaching the walled city. But there still remained the heavily fortified harbor, and through it the British were able to use their ships to bring in supplies of food to the besieged city.

  Soon after the siege began, the French decided to build a dike across the entrance to the inlet leading to the harbor. The aim of the French forces was to block completely the entrance to the harbor and thus prevent the British fleet from entering it. In March 1628, the king left for Paris and made Cardinal Richelieu the lieutenant general of the army. The cardinal began to construct the dike across the inlet, and when the king returned in April and reassumed command of half the French forces, there had already been much progress. The French had moved ships laden with rocks and other heavy loads and sank them in a line stretching across the inlet. Eventually, enough ships were sunk so that the besieging forces were able to build a wooden dike using the sunken ships as a base. The French used thirty-seven large vessels, their prows facing out to sea, to construct the dike. To these were added fifty-nine smaller boats. They built two forts along the ends of the dike, and one large triangular wooden fort was constructed in the center. According to Baillet, this dike was the most elaborate wartime construction the French had ever undertaken, and the siege was a most impressive spectacle. This attracted many curious onlookers—young adventurers from the French nobility who wanted to observe the siege. Among them was the ever-curious Rene Descartes.

  Descartes arrived outside the besieged city at the end of August 1628. Baillet, who is our only source on this event in the life of Descartes, tells us that Descartes came with the sole purpos
e of observing the military operation, as many other young men of his age were doing. He did not want to be a volunteer in the fighting or to take any part in it. He came to this battle more as an observer than he had done to any previous battle in his life. He was now the scientist—not the soldier. Descartes was especially interested in the physical properties of the great dike that Richelieu was building.

  Descartes studied mathematically everything he saw during the siege. He spoke with the engineers who were building the dike, learning from them the technical details of the construction. He also met the French mathematician Desargues, who was “an expert on mechanics and was well appreciated by Cardinal Richelieu.”

  Besides fortifications and communications, Descartes was interested in the trajectories of cannonballs. Following his admired Galileo, Descartes wanted to learn about gravity and how it made objects fall to the ground, and he studied the curves the cannonballs were making in the air as mathematical functions.

  With the huge construction project over, the harbor of La Rochelle was now completely blocked. No supplies could be brought in, and the people of the city began to starve. The fall of La Rochelle happened in stages. On September 10, the citizens sent a delegation out to meet the king at the fort in the center of the dike that was starving them out. The delegates threw themselves at the king's feet, and it seemed that an arrangement could be made. But a few days later they decided that the British fleet might still save them, and the deal was off. The French forces tightened their grip around the city, furious at the new development. In October, the British fleet indeed attempted to break through the blockade, encouraged by favorable winds. But the fleet was defeated by the French forces at the dike, and the British sought and obtained a fifteen-day cease-fire with the French. It now seemed that the city was doomed, and the two sides met to arrange a retreat of the British fleet. As British officers discussed terms of the cessation of hostilities with their French counterparts, Descartes was there, meeting the British officers. This was his first meeting with English people. He knew that there were excellent scientists in Britain and hoped to meet them. According to Baillet, two years later, in 1630, Descartes would travel to London for a short visit, and there would carry out observations of the curvature of the earth.