In this sense, then, one could view the distance extending to the right of the bottom left corner as 1, and the distance extending up from that corner also as 1, measured in the perpendicular direction. This gave Descartes the idea to formalize the observation of the ancient Greeks to create a coordinate system. Descartes realized that any point in the plane could now be described in terms of both its x-coordinate and its y-coordinate.
This breakthrough opened up a new world for Descartes—and would create a new world for science. But in particular, Descartes now knew what was constructible with the ancient tools of straightedge and compass, and how to construct it.
Descartes observed that a number, a, is constructible if we can construct two points that are a distance of a units apart. In his new coordinate system, a number a is constructible if Descartes can construct either the point (a, 0) or the point (0, a). Descartes observed that if a and b are constructible numbers, then so are the numbers a+b, a—b, ab, and alb. These properties are demonstrated in the figures below.
This was a great advance. The creation of the Cartesian coordinate system immediately taught Descartes much about numbers that could be constructed with straightedge and compass. But as we will see, more could be done with these instruments: the numbers that can be constructed are more extensive than just sums, differences, products, and quotients of constructible numbers.
Having used his new coordinate system to produce such fruitful results, Descartes took one big step forward. He was able to show that it is possible to use straightedge and compass to construct the square root of a number. The reason this is such an important, and perhaps unexpected, result has to do with the fields of numbers we are considering.
The rational numbers form a field. This means that if you start with rational numbers you stay with rational numbers: you stay within the same system, the same field, by taking sums and products and inverses. Every rational number—that is, either an integer or a ratio of two integers—has an inverse that is also a rational number. Simple examples are 7, whose inverse is 1/7. Or —15, whose inverse is —1/15. Or 3/19, whose inverse is 19/3. But generally, square roots are outside the field of rational numbers (except for the trivial cases: for example, the square root of 4 is an integer, 2). The square root of 2, for example, is outside the field of rational numbers, since no simple arithmetical operation on rational numbers (fractions whose numerators and denominators are integers) will produce the square root of 2.
But Descartes was able to show that constructions with straightedge and compass can still lead to square roots of numbers. This was one of his greatest achievements in mathematics, and the proof appears on the second page of his Geometrie. In providing this amazing proof, which would have stunned the ancient Greeks since they could only construct much simpler things, Descartes showed that the field of all the numbers that can be constructed with straightedge and compass is greater than the field of rational numbers since it now also includes square roots. Descartes was not able to show that this field includes cube roots of numbers, or any higher-order roots.
In fact, it would be shown two centuries after Descartes, through the work of the tormented genius Evariste Galois, a French mathematician who died in a duel at the age of twenty, that cube roots are not constructive, and that neither are any higher-order roots.
Descartes understood that it was this property—namely, that the straightedge and compass can go as far as square roots, but not far enough to cube roots—that made the Delian problem of the doubling of the cube impossible. It is important to note, again, that Descartes made a great stride forward by proving that square roots are constructive, but that he did not prove that cube roots are not. He “understood” that they were not, but the actual proof would require Galois Theory.
In a sense, the result could be seen as intuitive: since straightedge and compass are devices that work on the phne, they allow us to take square roots (recall that the square root of a square—a figure that lies in the phne—is the side of the square, also lying in the plane), but not cube roots. A cube naturally lives in three-dimensional space, and the cube root of the cube is the side.
Here is how Descartes proved that square roots are constructible with straightedge and compass: Descartes constructed, with the Greek straightedge and compass, the figure on page 166.
He now used the Pythagorean theorem three times. He obtained— from observing the three right triangles in the figure,
Opening up parentheses and substituting the second equation into the third, he got
a2+2a+l=c2+l2+b2
Now substituting for c2 the sum al+b2 using the first equation, he got that the equation above is
a2+2a+l=a2+b2+l2+b2
That gave him
2a=2b2
Or b=√a. Thus, using straightedge and compass, one can construct square roots. Voila!
Descartes realized the following: doubling the cube was an operation in three-dimensional space, and hence the straightedge and compass— inherently plane, or two-dimensional, instruments—could not provide a solution. Equivalently, using algebra, he noted that to double the cube was equivalent to constructing the cube root of 2. He had proved that the square root of 2 was constructible, but he understood (although his proof was inadequate, and the correct one would come two centuries later) that the cube root was not. Descartes began to think about higher dimensions—he was captivated by the mathematical properties of the cube, and by the mystical aspects the Greeks attached to this perfect three-dimensional object.
Chapter 16
Princess Elizabeth
WHILE WORKING OUT THESE GREEK problems, Descartes continued his wanderings through Holland. He lived in Eg-mond, then in Santpoort, and then near Haarlem. Throughout his travels, he received mail that was forwarded to him and followed his trail. One day Descartes received a letter about a princess. Princess Elizabeth of Bohemia was also living in Holland—in exile. As a young child she escaped from Prague with her parents; as we've seen, her father, Frederick V, had been deposed as king of Bohemia right after Descartes and the victorious Bavarians and imperial troops stormed into the defeated city in 1620.
Frederick then died of the plague in Mainz in 1632, at the age of thirty-six, leaving a widow and nine children, four princesses and five princes. Princess Elizabeth was the oldest of these children. Since the deposed king's mother was the sister of Prince Maurice of Nassau (Descartes' former military employer), his widow and children had a right to seek refuge in Holland. To her dying day in exile, Elizabeth kept the title of queen of Bohemia. Her grandson would become King George I of Great Britain.
Life in exile was not easy for Elizabeth and her family. While he was still alive, Elizabeth's father, Frederick, tried from time to time to enjoy some of the pleasures he was once accustomed to as king, albeit on a much smaller scale. One day, he took his dogs and horse and went on a hunt. He was chasing a hare through the countryside when his dogs led him across a cultivated field. Before he knew it, an irate hulking peasant came out, wielding a pitchfork. “King of Bohemia, king of Bohemia!” he bellowed, apparently recognizing the exiled monarch, “you have no right to trample on my turnips like that! I'll have you know that I have worked hard to sow them.” The deposed king apologized and quickly moved off the field, saying that his dogs had brought him there despite all his good intentions. J.-M. and M. Beyssade, who recount this story in Descartes: Correspondance avec Elizabeth., point out that in other circumstances, the rustic would have been severely punished for his insolence: in France, they surmise, he would have been put in irons; and a German prince would certainly have set his dogs on the impudent peasant.
The young Elizabeth had a thirst for knowledge, ever seeking to improve herself. She had read a Latin translation of Descartes' book the Discourse on the Method and wanted to learn more about his philosophy. Elizabeth was interested in all the philosophical questions that Descartes had written about. She wanted to find answers to questions about his metaphysics; she was intere
sted in the relationship between body and soul; and she wanted to know more about his proofs of the existence of God. In addition, Elizabeth was interested in mathematics, and in particular wanted to learn how Descartes solved problems in Greek geometry and to try her hand in solving such problems using his methods.
Princess Elizabeth was acquainted with a man whose origins were in Piedmont, named Alphonse Pollot (originally Pallotti), who had known Descartes and had renewed his friendship with the philosopher after reading Descartes' book himself. Pollot wrote to Descartes that the princess was interested in meeting him. At that time, Descartes lived not far from the deposed royals' abode. Flattered by the interest of a princess, Descartes agreed to meet her. He wrote to Pollot that he would come to her town (coincidentally named La Haye, the name of his own birthplace in France) and would “have the honor to bow to the princess and receive her orders. As for that which I hope would happen next…” Evidently, the aging philosopher did entertain hopes of something more.
Princess Elizabeth was twenty-four years old when she met Rene Descartes in 1642. At forty-six, he was almost twice her age. His nascent relationship with the princess made Descartes leave his more remote dwellings in Holland and move to Leyden and its vicinity so he could be close to her. Elizabeth became a student of Descartes' philosophy. According to Baillet, “Never had a master profited more from the penetrating mind and solidity of spirit of a disciple. Elizabeth was capable of profound meditation on the greatest mysteries of nature as well as geometry.”
Elizabeth spoke perfect German like her father, perfect English like her mother, was proficient in French, and had learned Italian and Latin. She was educated in the sciences as well, and had good ability and great interest in mathematics and physics. She was described as beautiful, and looked even younger than her twenty-four years. In his letters, Descartes would describe her as an angel. She would end all her letters to the philosopher with “Your most affectionate friend to serve you.”
There developed a tender relationship in which the two of them exchanged ideas. Many letters between Descartes and Elizabeth survive, and they paint a picture of a very lively and eager young woman, interested in learning from the older philosopher. Elizabeth was an excellent mathematician and understood Descartes' science and philosophy equally well. Descartes once said to her: “Experience has shown me that the majority of people who have the ability to understand the reasoning of metaphysics cannot conceive those of algebra; and reciprocally, that those who understand algebra are ordinarily incapable of understanding metaphysics. And I see only Your Highness as a person for whom both disciplines are equally easy to understand.”
Princess Elizabeth
Their letters were affectionate, but they give no hint of the true nature of their relationship. The reason for the ambiguity of the letters is that Descartes and Princess Elizabeth would meet and talk face-to-face, and most of the letters between them were only written later, once she was forced to leave Holland. When that happened, her letters always passed through the hands of her siblings, and therefore intimate details could not be included.
Later events, however—having to do with Descartes' decision to move to Sweden to become tutor to Queen Christina—hint that Elizabeth may have been jealous of his new interest. Such jealousy, which was mentioned in Descartes' letters, may be an indication of a deeper attachment. Descartes guarded his privacy, making it impossible to determine the true nature of his relationship with the princess. One of his biographers, at least, did claim that Descartes and the princess had an intimate relationship. Descartes had been unable to publicly marry Helene because she was a servant and thus socially beneath him. Equally, Princess Elizabeth was above him and most probably could not marry him. For this reason, the two kept the nature of their relationship secret. But as friends, at least, they were exceptionally close.
When Descartes moved farther north in Holland, in 1644, and was now a day's journey, rather than two hours', from La Haye, Elizabeth wrote him lamenting the distance that now separated them. But Descartes moved more and more frequently now. He felt uncomfortable staying at one location for long—perhaps experiencing the feelings of persecution from the onset of the academic controversy that would later erupt into what became known as the “Quarrel of Utrecht.” And perhaps he was eager to hide from the world the true nature of his relationship with Elizabeth, and staying close to her would have betrayed the secret.
In May 1644, Descartes returned to France for an extended stay— his first visit to his native land in sixteen years. He lodged with a friend, Abbe Picot, in the Marais—in the rue des Ecouffes between the rue du Roi de Sicilie and the rue des Francs-Bourgeois. Later, and during two other stays in Paris, Descartes rented an apartment just behind today's Place de la Contrescarpe.
Elizabeth wrote him letters to Paris, inquiring about physics and mathematics. From there, Descartes went south to the lands of his birth, visiting Blois (near Tours), Tours, Nantes, and Rennes, staying with his older brother, visiting his half-brother Joachim and his brother-in-law Roger, the widower of Rene's sister Jeanne. From there he wrote to the princess, promising her: “I hope, in three or four months, to have the honor of paying you a visit in La Haye.”
Descartes would keep returning to visit Elizabeth, and while away from her, he would write often. But soon an invitation would come that would take him away from her, despite her protestations, and bring him to the court of a queen. There, he would take all his secrets. In his biography of Descartes, Stephen Gaukroger reports the hypothesis that Descartes left Holland for Sweden in order to intercede on Elizabeth's behalf with Queen Christina. According to this theory, Descartes was in love with Elizabeth and was heartbroken by her poverty in exile, so unfitting for a princess. Supposedly, Descartes was hoping to convince the queen of Sweden to take care of a fellow royal in distress.
Suddenly, Elizabeth had to leave Holland and seek refuge in Germany. Two of her brothers had moved to England, to stay with the royal family, their uncle and aunt. A third brother, who remained in La Haye, got involved in a personal dispute with a Frenchman from Touraine, a M. d'Espinay, who was hiding there as the result of a love scandal back in France. Elizabeth's brother and the young man got into a fight in the spice market in town, and as a result the Frenchman died. Elizabeth's mother was furious and blamed Elizabeth for inciting her brother, a charge she vehemently denied. But the mother said, “I never want to see either of you again,” and the two siblings left for Germany. What was to be a relatively short stay eventually became permanent exile.
King Wladyslaw IV of Poland asked Elizabeth to marry him after his wife suddenly passed away, but she flatly rejected the royal offer. She was “in love with Descartes' philosophy,” she said, and wanted to devote her life to studying it. From Berlin, Elizabeth wrote even more frequently to Descartes. These letters were exchanged through an intermediary, her younger sister Sophie. These particular letters are now lost. We do know, however, that one letter in particular contained sensitive information, for in a subsequent letter, which survives, Elizabeth asked the philosopher to burn it.
Elizabeth lived with various members of her large family, and often moved from castle to castle in Germany. For a time she lived with her brother Charles Louis, who had become an imperial elector, in his castle in Heidelberg. She also lived for a time in Brandenburg with another royal relative. She would often go with friends and relatives to Berlin to listen to music or see a play. But her greatest occupation was pursuing Descartes' philosophy—something she would continue to do after Descartes' death and her own entry into a convent in Westphalia, where she would end her days. At the convent, Elizabeth would establish a salon for Cartesian philosophy and would tell her guests that she had known the philosopher well.
Some years later, Elizabeth's sister Princess Sophie also moved to the castle in Heidelberg to live with her brother the elector. She stayed there until her marriage to the duke of Hanover, who would become Leibniz's patron. Through this connecti
on, Sophie would develop a close friendship with Leibniz.
Chapter 17
The Intrigues of Utrecht
IN 1647, DESCARTES BECAME INvolved in one of the most vicious academic confrontations in history. It is hard to understand why Descartes got into the deep trouble he did. Great forces that had opposed his philosophy were finally converging to attack him simultaneously in Jone coordinated pile-on. For six years, from 1641 to 1647, Descartes had been living peacefully in the Dutch countryside, working on two books, Passions de I'ame (“Passions of the Soul,” which would be published in 1649 and was concerned with the distinctions between body and soul) and Prindpes de philosophie (“Principles of Philosophy,” an extension of his philosophical ideas, published in 1647). But Descartes was increasingly being harassed by Dutch academies and others who were opposed to his philosophy on various grounds.
This philosophy became popular in the years following the publication of Descartes' Discourse on the Method, and Cartesianism began to be taught at universities in Europe. But since Descartes' ideas were clearly contrary to the accepted scholastic tradition—the legacy of the Middle Ages—with the interest in Cartesian principles also came a growing opposition from people who held on to the old beliefs.
Jean-Baptiste Morin (1583-1665), a mathematician, physician, and astrologer whom Descartes had befriended in Paris in the 1620s, turned against him and first attacked his work in 1638. Morin was a proponent of the geocentric theory of the universe endorsed by the church, and saw in Descartes' scientific work a dangerous form of thinking. Morin challenged Descartes' entire approach to science, questioning his results in physics. He wrote him saying that science based on mathematics must not “depend on any opinions drawn from physics.” Morin sought in this way to detach science from what he saw as the evil influence of Copernican ideas in physics, fearing that they could contaminate mathematics.
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