Recently studied German documents give incontrovertible evidence that Descartes and Faulhaber did indeed meet. In 1622, Faulhaber published a mathematical text titled Miracuh arithmetica. In this book, he provided methods for solving quartic (that is, fourth-degree) equations, and these methods are virtually identical to those that Descartes gives in his own book, the Geometrie, published in 1637. In his book published in 1622 Faulhaber wrote:
Since this noble and very knowledgeable sire, Carolus Zolindius (Polybius), my most favored sire and my friend, has let me know that he will soon publish, in Venice or in Paris, such tables…
Faulhaber clearly knew someone named Polybius. And Descartes' hidden manuscript, the Preambles, seen and copied by Leibniz as well as studied by Baillet, clearly stated that Descartes planned to write a book about a mathematical truth using the pseudonym Polybius the Cosmopolitan. So Polybius and Descartes were one and the same, and as Faulhaber's book mentions Descartes' pseudonym, this is strong evidence that Descartes and Faulhaber knew each other.
According to Edouard Mehl, a professor of history at the University of Strasbourg who has made a careful study of this issue, Descartes indeed published another book, titled Thesaurus mathematicus, under the pen name Polybius. Furthermore, Descartes was traveling to Paris regularly, and while he made it to Venice only in 1624, he had already decided to go there as early as 1620. Thus the claim that Carolus Zolindius (Polybius) planned to publish the mathematical tables mentioned by Faulhaber in either Paris or Venice accords well with Descartes' movements and planned travel during that period. Descartes' works would indeed be published in Paris, and Venice had been an important publishing center ever since the printing presses were invented, and was on Descartes' itinerary.
Mehl concluded that Faulhaber and Descartes were close friends. He argued that “Polybius” was known to Faulhaber as Descartes' pen name, and that Faulhaber was in the habit of calling him by another secret name, Carolus Zolindius, but in his writings also indicated Descartes' other pen name in parentheses. Dr. Kurt Hawlitschek of Ulm, a leading expert on Faulhaber, further notes in an article about Descartes' meeting with Faulhaber that “Polybius” can be seen to mean “Rene” (reborn), because the Greek root poly means “more,” and bios means “life.” This may have been why Descartes chose this particular pseudonym. When Descartes' secret notebook was analyzed, after Faul-haber's book had come to light, it became evident that Descartes' notebook included, in part, answers to questions raised by Faulhaber.
It is believed that Johann Faulhaber (1580-1635) was born in Ulm and was trained as a weaver. He studied mathematics and was very successful in doing mathematical work. The city of Ulm then appointed him city mathematician and surveyor. In 1600, he also founded his own school in Ulm. Faulhaber's work was in high demand because of his great mathematical skills, and he was often employed in fortification work in Ulm, as well as in Basel, Frankfurt, and other cities. He designed waterwheels and made mathematical and surveying instruments, especially ones with military applications. Faulhaber worked with compasses of various kinds. As mentioned earlier, Faulhaber knew Johann Kepler. He may have worked with him on a number of joint mathematical projects.
Faulhaber studied alchemy, a mystical pseudochemistry whose main purposes were to convert base metals into gold, to find a universal cure for disease, and to discover a way to prolong life indefinitely. He used alchemical and astrological symbols in the work he did in algebra. His work in algebra was extremely important: he studied sums of powers of integers, and his development of results in this area was affirmed by mathematicians who lived after his time.
A further confirmation that Descartes and Faulhaber met comes from the fact that Descartes learned to use the very same symbols Faulhaber had used, symbols commonly found in writings about alchemy and astrology. One of Faulhaber's special symbols appears in Descartes' secret notebook copied by Leibniz. It is the alchemical and astrological sign of Jupiter, shown below.
Faulhaber's sign of Jupiter found in Descartes' secret notebook was one of the stumbling blocks to understanding the content of his notebook. No one who studied Leibniz's copy could understand what the sign meant—not until the French scholar Pierre Costabel figured out Leibniz's key to Descartes' mystery. The discovery of Faulhaber's books in Ulm has confirmed that part of the notation used by Descartes had been adapted from Faulhaber's symbols.
We also know that some of the mathematical methods used by Faulhaber were also later employed by Descartes, which makes it virtually certain that the two men indeed met and exchanged mathematical ideas. Both men worked in the same area of algebra: they were interested in extending the work started in the previous century by a quarrelsome group of Italian mathematicians on the solution of cubic equations, that is, equations of the form axJ+bxl+cx+d = 0.
Faulhaber's interest in mathematics was propelled by his passion for mysticism. He was inspired by the Jewish mystical tradition of Kabbalah. The Kabbalists look at letters of the Hebrew alphabet, attaching to each of them a numerical value (thus aleph=l, beth=2, and so on). By summing the numerical values of all the letters in a word, the Kabbalists seek a hidden meaning by finding other words that have the same numerical sum. The Christian Cabbala is also concerned with numerical values and their symbolism. One key example is the search for the number 666, associated with the beast of the Apocalypse. Revelation 13:18 reads: “This calls for wisdom. If anyone has insight, let him calculate the number of the beast, for it is man's number. His number is 666.” Through his very advanced work in mathematics, Faulhaber was searching for significant biblical numbers such as 666. He tried to solve equations and carry out computations that would result in the number 666.
“The first person Descartes met in Ulm was Sire Johann Faulhaber,” Baillet tells us, describing the meeting between Descartes and Faulhaber. Descartes came to Faulhaber's house and the mathematician asked him: “Have you spoken of analysis and of the geometers?”
“I have,” said Descartes.
“Well, will you be able to solve my problems ?”
He handed Descartes a copy of his book. Descartes took the book and looked at the problems of geometry Faulhaber had described in the book. He solved a number of the problems and handed the solutions to his host. Faulhaber laughed. He pointed out harder problems in the book, and Descartes solved those as well.
“Come, now,” said Faulhaber. “I want you to enter my study.”
As he walked in with his host, Descartes read above the doorway, in German, “Cubic Cossic Pleasure Garden of All Sorts of Beautiful Algebraic Examples.” Descartes came into Faulhaber's study, and his host closed the door. Descartes saw bookshelves all around him, overflowing with books. The two men discussed mathematics well into the night, and Faulhaber gave Descartes another book he had written, in German, about algebra. The book was filled with abstract questions with no explanations. Faulhaber asked Descartes for his friendship. Descartes agreed, and Faulhaber said: “I want you to enter a society of work with me.” Descartes found that he could not refuse this offer. “Good,” said Faulhaber. “Now I would like you to see a book that has been given to me,” he said, and he handed Descartes a book written by another German mystic-mathematician, a man named Peter Roth (or Roten). Descartes looked at the problems in Roth's book, and solved them as well. Roth had died a few years earlier. Descartes may not have been aware of it at the time, but Faulhaber and Roth were the two most able mathematicians whose works were associated with a mysterious society so secret that its members were known as “the Invisibles.”
Early in November 1620, Descartes and his valet left Ulm and traveled northeast to rejoin the rest of Duke Maximilian's troops, which by now were converging on the city of Prague. Before Descartes had a chance to further study the ideas of ancient Greek geometry and to try to solve the Delian mystery, or fully explore the stimulating problems Faulhaber had posed to him, he was finally called to his first battle. He was eager to see fighting, perhaps as eager as he was to di
scover truth through science and mathematics.
The forces of Maximilian of Bavaria—who led the German Catholics in this conflict—joined the other armies encircling Prague, all preparing to do battle with the forces of Frederick V, king of Bohemia, defending the city. Descartes and his fellow soldiers were quickly marshaled and made ready for an attack on the city. On November 7, some of the defending forces were able to slip through and regroup on the White Mountain outside the capital. The defenders of Prague had a force of fifteen thousand men, supported by artillery. The attacking force, which included Maximilian's Catholic League as well as imperial troops, was twenty-seven thousand strong. The great battle of Prague had begun.
The defenders enjoyed a quick first victory as their cavalry, supported by cannon, overcame some of the invading troops. But soon the tide turned as the much larger combined powers of the enemy overwhelmed them. By evening on November 8, two thousand men of the Bohemian army lay dead, while the Catholic attackers lost only four hundred. It was clear to the survivors that Prague would soon fall. Frederick V of Bohemia, with his wife, Elizabeth, and their family, hid in the Old Town of Prague, and the king made a hasty plan to smuggle them all out of Bohemia and seek refuge in Silesia.
In the evening, the attacking armies brought their cannon and infantry close to the walls of Prague, after the villages outside the city walls had all surrendered to the Bavarian and imperial armies. On November 9, Descartes and the victorious armies entered the city of Prague. This was the young man's baptism by fire, although Baillet tells us that Descartes did not take part in the actual fighting since he was a volunteer. As Descartes and the other soldiers entered the defeated city, a carriage passed them leaving the city in a hurry. Aboard it was King Frederick V—who would derisively be remembered as “the Winter King” because he lasted only one season—with his family. The departure of the king and queen of Bohemia was so humiliating that they were not even able to take any of their possessions with them. They would be reduced to poverty for the rest of their days and would be treated with contempt both by their enemies and by their erstwhile supporters who had held such high hopes for their reign. Thereafter, only a Habsburg would sit on the throne of Bohemia. One of the members of the fleeing royal family was a two-year-old girl who, like her mother, was also named Elizabeth. Descartes and Princess Elizabeth, unknowingly passing each other in the night, would meet twenty-three years later, and she would become one of the most important people in his life.
Descartes was in Prague with the celebrating troops the next day, November 10, the anniversary of the night of his dreams in the oven, the year before, as well as of his meeting with Beeckman, two years earlier, and of his law thesis, four years earlier. As fate would have it, on this very anniversary of the three watershed events in his life, a fourth event of great importance for Descartes was to take place—most likely inside the city of Prague. While Descartes was walking the streets of this medieval walled city with its ancient towers, majestic bridges on the Vltava River, and magnificent churches, he had a revelation. For it was on the next day that Descartes wrote in the lost Olympica: “November 11,1620.1 began to conceive the foundation of an admirable discovery.”
What was this discovery? And how was it related to the discovery he began to make a year earlier, in 1619? In her definitive biography, Descartes, Genevieve Rodis-Lewis attempts to identify the nature of the achievement about which Descartes is silent. She believes that Descartes' discovery was initiated in 1619 but completed in 1620, and that it is unlikely that he refers here to any of the material he would later include in the Discourse on the Method (1637) or its scientific appendixes, since those developments were more involved than any single discovery. His work on unifying algebra and geometry could hardly be traced to a single moment of revelation. Rather, Descartes' rapturous discovery after the heat of battle and the intoxicating influence of victory in Prague led to knowledge he decided to hide—knowledge he inscribed only in his private notebook, written in Latin. The mystical nature of the secret notebook had to have been derived from the influence of Johann Faulhaber on Descartes, especially as reflected in the symbols of alchemy and astrology.
Descartes stayed in Prague until December of that year. The baron of Tilly was left behind in Prague with a garrison of six thousand men, while the rest of the Bavarian troops, led by the duke of Bavaria, left the city. Descartes was there with Maximilian's troops moving to their new winter quarters in the extreme southern part of Bohemia. Six weeks in the capital of Bohemia was enough time for Descartes to learn about the city. While other soldiers pillaged, Descartes found interesting conversations and discussions with the curious and the savants of the city. His greatest pleasure during his stay in Prague was learning from local scholars about the work of the astronomer Tycho Brahe, who had worked in this city, and his greater former assistant, Johann Kepler.
Out of the city in the troops' winter quarters, Descartes once again found himself seeking solitude in his room, spending all his time meditating and studying. He resumed his analysis of geometry, but found himself also mulling his destiny and the road he should take in life. Descartes decided that he wanted to see more action, and more of the world. Staying with the stationary forces holding Bohemia was not to his liking, and at the end of March 1621, he quit the service of Maximilian of Bavaria. Descartes did not want to return to France because he knew that the plague was ravaging Paris; this plague would end only in 1623. So he took his time, traveling north to explore the parts of northern Europe he had not yet seen.
Descartes came back to Holland and visited his friend Beeckman. Isaac Beeckman had had several life-changing experiences in the intervening years. At the end of November 1619 he finally got his position as assistant principal of the Latin School of Utrecht, and five months later, having the stability of a job and an income, had married a woman of Middleburg, on April 20, 1620. Apparently he had been unable to meet someone suitable in Breda, as he had hoped to do two years earlier.
Descartes was happy to see his friend and congratulated him on his marriage. The two men resumed their joint work on mathematics, music, and mechanics. But Rene confided in Isaac that he had decided to keep some of his own, separate work on mathematics strictly secret. He had reasons for doing so.
Chapter 7
The Brotherhood
DURING THE TIME DESCARTES WAS in Germany and Bohemia, educated Europeans talked about nothing but the emergence in Germany of a secret society of savants known as the Brotherhood of the Rosy Cross. Books purportedly written by members of this society had begun to ap-pear in print just a few years earlier. Descartes' friends, who knew he was engrossed in science and dedicated to the pursuit of truth, naturally assumed that he was a member of this newly established brotherhood of scholars. According to Baillet, Descartes indeed wanted to get to know the members of this mysterious order devoted to knowledge, and to join their ranks.
“The solitude he endured that winter [of 1620] was always complete, especially with respect to people who were unable to help him progress in his ideas,” Baillet tells us. But, he continues, this did not exclude from Descartes' room people who could discuss with him the sciences or bring him news about literature. “It was through conversations with the latter that he learned about a fraternity of savants, which had been established in Germany for some time, under the name of the Brotherhood of the Rosy Cross. His new friends spoke admiringly, but in hushed tones, about this secret society. They told him that the brothers of this fraternity were men who knew everything. They were the masters of every science—they possessed all knowledge, they said: even knowledge that had not yet been divulged.”
Descartes saw in these conversations in his “oven” a sign that this was the direction God wanted him to take in order to pursue his destiny of unifying the sciences and searching for knowledge and truth. He yearned to meet these unknown scholars and to join their mysterious organization. Baillet reports that Descartes confided to a friend his view that the Brothers
of the Rosy Cross could not be impostors since “it would not be right that they should enjoy a good reputation as possessors of the truth at the expense of the good faith of the people.” He decided to make an effort to find them. But here Descartes came up against an insurmountable difficulty. By their very constitution, the Brothers of the Rosy Cross—also known as the Rosicrucians—were unidentifiable. People called them “the Invisibles.” They were no different in appearance and habits and customs and everyday behavior from the rest of the population. And their meetings were secret and closed to outsiders.
Despite all his efforts and the inquiries he made to everyone he knew, Descartes was unable to find a single person who would confess membership in the Brotherhood of the Rosy Cross, or who was even suspected of such membership. He was apparently completely unaware that he had already met, and developed a friendship with, one of the most prominent mathematicians associated with the Rosicrucian order— Johann Faulhaber.
The Brotherhood of the Rosy Cross was a secret mystical society of scholars and reformers established in Germany in the early part of the seventeenth century. The symbol of the society was a cross with a single rose in its center.
The story of the establishment of the Brotherhood of the Rosy Cross is truly fantastic. This story is told in the first Rosicrucian text, the Fama jraternitatis, published in 1614; it is repeated almost verbatim in Baillet (1691), as well as in several other sources such as Heindel and Heindel (1988). The original founder of the society was a German man of a poor family, but with noble origins, who was born in 1378. His name was Christian Rosenkreuz (in German, “Rosy Cross”), leading to the name of the society.
Descartes's Secret Notebook Page 7