The Burgi compass
Kenneth L. Manders of the University of Pittsburgh has studied the designs of compasses invented by Descartes and those of compasses described by Johann Faulhaber. According to Manders's interpretation of a particular Descartes text copied by Leibniz and then recopied in the nineteenth century by Count Louis-Alexandre Foucher de Careil of the Sorbonne, Descartes invented four kinds of compasses. Manders makes a striking observation. On December 19, 1620, Faulhaber advertised his skills in Ulm in an effort to attract students and obtain contracts for consulting jobs. A published description of Faulhaber's qualifications included the following:
In particular, four new proportional compasses, with which to find geometrically two mean proportionals between two given lines; just so, how to divide any angle on a circle geometrically into three equal parts; equally, how to obtain conic and cylindrical sections geometrically, a matter on which other authors have written large books. Moreover, how to show general rules for equations of arbitrary degree from the number 666.
Manders points out that Faulhaber's compasses are identical to Descartes' compasses, and since these are extremely unusual, special-purpose devices, it is certain that Descartes and Faulhaber met and exchanged very precise information. One of the compasses of Descartes (and of Faulhaber) can be used to solve one of the three classical problems of antiquity: trisecting an angle. But it should be noted that in the original Greek problem, one is to trisect an arbitrary angle using only a straightedge and a simple compass, which was available to the ancient Greeks, not the advanced device invented by Descartes, Faulhaber, Burgi, or any of them in collaboration. Note that, as also evidenced from Faulhaber's other works, his main interest in mathematics was motivated by his obsession with “biblical numbers” such as 666.
These interests make it almost certain that Faulhaber was indeed a Rosicrucian. In fact, in his book The Dream of Descartes, the French scholar Jacques Maritain says the following:
Now Faulhaber was a real Rosicrucian and a very ardent one, and one is justified in assuming, in spite of Baillet's denials, that Descartes found in him the man he was seeking, and that through him Descartes entered into direct contact with the intellectual atmosphere of the Rosicrucians. Might not such contact, however fleeting, have a deciding influence upon the moral lines and the aims of the philosopher's life? May we not even ask ourselves whether at its origin, Descartes' great idea did not permit the supposition that he intended—an intention that became hazier as time went on—fearlessly to transpose to the plane of everyday reason and of the most widespread common sense the design followed on the plane of alchemic mysteries by the naive Rosicrucians, and in so doing to render it much less “uplifted,” but much more efficacious—mathematics replacing the Cabala leading to universal knowledge, the hermetic sciences and their occult qualities giving place henceforth to geometric physics and the art of mechanics, as the elixir of life to the laws of rational medicine ?
According to the world's leading expert on Johann Faulhaber and his work, Kurt Hawlitschek of Ulm, Faulhaber was clearly a Rosicrucian. Hawlitschek hypothesizes in his book on Faulhaber that the meeting between Descartes and Faulhaber did not take place by chance. When Descartes was in Frankfurt to witness the coronation of the emperor, before proceeding to join Duke Maximilian's army, he met Count Philipp of Hesse-Butzbach (1581-1643). The count was interested in mathematics and also had Rosicrucian connections. Philipp sent Descartes to Ulm, which was close to where Duke Maximilian's army was encamped, with the purpose of having him meet Faulhaber so they could discuss mathematics.
Two other mathematicians belonging to the Brotherhood of the Rosy Cross were involved in astronomical calculations and the invention and use of compasses. They were Benjamin Bramer (1588-1652) and Faulhaber's friend the mathematician Peter Roth. Both Bramer and Roth are mentioned in Descartes' secret notebook. Descartes most likely met them in Kassel when he traveled to Germany in 1619, and was likely inspired by their work on proportional compasses, which led him to invent his own, related devices.
Faulhaber was the author of a treatise titled Numerus figuratus sive arithmetica arte mirabili inaudita nova constans, published in Germany in 1614 When this text was analyzed, after Pierre Costabel had deciphered Leibniz's copy of Descartes' secret notebook, striking similarities in the content of the two manuscripts were discovered. Did Descartes share his “admirable discovery” with his mystic friend Faulhaber? Or was Descartes influenced in his discoveries by the work of the latter? Either way, Faulhaber had at least some knowledge of Descartes' most profound secret.
The Rosicrucians and other mystics were looking for hidden meanings in numbers and shapes: arithmetic and geometry. Descartes was an expert in both areas. He, too, was on a search for meaning and understanding to be found in the realm of numbers and shapes. And the assumption that Descartes was inspired in his secret search by the work of Faulhaber would shed light on the meaning of his writings in the secret notebook. Hiding its content by using mystical symbols assured Descartes that no one uninitiated could understand the writings in his notebook, should it ever come to light.
In January 1618, Faulhaber mildly denied a connection with the Brotherhood of the Rosy Cross, saying: “I will not spare my zeal for obtaining the best information about the very dignified society of the Rosy Cross, but I believe that divine will has not yet determined that I should be worthy of making their acquaintance.” But this vow was soon broken when in July of that year he made the acquaintance of several members of the brotherhood, among them an important officer in the order, Daniel Mogling.
Edouard Mehl claims in his book that Faulhaber's mystical mathematical text, the MiracuL arithmetica of 1622, was published with the help of the Rosicrucians, among them Daniel Mogling. In this book, Faulhaber writes: “It is as impossible to separate the breadth of divine power from the number 666 as it is to separate this divine power from the holy evangelist.”
Daniel Mogling even lodged for a time in Faulhaber's house and was influenced by his host's mathematical work. He then changed the focus of his activities from medicine and alchemy to mathematics and astronomy and became the brotherhood's person in charge of these disciplines. Mogling himself had close connections with Johann Kepler. These included a correspondence on astronomical and mystical matters, as well as joint scientific work. Kepler's astrological work, his friendship with Daniel Mogling, and his stated desire for universal reform—echoing Rosicrucian writings—all make it clear that he had at least some association with the Brotherhood of the Rosy Cross.
Faulhaber was proud of having been able to predict, as early as 1617, the future appearance of a comet in 1618. He claimed to have made the prediction based on numerology and Cabbala. Faulhaber computed an astronomical table, and was looking for the apocalyptic number 666. He noticed in his table that the longitude of Mars and the latitude of the moon would both be 3 degrees, 33 minutes, on September 11, 1618. Then he knew that he had found what he was looking for, since 666=333+333. This meant to him that something important would appear in the sky on September 11, 1618. So Faulhaber predicted that a comet would be seen on that exact date. In fact, three comets were seen that year, the first one appearing in mid-October. When Faulhaber tried to publish his prediction, after the appearance of the first comet, he was accused of having used Kepler's mathematical tables for this purpose, rather than his own numerology.
This quarrel led to a public confrontation in Ulm on October 18, 1619, between Faulhaber and John-Baptist Hebenstreit, the principal of the high school of Ulm, who was an associate of Kepler. Hebenstreit submitted to Faulhaber eight questions about his use of biblical verbs as numbers; and finally, he concluded that Faulhaber's answers were unsatisfactory and blamed him for “mixing the heavens with the earth.” Hebenstreit summed up his tirade by accusing Faulhaber of “cabalistic log-arithmo-geometro-mantica.” This, in fact, was also the title of a short treatise he published.
Kepler himself never got involved in this dispu
te, finding it beneath his dignity. He harbored no animosity toward Faulhaber, and perhaps shared a kinship with his fellow mathematician and mystic. Kepler and Faulhaber shared a secret. They were among very few people who knew that Daniel Mogling was the author, using the pseudonym Theophilus Schweighart, of an important Rosicrucian text, the Speculum sophicum (1618). Hebenstreit held a grudge against Mogling, and after his attack on Faulhaber had failed, in part because of Kepler's noncooperation, turned his venom against Mogling, attacking him publicly in a very personal, insulting way.
One of Hebenstreit's associates wrote a treatise condemning the Rosicrucians and Faulhaber, titled Kanones pueriles, which was purportedly authored by Kleopas Herenius. This name is nothing but an anagram of Kepler's name in Latin: Iohanes Keplerus. Mogling himself, like many Rosicrucians, was fond of anagrams, for they worked well as a device for hiding identities. In 1625, Mogling published a book about perpetual motion, titled Perpetuum mobile, which he authored using an anagram of his first name in Latin, Danielis. The jumbled-up letters of “Danielis” gave him the name Saledini, its resemblance to “Saladin” lending it an Eastern, anti-Crusader flavor. This book was found in the inventory of Isaac Beeckman's library, which was part of his journal. Given the fact that Beeckman discussed most of his research and readings with Descartes, the latter likely was aware of Mogling's book.
Descartes' published works, and his letters, make it clear that he was influenced by the achievements and ideas of the two mathematicians Faulhaber and Kepler. Kepler's discoveries about the nature of the orbits of the planets in the solar system were of great interest to Descartes and, according to some scholars, were at the heart of his “admirable science.” And Descartes' secret notebook, by its unique use of symbols and by its content, echoed the writings of Faulhaber. Descartes' writings testify to the fact that he had at least exchanged ideas with the Rosicrucians.
Chapter 10
Italian Creations
AFTER TWO MONTHS AND A FEW days in Paris, a period during which he pretended to all but his confidant Mersenne that he had renounced the study of mathematics—to avoid the trap of being labeled a Rosicrucian—Descartes left for Rennes. He arrived there at the beginning of May 1623. From there, Rene went to Poitou and stayed until July. He managed to sell, with the consent of his father, most of his holdings in this region. On July 8, 1623, Descartes sold a large estate he inherited from his grandparents, called “the land of Perron,” to Abel de Couhe, a nobleman of Poitou, and the sale was sealed by the notaries of Chatellerault. He took the cash with him to Brittany, and after bidding his family good-bye, returned to Paris.
Descartes was unable to decide how to use all the money he had brought with him from the countryside. Much of it went into a bank account; he apparently wanted to place some of the funds in investments, but found none to his liking. He decided to use some of his new money to support a long trip to Italy.
Descartes wrote a good-bye letter to his father, saying “A voyage beyond the Alps would be of great use to me, to instruct me in handling my affairs, acquire some experience in the world, and form new habits that I do not yet have. If I will not become richer, at least I may become more capable.” Perhaps the young man felt he needed to justify to his father taking an expensive trip to be paid for by what was family money, while before that time, when he was at least nominally associated with an army, he could explain the travel as part of his occupation.
Descartes crossed the Alps and continued east to Zurich. He walked down the wide, cobblestoned Neumarkt Street with its medieval mansions and the towering church at the center of the old town. He sought out scholars and savants, and finding them, discussed nature and mathematics.
Descartes continued east to Tirol, and from there he descended to the plains of northern Italy, arriving in Venice just in time to witness the ceremony of the wedding of Venice with the sea on Ascension Day. Descartes arrived at the Church of San Nicolo on the Lido to view the bucentoro, a specially outfitted, gilded galley carrying the doge as he was rowed out to sea from the Port of San Nicolo. Descartes sat close to Venice's leading families and envoys from foreign states who had come to witness this unique ceremony. When the galley had traveled some distance from the port, the doge cast a golden ring into the waters of the Adriatic Sea and proclaimed by this act that he was taking dominion of the sea as a husband over his wife.
Legend has it that the pope had given a ring, and with it lordship over the Adriatic, to the doge in 1177. According to this legend, the Venetians defeated an imperial fleet, following which the Holy Roman Emperor Frederick I Barbarossa (Italian for “Red Beard”) came to Venice to kiss the pope's feet. However, the victory and the battle never took place, and were pure fiction. But the ceremony of the wedding of Venice with the sea was celebrated nevertheless, and four and a half centuries after this custom had begun, Descartes witnessed it.
From Venice, Descartes headed south to Rome—a city he had always longed to see, since it was the heart of the Catholic world and an exciting, vibrant cultural center. He passed through Loreto and fulfilled his vow to visit this religious shrine. After completing his visit to Rome, he was ready to return to France, but decided to stop in Tuscany. Descartes had heard much about Galileo and admired him greatly. He was hoping to meet him at his home in Arcetri. Descartes made it to Tuscany—but to his great disappointment was never able to meet Galileo.
April 1624 found Descartes in the town of Gavi, observing the military maneuvers of the duke of Savoy. Then in May he went to visit the city of Turin. Descartes climbed up into the Alps that rise over the border region between France and Italy. He spent some time in the mountains observing the melting snows and taking note of how thunderstorms developed. During his Italian trip, Descartes also observed rainbows and parhelic circles: luminous circles or halos parallel to the horizon at the altitude of the sun. These “false suns” appeared in Rome while Descartes was visiting that city and caused great excitement in the population. Everyone wanted to know how this phenomenon developed. Years later, when his Discourse on the Method and its scientific appendixes were published, Descartes gave his explanations of the natural phenomena he had observed on his trip to Italy. Descartes' Italian trip was an important event in his life since it allowed him to study nature. But there were other benefits to this voyage as well. Through his travels in Italy, the young man matured and developed a clearer sense of who he was and of the scholarly goals he would pursue in his life.
Descartes continued to Rennes, enjoyed the company of his family, and then moved back to Paris. Throughout this period, Descartes was working on algebraic problems studied by Italian mathematicians who had lived during the century before his time in the plains of the Veneto—the region of Venice, which he had just visited.
The Babylonians understood simple concepts of equations: given a specified set of arithmetical conditions, they could solve an equation to obtain the value of an unknown quantity. For example, they knew how to find the length and width of a field that was to have a specified area, given some condition on the two measurements. The ancient Egyptians also knew how to solve such problems. Papyri that survive, for example the famous Ahmes papyrus, now kept at the British Museum and dated to approximately 1650 B.C., show the solution of simple equations. For example, problem 24 in the Ahmes papyrus asks for the value of a “heap,” if a heap added to a seventh of a heap gives 19. Ahmes gives the answer as 16+1/2 + 1/8, which is 16.625, as we would obtain today by solving the equation x+(l/7)x=19.
The ancient Greeks were also able to solve equations, so that by the close of the classical period of ancient Greece and Rome, people knew how to solve some quadratic equations, that is, equations with highest-order term being ax1. But no general method had been known for solving such equations, nor was anyone able to solve higher-order equations (equations with powers higher than 2—for example, equations containing a term ax3).
The word “algebra” comes from the first two words in the Arabic title of a book writ
ten about A.D. 825 in Baghdad: Al-Jabr wa-al-Muqabahh, by Muhammad ibn Musa Al-Khowarizmi. In his book—the first important book on algebra—Al-Khowarizmi presented complete methods of solution for the quadratic equation. In Descartes' notation as we use it today, such equations are written in this general form—
ax2+bx+c=0
—and every high school mathematician knows the general formula giving the two roots, or solutions, of this equation.
While people knew how to solve quadratic equations, no one knew how to solve a third-order equation:
ax3+bx2+cx+d=0
How to solve such an equation remained a mystery for another seven centuries beyond Al-Khowarizmi—until four Italian mathematicians, working one against the other, endeavored to solve it.
Who were these Italian developers of algebra? They were four mathematicians who lived in the sixteenth century in northern Italy—the same area Descartes visited in his Italian trip in 1623-24 And they were a somewhat shady and unsavory group of individuals.
Niccolo Fontana (1499-1557), known as Tartaglia (“the Stammerer,” in Italian) was born in Brescia, in the republic of Venice, in 1499. As a fatherless boy of thirteen, he almost died in 1512, when French forces looted his hometown and many people were killed. The boy was dealt severe facial wounds from a saber that cut his jaw and palate, and he was left for dead. His mother found him and nursed him back to health. As an adult he grew a beard to hide his scars, and he could speak only with difficulty, hence the nickname Tartaglia.
Tartaglia taught himself mathematics. Having an extraordinary ability, he earned his living teaching mathematics in Verona and Venice. As a mathematics teacher in Venice, Tartaglia gradually acquired a reputation by participating successfully in many public competitions.
Descartes's Secret Notebook Page 10