Descartes interpreted the melon in the first dream as the charm of solitude. He interpreted the violent wind of his first dream, pushing him hard against the church, as the evil spirit, bent on forcing him into a place to which he was going to go of his own free will. It was for that reason that God did not allow him to advance too far away from his destiny, even though the evil spirit was directing him into a holy place.
Descartes interpreted the great sound of the second dream, a lightning and thunder that turned into sparkles of light inside his room, to represent the spirit of truth that came to possess him. Descartes now had an answer to his question posed in the third dream by the first poem of Ausonius: Which road should I choose in life? And the answer was that his mission in life was to unify the sciences. Having had his first revelation, upon solving the Dutch puzzle, that mathematics was his gift, Descartes now understood that unifying the sciences meant work in mathematics. His philosophy—his search for absolute truth and his principle of doubt—which he would develop in the years to come, was his attempt to impose reason and rationality on the universe using the principles of logic and mathematics. His philosophy would thus be inexorably linked with his geometry. But first, then, Descartes' charge was to develop his geometry—to bring its ancient Greek principles to the seventeenth century, in which he lived, and ultimately to bequeath to the world the new science he would create: analytic geometry.
Descartes spent the entire following day reflecting on his dreams. He thought that the spirit in the dream, a “genie” as he called it, had inserted the dreams into his head even before he had gone to bed, and that human elements had no effect on anything that followed. His dreams had been completely predetermined. Descartes reflected long about the dreams, and asked God to make him know his will and to conduct him toward the truth. He vowed, in return, to make a trip to Italy for a pilgrimage to one of the most important religious sites in the land: the shrine of Loreto, believed to include the cottage from Nazareth in which the Holy Family had lived. He wanted to leave for Italy by the end of November, but in the end, he would travel there only four years later. Descartes remained in the solitude of his oven and made a vow to write a treatise and finish it by Easter 1620. According to Baillet, the Olympica was written at this time, but the biographer felt that there was little order in the mysterious fragments of this manuscript, so that Descartes' vow to write a text must have meant a more significant manuscript than Olympica, and that this larger work was merely announced in the Preambles and the Olympica.
On November 11, 1619, Descartes described in his parchment register the dreams he experienced the night before, and his interpretation of these dreams. In their essence, the copy made by Leibniz (now held with the rest of his papers in the archives of the Gottfried Wilhelm Leibniz Library in Hanover) and Baillet's account agree with each other. The Leibniz copy of Olympica was discovered in the Hanover archives by Count Louis-Alexandre Foucher de Careil (1826-91), a professor at the Sorbonne who was researching Leibniz's work. Foucher de Careil published his discovery of the Leibniz copy of Olympica in 1859 under the title Cogitationes privatae. Apparently, he understood well that Descartes considered this manuscript, and other papers he left behind in Stockholm, as private. As such, they were written in Latin. Treatises Descartes intended for publication were written in French, to afford them a wide audience in his native country. Descartes, who described his progress through life by saying, “I advance masked,” wanted to keep certain things hidden. He had reasons to maintain his secrecy.
According to the inventory made at the time of Descartes' death, Olympica (Item 1C) was “a small register in parchment, of which the inside cover bears the inscription: Anno 1619 Kalendis Januarii.” But Descartes began to write in the notebook only much later that year, in the month of November, when he wrote, before his night of dreams:
X. Novembris 1619, cum plenus forem enthousiasmo, & mirabilis scientiae fundamenta reperirem…
[November 10, 1619, as I was filled with enthusiasm, and I discovered the foundations of an admirable science…]
What was the great discovery Descartes made on November 10, 1619, which filled him with such enthusiasm? Scholars have contem plated the word “enthusiasm” used by Descartes, looking for hints about the nature of his discovery. Recently, researchers have found a striking similarity between the way Descartes reported his discovery and another one made a few years earlier by the brilliant German astronomer and mathematician Johann Kepler (1571-1630), the man who discovered the laws of planetary motion.
Did Descartes ever meet Kepler? The Kepler scholar Luder Gabe has conjectured that such a meeting indeed took place. On February 1, 1620, John-Baptist Hebenstreit, the principal of the high school of Ulm and an associate of Kepler, wrote to Kepler in Linz, inquiring whether Kepler had received the letters that a certain Cartelius was supposed to have brought him. Hebenstreit wrote: “Cartelius is a man of genuine learning and singular urbanity. I do not wish to burden my friends with ungrateful vagrants, but Cartelius seems a different kind of person, and truly worthy of your consideration.”
Gabe identified Cartelius with Descartes, whose name, in Latin, is Cartesius. In fact, scholars today still refer to Descartes as Cartesius. Kepler's editor Max Caspar noted that a long s could have easily been misread as an I. Cartesius (Descartes) could well have brought the letters in question to Kepler and made his acquaintance. Gabe has hypothesized that at some point in his travels, Descartes studied optics with Kepler in Germany.
Whether or not a meeting of the two great mathematicians ever took place, some of Descartes' ideas agree with those of Kepler. Descartes became aware of Kepler's work through his friend Beeckman. He knew all of Kepler's major works; and in Descartes' published work on optics, the Dioptrique, which would appear as an appendix to his Discourse on the Method in 1637, he would write that Kepler was “my first master in optics.”
When Kepler was twenty-three years old—exactly Descartes' age when he wrote the Olympica in 1619—he too wrote about a great discovery that filled him with enthusiasm. Kepler was looking for a mystical link between ancient Greek mathematics and cosmology. He made what he thought was a stunning connection, and published it in Mystenum cosmographicum (1596). Kepler wrote of his rapturous moment of discovery about the planets, again employing a word Descartes would later use, calling it an “admirable example of [God's] wisdom.” We know that at some point, Descartes read Kepler's book. Was there a connection between Descartes' own discovery, the one described in his secret notebook, and that of Kepler?
Kepler and his work were associated with a mystical, obscure figure living in southern Germany at that time: the mathematician Johann Faulhaber (1580-1635). Faulhaber's work on mathematics was of very high quality, but was intertwined with mysticism and the occult. Recently, several scholars have independently analyzed Faulhaber's books, copies of which have been discovered at the Stadtbibliothek Ulm, the municipal library of Ulm, the city in which Faulhaber lived. These researchers have found strong and puzzling connections between Faulhaber's work and Descartes' secret writings. Was Descartes' “admirable science” connected with the work of this mystic-mathematician?
Chapter 5
The Athenians Are Vexed by a Persistent Ancient Plague
IMPELLED FORWARD BY HIS THREE dreams and his interpretation of them, Descartes began to delve deeply into ancient Greek geometry. He spent most of his time alone in his “oven,” working out problems and developing ideas. The heart of knowledge was mathematics, but what was the essence of Greek geometry, which Descartes considered the most important part of mathematics? Descartes reviewed the ancient Greek principle of using straightedge and compass to solve all problems. Then he remembered a tantalizing story about Greek construction with straightedge and compass he had heard from his mathematics teachers at La Fleche. This was a story of an ancient mystery whose solution was unknown.
The island of Delos lies at the apparent center of the Cycladic Islands of the Aegean Sea. Since
early antiquity, this island has been inhabited, and has always been considered sacred ground. Delos was first settled in the second half of the third millennium B.C. And according to legend, this island is the birthplace of Apollo and Artemis—it has always been the center of worship of Apollo, and became a sanctuary to the god by the seventh century B.C. The Greek city-states of the Aegean competed with one another to build the greatest and most opulent monuments to Apollo. In the seventh century B.C., the Naxians built a terrace with stone lions by the entrance to Delos harbor. The lions can still be seen today, although they are now eroded by millennia of exposure to the sea wind. The island contains countless ruins of ancient temples and shrines. Every city-state in the Aegean Sea had its own temple to Apollo on the island. The Athenians began to influence Delos in 540 B.C. and, after the defeat of the Persians by the Greeks in 479 B.C., founded the Delian Confederacy of Greek city-states, ostensibly for defense against future Persian invasions, but in reality in order to dominate this coveted island.
A temple on the island of Delos
In 427 B.C., a plague ravaged Athens, killing a quarter of its population, including the great leader Pericles. In desperation, the Athenians sent a delegation to Delos, to entreat Apollo's oracle to beg the god to spare their lives. The oracle returned with the god's demand: Apollo wanted the Athenians to double the size of his temple on the island. The Athenians quickly set to work. They doubled the length, the width, and the height of the Athenian temple to Apollo. They decorated it opulently and lavished it with gifts, and soon the Athenian temple on Delos was the most magnificent on the island, or perhaps anywhere. The delegation returned to Athens with great hope, expecting that the god would now lift the curse. But the plague continued to ravage the city. So a second delegation left Athens for Delos. When its members met with the oracle, he surprised them by saying: “You have not followed Apollo's instructions!” The oracle continued: “You have not doubled the size of the god's temple, as he demanded of you. Go back and do as he had commanded you to do!”
Again, the Athenians set to work. They understood their mistake: they had doubled each of the dimensions of the old temple—the length, the width, and the height—and a calculation they now made showed them that they had actually increased the volume of the temple eightfold (2 × 2 × 2=8). Apparently the god wanted the volume to be doubled, not the dimensions.
Ancient Greek draftsmanship and geometry were always carried out using only a straightedge and a compass, so the Athenian architects did their best with these two tools. But they failed. As hard as they tried, they could not double the volume of the cubic structure that was Apollo's original temple—or for that matter, double the volume of any cube—with straightedge and compass alone.
According to Theon of Smyrna (early second century A.D.), the Athenian architects went to ask Plato for his help. Plato, who had established the Academy in Athens, in which the best mathematicians of his age worked, enlisted the help of the two great mathematicians Eratosthenes and Eudoxus in trying to solve this difficult problem. Eratosthenes was such a superb mathematician that he had been able to estimate with excellent precision the circumference of the earth by measuring the angle rays of light from the sun made at two different locations separated by a known distance. Eudoxus, on the other hand, was the great genius who could compute areas and volumes using methods that anticipated the calculus, which would only be codiscovered over two thousand years later by Leibniz and Newton. But neither Eratosthenes nor Eudoxus could solve the mystery of doubling the cube with straightedge and compass. Nor could anyone else, however fervently urged by Plato, who was desperate to help his countrymen rid themselves of the plague. Plato, who was not a mathematician himself but was called the Maker of Mathematicians because the best mathematicians studied and worked in his Academy, had such great interest in the cube and in other three-dimensional objects of perfect symmetry that such objects would eventually be named after him.
Why was it impossible to double the size of Apollo's temple? If the volume of the original temple was, say, 1,000 cubic meters (having length, width, and height 10 meters each), then the new volume should be 2 × 1,000=2,000 cubic meters (and not 8,000 cubic meters, as they had obtained on their first attempt by doubling the length, width, and height to 20 meters each). So in order to double the cube—in this case, to obtain a structure with volume 2,000 starting with a temple of volume 1,000—they would need to increase the length, the width, and the height by the cube root of 2 each. This is so because the cube root of 2, when cubed, gives us 2—the factor needed to multiply the volume. This way, each measurement would have to change from 10 meters to 10 × (cube root of 2), or approximately 12.6 meters. It turns out that no finite sequence of operations with straightedge and compass can turn a given length to a number that is the product of that length by the cube root of 2—or the cube root of any number that is not a perfect cube. The problem Apollo's oracle gave the Athenians was an impossible one to solve. It is important to note that the new temple had to remain in the shape of a cube: otherwise, simply doubling one of its dimensions, say the length, would have done the trick.
The Greeks of antiquity did not know that the Delian problem, as it has come to be known, is mathematically impossible, given their tools. An understanding of this problem would have to wait many centuries. In the meantime, they discovered two other problems they could not solve as well, and today we know that these two problems are also mathematically impossible to solve with straightedge and compass. One was the squaring of the circle—to use straightedge and compass to create a square with the same area as that of a given circle. The other was trisecting an angle—given an angle, to use straightedge and compass to divide it into three equal angles. This problem is solvable in special cases, but a general method—one that would work with any given angle— does not exist.
The ancient Greeks—Pythagoras, Euclid, and other great mathematicians of antiquity—were excellent geometers. But they did not have a well-developed theory of algebra. And algebra is needed in order to understand and properly address complex problems of geometry such as doubling the cube, squaring the circle, and trisecting an angle, collectively called the three chssical problems of antiquity.
Two millennia after the plague of Athens, Descartes found himself mulling over the Delian problem. He contemplated the cube. What were its properties? What was its secret? Why could the cube not be doubled using straightedge and compass?
Descartes asked himself the question that was at the heart of Greek geometry—and one that would eventually bring him to an understanding of the Delian problem of doubling the cube, as well as to his great breakthroughs in mathematics: What does it mean to construct something with straightedge and compass?
Descartes knew what the two instruments did. The straightedge drew lines and made perfect square corners. The compass drew circles and marked off distances. He asked himself: How do I construct something?
If two points on the plane are given, Descartes (and the ancient Greeks) could construct a line passing through these points. The straightedge is used here.
That was easy enough. Descartes also knew how to construct a circle centered at one point and passing through another. The compass is used here.
This too was a very simple operation. But more complicated things could also be done. Descartes knew how to use these two ancient instruments to construct a line perpendicular to a given line and passing through a set point. This is how it is done. Use the straightedge to draw the line, and the compass to draw two circles whose intersections are used (again with the straightedge) to draw the line passing through them. This construction is fun.
Descartes, and his distant predecessors of ancient Greece, also knew how to construct a line that is parallel to a given line and passing through a given point.
Descartes looked at the next figure for a long time and thought. The crisscrossed lines used in the construction were telling. If somehow they could be labeled by their numerical length, then
a system could be used to tie in the numbers with the geometrical constructions. This could potentially allow him to construct a lot more figures than the ancient Greeks were able to create. The use of numbers and figures in this way could really unleash the hidden power of mathematics. He would continue to think about how to do it.
Descartes would eventually unify geometry with algebra, bringing an understanding of the three classical problems of antiquity. He would solve several celebrated mathematical problems of ancient Greece, and would also show us the way to solving many more. Descartes' work would shed light on all of mathematics, bringing the wisdom of ancient Greece to our modern world, and would pave the way for the development of mathematics into the twenty-first century. But meanwhile, Descartes also became interested in the mystical aspects of mathematics, and this interest would have strong personal consequences for him.
Chapter 6
The Meeting with Faulhaber and the Battle of Prague
IN JULY 1620, DESCARTES DECIDED to leave the rest of the troops as they progressed northeast, and to visit the southern German city of Ulm for several months to learn about this part of the country. The first person Descartes met in Ulm was the mystic-mathematician Johann Faulhaber. As well as from Baillet, who gives us this itinerary, there is also a description of Descartes' meeting with this mathematician from Descartes' earlier biographer, Daniel Lipstorp.
Descartes's Secret Notebook Page 6