ALSO BY AMIR D. ACZEL
Entanglement: The Greatest Mystery in Physics
The Mystery of the Aleph: Mathematics, Kabbalah, and the Search for Infinity
Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem
Chance: A Guide to Gambling, Love, the Stock Market, and Just About Everything Else
The Riddle of the Compass: The Invention That Changed the World
Pendulum: Leon Foucault and the Triumph of Science
God's Equation: Einstein, Relativity, and the Expanding Universe
For Debra
I am extremely grateful to the John Simon Guggenheim Memorial Foundation for making me a Fellow of the Foundation. My Guggenheim Fellowship made this book possible, and receiving this award has been the greatest honor in my professional career. I thank the foundation and its officers for believing in me and in this project even before its acceptance by a publisher. My special thanks to Senior Vice President G. Thomas Tanselle for his interest in my work.
I thank the librarians of the Bibliotheque nationale de France in Paris for giving me access to many original manuscripts and documents on Descartes and the mystery of his notebook. Also in Paris, I am indebted to the Institut de France and its librarians for allowing me to use the institute's extensive archives, and to Professor Jean Dercourt, Perpetual Secretary of the French Academy of Sciences.
Many thanks to the Gottfried Wilhelm Leibniz Library in Hanover, Germany, and to its administrator, Birgit Zimny, for making available to me the manuscripts Leibniz copied from Descartes, which held the key to the mystery addressed in this book. My appreciation for the photographic processing work on the Leibniz images performed by Kevin Wool and Boston Photo Imaging.
I am grateful to Professor Jay M. Pasachoff of Williams College for providing me with his image of Kepler's model of the universe, and I thank Wayne G. Hammond of the Chapin Library at Williams College for making this figure available to me.
I want to express my many thanks to Jeff Weeks (www.geometry games.org) for providing me with his images of the Poincare dodecahe-dral space and other geometrical models of the universe, and for explaining to me his work in cosmology.
I thank the Descartes Museum in Descartes, Touraine, and its administrator, Daisy Esposito, for guiding me to a number of important documents on the early life of Rene Descartes. I thank the Protestant Museum in La Rochelle, France, for information on the siege of the city in the seventeenth century.
I am grateful to the photographer Tzeli Hadjidimitriou (www.odoiporikon.com) in Athens, Greece, for her photograph of the temple on the island of Delos.
I wish to express my gratitude to Professor Owen Gingerich and the department of the history of science at Harvard University for appointing me a visiting scholar at the department.
I acknowledge the Center for the Philosophy and History of Science at Boston University, where I spent a year as a Research Fellow while writing this book. My work was aided by a number of people at the center, including Alfred Tauber and Debra Daugherty, and at various branches of the Boston University Library.
My deepest gratitude to my agent and friend, John Taylor (Ike) Williams, of Kneerim and Williams in Boston, for guiding my writing career with so much patience and wisdom.
My heartfelt thanks also to Hope Denekamp of Kneerim and Williams for all her tireless help with this book, and with everything related to publishing.
I am very grateful to Gerald Howard, my editor at Broadway Books in New York, for his clear judgment, knowledge, and guidance in shaping a manuscript into a book, and for always keeping me on the right track in this complex endeavor. I thank Rakesh Satyal for all his work on this project, for his boundless energy, and for his care in making this project a reality.
Many colleagues and friends helped me in the undertaking of researching and writing this book. They include Judith Alvarez-Perreire, Dan Carey, Stephen Gaukroger, Kurt Hawlitschek, Richard Landes, Kenneth Manders, Michael Matthews, Jacob Meskin, Evelyne Patlagean, Arthur Steinberg, and Marina Ville. Thank you, everyone.
My deepest thanks and appreciation to my wife, Debra, for helping me edit and revise the manuscript, for photographic work, and for her many important ideas. This book is dedicated to her.
Acknowledgements
Introduction
Prologue: Leibniz's Search in Taris
1: The Gardens of Touraine
2: Jesuit Mathematics and the 'Pleasures of the Capital
3: The Dutch Puzzle
4: Three Dreams in an Oven by the Danube
5: The Athenians Are Vexed by a Persistent Ancient Plague
6: The Meeting with Faulhaber and the Battle of Prague
7: The Brotherhood
8: Swords at Sea and a (Meeting in the Marais
9: Descartes and the Ksicrucians
10: Italian Creations
11: Duel at Orleans, and the Siege of la Rgchelle
12: The Move to Holland and the Ghost of Galileo
13: A Secret Affair
14: Tescartes' Philosophy and the Discourse on the Method
15: Tescartes Understands the Ancient Delian Mystery
16: Princess Elizabeth
17: The Intrigues of Utrecht
18: The Qall of the Queen
19: The Mysterioust Death of Descartes
20: Leibniz's Quest for Tescartes' Secret
21: Leibniz Breaks Descartes' Code and Solves the Mystery
A Twenty first-Century Epilogue
Notes
Bibliography
Illustration Credits
Introduction
I HELD THE FRAGILE ANCIENT MANuscript in my hand. I opened it carefully, and read:
PREAMBLES
Fear of God is the beginning of wisdom. The actors, called to the scene, in order to hide their flaming cheeks, don a mask. Like them, when climbing on stage in the theater of the world, where, thus far, I have only been a spectator, I advance masked. At the time of my youth, witnessing ingenious discoveries, I asked myself whether I could invent all on my own, without leaning on the work of others. Henceforth, little by little, I became aware that I was proceeding according to determined rules. Science is like a woman: if faithful, she stays by her husband, she is honored; if she gives herself to everyone, she is degraded.
The manuscript continued further. After a few more pages, I read another fragment of text:
OLYMPICA
November 11, 1620. I began to conceive the foundation of an admirable discovery.
These were the enigmatic words of Rene Descartes (1596-1650). They were never intended for eyes other than his own. But the manuscript I now held in my hands was not written by Descartes. It was a copy of Descartes' secret writings made by none other than Gottfried Wilhelm Leibniz (1646-1716)—one of the greatest mathematicians of all time and the man who, only a few years after copying Descartes' notebook in Paris in 1676, would give us the calculus.
The idea for a book about Descartes occurred to me while lost in a snowstorm somewhere in eastern Ontario, hard by the Quebec border, around midnight in early January of 2002. We were on our way back home to Massachusetts from a holiday visit to relatives in Toronto, hoping to spend the last night of our trip in Montreal, when our car got mired in deep snow dropped by a sudden blizzard. I turned off the highway to look for a place to wait out the storm; but after making an unfortunate sequence of turns onto various country roads, I came to the conclusion that I had gotten us completely lost. The visibility was poor, there were no lights in sight, and we had no idea where to turn. I knew that if the fuel ran out, eventually we would start to freeze.
While driving, I glanced at my dashboard. And I remembered something this ca
r came equipped with—something I had never had the need to use before. On my dashboard, I recognized the small lit button, and I pressed it. There followed the sound of a telephone number being dialed automatically. “Good evening, Mr. Aczel,” said a pleasant voice from a thousand miles away. “How are you this evening? I see that you are heading south on Glen Donald Road, half a mile north of the intersection with County Road 27, outside of Cornwall, Ontario.”
“Aha…,” I said, trying to keep my voice from betraying that I had absolutely no idea where we were. “We are looking for a hotel…. It's snowing hard.”
“No problem at all,” was the response. “Where are you trying to go?”
“Montreal.”
“That's not hard,” reassured the voice from civilization. “Make a left turn at the next intersection, the one with County Road 27. Follow that road for two miles, and then turn right at the intersection, and you will see the on-ramp for Highway 401 to Montreal. Which hotel would you prefer? There's a Marriott just as you reach the city from this direction. I can make you a reservation, and I'll be back with more directions as you continue.”
The voice on this cell phone led us throughout the night: from a deserted country road in snowy Ontario all the way to a warm hotel room in Montreal, directing us at every turn along the way. The person giving us instructions from afar knew our position, accurate to within a dozen feet, during every minute of the trip. The technology that made all this possible is the Global Positioning System (GPS), which allows one to determine the location of a small radio receiver (called a GPS receiver) anywhere in the world. In this case, a device installed in my car, linked to an internal cellular phone, allowed the company that provides this service to determine the location of my vehicle wherever it might be. The amazing GPS technology works because of an invention made almost four centuries ago by the philosopher, scientist, and mathematician Rene Descartes.
Descartes gave us the Cartesian coordinate system, named after him—a system of crisscrossed parallel lines, in two, three, or more dimensions, that allows us to describe numerically the position of a point in space. In this case, the position of the point (my car) was described in terms of its latitude and longitude, which were then translated into a location on a map. The GPS system works in three dimensions as well: it can give you, in addition to your latitude and longitude, your altitude—and this makes it useful in directing an airplane.
But Descartes' coordinate system is used for a lot more than GPS. Every pixel on your computer screen is described internally by a pair of numbers: its horizontal and vertical coordinates. Thus all computer technology relies on Descartes' invention. Graphs and diagrams and maps of all kinds rely on the Cartesian system, and so do digital photographs, so popular these days, and pictures and documents sent on the Internet, and engineering designs, and space flights, and oil exploration.
And the applications go even further—beyond our three-dimensional intuition of pictures and graphs and shapes. When data are available on many variables—more than the visual three dimensions of everyday life—such data can still be analyzed using the Cartesian coordinates. Your bank, for example, may have data on your income, your assets, the number of years at your current employment, the number of people in your family, your age, your educational level, and so forth. These multidimensional data can be mapped on a many-variable scale using the Cartesian coordinates (even though such a “map” cannot be visualized and is meaningful only within the context of the computer analyzing the data) and through statistical analysis lead to a decision by the bank to approve or deny your loan application. Statistical and scientific algorithms that analyze data on many variables use the Cartesian system in the analysis. The number of applications of the Cartesian coordinate system in our daily lives is immense. Literally everything we do or see or use in our daily lives has something to do with Descartes' great invention.
Interestingly, Descartes' invention of the coordinate system that bears his name was a special outcome of a much grander design. Descartes achieved an immense advance in mathematics, launching modern mathematical theory four centuries ago, when he unified algebra with geometry by inventing analytic geometry: a way of connecting the equations and formulas of algebra with the figures and shapes of geometry. The Cartesian coordinate system was just the device he created in order to facilitate this unification.
Of course, Descartes' greatest fame comes not simply from his work in mathematics or in physics—in which he also made important discoveries, especially on gravity and falling objects, as well as in optics—but from his philosophy. Descartes' “Cogito, ergo sum” (I think, therefore I am), and the philosophy behind this statement, is a pillar of modern philosophy; and his rationalism—Cartesianism—is considered of great importance in the development of philosophical thinking. Descartes is often viewed as the founder of modern philosophy. In his Meditations, published in 1641, Descartes wrote: “It must be acknowledged that this pronouncement, I am, I exist, whenever I assert it or conceive it in my mind, is necessarily true.” In the introduction to the book they edited, Descartes and His Contemporaries, M. Grene and R. Ariew describe Descartes' statement above as signaling a turning point in Western thought: “Suddenly, we have reached a new level of awareness, at which we could ask reflective questions about ourselves: Can we reach out from consciousness to an external world? Who are we as minds in relation to our bodies?” Some scholars have even asserted that Descartes' philosophy, in its introduction of the self into human consciousness, inaugurated modern psychological theory. Descartes' method of reasoning thus created self-reflection, which incorporated into philosophy the elements of modern psychology. Descartes pioneered metaphysical investigations, and hypothesized about the relation between body and soul. He tried to use reason and logic to prove the existence of God (in whom he believed).
There is a direct link between Descartes' logical, rational approach to philosophy and his work in mathematics. The reason for this is that Descartes' philosophy is based on an ambitious attempt to found all of human knowledge on the same precise, strictly logical principles that the ancient Greeks had used in creating their enduring geometry.
I believe that Descartes' work in mathematics, physics, and philosophy, as well as in other areas this unique individual studied, such as biology, anatomy, and music theory, are unified by invisible links of logic. This recondite internal rationality is what Descartes was all about—for the hidden Descartes was fundamentally a supreme mathematician: a man who was so good at doing mathematics that he came to believe he could apply his mathematical skills and methods to every area of human study.
Descartes lived in one of the most tumultuous, and yet intellectually fecund, periods in history. Descartes' time, the first half of the seventeenth century, was the era of the Thirty Years War, in which Catholics and Protestants were mercilessly pitted against each other in a series of bloody battles. This period also saw the ruthless suppression by the Catholic Church of new scientific and philosophical ideas, as evidenced by the trial of Galileo by the Inquisition, the persecution by the church of other thinkers who supported the theory of Copernicus, and the burning of forbidden books. However, this was also a period of great intellectual revival—an extension of the Renaissance to science, mathematics, and philosophy. Classical ideas in these areas were being studied and extended by intellectuals throughout Europe. Descartes' work was both a product of this period and the vanguard, leading the way in the development of mathematics and philosophy right up to our time.
My favorite cafe in all Paris is Les Deux Magots—that icon of literary society made famous by Hemingway, Fitzgerald, Sartre, and Beauvoir— facing the ancient church of Saint-Germain-des-Pres. Six months after our Canadian trip in the snow, I found myself on a sunny day sitting at this cafe with my friend the historian Richard Landes. We were drinking iced coffee, and Richard was telling me about arrangements he wanted to make for me to meet the director of the Descartes center in Paris, now that I'd come to the Fr
ench capital to research the life of the philosopher and mathematician.
I was renting an apartment in the Marais—the oldest, medieval part of Paris, and the only section of the city that had not been razed in the nineteenth century by Baron Georges Haussmann as part of his plan to build the wide and elegant boulevards we associate with Paris today. This part of the city, with its narrow ancient streets, looks very much as it appeared in the days of Descartes. My apartment, on the rue du Bourg-Tibourg, was in a building erected in 1629. Both the building and the apartment, on which apparently little work had been done in the past few centuries other than necessary maintenance, looked very much as they must have looked when Descartes walked these very streets when he lived just around the corner from this address for a short time in 1644: on the rue des Ecouffes, between the rue du Roi de Sicile and the rue des Blancs-Manteaux.
The apartment consisted of one large room with a kitchen corner and a bath, and it had a high ceiling supported by the original dark brown wooden beams so characteristic of seventeenth-century French construction. A narrow, dark spiral staircase with small windows cut into the thick stone walls led up to the apartment. Looking at the building from outside, one could admire the tall Parisian windows and notice the protruding iron supports holding together the exterior walls of this ancient structure. Knowing that my building was there at the time Descartes walked these same streets lent my search an added sense of reality.
I spent my days at the libraries and archives of Paris, researching material on Descartes and his work; I traveled to the locations at which he stayed or lived throughout Europe—Descartes was a great traveler who saw most of the Continent—and I walked the perimeter of the Place des Vosges, under the ancient arches, exactly as Descartes did in 1647 discussing mathematics with Blaise Pascal. I held in my hands original letters Descartes had written to his friend Marin Mersenne; I perused countless manuscripts written over the centuries; I even bought an original book of Descartes', published in 1664 But sometime in the middle of my search, I made a surprising discovery: Descartes had kept a secret notebook.
Descartes's Secret Notebook Page 1