In 1987, Pierre Costabel published his definitive analysis of Leibniz's copy of Descartes' notebook. This time, the notebook yielded its secrets. Costabel had carefully studied the notes that Leibniz had made in the margin of his copy of the notebook, and understood that Leibniz had discovered Descartes' secret key, which revealed the true meaning of Descartes' writings. The key to the mystery was finding Descartes' rule for handling the sequences of numbers in his notebook. This rule was the gnomon, the ancient Greek term originally meaning a staff used for casting a shadow on the ground, the length and direction of which was used to estimate time. But in Greek mathematics, the term “gnomon” came to mean a rule that specified how one was to arrange given sequences of numbers.
Descartes had analyzed the ancient Greek regular solids—Plato's mysterious objects. And within these three-dimensional geometrical objects, Descartes discovered a coveted formula: a rule that governs the structure of all of these magnificent solids. It was the Holy Grail of Greek mathematics—something the Greeks had longed to possess. But Descartes would reveal to no one this hidden truth he had discovered. Some knowledge had to be kept secret. But why keep so secret a work of geometry?
Chapter 21
Leibniz Breaks Descartes' Code and Solves the Mystery
JOHANN KEPLER KNEW THAT THE earth rotated about its axis and orbited the sun. All his astronomical work pointed in the direction of the Copernican view of the universe—the theory that Descartes espoused as well, although not in a public way. Kepler, who a few years later was to derive the laws of planetary motion—still used today in astronomy and space flights —wanted to discover the cause for the regularity he was observing in the orbits of the planets in our solar system. In doing so, while still teaching at a high school in 1595, Kepler hypothesized that there was a connection between the Greek discovery of the existence of the five regular solids and the regular orbits of the six planets known at his time (Neptune, Uranus, and Pluto were yet to be discovered).
Kepler knew from his study of Euclid's remarkable theorems in book XIII of the Elements that each one of the five regular solids could be perfectly inscribed inside a sphere. Motivated by his search for harmony in the creation of the solar system, he suggested the existence of celestial regular solids, whose spheres were nested. Each regular solid was inscribed in a larger sphere containing all previous regular solids and the spheres containing them. The five Platonic solids were thus placed within a sequence of nested spheres. Kepler believed that the orbits of the five other known planets (Mercury, Venus, Mars, Saturn, and Jupiter) and Earth could be viewed as circles on the surfaces of nested spheres, and he knew from Greek geometry that the five regular solids could each fit tightly inscribed inside one of these nested spheres. Kepler published this model of the solar system in his book Mystenum cosmographicum (“Cosmological Mystery,” 1596) and considered it one of his greatest achievements, and a divine confirmation, using pure geometry, of the Copernican theory. Each sphere contained on its surface the orbit of a single planet and held inside it a regular solid. The order of these planets and solids was as follows: Mercury, octahedron, Venus, icosahedron, Earth, dodecahedron, Mars, tetrahedron, Jupiter, cube, Saturn.
The figure on page 224, from Kepler's Mystenum cosmographicum, shows Kepler's cosmological model using the five Platonic solids and the planets nested between them. The sun is at the center of all the spheres containing the Platonic solids and the planets.
Seeking a cosmic formula governing the Platonic solids that Kepler used to explain the universe and support the Copernican theory, Descartes began to study the mathematical properties of these ancient mystical three-dimensional geometrical objects. His purely mathematical work could thus lend theoretical support for the forbidden Copernican theory of the universe. One of the first items in Descartes' secret notebook was a theorem about placing the regular solids inside spheres, a property known to the ancient Greeks. But then Descartes went much further.
Kepler's model of the universe (from Mysterium cosmographicum, 1596)
Descartes was looking for a transcendent truth that would describe all the regular solids. Later, he would discover that the formula he was after would describe not only the five regular solids but every three-dimensional polyhedron—regular or not. Descartes was interested in capturing numerical properties of these solids. He would then apply his overarching theory of analytic geometry to draw a link between the algebraic properties and the geometric structure of these solids. Descartes understood that the direct connection between the regular solids of ancient Greek geometry and Kepler's model of the universe could cause his work on these solids to be viewed as supporting the forbidden Copernican theory. He had to hide his work for fear of the Inquisition.
Leibniz looked at the mysterious sequences of numbers:
4 6 8 12 20 and 4 8 6 20 12
What was the meaning of these sequences? Leibniz saw it.
Descartes started by counting the number of faces of the five regular solids. He got the following sequence of numbers:
4 (tetrahedron), 6 (cube), 8 (octahedron), 12 (dodecahedron), 20 (icosahedron)
Then, for each one of the five regular solids, Descartes counted the number of vertices, or corners. This gave him, in order:
4 (tetrahedron), 8 (cube), 6 (octahedron), 20 (dodecahedron), 12 (icosahedron)
An inspection of the figures of the regular solids can verify these numbers. And indeed Leibniz understood that the obscure figures on the other side of the page he was looking at stood for the five regular solids.
The key to the mystery was to know what to do with the two sequences of numbers. This was Descartes' code. The key, the gnomon, told Leibniz exactly what to do with Descartes' two sequences. The rule was embedded in the way Descartes transformed and disguised other sequences of numbers in his text. Leibniz discovered the gnomon and noted it in the margin of the copy he was making. The rule told him to arrange Descartes' two sequences of numbers in an array—the second sequence below the first:
4 6 8 12 20
4 8 6 20 12
The page from Leibniz's copy of Descartes' secret notebook
But that's where Descartes' great discovery came in. Descartes then counted the edges of each of the five regular solids. Let's add these as the bottom row of the array above. This gives us the following table.
Having written down the array, Descartes made his discovery. He noticed something very interesting about this array of numbers. It had to do with the sum of the first two rows as compared with the third row. (Can you see it?)
What Descartes found was that for every one of the regular solids, the sum of the number of faces and vertices minus the number of sides equals two. Or as a formula:
F+V-E=2
Then Descartes found that his formula worked for any three-dimensional polyhedron—regular or not. Let's check it for a square-based pyramid (not one of the five regular solids, since it has one square face and four triangular ones).
We have F + V − E = 5 + 5 − 8 = 2
Descartes' formula was never attributed to him. His analysis of the three-dimensional solids would have advanced the study of geometry very much if he had published it. But because he feared the Inquisition, this important discovery remained hidden.
Descartes' formula F+V—E=2 is the first topological invariant found. The fact that the number of faces plus the number of vertices, minus the number of edges, equals 2 is a property of space itself. In deriving this formula, Descartes had thus inaugurated the immensely important mathematical field of topology. Today, topology is one of the major areas of research in mathematics, and it has important applications in physics and other fields. But since Descartes kept his discovery secret, he is credited with launching the field of analytic geometry, wedding algebra with geometry, inventing the Cartesian coordinates, and making other important advances in mathematics, but not with the founding of topology—the study of the properties of space. Others would receive the accolades for founding this field.
The Swiss mathematician Leonhard Euler (1707-83), who was born in Basel, was one of the greatest mathematicians of his century. Euler made many contributions to modern mathematics, spanning different areas within this field. Sometime after he moved to Russia to work at the Academy of Saint Petersburg, Euler discovered the magic formula that governs the structure of all three-dimensional solids: F+V—E=2. This equation became known as Eider's forrrada—although it might easily, as we now know, have been Descartes' formula. There is a tantalizing footnote to this story. On his way from Basel to Saint Petersburg in 1730 to assume the chair of natural sciences at the Saint Petersburg Academy, Euler passed through Hanover, Germany. He is known to have spent some time reading Leibniz's manuscripts at the Hanover archives. Whether he perused Leibniz's copy of Descartes' notebook is unknown. What has been known as Euler's theorem and Euler's formula for over two and a half centuries is now increasingly often referred to—following the decipherment of Descartes' secret notebook by Pierre Costabel in 1987—as the Descartes-Euler theorem and the Descartes-Euler formula. But this practice is not universally followed, and many mathematicians still refer to the important property, stated as a theorem or a formula, as belonging to Euler. Had Descartes not guarded his precious knowledge so zealously, his name alone would have been attached to this discovery.
Descartes lost more than his priority in making a great mathematical discovery that founded an entire field. Despite Descartes' meticulous care during his lifetime to avoid any controversy with the church, thirteen years after his death, in 1663, his writings were placed on the Index of Prohibited Books. And in 1685, King Louis XIV banned the teaching of Cartesian philosophy in France. In the time of Euler, the eighteenth century, Descartes' philosophy nearly disappeared. In 1724, the Libraires Associes published the last of the older French editions of Descartes' works. For a hundred years, no new editions of Descartes' works were published in France, and the philosopher and his work were all but forgotten as new ideas emerged and philosophy was developing further. Only in 1824, exactly a century later, did his work reappear in print, and he was again recognized for his greatness as a philosopher, scientist, and mathematician. And Pierre Costabel's definitive analysis of Descartes' secret notebook a century and a half later has finally restored to Descartes his credit for founding the field of topology.
Tantalizingly, just a few decades after he died, Descartes came close to receiving the recognition he deserved for this discovery. While researching his biography of Descartes, which would appear in 1691, Baillet was trying to understand Descartes' mathematical writings, including the secret notebook, lent him by Abbe Legrand. But he could not understand any of the mysterious symbols and drawings. When he asked him, Legrand told Baillet that some years before Clerselier died, he had been visited by a young German mathematician who copied Descartes' work and may have understood the writings in the mysterious parchment notebook. Baillet then made contact with Leibniz in Hanover. Leibniz complied and explained to Baillet Descartes' mathematics. But Baillet, not being a mathematician, failed to discuss Descartes' discovery in his biography. However, he did acknowledge in his preface the help rendered him by “Monsieur Leibniz, a German mathematician.”
Leibniz himself remained ambivalent in his relationship to the late French philosopher and mathematician. The man who deciphered Descartes' secret writings would grudgingly praise his work, and from some of the things he later said about Descartes, it is evident that he continually compared his abilities with those of the French genius with perhaps a degree of envy. In 1679, three years after he copied and analyzed Descartes' notebook, Leibniz wrote:
As for Descartes, this is of course not the place to praise a man the magnitude of whose genius is elevated almost above all praise. He certainly began the true and right way through ideas, and that which leads so far; but since he had aimed at his own excessive applause, he seems to have broken the thread of his investigation and to have been content with metaphysical meditations and geometrical studies by which he could draw attention to himself.
Leibniz remained obsessed with Descartes and his work for the rest of his life. He knew that Descartes had been crucial in laying down the foundations of modern science and mathematics, but he continued to argue that Descartes had stopped at a certain point in his development, believing that he, Leibniz, had gone much further. Descartes' work was clearly influencing him, and modern scholars would identify in Leibniz's philosophy both Cartesian and anti-Cartesian elements at the same time.
Leibniz remained in contact with Descartes' friends and followers. The most prominent Cartesian philosopher of the late seventeenth and early eighteenth centuries was Nicolas Malebranche (1638-1715). Malebranche first read Descartes' philosophy in a manuscript published by Clerselier in 1664 Upon reading Descartes' ideas he became so excited that he suffered from palpitations of the heart and had to be confined to bed rest. Ten years later, Malebranche wrote his own treatise on Cartesian philosophy, titled Recherche de la vérité (“The Search for Truth”). Leibniz exchanged letters with Princess Elizabeth, who was now sixty-one years old. He knew her through her sister Sophie, who was married to the duke of Hanover. On January 23, 1679, Leibniz wrote to Malebranche:
Through the favor of her highness, the Princess Elizabeth, who is celebrated as much for her learning as for her birth, I have been able to see [your Cartesian treatise]…. Descartes has said some fine things; his was a most penetrating and judicious mind. But it is impossible to do everything at once, and he has given us only some beautiful beginnings, without getting to the bottom of things. It seems to me that he is still far from the true analysis and the general art of discovery. For I am convinced that his mechanics is full of errors, that his physics goes too fast, that his geometry is too narrow, and that his metaphysics is all these things.
Why such biting and clearly unjustified criticism from the man who came so close to Descartes' monumental ideas and pursued his hidden writings ? The reason was the calculus.
Before he saw Descartes' secret notebook, in the years prior to his visit to Clerselier in 1676, Leibniz was already developing his differential and integral calculus. The differential calculus is a mathematical method of finding the slope—the instantaneous rate of change—of a mathematical function. Descartes' published writings contain elements that lead in this direction. More precisely, Descartes could find the slopes of particular curves, but lacked a general, systematic method of finding slopes of functions. The integral calculus consists of an operation that is the opposite of finding slopes—integrating a mathematical function means finding the area under the curve. The ancient Greeks, especially Archimedes and Eudoxus, made advances in this area, but Leibniz found the general method.
In 1673, Leibniz traveled from Paris to London, where he met mathematicians. He so impressed British scientists and mathematicians with his work that they elected him member of the Royal Society. Eleven years later, in 1684, Leibniz published his theory of the differential calculus, followed two years later by the integral calculus. Newton, on the other hand, possessed his results on the calculus as early as 1671, although his work was not published until 1736. Leibniz's own, independent work on the calculus was completed in Paris in October 1675—before he saw any of Descartes' hidden writings.
But the calculus was not a single development—it consisted of methods and techniques that had been developed over centuries: from the ancient Greek mathematicians Archimedes and Eudoxus to Galileo, Descartes, Fermat, and others. The final glory, a unified general approach to the solution of calculus problems, was developed by Leibniz and by Newton. It therefore happened that, since Leibniz had been discussing mathematical ideas with English mathematicians as particular results were being developed, he was accused of using the ideas of others. We know today that this was not the case, and that Leibniz developed his theory of the calculus all on his own. But at the time, a controversy raged in intellectual circles in England and on the continent about the priority of relate
d important discoveries in mathematics. For it was known even before 1736 that Isaac Newton had developed a theory of the calculus, and some had asserted that perhaps during his visit to London in 1673, Leibniz came in contact with Newton's ideas on the calculus.
Leibniz's being pressed to prove that his important discoveries were made alone and without knowledge of Newton's work also made him sensitive to any suspicion that his ideas might have been influenced by those of others—prime among them Descartes. In particular, in May 1675, English mathematicians made the claim that some of Leibniz's works in mathematics had been “nothing but deductions from Descartes.” Then Leibniz received a letter in 1676 stating that “Descartes was the true founder of the new mathematical method and the contributions of his successors were only a continuation and elaboration of Descartes' ideas.”
At this point, Leibniz realized that he had no choice: he had to see everything that Descartes had written—the published and the hidden, which might some day appear in print—in order to be able to defend himself and his theory of the calculus, once it was published, from any criticism whatsoever. This gave him the burning urge to search out all of Descartes' hidden works, find Clerselier, who owned these writings, and copy and understand as much as he could of all the discoveries of Descartes. He had to make sure that nothing in Descartes' writings looked too much like his own work on the calculus—or else the accusation of plagiarism could stick. It was this urgent need, and the fact that he was being attacked as having simply elaborated on the work of Descartes, that he explained to Clerselier when he came to see him in June 1676.
But the English continued to harass Leibniz with accusations of plagiarism. In August 1676, Newton wrote to Leibniz through a German interpreter, accusing him of using his work. The letter was delayed, and by the time Newton received Leibniz's answer, he was infuriated, thinking that Leibniz had taken six weeks to answer him, and hence that he was guilty. In fact, Leibniz had only a day or two to answer Newton's complaints. And he proved that his results were independent of those of Newton. He did this by showing that he had been communicated only some of Newton's specific results, and not a general method of solution. Since his calculus (and Newton's) was a very general method for the solution of a wide variety of mathematical problems, Leibniz could not have deduced it from separate, specific results that had been communicated to him by English mathematicians with whom he had ties.
Descartes's Secret Notebook Page 20