The last of these held a special attraction for Leibniz. The young student was fascinated by Cartesian logic and philosophy, but at the same time developed his own ideas, which were often at odds with those of Descartes. According to the great English philosopher and mathematician Bertrand Russell (1872-1970), Leibniz's thinking was formed in the scholastic tradition, and he was steeped in Aristotelian-scholastic ideas about the universe. He broke away from this philosophy only through work in mathematics later in life. These beliefs may have kept Leibniz from accepting Descartes' philosophy. Additionally, Leibniz's personality was such that he could never enter any area of scholarship without being an innovator and stamping on it his own unique impression.
Leibniz had a strange mix of attraction and repulsion toward the work of Descartes, a love-hate relationship with the legacy of the late French philosopher. At that time, the opposition to Cartesian philosophy in all German universities was so high that any professor who tried to defend Cartesian ideas would be in jeopardy of losing his academic position.
Leibniz held that Descartes' principle of doubt—one of the cornerstones of Cartesian philosophy—was flawed. He wrote:
That which Descartes says about the necessity of doubting everything of which there is the least incertitude can be made more satisfying and more precise: For every concept, we must consider the degree of approval or reservation it merits. Or, more simply, one must examine the reasons for each assertion. This way we can do away with the flaws in the Cartesian doubt.
Leibniz then gives a number of examples of the flaws in Descartes' absolute doubt. If we see a combination of the colors blue and yellow, he asks in one example, can we completely doubt that the color we perceive is green? There must be a degree to such a doubt, he concludes, since when the two colors are mixed very well, the result is indeed the color green. Similarly, he asks, if one hand feels cold and the other warm, which hand should we believe ? Should we completely doubt the feeling in either hand?
Leibniz wanted to learn as much as he possibly could about Descartes and was looking for more than just the published works. Years after his graduation, in 1670 and 1671, he would purchase original unpublished writings of Descartes, including letters; Descartes' Reguhe in manuscript, purchased in Amsterdam; a manuscript entitled Calcul de Monsieur Des Cartes, published in 1638 as a new introduction to Descartes' Geometrie; and the Latin manuscript Cartesii opera pMosoph-ica. But he wanted more.
Leibniz wrote his master's thesis in philosophy about the relationship between philosophy and the law, and received his degree in 1664 Nine days later, his mother died. Following a graduation tinged with the sadness of losing his remaining parent, Leibniz returned to the university to study law. In 1666, he received his doctorate in law from the University of Altdorf.
Like Descartes before him, Leibniz was attracted to the work of the thirteenth-century mystic Ramon Lull. Lull's “Great Art” of combinations, which used rotating wheels within wheels to create a large number of combinations of concepts coded by letters on these wheels, took a new and deeper meaning in Leibniz's eyes. Leibniz saw in these efforts more than just mystical play, but rather a mathematical attempt to study combinations. Leibniz developed these same concepts into a mathematical theory, and published it in a 1666 treatise titled De arte combinatoria. This work developed the mathematical foundations of combinations, although it contained elements that had been independently discovered by Pascal in France.
Shortly afterward, in Nuremberg, Leibniz joined an alchemical society. According to Leibniz's secretary and first biographer, Johann Georg Eckhart, Leibniz wrote a letter, using obscure alchemical terminology, to the president of the alchemical society of Nuremberg. The latter was duly impressed with the writer's familiarity with alchemical secrets and offered Leibniz admittance to the society.
Leibniz began to feel that the defeated and depressed Germany of his birth could never offer him the opportunities to advance intellectually as much as other countries could, especially France—the land of high culture and advanced ideas at that time. He longed to find a way to go to Paris. His keen sense of politics would be his ticket out of Germany. Shortly after receiving his law degree, Leibniz found a patron in the distinguished German statesman Baron Johann Christian von Boineburg.
Boineburg sent Leibniz to Paris with a special mission: to try to deflect the French king, Louis XIV, from his schemes of conquering Europe. This aim was to be achieved by means of a paper Leibniz had written with Boineburg's help suggesting that France should embark on a military adventure to Egypt. It is hard to see why a French king would want to be convinced by two Germans that he should attack Egypt, but Boineburg thought that this was a possibility, and he had the money to support Leibniz's trip to Paris. Boineburg had a personal financial interest in this scheme as well: he was owed rents for his properties in France and hoped that Leibniz's mission could earn him favor with the French royal court, and that as a result he would be paid what he was owed.
Leibniz arrived in Paris at the end of March 1672, but King Louis XIV would not meet with him. Still, the young man settled down, and he met others in the French capital who had some interest in his ideas, although no progress was made on his and Boineburg's diplomatic scheme. Leibniz's own political ideas revolved around the concept of unifying Europe by reconciling its religions. He made connections with influential people and attempted to gain support for his political initiatives.
Leibniz fell in love with Paris, and over the next four years made every effort possible to make it his home. Boineburg supported him for a while, ever hopeful that Leibniz would eventually get attention in the French court. In the meantime, Leibniz was employed with tutoring Boineburg's son, a young man who was also living in Paris. But soon Boineburg died, and Leibniz needed to find other sources of income if he was to stay in Paris. Before leaving Germany, he had made contact with the duke of Hanover, whom he had sent some of his philosophical writings. He now renewed the contact, and the duke agreed to sponsor him for a while, and even gave him a letter of recommendation to help him with his endeavors. But the duke insisted that Leibniz make plans to return to Germany and serve him as librarian in Hanover. Leibniz knew that time was running out on him. There was so much he had to do, and he hoped to be able to turn down the duke's offer and stay in Paris.
Leibniz again turned his attention to mathematics. He deepened his understanding of mathematical ideas and began to develop new ones. Among these was a mechanical way of carrying out computations, which allowed him to construct a rudimentary calculating machine. Leibniz's achievements would have warranted acceptance to the French Academy of Sciences, which would have ensured his ability to stay in the French capital. But the French declined to offer him this opportunity. They already had two foreign members in their academy: the Italian astronomer Cassini, and the Dutch physicist and mathematician Christiaan Huygens. The latter, nevertheless, became a friend of Leibniz.
Leibniz was on an intense mathematical quest. He was developing an immensely important theory, and in October 1675, he completed that theory. Leibniz indulged his fascination with Descartes by reading more of the philosopher's works. But he longed to learn ever more: he had a pressing reason to see everything he could find that had been written by Descartes.
In the spring of 1676, after he had been in Paris for over three years, Leibniz's attempts at political work collapsed completely. He no longer saw any hope of achieving success on his project of religion and diplomacy. Knowing that he would soon have no choice but to return to Germany and serve the duke of Hanover as librarian gave Leibniz's search for Descartes' manuscripts an air of urgency. He asked everyone he met whether they knew where he could find more of Descartes' writings. Finally, Christiaan Huygens told him about the inventory of Descartes' unpublished manuscripts, and gave him the name of Claude Clerselier.
So on June 1, 1676, Leibniz came to see Clerselier. He told him his story and pleaded with him to allow him to see Descartes' hidden writings. Reluctant
ly, the old man agreed, and Leibniz sat down and began to work.
Reading the Preambles, Leibniz saw Descartes' words:
Offered, once again, to the erudite scholars of the entire world, and especially to G. F. R. C.
And in the copy he made of the manuscript, Leibniz added parenthetically the Latin name for Germany, making it read:
G. (Germania) F. R. C.
Leibniz knew very well the acronym “F. R. C.” It stood for “Fraternitas Roseae Crutis.” Leibniz was very familiar with all the Rosicrucian writings. He knew the Fama fraternitatis very well, and had even discussed its finer points at length in letters. And in Leibniz's own works there is a strong Rosicrucian element that seems to be taken right out of the Fama fraternitatis. In 1666, in Nuremberg, Leibniz had joined the Brotherhood of the Rosy Cross. According to some sources, Leibniz was even elected secretary of the order. The order of the Rosy Cross was the parent organization of the alchemical society of Nuremberg.
We know today that the Preambles and Olympica were closely associated with the secret notebook. The contents of the lost notebook, whose title was De solidorum elementis, were revealed only after researchers made the decisive discovery that Leibniz had made over three centuries earlier: These private writings, which Descartes never intended to publish or share with anyone, were not separate entities. They were all pieces of a larger puzzle—the puzzle of life that Descartes wanted to solve.
Descartes' philosophy was an attempt to place human learning on a rational basis modeled on geometry. Descartes wanted people to reason in everyday life the way one reasoned in solving a mathematical problem. Within this context, the secret notebook was his crowning glory, for it contained the next level in geometry, encompassing the mystery of the universe as Descartes saw it.
Leibniz opened Descartes' secret notebook, De solidorum elementis, and scrutinized the writings in front of him. He didn't have much time. There were sixteen folio pages in this notebook. Either he knew that Clerselier did not want him to copy the notebook, or Clerselier had imposed very strict rules on the copying. Leibniz had to use all the mathematical skills available to him. But he had the tools needed for breaking codes—he was the expert on combinations and decoding. If anyone could break Descartes' code, it was Leibniz.
Leibniz looked at the page of Descartes' secret notebook. On one side were densely drawn figures that Descartes had sketched. It was hard to make out exactly what these were. On the other side of the page were formulas and symbols Leibniz could not immediately decipher. He quickly turned back his glance to the figures. Then he understood what these figures were meant to represent: a cube, a pyramid, and an octahedron (two square-based pyramids joined at their bases).
The cube has six faces, Leibniz knew. He also knew, without having to count, since he had an immeasurably fast mind, that the pyramid has six edges. The octahedron has six vertices (or corners).
Descartes must have been on the occult search for the beast of the Apocalypse. Each of the three figures gave him a 6, and together they spelled 666. So this was Descartes' secret search—his Rosicrucian journey to mystical power. Then Leibniz turned the page.
Descartes had studied the cube, a symmetric three-dimensional object whose doubling using straightedge and compass stumped the ancient Greeks. From his contacts with Faulhaber, Descartes knew that the pyramid was associated with mystical powers. He wanted to learn more about these mysterious objects. Euclid's Elements, translated into Latin, provided him the opportunity to do so.
Euclid wrote his Elements in thirteen volumes. In these volumes were the important works of Pythagoras, including his famous theorem about right triangles, work on prime numbers, and theorems about plane geometry, as well as the properties of triangles and circles. But in his thirteenth volume, Euclid devoted a major part of his writing to the reg-ulai solids, also called the Phtonic solids after Plato, who knew them. There are five Platonic solids:
The TETRAHEDRON, which is a pyramid with triangular faces
The CUBE, which has square faces
The OCTAHEDRON, which has triangular faces
The DODECAHEDRON, which has pentagonal faces
The ICOSAHEDRON, which has triangular faces
These solids are called reguhr because their faces are all the same and are congruent, and the face angles are all equal. Plato knew that there are only five such three-dimensional polyhedra (solids formed by plane faces). This fact made the ancient Greeks attach mystical properties to these solids, and they were believed to possess supernatural powers and to explain nature. And indeed, regular solids appear in nature. Many kinds of natural crystals are perfect (or close to perfect) regular solids. Euclid proved in volume XIII of his Elements many theorems about inscribing regular solids inside spheres. For example, a cube can be placed snugly inside a sphere, so that its eight vertices touch the inside surface of the sphere. The same can be done with the other regular solids. This fact proved of great importance in the late 1500s to the work of Johann Kepler.
In fact, the regular solids were known before Euclid (third century B.C.) and before Plato (fifth century B.C.). The cube, the tetrahedron, and the octahedron were known to the ancient Egyptians, whose culture predates that of the Greeks by at least a millennium. And a dodecahedron made of bronze has been found and dated to several centuries before Plato. These solids were very important in Greek mathematics, which is why sophisticated discussion and complicated theorems concerning them were placed in the last volume of Euclid's Elements. The regular solids were seen as the culmination of Greek geometry and its extension into three dimensions. These solids were believed to contain the secrets of the universe.
Plato visualized the five regular solids as the four elements earth, water, wind, and fire, as well as a fifth: the universe as a whole. This was a manifestation of the mystical qualities the ancient Greek mathematicians and philosophers ascribed to mathematical concepts and entities, and their view that God and the universe were mathematical. Plato visualized the five elements as in the diagram below (the fifth is the universe).
Descartes had studied the Euclidean theorems about the regular solids. But he strove to go far beyond Euclid and the ancient Greeks. The man who wedded geometry with algebra was looking for a formula that would unify all the Platonic solids, thus allowing him to extract from them a divine truth about mathematics, and perhaps about nature. Such a crowning mathematical glory could augment and cement his philosophy.
When he turned the page, Leibniz found the remaining Platonic solids—Descartes' secret notebook contained all five of them. The number 666 was clearly not Descartes' aim. So what was it that Descartes was looking for in the Platonic solids? We know that Leibniz copied the manuscript very hastily. For a page and a half of copying, he did not see the pattern. Then suddenly he understood everything. Leibniz had found the key.
He did not need to continue the copying. All he needed to do now was to add one small marginal note—a note that none of the analysts who later read Leibniz's writing over the centuries understood, until Pierre Costabel. The mystery was now solved—there was no need to see the remaining pages of Descartes' original notebook. Leibniz now knew exactly what Descartes had discovered. And so did Pierre Costabel, the French priest and mathematician who had spent many years deciphering Leibniz's copy of Descartes' notebook, finally breaking the code in 1987.
Two decades after Leibniz copied Descartes' pages, the original notebook disappeared. On Leibniz's death in 1716, his papers were given to the archives of the Royal Library of Hanover (today's Gottfried Wilhelm Leibniz Library). Since Leibniz had left an immense amount of material, his pages copied from Descartes' notebook escaped attention for almost two centuries.
In 1860, Count Louis-Alexandre Foucher de Careil of the Sorbonne was searching through Leibniz's papers at the Hanover archives when he came upon the copy of Descartes' notebook. Foucher de Careil was not a mathematician, and he lacked the ability to decipher the secret key Descartes had used to disguise his work.
In addition, he was confounded by Descartes' bizarre notation, and mistakenly substituted for Descartes' mystical symbols, as transcribed by Leibniz, the numbers 3 and 4 This made his work even more flawed. Consequently, Foucher de Careil's report of his discovery of the notebook, published that year, was found to be useless. Foucher de Careil's work has even confused later scholars who tried to use it over the years, leading them further away from Descartes' hidden meaning. A similar fate awaited the work of two French scholars, E. Prouhet and C. Mallet, who independently attempted to decipher Descartes' secrets later that same year.
In 1890, the French Academy of Sciences prepared to republish Leibniz's pages with an explanation based on new research on the manuscript by Vice Admiral Ernest de Jonquieres. But like Foucher de Careil before him, Ernest de Jonquieres lacked the mathematical ability to break the secret code needed to understand Descartes' work as transcribed by Leibniz. The academy had to abandon the project.
Almost eight decades later, in 1966, new work on deciphering the notebook was done by a research group that analyzed it in conjunction with information drawn from a collection of Descartes' works compiled by Charles Adam and Paul Tannery in 1912. But yet again the notebook refused to surrender its secrets. The true meaning of the strange symbols, sequences of numbers, and unusual drawings remained an enigma.
Descartes's Secret Notebook Page 19