“Now they are in a heap, later they shall be on shelves—either way, how do you find what you want?” Leibniz asked.
“I believe you are now questioning me in a Socratic mode.”
“And you may answer in any mode you like, Monsieur Fatio, provided that you do answer.”
“I suppose one would go by the numbers. Supposing that they were shelved in numerical order.”
“Suppose they were. The numbers merely denote the order in which the Duke acquired, or at least cataloged, the volumes. They say nothing of the contents.”
“Re-number them, then.”
“According to what scheme? By name of author?”
“I believe it would be better to use something like Wilkins’s philosophical language. For any conceivable subject, there would be a unique number. Write that number on the spine of the book and shelve them in order. Then you can go directly to the right part of the library and find all books on a given subject together.”
“But suppose I am making a study of Aristotle. Aristotle is my subject. May I expect to find all Aristotle-books shelved together? Or would his works on geometry be shelved in one section, and his works on physics elsewhere?”
“If you look at it that way, the problem is most difficult.”
Leibniz stepped over to an empty bookcase and drew his finger down the length of one shelf from left to right. “A shelf is akin to a Cartesian number-line. The position of a book on that shelf is associated with a number. But only one number! Like a number-line, it is one-dimensional. In analytic geometry we may cross two or three number-lines at right angles to create a multi-dimensional space. Not so with bookshelves. The problem of the librarian is that books are multi-dimensional in their subject matter but must be ordered on one-dimensional shelves.”
“I perceive that clearly now, Doctor,” Fatio said. “Indeed, I am beginning to feel like the character of Simplicio in one of Galileo’s dialogs. So let me play that rôle to the hilt, and ask you how you intend to solve the problem.”
“Well played, sir. Consider the following: Suppose we assign the number three to Aristotle, and four to turtles. Now we must decide where to shelve a book by Aristotle on the subject of turtles. We multiply three by four to obtain twelve, and then shelve the book in position twelve.”
“Excellent! By a simple multiplication you have combined several subject-numbers into one—collapsed the multi-dimensional space into a uni-dimensional number-line.”
“I am pleased that you favor my proposal thus far, Fatio, but now consider the following: suppose we assign the number two to Plato, and six to trees. And suppose we acquire a book by Plato on the subject of trees. Where does it belong?”
“The product of two and six is twelve—so it goes next to Aristotle’s book on turtles.”
“Indeed. And a scholar seeking the latter book may instead find himself with the former—clearly a failure of the cataloging system.”
“Then let me step once again into the rôle of Simplicio and ask you whether you have solved this problem.”
“Suppose we use this coding instead,” quoth the Doctor, reaching behind the bookcase and pulling out a slate on which the following table had been chalked—thereby as much as admitting that the conversation, to this point, had been a scripted demo’.
2 Plato
3 Aristotle
5 Trees
7 Turtles
2×5=10 Plato on Trees
3×7=21 Aristotle on Turtles
2×7=14 Plato on Turtles
3×5=15 Aristotle on Trees
[etc.]
“Two, three, five, and seven—all prime numbers,” remarked Fatio after giving it a brief study. “The shelf-numbers are composites, the products of prime factors. Excellent, Doctor! By making this small improvement—assigning prime numbers, instead of counting numbers, to the various subjects—you have eliminated the problem. The shelf position of any book may be found by multiplying the subject-numbers—and you may be assured it will be unique.”
“It is a pleasure to explain it to one who grasps the principle so readily,” Leibniz said. “Huygens and the Bernoullis have both spoken highly of you, Fatio, and I can see that they were by no means insincere.”
“I am humbled to hear my name mentioned in the same sentence with theirs,” Fatio returned, “but since you have been kind enough to so favor me, perhaps you will indulge me in a question?”
“It would be my privilege.”
“Your scheme is a fine way to build a library. For the correct position of any book may be found by taking the product of the several primes that correspond to its subjects. Even when those numbers grow to several digits, that presents no great difficulty; and in any event it is well known that you have invented a machine capable of multiplying numbers with great facility, which I now perceive is just one element of the immense knowledge engine you have proposed to build.”
“Indeed, all of these are of a piece, and may be considered aspects of my Ars Combinatorica. Did you have a question?”
“I fear that your library, once built, will be difficult to understand. You are seeking the help of the Emperor in Vienna, are you not?”
“It cannot be accomplished without the resources of a great kingdom,” Leibniz said vaguely.
“Very well, perhaps you are in communication with some other great prince. At any rate, it would seem, then, that you wish to make your Knowledge Engine on a colossal scale.”
“Marshalling resources is a continuing problem,” the Doctor said, still treading gingerly.
“I predict that you will find success, Doctor Leibniz, and that one day there will rise up, in Berlin, Vienna, or even Moscow, a Knowledge Engine on a titanic scale. The shelves will extend for countless leagues and will be crowded with books all arranged according to the rules of your system. But I fear that I could very easily become lost in the bowels of that place. Looking at a shelf I might see some number, eight or nine digits long. I would know this to be a composite number, the product of two or more primes. But to decompose such a number into its prime factors is a notoriously difficult and tedious problem. There is a curious asymmetry about this approach, in other words, lying in the fact that to its creator the structure and organization of the great library will be clear as glass—but to a solitary visitor it will seem a murky maze of impenetrable numbers.”
“I do not deny it,” Leibniz answered without hesitation, “but I find in this a sort of beauty, a reflection of the structure of the universe. The situation of the solitary visitor, as you have described it, is one with which I am familiar.”
“That is odd, for I conceive of you as the creator who stands with his hand on the Bücherrad and comprehends all.”
“You should know this about me. My father was a learned man who owned one of the finest libraries in Leipzig. He died when I was very small. Consequently I knew him only as a jumble of childish perceptions—between us there were feelings but never any rational connection, perhaps somewhat like the relationship that you or I have with God.”
And he related a story about how he had, for a time, been locked out of his father’s library, but later re-admitted.
“So I ventured into that library which had been closed up since the death of my father and still smelled like him. It might seem funny for me to speak of the smell, but that was the only connection I could draw at the time. For the books were all written in Latin or Greek, languages I did not know, and they treated of subjects with which I was completely unfamiliar, and they were arranged upon the shelves according to some scheme that must have been clear to my father, but to me was unknown, and would have been beyond my ken even if someone had been there to explain it to me.
“Now in the end, Monsieur Fatio, I mastered that library, but in order to do it I first had to learn Greek and Latin, and then read the books. Only when I had done these things was I finally able to do the most difficult thing of all, namely to understand the organizing principle by which my father had arrange
d the books on the shelves.”
Fatio said: “So you are not troubled by the plight of my hypothetical scholar, a-mazed in the penetralia of your Knowledge Engine. But Doctor Leibniz, how many persons, dropped into a library of books written in unknown languages, could do what you did?”
“The question is more than just rhetorical. The situation is not merely hypothetical,” Leibniz answered. “For every human being who is born into this universe is like a child who has been given a key to an infinite Library, written in cyphers that are more or less obscure, arranged by a scheme—of which we can at first know nothing, other than that there does appear to be some scheme—pervaded by a vapor, a spirit, a fragrance that reminds us that it was the work of our Father. Which does us no good whatever, other than to remind us, when we despair, that there is an underlying logic about it, that was understood once and can be understood again.”
“But what if it can only be understood by a mind as great as God’s? What if we can only find what we want by factoring twenty-digit numbers?”
“Let us understand what we may, and extend our reach, insofar as we can, by the making of engines, and content ourselves with that much,” Leibniz answered. “It will suffice to keep us busy for a while. We cannot perform all of the calculations needed without turning every atom in the Universe into a cog in an Arithmetickal Engine; and then it would be God—”
“I think you are coming close to words that could get you burnt at the stake, Doctor—meanwhile, I turn to ice. Is there a place where we could strike a balance between those two extremes?”
THE DOCTOR HAD CAUSED a large shed to be scabbed onto the outer wall of the stable and filled with the books and papers most important to him. In one corner stood a black stove having the general size and shape of the biblical Tower of Babel. When they arrived it was merely warm, but Leibniz wrenched open several doors, rammed home half a cord or so of wood, and clanged them to. Within seconds, ears began to pop as the mickle Appliance sucked the air from the room. The iron tower began to emit an ominous rumbling and whooshing noise, and Leibniz and Fatio spent the rest of the conversation nervously edging away from it, trying to find the radius where (to paraphrase Fatio) being burnt alive was no more likely than freezing to death. This zone proved surprisingly narrow. As Leibniz fussed with the stove, which had taken up a kind of eerie keening, Fatio stepped back a pace, and let his eye fall on a sheet of paper—the topmost of several that were sticking out of a book. A few lines of printing were visible at the top of the page, written in Leibniz’s hand:
DOCTOR
THE RECENT EVENTS IN THE BALLROOM OF THE HÔTEL ARCACHON WERE OF SUCH A DRAMATICK NATURE THAT I CANNOT BUT THINK YOU HAVE ALREADY HAD ACCOUNTS OF THEM FROM DIVERSE SOURCES HOWEVER MY VERSION FOLLOWS…
Beyond that point all was swallowed up between the pages of the enclosing book, which was expensively bound in red leather, ornately gilded with both Roman and Chinese characters.
“Any possibility of tea?” Fatio inquired, spying a kettle that had been left on one of the steps of the flaming ziggurat. Shielding his face in the crook of one arm, Leibniz ventured closer, seized a poker, and lunged like a fencing-master at the kettle to see whether it contained any water. Meanwhile Fatio peeled back the topmost sheet to reveal a letter written on different paper, in a different hand: Eliza’s!
To G. W. Leibniz from Eliza, the Marrying Maiden
Doctor,
You will want to know everything about the dress I was married in. The stomacher is made of Turkish watered silk decorated with several thousand of the tiny pearls that come from Bandar-Kongo on the Persian Gulf…
Leibniz had been rummaging in a drawer. He pulled up a black slab about the size of a folio book, impressed with a single huge Chinese character, and snapped off a corner. “Caravan tea,” he explained. “Unlike your English and Dutch tea, which comes loose off of ships, this stuff was brought overland, via Russia—it is a million dried leaves pressed together into a brick.”
Fatio did not seem to be as fascinated by this as Leibniz had hoped. Leibniz tried another gambit: “Huygens wrote to me recently, and mentioned you had come over from London.”
“Monsieur Newton and I devoted the month of March to reading Mr. Huygens’s Treatise on Light and were so taken with it that we agreed to divide forces for the year—I have been studying with Huygens—”
“And Newton toils at his Alchemy.”
“Alchemy, theology, philosophy—call it what you will,” Fatio said coolly, “he is close to an achievement that will dwarf the Principia.”
“I don’t suppose it has anything to do with gold?” asked Leibniz.
Fatio—generally so birdlike-quick in his answers—allowed some moments to pass. “Your question is a bit vague. Gold is important to Alchemists,” he allowed, “as comets are to astronomers. But there are some, of a vulgar turn of mind, who suppose that Alchemists are interested in gold only in the same sense as bankers are.”
“C’est juste. Though there is a troublesome banker, not far from here, who seems to value it in both the monetary and the Alchemical sense.” Leibniz, who until this point in the conversation had been the embodiment of good cheer, deflated as he was saying these words, as if he had been reminded of something very grave, and his eye strayed over to the outlandish red-leather book. This topic had had the same effect on his spirits as a handful of earth tossed into a fire. Again, Fatio allowed some moments to pass before he responded; for he was studying Leibniz carefully.
“I think I know who you mean,” Fatio said finally.
“It is most curious,” Leibniz said. “Perhaps you have heard some of the same stories concerning this as I have. The entire controversy, as I understand it, revolves around a belief that there is a particular sample of gold, whose precise whereabouts are unknown, but that possesses some properties that make it more valuable, to Alchemists, than ordinary gold. I would expect a banker to know better!”
“Do not make the error of believing that all gold is the same, Doctor.”
“I thought Natural Philosophy had proved at least that much.”
“Why, some would say it has proved the opposite!”
“Perhaps you have read something new in London or Paris that I have not seen yet?”
“Actually, Doctor, I was thinking of Isaac’s Principia.”
“I have read it,” Leibniz said drily, “and do not recollect seeing anything about gold.”
“And yet it is clear enough that two planets of equal size and composition will describe different trajectories through the heavens, depending on their distances from the sun.”
“Of course—that is necessarily true, by the inverse-square law.”
“Since the two planets themselves are equal in every way, how can this difference in their trajectories be accounted for, unless you enlarge your scope of observations to include the difference in their situations vis-à-vis the sun?”
“Monsieur Fatio, a cornerstone of my philosophy is the identity of indiscernibles. Simply put, if A cannot be discerned from B, then A and B are the same object. In the situation you have described, the two planets are indiscernible from each other, which means that they ought to be identical. This includes having identical trajectories. Since they are obviously not identical, in that their trajectories differ, it follows that they must in some way be discernible from each other. Newton discerns them by assigning them differing positions in space, and then presuming that space is somehow pervaded by a mysterious presence that accounts for the inverse-square force. That is, he discerns one from the other by appealing to a sort of mysterious external quality of space…”
“You sound like Huygens!” Fatio snapped, suddenly annoyed. “I might as well have stayed in the Hague.”
“I am sorry if the tendency of me and Huygens to agree causes you grief.”
“You may agree with each other all you like. But why will you not agree with Isaac? Can you not perceive the magnificence of what he has achieved?”
&n
bsp; “Any sentient man can perceive that,” Leibniz returned. “Almost all will be so blinded by its brilliance that they will be unable to perceive its flaws. There are only a few of us who can do that.”
“It is very easy to carp.”
“Actually it is rather difficult, in that it leads to discussions such as this one.”
“Unless you can propose an alternative theory that mends these supposed flaws, I believe you should temper your criticisms of the Principia.”
“I am still developing my theory, Monsieur Fatio, and it may be a long time before it is capable of making testable predictions.”
“What conceivable theory could explain the discernibility of those two planets, without making reference to their positions in absolute space?”
THIS LED TO AN INTERLUDE in the snow outside. Doctor Leibniz packed two handfuls of snow together between his hands, watched warily by Fatio. “Don’t worry, Monsieur Fatio, I’m not going to throw it at you. If you would be so helpful as to make two more, about the size of melons, as like to each other as possible.”
Fatio was not quick to warm to such a task, but eventually he squatted down and began to roll a pair of balls, stopping every couple of paces to pound away the rough edges.
“They are as close to indiscernible as I can make them under these conditions—which is to say, in twilight with frozen hands,” shouted Fatio towards Leibniz, who was a stone’s throw off, wrestling with a snowball that weighed more than he did. When no response came back, he muttered, “I shall go in and warm my hands if that is acceptable.”
But by the time Nicolas Fatio de Duillier had got back to Leibniz’s office, his hands were warm enough to do a few things. He took another look at the papers stuck into the Chinese book. The letter from Eliza was inordinately long, and appeared to consist entirely of gaseous chatter about what everyone was wearing. Yet on top of it was the other document, addressed to the Doctor but written in the Doctor’s hand. A mystery. Perhaps the book was a clue? It was called I Ching. Fatio had seen it once before, in the library of Gresham’s College, where Daniel Waterhouse had fallen asleep over it. The sheaf of papers had been used to mark a particular chapter entitled: 54. Kuei Mei: The Marrying Maiden. The chapter itself was a bucket of claptrap and mystickal gibberish.