The Confusion

Schlo? Wolfenbuttel, Lower Saxony
DECEMBER 1690

It seems to us indeed that this block of marble brought from Genoa would have been exactly the same if it had been left there, because our senses make us judge only superficially, but at bottom because of the connection of things the whole universe with all of its parts would be entirely different, and would have been another from the beginning, if the least thing in it went otherwise than it does.

–LEIBNIZ

THE FORMAL INTRODUCTIONS HAD PLAYED against the backdrop of a fireplace large enough to burn a small village. For half an hour or so, Nicolas Fatio de Duillier and Gottfried Wilhelm Leibniz had inched towards each other as if conjoined by an invisible spring stretched through the middle of the muttering swarm of Freiherren and Freifrauen. When finally they drew within hailing distance of each other they switched over to French and launched into an easy chat about involutes, evolutes, and radial curves. Leibniz moved on into a tutorial about a new notion he had been toying with in his spare time, called parallel curves, which he illustrated by drawing invisible lines on the hearth with the toe of his boot. Petty nobles of Lower Saxony who trespassed on these were politely asked to move, so that Fatio could draw several invisible lines and curves of his own. Then he managed, in a single grammatically correct sentence, to make reference to Apollonius of Perga, the Folium of Descartes, and the Limacon of Pascal.

The walls of the room were decorated with impossibly optimistic paintings, two to three fathoms on a side, of sowers, reapers, and gleaners plying their respective trades in sun-gilded fields. Fickle light was shed on these by flames burning in bronze baskets carried on the heads of naked, muscle-bound, bronze blackamoors planted on ten-ton pedestals in the corners.

Fatio looked significantly at his watch. “The sun rose-what-two hours ago? At this latitude, we have-say-two hours of daylight remaining?”

“A bit more, sir, by your leave,” answered Leibniz with a wink, or perhaps a cinder had flown into his eye. But that was all that needed to be said. Both men turned their backs to the fire for a last helping of warmth, then marched towards the room’s exit, groping through darkness and smoke for the door.

They were blinded by powerful bluish light. The Schlo?’s galleries-which served not only as connecting passages, but also as a sort of perimeter defense against the climate-ran around its exterior wall, and had plenty of windows. The low light of the heavy winter sun ricocheted off the ice-crusted snow that covered the dead gardens, filling these corridors with chilly brilliance. An indignant servant slammed the doors behind them to keep the heat in. Leibniz and Fatio began to match each other’s pace down the length of the gallery, moving just short of a sprint. The cold seemed to have dissolved their stockings. It was imperative to keep the knees and calves working.

“Some family,” Fatio ventured. “One hears of them but does not meet them.”

“They grow into the interstices left between other families,” Leibniz admitted. “You would find the Hanover crowd more interesting.”

“They do seem impossibly fecund,” Fatio said. “The Winter Queen left children strewn all over the place, and Sophie, at one time or another, has given birth to nearly everyone.”

“Sophie married in to this lot,” Leibniz said, glancing back.

“And that is how you became her librarian?”

“Privy Councillor,” Leibniz corrected him.

“Sir! I beg you to accept my apologies and my congratulations!” announced Fatio, faltering and reaching for his hat so that he could bow; but Leibniz caught his elbow and pulled him along.

“Never mind, it happened quite recently. In brief, the family of Dukes whose ancestral home is this Schlo? put on a tremendous spate of baby-making round the time of the Thirty Years’ War, probably because they were besieged here for ?ons by Danes, Swedes, and God knows who else, and had nothing to do but fuck. Four brothers were born in an interval of eight years! All survived!”

“Calamity!”

“Indeed. Through the 1650s the lads ran riot through the courts of Christendom, trying to mitigate the unnatural surplus of virgins that had built up during the War. All of them wanted Sophie. One of them was too fat and, in any event, Catholic. One was too drunk and impotent. One was famously syphilitic. But the youngest-Ernst August-was, as the F?ry Tale has it, just right! Sophie married him.”

“But my dear Doctor, how did the youngest brother end up in the best position?”

They came to a corner of the Schlo? and turned into another endless gallery.

“In 1665 the drunk one died. Ernst August and Georg Wilhelm-the syphilitic-were off sowing their wild oats. So John Frederick-”

“By process of elimination, he would be the obese Catholic?”

“Yes. He appropriated the Duchy and raised an army to defend it. By the time news of this coup de main had made its way to the Venetian brothel where Ernst August and Georg Wilhelm had set up their headquarters, ’twas a fait accompli. Later, like good brothers, they worked out a settlement. John Frederick got the great prize, and was made Duke of Hanover. Georg Wilhelm became Duke of Celle. Ernst August-despite being a Protestant-remained the Bishop of Osnabruck. The odds and ends of the clan ended up here in Wolfenbuttel-you have just met them. Now, Ernst August and Sophie had already resolved to make their little fiefdom into a Parnassus, a kingdom of Reason-”

“So they hired you, naturally.”

“No, actually, there was a lot of that going round at the time. John Frederick wanted to do the same at Hanover.”

“It must have been a good time to be a savant.”

“Indeed, one could name one’s price. John Frederick had more money and a vast library.”

“Right, now I am starting to remember it. Huygens told me that after he taught you everything he knew concerning mathematics-which would have been round about the early 1670s-you had to leave Paris and take a job in some cold bleak place.” Fatio looked significantly out the window.

“’Twas Hanover actually-a distinction without a difference, as to you it would seem very like Wolfenbuttel.”

Leibniz ushered Fatio into an entrance hall dominated by frighteningly massive staircases.

Sounding a bit perplexed, Fatio said, “Rather a lot of people must have died then, for Ernst August to become Duke of Hanover-”

“John Frederick died in ’79. Georg Wilhelm still lives. But it was Ernst August who became Duke of Hanover, by dint of this or that sub-clause in the agreement made between him and his brothers-I’ll spare you details.”

“So Sophie got to merge her Parnassus with John Frederick’s-of which you were the crowning glory-”

“Really you do flatter, sir.”

“But why did I have to come down here to meet you? I’d expected to find you at Hanover.”

“The Library!” Leibniz answered, surging past the younger man and hurling himself against an immense door. There was a bit of preliminary cracking and tinkling as ice shattered and fell from its hinges. Then it yawned open to afford Fatio a view across several hundred yards of flat snow-covered ground to a dark uneven mountainous structure that was a-building there.

“No fair making comparisons with the one Wren’s building at Trinity College,” Leibniz said cheerfully. “His will be an ornament-not that there is anything wrong with that-mine will be a tool, an engine of knowledge.”

“Engine?” Fatio, who was well-shod, pranced out into the snow in pursuit of Leibniz, who had given up any hope of preserving his boots and shifted to a sort of plodding, stomping gait.

“Our use of knowledge progresses through successively higher levels of abstraction as we perfect civilization and draw nearer to the mentality of God,” Leibniz said, as if making an off-handed comment about the weather. “Adam named the beasts; meaning, that from casual observations of particular specimens, he moved to the recognition of species, and then devised abstract names for them-a sort of code, if you will. Indeed, if he had not done so, Noah’s task would have been inconceivable. Later, a system of writing was developed: spoken words were abstracted into chains of characters. This became the basis for the Law-it is how God communicated His intentions to Man. The Book was written. Then other books. At Alexandria the many books were brought together into the first Library. More recently came the invention of Gutenberg: a cornucopia that spills books out into specialized markets in Frankfurt and Leipzig. The merchants there have been completely unreceptive to my proposals! There are too many books in the world now for any one mind to comprehend. What does Man do, Fatio, when he is faced with a task that exceeds the physical limits of his body?”

“Harnesses beasts, or makes a tool. And beasts are of no use in a Library. So-”

“So we want tools. Behold!” Leibniz proclaimed, taking his hands from his coat-pockets just long enough to direct a sort of shoveling gesture at the looming Pile. “It must be obvious to you that this was a stable* until quite recently. I will stipulate that this is a mean beginning for a library and that you will be able to elicit howls of laughter from the Royal Society and from any salon at Versailles by describing it to them…”

“On the contrary, Doctor! When I bid you adieu I shall go straight back to the Hague to resume my studies with Mr. Huygens. ’Twill be a year or more before I address the Royal Society on any matter. Wren’s library remains half-built for want of funds.”

“Very well, then,” Leibniz muttered, and led Fatio through a temporary door of rough planks and into the stable. It was unheated, but it got their faces out of the wind, and their feet out of the snow. The foundation and ground-floor walls were made of large blocks of undressed stone. Everything above those was built of timbers. So far, the temporary scaffolding was more substantial than the structure itself, which was only sketched in with a few posts and beams. Fatio was baffled, and soon turned his huge eyes downwards to the several tables arrayed in the center of the main floor.

“One day we shall haul out these rude workbenches and replace them with polished desks where scholars will go to their work, illuminated by sky-light from a high fair cupola above,” Leibniz said, craning his head back so that his wig shifted, and thrusting his index finger up through the cloud of vapor that had veiled his words.

“The cupola is a fine innovation, Doctor. Getting adequate light is ever a problem in libraries. Sometimes I am tempted to burn one page to shed light on the next.”

“It is but one instance of the principle.”

“Which principle?”

“I told you I am trying to build a tool, an engine.” Leibniz sighed out a vast cloud of steam.

The tabletops supported divers forms of industry. Each was a gelid still-life unto itself. About half were given over to the library-building project (drawings weighed down with stones and wood-scraps, ragged quills projecting from frozen ink-wells, half-completed ledgers, clamped timbers surrounded by knee-high piles of wood-shavings) and half to whatever the Doctor was interested in at the moment. Fatio drew up short in front of a table strewn with what appeared to be stones; but as he did not fail to note, each stone was impressed with the skeleton of a leaf, insect, fish, or beast-some of them utterly unfamiliar.

“What-”

“I have laid off of Dynamics for the time being and am trying to finish another book Proto-gaea, whose subject matter you may know from the title. But let us refrain from digressions,” said Leibniz, shuffling carefully across the cluttered room. He paused to regard a colossal piece of furniture. “You see I am not the first to contemplate the making of a knowledge engine.”

Fatio closed his eyes, which was the only way to pry his attention from the outlandish skeletons, and stepped back, then maneuvered over to join Leibniz.

“Behold, the Bucherrad!”* Leibniz said.

Viewed end-on, the Bucherrad was hexagonal, and nearly as tall as Fatio. When he worked his way round to the front, he saw that it consisted mostly of six massive shelves, each one a couple of fathoms long, bridging the interval between hexagonal end-caps that were mounted on axles so that the whole apparatus could be revolved. But each of the six shelves was free to revolve on an axis of its own. As the Bucherrad spun, each of those shelves counter-rotated in such a way that it maintained a fixed angle with respect to the floor, and did not spill its load of books.

Going round to the other end, Fatio was able to see how it worked: a system of planetary gears, carven from hard wood, spun about the central axle-tree like Ptolemaic epicycles.

Then Fatio turned his attention to the books themselves: curious folio volumes, hand-written, all in the same hand, all in Latin.

“These were written out personally by one Duke August, a forerunner of that lot you just met. He lived to a great age and died some twenty-five years ago. It was he who assembled most of this collection,” Leibniz explained.

Fatio bent slightly at the waist to read one of the pages. It consisted of a series of paragraphs each preceded by a title and a long Roman numeral. “It is a description of a book,” he concluded.

“The process of abstraction continues,” Leibniz said. “Duke August could not keep the contents of his library in his memory, so he wrote out catalogs. And when there were too many catalogs for him to use them conveniently, he had woodwrights make Bucherrads-engines to facilitate the use and maintenance of the catalogs.”

“It is very ingenious.”

“Yes-and it is threescore years old,” Leibniz returned. “If you do the arithmetick, as I have, you may easily demonstrate that to hold all the catalogs needed to list all the world’s books would require so many Bucherrads that we would need some Bucherradrads to spin them around, and a Bucherrad-rad-rad to hold all of them-”

“German is a convenient language that way,” Fatio said diplomatically.

“And so on with no end in sight! There are not enough woodwrights to carve all of the gears. New sorts of knowledge-engines will be demanded.”

“I confess you have lost me, Doctor.”

“Observe-each book is identified by a number. The numbers are arbitrary, meaningless-a kind of code, like the names Adam gave to the beasts. Duke August was of the old school, and used Roman numerals, which makes it that much more cryptickal.”

Leibniz led Fatio away from the center of the floor toward the rugged stone walls, which were mostly barricaded by high thick ramparts covered in canvas tarpaulins. He peeled up the edge of one and flung it back to reveal that the rampart was a stack of books, thousands of them. All of them had been bound in the same style, in pigskin (for like many noble bibliophiles Duke August had bought all his books as masses of loose signatures and had them bound in his own bindery, by his own servants). The newest ones (say, less than half a century old) were still white. More ancient ones had turned cream, beige, tan, brown, and tar-colored. Many bore scars of long-forgotten encounters between pigs and swineherds’ cudgels. The titles, and those long Roman numerals, had been inscribed on them in what Fatio now recognized as Duke August’s hand.

“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 role 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 role 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 Bucherrad 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 arranged 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: