"Your majesty," said Eliza, "I grieve for le duc. le duc. I trust he has gone to his reward. I pray that I trust he has gone to his reward. I pray that L'Emmerdeur L'Emmerdeur will get what he deserves for what he has done. But I cannot, I will not, allow the so-called King of the Vagabonds the additional satisfaction of disrupting the peaceful conduct of your majesty's household, that is to say will get what he deserves for what he has done. But I cannot, I will not, allow the so-called King of the Vagabonds the additional satisfaction of disrupting the peaceful conduct of your majesty's household, that is to say La France; La France; and so, notwithstanding my feelings of shock and grief at this moment, I beg your leave to accept the proposal of marriage that was tendered to me earlier this evening by etienne de Lavardac-now, duc d'Arcachon." and so, notwithstanding my feelings of shock and grief at this moment, I beg your leave to accept the proposal of marriage that was tendered to me earlier this evening by etienne de Lavardac-now, duc d'Arcachon."

"Then marry him with all the blessings a King can bestow," the King answered.

And in this moment Eliza was startled by a most unexpected rush of sound from all about her. In any other circ.u.mstance she'd have recognized it instantly. But here, given all that had happened, she had to look about and verify it with her eyes: the guests were applauding. It was not, of course, a raucous ovation. Half of them were openly weeping. Many of the ladies had fled the room. Madame la d.u.c.h.esse d'Arcachon was being carried out unconscious, and Eliza's unwitting fiance only remained in the room because someone was obliged to greet Madame la marquise de Maintenon. But for all that, the remaining guests produced a spontaneous patter of applause. It was not that they had forgotten the Duke's head-that was unlikely-but that they found something stirring in how this scene of shock and horror had been adroitly reversed. The applause was an expression of defiance. Eliza, understanding this belatedly, acknowledged it with a diffident curtsey. Presently etienne drifted to her side-someone had explained matters to him-and took her hand, and then the applause welled up again, for just a moment. Then it died abruptly and was replaced by altogether more fitting sounds of sobbing, wailing, and praying. Eliza was distracted for a moment by a glimpse of a rider out in the courtyard wheeling his mount around with great panache, and galloping out into Paris. It was the Earl of Upnor. was unlikely-but that they found something stirring in how this scene of shock and horror had been adroitly reversed. The applause was an expression of defiance. Eliza, understanding this belatedly, acknowledged it with a diffident curtsey. Presently etienne drifted to her side-someone had explained matters to him-and took her hand, and then the applause welled up again, for just a moment. Then it died abruptly and was replaced by altogether more fitting sounds of sobbing, wailing, and praying. Eliza was distracted for a moment by a glimpse of a rider out in the courtyard wheeling his mount around with great panache, and galloping out into Paris. It was the Earl of Upnor.

Then she attended to the King, who was speaking: "Father edouard. We came together here for a small celebration. But the only celebration that is fitting, on an evening such as this one, is that of the Ma.s.s."

"Of course, sire."

"We will observe a funeral Ma.s.s for Monsieur le duc d'Arcachon. Following that, a wedding for the new duc and Mademoiselle la comtesse de la Zeur."

"Yes, sire," said de Gex. "By your majesty's leave, the family chapel has already been made ready for a wedding; shall we perform the funeral here, where there is more room, and move to the chapel thereafter?"

King Louis XIV made a tiny nod of a.s.sent, and then turned his gaze on d'Avaux, who had not yet been dismissed. "Monsieur le comte," said the King, "you were about to voice an opinion as to the ident.i.ty of the woman who inspired the heinous murder of my cousin?"

"By your majesty's leave," d'Avaux said, "If we interpret L'Emmerdeur L'Emmerdeur's statement literally, literally, it will only amount to something ba.n.a.l. I have no doubt that he was merely trying to impress some it will only amount to something ba.n.a.l. I have no doubt that he was merely trying to impress some wh.o.r.e wh.o.r.e he met once in Paris." And he could not prevent his eyes from flicking at Eliza for just a moment as he said this; but then he returned his attention to the King. "I was, rather, attempting to make a more he met once in Paris." And he could not prevent his eyes from flicking at Eliza for just a moment as he said this; but then he returned his attention to the King. "I was, rather, attempting to make a more general general statement about statement about all all the enemies of France, and what moves them." He backed away one step, turned, and swept his arm up and out towards a corner of the painted ceiling, where Pandora was opening up her Box (in-come to think of it-an odd reminder of the box-opening scene that had just played out on the ballroom floor) to release a flood of demonic Vices. Pandora had been painted, as everyone knew, to resemble Mary, the usurper Queen of England. The foremost of the Vices rushing out of her box was green-eyed Envy, who had been made to resemble Sophie of Hanover. It was to Envy that d'Avaux now drew the King's attention. "That, your majesty, is the lady love, not only of the enemies of France, and what moves them." He backed away one step, turned, and swept his arm up and out towards a corner of the painted ceiling, where Pandora was opening up her Box (in-come to think of it-an odd reminder of the box-opening scene that had just played out on the ballroom floor) to release a flood of demonic Vices. Pandora had been painted, as everyone knew, to resemble Mary, the usurper Queen of England. The foremost of the Vices rushing out of her box was green-eyed Envy, who had been made to resemble Sophie of Hanover. It was to Envy that d'Avaux now drew the King's attention. "That, your majesty, is the lady love, not only of L'Emmerdeur L'Emmerdeur-who is after all a n.o.body-but also of all the Dutch and English. Envy is what inspires their chivalrous chivalrous acts." acts."

"You powers of observation are as keen as ever, monsieur," said the King, "and I have never been more pleased to number you among my subjects."

At this d'Avaux bowed very deeply. Eliza could not help but think that, for all the frustration and defeat d'Avaux had suffered here, this immense compliment from le Roi le Roi was more than compensation enough. It made her wonder: was more than compensation enough. It made her wonder: Did the King know everything? Did the King know everything?

The King continued: "Monsieur le comte d'Avaux has, as usual, spoken wisely. It follows that if we are to baffle Envy's devotees, we should celebrate all that is magnificent in this Realm: with funerals, the magnificence that has pa.s.sed, and with weddings, the magnificence that is yet to come. Let it be so."

And it was was so. so.

Most of the guests went home following the funeral in the ballroom, but enough remained to fill the chapel for the wedding. After that, they went directly into a second second funerary ma.s.s; for Madame la d.u.c.h.esse d'Arcachon had not recovered from the sight of her husband's head pulled from the box. What everyone had taken for a swoon, had in fact been a stroke. One side of her body had already gone lifeless by the time they had carried her to her bedchamber, and during the subsequent hours, the paralysis had spread to engulf the other side as well, and finally the heart had stopped. And so, by the time the newlyweds emerged from the doors of the Hotel Arcachon, around midnight, and climbed into a borrowed carriage (for the white seash.e.l.l-coach was both fouled and broken), both of etienne's parents were dead, and being made ready for shipment to consecrated ground at La Dunette. etienne was duc, and Eliza was d.u.c.h.esse, d'Arcachon. funerary ma.s.s; for Madame la d.u.c.h.esse d'Arcachon had not recovered from the sight of her husband's head pulled from the box. What everyone had taken for a swoon, had in fact been a stroke. One side of her body had already gone lifeless by the time they had carried her to her bedchamber, and during the subsequent hours, the paralysis had spread to engulf the other side as well, and finally the heart had stopped. And so, by the time the newlyweds emerged from the doors of the Hotel Arcachon, around midnight, and climbed into a borrowed carriage (for the white seash.e.l.l-coach was both fouled and broken), both of etienne's parents were dead, and being made ready for shipment to consecrated ground at La Dunette. etienne was duc, and Eliza was d.u.c.h.esse, d'Arcachon.

The new Duke and d.u.c.h.ess consummated their union under many blankets in a carriage en route to Versailles, and arrived at La Dunette in the darkest and coldest hours before dawn. Fresh hoof-prints in the snow on La Dunette's gravel paths told them that they were not the first to come this way since the snow had ceased to fall. When they reached the chateau, they found the servants already awake and dressed, and red around the eyes. The doyenne of the maidservants took Eliza to one side, and let her know that she must go down to the Convent of Ste.-Genevieve immediately, for there was dreadful news. Eliza, unwilling to wait for preparations to be made, straddled the first horse she could get to-it was an albino mare-and rode it bareback down to the little convent full of weeping and praying nuns. She went directly to the room where Jean-Jacques slept. She knew already what she would see there, for she had seen it before in nightmares, as every parent does: the shattered window, curtains riven, muddy bootprints on the sill, and the empty cradle. The blankets had been taken; that was a comfort to her, as it suggested that wherever Jean-Jacques might be, he was at least not freezing to death. Left in the little bed was a note, addressed to the Countess de la Zeur; for whoever had penned it had not got the news of her new rank and t.i.tle. It read: Fraulein! You and your Vagabond have something of mine. I have something of yours. You and your Vagabond have something of mine. I have something of yours.-L

Schlo Wolfenb.u.t.tel, 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 ill.u.s.trated by drawing invisible lines on the hearth with the toe of his boot. Petty n.o.bles of Lower Saxony who trespa.s.sed 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. 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 ill.u.s.trated by drawing invisible lines on the hearth with the toe of his boot. Petty n.o.bles of Lower Saxony who trespa.s.sed 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 lat.i.tude, 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 pa.s.sages, 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 hears of them but does not of them but does not meet meet them." them."

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

"They do do seem seem impossibly 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 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. everyone."

"Sophie married in to this this lot," Leibniz said, glancing back. 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 aeons by Danes, Swedes, and G.o.d knows who else, and had nothing to do but f.u.c.k. 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 Faery Tale has it, just right 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 coup de main had made its way to the Venetian brothel where Ernst August and Georg Wilhelm had set up their headquarters, 'twas a had made its way to the Venetian brothel where Ernst August and Georg Wilhelm had set up their headquarters, 'twas a fait accompli 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 Wolfenb.u.t.tel-you have just met them. Now, Ernst August and Sophie had already resolved to make their little fiefdom into a Parna.s.sus, a kingdom of Reason-"

"So they hired you, naturally."

"No, actually, there was a lot lot of that going round at the time. John Frederick wanted to do the same at Hanover." 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 Wolfenb.u.t.tel."

Leibniz ushered Fatio into an entrance hall dominated by frighteningly ma.s.sive 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 her Parna.s.sus with John Frederick's-of which you were the crowning glory-" Parna.s.sus 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 here to meet you? I'd expected to find you at Hanover." 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- will be an ornament-not that there is anything wrong with that-mine will be a tool, an engine of knowledge." 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 G.o.d," 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 G.o.d 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 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 salon at Versailles by describing it to them..." at Versailles by describing it to them..."

"On the contrary, Doctor! When I bid you adieu 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 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 any matter. Wren's library remains half-built for want of funds." 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-sc.r.a.ps, 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 Dynamics for the time being and am trying to finish another book for the time being and am trying to finish another book Proto-gaea, Proto-gaea, whose subject matter you may know from the t.i.tle. 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." whose subject matter you may know from the t.i.tle. 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. 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 ma.s.sive 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 a.s.sembled 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 t.i.tle and a long Roman numeral. "It is a description description of a book," he concluded. 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 n.o.ble bibliophiles Duke August had bought all his books as ma.s.ses 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 t.i.tles, 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. were. The numbers merely denote the order in which the Duke acquired, or at least cataloged, the volumes. They say nothing of the contents." 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 a.s.sociated with a number. But only one one number! Like a number-line, it is one-dimensional. In a.n.a.lytic geometry we may cross two or three number-lines at right angles to create a multi-dimensional s.p.a.ce. 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." number! Like a number-line, it is one-dimensional. In a.n.a.lytic geometry we may cross two or three number-lines at right angles to create a multi-dimensional s.p.a.ce. 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 a.s.sign 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 s.p.a.ce into a uni-dimensional number-line."

"I am pleased that you favor my proposal thus far, Fatio, but now consider the following: suppose we a.s.sign 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 this problem." problem."

"Suppose we use this 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 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'. demo'.