"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 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?"