“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 r?le 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 r?le 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.]