n 1939, employed in a small municipal library of Buenos Aires where the oafishness of his colleagues made him weep with daily frustration, the 40-year-old (and still largely unknown) Jorge Luis Borges collected a few of the reading notes he had made on the streetcar to and from work, and pieced together a short text which, under the title "The Total Library," he sent to the magazine Sur, where it appeared in the August issue. The essay, which links the names of Democritus, Lewis Carroll, Cicero, and the forgotten German fantasy writer Kurd Lasswitz, was developed a couple of years later into another, slightly longer one, "The Library of Babel," which Borges eventually included in his collection "The Garden of Forking Paths" (later expanded under the title of "Ficciones"). The end-paper pages of William Goldbloom Bloch's "The Unimaginable Mathematics of Borges' Library of Babel" (Oxford University Press, 192 pages, $19.95) reproduce the first and last pages of Borges's manuscript, showing that it was written (in what Borges called "the handwriting of a dwarf") on accounting sheets with the heading Haber, or "Credit," in Gothic letters that magically make the "H" look like a "B" and the "r" like an "l."

Kristofer Porter

Reading is an erratic craft. Judging Borges's book "an exotic and decadent work," the jury for the Argentinean National Prize in Literature refused to give him a prize and, in spite of the support of a handful of intelligent friends, it was decades before the extraordinary importance of the work was recognized. Several are the pieces in "The Garden of Forking Paths" (later included in "Ficciones") that have attained the stature of classics; "The Library of Babel" in particular has given rise to a small critical library of its own. Barely nine pages long, "The Library of Babel" is nothing less than an attempt to describe the chaotic order and meaning of the universe, building on the ancient notion of the world as a book (or a book itself divided into an almost infinite number of books) in which we ourselves are written, and which we also attempt to read.

Mr. Bloch, professor of mathematics at Wheaton College, has woven an elegant, ingenious, scholarly interpretation of Borges's text that contradicts the disingenuous "unimaginable" of his title. In 1967, Borges told the French critic Georges Charbonnier that he had kept two ideas in mind when writing "The Library of Babel." The first was a commonplace, an exposition of the combinatory art that has enthralled mathematicians from Archimedes onward, and a conceit amusingly described by Lewis Carroll in "Sylvie and Bruno": that since the number of words in any given language is finite, their possible combinations  i.e., books  are finite also, and that therefore, in the near future, writers will no longer ask, "What book shall I write?" but, "Which book shall I write?"

Borges confessed that, beyond this abstract idea, he was also describing the troubling feeling of being lost in the universe, and of not being able to understand it. "In my story," he told Charbonnier, "there is an intellectual component, and another, of greater importance, I think, that has to do with my sense of loneliness, anguish, uselessness, and of the mysterious nature of the universe, of time, and more importantly, of ourselves. Or rather, of myself." Sensibly, Mr. Bloch has busied himself with the first idea, and left the second one to the empathy of each individual reader.

Mathematics was one of Borges's lifelong passions; he considered it, with theology, a branch of fantastic literature. In his early childhood, Borges had been taught by his father the paradoxes of Zeno and the rudiments of algebra, and his writing abounds in references to magical mathematical imaginings, such as Leibniz's binary notation or Brouwer's map, which, as Guillermo Martinez demonstrated in his "Borges and Mathematics," lent Borges a framework or scaffolding for many of his fictions, most notable among them "The Library of Babel."

In Borges's imagining, the Library of Babel itself is a building composed of an indefinite number of hexagonal galleries. A ventilation shaft in the center of each allows the visitor to see the floors above and below, in endless sequence. Each wall of each hexagon holds 32 books of identical size; each book has 410 pages; each page, 40 lines; each line, approximately 80 letters. All possible combinations of the 25 orthographic symbols make up the books; therefore, every conceivable book must exist in the monstrous library. In his story, Borges gives just a few examples of what might be found here: "the detailed history of the future, the autobiographies of the archangels, the faithful catalogue of the Library, thousands and thousands of false catalogues, the proof of the falsity of those false catalogues, the proof of the falsity of the true catalogue ..."

The numbers he chose for the shelves and books in his story, Borges later confessed, were simply those of the municipal library in which he worked  and which he himself found so horrible. "Learned critics," he noted later, with some evident pleasure, "studied these figures and generously lent them a mystical significance." Mr. Bloch, with similar generosity, and in an exercise he himself describes as "tedious, uninspired, but straightforward" if carried out in full, asks whether our entire universe could in fact contain this dizzying number of books. Even if it could (if the size of the library, as Borges suggests, coincided with that of the universe), the inconceivably vast space would make it impossible for a human librarian to even barely begin its exploration. Walking 60 miles a day for 100 years, notes Mr. Bloch, our vigorous librarian would only travel a distance slightly less than that which light covers in two minutes. "To cross our universe, which is incomprehensibly dwarfed by the Library, light would need to travel for a least 15 billion years." It would take a librarian, moving at the leisurely pace of a connoisseur, considerably longer  a mathematical certainty that mirrors the nightmarish vision Borges said he wished to convey.

Though I confess that my mathematical illiteracy made it difficult for me to follow many of his formulas and graphs, the lucidity of Mr. Bloch's arguments enlightened me nevertheless, in conclusions such as this: "The librarian's life and the Library together embody a Turing machine [a rudimentary computer], running an unimaginable program whose output can only be interpreted by a godlike external observer." This fits precisely with Borges's intuition that the world is its own representation, a notion repeated many times throughout his writings: As a map that coincides with the geography it surveys ("The Cartographers' Empire"), as a world encyclopedia whose entries already are the very world it describes ("The Congress"), as a point of light in which everything in the universe is assembled ("The Aleph"), and as the goings and comings of one man incarnated in that man (the epilogue to "Dreamtigers").

Mr. Bloch notes in his preface that the ideal reader of his book is Umberto Eco. Unworthy as I am to aspire to the condition of the great polymath (with whom I share nothing but the girth and the beard), I was delighted and instructed by the book's wit and wisdom, and grateful for the guided tour through the mathematical foundations on which both the Library of Babel and its mirror, our universe, are so delicately balanced.

Mr. Manguel is a critic, translator, essayist, and author. Among his many books are "A History of Reading," "With Borges," and "The Library at Night."