“Mathematical notions are propositions constructed by our intellect in such a way that they function always as truths, either because they are innate or because mathematics was invented before the other sciences. And the library was built by a human mind that thought in a mathematical fashion, because without mathematics you cannot build labyrinths. And therefore we must compare our mathematical propositions with the propositions of the builder, and from this comparison science can be produced, because it is a science of terms upon terms. And, in any case, stop dragging me into discussions of metaphysics. What the Devil has got into you today? Instead, you who have good eyes take a parchment, a tablet, something you can make signs on, and a stylus…. Good, you have it? Good for you, Adso. Let’s go and take a turn around the Aedificium, while we still have a bit of light.”

So we took a long turn around the Aedificium. That is, from the distance we examined the east, south, and west towers, with the walls connecting them. The rest rose over the cliff, though for reasons of symmetry it could not be very different from what we were seeing.

And what we saw, William observed as he made me take precise notes on my tablet, was that each wall had two windows, and each tower five.

“Now, think,” my master said to me. “Each room we saw had a window….”

“Except those with seven sides,” I said.

“And, naturally, they are the ones in the center of each tower.”

“And except some others that we found without windows but that were not heptagonal.”

“Forget them. First let us find the rule, then we will try to explain the exceptions. So: we will have on the outside five rooms for each tower and two rooms for each straight wall, each room with a window. But if from a room with a window we proceed toward the interior of the Aedificium, we meet another room with a window. A sign that there are internal windows. Now, what shape is the internal well, as seen from the kitchen and from the scriptorium?”

“Octagonal,” I said.

“Excellent. And on each side of the octagon there could easily be two windows. Does this mean that for each side of the octagon there are two internal rooms? Am I right?”

“Yes, but what about the windowless rooms?”

“There are eight in all. In fact, the internal room of every tower, with seven sides, has five walls that open each into one of the five rooms of the tower. What do the other two walls confine with? Not with rooms set along the outside walls, or there would be windows, and not with rooms along the octagon, for the same reason and because they would then be excessively long rooms. Try to draw a plan of how the library might look from above. You see that in each tower there must be two rooms that confine with the heptagonal room and open into two rooms that confine with the internal octagonal well.”

I tried drawing the plan that my master suggested, and I let out a cry of triumph. “But now we know everything! Let me count…. The library has fifty-six rooms, four of them heptagonal and fifty-two more or less square, and of these, there are eight without windows, while twenty-eight look to the outside and sixteen to the interior!”

“And the four towers each have five rooms with four walls and one with seven…. The library is constructed according to a celestial harmony to which various and wonderful meanings can be attributed….”

“A splendid discovery,” I said, “but why is it so difficult to get our bearings?”

“Because what does not correspond to any mathematical law is the arrangement of the openings. Some rooms allow you to pass into several others, some into only one, and we must ask ourselves whether there are not rooms that do not allow you to go anywhere else. If you consider this aspect, plus the lack of light or of any clue that might be supplied by the position of the sun (and if you add the visions and the mirrors), you understand how the labyrinth can confuse anyone who goes through it, especially when he is already troubled by a sense of guilt. Remember, too, how desperate we were last night when we could no longer find our way. The maximum of confusion achieved with the maximum of order: it seems a sublime calculation. The builders of the library were great masters.”

“How will we orient ourselves, then?”

“At this point it isn’t difficult. With the map you’ve drawn, which should more or less correspond to the plan of the library, as soon as we are in the first heptagonal room we will move immediately to reach one of the blind rooms. Then, always turning right, after two or three rooms we should again be in a tower, which can only be the north tower, until we come to another blind room, on the left, which will confine with the heptagonal room, and on the right will allow us to rediscover a route similar to what I have just described, until we arrive at the west tower.”

“Yes, if all the rooms opened into all the other rooms…”

“In fact. And for this reason we’ll need your map, to mark the blank walls on it, so we’ll know what detours we’re making. But it won’t be difficult.”

“But are we sure it will work?” I asked, puzzled; it all seemed too simple to me.

“It will work,” William replied. “But unfortunately we don’t know everything yet. We have learned how to avoid being lost. Now we must know whether there is a rule governing the distribution of the books among the rooms. And the verses from the Apocalypse tell us very little, not least because many are repeated identically in different rooms….”

“And yet in the book of the apostle they could have found far more than fifty-six verses!”

“Undoubtedly. Therefore only certain verses are good. Strange. As if they had had fewer than fifty: thirty or twenty … Oh, by the beard of Merlin!”

“Of whom?”

“Pay no attention. A magician of my country … They used as many verses as there are letters in the alphabet! Of course, that’s it! The text of the verse doesn’t count, it’s the initial letters that count. Each room is marked by a letter of the alphabet, and all together they make up some text that we must discover!”

“Like a figured poem, in the form of a cross or a fish!”

“More or less, and probably in the period when the library was built, that kind of poem was much in vogue.”

“But where does the text begin?”

“With a scroll larger than the others, in the heptagonal room of the entrance tower … or else … Why, of course, with the sentences in red!”

“But there are so many of them!”

“And therefore there must be many texts, or many words. Now make a better and larger copy of your map; while we visit the library, you will mark down with your stylus the rooms we pass through, the positions of the doors and walls (as well as the windows), and also the first letters of the verses that appear there. And like a good illuminator, you will make the letters in red larger.”

“But how does it happen,” I said with admiration, “that you were able to solve the mystery of the library looking at it from the outside, and you were unable to solve it when you were inside?”

“Thus God knows the world, because He conceived it in His mind, as if from the outside, before it was created, and we do not know its rule, because we live inside it, having found it already made.”

“So one can know things by looking at them from the outside!”

“The creations of art, because we retrace in our minds the operations of the artificer. Not the creations of nature, because they are not the work of our minds.”

“But for the library this suffices, doesn’t it?”

“Yes,” William said. “But only for the library. Now let’s go and rest. I can do nothing until tomorrow morning, when I will have, I hope, my lenses. We might as well sleep, and rise early. I will try to reflect.”

“And supper?”

“Ah, of course, supper. The hour has passed by now. The monks are already at compline. But perhaps the kitchen is still open. Go look for something.”

“And steal it?”

“Ask. Ask Salvatore, who is now your friend.”

“But he will steal!”

“Are you perhaps your brother’s keeper?” William asked, with the words of Cain. But I saw he was joking and meant to say that God is great and merciful. And so I went looking for Salvatore and found him near the horses’ stalls.

“A fine animal,” I said, nodding at Brunellus, as a way of starting a conversation. “I would like to ride him.”

“No se puede. Abbonis est. But you do not need a pulcher horse to ride hard….” He pointed out a sturdy but ill-favored horse. “That one also sufficit…. Vide illuc, tertius equi….”

He wanted to point out to me the third horse. I laughed at his comical Latin. “And what will you do with that one?” I asked him.

And he told me a strange story. He said that any horse, even the oldest and weakest animal, could be made as swift as Brunellus. You had only to mix into his oats an herb called satirion, chopped fine, and then grease his thighs with stag fat. Then you mount the horse, and before spurring him you turn his face eastward and you whisper into his ear, three times, the words: “Nicander, Melchior, and Merchizard.” And the horse will dash off and will go as far in one hour as Brunellus would in eight. And if you hang around his neck the teeth of a wolf that the horse himself has trampled and killed, the animal will not even feel the effort.

I asked him whether he had ever tried this. He said to me, coming closer circumspectly and whispering into my ear with his really foul breath, that it was very difficult, because satirion was now cultivated only by bishops and by their lordly friends, who used it to increase their power. Then I put an end to his talk and told him that this evening my master wanted to read certain books in his cell and wished to eat up there.

“I will do,” he said, “I will do cheese in batter.”

“How is that made?”

“Facilis. You take the cheese before it is too antiquum, without too much salis, and cut in cubes or sicut you like. And postea you put a bit of butierro or lardo to rechauffer over the embers. And in it you put two pieces of cheese, and when it becomes tenero, zucharum et cinnamon supra positurum du bis. And immediately take to table, because it must be ate caldo caldo.”

“Cheese in batter it is, then,” I said to him. And he vanished toward the kitchen, telling me to wait for him. He arrived half an hour later with a dish covered by a cloth. The aroma was good.

“Here,” he said to me, and he also held out a great lamp filled with oil.

“What for?” I asked.

“Sais pas, moi,” he said, slyly. “Peut-être your magister wants to go in dark place esta noche.”

Salvatore apparently knew more things than I had suspected. I inquired no further, but took the food to William. We ate, and I withdrew to my cell. Or at least, so I implied. I wanted to find Ubertino again, and stealthily I returned to the church.