Filomena having told her story, the queen commended her, and bade Gabriele speak. He said it was his purpose to tell us of how what may seem to be chance is no such thing, and is often within our ken. Asking for our permission to tell a story of a time of pestilence, such as ours was, he forthright began thus, in the dry and simple manner that befitted him, as he was not from Florence, but from another place and another time.

The first one was the one who lived just some way to the left. The signs were obvious only in retrospect: a tell-tale weariness around the eyes, an uncertain gait, hoarseness. O. was not sure he was truly remembering those signs, given how much others talked about them later. He was fairly certain he had not seen him past the initial stage.

The next ones were his neighbor, who shared a small cottage with his dog, and his friend in number 5, who visited him often. (They never seemed to talk much; they mainly sat to drink tea and play chess on the courtyard, with a desultory air about them.) Nobody thought it important, or rather O. did not; he overheard their situation being mentioned when he went to get bread.

The season had been even drier than usual; there was barely a spot of green on the slopes, save for the olive trees, and dust seemed to be everywhere. But then, in that place, dust was even more of a feeling than a thing, almost a moral condition. Otherwise it would not have seemed less present in places that were opposed to it, as ideas: the canal, a town clock, the street in front of a well-kept place. It was also less of an issue if one just did not care, or had long decided to refuse to be bothered by it.

Then it was the third house, and the place next to 5, predictably, but also number ten. At that point it seemed to be all that people could be heard talking about. Some things transmit by vicinity and by affinity, thought O., but then of course they would; people who do the same things tend to meet, even if one of those things is taking walks alone. At least that is how O. pictured 10 and the man with the dog now, walking up by a crag with no apparent purpose. Or perhaps it was only 10 that he had seen that way, once; still, he felt as if there was something else that the two men both did or had. They were middle-aged and thin, with wrinkled faces.

It was around then that he who fell ill first died. People came from out of town to take him away — relatives perhaps, and someone who looked official.

Then it was 4, and the old woman in 15, and 7; or perhaps 11 and 12 were first. It was all predictable. People should have tried to leave before then. The bridge down the road was now closed, and gendarmes had come to set up a barrier on the uphill route. Of course people should or could still have left in some other way, had they wanted to. Number 12 was friends with 10, and was said to have gone to cook for him and help him. Three and 15 were related in some fashion — 15 was an old aunt, though others said she had really been 3’s mother’s housekeeper. Perhaps 3 had paid a call before taking his cough seriously, or whatever it was that went wrong with him first.

O. did not himself have relatives in town, or anywhere in the province. He also had no friends, or anybody he particularly cared for in one way or the other. He had come years ago, and had imposed on himself a certain discipline. He would read and take care of his own needs. Above all, he had wanted to be left to his own thoughts, and to have fewer of them, letting the less relevant and more futile ones die out. He kept a diary and tended to a small garden. He had no visitors. If he had anything in common with anybody else, he did not know it; even if there were such a thing as transmission by affinity, rather than contact, the only person he would get something from, he thought, was himself.

The curfew was imposed after the beginning of the next wave. He first heard of the case of 16. The curfew was strict, and obeyed for the most part, though nobody from elsewhere stayed to enforce it. After 16 came 13. Maybe that was just the order in which others said they knew about it. (O. went to town to buy chickpeas and more bread; that, he was told at the shop, was still allowed.) Also 17, of whom O. knew nothing, save that, like 11, he had a pet canary, whose cage he would hang outside when it was sunny. Or sunny but not all too sunny. Perhaps 17 had been ill before 16, or around the same time.

Then nothing seemed to happen. There must have been few people left to ask. O. stayed indoors and rarely looked outside the window. He recorded what he knew in his diary; then he would read, or sit at the desk and do nothing at all. Time seemed to plod on with deaf dullness; the walls were ochre and dull, the bare tree outside was dull, silence and his mind were also becoming dull.

The next time he went to town, a good number of days later, everything looked bare and uninhabited. There were signs of abandonment; wild plants seemed to be just starting to grow roadside and in cracks. A faint hint of a familiar aromatic herb wafted about the old sundial.

Walking across the empty town square, O. saw 9, and talked to her from a distance. All was well with her, said 9, but everybody else was gone. Perhaps 14 was still there; all others were gone. 18 had fallen very ill more than a week ago; he had got it from 15, most likely.

At that point, O. knew that his turn had come; time was up for 9, time was up for himself, and nobody would be left, not that it mattered. Of course it had been fairly quick, it had to be fairly quick, and of course he could have foreseen it.

Such was the story told by Gabriele. While Filomena found its conceit clear and simple, others were not of the same mind. When the queen asked her to explain it, she began to speak on this wise, saying it was only seemly that some facts that ought to be known to all first be set out, so that all could exercise their wit in seeing by their own devices what was already clear to some.

Modular arithmetic for neophytes.

Consider the hours on a clock. (It may help to label noon/midnight as 0 rather 12, but in the end it makes no difference.) Three hours after 2 equals five o’clock: 3+2 = 5. However, we do not say that three hours after 10 is thirteen o’clock, but one o’clock. What we have here is a system where 3+10 = 1. Substraction works in the same way: three hours before 4 is one o’clock, but three hours before 1 is ten o’clock: 4-3=1, 1-3=10. We can even multiply in this context: setting the clock forward by five hours four times is the same as setting it forward by 8 hours (54 = 8), and that is in turn the same as setting the clock backwards by 4 hours (54 = 8 = -4).

More formally and more generally, we can define arithmetic modulo m, where m stands for a positive integer of our choice. (In the case of a clock, m = 12, or m = 24 if you have a 24-hour clock.) The idea is that two integers are put in the same equivalence class — that is, are declared to be “the same” — if their difference is divisible by m. (“Divisible” means “divisible without leaving a residue”: 36 is divisible by 12, and so are 0 or -12, but 35 and 37 are not divisible by 12, and neither are -35 nor -37.) We go back to our examples: the difference between 1-3 (= -2, in the usual arithmetic) and 10 is -12, which is divisible by 12; the difference between 5*4 and 8 is 12, which is also divisible by 12.

The usual laws of addition and multiplication (e.g., commutativity, i.e., “the order of the summands (or factors) does not affect the result”, etc.) still hold. A number does not change if multiplied by 1, or if 0 is added to it. Zero times any number is still zero.

G. A. Gotthelf
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