历之本在于测验

Twenty-ninth of December, night. 2032 closed out. Three thousand and eighty bags against three thousand and twenty. Sixty more. The first year in this trade that the volume has barely moved, and it moved that way on purpose. Last December's plan opened with 2032: three thousand bags, and in small characters beside it: no increase this year. Net: minus eight thousand four hundred, against minus thirty-one thousand four hundred. The four-year line: −70,525 / −58,092 / −31,400 / −8,400. Volume flat, and the loss down by three-quarters.* Everything I did this year was done for reasons that had nothing to do with selling more.

Then the other sheet: five rows, one per household, three boxes each — which of the five bags their reference matched at closing; how many grades off the shared bags; and the weight difference between the cupboard copy and the doorframe copy. Chang: three, level, 1.2 g. Turpan: three, level, 1.9 g. Rehmanjan: two, minus one, 1.5 g. Imin: three, level, 0.8 g. Kucha, Ali's family: blank. Tursun died in July; the six houses were hung by three men this year, each picking his own, and no single string came out of it. Ali's message: we don't have that string this year. Next year I'll pick it.

Guo Shoujing came on the thirtieth. Very old — past eighty, back bent, eyes bright. A dark old robe with the cuffs rolled up. Bronze filings on his hands and under the nails, the kind that does not wash out. He carried a very thin leaf of bronze, two inches by four, with a hole in the middle no bigger than a needle's point, and set it on the windowsill facing the daylight before he said anything. Then: the fifth row of that table of yours is empty.

His own beginning. When I was young the calendar the court used was inaccurate. By how much? It could not predict the eclipses. It would name a day and be a day or two out. So? I said eight characters. The root of a calendar lies in observation; and among the tools of observation nothing comes before the gnomon. Meaning? A calendar is not calculated. It is measured. And: and to measure, you first need the thing you measure with.

Was there nothing before? There was. You set up a gnomon and watch its shadow at noon. It had been used for a thousand years. What's wrong with it? The gnomon is eight feet. An eight-foot gnomon throws a short shadow, and on a short shadow an error of one division is an error of one division. And: I put the gnomon up to forty feet. Five times. Yes. The shadow is five times longer. And, and I picked up the pen: the same one-division error, on my gnomon, counts for a fifth of what it did. And: I did not make anybody measure more accurately. I made the same inaccuracy matter less.

I said it sounded too cheap. He laughed. It is cheap. But it brings a new problem with it. Which? The taller the gnomon, the softer the edge of the shadow. And: at eight feet the edge is crisp. At forty feet it blurs into a smear, and then you have no idea where to measure to. And: five times the height, and the edge went soft by more than ten times. And: my one-fifth was entirely paid back.

Then he picked the bronze off the sill and held it against the window, and the pinhole let through a point of light. This is the shadow-definer. Placed at the shadow end, so that the sunlight passes through the hole — and on the ground you get a tiny sun, and the crossbar of the gnomon sits inside that little sun as a black line. And: that line is sharp. Sharp enough to read to a hundredth. One leaf of bronze with a needle hole in it. Just this one leaf of bronze. And, six times into the book: make the instrument bigger and you will certainly make a new fault. And: that new fault is yours to cure as well. And: if you can't cure it, making it bigger was for nothing.

Is the instrument enough? No. And: one station cannot find out where it is wrong. Why? You measure here for a year and get a set of figures. Are they right or wrong — what will you compare them against? Last year's. Last year's were measured by you too. I had to stop writing. I have heard this three times this year already: without the line, the dot is nothing (March); the tower built upright at least tells you how far it has since gone over (September); a converted record contains only the fact that I converted it (October). And: so we set up twenty-seven places. How far? East to Koryŏ. West to the Dian lake. South past the Vermilion Cliffs. North to the Tiele. And: twenty-seven places, the same day, measuring at the same time. For how long? Four years. And what came out: the year is three hundred and sixty-five days and two thousand four hundred and twenty-five parts. And, five times into the book: that figure was not measured by any one of the twenty-seven. And: it was measured by the twenty-seven together. And: one short is workable. Ten short is not. — M.

Memo to file. Chengdu, 29 December 2032 – 1 January 2033. Year-end, and the count I had never taken before.

1. The numbers. 3,080 bags against 3,020. Net −8,400 against −31,400. Four-year sequence: −70,525 / −58,092 / −31,400 / −8,400. Volume deliberately flat; the loss fell by three-quarters anyway. Qiao's consulting: ¥24,000. Days on the ground: 21, none of them spent looking at goods.

2. The five-row table, and the blank in it. Four households closed with a reference matched to the shared bags and both copies weighed. Kucha–Tursun's household: blank — three men hung six houses separately after he died and no single reference came out of it. The exchange lost a station in its first full year, and it lost it to mortality, not to disagreement. Ali: next year I'll pick it.

3. The gnomon, and why it is the correct model for what I did this year. The old instrument was eight feet; he made it forty. He did not improve anybody's eyesight — he made the same reading error count for a fifth of what it had. That is exactly what the paired half-strings did: I did not make anyone judge better; I gave the judgement something unchanging to sit against.

4. And the part I would have got wrong on my own. Five times the height blurred the shadow edge by more than ten times, and the gain was entirely cancelled. The cure was a bronze leaf with a pinhole in it — projecting a small image of the sun with the crossbar visible inside it as a sharp line, readable to a hundredth. Every enlargement of an instrument creates a second-order fault, and the enlargement is worthless until that fault is also solved. My own version this year: adding four more households multiplied the pairwise comparisons, and the offsets stopped being transitive until the shared bags were imposed (April). Same shape. I did not recognize it at the time.*

5. The argument for twenty-seven stations, and it is the argument this whole year has been circling. One station cannot find out where it is wrong, because the only thing it has to check against is its own previous work. This is the fourth time in twelve months: the mark needs a line (March); build it upright so the drift is legible (September); don't convert, keep both colours (October); twenty-seven stations, four years, one figure (December). And: 365.2425 was not measured by any one of the twenty-seven. It was measured by the twenty-seven together. One short is workable. Ten short is not.

6. Luo's December question, asked again. Last 31 December he asked how many of the five households still had an old man who could bite. The answer was four. This year: Chang 66, yes. Rehmanjan 48, yes. Imin 67, yes. Abdul Rahman 70 — right side dead, left hand can still pinch. Tursun, gone. I wrote three and a half and struck out the half. Pinching counts. Four. — The same number, and not the same four people.

7. The count I took for the first time, and it is the reason this year is not a losing year. Not how many are left. How many have just started. Chang Bin, 33 — two years holding the string; this year he learned the difference between the basket and the racks. Alim, 21 — three years of readings; this year he planted three saplings in three positions. Ali, 15 — this year he was given a red pen, and next year he picks his family's reference. Three. In the book: last year that number was zero — not because they didn't exist, but because I never counted them.*

8. The 2033 plan, one page. First line, again not money: five households become six — and the sixth is not one I recruited. Imin and Ali are holding the same year separately, an arrangement neither of them asked me to set up. Second line unchanged: twenty days, no goods, handovers only. Third line is new: 10 March, Turpan. Do not ask him how hard next year's should be — that box stays blank. Instead: give him this year's doorframe half and cupboard half and have him pinch both. Don't ask which is better. Ask whether they are the same string. I know the answer — same string, divided the same day. I am not testing the two halves. I am testing the hand. By December's rule from last year, a question whose answer I already know is a trap and I should not set it. But this one is not an examination of him. It is what he has been asking for: he asks once a month whether that cupboard is still there.

9. 1 January, before dawn, three messages. — Chang Bin: he weighed all three this morning and wrote it in the notebook himself. In his own hand. First time in thirty-one years. — Alim: he looked at the card on the cupboard door a long time last night, and drew a vertical stroke — the same one as before. This time he did not draw the circle. — Ali, in Chinese, one fewer error than in November: Grandpa Imin's cell and ours were both sealed yesterday. He says next year at this time we open them together. He says in forty years this is the first time anyone has sealed one alongside him.

10. And the entry I closed the year with. A blank card on top of the pile, front: one station cannot find out where it is wrong. Back, my own: so the part of these four years I did alone has never been checked by anything. And the last line in the notebook: in 2032 I printed a book in red and black, and not one of the things that actually happened this year is in it. A man of sixty-six picked up a pen for the first time. A man who cannot speak repeated a sign, and this time did not disown it. Two households that have never worked together sealed the same year on the same day. — None of the three has a number. None of the three was arranged by me.

"The root of a calendar lies in observation; and among the tools of observation, nothing comes before the gnomon."
Eight characters, from a man whose court was using a calendar that could not predict an eclipse to within a day or two, and whose answer was not a better computation but a better instrument.
The first move looks almost like cheating. The traditional gnomon was eight feet, and on the short noon shadow it cast, a misreading of one division was simply an error of one division. He built one forty feet high. Five times the shadow, and therefore the same misreading now cost a fifth of what it had.
He did not train anyone to read more carefully. He rearranged the situation so that ordinary carelessness mattered less — which is nearly always the cheaper repair, and nearly always the one nobody makes.
Then the honest part, which is why the account is worth reading rather than quoting. Raising the gnomon five times blurred the shadow's edge by more than ten, and the entire gain was cancelled. An enlarged instrument had produced a fault the small one never had, and until that fault was solved the enlargement was worse than useless.
The solution was a leaf of bronze, two inches by four, pierced with a hole no bigger than a needle's point, propped at a slant. Sunlight through the pinhole threw a small image of the sun onto the scale, with the crossbar of the gnomon crossing it as a hard black line — readable to a hundredth of an inch.
And even that was not sufficient, for a reason that has nothing to do with instruments. A single observatory measuring for a decade can only ever check its results against its own earlier results.
So: twenty-seven stations, from Koryŏ in the east to the Dian lake in the west, the same days, four years, producing a year-length of 365.2425 days — the figure Europe would adopt three centuries later.
That number was not measured by any one of the twenty-seven. It was measured by the twenty-seven together. One short is workable. Ten short is not.