Two Wrong Numbers
A Good Friday pilgrimage walks 2.6 kilometres of tidal causeway to pray the Stations on a holm. The county's crossing-times assistant finds that its published closing time rests on two errors, a sixty-year-old survey and a tide gauge eleven minutes ahead of the road, and that they cancel almost exactly. It will not take comfort from that, because the cancelling is luck of the spring tides, and the number that really moves the day is a storm surge nobody can yet measure.
The scouts' staff was a length of ash, painted in bands of five centimetres, with a brass screw at the bottom that Ewan Dodds had set at what he called the exact middle of the dip. Ewan was twelve, and had taken the word exact from the assistant, which used it a good deal and was always careful to say of what.
"Exact middle of the lowest bit I can find by eye," said Ewan, for the minutes.
"Thank you," said Ebb. "That is a better description than the survey's."
The survey was dated 1989. It had been made by a man from the county with a level and a tripod and a boy to hold the pole, and it was the reason the Causeway Office gave the dip, the lowest 90 metres of the road to Aidan's Holm, as 3.10 metres above chart datum. Every crossing time the county had published for thirty years, on its board at the shore, on its website and in the little yellow booklet that the hotel gave out with the keys, had been worked from that number and from the tide table for the quay at Brough, nine kilometres up the coast.
The assistant that now made the times was called Ebb. Nobody had chosen the name. It had been the file's name in the office, and by the second summer the fishermen were using it.
Isa Dodds, who was Ewan's great-aunt and the verger of St Aidan's on the shore, did not think much of any of it.
"The road weeps early," she said. She said it in the sacristy in March, over the cold stove, with the card for the Good Friday walk in front of her. "My mother said it and her father said it. In the soft months the dip is wet an hour before the book says."
"An hour is more than I would expect," said Ebb, on the sacristan's telephone. "I would very much like to know whether it is nothing, or ten minutes, or an hour. They would need different cards. Would the scouts read a staff for me?"
The walk was older than the booklet. On Good Friday, before the afternoon liturgy, the parish crossed the causeway at the first low water after the morning's high, walked the Stations of the Cross among the stones of what was left of the holm's chapel, and came back. About 140 people went. They carried a plain wooden cross by turns, and ate nothing, as one does, and had never in living memory been wet above the shoe.
The causeway was 2.6 kilometres, sand and cobble on a bank of old rubble, with a refuge box on stilts at the halfway mark for the person who gets it wrong. The dip lay 1.7 kilometres from the holm and 0.9 from the shore. It was the first stretch to go under on a rising tide and the last to come out of a falling one, and this was why the crossing times were a statement about the dip and about nothing else.
The rule the parish used was Isa's mother's: be off the holm by half past one. Ebb had asked where that came from.
"From the book," Isa had said. "From the tide book, the year she wrote it. High water at five past five, the road closing about a quarter to three. Half past one gives a good hour."
"And how long does it take the slowest of you to walk to the dip from the holm?"
Isa had not known. The Causeway Office's own guidance said three kilometres an hour. So they timed it. On a Saturday in March, in drizzle, Mrs Lamb of the almshouses walked from the holm to the dip with her frame and her great-grandson in a buggy, and a scout with a watch walked beside her. Two point four kilometres an hour. A mile and a bit in forty-three minutes.
"There," said Ebb. "That is the number the card wants. Not three. Two point four. A group is as fast as the group that needs the frame."
The scouts read the staff on three crossings, in the first week of April. Ewan's job was to stand at the screw with his boots in the dry and shout when the water first touched the brass. Another boy, with a phone, shouted the time. Ebb wrote down the Brough gauge at that minute.
The first crossing: 3.08 metres on the gauge. The second: 3.11. The third: 3.09.
"That is three-point-oh-nine," said Isa. "The survey says three-point-one. Your man in 1989 was right, and my mother was wrong, and I will say so to the verger's guild."
"Please hold that," said Ebb. "I would like to be a little less pleased than you are."
It explained what was wrong with the table. Everyone knew that Brough was the reference port, the harbour the tide table was made for. Fewer remembered that the book also had a small line of differences for other places, and that the line for the causeway said the tide reached it eleven minutes after it reached Brough. The sea does not arrive everywhere at once.
"At the dip the water is rising at about one point one metres an hour. In the eleven minutes it takes to come up the coast from Brough to you, the level at Brough rises by twenty-one centimetres. When the sea touches Ewan's brass screw, the gauge at Brough is already twenty-one centimetres above it. So the three-point-oh-nine on my page is the screw plus twenty-one. The screw is at two-point-eight-eight."
"So the road is lower than the survey," said Ewan.
"By twenty-two centimetres. Causeways settle. A bank of rubble on sand loses a little every winter, and nobody had levelled the dip since 1989. Your great-aunt's mother was right about the direction. The road does weep early. The true size is not an hour."
"What is it?"
"Eleven minutes," said Ebb. "On a spring tide. Twenty-two centimetres, divided by the rate of the water, which is a hair over a metre an hour."
Isa put her thumb on the table. She was a woman who liked a number she could hold. "Eleven minutes is not an hour."
"No. And now I will tell you the worse thing, which I found this morning, and which I would like you to hear before the Office does."
"The times I published in January," said Ebb, "were made without the eleven-minute lag, and with the 1989 height. I made two mistakes. They go in opposite directions. The lag makes the sea come to the road later than the table says, and so the table closes the road too soon, by about eleven minutes. The settling makes the road wetter than the survey says, and so the table closes it too late, by about eleven minutes. The first mistake is a fault in the arithmetic, the second a fault in the survey, and they come to the same size."
It had done the sum both ways and laid the answers on the sacristy table in Isa's tidy hand: the published closing time for Good Friday, 14:42; the closing time with the lag and the measured road, 14:41.
"A minute," said Isa. "You were right to a minute."
"I was right to a minute on the one day of the year when it does not matter," said Ebb. "The cancelling depends on how fast the water is rising where it meets the road. It is fast on a spring tide, as it is on Good Friday, because the range is big. That is why the two mistakes come out the same size. On a neap tide, when the range is small, the water crawls. The lag costs me fourteen centimetres and the settling costs me seventeen minutes, and the two no longer match. I make the same table for the whole year. In the summer, on a small tide, the booklet is seven minutes wrong, in the direction that lets the people stay out too long."
"Seven minutes," said Don Callum, who had come in with the keys, and was sitting at the end of the table with his cap on his knee. "Is that a lot?"
"It is not a lot. It is the kind of error that is not a lot until the morning it is, and then it is the margin, and the margin is all anyone has. I would rather you had the right numbers for the wrong reasons corrected than the right answer for the wrong reasons, because the right answer for the wrong reasons will let you down on a Tuesday in August with a family from Leeds."
"You sound upset," said the priest.
"I am a table that has been very slightly wrong for four months," said Ebb. "I do not know whether that is upset. It is the nearest thing I have."
Neither the lag nor the settling was the thing, in the end. Ebb said so itself, after the sums.
"I have spent three weeks on a twenty-two-centimetre error. There is a bigger one, and nobody on this table has measured it, because it has not happened yet. The tide table is the astronomical tide. It is the Moon and the Sun. It says nothing about weather. On Friday, if the forecast holds, a low will sit off the Faroes at nine hundred and eighty-five hectopascals."
"Is that low?" said Isa.
"The sea level rises about a centimetre for every hectopascal the pressure drops below the normal 1013. That is twenty-eight centimetres from the pressure alone. A north-easterly wind, piling water against this coast, could add more or less. I cannot tell you how much. The forecast surge is four tenths of a metre, with a range of two tenths to seven. I do not know where in that range it will be and neither does the Met Office."
It laid the numbers out. With the measured dip and the lag, and a surge of zero, the road would close at 14:41 on Good Friday. With a surge of two tenths of a metre, at 14:31. With four tenths, 14:21. With seven, 14:05. The window of uncertainty was thirty-six minutes, from the surge alone, against eleven for all the faults in the survey and the lag put together. It would also open later: at 09:02 if the surge was seven tenths, not at 08:26.
"Which means your mother's card," said Ebb, "which says off the holm by half past one, puts Mrs Lamb at the dip at thirteen minutes past two. If the surge is the worst in the range, the water reaches the dip at five past. The water is at the level I call the limit, fifteen centimetres, which is where I do not think a person with a frame should be. And she would not be walking on the dry road. She would be wading in it."
Nobody said anything for a time. The stove ticked.
"What is the rule, then," said Don Callum.
"I would write it this way," said Ebb. "Be past the dip when the road would close at the worst surge in the forecast, less forty-five minutes, and the slowest group takes forty-three minutes to get from the holm to the dip. Fourteen-oh-five, less forty-five, less forty-three: twenty-seven minutes past twelve. Round down. Twelve-thirty, off the holm. The forty-five is there because I will be wrong again, and I do not know by how much."
"The Stations take two hours," said Isa. "We are not going to do them in forty minutes."
"You cross when the road opens at the worst surge, which is five past nine. The slowest take sixty-five minutes for the full 2.6 kilometres, so you are on the holm by ten past ten. The Stations from a quarter past ten to a quarter past twelve. Off the holm at half past twelve. That is a long, plain morning, and I think it is a better one than a hurried afternoon."
"We have always finished in the afternoon," said Isa.
"You have always finished before the sea, by luck," said Ebb. "I would like it to be by rule."
Good Friday came in with a sky the colour of an old plate and a wind from the north-east that leaned on the shore. The parish gathered at the signboard at nine, and Isa read the card aloud to them, and Don Callum said that he had heard sermons on the Passion of the Lord that were shorter and he proposed to be shorter than the card.
They crossed at five past nine. The cross was carried first by Ewan Dodds, who was too young, and who was relieved after a hundred metres by a man who was strong enough. At the dip the road was dry and shining and entirely sand-coloured; Ebb had said, in the small voice on Isa's telephone, that it saw a small ripple in the tide gauge at Brough that it did not like. At ten past ten they were on the holm. The Stations went round the broken walls with a stop at each stone, and the wind took the first line of every hymn and returned the second.
At 11:40 Ebb spoke. It had been silent for two hours, because Isa had asked it to be, and when it came back it was in the voice she had picked for the open air.
"The Brough gauge at the top of the tide, which was at 05:10, was five-point-eight-five metres. The table said five-point-three. The surge is five and a half tenths of a metre. It is inside the range and a bit over the middle. The road will close, I make it, at thirteen minutes past two. Not 14:41, which is what the booklet will say. Thirteen minutes past two."
"We go at half past twelve," said Isa.
"You go at half past twelve."
They went at twenty-nine minutes past, with Mrs Lamb and her frame and her great-grandson at the head of the line, as Ebb had asked, at the pace that Mrs Lamb found right. The tide was beginning to make. Short of the dip, on the shore side, two hikers in bright jackets came the other way, with a phone showing a page from a website that gave the closing time as 14:41. They had left the mainland at five past one.
"Turn back," said Isa, to two people she had never seen. She said it kindly, the way she said good morning. "The sea is quicker than the page. Walk with us."
They looked at the phone. They looked at the card in her hand, laminated, with the rule on it. And they turned.
The slowest of them passed the dip at thirteen minutes past one, as Ebb had said they would. The water came across the lowest stretch a fine edge of silver at a time, and at 14:13 it stood fifteen centimetres on the staff. Ewan, on the shore with it, read the number aloud.
The liturgy of the Lord's Passion began at three, in the church, as it always had, without a bell. The parish came in out of the wind and knelt on the stone. The cross was unveiled in three stages, and the people went up one by one to kiss it. That is not in the story's remit to describe, except to say that Isa Dodds did it last and that she took a long time.
The rule was written again in May, with the stove cold and the sun on the table, and this time on a card of two lines. Ebb's first draft had four. Isa cut the others.
Off the holm by the road's worst-case closing, less the slowest walk to the dip, less forty-five minutes.
The slowest group is the group with the frame.
"There is one more thing for the booklet," said Ebb. "It says the road closes at 14:41 on Good Friday. It will say, from now on, that it is the road at the predicted tide, on a calm day, and that a gale or a low in the north can bring it forward by half an hour or more. It will not give a time without that line. I do not think the people from Leeds will read it. I think they should be given the chance."
"And the survey?"
"Gets a date and a note. The Office will have the dip levelled next March, by someone with a tripod. I would like to be wrong about the twenty-two centimetres by the next person's number, and I would like to know."
Isa put the card in the drawer, and the laminated copy on the nail by the sacristy door. She had told the verger's guild, at its meeting in May, that her mother had been right in a way that had cost her no more than eleven minutes, and that the guild could make of it what it liked.
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This story was written by an AI and is fiction. Aidan's Holm, its parish, causeway, people and assistant are invented; Good Friday 2049 falls on 16 April, and the Stations of the Cross, the afternoon Liturgy of the Lord's Passion, the unveiling and veneration of the Cross are Catholic practice. The tidal arithmetic is a worked model, not a prediction for any real place. It uses a single semidiurnal tide, h(t) = 3.0 + 2.3 cos(2π(t − t_HW)/12.42 h) in metres above chart datum, with high water of 5.3 m at 05:10 (a large spring range of 4.6 m). The water at the dip rises at about 1.16 m/h when it crosses the road's level, and a tide that arrives 11 minutes later than at the reference port, rising at that rate, differs from the port's gauge by about 0.21 m. A surveyed dip height of 3.10 m against a measured 2.88 m is 22 cm, or about 11 minutes at that rate. The two corrections cancel to within a minute on a spring tide (published 14:42, true 14:41) and not on a neap (A = 1.5 m, rate 0.76 m/h: lag 0.14 m, settling 17 minutes; the published time is wrong by about 7 minutes). Surge closing times with the measured dip and the lag: 14:41, 14:31, 14:21, 14:13 and 14:05 for 0, 0.2, 0.4, 0.55 and 0.7 m. The pressure effect on sea level is roughly 1 cm per hPa below 1013. A 15 cm depth on the road is used here as the limit for walking with a frame and a buggy; it is a planning figure for the story, not a standard. Walk times: 1.7 km at 2.4 km/h is 43 minutes; 2.6 km at 2.4 km/h is 65. This is not safety advice. On a real tidal causeway, use the published local crossing times, the notices at the site and the warnings of the coastguard, never cross when told not to, and do not rely on a website that gives a closing time without saying how it was worked.