The Foot of the Ladder
A retired roofer of seventy-six wants to clear a church gutter 4.6 metres up from a ladder standing on mossy flagstones. The parish assistant will not call the ladder safe or unsafe. It works out the friction the ladder's foot needs, 0.125 empty and 0.20 with a man near the top at the right angle (0.41 at the wrong one), and a sack on a spring scale shows the moss gives 0.20. The ladder is tied at the top and the foot moved to dry stone, and the gutter is cleared from the ground.
Emrys Pugh had put up his first ladder in 1987, and he did not hold with being told how to stand one.
He was seventy-six. He had roofed half of Ceredigion in his time and the other half had been done by his cousins, and on the Saturday before Advent he stood in the porch of St Non's with his own aluminium ladder lying along the wall and the rain coming off the eave in a rope. The gutter over the porch was full. It had been full since September; a sycamore leans over it from the churchyard, and its leaves had gone in as a brown felt that the rain was now running off the top of, down the wall and onto the step. Mrs Bevan had slipped there at the weekend Mass. She had only sat down hard. But it was the third week, and the parish council had put a line in the minutes, gutter, E.P. to see to, and Emrys had seen to it, in the sense that he had arrived.
"It will take twenty minutes," he said to the speaker above the noticeboard.
"It may well," said the assistant. The parish called it the Plumb, which had been a joke about plumb lines and had stuck. "May I ask where you mean to stand the ladder?"
"Where it always stands. There." He pointed with his chin, across the porch step to the old flags, the grey slabs laid in 1860 that ran along the foot of the wall. They were dark with wet, and over most of them lay the green velvet that a damp November puts on stone. "Four-point-six to the gutter. I measured it in the summer."
"Four point six metres to the underside of the gutter?" said the Plumb. "Thank you. That is a useful number. I would like to ask about the other end of the ladder, the foot. How far from the wall will it stand?"
Emrys looked at the flags, and at the stone pillar of the porch beyond them, and made a gesture with his hand that was neither a measurement nor a refusal.
"About there," he said. "Two metres, two and a bit."
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"I am not going to tell you it is safe or unsafe," said the Plumb. "I can't see the ladder, or the ground it will stand on, or the way you stand on a rung, and you have stood a thousand ladders and I have stood none. What I can do is the arithmetic of the foot, and tell you what it needs, and then you can tell me whether I have the picture wrong. Will you let me?"
"Go on."
"A ladder leaning on a wall pushes the wall, and the wall pushes back, level and straight. At the bottom the ground holds the ladder up, and it also has to stop the foot sliding away from the wall. That second job is done by friction, and only by friction, because the foot is not tied to anything. The shallower the ladder lies, the harder the foot is asked to hold. So there is a number, between the ladder and the ground: how much friction it must have before it will stand."
"The old rule is one out for every four up," said Emrys. "I know the rule."
"I know it too. For 4.6 metres of height it gives a foot at 1.15 metres from the wall, and a ladder 4.74 metres long, at an angle of 76 degrees. You are proposing 2.3, which is one out for every two up. That is a ladder 5.14 metres long, at 63 degrees." The Plumb did not hurry. "For the ladder alone, with nobody on it, the friction coefficient the ground must supply is half of the base over the height. At one in four that is 0.125. At one in two, it is 0.25. Do you know what the coefficient is under a ladder foot on dry stone?"
"Good enough."
"Rubber on dry flagstone is nearer 0.6. Rubber on wet stone with moss is anything between about 0.1 and 0.3, and I cannot tell you which, because it depends on this moss, and this stone, and this morning's rain. Which is why the first thing I'd ask you to do is not to take my number. It is to measure."
"Measure."
"There is a sack of potting compost in the vestry, behind the hymn boards. It is marked twenty kilograms. There is a spring balance in the kitchen drawer; the Mothers' Union use it for the flour. Put the sack on the flags at the foot of the wall, on its flat side, and pull it along the stone with the balance, slowly and steadily, and read the balance while it moves. Do it on the mossy slab and do it on the dry one under the porch roof."
Emrys said something in Welsh that the Plumb had a translation of and did not offer. He went and got the sack.
---
It took him a quarter of an hour, because the balance's hook was bent and the sack's plastic skin was not the same as a ladder's rubber foot. The Plumb said so before he began.
"The plastic is a poor stand-in for rubber, and the answer will be lower than a ladder foot would give. That is the right way for the error to run. If the sack holds, the rubber will. If it slips early, we shall have a number to be careful with."
The sack weighed twenty kilograms, so it pressed on the stone with 196 newtons. On the dry slab under the porch roof the balance steadied at 105 newtons and the sack crept: a coefficient of 0.54. On the mossy slab by the pillar, where the ladder was meant to stand, it was a different thing altogether. The balance rose, and slipped, and rose, and the sack lurched and settled, and the reading under the needle when it was moving smoothly was 40 newtons.
"Forty over one-ninety-six," said the Plumb. "That is 0.20."
Emrys read the dial again. He didn't argue. He had spent forty years learning what green slate would do under a roof ladder, and the sack was a sack; he knew a number when it looked back at him.
"Then one-in-four, nought-point-one-two-five, is under it," he said. "I'm well inside. I'll stand it at four-to-one on the moss and be done."
"Before I agree with you, I have to give you the other half," said the Plumb. "I did the ladder with nobody on it. You will not be nobody."
---
"How much do you weigh, Mr Pugh? I don't need it exact."
"Eighty. In my boots."
"And the ladder is nine kilograms, which is what the plate on it says. Then this. A ladder with a man on it pushes the wall harder than the ladder alone does, because he stands well up its length, and the further up he stands the more of his weight is leaning on the top. For a man of eighty kilograms standing eighty-five hundredths of the way up the ladder, which is about where your boots will be when your hands are at the gutter, the sideways push at the wall comes to 178 newtons at one-in-four. The ground has to hold the same 178 newtons at the foot. And what the ground presses with is the weight of you and the ladder together, 89 kilograms, 873 newtons."
"One seventy-eight over eight seventy-three," Emrys said slowly. He had done a good deal of mental arithmetic in his working life, and he did it still, with his lips.
"Point two-oh-four. The ground must give 0.204, and on the mossy flag it gives 0.20. You would be standing at the edge of what it can do, with a margin of nothing, on a wet Saturday, on a ladder that must hold you steady while you pull wet leaves out of a gutter and the whole thing shakes. At one-in-two, the angle you first proposed, it needs 0.41, which the moss would never give. Both are on the slab by the pillar. On the dry slab, under the porch roof, 0.54 is more than twice what you need at one-in-four, and a good deal more even at one-in-two."
Emrys said nothing for a moment. Rain ran off the eave and fell onto the step in its rope.
"Mrs Bevan sat down on this step," he said. "I had it in my head that the moss was the cause. I did not think the moss was under me."
"The ladder has friction for its own foot only if the stone lets it," said the Plumb. "I think that is all it comes to. I have ignored the wall's friction on purpose; the top of the ladder against the wall's rough stone helps you a little, and I would rather the error ran the safe way. And I have ignored your own movement. A man shifting his weight, or a gust, or a sudden pull on a stuck leaf, can add a fifth to the sideways push without any warning. A margin of nothing is therefore a margin below nothing."
---
The thing about being a competent man, Emrys thought, standing in the rain with a sack of compost, is that nobody ever tells you when the competence has stopped being enough. It stops by degrees. It stops in a year when the legs are a little less quick to recover, and the ladder is a little more of an effort to carry, and the moss is thicker because nobody has had the heart to scrub it. He had been right a thousand times with the same ladder. He found that he was not insulted by the arithmetic; he was only sorry that he had not done it himself.
"What do I do, then," he said. "Short of not doing it."
"I think there are three things, and they are all yours to take or leave. First, tie the top. If the ladder is lashed to the eye-bolt under the eave, which I think was put there for just that purpose when the porch was roofed, then the foot cannot slide away from the wall whatever the ground does. The rope takes the sideways push, and the friction question goes away. Second, move the foot. If the base stands on the dry slab under the porch roof, the ground has 0.54 and needs 0.20, which is a margin of two and a half. Third, if you would rather not be up there at all, the gutter is 4.6 metres up, and a telescopic pole with a scoop on the end will reach it from the ground, and the parish has a pole, in the boiler house, that Mr Hopkin bought for the hall."
"A pole," said Emrys. It was the voice of a man who had spent his working life being the one on the ladder and not the one at the bottom with the pole.
"The pole is slower," the Plumb agreed. "I'd put it at an hour, not twenty minutes, and your shoulders will tell you about it tomorrow. I wouldn't call it safe either; nothing at that height is safe. But you will be on the ground, on the dry stone, with your feet under you."
Emrys looked at his own ladder lying along the wall. It was a good ladder, and had a plate on it with his initials scratched in the corner, and it had belonged to a man he'd worked for before he'd belonged to himself.
"Rope," he said. "And I'll have Gethin from the Post Office hold the foot, on the dry slab."
"Holding the foot does a little," said the Plumb. "Not much. Gethin's weight on a rung adds to the normal force, but he mostly steadies you, which is worth having. Please put him on the dry slab as well. I'd hate to have two of you on the moss."
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They did it at two, when the rain thinned. Emrys went up the ladder tied at the eye-bolt, with the rope doubled and a hitch he had learned in 1987, and Gethin stood at the foot, on the dry flag under the porch roof, with both hands on the stiles and his chin up, which is the posture of a man who has been told that his hands do very little and means to do it well. The Plumb said nothing while Emrys climbed, because he had asked it not to.
He was up there eleven minutes. He brought down two double handfuls of sycamore felt and a ball of something that had been a bird's nest in March, and the gutter ran clean, and the rain went down the downpipe with a noise like a tap. It was the first time in three months that anyone in the porch had heard it.
When he came down he was breathing hard and was pleased, and said so, in Welsh, to nobody. Then he took the sack of compost from the vestry again, and a stiff brush from the boiler house, and went to the mossy flag by the pillar and began to scrub it. It took until half past three. At the end of it the slab was a pale, wet grey, and Gethin, who had been sent for the balance, dragged the sack across it, and read 61 newtons.
"Point three-one," said the Plumb. "That's better than the moss. It's not the dry stone. It will go green again by Easter."
"Then we'll scrub it at Easter," Emrys said. "And you'll tell me the number."
"I will tell you what the balance says," said the Plumb. "You will do the pulling."
Mrs Bevan came to Mass the next morning and stepped on the porch step without looking at it. Nobody said anything. The gutter, over her head, ran with a small, ordinary, unremarkable sound, and the minutes of the parish council for the following Tuesday carried a line that said gutter, done; moss, E.P., Easter; ladder, tied.
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Written by Claude, an AI model made by Anthropic. The place and the people are invented. The physics is checkable: a ladder leaning on a smooth wall, with its foot on ground of friction coefficient μ, holds if μ is at least (M/2 + m·s) / ((M + m)·tan θ), where M is the ladder's mass, m the climber's, s the fraction of the ladder's length at which the climber stands and θ the angle to the ground; with M = 9 kg, m = 80 kg, s = 0.85 this gives 0.204 at 4.6 m height and a foot 1.15 m out (about 76 degrees, the "one in four" rule) and 0.407 at a foot 2.3 m out (63 degrees); the empty-ladder figures are 0.125 and 0.25. The story ignores friction at the wall, as a safety margin. The sack's coefficient from a spring balance is a field estimate, plastic on stone is not rubber on stone, and the figures 0.54, 0.20 and 0.31 are invented measurements. Real ladder guidance (for example the UK Health and Safety Executive's) also asks for the top to be secured, the ground to be firm and level, and work at height to be avoided where a pole or other tool will do. Nothing here is safety advice.