Floating

The summer before I went to Stanbury, I couldn’t float.

I could swim. I’d been swimming since I was six, lengths and lengths of it, and I had a badge with a dolphin on it to prove so. But ask me to lie still on my back and do nothing, and I sank: slowly, feet first, like a ship in a film, until the water closed over my face and I came up spluttering. Ada could lie on the sea like a lilo. So could Dad. So, it turned out, could most people on the beach below Granny’s house in Cornwall, including a toddler in armbands, which didn’t count, and an old man reading a newspaper, which did.

It was the last week of August. Ada had her notebook on her towel, in a freezer bag, with a biro clipped to the cover, and every so often she’d stop reading whatever she was reading and write something in it, and every time I came near she put her hand flat on the page. I had given up asking.

At lunch there was one flapjack left in the tin, and we both said “mine” in the same instant, which in our family starts a procedure rather than a fight.

“Cut and choose,” said Mum, without looking up from her book.

It is the oldest rule in our house. One of you cuts; the other picks first. Ada took the plastic knife and cut that flapjack the way she does everything, with the tip of her tongue between her teeth, measuring against the edge of the biro, because she knew that if one half came out a crumb bigger, the crumb would be mine. I took the left one. They were the same. They always are, when Ada cuts.

“Why do I never get to cut?” I said, with my mouth full.

“You don’t want to cut. Cutting’s the worse job.” Ada licked her fingers. “Cut perfectly and you get exactly half. Cut badly and you get less. The one who picks can never get less than half. So you want to pick, and so do I.”

“So who cuts?”

“That’s what the coin’s for,” said Mum, and held up the 10p that lives in the pocket of the beach bag for exactly this. When you can’t agree who gets the worse job, you toss for it, and nobody has ever won an argument with a coin, because a coin isn’t on anybody’s side.

Then Mum, who has read about nearly everything, told us that cut-and-choose is in a Greek poem nearly three thousand years oldHesiod’s Theogony, written down in Greece around 700 BC. Prometheus divides an ox between the gods and people, hides the bones under a layer of shining fat and the good meat inside the ox’s stomach, and Zeus chooses the fat. Hesiod says Zeus saw through the trick and chose the fat anyway, so as to have a reason to be angry, which is the sort of thing gods do in Hesiod., and that the one doing the cutting cheated: he hid the bones under a handsome layer of fat and the good meat inside the stomach, and the chooser took the fat. “Which is why it only works if you can see what you’re choosing,” said Mum. “Hide something, and it isn’t a cut any more. It’s a lie.” Ada said that for two people the rule is perfect, and mathematicians have proved it, and that for three it gets so hard that a fair way for any number of people wasn’t found until 1995For two people, cut-and-choose gives each a share she thinks is at least half, so neither can envy the other’s. The Polish mathematician Hugo Steinhaus set the problem for three people in 1944, while hiding from the German occupation; a method that works for any number of people, with no envy, was published by Steven Brams and Alan Taylor in 1995., and Mum said that was quite enough maths for a beach, and I went back in the water.

“Watch,” I said, from the water, about the ninth time. “I’m doing exactly what you do.”

Ada looked up. Then she put the notebook down, which she never did, and came in.

“You’re not,” she said. “You’re breathing.”

“You have to breathe.”

“Not like that. Watch my chest.” She lay back, took a breath so big that her ribs stood out, held it, and lay there on top of the sea, perfectly still, with the swell lifting her and putting her down again. Then she let the breath go, slowly, all of it, until her cheeks went hollow, and as it went, she sank. Not fast. Her chest went first, then the rest of her, a few centimetres, then more, until only her face was out, and then not even that. She came up blowing.

“Again,” I said.

She did it again. Breath in: she rose. Breath out: she sank. Like a lift.

“How?”

“You’ve got two balloons,” said Ada, wiping her eyes. “Lungs.” She stood up, waist-deep, with the waves bumping into her. “Right. Everything in water gets pushed up by it. And the push is exactly the weight of the water you’ve shoved out of the way, to be where you are.”

“Who says?”

“Archimedes. A Greek mathematician, two thousand two hundred years ago, and he was right.” She said it the way she says all of those things, as if it were a personal insult. Ada is permanently amazed that people worked something out that long ago, with no electricity, no computers and no proper paper, and wrote it down, and proved it, and that you could still walk up and down this beach asking and not find three people who knew. “Two thousand two hundred years,” she said. “We’ve had two thousand two hundred years to tell everybody, and most of them still think you float because you’re light. There’s a story about him jumping out of his bath shouting Eureka! and running down the street with no clothes on, but somebody wrote that two hundred years after he’d died, so it’s probably made up. The bit about the push isn’t. People have tested it every way you can think of, and ships are built on it.” She pressed her hand down flat into the water and let it come back up. “So it’s a contest. Your weight pulls you down. The water you’ve pushed aside pushes you up. Weigh more than that water, and you sink. Less, and you float.”

“I weigh more than water?”

“About the same, which is the annoying part. You’re mostly water, plus bone and muscle, which are heavier than water, and a bit of fat, which is lighter. Most people come out almost exactly the same as water, a little bit either side.” She tapped her ribs. “Except for these. Take a really big breath and your chest gets bigger by two or three litres. Call it three. Three more litres of you under the sea means three more litres of sea shoved out of the way, and three litres of sea water weighs about three kilos. So that’s about three kilos more push.”

“And the air weighs...”

“Nearly nothing. A litre of air is about a gram.”?A litre of air weighs about 1.2 grams, so three litres is about 3.6 grams. A litre of sea water weighs about 1.025 kilograms, so three more litres of you under water means about 3 × 1.025 ≈ 3.1 kilograms more push up. Three kilograms more push, for less than four grams more weight. She grinned. “So a big breath makes you three litres bigger and about four grams heavier. Same you, more room. That’s what tips it. Breathe in, and you take up more room than your weight’s worth of water: you float. Breathe right out, and you’re back to being a tiny bit heavier than the water you’re in: you sink. You keep breathing out to relax. You’ve been letting the air out of your own lilo.”

I tried it. Big breath, hold it, lie back. And I floated, properly, for about four seconds, the sea holding me up like a hand under my back, and then my legs sank, and took the rest of me down with them.

“See?” I said, when I’d finished coughing. “I’m broken.”

“You’re not broken. You’re a see-saw.” Ada looked at me, and I could see her deciding whether to explain or go back to her notebook. “The air’s all at one end of you. The heavy bits are at the other. That’s the next lesson.”

“Tell me now.”

“No. It’s better in a pool. And I want to see your face when you get it.” She waded back up the beach towards her towel, and said over her shoulder, “Also, the sea’s cheating. It’s got salt dissolved in it, and that makes every litre of it about two and a half per cent heavier than the fresh water in a pool, so it pushes you up harder. It won’t be this easy at school.” She turned round at the top of the beach, walking backwards. “The Dead Sea’s so salty that a litre of it weighs a kilo and a quarter. You can’t sink in it if you try: people sit in it like an armchair and read the newspaper. And if you lay down on a pool of mercury, which is thirteen and a half times as heavy as water, hardly any of you would go under at all: about a fourteenth. You’d lie on it like a lilo on a lilo.”?A body floats with just enough of itself under the surface to push aside its own weight of liquid. You weigh about the same as water, so in mercury, 13.6 times heavier, you’d need only 1 ÷ 13.6 ≈ 0.07 of your volume under: about a fourteenth.

“Can I go?”

“To the Dead Sea? It’s in the Middle East, about four thousand kilometres away. Ask Dad.” She sat down by her notebook. “You’ll still sink faster at school.”

She was right about that. I found it out on my first morning at Stanbury, in lane one, at six o’clock. The next lesson took until my fourth week, and Ada ducking under two lane ropes she wasn’t allowed to duck under.