The Angle in a Semicircle

Mrs Pruitt was wearing a red cardigan and a green one, the red one over the green one, so that the green showed at the cuffs and the collar, and when she came in she stood at the front of the room and looked down at herself and said, “Tuesday,” and then, “I know. It’s Christmas. It’s not my fault, it was dark,” and the class laughed, and she put her bag on the desk and opened her laptop and said, “Proofs. Pass them forward.”

They passed them forward. Frederick’s was two sides, in pencil, with the diagram drawn three times because he had got the first two wrong, and he had done it on Friday night after the cake and the folder and George, at the kitchen table, with Bridget’s radio playing violins, and he had understood every line of it then and not understood why it mattered. He passed it forward. Theo passed his forward on top. Theo’s was one side and had a mole drawn at the bottom in a waistcoat.

“Right,” said Mrs Pruitt, squaring the pile. “I’ll mark these tonight and I’ll be horrible about it. Now. Let’s do it properly, together, on the board, because I looked at your faces on Thursday and I know that at least half of you copied the last three lines off somebody who didn’t understand them either.”

She drew the circle.

She drew it freehand, with the board pen, in one go, the way she always did, and it was nearly a perfect circle, and the class made the small noise it always made when she did that. Behind it, very faint, on the whiteboard, the ghost of the quadratic equation from some other year floated where it had always floated, x squared plus something, in a permanent marker nobody had ever been able to get off, and her circle went round it and through it and enclosed it, so that for the rest of the period the old equation sat inside the new circle like something in a jar.

“Diameter,” said Mrs Pruitt, and drew it, straight across, through the centre. “A to B. And a point on the curve. Anywhere. Somebody give me anywhere.”

“There,” said Felix, pointing at nothing.

“There,” said Mrs Pruitt, and put a dot on the curve near the top, a little to the left, and labelled it C. “Join it up.” She drew the two lines, from A to C and from C to B, so that there was a triangle sitting inside the half-circle with its long side along the diameter and its top corner on the curve. “And the claim is?”

“The angle at C is a right angle,” said Priya.

“Always?”

“Always.”

“Wherever I put C?”

“Wherever.”

“Prove it,” said Mrs Pruitt, and held out the pen, to nobody, and to everybody.

---

The clock on the wall above the door said a quarter to two.

It was not the clock in the wire cage, which was two minutes fast. It was a classroom clock, round and white with black hands, and it was right, because the school’s clocks were all set from the office, all except the one in the wire cage, which had been put up before the system and never connected to it. Frederick had checked this clock against Castellanos’s on the dashboard, once, in his first year, out of curiosity. It was right to the second.

He looked at it and then made himself not look at it.

Theo, beside him, was not looking at it either. Theo was looking at the board with an expression of tremendous concentration, the expression he wore in geometry when he had decided to understand something by force, and Frederick knew, without looking at him, that Theo had not looked at the clock once since they came in and was not going to, and that this was deliberate, and that it was for him.

“Who’ll start?” said Mrs Pruitt.

Devon started. Devon got up and went to the board and took the pen and drew a line from the centre of the circle to C, which was the right first move, and then stood there looking at it.

“Why did you do that?” said Mrs Pruitt.

“Because it’s what you do,” said Devon.

“Why is it what you do?”

Devon looked at the line. “Because then there’s two triangles.”

“Yes! Good. Two triangles. And what’s special about them?”

Devon did not know what was special about them. He stood there with the pen. Somebody at the back said, “They’re isosceles,” and Mrs Pruitt said, “Why are they isosceles?” and it went on, the way it always went on in Mrs Pruitt’s room, around the class, from one person to another, everybody putting in one piece, the pen going from hand to hand, Devon to Priya to a girl called Ana who never spoke and spoke now, quietly, and was right, and Mrs Pruitt saying yes and why and yes and go on, so that the proof built itself up on the board out of nine or ten people’s pieces, one line at a time, the way a wall goes up.

Frederick watched it go up.

He did not put his hand up. He knew the proof. He had known it since Friday. He watched the lines go up on the whiteboard in different people’s handwriting, Devon’s big and Priya’s neat and Ana’s very small, inside the circle, around the ghost of the old equation, and he thought that if he put his hand up now he would have to stand up and go to the front and take the pen, and he would be standing at the board with the pen in his hand when the clock went round, and he did not want that. He wanted to be sitting down. He wanted to be in his place by the window, in the second row, next to Theo, because of the alphabet. He wanted it to happen to him sitting down, the way it was happening to everybody else in the room, who were all also sitting down, and who did not know that they were sitting in the room where it was going to happen, or did know, and were doing geometry anyway.

The clock said six minutes to two.

“So the angles at the bottom of this one are equal,” Priya was saying, at the board, “because it’s isosceles, because both of those sides are radii. Call them x. And the angles in this one are equal too. Call those y.”

“Good,” said Mrs Pruitt. “And?”

“And all three angles of the big triangle add up to a hundred and eighty.”

“Which are?”

Priya wrote them. x plus x plus y plus y.

The clock said two minutes to two.

“So two x plus two y is a hundred and eighty,” said Priya, and wrote it. “So x plus y is…”

She stopped, with the pen on the board.

“Ninety,” she said. And then, slowly, as if she had only just seen it, as if it had come up out of the board at her: “Oh. And x plus y is the angle at C. So the angle at C is ninety. Wherever C is.” She turned round. “It doesn’t matter where C is.”

“It doesn’t matter where C is,” said Mrs Pruitt. “That’s it. That’s the whole thing. You can put the point anywhere you like on the curve and it makes no difference at all. Isn’t that the most beautiful fact in elementary geometry?”

“No,” said Theo.

The class laughed.

The clock above the door said two o’clock.

---

Nothing happened.

He had known that nothing would happen. He had been told, by his mother, by Dame M, by Rosalind Aird, by the order paper and the radio, by the whole careful arithmetic of four hours and six in London and the day the last of them acts. He had known for a day exactly what would happen at two o’clock, which was nothing that anybody could see. And still, when the black minute hand on the white clock above the door moved onto the twelve, with a small tick he could hear because the room was quiet, he found that some part of him had been waiting for something. A sound. A light. A change in the air, the way the air changes before a storm. Something.

There was nothing. The clock went on. Priya was putting the cap back on the pen. Theo was writing x + y = 90 in his book under the mole. Through the window, in the schoolyard, the man in the orange jacket had stopped blowing leaves and was sitting on the low wall by the bike racks, eating an apple.

He was King.

He did not feel it. He sat in the second row by the window, with his hands on the desk, and tried to feel it, and could not find anywhere in himself where it was. He was exactly the same as he had been at one minute to two. He was a boy in a geometry class who knew the proof and had not put his hand up. He was fifteen. His feet were cold, because the radiator under this window didn’t work either. He thought that if he had been in the room on the sixth floor with his father, at this moment, his father would have been exactly the same too, in the chair by the window, under the blanket, and that neither of them would have been able to find it, and that this was the most important thing he had learned all autumn and he did not know what it meant.

On the desk at the front, Mrs Pruitt’s laptop made a noise.

It was the noise it made when the office sent something, a soft two-note chime, ding-dong, like a doorbell in a small house. Everybody in the room knew the noise. It meant a fire drill, or a change to the bus timetable, or a reminder about the Fall Formal. It meant nothing. It meant the office.

Mrs Pruitt glanced at the screen.

She read it. Frederick watched her read it. He watched her read it the way he had watched his mother read his statement at the kitchen table on Wednesday, once fast and once slowly, and her face did not do anything for the first time and did something for the second, which was to go a bit pink, from the collar of the green cardigan upwards, slowly, like a glass filling.

She closed the laptop. She did not close it fast. She closed it the way you close a door on a sleeping child.

Then she looked up, across the room, over all the heads, at the second row, by the window.

“Frederick,” said Mrs Pruitt, “could I have a word outside?”

It was what she said when you hadn’t done the homework. Everybody in the class knew it. She said it in exactly that voice, light and dry and a little bored, could I have a word outside, and three people at the back had heard her say it to them in September, and they knew it, and they knew that he had done the homework, because it was on the top of the pile on her desk, two sides in pencil with the diagram drawn three times.

Nobody laughed.

He stood up.

He walked to the front, between the desks, and he heard it begin behind him before he had got to the door. It was not loud. It was the sound a room makes when it has been holding something for a whole period and somebody has put a pin in it: a breath, and a chair, and a whisper at the back, what’s going on, and another, it’s two o’clock, and somebody saying shh, and somebody else saying you shh. He did not turn round. Mrs Pruitt held the door for him, and he went through it, and she came after him.

---

The hall was empty and very bright. It was the long corridor of the math wing, with the lockers down one side and the windows down the other, and the sun had come round onto the windows and was lying across the floor in long yellow bars, and there was nobody in it at all. He could hear, from the room next door, somebody else’s class, a man’s voice explaining something about fractions. From the end of the corridor, faintly, the sound of the music rooms, somebody practising scales on a piano, badly, the same five notes over and over.

Mrs Pruitt shut the door behind them.

It did not shut the room out. It was an old door with a glass panel at the top, wired, and a gap underneath it where the floor had sunk, and through it, the moment it closed, he heard the room go up like a pan coming to the boil: twenty-six voices at once, low and then less low, a chair going over, somebody laughing too high and stopping, and Devon, unmistakably Devon, saying, I told you, it’s the homework, she does that when you haven’t done the homework, and somebody, Priya, he thought, saying, He’s done the homework, you idiot, it’s on the top of the pile, and a third voice, that he did not know, saying, quite clearly, it’s two o’clock, and then a great many people saying shh.

Mrs Pruitt looked at the door, and then at him, and did not go back in to stop them.

She stood in the hall in her two cardigans with her back to the classroom door and looked at him. She was not very tall. She was about his height. He had not noticed that before, because she was always moving.

“The school’s just had an email,” said Mrs Pruitt. “It’s two o’clock. I’m supposed to tell you.”

He nodded.

He waited for her to say the rest of it. There was a rest of it. There was the whole of it: your father, and the Crown, and all fourteen, and King, and Your Majesty, and George, and whatever it was that people said. He waited for her to say any of it.

She didn’t. He saw her not say it. He saw her open her mouth to say something else, and look at him, and close it again. He saw her decide, in the bright empty hall, with the sun on the floor and somebody’s scales going wrong at the end of the corridor, that she did not know what else to say, and that she was not going to pretend to. He saw that she was a geometry teacher from a school in Baltimore who had been at Ellicott since September and had been sent an email at two o’clock by the principal, and that was the whole of what she was, and that she was not going to make herself into anything else in front of him by saying something she did not mean.

He did not know what else to say either.

They stood in the hall.

Behind the door the room had settled into a kind of fierce whispering, the sound of a great many people arguing as quietly as they could. He caught pieces of it through the gap. Is it his dad. It’s not his dad, it’s the thing, the six o’clock thing. What six o’clock thing. Are you serious. Do we stand up. What? When he comes back in. Do we have to stand up. And then a voice he did know, Theo’s, low and very fierce, saying something he could not hear at all, and after it, for a moment, nothing.

“Do you want to come back in?” said Mrs Pruitt.

“Yes,” said Frederick.

He said it at once. He did not have to think about it. He said yes because they were in the middle of the proof, and because Priya had just got it, and because the angle at C was ninety wherever you put it, and because the radiator under his window didn’t work, and because Theo was in there with a mole in a waistcoat, and because it was sixth period on a Tuesday and there were twenty-five minutes left, and because he wanted, more than he had ever wanted anything, to go back into the room and sit down.

Mrs Pruitt nodded, once. She did not say good, or are you sure, or of course. She turned round and opened the door and held it, and stood aside, the way Dame M had stood aside for him in the passage on Wednesday, and he went in first.

---

The room stopped.

It stopped the moment the door opened, all at once, the way a radio stops when you pull the plug out of the wall: a roar of whispering and then nothing, so sudden that the nothing rang. Everybody was sitting where they had been sitting, more or less. Somebody had picked the chair up. Nobody was talking. Nobody was looking at the door. Twenty-six people were looking at their books, or at the board, or at the window, or at their hands, with tremendous concentration, the way Theo had looked at the board, the way Castellanos looked at the ceiling, and he understood, walking back across the front of the room to the second row, that every one of them was doing it on purpose, and that it was for him, and that no teacher had told them to.

He sat down.

Theo looked up.

It was a quick look, sideways, over the mole and x + y = 90, a question with no words in it, the look you give somebody coming back from the office, what did they want.

Frederick nodded.

Theo looked at him for one second more. Then he went back to his book.

That was all. That was the whole of it. Theo went back to his book, and picked up his pencil, and wrote something under the mole, and Frederick looked at the board, at the circle with the old equation inside it like something in a jar, and the point C up on the curve near the top, a little to the left, where Felix had put it, at random, because you could put it anywhere.

Mrs Pruitt went back to the front.

She did not sit down. She stood beside the board with the pen in her hand and looked at the proof for a while, Priya’s line and Ana’s line and Devon’s, as though she were checking it.

And then there was a click from the loudspeaker in the corner of the ceiling, the grey box above the door that nobody ever looked at, which was used for fire drills and for snow, and a breath, and a voice.

“Good afternoon, everybody. This is Dr Brooks.”

The whole room went still, a different still.

“At two o’clock this afternoon the abdication of King Ernest took effect in all of his realms. One of our own students is King. He has gone back to his class. I would like every one of you to stay in yours. Thank you.”

Click.

That was all. Four sentences. Five, with thank you. In the dry, plain, unhurried voice that said the buses will be twenty minutes late and the building will close at noon because of the weather and would the owner of a blue bicycle please come to the front office. She had not said his name. She had not needed to.

Nobody moved.

Nobody turned round. Nobody looked at the second row. In the next room the man explaining fractions had stopped. At the end of the corridor the piano had stopped. The whole of the math wing, the whole building, eleven hundred people, he thought, sat in their rooms for a moment in a silence that had a shape to it, and stayed in them, as they had been asked.

Then, from somewhere a long way off, from the music rooms, very faintly, the piano started again. The same five notes. Wrong.

Mrs Pruitt looked at the board.

“Right,” she said. “Where were we.”

“C,” said Priya.

“C,” said Mrs Pruitt. “The point on the curve.” She put the tip of the pen on the dot near the top, a little to the left. “Which can go anywhere. Watch.” And she rubbed it out with the side of her hand, the dot and the two lines going down to it, and drew another one, lower down, on the right, and joined it up, and it was a different triangle, long and thin, and the angle at the new C was still a right angle, and she drew the little square in the corner to show it. And then she rubbed that one out too and drew another, right at the top, so that the triangle was the same on both sides, and that one was a right angle too.

“Anywhere,” said Mrs Pruitt. “It makes no difference. That’s the point. That’s the whole point.”

She drew it six more times before the bell, all round the curve, and every time it was ninety, and nobody in the room said a word except to answer her, and Frederick watched the point go round, and wrote it down, all six, in his book, in pencil, very neatly, and when the bell went for seventh period he closed the book and put it in his bag and stood up, and Theo stood up beside him, and they went out together into the bright hall, into the crush.

“Theo.”

“What?”

“What were they saying? When I was outside.”

Theo shouldered his bag. He looked straight ahead, down the hall, at the clock in the wire cage at the far end, two minutes fast.

“Devon said you hadn’t done the homework,” said Theo. “Priya said you had, it was on the pile. Felix said maybe there was a helicopter on the roof. Ana said there’s no helicopter, she’d have heard it. Somebody at the back said it was your dad and everybody told him to shut up. Then Luis at the back said, it’s two o’clock, it’s on the radio, it’s the six o’clock thing, it’s happened, and everybody went quiet, and then everybody started again, worse.” He stepped round a boy with a plastic axe through his head. “And then Marcus asked if they had to stand up when you came back in.”

“What did you say?”

“I didn’t say anything,” said Theo. “Priya said no. She said nobody stands up, he’d hate it, he’d want to die. And Marcus said but it’s the law, isn’t it, you have to stand up for the King, and Priya said it’s not the law, it’s manners, and he’d hate it, so it would be bad manners, so you don’t.” Theo looked at the clock. “And then Devon said what if he wants us to, and everybody looked at me.”

“And?”

“And I said if anybody stood up when you came in I’d sit on them,” said Theo. “For the rest of the year. In geometry. Every day.” He shrugged. “So nobody did.”

They walked down the hall towards the gym. Somebody banged a locker. Somebody shouted walk.

“Thank you,” said Frederick.

“Don’t,” said Theo. “Priya did most of it. I only did the sitting.” He thought about it. “Marcus would have stood up. Marcus would have played something. He’d got his hand in his bag.”

Frederick laughed, in the hall, in the crush, as though it were any Tuesday, because it was.