Stride and Echo

Stride looked at the clock. The minute hand was on the forty-two. The hour hand was on the seven. The second hand was not moving.

He pulled out his pocket-watch and held it up next to the clock face. The watch's second hand swept smoothly. The clock's second hand did not.

"It is stopped," Stride confirmed. "When did it stop?"

Echo thought for a moment. "I do not know. I do not look at the clock. But it must have stopped recently. The angle is still natural."

"What do you mean, the angle is still natural?"

"The minute hand has not drifted. If the clock had been stopped for a long time, the hands would have settled to a slightly different position. The mechanism is friction-tilted. These hands are still in their last running position. The clock stopped between yesterday seven-twelve and now seven-twelve."

Stride raised both eyebrows. He had not known that about old village clocks. He had also not known that Echo had known that about old village clocks.

Echo, who had spent his whole childhood watching things that did not change, had noticed many small features of things that did not change very often. He noticed clocks. He noticed where they pointed. He noticed when they didn't.

"My father will be very upset," Echo said. "He wound it on Sunday. It should not have stopped this week."

He did not sound upset himself. He sounded thoughtful.

Stride looked at the stopped clock. Then he looked at his pocket-watch. Then he looked back at the stopped clock.

"Echo," he said. "This is a chapter."

The two of them stood under the bakery's awning for a moment.

"A chapter of what?" Echo asked.

"Of the lessons-layer," Stride said. "Look at the clock. The clock face says seven forty-two. It will say seven forty-two in five minutes. It will say seven forty-two in five hours. It will say seven forty-two tomorrow. Until someone winds it, the clock face is a constant function."

Echo's expression did not change. But he was paying attention.

Stride pulled out a small notebook from his coat pocket. He drew a horizontal line. He labeled the horizontal axis real time. He labeled the vertical axis what the clock says. He drew a flat horizontal line at seven-forty-two.

"The clock-face value," he said, "is now y equals seven-forty-two. Regardless of x — regardless of how much real time passes — the output of the clock is seven-forty-two. That is a constant function. You are standing under a constant function."

Echo considered this. He looked up at the clock again. Seven forty-two.

"It is a sad constant function," Echo said. "Normally the clock is not a constant function. Normally the clock is a linear function. For every minute of real time, the minute hand advances one tick. The output increases steadily."

"That's the whole point," Stride said. He flipped the page in his notebook. He drew a sloped line. He labeled it clock running. Underneath it, he drew the flat line again. He labeled that one clock stopped.

"This," he said, pointing at the sloped line, "is the clock when it works. It is linear. Slope: one tick per minute. It is keeping pace with real time. Plot the time it tells against real time, and you get a straight line. Forty-five degrees of slope. y equals x. That is a linear function."

"And the other one," Echo said, "is what the clock does now."

"Slope: zero. The output never changes. y equals seven-forty-two. That is a constant function. A constant function is a special case of a linear function. It is a linear function whose slope is zero — whose rate of change is zero — so its graph is a horizontal line. Same equation form, y equals mx plus b. Just with m equal to zero."

"So a constant function is flat," Echo said. "And a linear function is steady. The difference is the slope."

"The difference is the rate of change," Stride said. "Constant: no change per unit of x. Linear: a fixed amount of change per unit of x. Same family. Different m."

Marn the baker came out the front door, wiping his hands on his apron. He looked up at the clock. He stopped wiping his hands.

"Stopped," he said.

"Stopped," Echo confirmed. "Recently. Hands haven't drifted."

Marn squinted. Then he saw Stride and the notebook.

"You're teaching off my broken clock," he said.

"We are," Stride admitted. "It's a very good clock for teaching. We were just explaining how it has changed from a linear function into a constant function."

"It has what?"

Echo translated. "It used to tick. Now it doesn't. Every minute used to add one tick. Now it doesn't. So it used to change by a known amount per minute, and now it changes by nothing per minute. That is the slope of the function the clock is performing. The slope used to be one. Now it is zero."

Marn looked from his son to Stride to the clock and back. He was a baker. He did not, on most mornings, get this much functional-analysis attention before breakfast.

"It needs winding," he said. "Don't make a story out of it."

"Too late," Stride said cheerfully.

Echo helped his father carry a small step-stool out to under the clock. Marn climbed up, opened the casing, and wound it. The second hand began sweeping. The minute hand jumped one tick. The clock began being a linear function again.

Marn climbed down. He looked at Stride. "Now what."

Stride showed him the notebook page with the two lines. "Sloped: clock running. Flat: clock stopped. Same family of equations. Different m. Your clock has switched back into the sloped one and is now running again."

Marn studied it for a long moment. Then he said, slowly: "So if it stops once a year, that's a problem. If it stops every Tuesday morning, that's a pattern."

"That's a higher-order function," Echo said, almost smiling. "We are not ready for that one."

"Don't be cute," Marn said. But he was almost smiling too.