An Elegant Solution(88)

And so I came to the door of the Boot and Thorn, and stroked Charon for passage, and looked in at the smoke and dice and flagons and fire and thought how this intersection of minute man and immense man was all in God’s image.

I paused and knew that Daniel and Nicolaus were waiting for me. And I was met by someone else.

“Leonhard.”

It was Gottlieb. I thought a moment that I was meant to bring my paper and pen to another questioning, but for this time it was I who was to be questioned.

“Have you heard?” he asked. “The proof?”

“Yes. Daniel found me earlier.”

He nodded to the windows. “He’s in there?”

“I think so, and Nicolaus.”

“What do they want of you?”

“They just want to show me. It’s generous of them.”

“None of that,” Gottlieb said. “There’s no generosity there, not likely in this whole pile of a building. All right, then, go in and we’ll see what they really are after.”

Together we went in. Daniel was indeed there, at a table near the fire where the light was best, and Nicolaus with him.

“What! Cousin?” he said as he saw us. “You’re here again? Is there another Inquiry?” Daniel teased as he always did, but he also seemed somehow welcoming.

“Yes, there is,” Gottlieb answered. “Into that.” And of course that was the set of papers on the table between them.

“Then let’s get to it,” Daniel said. “Sit down, Cousin, sit down, Leonhard, sit and tell me what you see here.”

“We’re not all here.” That was Nicolaus.

“What? Who?” his brother asked, and Nicolaus crooked a finger to beckon to the door. Pale even in the dark and red, we were joined by the one other: Little Johann. “Come, come!” Daniel said. “Welcome and plant yourself; you’re right, Nicolaus, we need the full measure. We’re not all unless we have our best.”

“I thought you’d be here,” the newcomer said.

“I’m glad you’re here, too,” I said. “Not your father, also?”

“Full measure,” Daniel answered. “Not running over.”

“Come on, look at the pages,” Gottlieb said. “That’s what we’re here for.”

And so we were.



I quickly saw that it was indeed my proof. The words were mine, the equations just as I’d written them. I’d known it would be. I hardly knew what to say. My first concern was to not betray Master Johann’s confidence. He’d chosen to keep the proof anonymous.

“Pi squared, and divided by six?” Gottlieb said to start. “And the reciprocal squares? Are they even nearly the same value?”

“They are,” Nicolaus answered. “I’ve calculated them both to the fifth decimal.” I was amazed. That was an entire day’s work!

“That proves nothing,” Gottlieb replied. “It could be in the sixth place they diverge. Or the tenth.”

“It doesn’t prove,” Daniel said, “but it does convince. Now look at this.” He was on the third page of the proof. “This is at least obscure. What’s the reason for the sine? It comes out of pure air.”

“It’s to make the polynomial,” Nicolaus said.

“But the polynomial could stand on its own. Anyone could write it.”

“Would they, though?” Gottlieb said. “An infinite polynomial? That’s what’s of pure air.”

“No, I’ll take the polynomial,” Daniel said. “But the factoring of it. Now that’s the worst of all.”

“The sine’s for knowing the factors. That’s what it’s for.”

“But how could it be infinite? An infinite angle?”

I could hardly breathe, listening to them.

“Draw me that triangle, then. It’s absurd.”

“But it’s not meant to be a triangle.”

“A sine without a triangle?”

Oh, it was torture to hear them argue, with each other and with the papers!

“But see what it means. The hypotenuse becomes the radius of a circle.”

“Then the polynomial derives from a point on the circle as the radius rotates.”

“And the roots are periodic. I see . . .”

“But the infinite factoring?”

“An infinite polynomial for an infinite series. It’s clever, that I’ll say. Very clever. Elegant.”

“What do you say, Leonhard?” It was Nicolaus who asked.

Despite that I’d known I’d be asked, I was still lost for an answer. “How do you think your father came on to this?”

“I’ll say MacLaurin,” Gottlieb said. “See how the Taylor Series is used? He’d be first to try an infinite series for a sum.”