An Elegant Solution(89)

“Never,” Daniel said. “First, it’s nothing like his series. And second, the trigonometry. He’d have to have that idea from someone else. And more than those, he’d never send it to Basel.”

“He might for malice. I’ve heard he’s vindictive as any of us.”

“That’s not possible. And he’d have to know it would be stolen, too. Whoever’d show this to the Brute without publishing it in their own name first is a knave.”

“It might be already published in England.”

“Then it wouldn’t have been mailed here.” This was Nicolaus. “No, it’s someone who wants it validated before it’s published.”

Daniel said, “I claim it’s Newton.”

“No!” Even Little Johann joined in the denials. And that young man added, “Not after fifty years of him trying. Nicolaus is right. It’s someone new at it who wants to know that it’s true.”

“An unknown? A novice? The Brute wouldn’t waste opening his letter.”

“He might,” Gottlieb said. “Or it’s someone he knows.”

If before had been torture, this was torment beyond it.

“Who would he know?” Nicolaus asked. And it had perhaps been inevitable.

“Leonhard,” Little Johann said.

They all four rounded on me like hounds on a deer. “What, is it?” Daniel asked, and right away he answered, “Yes, it is! I see it in your face. Of course it is!”

“No,” Nicolaus disagreed. “He’s clever, we all know, but—”

“He’s genius,” Daniel said.

“But this is past genius.” Nicolaus cocked his head. “It’s greatness. It’s pure elegance. Isn’t it? Who’s the greatest Mathematician in Europe? Newton?”

“The Brute, I’d say,” Daniel said. “And this isn’t his.”

“I’ll still say it’s MacLaurin,” Gottlieb said. “He’s young. He’s a novel thinker.”

“I think it’s Leonhard,” Little Johann said. “Ask him and let him answer.”

“All right then,” Daniel said. “Here it is, Leonhard. Is it yours?” He pointed to the papers. “Answer us. If you don’t, Nicolaus will ask your grandmother and she’ll tell us.”

There was no escape. Why wouldn’t I want to claim it? But I was overwhelmingly reluctant. It wasn’t for fear of betraying Master Johann. It seemed instead that I was at a gate, an Ash Gate, that could only be entered once; it was ten times the weight of being given a tricorne, or a hundred times!

“Yes,” I said. “It’s mine.”

“I knew it was!” Daniel crowed it like a rooster.

“I’m not convinced,” Nicolaus said.

“He’d lie?” Gottlieb asked. “He had it from someone else?”

“Let him explain it,” Little Johann said.

“All right,” Nicolaus said. He pushed the papers toward me. “Show us this proof.”

I pushed the papers back to him.

“I’ll explain it,” I said. “Where’s paper and ink? Blank paper. And more light.”



We brought candles and paper, and swept the table of crumbs, and I readied myself.

“Here’s the start, with the meaning of sine. It’s as you said, to make a circle. It’s not a mere ratio as it’s used in triangles. It’s a true function. I understood it more when I wanted equations for waves.”

And so I passed the gate. We went for hours, I think. I took them through the infinite polynomial made by a radius that circles endlessly, and what the roots of it would be.

“Though what is an infinite polynomial?” Nicolaus asked. “How do you write it?”

“Think of the wave on water,” I said. “But every rise and fall is a root.”

And then, how the polynomial would appear on a plain of Descartes, and then how its infinite factors were derived from its infinite roots. And then, how the pairs of roots could be combined. And then, what the coefficients must be when all the pairs were multiplied together.

“But there are infinite other terms! And each term is an infinite sum.”

“But each term, on its own, must have a particular value,” I said. I showed them the expansion of the sine function, which Mr. Taylor in England had proposed. “And this sum must equal six, which is the factorial of three. And if the equation is divided by the cube of x, and multiplied by Pi, then the proof is complete.”

And they, as their father also, considered the proof was far from complete. They disputed and fought every step, with me, with each other. It was as Saturday with Master Johann had been, but in four directions and fiercer questioning, and Little Johann as sharp as any of them.