Master Johann was the center, never moving, but with everything orbiting him. Caiaphas was a meteor from some far end of the cosmos appearing in flame.
What was a Chair worth? What would anyone give for it? Or do for it?
And then I understood. I knew. I was Saul with my face toward Damascus. I didn’t deduce it myself, but instead I was told. It was given to me. I just knew. I took my paper and ink and began writing. It was so elegant.
11
THE RECIPROCAL SQUARES
I awoke Saturday morning, first doubting, then sure, then doubting. But no, I was sure. The whole of the story was laid out in my mind unchanged from the night before.
I worked my chores, but said nothing to my grandmother. It would need to be said first to Master Johann. At that thought, I trembled again. He would confront me, attack every point, doubt, accuse. I knew it all. I loved my Master and all his family, and I knew it would be a strike against them all. But it was all so plain and simple and elegant.
No other assignments or errands were waiting. I set out to read. I cleared my mind of the whole chain of logic, for I knew it would drive me mad to concentrate on it more. I read MacLaurin, then Taylor, but when two o’clock tolled I put them down. It was impossible to keep my mind closed to what was stored in it. I opened that treasure chest and there it all was, like the mountains outside a window that were there whether the window was open or not. Every line of the story sprang back into its place, more sure than ever.
So I just sat at my desk and waited and waited for the longest hour to pass, and finally I dressed and presented myself to Grandmother and then walked the short blocks to my Master’s door.
I arrived at three thirty, as usual. And all the forms were as usual: the solemn door opening and the silent stair climbing and the grave single knocking and the summoning. Even the candle on the table attended to its proper place. But beyond the visible, behind it, beneath it, was a difference.
“Good afternoon, sir,” I said.
“Yes, good afternoon.” He examined me very closely. “And what have you studied this week? What exercises have you done? Have you had any time for studies?”
I answered with always perfect respect. But my heart raced as it never had before. I’d rehearsed the words in every way, experimenting between candor, circumspection, and innocence. “Yes, Master Johann,” I said. “I have had time.” And now I chose to be forthright. It was no longer possible to hold back and ignore what I’d discovered. I knew his reaction would be swift and merciless, and the risk to myself was great. Yet my knowledge was so sure and undeniable, I spoke the fateful words anyway:
“Master Johann,” I said. “Sir.”
“Yes, Leonhard?”
“I have solved the Reciprocal Squares problem.”
“You have . . . what?”
He gaped, open mouthed at me. It was the first I’d ever seen him confounded! Then he frowned, and frowned deeper, and I saw the storm gather, and strike. “I will see a proof, then.” He was outraged. Alexander had besieged Tyre over a milder insult; Tamerlane threw down Isfahan for less an affront. Augustus couldn’t have been more stern, and Nebuchadnezzar couldn’t have been more menacing. But I saw it, too, in his eyes, that the chance of a solution was irresistible to him.
“Yes, sir. I have a proof.”
“Tell me first, what is the value?”
No one else knew the proof besides me, among all Mathematicians, among all ages, and now I would give away my secret knowledge. I might have seen jealousy in Master Johann’s stare, and maybe greed, and maybe even scheming and betrayal. But I decided I knew him better than that, that he was better than that. And I decided that I could presume to read the thoughts of a much greater man.
“The value pi, which is the ratio of the circumference of a circle to its diameter—”
“Yes?”
“The infinite sum of Reciprocal Squares is equal to that value of pi, squared, and divided by six.”
He leaned forward, closer, I think, than I’ve ever been to him, and his mouth open and his eyes wide open.
“It is . . . what?!”
And even now I faltered and nearly failed under his extreme intensity. I was so unsure. But then I remembered the vision. I was sure. “I’ve conjectured . . .”
He was still leaning toward me. “Yes? Yes? What?”
“I imagine a polynomial and its roots, and it continues and continues, always crossing its axis.”
“How many times?”
“Many, many. An infinity of times.”
“An infinity of roots? Then it would have an infinity of terms, and an infinity of order.”
“Yes, sir. A polynomial of infinite order, and whose roots are every integer multiple of pi, positive and negative and zero.”