We had our Sunday dinner.
“Grandmother,” I said, “I think highly of Master Desiderius.”
“He seems a pious man.”
“He has a Chair at the University, yet he doesn’t seem proud. I believe he’s humble about it.”
She looked at me shrewdly. “Yes, Leonhard. It is possible to have an eminent position and not be brought down by pride. But it’s rare. Pride may be slow to increase but it always does.”
“I’ve never seen any arrogance in Master Desiderius. Am I mistaken?”
“I’ll speak no evil of him.”
“Daniel is hoping to win Master Huldrych’s Chair.”
“You know Daniel well enough.”
“I do. I think he’s very full of pride. And I am, too. I try not to be.”
“At least you try, Leonhard. Does Daniel expect to be nominated to the Chair?”
“Oh, of course he does. He’s very sure he will be. And he should be nominated. He’s already famous.”
“We’ll know soon.”
“Everyone says it’ll be weeks before the Election starts.”
“It will be sooner than that,” she said. “Much sooner.” I didn’t question her. I’ve always been surprised at how she knows much more of the University than I’d think she would. I believed that, as I would see the invisible, she would hear the inaudible.
I was prepared on Monday. I came to Mistress Dorothea’s kitchen in brown but as neat and respectably as I could. Mistress Dorothea was solemn and severe and took me upstairs to the door. In a shadow in the hall I saw a darker, paler shadow, and that was Little Johann watching me as I knocked.
“Come,” and I opened the door. “Good morning,” he said, and I could see immediately what hadn’t moved and what had. Two books that had not been on his shelf before were set about on his desk: MacLaurin’s Geometrica Organica and Taylor’s Methodus incrementorum directa et inversa. The papers on the desk were mostly changed. Some few were only moved, but most were new.
“Good morning, sir,” I said. The books on his desk were set atop the papers, I thought, to obscure them. But I could still see a few edges.
“Do you have a drawing from the stonecutter?” On the exposed edges of the papers were equations, and I recognized parts. They were his own experiments with infinite polynomials.
“Yes, sir, I do.” And also, another letter had been moved. It was the formal statement from Paris, of the Reciprocal Square challenge. It was also open on his desk. I held out, from my pocket, the sheet that Lithicus had given me.
“Thank you,” he said, and unfolded it. He studied it briefly. “And a price?”
“He says thirty florins.”
“Reply to him that he’ll be paid a hundred.”
“One hundred?” It was a huge sum.
“And tell him I want an additional line added.”
“Yes, Master?”
He took ink and a quill and wrote on the back, INLUSTRIS MORBO CHRONICO MENTE AD EXTREMUM INTEGRA.
“That.”
“Yes, sir.” I left him there with my mind reeling. It was an extreme surprise to me that my Master Johann should have been reading MacLaurin, and especially Taylor. It was prideful of me to think it, but it seemed the only reason was that he was comparing them to my proof.
I was surprised that Master Johann would have been reading MacLaurin, because the Scotsman was an ardent supporter of Mr. Newton. But this Scotsman had also written on infinite series. I’d read all his books and eagerly awaited the others I expected him to publish. Only four years ago he was awarded a prize by the Paris Academy.
But the spectacle of the Master of Basel with a book by Taylor on his desk would have wagged tongues from Paris to London. They were terrible enemies. When Master Johann answered a challenge raised by Mr. Taylor some ten years ago to integrate a peculiar differential, the Englishman disputed my Master’s solution. The dispute has continued unresolved, even to the point of threats and hostile wagers against each other in their various publications. But the Methodus incrementorum greatly extended the theory of writing differentials as infinite series. It was precisely the book in which to seek an answer to questions that my Reciprocal Squares proof raised.
I felt that I should hurry to find Lithicus, but I was also hesitant. Every mention of Master Johann seemed more fretful to him. Though the new and higher payment might hearten him, I’d want caution and mildness speaking with him.
“You!” a voice spoke from behind me. I turned and it was Daniel, of course. It was another chance encounter in Basel’s streets.
“Me?”
“To the Boot. I’m getting my horse. But you’ll do for now.”
“I’ll do?”
“Though you’re a poor substitute for my Coal.”
“I’ll try my best.”
“You always do, Leonhard, and it’s credit to you. But now, this is why I found you. I’ve a use for you that even a horse can’t match.”
“What is it?” I asked.
“There’s fire and fury back under my Brutus’s roof.”
“I don’t have my water buckets.”
“This fire won’t be put out with water, and I don’t want it out anyway. Have you heard of the Reciprocal Squares?”
“Yes . . . just recently. I’ve heard of the challenge from Paris.”
“Well, Brutus has a proof for the Reciprocal Squares.”
We’d just come to the Barefoot Square and I tripped on the first paving stone. “He has?”
“He has, and it’s stunner.”
“Is it his own proof?”
“Someone’s sent it to him, I think. I don’t think he’s come up with it himself.”
“Have you seen it?”
“I have and it’s written in his own hand. But it’s just that he copied it.”
“But is it valid?”
“That’s what I want you for. I want you to look at it. There’s some of it I can follow and some I can’t.”
“Why did your father show it to you?”
Daniel laughed. “He’s under torture. He’d die rather than allow that someone else solved it before him. And that’s worth it being valid just by itself.”
“Daniel!”
“He’s desperate to know if it’s valid, and he’s not sure himself! He just had to show it to me, and Nicolaus, and Gottlieb, too. It’s Mathematics, Leonhard! He wants so much to find a flaw in it he’ll even show it to us! Oh, it’s delicious, it is. And if there’s something to be found in that proof that Brutus can’t find, I’ll ask anyone. Anyone. And I’d give about anything.”
“It’s not worth that much.”
“I’d trade anything I had for it.” He wrinkled his nose, suddenly thinking. “Maybe that’s what Brutus has done. What do you think? What’s nefarious? He’s got a proof he’s always wanted, just out of air.”
“You make it sound like Faust,” I said, and Daniel pulled back.
“Don’t say that.” He said it vehemently.
“I won’t, then.”
“It’s not a joke.”
“I won’t say it again.”
He breathed deep. “You look at the proof, Leonhard. You see things we don’t, we all know that.”
“I . . . I won’t help you humiliate your father.”
“Oh, that’s the small part of it. The real part is whether it’s valid. That’s what we most want to know.”
“Well, then bring it to the Inn tonight.”
“I will.”
“And are you really all working together on it, Daniel? You and your father together?”
“It’s Mathematics, Leonhard. Of course we are.” We’d come to the Inn and I followed Daniel through the tunnel to the stables. “Where’s my black?” he asked Willi.
“Shoeing. Gustavus has him in the smithy.”
When the Olympic gods had been overthrown by the true Church, it was Gustavus who took Hephaestus’ forsaken anvil and hammer for his own. Only Gustavus could ever move those weigh
ts of iron, and the sparks they made were Zeus thrown lightnings.
The smith shop of the Boot and Thorn was in another far corner of that many-cornered building, near but beyond the stables. As with all the corners, there was fire. This flame was in a kiln-like oven, charcoal fed and white hot, the hottest fire in Basel.
We watched Gustavus form a horse’s shoe. No metal could withstand, between that continent of an anvil and that mighty hammer wielded as the earth wields mountains. It was a place to wonder about nefarious purposes. Gustavus in his black apron struck the shoe with his sledge, and I thought the sparks flew into it instead of out, to add fire to the horse’s speed.
The smithy was more a cave than a room. The walls were rock and the oven was in the rock, with a chimney bored straight up to the air above. There was water in a pit carved into the floor. When the shoe, still glowing, was dropped into that pool, the water was barely able to cool it. Water was always unwelcome by the fires that ruled that inn.
When Gustavus nailed the shoe on, the black horse suffered him gladly to do it.
“He’s ready, there?” Daniel asked.
“He’ll take you well, now,” Gustavus said.
“I’ll let him!”
The room was so dark compared with the white fire that everything in it was invisible.
Outside the inn, I was quick to find Lithicus on his scaffolding using the bright light of day to reach the shadow high and deep in the arch of the Coal Gate.
“What does he say?” he asked, indignant as sharp gravel and anxious, when I told him what Master Johann had said. “More lines? Show me the words.”
He didn’t climb down, so I put my foot on the first cross piece of the wood frame, then the second, to hand it up to him. He squinted in the poor light. “I know that line,” he said. “I’ve used it before. Most with University men.”
I said, “It means, Despite his illness, his distinguished mind kept its integrity until the last.”
“His mind? What’s that to anyone? The merchant, he’s proud to keep his money to the end, and the churchman his piety and the wife her family to the end. But the end comes and they all lose all. And tell him, I’ll need a new slab. The other one’s not big enough.”
“What will you do with the other one?”
“It’ll be for someone else who kept nothing to the end and doesn’t need the extra space.”
“There’ll be an Election soon to replace Master Huldrych,” I said.
“I’ve no part of that.”
“But you are part, Lithicus! You made the stones they use to cast lots.”
“I made them. And that was all I made.”
“I’ve only seen them from across a large room. What are the symbols on them?”
“Why are you asking that? What would it matter to you?” He got hold of his anger, but kept it ready in hand. “What would you do with it if I told you?”
“I only want to know. I was curious.”
“I won’t tell you.” He’d come calm, but was all suspicion. “There’s no good you’d do from knowing.”
“Then I wouldn’t want to know. I didn’t know they were secret.”
“They are or aren’t. I don’t speak of them.”
“Gustavus said it was twenty years ago. You wouldn’t need to have even remembered.”
“I remember.”
“And you’ll never need carve them again.”
I’d meant to be agreeable and calming to him, and he had been more at ease, but he suddenly was angry again.
“I’ll never carve them! That I’ll never do again!”
“Of course you wouldn’t.”
“What do you mean by that! What do you mean?” He was nearly yelling at me. “Who’d say they needed to be done more than once? Who said it?”
“No one!” I dropped back to the paving stones. His hammer was waving too wild to be close to him. “I don’t know why they would be.”
“Tell that man he’ll have his stone, and fast as it can be done. I’ll need a new slab. That will cost more”
“And he’ll pay you more,” I said. “He said he’d pay you one hundred florins!”
But his reaction was only worse and worse. “I don’t want any! Tell him I won’t take any payment! I’ll just be done with the stone and never any more!”
“No payment? But he’s willing to pay.”
“What do you mean at that? Willing? You don’t know how willing. No more. He’ll have it and nothing else. Thirty was enough for Judas, and he thinks I’ll take more?” In a quick motion, he put his hand against an arch stone above his head. “Don’t slip out, you!” It wasn’t me he was talking to, but the stone. “You’ve bothered me,” he said, and that was to me, “and now I’m addled! I’ll drop a stone and lose the day’s work.”
I couldn’t think of anything else to say, as everything I did say seemed to make him more angry and alarmed. There were even a few others in the Square who were turning to look. Gustavus had finished his shoeing and was standing with the horse and with Daniel at the stable tunnel. I backed away to them.
“What’s afflicting the stoneman?” Daniel asked.
“He’s just touchy,” I said.
“Touchy and with a chisel and hammer, that’s bad! What is it about?”
“I was asking him about the lot stones. They’ll be used in the Physics Election.”
“And soon, so I’ve heard. The University’s convening Wednesday.”
“For the Election?”
“In two days. The Chair’s to be filled.”
“The Chair’s only empty ten days,” I said.
“Brutus commands, the plebs obey. Tomorrow the committees will be chosen. It’s the first step. And Leonhard, I want you here at the Inn tonight. I want to show you this proof.”
“I’ll be there to see it.”
“Is it gossip or trouble you’re pursuing tonight?”
I was glad to answer as we finished supper. “Neither, Grandmother. It’s Mathematics.”
Then I was out in the last dusk light. My heart was full of both joy and caution. The prospect of Daniel and Nicolaus and good round dispute over a proof, whatever proof it was, was tonic. Even more, that it would be the Reciprocal Squares! But which proof? Mine, or another . . . that was worth caution. And though it was likely impossible, I wanted to be no cause of strife.
The stars were vast, but their infinite sum still was only a finite portion of the sky. They were vastly far away, and who would know their bright essence? I knew I was very small on the great planet, beneath the greater heavens, but it was within me to comprehend them and know how they were governed. What could it mean that God had put in finite man the chance to study the infinite?
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 some
how 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.”
An Elegant Solution Page 23