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.”
I proceeded. I described the steps, from one statement to the next. I was building a castle, or a palace, or a mountain. It seemed like each of them. Every part was just a breath of air and a few sounds that touched the room, then were gone. The castle was the thinnest unsubstantial thing ever built, yet it was as hard as obsidian and adamant. Nothing in creation could break it. And when I reached the end it was irrefutable and impervious.
“There are errors,” Master Johann refuted. He began stating them.
The errors he claimed were complex, where the foundations of the walls rested on untested rocks, and I’d known just where he’d mount his attack. But they weren’t errors. They were proofs yet to be found, but they were truths and I knew the proofs would eventually yield.
“An infinite product,” he said, “and you claim to still know the individual finite terms. That is unproven.”
“But they must be,” I said. “It can be proven that they’re nothing else. There would be no other part to them.”
He circled, he probed, he thrust and I parried. He challenged every line. But he was hesitating and pausing, and then he began answering his own attacks. Then finally he shook his head.
“This will take more study,” he said. “It is intriguing, Leonhard.”
“Yes, sir.”
“But it is far from convincing.”
But I was convinced.
“Have you shown this to anyone else?” he asked.
“No, Master. It only came to me yesterday.”
“Do not discuss it with anyone. I will continue to study it.”
It was very unusual for him to not give me work for the coming week. But it was plain that nothing so mundane would be taken up in the rest of our session. Indeed, the session was over. I heard the bell in the Munster tower.
“Yes, go on, Leonhard,” he said as I stood. “It will take a great deal of study.”
“Thank you, sir.”
But instead of dismissing me, he paused. I waited. He had some other subject that had come to him. “Leonhard.”
“Yes, Master?”
“You’ll be finishing your studies soon?”
“I hope to present my dissertation by next year.”
“Yes. I have expectations for it. Leonhard.”
“Yes, Master?”
“You said that this proof came to you.”
“Just last night.”
“Not that you solved it, but that it came to you.”
“It did, sir. I don’t believe I could have solved it myself.”
“I see. Then who would you say did solve it?”
“I . . . don’t know. Do you understand what I mean, Master?”
“Yes. Very much.” And he looked into me in a deep, searching way, seeing something in me he recognized. “Yes, I understand.” And he seemed to see something else in me. “If the proof is true,” he said, “it will be an elegant solution.”
“Thank you, sir.”
“To many things. But only if.”
“What does it mean,” my grandmother asked, “that you’ve proven your answer but Master Johann doesn’t believe your proof?”
“There are things that I believe are true, and he doesn’t.”
“Are they true, then? You’ve said that in Mathematics a thing is true or not, whether it is believed or not.”
“There are some parts of Mathematics that aren’t understood well enough to be sure what is true.”
“As with God,” she said.
“The two are very close. Sometimes I think Mathematics is the thing God made that is most like him.”
“What will become of your proof?”
“I want to write to Paris.”
“You? Is it acceptable for you to send letters to this Academy?”
“I think not, Grandmother. I’ll need to ask Master Johann to send a recommendation.”
After that afternoon I felt as if my thoughts had been swept clean. Everything had ebbed away that had occupied my brain and it was like a hunting dog asleep and twitching on its rug. I was exhausted but nervous and edgy. I needed something to fill my empty head again and so I thought of dust, and I went to see Lithicus.
His yard was mostly as before, but somehow dustier. The stonecutter was mostly as before, as well. “And it’s you?” he greeted me. “Here to question me?”
“You told me to come. And to ask for my Master if there’s progress for him.”
“Progress, there’s progress. Does he think I’m idle?”
There’d been little rain or wind in the last few days, though I thought that yard would always be filled with dust. Of dust was man made and to dust man returned. Here the carving of epitaphs and memorials left dust which was surely a part of a man’s return.
“I don’t know what he thinks,” I said.
“Nor anyone does. And there’s progress. It’s this.” From his leather apron he’d found a folded page of paper, and unfolded it was a sketch of scroll bordered with the folds of a robe and headed with insignia of the University. “That’s what I’ll do, meeting the Master’s approval.” The sketch was drawn with skill and art by a charcoal stick. The words were centered, and all the lines balanced. Master Johann’s text had seemed short, but laid out in lines and capitals it filled the scroll. “And the slab is this one.” He pointed his hammer to a rough flat oblong of clean gray with one bright vein branched across it, like a lightning on an empty twilight sky.
“I think it’s fitting,” I said. “I’ll show him the paper.”
“But no other symbols or dabbles. That’s all.”
“I’ll tell him.”
“No, don’t tell him! He’ll have me wrung. You—” and he aimed his hammer at me, “—you keep him from asking.”
“I will,” I said.
“And thirty florins.”
“I’ll tell him that’s the price.”
“Aye, thirty pieces of silver. Tell him that.”
Gustavus had taken note that I had become a regular patron of his Common Room, where I’d been now more times in three weeks than in three months before. Also, my status had changed with him: Now I was always a Master.
The talk in the room was of the Physics Election. There was speculation of when it would begin, whether in two weeks or two months or two years. It was an eager conversation. In black and white there were students and a few lecturers, for whom the Election might have a real effect, and a merchant and guildsman or two, but the larger number were in brown. Craftsmen, laborers, peddlers and farmers, all were far removed from the inner parts of the University and would never set foot in a lecture or understand a word of Latin. But in Basel, the University was owned by all, in that it was a part of the city, and all of the city was one. And there were many questions about how the Election was conducted, and many answers.
There were three stages to an Election. The first was the selection of the three committees of six members each; the second was the announcement of each committee’s candidate; and last, after each candidate was allowed an opportunity to give a guest lecture, one of the three names was drawn at random. Of course it was the second, and even more the third, of these which caused the greatest excitement; but it was the first, which was accomplished through a Convening of the University, that had the greatest pageantry.
The lengths of time between the three were also variable, as it might take weeks or months for the committees, meeting in secret, to choose their nominees, and then, if any of these men were distant, it might be months more before each of them would have arrived and lectured, and it was traditional that at least one candidate not be from Basel. If the drawing was later still, the men were unlikely to have remained in Basel for it.
A small, iron casket was kept by the Provost for the drawing. It had a lock to which he kept the only key. Within the casket were ten small carved stones, all about an inch square and a half inch deep, so that any two could be put together as a one inch cube. The stones were smooth and plain on all their sides but for a specific exception, that seven, each on a square side, had a symbol carved into them, so that there were seven different symbols. The other three stones were all plain.
As the names were announced, each new candidate, or his committee, would choose a symbol stone. This stone would become that candidate’s lot. A clerk would take the stone, make note of which symbol would now stand for that candidate, and then seal the lot.
The sealing was done by dripping wax onto the symbol itself, filling it and more, then fixing one of the three plain stones to that side. Once the wax had dried, the lot would be a simple cube with no outward sign, the symbol hidden inside. When all three candidates had been named, the three sealed lots would be kept in the casket. The unused stones would be set out. The casket was locked and set on the lectern, where it would reside until the final choice.
It was an odd ritual, part tradition, part compromise, like Basel itself laid out on older patterns of purposes no longer remembered.
The three candidates were then invited to give their lectures. A very poor lecture might disqualify a candidate; otherwise, the lectures were an opportunity for the University to hear these eminent scholars. Some men from distant cities and universities would decline the offer, as a one-third chance of a Chair wasn’t worth the journey. Occasionally the University would accept a written lecture, to be read by a member of the nominating committee, but more often would take the refusal of the lecture as pertaining to the entire candidature, and nominate a replacement who was more appreciative of the honor.
Once the lectures had been given, the University would meet a third and last time. The casket would be unlocked and the Provost, humbly submitting his high position to the ignominy of a blindfold, would choose a single sealed lot from the three. The seal would be broken and the symbol revealed, and the new Chair congratulated and presented to the city.
In the Common Room, all the details of the ritual were discussed.
The casket is left out on the lectern, without guard? Could it be pried open and the stones exchanged?
Gustavus, as blacksmith, had made the casket. “That casket will never open unless the lock is turned,” he said.
Could a pick-lock turn the lock?
“Keppel the locksmith made the lock. I told him to make it safe against picks.” And if Gustavus had told him, then it would have been done.
But where were the keys?
“There is only one, and the Provost has it.”
Before the casket was made, what had been used then? A previous casket?
“The old one was lost in the river, twenty years ago.”
But the stones? Where had they come from?
“Lithicus carved them when the new casket was made.”
And the Election itself? When will it be held?
To that, Gustavus had no answer, and Daniel’s was morose: “The Election will begin when Brutus says it will,” he said. “He’s doing all the choosing now, and when he’s told everyone their parts, he’ll let it start.”
No one had a reply. Basel had great faith in the integrity of the University’s Election, and there might have been a protest. But Master Johann also had a place in the city’s beliefs. No one would claim surety of what that man might do.
And what of the unopened stones?
The Senior Chair of the College owning the Chair would take the casket and verify the stones, then return it to the Provost. That had been Huldrych; now it would be Johann.
I walked early Sunday morning by the Rhine.
There was a moment, as a child, when I realized numbers were infinite. I didn’t then yet know the names of Thousand and Million. I may not have known even Hundred. I was watching the raindrops falling on the river. It was even before my father moved us to Riehen to take the pastorate of that village’s church. I could have only been four or five years old. My thought, walking with my father on the riverbank, where we were caught in a shower, was that the river was made of all the drops of water, all the rain. I’d looked at the wide surface, which was vast to me then, and considered how very, very many drops of water there were: innumerable, then no, they could be counted. It would only take a very long time. Perhaps all day! in my childish calculation.
But I watched more drops fall. We were under a tree, father and I. We’d had to run to it. I remember that well, laughing and running, how we both loved to run. The rain decreased and the raindrops lessened, but I was fascinated at those small beads crashing into the river and being absorbed by it, and my father let me just be and watch. One branch over us tilted steeply down, so I could see its last leaf just inches from the surface, and finally after minutes of staring, the drips from that leaf had slowed to only one, by one, by one, each falling across the last space to their sum. And then I knew, that whatever their sum was, it could always be one more, and if always more, then never to end. For any number, there was one more beyond. Always.
I had only one way to comprehend that. The Mathematics of infinity was still beyond me. But my father’s preaching was already deep in everything I would know about my world. From him, I knew a word for something that was beyond everything else: heaven; a place where “one more beyond, always” did reach its end. So I had always understood the infinite end of all numbers as God showing himself in his creation. Everything he made had his image, and part of his image in Mathematics, was infinity. It was invisible because it was far past the end of sight. It was the greatest elegance.
Then later that morning at Saint Leonhard’s, with my grandmother, all my thoughts were on infinity and the infinite sum of infinite things. We were instructed in the sermon that the gulf between ourselves and God was vast and unbridgeable, which Mathematically would be infinite. Yet, we were reminded, it was bridged, by sacrifice.
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.”
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 weights 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 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.”
“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.
But in the end they were convinced. “A new lion,” Daniel said at last.
“And what does it mean that there is?” Gottlieb said. “A new rival.”
“We could end him here!” Daniel said. “The four of us. And we’d have the proof for our own!”
“Father knows where it came from,” Nicolaus said.
“Cut him in on the spoils, then. We’ll publish under the whole family’s name. Or do away with the Brute, too! Then we’d have the Chair open, too.”
“Only one can have it,” Gottlieb said.
“Then watch your own throat.”
“Those are poor jokes,” I said, but I laughed. I felt light-headed from the long debate.
“He’s not joking,” Nicolaus said. “It’s Mathematics.”
I answered,“I don’t have fear of any of your family.”
Daniel said, “But on to other matters. There’s a propriety to answering a public challenge from the Paris Academy. It’s meant for men of reputation and position. You’ll need someone to write a letter for you.”
“I mean to ask your father.”
“And he’ll sneer, won’t he? That will be a lesson in derision. Even to you, dear Leonhard.”
“He won’t sneer at Leonhard,” Nicolaus said. “He’ll be civil. But he’ll still turn him down.”
“And if he doesn’t,” Daniel said, “Then that will be near as interesting as the proof itself.”
“What do you mean?” Gottleib asked him.
Daniel only shook his head. “Cheers to you, Leonhard. Remember this night, when the Reciprocal Square was solved and proven. We’ll leave the intrigues for tomorrow. This night is yours.”
I let it be. So we went on to talk of other things. I asked about Italy, where the three of them all had lived and had Chairs. The conversation was affable and the only jabs were good natured. Little Johann asked more questions than I did. We talked about other Universities without sarcasm or bitterness. If the night was mine, their gift to me for it was conviviality. We talked of Paris and Holland and Heidelberg. We talked of kings and princes, the courts of Prussia, Austria, France, Hanover, Saxony, and even England, and which would be more advantageous in which to gain a position, which would be more cultured, which would be more lucrative. Then we talked of Empires and Kingdoms, the present wars and recent wars and Basel’s place among them all. It was well that Louis the Sun King of France had died when he did, for Basel would have been his next cherry to pick.
Then at the end we parted, all close friends. The forces that pulled together were stronger than the forces that pushed apart.