An Elegant Solution

by Paul Robertson

  “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 court
s 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.

  12

  The Physics Election

  On Monday, Daniel had said the University would convene on Wednesday. On Tuesday, everyone in Basel had heard. But the true Announcement was yet to come, and once Basel knew that the University would Convene, Basel knew that it would be Announced.

  After the recent fearful days of the plague, the news was like fresh water. The announcing and convening of the University was a ritual as old as the University, and in austere Basel, where any feather of pageantry was suspect, only the academic and ancient in alliance justified a display. And as long as it was justified, the citizens would gladly spectate a spectacle.

  It had been two years since the last convention, for the Election to Logic, and the elevation of Gottlieb and the departure of Daniel. I knew the ritual fairly well from watching it then. It was all pompous pomp on the outside, Basel’s gaudiest rite, as ritual as a coronation; as ponderous as a planet’s orbit, and as full of robes and regalia as a cathedral choir and dressmaker’s shop together. To Italians or French, it would seem just as black and white as everything else in a Protestant city. Yet color was measured by contrast, and in Basel, bright colors were kept in their proper place.

  But there was what was seen, and there was what had substance. It was the invisible part of the convention that had the most meaning: the quiet conversations before the loud gathering, the two heads leaned close across a table in a dim room, the sparse written note then thrown in the fire. Daniel claimed that when finally the University sat to deliberate, its deliberations had already been done.

  Master Johann was now the senior Chair of the college and had a critical role in the first meeting. What would happen in the conclave, and what had been agreed beforehand between the most influential members, would never be known outside.

  The carriage returned that evening from Strasbourg and Freiburg. I was in the Barefoot Square at the time, just coming out from the Barefoot Church. All day the rumors and gossip had been pouring out the Inn and every other door about the coming Convention and Election. When the coach arrived, it seemed a part of the disturbance. It was surrounded by hounds, and wolves, and hunting dogs, and then I saw Jehu in the seat next to Abel. That King of Israel looked up to the windows of the Inn, and there, Jezebel looked down at him. He shook his fist at her and she jeered back at him, and all the dogs bayed and howled back at the face in the window.

  When I came home that night, my grandmother met me with a note that had been sent from the Dean of the College of Arts. It instructed me to report to his own home the next morning at nine o’clock. I knew what this meant. “It’s part of summoning the Faculty,” I said.

  “You’ll call on Master Johann?” Grandmother asked.

  “I think it will be Master Desiderius. He told me I’d be asked to do something for him.”

  Wednesday morning I was up early and quickly done with chores. I dressed in my immaculate black and white, and my grandmother sent me out the door to the Dean’s house. The Convening of the University was to begin.

  The first spark always would come from the Provost. Four students in their brightest black would issue from his front door. They would be mature and responsible young gentlemen but still sprightly, to prance to the homes of the four Colleges’ deans. There, they would rap smartly on the doors. A great part of the ritual was this knocking. Neighbors would step out of their homes to watch it. Through the morning, as the summons would unfold, great men would be answering their own doors to receive the news: the University is Convening.

  Then the sparks would spread into flame. The deans would acknowledge the call and send out their own. They must step back from their door, apparently to summon their own messengers, but in truth to not be knocked over. From their doors would burst a flock of students sent to summon their professors, and as the most respectable students were already serving the Provost, and there were more professors and lecturers than mature students anyway, this herd of flapping, flying black robes would be worth seeing and worth being out of their way.

  I was to be one of them. Master Johann as Senior Chair of his College would have to be summoned by the same messenger sent from the Provost to the Dean. I was meant for Desiderius.

  I ran! The young men chosen were favorites of their teachers, and high spirited though they were, they were counted on to play no pranks on a serious occasion. We raced through the streets, crossing paths between deans’ and professors’ houses. In all, there were some fifty. Most went for the lecturers and officials of the University, but fifteen or so who could make a good bow and had the nicest wigs were sent to the Chairs.

  These several would carry tokens of authority. The Dean of Theology sent hourglasses to the Chairs of his college, that life was measured and would end. The Dean of Law sent quills, that words were the structure of authority. The Dean of Medicine sent pestles, that man was mixture of soul and body. And the Dean of Arts, who had the largest college, sent candles, because knowledge was light. The tokens signified an important notion. The Deans were between master and servant of the Chairs. They were a higher position, but not a higher rank. They were often former professors of whom it was thought best that they no longer profess, the moon moved aside to make way for the sun. They had an important role and were usually men of substance, but not always of eminence. So the tokens were a command and a plea to the Chairs, who were Great Men not to be called as if they were Less.

  But of course they would come. Their doors were knocked upon and they opened them and stared out into the street. There were many houses in Basel on many streets, but not so many of either that provided such high residence: the Chairs furnished only certain areas. They received the token and the summons and they stepped out their doors into the light. As if it was their daily habit, they were in their finest robes. Often more than one would be in sight at a time.

  The people were watching. Basel was proud of its University and viewed the great men with satisfaction. They appeared in the streets, rounding the corners, enrobed in black and scarlet, black and emerald, black and azure, black and sienna, black and maroon, black and canary. The striping was in corvettes, diagonals, diamonds, and arcs. Their vast wigs curved, curled, and coiled like rioting ivy and wide mountain waterfalls. Atop the wigs were triangular, square, rectangular, trapezoidal, and pentagonal velvet caps, and each color and shape imbued by tradition with centuries of meaning.

  As streams to the Rhine, the Professors would flow into the University. They would come from all directions, though none across the bridge, for no professor lived in Small Basel. They would come on foot, slowing as they approached the portal, gathering like an army and then entering their Fortress, the University Building itself.

  As part of the flood, I arrived at the House of Desiderius and knocked on the door. All the children of the street were out to watch, and even a few of the wives had their curtains drawn aside. The door opened and the Master himself, brow furled at my completely unexpected appearance, leaned out into the street.

  “What, Leonhard?” he said. “How is it you’re here?”

  “I’m sent by the Dean,” I said. “I’m to bring you to the University.”

  “To the University? Why, what is it? I’m being summoned?”

  “The University is convening, Master. And I’ve a token for you of the seriousness of the matter.” And I held up the candle.

  Master Desiderius took it carefully and examined it thoroughly. “Then I’ll come,” he said. And of course it was no signal at
all that I was expected, that he happened to have on his University robe and cap!

  So we set off. Master Desiderius was not too senior a Chair to keep his own lively pace through the Basel streets. And when we turned one way or the other in our path that wasn’t the straightest route, I thought it was that he even wanted to be seen by some acquaintances. While Master Johann would surely be stepping his slow, stately beat, only hurrying at all because it would be tiresome to be a spectacle for the common people, Master Desiderius knew that he was on display and made well of it. His robe flowed behind him like an exultation of larks.

  His appearance in the Barefoot Square was especially meant to be jubilant. We came to the Coal Gate and saw the Square beyond, over full as people had planned their marketing to coincide with the Convening. We even paused a moment in the shadow of the gate in anticipation of our entrance. And as we stood in that dark spot I had a sudden sense of something, motion or sound, but really of rending. I took hold of Master Desiderius’s arm, and in the instant felt him also gripping mine, and then I lunged, pulling and being pulled. Above me, and then as I fell headlong forward, behind me, there was weight, and force, and a terrible plunging, and collapse. All the stones of the arch came crashing into the space they’d stood over, with roaring and tumbling clamor, piling in an instant into a mountain of rock, a blockade, and a destruction. I was on my face on the paving of the Square, and Desiderius beside me, and we turned to see the tons of gate where we’d stood three seconds before, while still more stones were falling, and finally the last did.

  And a great cloud of dust rose from the pile.

  In a few seconds more we were being helped to our feet. Master Desiderius was unscathed; I think I’d tripped and he’d been pulled more gently down by his hold on me. But I was battered and scraped and felt unsteady from unexpected pains as I stood. Small, bright red lines appeared from the brown dirt on my hands. And I was covered with dust. I stared at the great wind of dust that wrapped the pile of stones.