An Elegant Solution

by Paul Robertson

  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.”

  He sat back. “Indeed.”

  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 Thousan
d 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.