An Elegant Solution

by Paul Robertson

  “Well . . . if I’d been deceived. I think I could then.”

  “That’s what might be. You’d say I’d been deceived, I think. I’m sure you would.”

  “Tell the man, then. Tell him how you think he’s misled you, and you want the bargain off. Is there one of you who acts first?”

  “He already has. Part of it. But a big part. Maybe he has. I don’t know whether he has.”

  “What was it, Daniel?” I asked.

  “I won’t say.”

  The next pause was mine. “Was it anything evil?”

  “Evil . . . ?”

  “That you agreed to.”

  “Not that I agreed to, no. I wouldn’t have. And that’s a point, too, that I didn’t. And it’s done. Even whether he did it or not. It might not have been.” He tried to laugh. “It wasn’t anything a man would be thrown into the river for! Not what I agreed.”

  “What was it?” I asked again.

  He was sober again, and finished. “Leonhard, it’s best not said. There’s much to think about and I will.”

  “I will, too.”

  “Don’t. There’s no use.”

  I think neither of us was much satisfied. Mistress Dorothea had asked me to watch for regret and remorse. I wasn’t sure yet whether I’d seen any.

  I sat late, through two candles, writing in silence about sound and waves. I’d been writing this thesis for very long now: two years. The scratch of my pen came to seem like thunder to me in the quiet night. With every sentence I also heard Huldrych’s arguments against my ideas, and I wished I could answer him. Only when I could argue back did I know that there was some strength in my ideas. He was like the hammer that put force behind the chisel, forcing elegance out of my coarse and ill-formed proposals.

  Finally I put the quill aside and closed the ink. I dressed for bed. I didn’t know the time. Last, I looked across my bookshelf for which volume I’d have for my Saturday. It was hard. I was still diminished. It would take time to re-grow. I picked the Ars Conjectandi and extinguished the light.

  My sleep was short but restorative and I woke to my Saturday morning more at peace. I was out to the well in good time, and I chose a different fountain than the Barefoot. At home again, I felt the Saturday morning buoyancy lifting me. It was like I had hold of some hourglass that didn’t whip with the waves.

  My grandmother found me more talkative, and she was, also. “What were you writing last night?” she asked, and that was a sign that she wanted to just hear me chatter. I was always writing and to her, one thing was mostly the same as another.

  “I was disserting,” I said. “It’s still on sound and waves in the air.”

  “And what in particular?”

  “On what a wave is.”

  And I said it eagerly enough that she had to ask, “And what is it, Leonhard?”

  “I think it is a Mathematical equation.”

  “And how can it be?”

  “I don’t know if the wave is the equation or the equation is the wave. But this is what I’ve written, that the equation is the law that the wave must obey.”

  “Sound must obey laws? Like men must? And who gives laws to the air?”

  “I think God does.”

  “And do they follow His laws?”

  “Yes,” I said, and firmly. “In every circumstance they do. They aren’t unrighteous.”

  “Where do they learn the laws?” She was teasing my words, but I think they were interesting to her.

  “They don’t learn. They just follow. The laws are invisible.”

  “You see so many invisible things, Leonhard.”

  “It’s only because there are so many,” I said.

  Then I was in my room and I read Uncle Jacob.

  The subject was conjecture, chance, probabilities. To throw dice, what was expected? The cube would fall on one side of six, and any was as likely as any other, though only one would land upright. And throwing it again, there was still the same chance. Landing a three twice made the chance of another three no greater or less. But it was still unlikely to land threes thrice. What did chance mean? And how could chance be, when all the universe was ordered and clockwork? Did God choose which side would land?

  I’d read the Ars Conjectandi a dozen times at least. I could have written out good stretches of it from memory. And now, though it’s prideful of me to say, I could place it only in the middle of Mathematical writing. Leibniz, DesCartes, Newton, and others were still above it. But it had hints. The book couldn’t have included all that Master Jacob thought, or even wrote. So I looked, as I read, for what had been written but not in these pages; I looked for the invisible writing.

  Master Gottlieb had written the book from Master Jacob’s notes. Perhaps he had them still. It would have been more likely that Master Johann had those notes, but that also seemed most unlikely. It would not have been Jacob’s wish that his rival brother come into possession of them.

  Instead, my thoughts turned in a spiral to another place the papers might have been. Most likely they had been gathering dust. Great amounts of dust, for many years. And then they were somewhere else.

  And for a few moments, they may have been in Mistress Dorothea’s kitchen.

  At the dogmatic three thirty I put my knuckles to Master Johann’s door, and it was opened by Mistress Dorothea. That sequence at least was as absolutely unchangeable as anything in Mathematics. There was no chance to it.

  There was a change in me, though. I was not fearful, or not as much. But there was one obstacle between myself and my proper place, and I couldn’t overcome it. It was my hat.

  I had no other hat! Gottlieb still had my humble student hat with the simple roll of the brim on either side, and I hadn’t had the opportunity or purse to buy a new one. I had only this tricorne. Would it be monstrous of me to wear it into my Master’s presence? But I couldn’t have come without a hat, either. It had to be doffed in respect and set on the table. And there it would be! As I approached the dull door, I was in a sweat and un-confident. But not fearful. How could I fear anything when I was wearing a gentleman’s hat?

  What pure dilemma! And all my musings in Mathematics and Physics, Theology and Greek gave me no guidance at all for solving the problem. As the Mistress knocked on the final door, only Logic could help me. There were two choices, Hat or Not Hat, and Not Hat was impossible. So Hat had to be, so Hat was. The knock was answered, I opened the door, and not just a student but a gentleman went in.

  I had a worry, as I came into the room and studied his face and mood, that our other meeting in that place on Thursday morning would not be forgotten. But I didn’t see anything. Master Johann was seated at his table, as always, with his candle, his paper, pen and ink. His stare, as always, was just past my shoulder as if he still was in his previous thought. His wide face seemed at its least alert, which I’d learned meant just the opposite. I felt it would be a taut and grueling afternoon. He saw my hat. Surely it would only be a goad to him, to challenge me even more fiercely.

  “Good afternoon,” he said, and I answered the same. “Sit down,” he said, and I did. This was the formula. I took my seat, and set my hat on the table. It was a little more between us than to the side where I would usually position it.

  “Have you done your exercises?” He always asked this, and even the turmoil of the last week didn’t change what would be asked, or that I’d done the work. I handed him my papers and he looked at them critically. If there was ever an error, he would see it immediately and tell me with no attempt at gentleness. I would want no attempt; it was my fear of his rebuke that drove me to perfection. At least, that was one motive: I was driven by the rebukes I gave myself just as strongly.

  “Yes, that is correct,” he said after his study. “What have you been reading this week?” It was as if he knew. To tell him Uncle Jacob had been in my hands just an hour previous would have been effrontery.

  “Boccaccio. Master Desiderius lent it to me.”

  “When?”
r />   “Tuesday morning.” Before the coach arrived, and before the Town Council meeting.

  “It’s satire. What did you learn from it?” He sometimes asked such questions, for him to learn about me.

  “About death, sir. Black Death’s not a subject for satire this week.”

  “It is not.” That was the first acknowledgment he’d given of the events of the week. “At another time you might find that the book is much about life.” He paused, and searched me. “You grieve for Master Huldrych, don’t you?”

  “Yes, sir. Very much.”

  “I do also,” he said, and that was the completion of our beginning. But I knew him this well, that those three words were complete truth, and his grief was as deep as mine; but he was a deeper man than me. “We will begin a new discussion today.”

  Those were words that were even more wondrous than the offer of a book by Master Desiderius, or the prospect of a long, empty road begging to be run. To my great discredit, the thought of Master Huldrych stood back out of the light. But perhaps that was Master Johann’s intent, to help me with my grief.

  So I set my pen at the ready. How I loved to write! White paper was a heaven for me, and the beginning of a new Mathematic subject was like angels singing. I wasn’t irreligious at all thinking this. God would be worshipped sublimely in sublime things.

  “Consider a polynomial of the fifth degree.”

  “Yes, sir. A specific one?”

  “That it was created from five known roots.”

  And we were off. He would ask questions and I would answer if I could. He would push until I could not. “In what way does it inflect? What is a description of its differential polynomial? How does Leibniz find the maximum values?” And here, already reeling and breathless, I had to also remember to use only Leibniz’s words and none of Newton’s.

  “The maximum occurs where the ratio of infinitesimals is zero,” I answered.

  “What is the meaning of that ratio?”

  “It is how the value of the polynomial changes as the independent varies. When the ratio is zero, the polynomial is neither increasing nor decreasing, and it has reached a point between those two. If it is increasing to the point, and decreasing after it, it must be a maximum.”

  “Must be? In every case?”

  “It might also be a minimum.”

  “Every case is one or the other?” He was like a wolf with his jaws clenched on the neck of a sheep. I was the sheep.

  “Yes. It must be a maximum or minimum.” In that instant I knew I was wrong.

  “What if the ratio of infinitesimals is itself at a minimum or maximum, even at zero?”

  I was dizzy. “It would . . . the polynomial would increase to a point, come to level, but then increase again.” I took a deep breath. “Or decrease, and decrease,” I added hastily.

  “Yes.” He nodded, and also breathed. “And there are more cases where each following differential polynomial is itself such a leveling case.”

  There was a book on my shelf, Analyse des infiniment petits pour l’intelligence des lignes courbes, by Monsieur de l’Hopital of Paris, written nearly thirty years ago. Some ten years before that, the great Master Leibniz had published in the Acta Eruditorum his article Nova Methodus pro Maximis et Minimis, itemque Tangentibus, on the Calculus, the first ever published. Monsieur de l’Hopital, certainly a great Mathematician himself, failed to comprehend it and hired for himself a tutor, a young man then, my Master Johann, who began a correspondence with Paris and instructed his elder. He was likely one of only three men in the world who could have: who had both the understanding of the material, teased from Master Leibniz’s very obscure Latin, and also the ability to teach it. The other two were his brother, Master Jacob, and Mr. Newton in England.

  Monsieur de l’Hopital then himself published Master Johann’s notes, with just the barest attribution to their true author, as the first textbook in the world on the Calculus. I’d read Analyse des infiniment petits and I recognized it to be thoroughly Master Johann’s own work. Only after de l’Hopital’s death did Master Johann make his claim that the book was actually his. It was Daniel’s opinion that de l’Hopital had paid Master Johann a princely sum for his silence. If that were true, Monsieur de l’Hopital at least for his own lifetime had purchased, and Master Johann sold, a very great renown. Now, though, all of that fame and prestige has returned. And Master Johann has only increased his, and all the world’s, understanding of this vast new continent of Mathematics.

  And this was the man who was before me now, teaching me the Calculus. His explanations of it over the last years had always been so lucid and straight. It has all seemed so simple to me, but I knew that it was only because I had been taught so well. When I would describe the mysteries to another student, they seem to understand nothing of it. They would only shake their heads at my gibberish. I was a very poor teacher.

  On we went, and on and on, and as always I’d lost all track of time: of the clock and even of the calendar. Then there was always the sudden moment when he rubbed his hands and leaned back in his chair. This was when he would give me my assignment for the following week. I was already exhausted, but now had to pay the closest attention of all. But this time he didn’t tap my papers and show me what from them I was to work on. Instead, he pulled out a paper of his own, but didn’t show it to me.

  “Let us address an issue of a series of infinite numbers.”

  This was quite different from what we’d been discussing. He nodded to me and I took up my pen and ink again.

  “A sum,” he said. “One half, one fourth, one eighth, one sixteenth, one thirty-second, one sixty-fourth, and on. An infinite series. What is the sum?”

  ¹⁄₂ + ¹⁄₄ + ¹⁄₈ + ¹⁄₁₆ + ¹⁄₃₂ + ¹⁄₆₄ + . . .

  “Exactly one.”

  “And how is that? An infinite count of numbers, and they add to a finite sum?”

  “Yes, sir. Because they grow infinitely small.” This was very plain, and we had discussed it long ago. He was plotting something. He wouldn’t have asked such a simple question unless he had a difficult plan.

  “Then one half,” he said, “one third, one fourth, one fifth, one sixth, one seventh, and on. An infinite series. What is the sum?”

  ¹⁄₂ + ¹⁄₃ + ¹⁄₄ + ¹⁄₅ + ¹⁄₆ + ¹⁄₇ + . . .

  “The sum is infinite.”

  “But they also grow infinitely small?”

  “But for this series, not as quickly. Not as quickly as the sum grows infinite.”

  “That is correct.”

  Oh, he had my interest piqued entirely. I thought he might really be a show-man, the way he drew out a puzzle and pulled his student into it.

  “How do you know that the first series added to exactly one?”

  “There is a method,” I said. This was still all very simple. He knew the method, of course. “If the series is multiplied by two, it becomes one, one half, one fourth, one eighth, and on. If the two are subtracted, the infinity of terms is cancelled, and the remaining value is one. So, the series subtracted from twice the series is the value of the series, which is one.”

  “Does this method work for any infinite series?”

  “No, sir. Only for this geometric type.”

  “Now write these numbers.”

  Then I knew, from his voice, that this was to be the challenge. The first questions had only been to set his stage, and now he was ready to play his drama. “One, one fourth, one ninth, one sixteenth, one twenty-fifth, one thirty-sixth, and on. What are these?”

  ¹⁄₁ + ¹⁄₄ + ¹⁄₉ + ¹⁄₁₆ + ¹⁄₂₅ + ¹⁄₃₆ + . . .

  “They are one over the square of one, one over the square of two, one over the square of three, one over the square of four, and on. They are Reciprocal Squares.”

  “Yes, Reciprocal Squares.” It was roast veal and wine, exquisite, the way he said it. “They are Reciprocal Squares. Is the sum infinite? Or finite?”

  “Finite,” I said
, though I paused to think. “Yes. Finite. Besides the beginning one, the numbers are each smaller than the first series you listed. One fourth is smaller than one half, one ninth is smaller than one fourth, one sixteenth is smaller than one eighth, and on. So if the first sum was finite, this must be also.”

  “Yes. Finite. Very good, though that was simple.” And he paused, and his pause was perfect in length and depth and width. “And what is that finite sum of the infinite Reciprocal Squares?” he asked.

  “The sum . . .” I was bewildered. I stared at the numbers on my paper and tried to make sense. I looked at the pattern of them, at what they seemed to be adding to, at the other methods I knew, anything. They seemed very simple, as simple as the other sums we’d done. And finally I grasped that . . . “I don’t know.” I realized minutes had gone by. “What is the sum?”

  “What is it?” He rubbed his hands. “No one knows. It is a number, somewhat larger than one and a half, and less than two. It has been calculated to a close value. But no one knows what it really is. Perhaps it’s no particular number at all, just a number. But it should be something more important than that. A squared root, a cubed root, a ratio of important numbers. No one knows.”

  “It would be something . . . surprising,” I said.

  “Perhaps. Perhaps. And now, the Paris Academy has issued a challenge to anyone in Europe who might discover the true value of the Reciprocal Squares.” He’d kept the paper in his hand closed from me; now he opened it. “Monsieur Fontenelle and Monsieur de Molieres of the Academy are very great Mathematicians. I have instructed them myself. Their challenge is to all Europe.”

  I was reading. The page was in Latin, of course, and it was just as he’d said. The two men were members and directors of the Royal French Academy, and their announcement was as weighty as a mountain: to explain the meaning of the Sum of the Infinite Reciprocal Squares.

  Master Johann was a member of the Royal Academy; if he were not, the Academy would hardly have been worth anyone else’s membership. Therefore, he had received the first copy of the challenge, and soon it would go out to all the rest of Europe. And whoever first solved it, or proved it unsolvable, would instantly leap to the highest rank of Mathematicians, if he wasn’t already there. And I was being given this glimpse into their world.