“What?”
“I don’t think any of them truly understand it. Even Master Johann. When I explain it, they just touch it but they don’t grasp it. So I wonder if I’m so poor at even explaining.”
“How did you first conceive this proof, Leonhard? How did you invent a proof that exceeds great men’s understanding?”
“I don’t know. It was just there. It was given.”
“There might be a purpose,” she said.
And last, before sleep, I chose to read Leibniz in the morning. I would be put into the right mind for my lesson with Master Johann, and I would have opportunity to think on the question of how Mathematicians become great.
13
The Logarithmic Spiral
It had been three years before, when I was fifteen, that I wrote my thesis on Descartes. It was precocious of a child who knew so little to attempt so much. I blushed now to think of my gray-haired Masters, steeped in decades of long thought, regarding my chubby cheeks and naïve, earnest eyes as I, the dwarf, described my opinions of giants. And even more, that I compared Newton with Descartes, to the faculty of Basel, the Mathematical hotbed of anti-Newtonism, and the theological nemesis of Cartesians. It was a miracle that I even still existed! But those generous and patient professors allowed me past, and Master Johann, who could have sunk the whole battleship with one raised eyebrow or finger tap of annoyance, instead allowed me passage across his cannon, and himself signed the parchment of my Master’s degree. For a while afterward a few people called me Master, but I’d never wanted that. Even with a tricorne, I still felt I was no Master.
Between those two, Newton and Descartes, I believed Mr. Newton to be more correct; and three years later, I was more convinced. Many criticisms of Mr. Newton complained that his theories were too precise: they applied such a severe Mathematical exactitude to the beauty and gentle motions of nature. This was, besides, the completely invisible Gravity that he proposed, and others rejected as absurd. But the comparison was to Monsieur Descartes, who believed that the only truth was what we experienced, and could know with our own senses. If the universe was to be measured by Mathematic rules, or else by man’s experience, then I knew that Mathematics was superior. I believed that the universe would exist and follow its Mathematic orbits even if no man ever lived to see it. I even believed that if there was no universe, the laws of Mathematics would still exist. And I thought Mathematics was beautiful itself, and added to the exquisite harmony of creation rather than degrading it. And for Gravity, well, I very greatly believed in invisible things.
It was still early on Saturday, though already the day was warm, when I sat at my desk to read Herr Leibniz. I’d read Nova methodus pro maximis et minimis many, many times, but it still put me in the best frame for my meeting with Master Johann. The two had been correspondents and friends. I thought this was one real reason that Master Johann sided so passionately with Herr Leibniz over Mr. Newton on the discovery of the Calculus. His passion had carried all the Mathematicians in the continent with him, turning the whole continent of Europe against England, and England against Europe.
It was strange that a subject as perfect and rational as Mathematics could stir disagreements and conflict. Yet it did in two ways: first, in discovery, where a man’s pride would lead him to grasp a new theorem as a dog did a bone, and yield to no other that it was his original; and second, in proof, where claims might be shaky or unfounded, and the rigorous deduction to prove or disprove was beyond knowledge. So, the Newton camp and the Leibniz camp had warred over which champion first discovered the Calculus. Then, there was a civil war within each camp over what had actually been truly proven and what had not but would be, and what was actually false.
Then, there was a third hostility that arose from Mathematics, founded in the first two conflicts, and that was outright, which was the pursuit of Publications, of Eminence and Esteem, and most of all, of Chairs. So reading Leibniz, I thought much more about these qualities of humanity, and not as much about the qualities of polynomials.
On my dresser, beside my friend the wood head, and my marvel, the conch shell, and the small charred slat with the Logarithmic spiral, I had two other items, my bowls. They were pottery and seemed very normal. Each was only eight inches from rim to rim and somewhat shallow, just a few inches deep, with the sides becoming steep at their outer edge. They might even have seemed identically shaped, though they weren’t. In color they were plain but had interiors which were very polished. However, each had its own unique and amazing property: one was brachistochronic, the other was tautochronic.
The tautochrone was the easier to demonstrate. Set a small flat pebble that would slide on the smooth surface, or a very round pebble that would roll, near the rim and it would accelerate down and quickly travel the long distance to the center. Place it farther from the rim, closer to the center, where the slope was less, and it would accelerate more slowly, but would still reach the center quickly because it was closer. Indeed, place the stone anywhere in the bowl, and the combination of the steepness and distance from the center at any point would contribute to the stone’s reaching the center in the exact same time. Tautochrone meant identical time. Set two stones, or three, or as many as would fit and could be held, anywhere in the bowl from rim to nearly the center, and after they were all let go at the same time, they would all reach the center simultaneously.
That the second bowl was a brachistochrone was more difficult to demonstrate. In this bowl, the pebble set anywhere would reach the center faster than it would in a bowl of any other shape. If the side were steeper, the stone could accelerate more quickly but have a greater distance to travel; if it were more shallow, the reverse applied. Brachistochrone meant the fastest time.
These bowls showed another reason why Master Johann was so hostile to Mr. Newton.
The shape of either bowl could only be derived with a special set of the Calculus. The shapes were first sought by Galileo nearly a hundred years ago, but were beyond the Mathematics of the time. Only after Herr Leibniz published his Nova Methodus were the necessary theorems available. The equation of the tautochrone was fairly easy. But Master Johann, thirty years ago, and only a little older than I was now, published the first solution to the brachistrochrone, which was much more difficult. Yet only his solution was correct.
His proof was not. Often in Mathematics this would happen. It would be the same as stating that the source of the Barefoot Square fountain was the Birsig, which I believed was true but was not certain; and having given as my proof that it was water in the fountain, and it was water in the stream, so the one must feed the other. This would not be a valid proof, and neither were Master Johann’s calculations of the brachistochrone.
This soon came to light, and Master Johann’s proof was invalidated. A challenge was sent out to the Mathematicians of Europe in the journal Acta Eruditorum to find the correct proof, and also presented several variations to the problem. The challenge was actually posted by Master Johann himself, but anonymously, as he had a scheme to remove the stain on his reputation.
Only five men among all then living could have solved the problem, and they each did. Herr Leibniz solved it, of course. Monsieur L’Hospital also sent in a solution, though it was apparent that he’d corresponded with Herr Leibniz and had mainly repeated that Master’s answer.
Two others able were the brothers Jacob and Johann. Master Jacob solved it in an original way, and as the two brothers were then still cordial, he showed his proof to Master Johann. Master Johann, expecting this, took his brother’s proof and attempted to pass it as his own.
All of these proofs were published in the Acta, anonymously as always, though each of the men quickly recognized the others’ work. Accusations flew and the brothers were permanently sundered. They’d all spent days and weeks at their calculations, each sure of his own unique genius.
There was, though, a fifth proof published in that same edition, superior actually to all the others. Mr. Newton in Eng
land, it was said, had solved the problem in a few moments the very night he received the journal, after a tiring day of work. Though the attribution was anonymous, the notations made it obvious that they’d all been bested by the Englishman.
As Master Johann said when he read it, “I recognize the lion by his paw.” The saying had persisted in his family. The feelings between them had also persisted.
After Leibniz, I read Newton. The Principia was a book that would stand forever: it bested all his peers, just as his proof had bested them. Whether Mr. Newton or Herr Leibniz first understood the Calculus, I did not know. I did know that Mr. Newton’s work reached farther, and his principal that Mathematics ruled, and also explained, the motions of both planets and pebbles, of both raindrops and rainbows, was a new beginning of history. He would be famous forever. Whenever he died, and he was now very old, his fame would only keep growing.
My Master Johann would also hold a place in history. Again whether Mr. Newton or Herr Leibniz first discovered the Calculus, it was without doubt that everyone who now knew the subject learned it through Master Johann’s explanation. His books were not on the same pinnacle as the Principia, but they would always be known. He had the renown that Master Faust sought.
Daniel and Nicolaus and Gottlieb would also be remembered. In Mathematics, their family had been a constellation. Daniel wanted to outshine his father and would have paid a dear price to do it, and Nicolaus had more ambitions than he showed. And, in some obscure journal, an author might even be remembered for the first proof of the Reciprocal Squares.
It seemed to be the warmest day yet of the spring, and the sky was cloudless. All sound was deadened.
When I was admitted into Master Johann’s house, there was a thickness to the silence. I could hardly hear my own footsteps in the stairs, and Mistress Dorothea’s knock on the inner door was like a leaf fall. The call from within sounded as from a grave.
“What have you been reading this week?” he asked, and he seemed much older and worn.
I was able truthfully to give an answer pleasing to him. “Herr Leibniz.”
Though less, he was still omniscient and omnipotent in his dark room. He’d known. “I have been, also,” he said. “Even now.”
“The Nova Methodus?” I didn’t see a book on the table, only some papers. But these were what he set his hand on as he answered.
“Our correspondence.”
“From himself?” I asked, awed. I knew, of course, that there had been many letters between them, but these were those letters themselves.
“I was younger then. Leonhard, why do you pursue Mathematics?”
“I can’t not.”
“How does Mr. Newton describe the study of Mathematics?”
This was disturbing. It may have been the first time I’d ever heard that name from my Master’s lips. It was dangerous, as well. To deny my respect would be foolish, as he certainly remembered my thesis of three years before. But to show too much admiration would also be unwise. “The Principia states that Mathematics explains the revolving of earth and the motion of water and the colors of light,” I said.
“How can Mathematics do this?”
“I believe there are deep laws that govern motion and substance.”
“So is that Mathematics?” he asked, and I sensed his dissatisfaction with my answer. “It is just a principal of natural philosophy?”
“No. It is more.”
“Then what?”
“I think that natural philosophy, and Physics, and all of such things are built on Mathematics like a castle is built on a mountain. There is more to the mountain than just the portion that holds the castle.”
“There is more of Mathematics,” he said, and now he was satisfied, “than is used in the earth and heavens.”
“I believe so. Master, what do you understand that Mathematics is?”
“It is an invisible world,” he answered. “Greater and deeper than anything we see.”
“I do see invisible things,” I admitted. “Sometimes.”
“Then you see that world,” he said, “also.”
“Where did it come from?”
“Where?” He hadn’t thought that question before. “It would always have been. The world we see is created on it.”
“But could it have always been? Even the invisible world must have been created, also.”
“The created world is as it was chosen to be,” he said. “Mountains and rivers are where they are, but they could have been otherwise. Mathematics can only be as it is. There is no other possibility. So, was it created?”
I tried to think of a Mathematics different from the one I knew. Could one and two add to something other than three? Or did they only because that was how they were created to be?
“That would mean they are beyond the Creator.”
“We will keep our discussion on Mathematics and Mathematicians.”
“Yes, Master. But then I have another question. What makes a Mathematician great?”
“If he discovers great things. Then he is known.” He looked at me a moment. “Is that what you mean?”
“No, sir. What would make him able to discover great things?”
“What are his qualities? Is that what you mean?”
“Yes, sir.”
He sat back into the silence of the room. “He sees.”
“Into the invisible.”
“Yes.” Then Master Johann laughed. “But then he must also publish and he must become known. A great Mathematician who is unknown is not great.”
So I plunged forward. “Then I wish to send my proof to Paris.”
“To Paris? Monsieur Fontenelle will deride and dismiss the work of a young student. Monsieur Molieres would glance through it and find every fault that I do.”
“It would be treated respectfully if you wrote a letter of recommendation for it.”
“Yes, it would.” He did not deride me. He even seemed unsurprised that I had asked. “But should I? I am not fully convinced of it myself.” The one word though, fully, showed that he might soon be. He considered me, his young supplicant, in the same way he had the last Saturday, and it seemed that he was seeing something new.
My mother’s father was pastor of Saint Leonhard’s for three decades. He died before I was born. My grandmother didn’t speak often of him, as she was not given to reminiscing. Mostly she used him as an example to me of a sound, pious, dedicated, and competent man. Everyone else who remembered him spoke more highly even than she did.
This pastor’s daughter, my mother, married a pastor. He was a young man of good family and humble means, unshakable in his faith and in every part of his life. Beneath his calm demeanor, though, his heart burned with a slow, steady fire so hot that all the flames of the Boot and Thorn would be just a sputtering candle beside it.
For several years he had the pulpit at Saint Leonhard’s. It was even now still warm from him. When his son was born, the child was named for his father’s parish. Then, instead of taking a higher perch at the Munster or beyond, this man took his wife and child out of Basel to pastor a village in the hills north of the city and settled into its smallness. Whether the cottages of Riehen ever knew what great spirit they had, I would not be certain. He was certain of the choice he was making. He was my father. I learned from him that God was not served by our greatness but by our humbleness.
On Sunday morning I took my grandmother to church, and in the preaching we were instructed on the very greatness of God: his righteousness, his love, and his sacrifice. I would never tire of that lesson.
Monday morning, Mistress Dorothea poured words as a fountain pours water. Basel’s fountains provided a greater flow than the citizens need, so that a great deal simply flowed over the basin and into the streets; and Mistress Dorothea had more to say than there were ears to hear.
But as I was finishing, the fountain stopped. “Leonhard.”
“Yes, Mistress Dorothea?” I said.
“Master Johann has instru
cted me that you are no longer to be obliged to perform chores in this house.”
“Not obliged?” I was dumbfounded.
“From this moment on.”
“But . . . Mistress Dorothea . . . why not?”
“I don’t question the Master’s reasons.”
I was trying to understand, and reeling, though I also did notice that she’d let me put in my full morning’s labor before she’d said this. “Then . . . am I no longer to come on Saturdays?”
“You’ll need to discuss this with Master Johann. He is expecting you upstairs.”
“Come.” This was the third Monday Master Johann had answered my knock on his door.
It seemed more difficult each time to concentrate on speaking. The room was too marvelous. “Mistress Dorothea said I was no longer obliged to do chores for her in the mornings.”
“That is true.” From where I stood and how he was seated, I couldn’t see his desk well. There were just a few papers visible at the edge.
“Then I’ll no longer come on Saturdays?”
“You will still come.”
“But, Master, how will I pay you for the time?”
“I’ve written a letter to your father. From the present, your lessons will be at my pleasure. They will not be in exchange for tuition or service.”
“Thank you, sir . . . but I can’t. It would be unearned. An imposition.”
He shook his head. “It is my decision.”
“Yes, sir. Master Johann?”
“Yes?”
“May I still be allowed to do chores here nonetheless?”
I saw no reaction. He just said, “You may request that of Mistress Dorothea.”
“Yes, yes, Leonhard,” was Mistress Dorothea’s answer to that request. “I’ll keep you for my kitchen, for the present. But there’s a time coming when it won’t be fitting.”
An Elegant Solution Page 27