“Several pages. I’ve thought more about waves and that they can go on without stopping. It was Master Johann’s puzzle of the Reciprocal Squares that led me to these ideas.”
“What does he think now of your solution?” She knew this wouldn’t be a simple answer.
“I believe it troubles him. He’s very intrigued by it. But doubtful, too, of course.”
“Are you still sure your solution is correct?”
“I’m sure.”
“If it is true, can it be proved false?”
“No. Mathematics doesn’t allow such a thing. All the principles of Mathematics are true once they’re proven, and Mathematicians accept proofs if they’re according to the principles. That’s part of the elegance of Mathematics. It is only made of proofs, and nothing else. There are a very few simple truths that are accepted as themselves, and all the rest is proved from them. I think there’s no end to it.”
“And will Master Johann be persuaded, then?”
“He will, I think.”
“Then Mathematics is very different from most other things,” she said, “if it can overcome a stubborn man’s beliefs. What troubles him with this proof?”
“It might be that it’s unlikely that a young, poor, un-noteworthy student would solve it correctly.”
“But if it is true, it couldn’t matter who first proved it.”
“It couldn’t. It may be that he is just jealous. But he’s too great a man to be resentful.” I tried to be sincere, but she knew he was very capable of resentment. “And Daniel accepts it. And Nicolaus and Gottlieb have doubts but still accept that it might be true. But also . . .” I was afraid to say it. It was such a boast on my part.
“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.