“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 + . . .