An Elegant Solution(101)

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.