The Challenge of the Brachistochrone
William Dunham

With this and so many other contributions, the Bernoulli brothers left a significant mark upon mathematics of their day. But one additional tale must be told of these cantankerous, competitive, and contentious brothers, a story that is surely one of the most fascinating from the entire history of mathematics.

It began in June of 1696 when Johann Bernoulli published a challenge problem in Leibniz’s journal Acta Eruditorum. Obviously, a legacy of public challenge remained from the days of Fior and Tartaglia. Although contests were now conducted in the sedate pages of scholarly journals, they retained their power to make or break reputations, as Johann himself observed:

… it is known with certainty that there is scarcely anything which more greatly excites noble and ingenious spirits to labors which lead to the increase of knowledge than to propose difficult and at the same time useful problems through the solution of which, as by no other means, they may attain to fame and build for themselves eternal monuments among posterity.

Johann’s particular challenge was a good one. He imagined points A and B at different heights above the ground and not lying one directly above the other. There is certainly an infinitude of different curves connecting these two points, from a straight line, to an arc of a circle, to any number of other wavy, undulating paths. Now imagine a ball rolling from A down to B along such a curve. The time it take to complete the trip depends, of course, on the curve’s shape. Bernoulli challenged the mathematical world to find that one particular curve AMB along which the ball will roll the shortest time. He called this curve the “brachistochrone” from the Greek words for “shortest” and “time”.

An obvious first guess is to take AMB as the straight line joining A and B. But Johann cautioned against this simplistic approach:

… to forestall hasty judgment, although the straight line AB is indeed the shortest between the points A and B, it nevertheless is not the path traversed in the shortest time. However the curve AMB, whose name I shall give if no one else discovered it before the end of this year, is one well-known to geometers.

Johann gave the mathematical world until January 1, 1697, to come up with a solution. However, when his deadline arrived, he had received but one solution, from the “celebrated Leibniz”, who

has courteously asked me to extend the time limit to next Easter in order than in the interim the problem might be made public … that no one might have cause to complain of the shortness of the time allotted. I have not only agreed to this commendable request but I have decided to announce myself the prolongation and shall now see who attacks this excellent and difficult question and after so long a time finally masters it.

At this point, Johann waxed enthusiastic about the rewards of solving his brachistochrone problem. Recalling that he himself knew the solution, one finds his remarks about the glories of mathematics a bit self-serving:

Let who can seize quickly the prize which we have promised to the solver. Admittedly this prize is neither of gold nor silver, for these appeal only to base and venal souls. … Rather, since virtue itself is its own most desirable reward and fame is a powerful incentive, we offer the prize, fitting for the man of noble blood, compounded of honor, praise, and approbation …

In this passage, it sounds as though Johann was setting himself up for another triumph over poor Jakob. But he had a different target more squarely in his sights. Wrote Johann:

… so few have appeared to solve our extraordinary problem, even among those who boast that through special methods … they have not only penetrated the deepest secrets of geometry but also extended its boundaries in marvellous fashion; although their golden theorems, which they imagine known to no one, have been published by others long before.

Could anyone doubt that the “golden theorems” he referred to were the techniques of fluxions, or that the object of his scorn was none other than Isaac Newton himself, a man who claimed to have known about calculus long before Leibniz had published it in 1684? Then, to leave no doubt about the explicit nature of his challenge, Johann put a copy in an envelope and mailed it off to England.

Of course, in 1697, Newton was deeply involved with matters of the Mint, and, as he himself admitted, he no longer felt the agility of mind that characterized his mathematical heyday. Newton was then living in London with his niece, Catherine Conduitt, and she picks up the story:

When the problem in 1697 was sent by Bernoulli — Sir I.N. was in the midst of the hurry of the great recoinage and did not come home till four from the Tower very much tired, but did not sleep till he had solved it, which was by four in the morning.

Even late in life and tired from a hectic day’s work, Isaac Newton triumphed where most of Europe had failed! It was a remarkable display of the powers of the great British genius. He had clearly felt his reputation and honor were on the line; after all, both Bernoulli and Leibniz were waiting in the wings to publish their own solutions. So Newton rose to the occasion and solved the problem in a matter of hours. Somewhat exasperated, he is reported at one point to have said, “I do not love … to be … teezed by foreigners about Mathematical things.”

Back in Europe, as Easter neared, a few solutions came into the hands of Johann Bernoulli. The curve that everyone was seeking — one that “is well-known to geometers” — was none other than an upside-down cycloid. As we have noted, this important curve was studied by Pascal and Huygens, but neither of these mathematicians had realized that it would also serve as the curve of quickest descent. Johann wrote with characteristic hyperbole, “… you will be petrified with astonishment when I say that precisely this cycloid … of Huygens is our required brachistochrone.”

On Easter, the challenge period had expired. All together, Johann had received five solutions. There was his own and the one from Leibniz. His brother Jakob came through (perhaps to Johann’s dismay) with a third, and the Marquis de l’Hospital added a fourth. Finally, there was a submission bearing an English postmark. Opening it, Johann found the solution correct, although anonymous. He clearly had met his match in the person of Isaac Newton. Although unsigned, the solution bore the unmistakable signs of supreme genius.

There is a legend — probably of dubious authenticity but nonetheless of great charm — that Johann, partially chastened, partially in awe, put down the unsigned document and knowingly remarked, “I recognize the lion by his paw.

From William Dunham, Journey Through Genius: The Great Theorems of Mathematics, Wiley, 1990, page 199-202.

 created: 2011-10-15 18:25 updated: 2013-05-08 13:30