8.3 The Helen of Geometers

The curve traced out by a point on the rim of a rolling circle is called a cycloid, and we've seen that this curve described gravitational free-fall, both in Newtonian mechanics and in general relativity (in terms of the free-falling proper time). Remarkably, this curve has been a significant object of study for almost every major scientist mentioned in this book, and has been called "the Helen of geometers" because of all the disputes it has provoked between mathematicians. It was first discussed by Charles Bouvelles in 1501 as a mechanical means of squaring the circle. Subsequently Galileo and his student Viviani studied the curve, finding a method of constructing tangents, and Galileo suggested that it might be a suitable shape for an arch bridge.

Mersenne publicized the cycloid among his group of correspondents, including the young Roberval, who, by the 1630's had determined many of the major properties of the cycloid, such as the interesting fact that the area under a complete cycloidal arch is exactly three times the area of the rolling circle. Roberval used his problem-solving techniques in 1634 to win the Mathematics chair at the College Royal, which was determined every three years by an open competition. Unfortunately, the contest did not require full disclosure of the solution methods, so the incumbent (who selected the contest problems) had a strong incentive to keep his best methods a secret, lest they be used to unseat him at the next contest. In retrospect, this was not a very wise arrangement for a teaching position. Roberval held the chair for 40 years, but by keeping his solution methods secret he lost priority for several important discoveries, and became involved in numerous quarrels. One of the men accused by Roberval of plagiarism was Torricelli, who in 1644 was the first to publish an explanation of the area and the tangents of the cycloid. It's now believed that Torricelli arrived at his results independently. (Torricelli served as Galileo's assistant for a brief time, and probably learned of the cycloid from him.)

In 1658, four years after renouncing mathematics as a vainglorious pursuit, Pascal found himself one day suffering from a painful toothache, and in desperation began to think about the cycloid to take his mind off the pain. Quickly the pain abated, and Pascal interpreted this as a sign from the Almighty that he should proceed to study the cycloid, which he did intensively for the next eight days. During this period he rediscovered most of what had already been learned about the cycloid, and several results that were new. Pascal decided to propose a set of challenge problems, with the promise of a first and second prize to be awarded for the best solutions. Roberval was named as one of the judges. Only two sets of solutions were received, from Antoine de Lalouvere and John Wallis, but Pascal and Roberval decided that neither of the entries merited a prize, so no prizes were awarded. Instead, Pascal published his own solutions, along with an essay on the "History of the Cycloid", in which he essentially took Roberval's side in the priority dispute with Torricelli.

The conduct of Pascal's cycloid contest displeased many people, but it had at least one useful side effect. In 1658 Christiaan Huygens was thinking about how to improve the design clocks, and of course he realized that the period of oscillation of a simple pendulum (i.e., a massive object constrained to moving along a circular arc under the vertical force of gravity) is not perfectly independent of the amplitude. Prompted by Pascal's contest, Huygens decided to consider how an object would oscillate if constrained to follow an upside-down cycloidal path, and found to his delight that the frequency of such a system actually is perfectly independent of the amplitude. Thus he had discovered that the cycloid is the tautochrone, i.e., the curve for which the time taken by a particle sliding from any point on the curve to the lowest point on the curve is the same, independent of the starting point. He presented this result in his great treatise "Horologium Oscillatorium" (not published until 1673), in which he clearly described the modern principle of inertia (the foundation of relativity), the law of centripetal force, the conservation of kinetic energy, and many other important concepts of dynamics - ten years before Newton's "Principia".

The cycloid went on attracting the attention of the world's best mathematicians, and revealing new and remarkable properties. For example, in June of 1696, John Bernoulli issued the following challenge to the other mathematicians of Europe:

Pictorially the problem is as shown below:

In accord with its defining property, the requested curve is called the brachistochrone. The solution was first found by Jean and/or Jacques Bernoulli, depending on whom you believe. (Each of the brothers worked on the problem, and they later accused each other of plagiarism.) Jean, who was never accused of understating the significance of his discoveries, revealed his solution in January of 1697 by first reminding his readers of Huygens' tautochrone, and then saying "you will be petrified with astonishment when I say that precisely this same cycloid... is our required brachistochrone".

Incidentally, the Bernoulli's were partisans on the side of Leibniz in the famous priority dispute between Leibniz and Newton over the invention of calculus. Before revealing his solution to the brachistochrone challenge problem, Jean Bernoulli along with Leibniz sent a copy of the challenge directly to Newton in England, and included in the public announcement of the challenge the words

It seems clear the intent was to humiliate the aging Newton (who by then had left Cambridge and was Warden of the Mint), by demonstrating that he was unable to solve a problem that Leibniz and the Bernoullis had solved. The story as recounted by Newton's biographer Conduitt is that Sir Isaac "in the midst of the hurry of the great recoinage did not come home till four from the Tower very much tired, but did not sleep till he had solved it, which was by 4 in the morning." In all, Bernoulli received only three solutions to his challenge problem, one from Leibniz, one from l'Hospital, and one anonymous solution from England. Bernoulli supposedly said he knew who the anonymous author must be, "as the lion is recognized by his print". Newton was obviously proud of his solution, although he commented later that "I do not love to be dunned & teezed by forreigners about Mathematical things..."

It's interesting that Jean Bernoulli apparently arrived at his result from his studies of the path of a light ray through a non-uniform medium. He showed how this problem is related in general to the mechanical problem of an object moving with varying speeds due to any cause. For example, he compared the refractive problem with the mechanical problem whose density is inversely proportional to the speed that a heavy body acquires in gravitational freefall. "In this way", he wrote, "I have solved two important problems - an optical and a mechanical one...". Then he specialized this to Galileo's law of falling bodies, according to which the speeds of two falling bodies are to each other as the square roots of the altitudes traveled. He concluded

Presumably his enthusiasm would have been even greater had he known that the same curve describes radial gravitational freefall versus proper time in general relativity, and one certainly sees here the beginnings of Maupertius least action principle. Interestingly, Fermat himself was much less philosophically committed to the principle that he himself originated (somewhat like Einstein with the quantums of light). After being challenged on the fundamental truth of the "least time" principle as a law of nature by the Cartesian Clerselier, Fermat replied in exasperation

Fermat was content to regard the principle of least time as a purely abstract mathematical theorem, describing - though not necessarily explaining - the behavior of light.