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By Root 755 0

should I conceal the truth?) to solve it in full…It is true that it cost me study that robbed me of rest for an entire night…but the next morning, filled with joy, I ran to my brother, who was still struggling miserably with this Gordian knot without getting anywhere, always thinking like Galileo that the catenary was a parabola. Stop! Stop! I say to him, don’t torture yourself any more to try to prove the identity of the catenary with the parabola, since it is entirely false…But then you astonish me by concluding that my brother found a method of solving this problem…I ask you, do you really think, if my brother had solved the problem in question, he would have been so obliging to me as not to appear among the solvers, just so as to cede me the glory of appearing alone on the stage in the quality of the first solver, along with Messrs. Huygens and Leibniz?

In case you ever needed proof that mathematicians are humans after all, this story amply provides it. The familial rivalry, however, does not take anything away from the accomplishments of the Bernoullis. During the years that followed the catenary episode, Jakob, Johann, and Daniel Bernoulli (1700–1782) went on not only to solve other similar problems of hanging cords, but also to advance the theory of differential equations in general and to solve for the motion of projectiles through a resisting medium.

The tale of the catenary serves to demonstrate another facet of the power of mathematics—even seemingly trivial physical problems have mathematical solutions. The shape of the catenary itself, by the way, continues to delight millions of visitors to the famous Gateway Arch in St. Louis, Missouri. The Finnish-American architect Eero Saarinen (1910–61) and the German-American structural engineer Hannskarl Bandel (1925–93) designed this iconic structure in a shape that is similar to that of an inverted catenary.

The astounding success of the physical sciences in discovering mathematical laws that govern the behavior of the cosmos at large raised the inevitable question of whether or not similar principles might also underlie biological, social, or economical processes. Is mathematics only the language of nature, mathematicians wondered, or is it also the language of human nature? Even if truly universal principles do not exist, could mathematical tools, at the very least, be used to model and subsequently explain social behavior? At first, many mathematicians were quite convinced that “laws” based on some version of calculus would be able to accurately predict all future events, large or small. This was the opinion, for instance, of the great mathematical physicist Pierre-Simon de Laplace (1749–1827). Laplace’s five volumes of Mécanique céleste (Celestial Mechanics) gave the first virtually complete (if approximate) solution to the motions in the solar system. In addition, Laplace was the man who answered a question that puzzled even the giant Newton: Why is the solar system as stable as it is? Newton thought that due to their mutual attractions planets had to fall into the Sun or to fly away into free space, and he invoked God’s hand in keeping the solar system intact. Laplace had rather different views. Instead of relying on God’s handiwork, he simply proved mathematically that the solar system is stable over periods of time that are much longer than those anticipated by Newton. To solve this complex problem, Laplace introduced yet another mathematical formalism known as perturbation theory, which enabled him to calculate the cumulative effect of many small perturbations to each planet’s orbit. Finally, to top it all, Laplace proposed one of the first models for the very origin of the solar system—in his influential nebular hypothesis, the solar system formed from a contracting gaseous nebula.

Given all of these impressive feats, it is perhaps not surprising that in his Philosophical Essay on Probabilities Laplace boldly pronounced:

All events, even those which on account of their insignificance do not seem to follow the great laws of nature, are a result of it