Is God a Mathematician_ - Mario Livio [56]
Continuing in Newton’s and Leibniz’s giant footsteps, mathematicians of the Age of Reason (the late seventeenth and eighteenth centuries) extended calculus to the even more powerful and widely applicable branch of differential equations. Armed with this new weapon, scientists were now able to present detailed mathematical theories of phenomena ranging from the music produced by a violin string to the transport of heat, from the motion of a spinning top to the flow of liquids and gases. For a while, differential equations became the tool of choice for making progress in physics.
A few of the first explorers of the new vistas opened by differential equations were members of the legendary Bernoulli family. Between the mid-seventeenth century and the mid-nineteenth century, this family produced no fewer than eight prominent mathematicians. These gifted individuals were almost equally known for their bitter intrafamily feuds as they were for their outstanding mathematics. While the Bernoulli quarrels were always concerned with competition for mathematical supremacy, some of the problems they argued about may not seem today to be of the highest significance. Still, the solution of these intricate puzzles often paved the way for more impressive mathematical breakthroughs. Overall, there is no question that the Bernoullis played an important role in establishing mathematics as the language of a variety of physical processes.
One story can help exemplify the complexity of the minds of two of the brightest Bernoullis—the brothers Jakob (1654–1705) and Johann (1667–1748). Jakob Bernoulli was one of the pioneers of probability theory, and we shall return to him later in the chapter. In 1690, however, Jakob was busy resurrecting a problem first examined by the quintessential Renaissance man, Leonardo da Vinci, two centuries earlier: What is the shape taken by an elastic but inextensible chain suspended from two fixed points (as in figure 31)? Leonardo sketched a few such chains in his notebooks. The problem was also presented to Descartes by his friend Isaac Beeckman, but there is no evidence of Descartes’ trying to solve it. Eventually the problem became known as the problem of the catenary (from the Latin word catena, meaning “a chain”). Galileo thought that the shape would be parabolic but was proven wrong by the French Jesuit Ignatius Pardies (1636–73). Pardies was not up to the task, however, of actually solving mathematically for the correct shape.
Figure 31
Just one year after Jakob Bernoulli posed the problem, his younger brother Johann solved it (by means of a differential equation). Leibniz and the Dutch mathematical physicist Christiaan Huygens (1629–95) also solved it, but Huygens’s solution employed a more obscure geometrical method. The fact that Johann managed to solve a problem that had stymied his brother and teacher continued to be an immense source of satisfaction to the younger Bernoulli, even as late as thirteen years after Jakob’s death. In a letter Johann wrote on September 29, 1718, to the French mathematician Pierre Rémond de Montmort (1678–1719), he could not hide his delight:
You say that my brother proposed this problem; that is true, but does it follow that he had a solution of it then? Not at all. When he proposed this problem at my suggestion (for I was the first to think of it), neither the one nor the other of us was able to solve it; we despaired of it as insoluble, until Mr. Leibniz gave notice to the public in the Leipzig journal of 1690, p. 360, that he had solved the problem but did not publish his solution, so as to give time to other analysts, and it was this that encouraged us, my brother and me, to apply ourselves afresh.
After shamelessly taking ownership of even the suggestion of the problem, Johann continued with unconcealed glee:
The efforts of my brother were without success; for my part, I was more fortunate, for I found the skill (I say it without boasting, why