The Calculus Diaries - Jennifer Ouellette [81]
Interest in Hooke’s contributions to science has revived in recent years, and these include a footnote in the history of calculus—specifically, the catenary and its importance to architectural arches. Thanks to his early apprenticeship to an artist, Hooke was a gifted draftsman, and his architectural bent proved useful when the Great Fire of London destroyed much of the city. It was in the process of rebuilding St. Paul’s Cathedral in 1671 that Hooke “rediscovered” the secret of the catenary.
Alas, Hooke was a bit too clever for his own good. He announced his solution to the problem of the optimal shape of an arch to the Royal Society, but he never published it. Instead, four years later, he published an encrypted solution in the form of a Latin anagram in the appendix to his treatise, Description of Helioscopes. It did not attract much notice. Finally, in 1705, the executor of his estate published the anagram’s solution: “As hangs the flexible chain, so inverted stands the rigid arch”—or, if you want to be all Latinate about it: Ut pendet continuum flexile, sic stabit contiguum rigidum inversum.
Had Hooke been less secretive about his discovery, he might have received credit sooner for his solution to the problem of the catenary. Instead, a German mathematician named Johann Bernoulli found the solution independently and announced it in 1691. Johann was one of eight gifted mathematicians and physicists in the legendary Bernoulli family. They were a virtual dynasty during this period. The Calvinist family originally hailed from Belgium but fled to Switzerland to escape Catholic persecution. There the family patriarch, Nicolaus, made his fortune as a spice merchant.
Nicolaus had intended that his son Johann take over the family business. Alas, Johann failed miserably as an apprentice in training and opted to study medicine at Basel University instead. In between his studies, he and his older brother, Jakob, began collaborating on the study of this shiny new mathematical tool called calculus and were among the first to apply it to various problems. Eventually, Johann switched from medicine to math, and thus began a series of nasty sibling rivalries that rippled through the Bernoulli family tree for decades.
The brothers Bernoulli were highly competitive, fought constantly—their letters to each other are filled with heated insults and strong language—and always sought to outdo each other when it came to posing mathematical challenges. (This practice of issuing challenges was all the rage back then among mathematically minded sorts.) The fact that Jakob had trained his younger brother made it difficult for him to accept Johann as an equal. Johann, in turn, hated to be outdone; he was even jealous of his own son Daniel, banning his offspring from the house when Daniel won a math contest at the University of Paris that Johann had also entered. Nor was he averse to a spot of plagiarism: He once stole one of Daniel’s papers, changed the name and date, and claimed it was his own work.
That constant bickering might have been ruinous to harmonious familial relations, but it seemed to fuel the Bernoulli brothers’ mathematical creativity. It was Jakob who set forth the problem of the catenary: determining the precise mathematical shape formed by a hanging chain. Nearly fifty years before, Galileo theorized that it formed a parabola, but this was disproved in 1669, leaving the matter open to debate.49 Johann Bernoulli, Leibniz, and Huygens all responded with their solutions within months, beating poor Hooke to publication. In modern calculus, it is possible to find the solution of the optimal shape for a hanging chain via a minimization problem, because the goal is to minimize tension. In contrast, finding the strongest shape for an arch is a maximization problem, because we wish to find the shape with the most compression forces.
There is yet another quirky feature of the catenary: It is related both to exponential growth curves