The Calculus Diaries - Jennifer Ouellette [9]
The problem with the method of exhaustion is that the process literally could go on forever. One would never be able to calculate the exact area under a given curve, because how can one draw an infinite number of rectangles or triangles? Managing infinity is a crucial achievement of calculus. The ancient Greeks had an imperfect understanding of the concept of infinity, as do most of us encountering calculus for the first time. It’s not something easily grasped by our finite human minds. So Archimedes’ methodology still fell short of actually inventing integral calculus. Perhaps he might have done so, had he not run afoul of that hotheaded Roman soldier. “Killing Archimedes was one of the biggest Roman contributions to mathematics,” Charles Seife drily observes in Zero: The Biography of a Dangerous Idea. “The Roman era lasted for about seven centuries. In all that time, there were no significant mathematical developments.”
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While European mathematics languished in the medieval wilderness, a veritable renaissance was brewing in the East—specifically the rise of Baghdad as a cultural mecca for science and mathematics in the ninth century. The driving force behind this intellectual rebirth was the caliph Hārūn ar-Rashīd, who ruled the Islamic Empire from 786 to 809. He insisted on translating the greatest ancient works on math and science from around the world into Arabic—not just the work of the ancient Greeks, but also the achievements of scholars in India, South Asia, and China. His successor, Abu Jafar al-Ma’mūn, went one step further and established the House of Wisdom (Bayt al-Hikma), a scholarly “think tank” to bring together the Islamic world’s greatest minds.
One of those minds belonged to Abu Jafar al-Kwarizmi, whom we can blame for the development of modern algebra. He dreamed up how to use an equation to describe an unknown, the original x factor. He’s the guy who invented that tedious exercise of “balancing” both sides of an equation by adding, subtracting, or dividing by the same amount on both sides, a plague for high school students to this day. He called his brainchild “comparing and restoring.” Since the Arabic word for “restoring” is al-jabr, today we know this discipline as algebra.
Al-Kwarizmi did this without the benefit of one little character, literally, that we’ve come to take for granted. The equal sign didn’t exist until the sixteenth century.7 He didn’t use modern algebraic notation, either. Instead, he expressed his unknowns in words rather than variables, and his equations in sentences. In essence, that is what a mathematical equation is: a sentence reduced to a symbolic shorthand so that the quantities can be more easily manipulated. Algebra is about symbols, while geometry is about shapes, yet they share a mathematical connection, even though it would take another several hundred years after al-Kwarizmi’s work before East and West merged. Two French mathematicians, Pierre de Fermat and René Descartes, definitively proved the geometry-algebra connection in the early seventeenth century, thereby forging a crucial link in the development of calculus.
The son of a leather merchant, Fermat was a lawyer by profession, working as a counselor to Parliament in Toulouse. He rose quickly through the ranks, aided by the high death rate of that era, when outbreaks of the plague swept through the city frequently. Fermat himself contracted the plague at one point, but proved to be one of the lucky few to survive. Eventually he became a judge near Toulouse, at a time when heretical priests were routinely burned at the stake. I’d surmise that the intellectually minded Fermat appreciated the fact that judges were discouraged from social interactions,