Infinite Powers: How Calculus Reveals the Secrets of the Universe

by Steven H. Strogatz

Given that cutting comes naturally before rebuilding, it seems sensible for a novice to learn differential calculus first. And indeed, that’s how all calculus courses begin today. They start with derivatives—the relatively easy techniques for slicing and dicing—and then work their way up to integrals, the much harder techniques for reassembling the pieces into an integrated whole. Students find it more comfortable to learn calculus in this order because the easier material comes first. Their teachers like it because the subject seems more logical this way. Yet, strangely enough, history unfolded in the opposite order. Integrals were already in full swing in ancient Greece in Archimedes’s work around 250 BCE, whereas derivatives weren’t even a gleam in anybody’s eye until the 1600s. Why did differential calculus—the easier side of the subject—develop so much later than integral calculus? It’s because differential calculus grew out of algebra, and algebra took centuries to mature, migrate, and mutate. In its original form in China, India, and the Islamic world, algebra was entirely verbal. Unknowns were words, not today’s x and y. Equations were sentences, and problems were paragraphs. But soon after algebra arrived in Europe, around 1200, it evolved into an art of symbols. That made algebra more abstract . . . and more powerful. This new breed, symbolic algebra, then coupled with geometry and spawned an even stronger hybrid, analytic geometry, which in turn begat a zoo of n...