Infinite Powers: How Calculus Reveals the Secrets of the Universe

by Steven H. Strogatz

From a modern perspective, there are two sides to calculus. Differential calculus cuts complicated problems into infinitely many simpler pieces. Integral calculus puts the pieces back together again to solve the original problem. 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