Infinite Powers.

I’m a sucker for a good book about math, but a lot of books about math aren’t that good – either they’re dry, or they don’t do enough to explain why any of this matters. (Sometimes it doesn’t matter, as in Prime Obsession, but the author did such a good job of explaining the problem, and benefited from the fact that it’s still unsolved.) Steven Strogatz’s Infinite Powers: How Calculus Reveals the Secrets of the Universe manages to be entertaining, practical, and also educational, as the author builds up the reader through some essentials of pre-calculus before getting into the good stuff, to the point that I recommended that my daughter check it out before next year when she takes calculus in school.

Calculus underlies everything in the universe; it is the foundation upon which the universe, and everything in it, functions. It is also one of humanity’s most remarkable discoveries, one that required multiple leaps of mathematical faith to uncover hidden truths about the universe. Physicist Richard Feynman quipped that it is “the language that God talks,” although he meant it in a secular sense, while mathematician Felix Klein said that one could not understand “the basis on which the scientific explanation of nature rests” without at least some understanding of differential and integral calculus.

 The story of how both Isaac Newton and Gottfried Wilhelm Leibniz simultaneously discovered calculus in the late 1600s, doing so both with their own remarkable insights and by building on the discoveries of mathematicians before them, going back to the ancient Greeks, would by itself be enough for an entertaining history. Strogatz does start with that, and uses the history as scaffolding to bring the reader up from algebra through geometry and trigonometry to the mathematics of limits, which is the essential precursor to calculus, before getting to the main event.

Or I should say “events,” as differential and integral calculus, while two sides of the same analytical coin, were discovered at separate times, with separate methods, and Strogatz tells their stories separately before bringing them together towards the end of the book. Differential calculus is what we learn first in schools, at least in the United States. It’s the mathematics of the rates of change; the rate at which a function changes is the derivative of that function. Acceleration is the derivative of velocity – that is, the rate at which velocity is changing. Velocity, in turn, is the derivative of position – the rate at which an object’s position changes. That also makes acceleration the second derivative of position, which is why you see a ² in the formula for the acceleration of an object falling due to Earth’s gravity (9.8 m/s²): a position might be measured in meters, so velocity is measured as the change in position (meters) by time (seconds), and acceleration is the change in velocity (meters per second) by time (seconds, again).

Integral calculus goes the other way – given an object’s acceleration, what is its velocity at a given point in time? Given its velocity, what is its position? But Leibniz and Newton – I expect to hear from Newton’s lawyers for listing him second – conceived of integration as a way to solve an entirely different problem: How to determine the area under a curved function. Those two didn’t think of it that way – the concept of a function came somewhat later – but they understood the need to find out the area underneath a curve, and came up, independently, with the same solution, which broke apart the space into a series of rectangles of known heights and near-zero widths, giving rise to the infinitesimals familiar to any student who’s taken integral calculus. They aren’t real numbers, although they do appear in more arcane number systems like the hyperreals, yet the sum of the areas of this infinitesimally narrow rectangles turns out to be a real number, giving you the area under the curve in question. This insight, which was probably Leibniz’s first, opened the world up for integral calculus, which turns out to have no end of important applications in physics, biology, and beyond.

Strogatz grounds the book in those applications, devoting the last quarter or so of Infinite Powers to discussing the modern ways in which we depend on calculus, even taking its existence for granted. GPS devices are the most obvious way, as the system wouldn’t function without the precision that calculus, which GPS uses for dealing with errors in the measurements of distances, offers – indeed, it’s also used to help planes land accurately. Yet calculus appears in even less-expected places; biologists used it to model the shape of the double helix of strands of DNA, treating a discrete object (DNA is just a series of connected molecules) as a continuous one. If your high school student ever asks why they need to learn this stuff, Infinite Powers has the answers, but also gives the reader the background to understand the author’s explanations even if you haven’t taken math in a few decades.

Next up: David Mitchell’s The Thousand Autumns of Jacob de Zoet.