Infinite Powers: How Calculus Reveals the Secrets of the Universe

by Steven H. Strogatz

In the preface to Harmonies of the World (1619), he wrote in ecstasy about finally seeing the pattern in God’s plan: “Now, since the dawn eight months ago, since the broad daylight three months ago, and since a few days ago, when the full sun illuminated my wonderful speculations, nothing holds me back. I yield freely to the sacred frenzy.” The numerological pattern that enraptured Kepler was his discovery that the square of the period of revolution of a planet is proportional to the cube of its average distance from the sun. Equivalently, the number T2/a3 is the same for all the planets. Here, T measures how long it takes a planet to go around the sun once (a year for the Earth, 1.9 years for Mars, 11.9 years for Jupiter, and so on), while a measures how far away the planet is from the sun. That’s a bit tricky to define, because the actual distance changes from week to week as a planet moves in its elliptical orbit; sometimes it’s closer to the sun and sometimes it’s farther away. To account for this effect, Kepler defined a as the average of the planet’s nearest and farthest distances to the sun. The