The Ballet of the Planets: On the Mathematical Elegance of Planetary Motion

It’s widely known that the geocentric view of the solar system was prevalent until the late 16^th century. Also well known is the fact that anyone who dared to recommend an alternative theory could invoke the wrath of the church. What is not so well known is that astronomers, who worked within this model, relied heavily upon the notion of an ‘epicycle’, which is a theme occupying the first half of this eloquently written little book. Although I say ‘little’ book (178pp), it is big with respect to the ideas upon which its author expatiates and, despite its catchy title, it is not an easy read.

The ancient Greeks regarded the circle as the embodiment of spatial perfection, so it was decreed that planetary motion must be circular — and their available methods of astronomical observation more or less supported their belief. But many irregularities, such as the retrograde motion of Mars, meant that this theory didn’t quite match reality. This Ptolemaic model endured for over 1000 years before it eventually yielded to the ideas of heliocentricity, and elliptical orbits, due to Copernicus, Galileo, Kepler and Newton.

A precise account of the work of these astronomers is provided here by Donald Benson. It includes an analysis of observational techniques, explains how theory emerged from the corresponding data, and outlines the mathematical principles that form the basis of planetary motion. Not only that, the narrative concludes by explaining how modern physics arose from Newton’s work on astronomy and his theory of universal gravitation

Although much space in this book is devoted to epicycles (epicycloids, astroid, cardioid, etc), the mathematical description of such curves is mainly confined to their construction by means of simple linkages. The only reference to the algebra of these curves consists of a parametric equation provided in the notes at the end of the book. On the other hand, as a prelude to coverage of heliocentric theories, chapter 7 introduces the reader to the ellipse and many of its relevant properties. Other than that, the mathematical ideas upon which the book depends should be accessible to good high school students and beyond.

Because this book is illustrated with nearly 80 diagrams, it would seem churlish to suggest that it needs even more. But planetary motion is three dimensional, and static 2d pictures aren’t always able to convey concepts from celestial kinematics. And it is even more difficult to convey such ideas in verbal written form. As a result, I found it beneficial to go online to gain a clearer idea of concepts such as retrograde motion, the meaning of ‘equinox’ with respect to the intersection of the ecliptic plane and the celestial equator, and so on. Of course, the more lavishly illustrated a book becomes, the higher the cost of publication.

At a time when we have pictures transmitted from Mars on an almost daily basis, this book is a most welcome addition to the literature on celestial mechanics. Of all the planets discussed by Donald Benson, Mars, because of its deviant behaviour, is the one that receives greatest attention. So don’t be put off by my limited capacity to translate words into pictures — buy a copy of this fascinating historical account of a journey from Ptolemy to Newton.

Peter Ruane is retired from a career spent in the provision of courses for teachers of (primary and secondary) mathematics.