The history of mathematical astronomy is hard to write. The technical requirements alone are daunting, and the diversity and depth of content to be covered forces difficult choices. How closely does one stick with the historical texts? To what extent can one expect the reader to follow the mathematical arguments? So, it's not really a surprise that it has been 150 years since this history was last written, and a pleasant surprise to see it tackled here, and very well, by a scholar who does not claim the title of historian.

Linton's choices will not agree perfectly with the preferences of any particular reviewer, so it is important to get them into the open at the outset. He states in his opening paragraph that "it is more important to understand what it is that was accomplished than precisely how it was achieved" — not a way to win friends and influence people in a crowd of historians, but defensible in a topic as difficult to approach as this one. Generally speaking, the goal is accomplished: the writing is clear, and the ideas accessible — as long as one has a degree in applied mathematics. However, Linton's principle is taken to an uncomfortable extreme on occasions, especially in the earlier material, where we find a modern mathematical analysis substituted for the historical accomplishment (for instance in Apollonius's work on epicyclic theory, or Kepler's oval orbits). Nevertheless the author is to be commended: usually, he wisely stays out of the way of his subjects, allowing the compelling story to tell itself.

As the title suggests, the book begins in ancient Greece. The near-omission of Babylonian (and Indian) astronomy is somewhat puzzling; what is more mathematical than Babylonian planetary theory? Nevertheless, the progress from Eudoxus's model of concentric spheres to a fully-realized and accurate epicyclic planetary model in Ptolemy's *Almagest* is covered faithfully, reflecting well the views of mainstream historians on the contributions of all figures, particularly Hipparchus and Ptolemy. It is refreshing to see that the actual mathematics (in modern transcription) is here as well: we find enough of Aristarchus's method for determining relative distances between the Earth, Moon, and Sun, for instance, to really understand it. To some extent the mathematics *is* the history; without it the book would have lost much of its purpose.

One of the great improvements of this book over its outdated predecessors is its chapter on Islamic astronomy, much of which has come to light in recent decades. As a specialist in this period, the reviewer can vouch for the fairness and veracity of the story told here. In a book of this detail there are bound to be some mistakes (for instance, I would love to see examples of the trigonometric tables "often computed to over twelve significant figures", p. 96), but these are relatively scarce, and seldom if ever cause the historical point to waver from the truth.

From Western Europe to Newton the book takes a "great men" approach (Copernicus, Tycho, Kepler, Galileo, Descartes, and finally Newton), allowing secondary figures to fade somewhat in order to delve more deeply into fundamental accomplishments. With Copernicus, for instance, we see much more than in standard accounts: a good explanation of his debts to his ancient and medieval predecessors, his theories of precession and of the Moon (including a comparison of the latter with the model of Ibn al-Shāṭir), and an account of his planetary models, including Mercury. The science of Tycho and Kepler is covered well without skipping over their astrological and mystical inclinations, which are described fairly and within their culture. Galileo gets fewer pages than one might expect, but his contributions to *mathematical* astronomy are not as great as the others. One feels palpably the gradual turning away from the ancient commitments to circular motion and geometrical reasoning to a new physics based on universal gravitation. The 40 pages on Newton and the *Principia* are both climax and turning point; Aristotle is finally behind us, and astronomy begins to base itself on mathematical rather than philosophical principles.

After Newton, the book shifts its approach from great men to great problems, most of which began as challenges to the principle of universal gravitation, and ended in triumphs. The magnitude of the motion of the lunar apogee was finally brought into accord with Newtonian physics by Alexis-Claude Clairaut in 1749. Lagrange's mathematically elegant work on the libration of the Moon (small periodic changes in the position of the Moon's surface relative to the Earth) led to another powerful confirmation by explaining an odd coincidence in the Moon's position that had been observed by Cassini: "once more, what had presented itself as an apparently arbitrary fact turned out to be a consequence of universal gravitation" (p. 323). Laplace's *System of the World* and his monumental *Celestial Mechanics* completed the case for gravitation so thoroughly that they presented a powerful argument for determinism. The discovery of Neptune in 1846 by pure theory — assuming that an unknown planet was causing Uranus to be disturbed from its expected path, determining where it would have to be, and pointing a telescope to that spot in the sky (although the story, told here, is more complex) — was a spectacular public success for gravitation, but as far as the scientific community was concerned, any controversy had ended many decades earlier.

The demise of universal gravitation, making way for general relativity, is told well as a gradual buildup towards Einstein. Poincaré's qualitative, geometric theory of differential equations transformed dynamics and gave birth to chaos theory, rendering Laplacian determinism obsolete even within Newton's physics. The anomalous advance of the perihelion of Mercury was thought to be yet another Newtonian triumph-in-waiting; various predictions and apparent discoveries were made of planets and asteroid belts within Mercury's orbit. However, it eventually became one of Einstein's key confirmations that his theory of general relativity was correct.

This thoroughly-researched book demands the reader's attention and effort, but rewards the investment richly. It will be an important reference work for decades to come; I hope it will also provide a model for more expositions of challenging mathematical topics to a wider audience.

Glen van Brummelen is associate professor of mathematics at Bennington College. He has published extensively on Islamic mathematical astronomy and is currently writing a history of trigonometry.