This is the second book on historical aspects of lunar theory to have been published by Springer within a twelve-month period. The first one (by Stephen Wepster) concerns the work of Tobias Mayer, who devised the most accurate lunar tables for navigational purposes available in the late 18^{th} century. In this second book, Curtis Wilson examines the later improvements to the foundations of lunar theory that were initiated by G. W. Hill in 1878 and completed by E. W. Brown in the period 1892 to 1919.

Between the work of Mayer and the emergence of the Hill-Brown theory, there is an historical gap of 130 years, and developments in lunar theory during that period are summarised in the second chapter of Wilson’s book. Included in that brief survey is a summary of the work of Delauny and Hansen, whose theories respectively represent mathematical elegance and empirical assiduity. Consequently, these two Springer publications jointly form a highly detailed history of lunar theory from the time of Newton to the present day.

To put it simply, lunar theory is the application of mathematical techniques for the purpose of defining the Moon’s orbit about the earth. The problems in doing this are due to the orbit not being exactly elliptical, because there are perturbations in its motion that are mainly caused by the gravitational effect of the Sun. Another factor is the tidal acceleration of the Moon, which entails a gradual slowing down of the Earth’s rotation.

The development of lunar theory came about due to the need to calculate longitude at sea, and its evolution was due to the work of mathematicians such as Newton, Euler, Clairaut, Laplace and Lagrange. Lunar theory was also regarded as a means of validating Newton’s inverse square law, which is the basis of his theory of gravitation. From such work arose the notion of the 3-body problem and various mathematical methods, including Euler’s use of infinite trigonometric series.

The theory developed by Hill was based upon Euler’s lunar theory of 1772 but, in the form in which Brown carried it to completion, it was semi-numerical, and the initial orbit was given by the dynamics of the simplified three-body problem. The numerical input was just the ratio of the mean motion of the Sun to the synodic motion of the Moon. This was used because it was one of the most accurately known constants in the whole of astronomy. The ensuing theory could therefore be based upon it with a higher degree of confidence.

Wilson’s book contains short biographies of both Hill and Brown. It describes Hill’s ideas on the lunar perigree, his notion of the variational curve and summarises views on his overall lunar theory. Parts I and II of the book give highly detailed information as to how Brown organized the long series of computations that were required to complete this jointly conceived theory. Part III gives further insight into the phenomenon of tidal acceleration and the transition from ephemeris time to the use of atomic clocks — in other words, it explains the demise of lunar tables as a basis for the compilation of nautical almanacs and for accuracy in the measurement of time.

The publisher says that the text is largely accessible without specialist knowledge of its central themes. To some extent, this is true, but there are very many celestial concepts at the heart of lunar theory, and the associated terminology is extensive. Unfortunately, no glossary is included, and the book is contains very few illustrations. This means that it will be hard going for readers with inadequate knowledge of astronomy.

The mathematical content of the book is summarised as being ‘some calculus and differential equations’. However, the approach is mainly computational and Brown’s treatise, for example, is said to embody ‘18 years of calculative labour’. Understandably, therefore, making sense of the mathematical dialogue in this book requires painstaking scrutiny.

Nonetheless, the foregoing comments are not meant to be points of criticism. They are made to support the view that this text is mainly one for specialists. I say ‘mainly’, because there is so much in the book that will inspire an interest among those who are equipped with only rudimentary knowledge of astronomy (myself, for example). In short, Curtis Wilson’s coverage of this aspect of the history of lunar theory introduces many fascinating ideas, and it refers very many creative mathematicians and astronomers. Consequently, the reader doesn’t have to labour over the mathematical intricacies to extract pleasure from this scholarly historical tome.

Peter Ruane** **spent most of his working life in the mathematics education of primary and secondary school teachers.