During the mid 18^{th} century, the development of nautical chronometers was in its infancy, and the method of lunar distances was the main alternative for calculating longitude at sea. The idea was to regard the Moon as the hour hand of a celestial clock that consisted of a pattern of stars forming the numbers on its face. Lines of longitude would then correspond to the relevant positions of the Moon on this “clock-face”

This method was eventually refined by Tobias Mayer, whose work on lunar motion is the subject of this book by Steven Wepster, which concentrates on the following themes:

- The development of Mayer’s lunar tables back to Newton’s lunar theory of 1702.
- The derivation of his dynamical lunar theory from Euler’s work on celestial mechanics.
- Mayer’s adjustment of available theoretical models to a set of available observations.

Lunar theory has a protracted history in which the work of Newton constitutes a pivotal moment. This aspect of his work wasn’t a theoretical structure in the usual sense, but rather a set of “rules and formulae for constructing diagrams and tables that would represent celestial motions and observations with accuracy”. He could calculate many general properties of lunar motion from his principles of gravitation, but perturbations in the Moon’s orbit, caused by the gravitational effect of the Sun, meant that this was a three-body problem which he could not solve on the basis of his laws of motion. Concerning exact solutions of this problem, Newton said “it exceeds, if I am not mistaken, the force of any human mind”.

This was a period when Newton’s theory of gravitation hadn’t yet gained universal acceptance, so this anomaly placed yet another question mark over its validity. Most of the irregularities of lunar orbits were subsequently explained by the combined efforts of Euler, d’Alembert, Laplace and Clairaut. Using gravitational analysis, they showed that lunar perturbations could be expressed in terms of certain angular arguments and coefficients.

But Euler’s name is particularly connected with the topic of lunar tables, and it is frequently mentioned throughout Wepster’s book. He was, in fact, the recipient of a minor portion (£300) of the Longitude Prize for his contribution to the lunar distance method (in aid of Mayer). Seven out of the ten main beneficiaries were rewarded for work on chronometers, while Mayer was posthumously awarded joint second prize for his work on lunar tables (his widow received £3000). Mayer’s more accurate astronomical measurements were used in conjunction with the theoretical model devised by Euler, which Mayer probably modified by use of ideas due to Clairaut.

Euler’s starting point was to express Newton’s second law in the form of ordinary differential equations, given in terms of spherical coordinates. The earth would be the origin and the plane of reference would be the plane of the ecliptic. With time as the independent variable, this could be done by selecting from a choice of over ten variables representing a range of orbital phenomenon. Among these were: true longitude of moon, true anomaly of the sun, scaled curtate distance of the moon, and so on. Also involved were 12 orbital constants, including those for mean lunar eccentricity, solar mean motion and the lunar mean motion relative to the apogee. In this context, Euler’s powerful use of infinite trigonometric series enabled him to construct reasonably accurate lunar tables.

To repeat, Mayer’s achievement is described in this book as emanating from his adjustment of Euler’s mathematical model to fit his own more accurate sets of data, but Wepster provides evidence to show that his work on lunar tables is also based upon Newton’s attempts. He also maintains that Mayer’s methodology constitutes the first known example of non-trivial model fitting, and that it was the harbinger of the method of least squares by half a century. Chapter 5 also provides a concise verbal summary of Mayer’s use of mathematics, which helps the reader to navigate the deeper mathematical analysis at its heart.

In keeping with the book’s title, most space is accorded to the structure (and use) of the tables themselves, and it is this aspect that I found most difficult to comprehend. It introduces a welter of astronomical concepts and much new terminology. For instance, the word “equAtion” (sic) refers to the periodic correction of mean coordinates — so it has no connection with its mathematical counterpart. Also, the word “inequAlity” refers to the measure of deviation from uniform motion, and these words therefore represent closely related concepts.

Mayer’s twenty-two tables come in either *multistep *form or *single-step *form. Using the multistep tables, formulae and computations are dealt with by adjusting the arguments in several steps. But the single-step tables use only mean motions and arguments throughout. Several tables were required for the calculation of a single position of the moon. Some tables represent mean position or mean motion. Others represent an equAtion, such as the periodic correction of mean coordinates that depend on the position of the sun.

The author allocates his own aliases to these tables, usually with Dutch words such as **rede, kil, plaat, gat, geer **etc. I had a recurrence of a recent attack of vertigo when I read the following sentence on page 210:

The development of **geer **via **gat, put** and **plaat **into **zwin**, **pas **and finally **kil **can be traced out by analysis from the position calculations in **cod**.

Perhaps another reviewer could more succinctly summarise Weptster’s portrayal of the technical aspects of Tobias Mayer’s work, but I think that the only way to gain insight into the intricacies of his lunar tables is to study this unique and interesting book. It certainly meets the publisher’s claim of being of interest to historians of mathematics and astronomers. Even without delving into its complexities, however, one can gain insight into the historical context from which Mayer’s work originated. There is certainly no other book that researches this topic so thoroughly as this one by Stephen Wepster.

Peter Ruane** **is retired after many years spent in primary mathematics education. He is now 71 years old, and is reluctant to accept the fact of human mortality.