How spoke Galileo? Untypically for a scholar of his day, he wrote largely in his native Italian and, to a lesser extent, in Latin, which the authors claim arose from his aim of disseminating his ideas to the wider population, as opposed to limiting their availability to elitist groups of 16^{th} and 17^{th} century academics. To further enhance the accessibility of his findings, he explained them in a Socratic form of dialogue, carried out between three fictional characters. These were Salviati (alias Galileo, the Copernican), Simplicio (his Ptolemaic foil) and Sagredo (a neutral arbiter, who invariably agreed with Salviati). Most of this book consists of such ‘conversations’, focussing upon mechanics, optics, material science and astronomy. Of this mode of communication, Galileo said:

*‘I thought it would be very appropriate to explain my ideas in dialogue form; for it is not restricted to the rigorous observation of mathematical laws, and so it allows digressions which are sometimes no less interesting than the main topic.’*

Although Galileo is commonly mentioned in the standard histories, he is associated with little innovatory achievement in the field of pure mathematics. In fact, his singular success in that sphere seems to be his proof of the equipotence of the set of natural numbers and the set of square numbers, which he discusses in the context of unresolved speculation on indivisibles. But his renown emanates from his pioneering work in the application the existing mathematics to the analysis of a wide range of physical situations. By such means, he founded the mechanics of falling bodies, the theory of elasticity and he provided vindication for the Copernican system. Struik [1] puts it this way:

*‘Above all, we owe to Galileo, more than any other man of his period, the spirit of modern science based upon the harmony of experiment and theory, with stress on the intensive use of mathematics.’*

So, this book forms an anthology of selected writings of Galileo, which should be of interest to those concerned with the history of science and applied mathematics. The first edition (1998) was in Italian, but this well translated English version came out in 2006. Its aim is ‘*to provide a representative selection of Galileo’s original writings, for the attention mainly of the young…’* and to shed light on the range of his work and his methodology.

Covering a wide range of Galileo’s investigations, there are twenty four chapters, nearly all of which are similarly structured. Each begins with a very brief abstract of chapter contents, followed by the authors’ deeper explanation of the main ideas. Next come the extracts of Galileo’s dialogues, culminating with the authors’ analysis that uses present day mathematics, which should within the grasp of many high school students. But, compared with other translated extracts (such as [2]), there is little indication of the actual mathematical methods used by Galileo himself.

However, I found the very first chapter of this book to be the most charming of all. It consists of a posthumous autobiography of Galileo, cleverly and convincingly compiled by the authors and setting the tone for what follows. Moreover, having just seen a performance of Brecht’s play about Galileo, I am struck by the similarity of the two portrayals of this heroic figure but I wonder if, had he written solely in Latin, whether the Church would have been so threatened by his findings.

**References**

[1] *A Concise History of Mathematics* , by Dirk J. Struik (Dover, 1967)

[2] *The World of Mathematics* , vol 2, ed James R. Newman (Simon & Schuster 1956)

Peter Ruane (ruane.pandh@blueyonder.co.uk) is retired from university teaching, where his interests lay predominantly within the field of mathematics education.