This book is an English version, with expansions and improvements, of the author’s *La sintassi logica della matematica greca* (2011, published online). Since this earlier work has received too little attention—perhaps partly for linguistic reasons—an English version is a welcome contribution to scholarship. That said, this book is still written for the specialist. It is directed to the relatively small number of scholars who study ancient Greek mathematical texts in the original language (p. xii), and Acerbi offers little or no help for the general reader. In fact, my overall impression is that this book is not an argument to be read cover-to-cover, but rather serves as a repository for many of the insights and views that the author developed over his years of working on these sources.

As Acerbi makes clear in the preface, this book is not a history of Greek mathematics. Indeed, it largely assumes a knowledge of the mathematical contents of the ancient sources. As the title of the book makes clear, it is a study of the linguistic practices of Greek mathematicians and the implications of these for the logical methods preserved in their texts. Acerbi treats Greek mathematics as a literary product of the Hellenistic period (p. x). In fact, his account can be taken as an extended argument against a common view that Greek mathematics fundamentally involves drawing a diagram and then making an argument about it (for example, Netz 1999, MR1683176).

The book is divided into five parts. The first part introduces Acerbi’s tripartite division of all Greek mathematical discourse into what he calls “stylistic codes”: demonstrative, procedural, and algorithmic. Linguistic cues for identifying the codes are listed and discussed. There are unfortunately relatively few examples of the procedural and algorithmic codes, and, although these are clearly differentiated from a linguistic perspective, it is not so clear how they mathematically differ from one another.

The second part discusses linguistic and structural templates that are used for validation. It begins by discussing noun phrases, the structure of clauses in sentences, sentence structures, and so on. Then larger scale templates are studied, such as the format of analysis and synthesis, and the validation of algorithms and procedures through a “chain of givens” (p. 68).

The remainder of the book focuses on the demonstrative code, which makes it clear that Acerbi considers this to be most worthy of attention. In my view, the third section is the most important for historians of mathematics and logic. In this section, Acerbi presents his solution to the vexed question of how Greek mathematicians produced general proofs of the claims they made. At the risk of oversimplification, his case that a Greek proposition is fully general throughout rests on three main claims. In the first place, the verb of being, εἶναι , is often, and particularly in the setting-out, used, not as a copula, but in the presential sense: “let there be...,” “there is,...” and so on. In the second place, the letter labels, which denote geometrical objects, refer not to objects in the diagram, as is sometimes argued, but rather serve as labels, or names, for the general objects that are under discussion. And finally, the generality of the mathematical objects under discussion is secured by the use of the indefinite construction whenever objects are introduced, and when references are made to previous results. This section contains a new discussion of epigraphical material that was not contained in *La sintassi logica* (2011), and which argues that even in the earliest attested forms of the denotive letters, such as ἐφ᾿ οὗ τὸ Α , the letter does not denote an object in the diagram near which it is written, but is simply a label, or name, for the general object mentioned in the text (pp. 95–96).

Part four treats the logical structure of the complete proposition, focusing on the classical division into parts: enunciation, setting-out, construction, proof, and so on. Acerbi also discusses the first section of the proof, called the “anaphora” by M. Federspiel, which refers back to what is supposed in the setting-out or introduced in the construction. A substantial portion of this part details the syntax of logical relations in the proofs, and is proceeded by a short account of Aristotle and Galen’s discussions of logical relations.

Part five, the longest in the book, is a somewhat loose collection of all of the other linguistic units that convey logical force, such as modals, conditionals, disjunctions, conjunctions, and so on. The longest section in this part bears directly on the material presented in part three and sets out the many ways in which generality is handled in Greek mathematical texts.

There are three useful appendixes: a list of all “problems” in the Greek mathematical corpus, a list of all sources in the corpus that pertain to analysis and synthesis, and list of the names of the Greek mathematicians with their works, if extant, or brief description of their reported work, with references. Throughout the book, there are also a number of sections that are only loosely related to the overall argument, in which Acerbi sets out his views on certain topics related to Greek mathematics—such as section 3.4, “Ontological commitment,” in which he discusses Greek approaches to the question of the ontological status of mathematical objects, with a focus on Archimedes’ Method.

This is an important book on the language of Greek mathematical texts, and scholars of these sources will find much that is useful in its pages. For the non-specialist, however, this book will probably present a number of challenges, because many of the author’s arguments are not presented in detail, and scholarship with which the author disagrees is often scornfully dismissed with no discussion. That said, the argument that Acerbi makes for his claim that Greek mathematical propositions do not involve universal instantiation is important and provides, in my view, the most linguistically faithful solution to the question of generality in Greek propositions that has been put forward to date.

Nathan Sidoli is a Professor of the History and Philosophy of Science at Waseda University, Tokyo. His research focuses on the Greek mathematical sciences and their development in Arabic sources.