This beautifully produced book provides a new edition of the Greek text of Iamblichus' In Nicomachi arithmeticam introductionem, accompanied by a facing French translation, an extensive introduction, a detailed commentary and very full indexes. The edition is based on a new study of the manuscript tradition. In the preceding edition, published by H. Pistelli (Teubner, 1894), it was assumed that all other MSS copies of the text depended on F (Laurentianus 86, 3). However, V. claims (pp. 55–65) that these copies depend on L (Laurentianus 86, 29), which derives in turn from a lost archetype on which F also depends. V.'s claim throws doubt on the received view and is of importance also to the question of the manuscript tradition of the three other books which precede the In Nic. in Iamblichus' Pythagorean synthesis and in the MSS: the De vita Pythagorica, Protrepticus and De communi mathematica scientia. I do not find the variants V. gives as evidence for his claim (pp. 62–3) conclusive (a variant at Pistelli 58.18 = IV.11.4 in V.'s numbering of the text seems to be incorrectly reported), a claim somewhat complicated by his allowing for the possibility of contamination between L and F (p. 64). At any rate, V.'s claim will need to be examined further by future editors of the other three volumes of Iamblichus' work on Pythagoreanism. V.'s edition of the Greek text seems to me on the whole to improve on Pistelli's edition and to replace it. In his introduction, V. proposes a revaluation of Iamblichus' In Nic., rescuing it from the neglect from which it has suffered in earlier scholarship, thus contributing to a re-evaluation which has already been advocated for other volumes of Iamblichus' Pythagorean synthesis, in particular for the De vita Pythagorica. V. suggests that the original title of the In Nic. was Περὶ τῆς Νικομάχου ἀριθμητικῆς (p. 15), arguing that the work is not a commentary on Nicomachus' Introductio arithmetica, but a new introduction to arithmetic, composed under Nicomachus' ‘patronage’ (p. 20): Iamblichus makes use of many other (‘Pythagorean’) materials (including other texts of Nicomachus), besides the Introductio arithmetica (p. 21). Iamblichus also introduces new terminology and a new philosophical approach, as compared with that of Nicomachus and Euclid (pp. 22–3, 37–8). He provides one of the earliest Greek witnesses to a Pythagorean interest in ‘magic squares’ (pp. 23–36); and he elaborated a concept of zero, V. claims (pp. 41–52), which would quickly be forgotten. This claim, it seems to me, is somewhat exaggerated. The relevant section in the text (2.32–42) is concerned, not with ‘Le concept du “rien”’ (the subtitle V. gives to the section), but with the particularity of the monad (which V. translates as ‘unité’), which is not a number and whose particularity is brought out by comparison with ‘nothing’ (τὸ οὐδέν). A contentious translation of the Greek (at 2.36.6–7 ‘le milieu, c'est-à-dire 5, est autant en-deçà qu'au-delà: il diffère donc de “rien”’; I would translate the phrase as follows: ‘the middle term 5 is deficient by as much as that by which it is in excess, therefore by nothing’ [i.e. there is no deficiency or excess]) shows this exaggeration. However, it does indeed seem that Iamblichus is prepared to introduce ‘nothing’ as the first term in a series which includes the monad and numbers, and that he can use it in showing how it shares specific properties with the monad in arithmetical calculations. V.'s French translation of the text does not always seem accurate enough, in particular regarding philosophical terminology (1.1.2 θεωρíα, here and elsewhere: not ‘examen’, but rather ‘knowledge’; 2.1 πολυειδής: not ‘hétérogène’; 3.2 ὑποθέσει: not ‘fondement’; 9.3 γόνιμον: not ‘authenticité’). An incomplete knowledge of Neoplatonic philosophy also affects V.'s interpretation and commentary. So, for example, at 1.5.1–4 Iamblichus is referring to his anti-Aristotelian theory that mathematical objects are not concepts derived by abstraction from sensible objects, a theory to be found just before, in the De communi mathematica scientia 19.19–21; 34.9–10; 89.5–6 (compare V.'s commentary, n. 7). The digressions on the implications of arithmetic for ethics (2.115) and physics (4.90–1) are not given much attention in V.'s commentary. I do not think that the metaphysical ineffable (ἄρρητος) in Damascius, the uncoordinated One beyond all determination, should be associated with the mathematical concept of the irrational (p. 45). However, V.'s commentary is very full and helpful on the more technical mathematical aspects of the arithmetic presented in Iamblichus' book. His work is a valuable contribution to the study of the integration of arithmetic in later Neoplatonic philosophy and to the history of arithmetic in Late Antiquity. His new edition of the Greek text must certainly be consulted, as must his commentary, by those interested in the history of Greek mathematics and in the history of philosophy in Late Antiquity.