*Jamblique: In Nicomachi Arithmeticam, introduction, texte critique, traduction française et notes de commentaire. Mathematica graeca antiqua, 3.*Pisa; Roma: Fabrizio Serra editore, 2014. Pp. 348. ISBN 9788862276160. €110.00 (pb).

Reviewed by John Dillon, Trinity College Dublin (jmdillon@eircom.net)

This is a very fine edition of a work of Iamblichus that has in general been much disdained but, as Vinel well argues, not entirely justly. It is generally, but superficially, presented as a commentary on the *Introduction to Arithmetic* of the second-century CE Neopythagorean Nicomachus of Gerasa, but in fact Iamblichus is not purporting to be composing anything like a formal commentary; instead, as is suitable to the context in which it is presented (which is his so-called 'Pythagorean Sequence' of ten treatises, setting out an introduction to the whole of Pythagorean philosophy), it is rather a meditation on the first principles of arithmetic based, rather loosely, on Nicomachus' introductory work, with various additions (including a number of passages from other volumes of the collection, specifically the *Vita Pythagorica* and the *De Communi Mathematica Scientia*).

A nice example of Iamblichus' adaptation of Nicomachus occurs at the beginning of §2 of the work, where he prefixes to Nicomachus' discussion of Quantity a series of (largely Pythagorean) definitions of number and the monad which are of considerable interest (and borrowed gratefully later by Syrianus in his *Commentary on Metaphysics M-N*) – all this excellently annotated by Vinel.

Appreciating this properly is of some importance for an accurate evaluation of Iamblichus' work, and it is this which Vinel undertakes, with an impressive degree of comprehensiveness. He provides first a copious introduction (1-66), in which, after a short account of the life of Iamblichus, and of the composition of the ten-volume 'Pythagorean Sequence' (of which the present work is the fourth, and the latest surviving, member – the last six being lost), he turns to an examination of the work itself. He starts (§3) with a discussion of the nature and title of the work, arguing, I think persuasively, that the title as handed down in the manuscripts, *Peri tês Nikomakhou arithmêtikês eisagôgês* is misleading, the last element, *eisagôgês*, being the work of an 'intelligent' scribe, who assumed that it was similar to later commentaries on Nicomachus, such as those of Philoponus or Asclepius of Tralles. If that is excised, the title describes the work pretty well. Vinel also provides a detailed analysis of the contents, dividing it up into chapters and sections in a way that previous editors, Tennulius and Pistelli, had omitted to do, and thus rendering it much more approachable.

He then embarks on a series of three sections, each covering a salient feature of the work, and all presenting much material of interest. In §4, 'L'arithmétique de la justice et les carrées dits "magiques"', we find an extended discussion of the tradition of numerical squares, or wheels, beginning with the so-called 'Square of Theon (of Smyrna)', representing the number 5 as a centre for the sequence of numbers 1-9, followed by a series of more elaborate squares, embodying more complex sets of numbers, having magical significance, including some found in Pompeii, which Iamblichus discusses in II.33-7 and 51-2.

In §5, 'Le topos du point et de la ligne réinventé', he focuses on IV.5-6, where Iamblichus adverts to the problem of postulating the point as first principle of the line (the problem, indeed, which led Xenocrates, in the Old Academy, to postulate the 'indivisible line' as such a first principle). In connection with which Iamblichus produces a sentence containing a fine sequence of hapax legomena: *all' êtoi psaustôn adiastasia estai ê diastantôn apsaustia*, duly celebrated by Vinel.

The third of these sections, however ('La naissance oubliée du concept de zéro'), is the most substantial, dealing as it does with what seems a notable innovation by Iamblichus in proposing the concept of *ouden*, or 'zero', in a numerical sense. This arises in II.32-52, where Iamblichus embarks on an extended discussion of the concept of *ouden*, in connection with his observation that, if the number 1, like all other numbers, is to be reckoned as the sum of either of its contiguous numbers divided by two, we will have to postulate, on either side of 1, 2 and 0. He then finds various other uses for 0. Vinel finds this quite notable, and I agree with him; I am less confident, though, about his interesting suggestion that Iamblichus may have been stimulated to postulate the zero by reason of his equally remarkable postulate of an 'absolutely ineffable' principle prior to the One! The innovation seems adequately motivated by the considerations adduced by Iamblichus in the context. However, his discussion of this topic is both full and valuable.

The text itself is a considerable improvement on that of Pistelli. Fortunately, there are only two manuscripts that need to be considered, Laur. Plut. 86,3, and Laur. Plut. 86,29, both in Florence (all others being *apographa* of the latter. But, as Vinel remarks, the discipline of having to translate the text (of which he does an excellent job) constrains one to adopt emendations which a mere editor of the Greek may simply toy with, and he has adopted many necessary ones. Apart from short notes at the bottom of the page, there follow the text and translation 65 pages of most useful supplementary notes, together with a copious Index Graecitatis, followed by much shorter indices of passages quoted, manuscripts, and proper names.

All in all, this is a very fine piece of work, which gives the *In Nicomachi Arithmeticam* the sort of edition it deserves.