Jamblique. In Nicomachi arithmeticam. Introduction, texte critique, tra- duction française et notes de commentaire by Nicolas Vinel Mathematica Graeca Antiqua 3. Pisa/Rome: Fabrizio Serra Editore, 2014. Pp. 356. ISBN 978–88–6227–616–0. Paper €110.00 Reviewed by Federico M. Petrucci Scuola Normale Superiore, Pisa federicofi[email protected] The relationship between Plato and ancient Pythagoreanism remains a mys- tery. Yet, one can be quite sure that a strong interlacement of the Platonic tradition and a conscious Pythagorean inspiration (at least in attention to num- bers and mathematics, and their applications in physics and metaphysics) characterized not only the Old Academy1 but also Middle Platonism and Neo- platonism.The excellent book by Nicolas Vinel contributes to our knowledge of this fact by providing scholars with a very useful and careful edition of a (usually disregarded) work by Iamblichus, a Neoplatonist deeply interested in the Pythagorean tradition. In the Old Academy, there was established a mathematical tradition that was grounded mainly in an arithmo-geometrical perspective. Although some of its elements were probably recovered to some extent and refashioned by Euclid,2 the arithmo-geometrical core of this tradition was somehow left aside by the ‘major’ stream of Greek mathematics. Nonetheless, one can easily find its re-appearance in the Imperial age: for example, in the extensive work by Moderatus of Gades in the early first century ad, who gathered the opinions (ἀρέϲκοντα) of ‘Pythagoreans’ in 10 books, of which only a 1 See Burkert 1972 which demonstrates how far post-Platonic accounts on Pythagore- anism are influenced by Platonic and Academic elements. A different approach to the history of Pythagoreanism is now proposed by Phillip Horky in several contri- butions: see especially Horky 2013. On the history of Pre-Platonic Pythagoreanism, see also Centrone 1995 and Huffman 2014. For a different perspective, see Zhmud 2012. 2 Bernard Vitrac and Fabio Acerbi have in many works indicated the need to go be- yond the traditional idea that Euclid’s Elements are only a summa of preceding discoveries. © 2014 Institute for Research in Classical Philosophy and Science issn 1549–4497 (online) All rights reserved issn 1549–4470 (print) Aestimatio 11 (2014) 342–353 343 Aestimatio few fragments remain, and in the writings by Nicomachus of Gerasa in the second.3 It has been demonstrated that the usual term for these authors, ‘Neopythagoreans’, is misleading since they do in fact work within the Pla- tonic tradition [see Centrone 2000]. Nonetheless, they show a specific interest in mathematics and appeal explicitly to the Pythagorean tradition.This stream, moreover, provides an effective example of a widespread tendency of Imperial Platonism that consists in associating Plato with the Pythagorean tradition from a more general point of view.This is the case, for instance, with Numenius of Apamea [see des Places 1973, fr. 24] and Plutarch’s De E ch. 7–16 (though this is not Plutarch’s own position, at least at the time that he wrote this work). Such a tendency of Imperial Platonism remained fundamental in the Platonic tradition, albeit to different extents.Thus, while Porphyry was generally interested in Pythagoreanism—he wrote a Life of Pythagoras—Iamblichus of Chalcis shows a peculiarly strong commitment to it.4 Indeed, besides his more traditional writings such as his commentaries on various Platonic dialogues,5 Iamblichus engaged in the ambitious project On the Pythagorean School (Περὶ τῆϲ Πυθαγορικῆϲ αἱρέϲεωϲ), which aimed to set out in 10 books an introduction to the whole of Pythagorean doctrine.6 3 In the same tradition, one should probably consider also other Platonists, the best known of whom is Thrasyllus [see Tarrant 1994]. The case of Theon of Smyrna is different, since his Expositio rerum mathematicarum ad legendum Platonem util- ium must be considered rather as a technical exegesis of the mathematical sections of Plato’s psychogony. Interesting papers on the relationship between Platonism and Pythagoreanism in the Imperial age are collected in Bonazzi, Lévy, and Steel 2007. I emphasize that it is necessary to suppose that a ‘Pythagorean’ tendency was some- how preserved also in the Hellenistic age: this is almost the unique historical condi- tion under which one can understand Moderatus’ work in the first century ad. It is worth noting, finally, that one among the ‘founders’ of post-Hellenistic Platonism, Eudorus of Alexandria, had a strong interest in the Pythagorean tradition, which he ‘used’ in order to sustain his new Platonic perspective: the so-called Pseudo- Pythagorica [see Centrone 2014] were probably produced, at least in the majority of cases, in the context of his school [see Bonazzi 2013]. 4 On the Neoplatonist interest in Pythagoreanism, see Macris 2014 and O’Meara 2014. O’Meara, especially, has contributed studies providing an authoritative basis for deeper inquiry. 5 The fragments are collected in Dillon 1973. 6 For a comprehensive account of Iamblichus, see Dillon 1987. Vinel supplies a brief sketch of him and his philosophical project on pp. 11–13. Federico M. Petrucci 344 His In Nicomachi arithmeticam belongs to this latter project, being the fourth treatise of the series. This outstanding work by Vinel, then, has the great merit of making available to scholars a new suitably critical edition of the Greek text, a good translation, and a very careful commentary. The book consists of an introduction [11–66] dealing with general interpretative problems in this text and focusing on its most important aspects; a French translation of a new critical text in Greek that is based on a complete collation of extant manuscripts and offers a useful apparatus of parallel passages along with an apparatus criticus [68–197]; a series of notes of commentary [199–265]; and an impressive set of indices, which make the book even more useful [267–344]. At the same time, Vinel aims to make clear the originality of Iamblichus’ In Nic. arith. by demonstrating that it is not a commentary on Nicomachus’ Introductio arithmetica (the fundamental treatise of Pythagorean-Platonic arithmetic in the Imperial age) but a work taking Nicomachus’ writings (both transmitted and now lost) as a point of departure in order to develop a new account of Pythagorean and Platonic arithmetic.7 Apart from some minor aspects, which I will discuss in due course, this goal is well achieved.8 The first among Vinel’s tasks, then, is to overcome the commonplace no- tion that has Iamblichus’ In Nic. arith. as only a sort of rearrangement of Nicomachus’ Intro. arith.9 He begins by focusing on the title and the de- 7 This point is convincingly achieved by referring to Iamblichus’ writing practice (usus scribendi) [14–15]: Iamblichus says that he will appeal to Nicomachus’ τέχνη, and at the same time he uses « τέχνη » to indicate a general field of interest or study [see also 200n10]. I am less inclined to agree with Vinel’s translation of « τέχνη » as ‘science’, since this somehow leaves aside the procedural aspects that are implied by « τέχνη ». 8 The idea that Nicomachus is the most important reference for Iamblichus, who, however, tries to supplement his doctrines, raises the issue of Iamblichus’ relation to the Euclidean tradition. Vinel emphasizes (quite briefly, though: see 23 and the notes at 216 ff.) that Iamblichus criticizes Euclid. Here it would have been helpful toexplore whether Iamblichus deliberately obscures other interpretations of topics dealt with by Nicomachus (e.g., Geminus’ account of the classification of sciences, which was to some extent known in Neoplatonism) or whether he was not acquainted with them. 9 Vinel offers a valuable survey of this prejudice against Iamblichus’ originality [13] but also emphasizes that scholars still could not avoid noting, albeit in a non-systematic and inconsistent way, some originality in Iamblichus’ work. 345 Aestimatio clared aim of the work [14–15]. As a matter of fact, Iamblichus never says that he is producing a commentary on Nicomachus but only that he will offer an introduction (εἰϲαγωγή)10 to arithmetic by taking Nicomachus’ writ- ings (and not only his Intro. arith.) as a starting point. Nevertheless, Vinel demonstrates that the title offered by the manuscript tradition «Περὶ τῆϲ Νικομάχου ἀριθμητικῆϲ εἰϲαγωγῆϲ» should not be accepted. He proposes three alternatives, the last of which («Περὶ τῆϲ Νικομάχου ἀριθμητικῆϲ») is (quite reasonably) adopted in the critical edition [15]. Accordingly, this conclusion is confirmed by means of a survey of Nicomachean doctrines and other sources in Iamblichus’ text [19–23]: Nicomachus, it turns out, is a sort of fil rouge for discussion that Iamblichus sometimes follows and sometimes leaves behind as he introduces new doctrines and terms.This would be typical of Iamblichus’ way of dealing with sources. Vinel compares this approach to Iamblichus’ treatment of Porphyry’s commentaries [20].11 There is one point that I should like to stress. As Vinel correctly notes [201n13], Iamblichus explicitly criticizes innovation (καινοτομία) and prefers to appeal to a well–established tradition, i.e., to that of Pythagoras, which he takes to have been advanced by Nicomachus. Vinel then indicates that this would seem to be inconsistent with Iamblichus’ own innovations but leaves the matter without further discussion. However, in my view, this should be a 10 If this is the case, however, it would have been helpful to expand on the relationship between this work and the prolegomena to mathematical works [see Mansfeld 1998] and to the literary genre to which In Nic. arith. belongs. 11 While the analysis of contents and titles of the work is effective, the latter point concerning Iamblichus’ attitude towards his sources is a bit controversial. First, al- though a passage of Simplicius’ In Arist. cat.[Heiberg 1894, 2.9–13] seems to work as a confirmation, one cannot establish a strict parallel between Iamblichus’ methods, since the context and form of the fragments of Iamblichus’ commentaries are usu- ally puzzling.