This valuable book, the second issue in an important series devoted to Greek mathematics, provides a comprehensive study on Domninus, a careful critical edition of his works and a detailed commentary (with extensive indexes). Although this is the third translation of Domninus' work since 2000,1 its publication is welcome: as Riedlberger shows, Domninus' traditional (and fossilised) image as an "Euclidean maverick" is wrong, and the writings attributed to him needed to be re-edited (or edited for the first time in one case) and discussed, both in general terms (e.g. with respect to their authorship) and by addressing specific issues. Riedlberger's book meets our expectations: Domninus (a contemporary and fellow student of Proclus) "emerges as a fairly standard late antique Platonic philosopher" (p.91-92).
The introduction consists of five parts. The first three provide an overall (but detailed) description of the philosophical framework (p.19-41) and a close analysis of both indirect witnesses on Domninus (p.43-64) and his works (p.65-90). The forth section is a general outline (p.91-92), the fifth (p.93-106) a philological praefatio. In the first part Riedlberger offers an overview of the history of Platonism in late antiquity, and in particular of the School of Athens, where Domninus received his education: this survey is conducted with precision and closely follows the primary sources. 2 Riedlberger's aim is to emphasise that studying and writing about mathematics (in the ancient sense of the word) were common pursuits in this context. He thus places Domninus squarely within the philosophical framework of late Neoplatonism, as opposed to treating him as an Euclidean maverick.
Riedlberger next discusses the (scarce) indirect witnesses. His study of these is of great importance, since Domninus' traditional image is based on them, and above all on Damascius' account (apud Suda Δ1355 Δομνῖνος, plus M815). Riedlberger's analysis represents a fresh start, since he argues that Damascius' picture of Domninus as a mere mathematician who did not follow a philosophical lifestyle and whom Proclus attacks for his philosophical incompetence, is misleading. While the quarrel between Domninus and Proclus can hardly be Damascius' own invention, his negative characterisation may be regarded as both "standard" and tendentious. On the other hand, Riedlberger presents some points that go against the scholarly consensus (e.g., Domninus was not Jewish [pp.53-55]); he confirms that the philosopher was a contemporary and fellow student of Proclus, and later a student of Syrianus, and that what lay at the basis of the disagreement between Proclus and Domninus was a "normal" philosophical polemic. This view is confirmed (p.57-60) by an incidental reference in Marinus (26,1-14) and above all by Proclus' Commentary on Plato's Timaeus (p.60-63), where he ascribes to his ἑταῖρος "physicalist/mathematical" interpretations (In Tim. I 109,30-110,23 and 122,18-123,4), probably delivered in oral discussions. In these pages of Proclus' Commentary, which was written before Domninus' departure from Athens (to Syria), it is not possible to detect any grudge, although it is worth noting that Proclus does not quote Domninus anywhere else. All in all, Riedlberger's conclusions are remarkable for their sobriety (p.63-64): "Domninus does not appear as an exceptional figure" in this context, and there is nothing which suggests he was a "scientific outsider amongst mystical-minded philosophers".
At this point, Riedlberger refashions Domninus' image by referring to "his" works, starting from those which are lost (p.65-72). First, Riedlberger discusses the possibility that the philosopher may have composed a Commentary on Plato's Timaeus (and seems to conclude that it is unlikely). By contrast, we can be sure that he composed a Commentary on Aristotle's Sophistical Refutations: a remarkable inquiry into catalogues and Humanistic sources indicates that this work was contained in a manuscript brought to Italy by John Lascaris, and was preserved in another manuscript in the Escorial up until the fire of 1671. Finally, we cannot know whether Domninus composed the Arithmetical Stoicheiosis which he announces at the end of the Encheiridion, and which in any case was meant to be something like a "technical" commentary on the Timaeus (p.195-198). Albeit rather speculative, this conclusion is quite plausible; as a parallel, however, it might be best to refer to the tradition of technical exegeses which started in Middle Platonism (with Theon,3 Aelianus, Adrastus), than to the lost appendix of Proclus' Commentary on the Timaeus (to which Riedlberger draws the reader's attention).
Since, as Riedlberger demonstrates, only the Encheiridion is ascribable to Domninus, this work alone can be considered a direct witness (p.72-77). It consists of a series of arithmetical definitions and related examples, which always find parallels elsewhere (except in §40). In particular, Riedlberger identifies the following sections (p.74): the definition of monad and number and the classification of numbers (§1-9); the Greek system of number representation (§10-14); relatively prime and composite numbers (§17-18); relations of equality and inequality (§20-31); perfect numbers (§32); being relatively composite or prime as a property of composite or prime numbers (§344); means and proportions (§35-41); figurate numbers (§42-55); summing up and outlook (§56-57). Riedlberger's most remarkable claim concerns the relation between this work and Euclid5: while scholars have traditionally regarded the Encheiridion as a Euclidean work, Riedlberger conclusively indicates that Euclid is a source for Domninus, yet not the most important one, for that probably is Nicomachus' Introductio Arithmetica. On the one hand, Euclidean notions are integrated within a scientific "context" that is very reminiscent of that of Nicomachus; on the other hand, "there is not a single proof within the Encheiridion" (p.75-76). This claim, which is demonstrated passim in the commentary, overturns the traditional image of Domninus, which Riedlberger replaces with a quite different one: Domninus' work -- he suggests – was an epitome of Nicomachus' Introductio. This would also be testified by the title, Ἐγχειρίδιον ἀριθμητικῆς εἰσαγωγῆς, where ἀριθμητικῆς εἰσαγωγῆς, which would refer directly to Nicomachus' Ἀριθμητικὴ εἰσαγωγή, as well as by several other (implicit) references to Nicomachus. This point seems controversial to me, although it does not invalidate Riedlberger's general thesis. First, the only further argument adduced (p.77), that Nicomachus' work was the only technical treatise (besides the Elements) which was definitely used in the School of Athens, is imprecise or at least questionable: Theon of Smyrna (albeit not his Expositio), for example, is also quoted by Proclus (In Tim. I 82, 14-15 Diehl). In any case, if this assumption were correct, it would make Nicomachus' work the only introduction to arithmetic available to members of the School, in which case Domninus must have conceived of his work more as an epitome rei than an epitome auctoris. In addition, although Domninus constantly refers to Nicomachus, he sometimes (§96 and §17-18) seems to adopt an Euclidean perspective more than a Nicomachean one. In this case too, the Encheiridion would appear to be an epitome rei, albeit one chiefly grounded on Nicomachus. However, since we are unlikely to demonstrate conclusively that the Encheiridion is an epitome auctoris rather than an epitome rei, it might be safer to avoid specific claims.
Two further works are edited and commented upon, How to remove a ratio from a ratio and Scholia to Nichomachus. Riedlberger conclusively demonstrates that both are spurious. The first work (p.77-83) is a step-by- step guide to the division of a fraction by a fraction, which is developed by referring to four different methods (p.207- 210). Riedlberger emphasises the similarities between this work and passages of introductory writings to classical mathematical works (p.79-83), and concludes (if only speculatively) that it was composed in the context of a philosophical school in the 5th-6th century. Its attribution to Domninus is only due to his (previously unclear) textual tradition, on which Riedlberger throws a remarkable amount of light (p.77-78 and 103- 104). Scholia to Nicomachus (p.83-88) is a heterogeneous collection of notes on Nicomachus' Introductio which is transmitted (without any attribution to Domninus) by only one manuscript (S = Par. Gr. 2531) after the Encheiridion and the How to, and which is edited here for the first time. These notes probably belong to the same chronological milieu as the How to, and its content is not consistent with Domninus' perspective (p.229-232).
The Prolegomena to the edition describe the manuscripts (some of which are here examined for the first time) and convincingly establish stemmatic relations (p.93-106). Even though the dating of the manuscripts, the most ancient of which dates back to the 14th century, is unproblematic, one would have expected more detailed palaeographic (e.g. ductus and accuracy) and codicological (e.g. watermarks) descriptions. These would have provided not only useful evidence and indications of the aims of the scribes – i.e. on the intended use of each manuscript – but also helpful remarks for an evaluation of minor textual divergences. Regardless of this, Riedlberger far outdoes all previous editions. The philological notes, which are widespread throughout the commentary, are extensive and conclusive (see e.g. p.161-162, 164-165, 172-174, 174-178). The edited text is much improved, as is the critical apparatus. Only in a single case (p.203) one wonders whether the editor's conclusions are well grounded. The scribe of Chalc. Pan. 157 (K, containing How to) is said to have modified the text of §1 by conjecture, since we find an inversion there, λόγον ἐκ λόγου ἀφελεῖν for λόγον ἀφελεῖν ἐκ λόγου, and μετὰ τοῦτο instead of μετὰ τοῦτον; from that point onwards, as Riedlberger admits, K faithfully copies his model. However, given that the first phrase reproduces the wording of the title and the second is "more natural at first glance" (as the author acknowledges), and considering that no other conjectures occur, these divergences are likely to be involuntary.
The valuable running commentary consists of philological and lexical notes, detailed explanations of mathematical concepts (both technical and historical, with rich lists of parallels: e.g. p.151-156 on the system of numbers, p.178-185 on means and proportions, p.185-195 on figured numbers), and careful explanations of difficult passages.
From now onwards Riedlberger's "new Domninus" must be considered the only authoritative book on Domninus and his writing(s), as well as a reference book for the mathematical themes which Domninus (and others) dealt with.
1. In 2000 F. Romano (who is repeatedly criticised by Riedlberger) published a monograph on Domninus with an edition and translation of the Encheiridion, and P. Brown published an English translation with notes of the same work (The Harvard Review of Philosophy VIII, 82-100).
2. In passing, I wish to note some minor shortcomings. First, Riedlberger says (p.20) that he would rather avoid the misleading categories of Middle Platonism and Neoplatonism; however, he does not refer to the debate on this topic initiated by M. Frede (‘Numenius’, in ANRW II 36, 2 (1987), 1034-1075, at p.1040ff.) and P.L. Donini (‘Medioplatonismo e filosofi medioplatonici. Una raccolta di studi’, Elenchos 11 (1990), 79-93, at p.81-83). Secondly, Riedlberger's claim that "Late antique Platonists found in Plato invariably that which they wanted to find" (p.29-30) appears imprecise, since it does not consider the "internal" perspective adopted by Platonists when dealing with Plato. Thirdly, it is too strong to claim that according to Neopythagoreans Plato was a "plagiarist" of Pythagoras (p.32); rather, Neopythagoreans thought of themselves as Platonists and believed that Plato had ameliorated and systematised Pythagorean doctrines.
3. As concerns Riedlberger's account of Theon, I must at least note that it seems very doubtful that Theon's Expositio has not been transmitted in its entirety (p.33). Furthermore, Theon's Expositio should be considered a technical exegesis of the Timaeus' psychogony more than a (more or less satisfying) technical treatise.
4. §33, not considered by Riedlberger in this summary, is in fact a very short transitional paragraph.
5. Another important aspect of Domninus' method, which is well discussed by Riedlberger (p.149-151), is the quite systematic use of the distinction of numbers according to the matrices "by themselves/with regard to one another" and "by type/ by multitude of monads".
6. Here (148-149, on §9) Riedlberger claims that Domninus' sentence διὸ καὶ ἡ δυὰς τῶν πρώτων καὶ ἀσυνθέτων ἀριθμῶν εἶναι δοκεῖ is meant to uphold Nicomachus' position (that the dyad is not a prime number) against Euclid's. However, it seems more natural that Domninus is mixing both perspectives, and then somehow "correcting" Nicomachus.