Dominic j. O'Meara

In the preceding two chapters the overall intentions and structure of Iamblichus' work On Pythagoreanism have been examined as they appear in the first four extant books and in so far as they can be discovered in Psellus' excerpts from Books V-VII. It remains to review briefly the results achieved and to attempt some assessment of the significance of the work as a whole, in relation both to Iamblichus' other works and interests and to the earlier versions of Pythagoreanism sketched above in Chapter 1. The importance of Iamblichus' Pythagoreanizing programme for the later history of Greek philosophy will be the subject of Part II of this book.

But have we reached, even with the help of Psellus' excerpts, an adequate view of the work On Pythagoreanism as a whole? There were after all three further books (Books VIII-X) concerning which next to nothing is known. Might they not, were they to resurface some day, alter our general picture of the work? The last three books concerned geometry, music, and astronomy 'according to the Pythagoreans'. In the general scheme of the work they occupied positions comparable to that of Book IV, the Nicomachean introduction to arithmetic. We can therefore expect that they would have been similar in function and character: elementary introductions to the remaining three mathematical sciences, using 'Pythagorean' sources.

But I think we can go further than this. Iamblichus might have composed, for Books VIII-X, new elementary introductions to geometry, music, and astronomy, on occasion citing 'Pythagorean' authorities. It is more likely, however, that he would have reissued 'Pythagorean' introductions to these sciences if such were available, as he did in the case of Nicomachus' Introduction to Arithmetic (= Book IV). And certainly such introductions were available to him. But one can only speculate about precisely which ones he would have used. Given his high standing as a Pythagorean in Iamblichus' eyes, Nicomachus would seem the most likely author he would have chosen and there is

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some evidence that Nicomachus wrote introductions to mathematical sciences other than arithmetic. Nicomachus himself refers to an Introduction to Geometry in his Introduction to Arithmetic (83, 3 ff.). His brief Manual of Harmonics refers to a larger work on harmonics or music which is in all probability one of the major Greek sources of Boethius' De Musica. And finally Simplicius refers to a Pythagorean discovery in astronomy reported by Iamblichus 'following Nicomachus'. ¹ 

¹ Simplicius, In De caelo 507, 12-14 = Iamblichus, fr. 154 Larsen. Cf. Burkert (1972), 101, Tarán (1974) on Nicomachus' works. The language in Syrianus, In met. 103, 6 suggests use by the Neoplatonists of Nicomachean introductions to mathematical sciences other than arithmetic. However for geometry Syrianus and Proclus used Euclid (below, Ch. 8).

It is not certain however that Simplicius is inspired, directly or indirectly, by On Pythagoreanism X, or that his report means that Nicomachus wrote an Introduction to Astronomy which Iamblichus used.

Whatever Pythagorean materials Iamblichus had recourse to in On Pythagoreanism VIII-X, I think we can safely say that these volumes were elementary introductions to the three mathematical sciences subordinate to arithmetic. It is thus not likely that they would have significantly altered the system of Pythagorean philosophy or the Pythagorean structure of reality as these are presented in Books I-VII. From the point of view of Iamblichus' reader, the highest stage in his philosophical progress is reached, not in the final books, but in Book VII where a transition is prepared from the realm of mathematical science to the highest levels of Pythagorean philosophy.