71 In met. 140, 10-15 (cf. Psellus' excerpts, 73-4). For the intelligible/intellectual distinction in Porphyry as compared to Iamblichus cf. P. Hadot (1968), I 98-101.

Thus in On Pythagoreanism VII Iamblichus was more precise than in the earlier books concerning the difference between levels of reality: above the world of enmattered forms (physical numbers) were to be found 'hypostatic number' (?), self-moved being (or number), intellectual being (or number), intelligible or pure being or the Forms (ideal number), and, transcending all, the gods (or divine number).

There is another, more striking difference between the passage in Book III and Psellus' excerpt at 53-8. The former text calls for an extrapolation upwards of mathematics, 'according to an appropriate

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assimilation ( π ικασ α)', towards the divine, whereas the text of Psellus presents the hierarchy of sorts of number with the purpose of stressing the transcendence of the divine and thus of denying such extrapolations (ο τ ναλογ αις π ικ ςων, 55). The conflict can be resolved by reference to Psellus' excerpts at 81-7: assimilation upwards to the divine unities ( ν σ ις) of the 'natural sequence' (or 'flow') of numbers is an inferior approach, an analogy. A more accurate approach would consist in starting, not from below, but from the divine itself; this method depends, it appears, not on analogy, but on 'higher insight'. ⁷² 

72 Κατ κ ττους ννο ας (87). Cf. Above, p. 47.

This difference in approach seems to be followed in Psellus' excerpts. Having emphasized the transcendence of divine number (54-8), they discuss divine number on the basis of the inferior mathematical analogies (64-80), and then refer to the superior, direct insight into divine number (84-7). What has already been learnt from Psellus' excerpts (81-7) about the approach by analogy points to an agenda: relating the numbers of the natural sequence of numbers (i.e. from the monad to the decad) to the gods. This programme is also followed in the excerpts relating to mathematical analogy: they discuss successively the monad (70-5), dyad (75-8), and triad (78-80). Had Psellus cared to excerpt more, we would in all likelihood find the subsequent numbers of the decad also represented.

I believe that we are now in a position to draw some tentative conclusions about the structure of On Pythagoreanism VII. Iamblichus may have treated in this work of ideal or intelligible number, at least to the extent that he may have responded in some measure to Aristotle's attack on theories of separately existing Forms or ideal numbers in Metaphysics MN. Although Iamblichus clearly believed in the distinct reality of ideal numbers, he was especially concerned in On Pythagoreanism VII with a higher sort of number, 'divine number'. To deal with this subject he took up again the traditional Pythagorean decadic order of exposition, such as was used by Nicomachus in his Theologoumena arithmeticae, fitting each number, from monad to decad, to the gods. If we wish to speculate about what works Iamblichus would have used here, we can say that they certainly did not include Aristotle's Metaphysics, but probably were traditional Pythagorean works on the decad such as are reflected in Anatolius' piece and used by Nicomachus' Theologoumena. One of these might have been the Pythagorean Sacred Discourse to which Syrianus refers in connection

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with his mention of Iamblichus' On Pythagoreanism VII. ⁷³ 

⁷³ In met. 140, 16-18, and above, p. 20, below, p. 93.

However, Iamblichus considered the method by mathematical analogy inferior to direct insight into the gods, a subject on which he also touched to some degree.

At the same time a certain amount has been learnt concerning the structure of reality presented in On Pythagoreanism VII. A more precise picture of this structure emerges than that found in earlier books of the work. The earlier lack of clarity about whether or not the divine and pure being are distinct is removed: ⁷⁴ 

⁷⁴ Cf. above, Ch. 2 n. 46.

the level of pure being (or the intelligible) is inferior to the level of the gods which is beyond being. So also does a distinction between the intelligible and the intellectual levels, scarcely drawn earlier, clearly emerge in On Pythagoreanism VII. This structure of reality is not, however, complete. In what follows I shall attempt to fill it out further, to the extent that Psellus' excerpts permit.

(III) The Approach by Analogy

Most of Psellus' theological excerpts (64-80) relate to this subject. They are not for all that particularly representative or informative. Their defectiveness and the amount of omission will become clearer as they are studied in detail.

The excerpts first suggest that Iamblichus presented a broad analogy between what is highest in numbers (the one, limit, equality) and the divine (64-6). This corresponds to the broad assimilations between numbers and enmattered forms (On Phys. Numb. 8-12), and between numbers and virtue (On Eth. and Theol. Arith. 24-30) which introduced the more detailed accounts of physical and ethical arithmetic in On Pythagoreanism V-VI.

This is followed in Psellus' excerpts by some lines (66-70) having to do with the cause or principle ( χ ) of divine number. The correspondence between this and the identification of causes and first principles in physical and ethical arithmetic is pointed out. Thus prior to unified divine number is to be found a 'uniform unity' (μονο ιδ ς νωσις). The language used here implies that despite its unity, divine number (or the divine) ⁷⁵