Dominic j. O'Meara

In this chapter Proclus' attitude to mathematics will be discussed for the most part as it is expressed in his Commentary on Euclid. His recourse in other works to mathematics in relation to physics and metaphysics will be examined in the next two chapters. In these chapters I point also to continuities between Proclus' general account of mathematics in the Commentary on Euclid and his use of mathematics in his other works. This presupposes, however, an account of the philosophy of mathematics of the Commentary on Euclid in particular as compared to the philosophies of mathematics of Nicomachus and of Iamblichus, the subject of the following pages.

What survives of Proclus' Commentary on Euclid may be divided into four parts: the first prologue (on general mathematics); the second prologue (on geometry); the exegesis of the definitions, postulates, and axioms at the head of Euclid's Elements I; and commentary on the theorems of Euclid Book I. ¹ 

¹ Proclus certainly intended to go beyond Elements I in his commentary, although he may not have covered the whole; cf. Heiberg (1882), 164-8; Morrow (1970), xxxi. Marinus' introduction to the Data can also be read as evidence of the study of Euclid in Proclus' school.

With the exception of the Elements of Theology, no other work of Proclus has been so often translated and studied. In particular the first prologue has attracted much interest and admiration, from Kepler, who inserted large extracts from it in his Harmonice Mundi, to Nicolai Hartmann, who used it as an important and relevant mathematical philosophy. ² 

² For an example of unbridled enthusiasm for the Commentary on Euclid cf. Steck (1945).

As the contents of the first prologue have been summarized often enough, ³ 

³ Hartmann's account (1909) can be used, allowance being made for his neo-Kantian approach.

I will not attempt another summary here. There remain, however, some basic issues connected with the first prologue, the resolution of which will require detailed analysis leading, as I hope to show, to some new insights into Proclus' assumptions in the first prologue.

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1. General Mathematics

(I) Iamblichus' On Pythagoreanism III and Proclus' On Euclid Prol. I

It has long been known that Proclus' first prologue substantially corresponds to the third book of Iamblichus' On Pythagoreanism, On General Mathematical Science. ⁴ 

⁴ Cf. Merlan (1960), 8 ff.; Mueller (1987b) for detailed comparisons. The correspondences between the chapters in Proclus (the first number in each set) and in Iamblichus (the following number[s]) are: 1: 1-2; 2: 3; 3-4: 5; 5: 8; 7: 11-12; 8: 15-16; 9: 26; 11: 27; and compare 12: 7; 15: 11 and 34. (I use the chapter divisions introduced by Morrow in his translation of Proclus, and his translation much revised.)

The most natural inference from this would seem to be that Proclus used Iamblichus' text extensively in writing his prologue. Proclus after all was Syrianus' prize pupil; he had read with Syrianus Aristotle's Metaphysics (Marinus, Vita Procli, ch. 13); and it has been shown in Chapter 6 above that Syrianus, in his treatment of the Metaphysics, referred the student to Iamblichus' On Pythagoreanism as a source-book for true (i.e. Pythagorean) mathematics, summarizing in particular the contents of Book III of Iamblichus' work.

This inference has not, however, been favoured in what little discussion there is of the question. It has been felt rather that Proclus is using, independently of Iamblichus, an older source which Iamblichus also used. ⁵ 

⁵ Proposed by Festa (without argument) in Iamblichus Comm., IX; intimated by Morrow (1970) in a note, 344-5; and developed by van der Waerden (1980). Under the influence of this idea Mueller (1987b) seems disinclined to conclude that Iamblichus is Proclus' source, although he does not seem to have good grounds against such a conclusion.

It has been claimed, in support of this position, that there are few verbal parallels between the texts of Proclus and of Iamblichus. The fact that this claim is manifestly untrue ⁶ 

⁶ Compare Procl. 6, 13-15, Iambl. 13, 18-21; Procl. 7, 20-7, Iambl. 18, 26-19, 6; Procl. 8, 25-9, 2, Iambl. 20, 1-5; Procl. 20, 25-21, 4, Iambl. 55, 12-16; Procl. 25, 1, Iambl. 57, 23; Procl. 25, 4-5, Iambl. 57, 24-5; Procl. 25, 23-26, 3, Iambl. 79, 20-4; Procl. 27, 27-8, Iambl. 82, 11-13; Procl. 32, 21-3, Iambl. 7, 10-13; Procl. 34, 20-35, 5, Iambl. 86, 26-87, 5.

is not, however, enough to disprove the position it is supposed to support. More decisive appears to be the fact that Proclus' text is better organized, clearer, more consistent than Iamblichus'. The greater clarity, it is assumed, derives from the original source and therefore testifies to the independence of Proclus' text with respect to Iamblichus. The original source has even been identified: Geminus, a mathematician of

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the first century ad . Chapters 1-7, 9, 11, and 13 of Proclus' prologue are taken, it is claimed, from Geminus, whose work reappears in a corrupt and confused form in Iamblichus. ⁷ 

⁷ Van der Waerden (1980). On Geminus' date, cf. Aujac (1975a), xix-xxiv. For a detailed critique of van der Waerden cf. O'Meara (1988), the major points of which are summarized in the following pages. Cf. also Mueller (1987a).

The assumption that the greater clarity and organization of Proclus' text argues against his use of Iamblichus' somewhat confused text, and for his proximity to an even earlier original, is admissible if one does not consider Proclus' other works. The implied denial that Proclus himself is the source of the clarity and organization of the chapters of the first prologue of his Commentary on Euclid becomes implausible, however, if one reflects, for example, on the skill Proclus demonstrates in organizing the whole tradition of Platonic metaphysics in his Elements of Theology. In comparison to this, the task of producing a clearer version of Iamblichus would have been a simple matter. In what follows I would like also to show that the thesis that Proclus' text comes in large part from Geminus, not from Iamblichus, entails anachronism. I shall then show how a chapter in Proclus' prologue (ch. 8) which is not claimed for Geminus—for it quotes Plotinus—represents a revision of the corresponding chapter in Iamblichus, and provides information on Proclus' modus operandi in composing his prologue, information that yields an explanation of various differences between Proclus' prologue and its Iamblichean source.

The chapters claimed for Geminus in Proclus' prologue contain matter which cannot be dated without anachronism to the first century ad . For example the terminology of the 'Chaldaean Oracles' (second century ad ) assimilated into a Neoplatonic context (first possible with Porphyry, then fully worked out by Iamblichus) can be found in ch. 2 (6, 9) in the reference to intelligible and 'hidden' principles (κ υφ ων χω ν). 'Hidden' is a word derived by the later Neoplatonists from the 'Chaldaean Oracles' and used as a technical term for the highest intelligible causes. ⁸