heart and soul, becoming one of its most prominent exponents. He speaks everywhere with the highest respect of his masters, and was in turn regarded with extravagant veneration by his contemporaries, as we learn from Marinus his pupil and biographer. On the death of Syrianus he was put at the head of the Neo-Platonic school. He was a man of untiring industry, as is shown by the number of books which he wrote, including a large number of commentaries, mostly on the dialogues of Plato. He was an acute dialectician, and pre-eminent among his contemporaries in the range of his learning1; he was a competent mathematician; he was even a poet. At the same time he was a believer in all sorts of myths and mysteries and a devout worshipper of divinities both Greek and Oriental. Though he was a competent mathematician, he was evidently much more a philosopher than a mathematician. This is shown. even in his commentary on Eucl. I., where, not only in the Prologues (especially the first), but also in the notes themselves, he seizes any opportunity for a philosophical digression. He says himself that he attaches most importance to "the things which require deeper study and contribute to the sum of philosophy""; alternative proofs, cases, and the like (though he gives many) have no attraction for him; and, in particular, he attaches no value to the addition of Heron to I. 47', which is of considerable mathematical interest. Though he esteemed mathematics highly, it was only as a handmaid to philosophy. He quotes Plato's opinion to the effect that "mathematics, as making use of hypotheses, falls short of the non-hypothetical and perfect science"".. "..."Let us then not say that Plato excludes mathematics from the sciences, but that he declares it to be secondary to the one supreme science"." And again, while "mathematical science must be considered desirable in itself, though not with reference to the needs of daily life," "if it is necessary to refer the benefit arising from it to something else, we must connect that benefit with intellectual knowledge (voepàv yvwow), to which it leads the way and is a propaedeutic, clearing the eye of the soul and taking away the impediments which the senses place in the way of the knowledge of universals (TV ὅλων)7.” We know that in the Neo-Platonic school the younger pupils learnt mathematics; and it is clear that Proclus taught this subject, and that this was the origin of the commentary. Many passages show him as a master speaking to scholars. Thus "we have illustrated 1 Zeller calls him "Der Gelehrte, dem kein Feld damaligen Wissens verschlossen ist." 2 Van Pesch observes that in his commentaries on the Timaeus (pp. 671-2) he speaks as no real mathematician could have spoken. In the passage referred to the question is whether the sun occupies a middle place among the planets. Proclus rejects the view of Hipparchus and Ptolemy because “ỏ бeoupyós” (sc. the Chaldean, says Zeller) thinks otherwise, "whom it is not lawful to disbelieve." Martin says rather neatly, "Pour Proclus, les Éléments d'Euclide ont l'heureuse chance de n'être contredits ni par les Oracles chaldaïques, ni par les spéculations des pythagoriciens anciens et nouveaux...... 3 Proclus, p. 84, 13. 5 ibid. p. 31, 20. • ibid. p. 429, 12. 6 ibid. p. 32, 2. 7 ibid. p. 27, 27 to 28, 7; cf. also p. 21, 25, pp. 46, 47. and made plain all these things in the case of the first problem, but it is necessary that my hearers should make the same inquiry as regards the others as well," and "I do not indicate these things as a merely incidental matter but as preparing us beforehand for the doctrine of the Timaeus." Further, the pupils whom he was addressing were beginners in mathematics; for in one place he says that he omits "for the present" to speak of the discoveries of those who employed the curves of Nicomedes and Hippias for trisecting an angle, and of those who used the Archimedean spiral for dividing an angle in any given ratio, because these things would be too difficult for beginners (Svo@ewpηTOVS TOîs eio ayoμévois). Again, if his pupils had not been beginners, it would not have been necessary for Proclus to explain what is meant by saying that sides subtend certain angles', the difference between adjacent and vertical angles" etc., or to exhort them, as he often does, to work out other particular cases for themselves, for practice (yvuvaoías eěveka)®. The commentary seems then to have been founded on Proclus' lectures to beginners in mathematics. But there are signs that it was revised and re-edited for a larger public; thus he gives notice in one place? "to those who shall come upon" his work (TOîS ÉVTEUķoμévois). There are also passages which could not have been understood by the beginners to whom he lectured, e.g. passages about the cylindrical helix, conchoids and cissoids. These passages may have been added in the revised edition, or, as van Pesch conjectures, the explanations given in the lectures may have been much fuller and more comprehensible to beginners, and they may have been shortened on revision. In his comments on the propositions of Euclid, Proclus generally proceeds in this way: first he gives explanations regarding Euclid's proofs, secondly he gives a few different cases, mainly for the sake of practice, and thirdly he addresses himself to refuting objections raised by cavillers to particular propositions. The latter class of note he deems necessary because of "sophistical cavils" and the attitude of the people who rejoiced in finding paralogisms and in causing annoyance to scientific men1o. His commentary does not seem to have been written for the purpose of correcting or improving Euclid. For there are very few passages of mathematical content in which Proclus can be supposed to be propounding anything of his own; nearly all are taken from the works of others, mostly earlier commentators, so that, for the purpose of improving on or correcting Euclid, there was no need for his commentary at all. Indeed only in one place does he definitely bring forward anything of his own to get over a difficulty which he finds in Euclid"; this is where he tries to 1 Proclus, p. 210, 18. 3 ibid. p. 272, 12. 2 ibid. p. 384, 2. 4 ibid. p. 238, 12. 8 ibid. p. 105. 10 ibid. p. 375, 8. prove the parallel-postulate, after first giving Ptolemy's attempt and then pointing out objections to it. On the other hand, there are a number of passages in which he extols Euclid; thrice1 also he supports Euclid against Apollonius where the latter had given proofs which he considered better than Euclid's (I. 10, II, and 23). Allusion must be made to the debated question whether Proclus continued his commentaries beyond Book I. His intention to do so is clear from the following passages. Just after the words above quoted about the trisection etc. of an angle by means of certain curves he says, "For we may perhaps more appropriately examine these things on the third book, where the writer of the Elements bisects a given circumference?." Again, after saying that of all parallelograms which have the same perimeter the square is the greatest "and the rhomboid least of all," he adds: “But this we will prove in another place; for it is more appropriate to the (discussion of the) hypotheses of the second book"." Lastly, when alluding (on I. 45) to the squaring of the circle, and to Archimedes' proposition that any circle is equal to the right-angled triangle in which the perpendicular is equal to the radius of the circle and the base to its perimeter, he adds, "But of this elsewhere"; this may imply an intention to treat of the subject on Eucl. XII., though Heiberg doubts it. But it is clear that, at the time when the commentary on Book I. was written, Proclus had not yet begun to write on the other Books and was uncertain whether he would be able to do so: for at the end he says", "For my part, if I should be able to discuss the other books' in the same manner, I should give thanks to the gods; but, if other cares should draw me away, I beg those who are attracted by this subject to complete the exposition of the other books as well, following the same method, and addressing themselves throughout to the deeper and better defined questions involved” (τὸ πραγματειώδες πανταχοῦ καὶ εὐδιαίρετον μεταδιώκοντας). There is in fact no satisfactory evidence that Proclus did actually write any more commentaries than that on Book 1.8 The contrary view receives support from two facts pointed out by Heiberg, viz. (1) that the scholiast's copy of Proclus was not so much better than our 1 Proclus, p. 280, 9; p. 282, 20; pp. 335, 336. 3 ibid. p. 398, 18. Heiberg, Euklid-Studien, p. 165, note. 2 ibid. p. 272, 14. 7 The words in the Greek are : εἰ μὲν δυνηθείημεν καὶ τοῖς λοιποῖς τὸν αὐτὸν τρόπον ἐξελθεῖν. For ἐξελθεῖν Heiberg would read ἐπεξελθεῖν. 8 True, a Vatican Ms. has a collection of scholia on Books 1. (extracts from the extant commentary of Proclus), 11., V., VI., X. headed Εἰς τὰ Εὐκλείδου στοιχεία προλαμβανόμενα ἐκ τῶν Πρόκλου σποράδην καὶ κατ ̓ ἐπιτομήν. Heiberg holds that this title itself suggests that the authorship of Proclus was limited to the scholia on Book I.; for poλaμßavóμeva ék Tŵy IIpókλov suits extracts from Proclus' prologues, but hardly scholia to later Books. Again, a certain scholium (Heiberg in Hermes XXXVIII., 1903, p. 341, No. 17) purports to quote words from the end of "a scholium of Proclus" on x. 9. The words quoted are from the scholium x. No. 62, one of the Scholia Vaticana. But none of the other, older, sources connect Proclus' name with x. No. 62; it is probable therefore that a Byzantine, who had in his MS. of Euclid the collection of Schol. Vat. and knew that those on Book 1. came from Proclus, himself attached Proclus' name to the others. MSS. as to suggest that the scholiast had further commentaries of Proclus which have vanished for us1; (2) that there is no trace in the scholia of the notes which Proclus promised in the passages quoted above. Coming now to the question of the sources of Proclus, we may say that everything goes to show that his commentary is a compilation, though a compilation "in the better sense" of the term. He does not even give us to understand that we shall find in it much of his own; "let us," he says, "now turn to the exposition of the theorems proved by Euclid, selecting the more subtle of the comments made on them by the ancient writers, and cutting down their interminable diffuseness...": not a word about anything of his own. At the same time, he seems to imply that he will not necessarily on each occasion quote the source of each extract from an earlier commentary; and, in fact, while he quotes the name of his authority in many places, especially where the subject is important, in many others, where it is equally certain that he is not giving anything of his own, he mentions no authority. Thus he quotes Heron by name six times; but we now know, from the commentary of an-Nairīzī, that a number of other passages, where he mentions no name, are taken from Heron, and among them the, not unimportant addition of an alternative proof to I. 19. Hence we can by no means conclude that, where no authority is mentioned, Proclus is giving notes of his own. The presumption is generally the other way; and it is often possible to arrive at a conclusion, either that a particular note is not Proclus' own, or that it is definitely attributable to someone else, by applying the ordinary principles of criticism. Thus, where the note shows an unmistakable affinity to another which Proclus definitely attributes to some commentator by name, especially when both contain some peculiar and distinctive idea, we cannot have much doubt in assigning both to the same commentator*. Again, van Pesch finds a criterion in the form of a note, where the explanation is so condensed as to be only just intelligible; the note is that in which a converse of I. 32 is proved", the proposition namely that a rectilineal figure which has all its interior angles together equal to two right angles is a triangle. It is not safe to attribute a passage to Proclus himself because he uses the first person in such expressions as "I say" or "I will prove" -for he was in the habit of putting into his own words the substance of notes borrowed from others-nor because, in speaking of an While one class of scnoua (Schol. Vat.) have some better readings than our MSS. of Proclus have, and partly fill up the gaps at 1. 36, 37 and 1. 41-43, the other class (Schol. Vind.) derive from an inferior Proclus MS. which also had the same lacunae. 2 Knoche, Untersuchungen über des Proklus Diadochus Commentar zu Euklid's Elementen (1862), p. 11. 3 Proclus, p. 200, 10-13. Instances of the application of this criterion will be found in the discussion of Proclus' indebtedness to the commentaries of Heron, Porphyry and Pappus. 5 Van Pesch attributes this converse and proof to Pappus, arguing from the fact that the proof is followed by a passage which, on comparison with Pappus' note on the postulate that all right angles are equal, he feels justified in assigning to Pappus. I doubt if the evidence is sufficient. H. E. 3 objection raised to a particular proposition, he uses such expressions as "perhaps someone may object” (ἴσως δ ̓ ἄν τινες ἐνσταῖεν...): for sometimes other words in the same passage.indicate that the objection had actually been taken by someone1. Speaking generally, we shall not be justified in concluding that Proclus is stating something new of his own unless he indicates this himself in express terms. As regards the form of Proclus' references to others by name, van Pesch notes that he very seldom mentions' the particular work from which he is borrowing. If we leave out of account the references to Plato's dialogues, there are only the following references to books: the Bacchae of Philolaus, the Symmikta of Porphyry3, Archimedes On the Sphere and Cylinder, Apollonius On the cochlias", a book by Eudemus on The Angle, a whole book of Posidonius directed against Zeno of the Epicurean sect', Carpus' Astronomy, Eudemus' History of Geometry, and a tract by Ptolemy on the parallel-postulate". Again, Proclus does not always indicate that he is quoting something at second-hand. He often does so, e.g. he quotes Heron as the authority for a statement about Philippus, Eudemus as attributing a certain theorem to Oenopides etc.; but he says on I. 12 that "Oenopides first investigated this problem, thinking it useful for astronomy" when he cannot have had Oenopides' work before him. It has been said above that Proclus was in the habit of stating in his own words the substance of the things which he borrowed. We are prepared for this when we find him stating that he will select the best things from ancient commentaries and "cut short their interminable diffuseness," that he will "briefly describe" (σνvтoμws iσтoρñσai) the other proofs of I. 20 given by Heron and Porphyry and also the proofs of 1. 25 by Menelaus and Heron. But the best evidence is of course to be found in the passages where he quotes works still extant, e.g. those of Plato, Aristotle and Plotinus. Examination of these passages shows great divergences from the original; even where he purports to quote textually, using the expressions "Plato says," or "Plotinus says," he by no means quotes word for word". In fact, he seems to have had a positive distaste for quoting textually from other works. He cannot conquer this even when quoting from Euclid; he says in his note on I. 22, "we will follow the words of the geometer" but fails, nevertheless, to reproduce the text of Euclid unchanged". We now come to the sources themselves from which Proclus drew 1 Van Pesch illustrates this by an objection refuted in the note on 1. 9, p. 273, 11 sqq. After using the above expression to introduce the objection, Proclus uses further on (p. 273, 25) the term "they say" (paσiv). Proclus, p. 22, 15. 4 ibid. p. 71, 18. 6 ibid. p. 125, 8. 8 ibid. p. 241, 19. 10 ibid. p. 362, 15. * ibid. p. 56, 25. 5 ibid. p. 105, 5. 9' ibid. p. 352, 15. 11 See the passages referred to by van Pesch (p. 70). (p. 21, 19) where he quotes Plotinus, using the expression The most glaring case is a passage "Plotinus says... Comparison with Plotinus, Ennead. 1. 3, 3, shows that very few words are those of Plotinus himself; the rest represent Plotinus' views in Proclus' own language. 12 Proclus, p. 330, 19 sqq. |