(3) Pappus gave a pretty proof of I. 5. This proof has, I think, been wrongly understood; on this point see my note on the proposition. (4) On I. 47 Proclus says: "As the proof of the writer of the Elements is manifest, I think that it is not necessary to add anything further, but that what has been said is sufficient, since indeed those who have added more, like Heron and Pappus, were obliged to make use of what is proved in the sixth book, without attaining any important result." We shall see what Heron's addition consisted of; what Pappus may have added we do not know, unless it was something on the lines of his extension of I. 47 found in the Synagoge (IV. p. 176, ed. Hultsch). We may fairly conclude, with van Pesch, that Pappus is drawn upon in various other passages of Proclus where he quotes no authority, but where the subject-matter reminds us of other notes expressly assigned to Pappus or of what we otherwise know to have been favourite questions with him. Thus: I. We are reminded of the curvilineal angle which is equal to but not a right angle by the note on I. 32 to the effect that the converse (that a figure with its interior angles together equal to two right angles is a triangle) is not true unless we confine ourselves to rectilineal figures. This statement is supported by reference to a figure formed by four semicircles whose diameters form a square, and one of which is turned inwards while the others are turned outwards. The figure forms two angles "equal to" right angles in the sense described by Pappus on Post. 4, while the other curvilineal angles are not considered to be angles at all, and are left out in summing the internal angles. Similarly the allusions in the notes on I. 4, 23 to curvilineal angles of which certain moon-shaped angles (unvoeideîs) are shown to be "equal to" rectilineal angles savour of Pappus. 2. On I. 9 Proclus says that "Others, starting from the Archimedean spirals, divided any given rectilineal angle in any given ratio." We cannot but compare this with Pappus IV. p. 286, where the spiral is so used; hence this note, including remarks immediately preceding about the conchoid and the quadratrix, which were used for the same purpose, may very well be due to Pappus. 3. The subject of isoperimetric figures was a favourite one with Pappus, who wrote a recension of Zenodorus' treatise on the subject'. Now on I. 35 Proclus speaks about the paradox of parallelograms having equal area (between the same parallels) though the two sides between the parallels may be of any length, adding that of parallelograms with equal perimeter the rectangle is greatest if the base be given, and the square greatest if the base be not given etc. He returns to the subject on I. 37 about triangles. Compare' also his note on I. 4. These notes may have been taken from Pappus. 1 Proclus, p. 429, 9-15. 2 Van Pesch, De Procli fontibus, p. 134 sqq. 4 Pappus, v. pp. 304-350; for Zenodorus' own treatise see Hultsch's Appendix, pp. 1189 -1211. • Proclus, pp. 396—8. • ibid. pp. 403—4. 3 Proclus, p. 272, 10. 7 ibid. pp. 236—7. 4. Again, on I. 21, Proclus remarks on the paradox that straight lines may be drawn from the base to a point within a triangle which are (1) together greater than the two sides, and (2) include a less angle provided that the straight lines may be drawn from points in the base other than its extremities. The subject of straight lines satisfying condition (1) was treated at length, with reference to a variety of cases, by Pappus', after a collection of "paradoxes" by Erycinus, of whom nothing more is known. Proclus gives Pappus' first case, and adds a rather useless proof of the possibility of drawing straight lines satisfying condition (2) alone, adding that "the proposition stated has been proved by me without using the parallels of the commentators." By "the commentators" Pappus is doubtless meant. 5. Lastly, the "four-sided triangle," called by Zenodorus the "hollow-angled," is mentioned in the notes on I. Def. 24-29 and I. 21. As Pappus wrote on Zenodorus' work in which the term occurred, Pappus may be responsible for these notes. IV. Simplicius. According to the Fihrist, Simplicius the Greek wrote "a commentary to the beginning of Euclid's book, which forms an introduction to geometry." And in fact this commentary on the definitions, postulates and axioms (including the postulate known as the ParallelAxiom) is preserved in the Arabic commentary of an-Nairizi®. On two subjects this commentary of Simplicius quotes a certain "Aganis," the first subject being the definition of an angle, and the second the definition of parallels and the parallel-postulate. Simplicius gives word for word, in a long passage placed by an-Nairīzī after I. 29, an attempt by "Aganis" to prove the parallel-postulate. It starts from a definition of parallels which agrees with Geminus' view of them as given by Proclus', and is closely connected with the definition given by Posidonius. Hence it has been assumed that "Aganis" is none other than Geminus, and the historical importance of the commentary of Simplicius has been judged accordingly. But it has been recently shown by Tannery that the identification of "Aganis" with Geminus is practically impossible. In the translation of Besthorn-Heiberg Aganis is called by Simplicius in one place "philosophus Aganis," in another "magister noster Aganis," in Gherard's version he is "socius Aganis" and "socius noster Aganis." These expressions seem to leave no doubt that Aganis was a contemporary and friend, if not master, of Simplicius; and it is impossible to suppose that Simplicius (fl. about 500 A.D.) could have used them of a man who lived four and 1 Pappus 111. pp. 104-130. Proclus, p. 165, 24; cf. pp. 328, 329. 5 Fihrist (tr. Suter), p. 21. • An-Nairizi, ed. Besthorn-Heiberg, pp. 9-41, 119-133, ed. Curtze, pp. 1-37, 65-73. The Codex Leidensis, from which Besthorn and Heiberg are editing the work, has unfortunately lost some leaves so that there is a gap from Def. 1 to Def. 35 (parallels). The loss is, however, made good by Curtze's edition of the translation by Gherard of Cremona. 8 ibid. p. 176, 7. 7 Proclus, p. 177, 21. Bibliotheca Mathematica, II, 1900, pp. 9-11. a half centuries before his time. A phrase in Simplicius' word-forword quotation from Aganis leads to the same conclusion. He speaks of people who objected "even in ancient times" (iam antiquitus) to the use by geometers of this postulate. This would not have been an appropriate phrase had Geminus been the writer. I do not think that this difficulty can be got over by Suter's suggestion' that the passages in question may have been taken out of Heron's commentary, and that an-Nairīzi may have forgotten to name the author; it seems clear that Simplicius is the person who described "Aganis." Hence we are driven to suppose that Aganis was not Geminus, but some unknown contemporary of Simplicius. Considerable interest will however continue to attach to the comments of Simplicius so fortunately preserved. Proclus tells us that one Aegaeas (? Aenaeas) of Hierapolis wrote an epitome of the Elements; but we know nothing more of him or of it. 1 Zeitschrift für Math. u. Physik, XLIV., hist.-litt. Abth. p. 61. 2 The above argument seems to me quite insuperable. The other arguments of Tannery do not, however, carry conviction to my mind. I do not follow the reasoning based on Aganis' definition of an angle. It appears to me a pure assumption that Geminus would have seen that Posidonius' definition of parallels was not admissible. Nor does it seem to me to count for much that Proclus, while telling us that Geminus held that the postulate ought to be proved and warned the unwary against hastily concluding that two straight lines approaching one another must necessarily meet (cf. a curve and its asymptote), gives no hint that Geminus did try to prove the postulate. It may well be that Proclus omitted Geminus' "proof" (if he wrote one) because he preferred Ptolemy's attempt which he gives (PP. 365-7). 3 Proclus, p. 361, 21. CHAPTER IV. PROCLUS AND HIS SOURCES1. IT is well known that the commentary of Proclus on Eucl. Book I. is one of the two main sources of information as to the history of Greek geometry which we possess, the other being the Collection of Pappus. They are the more precious because the original works of the forerunners of Euclid, Archimedes and Apollonius are lost, having probably been discarded and forgotten almost immediately after the appearance of the masterpieces of that great trio. Proclus himself lived 410-485 A.D., so that there had already passed a sufficient amount of time for the tradition relating to the pre-Euclidean geometers to become obscure and defective. In this connexion a passage is quoted from Simplicius' who, in his account of the quadrature of certain lunes by Hippocrates of Chios, while mentioning two authorities for his statements, Alexander Aphrodisiensis (about 220 A.D.) and Eudemus, says in one place3, "As regards Hippocrates of Chios we must pay more attention to Eudemus, since he was nearer the times, being a pupil of Aristotle." The importance therefore of a critical examination of Proclus' commentary with a view to determining from what original sources, he drew need not be further emphasised. Proclus received his early training in Alexandria, where Olympiodorus was his instructor in the works of Aristotle, and mathematics was taught him by one Heron' (of course a different Heron from the "mechanicus Hero" of whom we have already spoken). He afterwards went to Athens where he was imbued by Plutarch, and by Syrianus, with the Neo-Platonic philosophy, to which he then devoted 1 My task in this chapter is made easy by the appearance, in the nick of time, of the dissertation De Procli fontibus by J. G. van Pesch (Lugduni-Batavorum, Apud L. van Nifterik, MDCCCC). The chapters dealing directly with the subject show a thorough acquaintance on the part of the author with all the literature bearing on it; he covers the whole field and he exercises a sound and sober judgment in forming his conclusions. The same cannot always be said of his only predecessor in the same inquiry, Tannery (in La Géométrie grecque, 1887), who often robs his speculations of much of their value through his proneness to run away with an idea; he does so in this case, basing most of his conclusions on an arbitrary and unwarranted assumption as to the significance of the words οι περί τινα (e.g. "Ηρωνα, Ποσειδώνιον etc.) as used in Proclus. 2 Simplicius on Aristotle's Physics, ed. Diels, pp. 54-69. • ibid. p. 68, 32. Cf. Martin, Recherches sur la vie et les ouvrages d'Héron d'Alexandrie, pp. 240-2. 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-Piatonic 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 yvwov), 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 "ò eoupyós" (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...... Proclus, p. 84, 13. ibid. p. 31, 20. ▲ ibid. p. 429, 12. • ibid. p. 32, 2. 7 ibid. p. 27, 27 to 28, 7; cf. also p. 21, 25, PP. 46, 47. |