4. Again, on 1. 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 Pappus1, 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 commentators2.” By “the commentators” Pappus is doubtless meant.

5. Lastly, the “four-sided triangle,” called by Zenodorus the “hollow-angled,” 3 is mentioned in the notes on 1. Def. 24-29 and I. 21. As Pappus wrote on Zenodorus' work in which the term occurred4, Pappus may be responsible for these notes.

IV. Simplicius.

According to the Fihrist5, 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-Nairĩzĩ6. 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 1. 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 Proclus7, and is closely connected with the definition given by Posidonius8. 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 impossible9 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.

2 Proclus, p. 328, 15.

3 Proclus, p. 165, 24; cf. pp. 328, 329.

4 See Pappus, ed. Hultsch, pp. 1154, 1206.

5 Fihrist (tr. Suter), p. 21.

6 An-Nairĩzĩ, ed. Besthorn-Heiberg, pp. 9-41, 119-133, ed. Curtze, pp. 1-37, 65-73. The Codex Leidensis, from which Besthorn and Heiberg's edition is taken, has unfortunately lost some leaves, so that there is a gap from Def. 1 to Def. 23 (parallels). The loss is, however, made good by Curtze's edition of the translation by Gherard of Cremona.

7 Proclus, p. 177, 21.

8 ibid. p. 176, 7.

9 Bibliotheca Mathematica, 113, 1900, pp. 9-11.