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 (1. 10, 11, and 23).
Allusion must be made to the debated question whether Proclus continued his commentaries beyond Book 1. 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 circumference2.”
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 book3.”
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 elsewhere4”
; this may imply an intention to treat of the subject on Eucl. XII., though Heiberg doubts it5. But it is clear that, at the time when the commentary on Book 1. 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 says6, “For my part, if I should be able to discuss the other books7 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 πραγματειῶδες 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.
2 ibid. p. 272, 14.
3 ibid. p. 398, 18.
4 ibid. p. 423, 6.
5 Heiberg, Euklid-Studien, p. 165, note.
6 Proclus, p. 432, 9.
7 The words in the Greek are: εἰ μὲν δυνηθείημεν καὶ τοῖς λοιποῖς τὸν αὐτὸν τρόπον ἐξελθεῖν. For ἐξελθεῖν Heiberg would read ἐπεξλθεῖν
8 True, a Vatican Ms. has a collection of scholia on Books I. (extracts from the extant commentary of Proclus), II., V., VI., X. headed Εἰς τὰ Εὐκλείδου στοιχεῖα προλαμβανόμενα ἐκ τῶν Πρόκλου .σποράδην καὶ κατ̓ ἐπιτομήν. Heiberg holds that this title itself suggests that the authorship of Proclus was limited to the scholia on Book I.; for προλαμβανόμενα ἐκ τῶν Πρόκλου 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 I. came from Proclus, himself attached Proclus' name to the others.
Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.
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