Complete Works of Euclid, Vol. 8 (Phenomena and Musical Writings)

they bear; the second part Vat. completes at p. 56, 10 ff. with the words φανερὸν δέ etc. On the contrary, in Prop. XI, after it has been shown in both recensions not dissimilarly at what time the one of the arcs of the zodiac, both equal and opposite, sets, the other rises, V omits the remaining part of the demonstration and holds it sufficient to add: ὁμοίως δὴ δείξομεν etc.¹⁾, whereas Vat. believes that even the other part of the proposition must be demonstrated in almost the same words as the former. The demonstrations of Prop. XII differ from each other to such a degree that V does not touch upon the rising of the arcs except at the end, where it joins to the conclusion at p. 78, 6: ὁμοίως δὴ δείξομεν, ὅτι καὶ ἐν ἴσῳ χρόνῳ ἀλλήλαις ἀνατέλλουσιν, but Vat. at p. 72, 25 ff. clearly teaches that arcs equidistant from the equinoctial rise in equal time. In the demonstrations of Prop. XIII we have a sufficiently great difference of phrasing; for example, V composes and compares six arcs comprehended with six equal and western arcs comprehended, whereas Vat. compares individual arcs of each order with individual ones; furthermore, V at p. 82, 3 ff. sets forth Prop. XII with all its words, while Vat. utilizes it silently. The demonstrations of the first part of Prop. XIV follow almost the same path, but differ not insignificantly in phrasing. At the end at p. 90, 15, Vat. undertook to demonstrate that one of the arcs under inquiry is closer to the contact of the summer tropic than the other. The second part of the demonstration at p. 92, ὡσαύτως etc., which Vat. provides in an expanded form, taking (no less than in the previous part) two positions of the zodiac, V handed down in a pressed and tightened form. The demonstrations of Prop. XV proceed by the same path and do not differ much in phrasing, although V does not care for the second part of the protasis at p. 100, 4 ἐν ᾧ χρόνῳ etc., which Vat. completes with the words at p. 102, 14 ὁμοίως δὲ δείξομεν etc. Finally, from the portion which V contains of the demonstration of Prop. XVI, it is permitted to conclude that it used a more pressed phrasing than the Vat. manuscript. We also find the discrepancy of the recensions subject to the eyes in the figures, and especially the figures of Props. II, XII, XIV bear a different perception and description.

I shall gather here very briefly the individual points in which V and Vat. differ from one another, some of which I mentioned above. And first, it must be established that the Vat. manuscript holds the true reading in not a few places against recension a and rather against V not as a corrected reading; see p. 4, 8; 6, 5, 28; 10, 8, 14, 15; 12, 25, 29; 14, 1; 16, 21; 18, 10, 15; 20, 29; 22, 17; 26, 21, 26, 28; 30, 3, 16; 32, 11; 34, 4; 36, 5; 38, 21. Regarding those things,

1. Cf. vol. IV p. 120, 2 ff.