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

(abbreviated) p. 128, 6 δύνει] added. p. 132, 4 ΥΞΝΜ] M added. 11 ἀφανές] follows: ὁμοίως δὴ δείξομεν, ὅτι καὶ ἐν ᾧ χρόνῳ ἡ ΘΖ ἐξαλλάσσει τὸ ἀφανὲς ἡμισφαίριον, ἐν τούτῳ ἡ ΕΗ τὸ φανερόν p. 146, 16 after Z add ἴση p. 153, 23 ΕΛ] ΕΚΛ 27 ΕΚΛ] Λ added. p. 154, 30 ἤπερ ἡ] in erasure. p. 155, 26 ΜΕΗ] Η in erasure. 28 ἤπερ ἡ ΜΕΖ] added. p. 156, 12 after θερινοῦ add τροπικοῦ. Furthermore, two recent hands of the Vat. manuscript appeared to themselves to correct it at their own discretion; see p. 72, 16 after δύνει add ὧν ἐπεὶ ἡ ΛΞ τῇ ΞΜ ἐν ἴσῳ χρόνῳ δύνει; p. 78, 8 above ἴσῳ write ἴσοις; p. 80, 8 after ΝΔ add περιφέρειαι, 10 ἴσω changed into ἴσοις, 11 ἀνατέλλει into ἀνατέλλουσι; p. 98, 10 after ΟΘ add περιφέρεια; p. 122, 10 schol. no. 122 inserted into text. I would add that at p. 98, 15 they deleted ἐξαλλάσσει τὸ φανερὸν ἡμισφαίριον, which I ought to have retained.

Now indeed let us see what ratio exists between the two recensions, or if you prefer, between their authors, V and Vat. First, the second demonstrations of Props. VI, XII, XIV, XV of recension b, which cannot be attributed to Euclid ¹), are not found in the manuscripts of rec. a, but it is memorable that the second demonstration of Prop. XII is very similar to the demonstration of rec. a, and likewise in the other part of the second demonstration of Prop. XIV a similarity with a appears. Next, the scholia 102, 127, 128, 129 2) of the Vat. manuscript, interposed by hand 1 to the text, do not exist in V. Conversely, the Vat. manuscript omits p. 8, 29 — 10, 2 and has what follows at p. 10, 3 — 10 instead as scholia; see schol. 106, 114. Indeed, the demonstrations of Props. I—VIII after the common proemium in both recensions proceed by the same path and method, such that the same theorems of Autolycus and Theodosius are utilized in the same places, the order of conclusions is the same, and most things are read expressed in almost the same words. But the other demonstrations, whether you look at the sum or consider the individual parts, differ from one another more or less. And in the case of Prop. IX, the protaseis themselves differ; for Vat. omits at p. 44, 13 ὅταν — 15 τροπικοῦ, which words I denied above to be redundant. The ectheseis of the first part, however, differ from one another in such a way that V provides the construction at p. 46, 22 ff. before λέγω δή etc., whereas Vat. at p. 48, 4 at the beginning of the demonstration. The ecthesis of the second part at p. 52 is missing in V; in Vat. at p. 54 the conclusion is wanting; the demonstrations themselves follow the same ratio. The demonstrations of the first part of Prop. X display a certain similarity;

1. V vol. V p. LXXIX and VI p. XLIX.
2. I omit scholium 122 here, since it exists in the text of the Vat. manuscript by a recent hand and does not pertain to recension b itself.