Euclid, Elements
Thomas L. Heath, Sir Thomas Little Heath, Ed.

Vat. (the lemma to X. 17=Schol. X. No. 106, and the lemmas to X. 54, 60 come in Schol. X. No. 328); and it is not possible to suppose. that these lemmas, if they were already in the text, would also be given as scholia. Of these three lemmas, that before X. 60 has already been condemned for other reasons; the other two, unobjectionable in themselves, must be rejected on the ground now stated. There were four others against which Heiberg found nothing to urge when writing his prolegomena to Vol. v., viz. the lemmas before X. 42, X. 14, X. 22 and X. 33. Of these, the lemma to X. 22 is not reconcilable with Schol. X. No. 161, which takes up the assumption in the text of Eucl. X. 22 as if no lemma had gone before. The lemma to X. 42, which, on account of the words introducing it (see p. 60 above), Heiberg at first hesitated to regard as an interpolation, is identical with Schol. X. No. 270. It is true that in Schol. X. No. 269 we find the words “this lemma has been proved before (ἐν τοῖς ἔμπροσθεν), but it shall also be proved now for convenience' sake (τοῦ ἑτοίμου ἕνεκα),” and it is possible to suppose that “before” may mean in Euclid's text before X. 42; but a proof in that place would surely have been as “convenient” as could be desired, and it is therefore more probable that the proof had been given by Pappus in some earlier place. (It may be added that the lemma to X. 14, which is identical with the lemma to XI. 23, condemned on other grounds, is for that reason open to suspicion.)

Heiberg's conclusion is that all the lemmas are spurious, and that most or all of them have found their way into the text from Pappus' commentary, though at a time anterior to Theon's edition, since they are found in all our MSS. This enables us to fix a date for these interpolations, namely the first half of the 4th c.

Of course Pappus had not in his text the interpolations which, from the fact of their appearing only in some of our MSS., are seen to be later than those above-mentioned. Such are the lemmas which are found in the text of V only after X. 29 and X. 31 respectively and are given in Heiberg's Appendix to Book X. (numbered 10 and 11). On the other hand it appears from Woepcke's tract¹ that Pappus already had X. 115 in his text: though it does not follow from this that the proposition is genuine but only that interpolations began very early.

Theon interpolated a proposition (or lemma) between X. 12 and X. 13 (No. 5 in Heiberg's Appendix). Schol. Vat. has the same thing (X. No. 125). The writer of the scholia therefore did not find this lemma in the text. Schol. Vat. IX. Nos. 28, 29 show that neither did he find in his text the alterations which Theon made in Eucl. IX. 19; the scholia in fact only agree with the text of P, not with Theon's. This suggests that Schol. Vat. were written for use with a MS. of the ante-Theonine recension such as P is. This probability is further confirmed by a certain independence which P shows in several places when compared with the Theonine MSS. Not only has P better readings in some passages, but more substantial divergences; and,