Six palimpsest folios – or, to be accurate, three bifolios – of the Verona manuscript, Biblioteca Capitolare XL (38), contain fragments of a Latin translation of Euclid's Elements: fols. 331v–r and 326v–r, 341r–v and 338r–v, 336r–v and 343r–v. The folios are dated to around a.d. 500, and the text is written in capital script in two columns. Unfortunately the folios have suffered severe damage from various chemical substances, which were used by nineteenth-century scholars in attempts to retrieve the underlying text. Nevertheless, an edition of the fragments finally appeared in 1964 by M. Geymonat.Footnote ¹

With modern techniques for deciphering palimpsests, it would perhaps in the future be possible to extract more text from the folios and to obtain more reliable readings of obscure passages. Ventures to this end are to be encouraged, and hopefully this paper will at least in some respect contribute to a renewed interest in the fragments among specialists in palimpsests. For the time being, however, Geymonat's edition will have to do as basis for philological studies of the fragments.Footnote ²

I. On three occasions in the fragments Geymonat read deliget:

Geymonat probably identified deliget as the verb deligere, for in his note on the deliget on fol. 331v he reports I. Cazzaniga's conjecture delige et with the additional comment ‘fortasse recte’,Footnote ³ and in his index verborum to the edition all three occurrences of deliget are recorded together under the same lemma: deliget.Footnote ⁴ I would suggest, however, that it is far more likely that <UI>DELIGET, <UI>(D)<E>(LI)GET, and <UI>DELIGET, that is, the adverb videlicet, should be read. On the first occasion (fol. 331v col. 1, line 11), the second half of the preceding line 10 is illegible. On the second occasion (fol. 336v col. 2, line 14), several letters are illegible between EST and (D)<E>(LI)GET. And, finally, the first half of the line in which the third deliget is found (fol. 336v col. 2, line 17) is illegible as well. Hence, in all three occurrences, the letters U and I may well once have been written before deliget.

At fol. 331v col. 1, lines 11–12, deliget is followed by the relative clause QUAE [sc. plana] ·ΑΓ·..·..· | ·ΕΒ·ΒΖ·ΕΖ· SUNT, which, as the corresponding passage in the Greek text of Euclid shows (Elements 11 Prop. 24 p. 70,19 Heiberg),Footnote ⁵ obviously specifies six planes by which a certain solid is comprehended. At fol. 336v col. 2, lines 14–15, deliget is most likely followed by a relative clause of similar, specifying type: (D)<E>(LI)GET QUAˉ | ·Γ.· (= deliget quam [sc. lineam?] Γ?). At fol. 336v col. 2, line 17, finally, deliget is followed by QUod, which probably also begins a relative clause of the aforementioned type. Now, consider the following passages, in which the words videlicet and scilicet are read: fol. 341v col. 1, lines 21–2 UIDELICET ILLAE [sc. lineae] QUAE | ·ΑΒ·, fol. 343r col. 1, line 10 ET UIDELICET QUEM [sc. angulum] ·(ΖΘ)Δ·, and fol. 343r col. 1, lines 13–14 ET SCILICET QUEM [sc. angulum] ·ΖΕΔ· TES | TAN(TUR). As these examples indicate, the obscure occurrences of deliget are most likely the remains of three original instances of videlicet.

The word videlicet is spelled with a C at fol. 331r col. 1, line 13, fol. 341v col. 1, lines 9 and 21, fol. 341v col. 2, lines 2–3, fol. 338r col. 2, line 22, fol. 343r col. 1, lines 3 and 10; cf. also fol. 341r col. 2, line 9 and fol. 343r col. 1, line 13, where the word scilicet is spelled with a C. Since videlicet is usually spelled with a C, I think that the three examples of the spelling with G are to be regarded as scribal errors; moreover, I have not found any other attestation of the letter G being used instead of an expected C in the fragments.

Occasionally, however, the letter C is found instead of an expected G. At fol. 343r col. 1, line 2, RELICABUNT is written, where religabunt would have been expected;Footnote ⁶ so also at fol. 336v col. 1, line 19, where R<E>LICABUNT (with the letters N and T written in nexus) is read. At fol. 341r col. 2, line 8, SECTIO(NE)S occurs, but a contemporary corrector has deleted TIO(NE)S and added menta above the line, producing the corrected reading secmenta, where segmenta would have been expected. At fol. 341r col. 2, line 23–fol. 341v col. 1, line 1, Geymonat reads TRIAN | <GULAS SEDES>; a contemporary corrector, however, has deleted AN and added conas above the top line of fol. 341v col. 1, which gives the corrected reading triconas, where trigonas would have been expected.Footnote ⁷

II. At fol. 341r col. 1, lines 3–4 (p. 22,3–4 Geymonat) the following is read:

These words end a short fragment of Euclid, Elements 12 Prop. 2. In his note ad loc., Geymonat compares the phrase expedire rationes with Vitruvius, De architectura 10.16.12:Footnote ⁸Quas potui de machinis expedire rationes pacis bellique temporibus et utilissimas putavi, in hoc volumine perfeci.Footnote ⁹ This comparison, however, is hardly apt, since quod fuit expedire rationes is, most probably, a Latin rendering of the Euclidean ὅπɛρ ἔδɛι δɛῖξαι, nowadays more commonly known in the Latin form quod erat demonstrandum (Q.E.D.), which is regularly placed at the end of a proof and signals its completion.Footnote ¹⁰ The quod fuit expedire rationes is, moreover, probably the earliest preserved Latin translation of the famous phrase. In the fragmentary Latin translation of Euclid's Elements ascribed to Boethius, the phrase quod oportebat facere is attested;Footnote ¹¹ there, however, the Latin is a translation of the similar phrase ὅπɛρ ἔδɛι ποιῆσαι, equivalent to the Latin form quod erat faciendum (Q.E.F.).

III. According to Geymonat's edition the following is read at fol. 341v col. 1, lines 5–7 (p. 24,5–7 Geymonat):

A comparison with the Greek text of Euclid indicates that the words after the initial sectiones ought to be rendering the following phrase: Τɛτμήσθωσαν γὰρ αἱ ΑΒ, ΒΓ, ΓΑ, ΑΔ, ΔΒ, ΔΓ δίχα (Elements 12 Prop. 3 p. 150,7–8 Heiberg). Hence, (SIUEDA)NTUR is probably either a corrupt or misread dividantur, which would correspond to τɛτμήσθωσαν.Footnote ¹² Furthermore, as I would argue, the reading QUO IURE does not make sense. It is, most likely, either a corruption or a misreading.Footnote ¹³ In either case, I suggest that QUO IURE should be corrected to duo illae. In capital script the letters D and Q (and O) are easily confused. Moreover, capital A may easily be mistaken for R, and the combination LL may be confused with U, especially if the lower parts of the letters LL are somewhat obscured and, as is the case with the script of the palimpsest folios, the letter ‘U approaches the uncial form’.Footnote ¹⁴

By reading dividantur quidem duo illae quae sunt ΑΒ, ΒΓ, ??, ??, ΔΒ, ΔΓ, a sensible correspondence with the Greek is beginning to emerge: dividantur = τɛτμήσθωσαν, quidem = γάρ, illae quae sunt = αἱ, duo ≈ δίχα. Granted the conjecture duo illae, however, a difficulty emerges: one would have expected the preposition in between quidem and duo. The preposition is undoubtedly indispensable; cf. fol. 341r col. 2, lines 3–4 and 21–2, where the verb dividere in the passive is construed with the prepositional phrase in duas pyramidas; but cf. especially the following examples of the verb dividere + in duo, where in duo (in the neuter) approaches the function of an adverb (like δίχα):

In addition, in the fragmentary Latin translation of Euclid's Elements ascribed to Boethius, the Euclidean δίχα is rendered in various ways: in duas aequales/aequas partes,Footnote ¹⁶per aequalia,Footnote ¹⁷in duo aequa. In the following example, the last expression is found used absolutely, as it were: Datam circumferentiam semicirculi in duo aequa dividere.Footnote ¹⁸

On the one hand, the expected in may have been erroneously left out by the scribe. If so, the intended text should be re-established thus: dividantur quidem <in> duo illae quae sunt ΑΒ, ΒΓ, ??, ??, ΔΒ, ΔΓ. On the other hand, the expected in may lie concealed in QUIDEM, which was read only by Mai.Footnote ¹⁹ Since Mai's transcript is not always trustworthy and since Mai, as is correctly reported in Geymonat's edition, doubted his own reading of the letters D and E in QUIDEM,Footnote ²⁰ it can, in my view, be suggested that QUIDEM is a misreading of quippe in (PP erroneously read as D, E correctly read, and IN erroneously read as M). If so, the intended text should be re-established thus: dividantur quippe in duo illae quae sunt ΑΒ, ΒΓ, ??, ??, ΔΒ, ΔΓ. Of course, it could also be the case that QUIDEM is indeed the transmitted text, but represents a scribal corruption of quippe in accomplished by the scribe. If accepted, this conjecture neatly restores the desired in, and γάρ is given a Latin rendering which is attested elsewhere in the fragments.Footnote ²¹