1. πρότασις (enunciation), ἔκϑεσις (setting-out) and διορισμός (definition/specification)

In L, proposition 3 is relatively well preserved up to the διορισμός (definition or specification). According to Geymonat's edition, the πρότασις (enunciation), the ἔκϑεσις (setting-out) and the διορισμός read thus:Footnote ²³

In Heiberg's edition the corresponding portions run as follows (significant deviations in the manuscripts are reported in the critical apparatus; for MS b, see p. 393.23–30 Heiberg):

A comparison between L and Heiberg's text will show that the Greek source text of the translation must have differed in several respects from Heiberg's printed text. In the following analysis of the Latin πρότασις, ἔκϑεσις and διορισμός I shall try to reconstruct the wording of the source text as far as possible; this will be done under consideration of the variant readings attested in the Greek manuscripts. Moreover, the analysis will be conducted in the form of a running commentary. (As the contemporary corrections in the Latin πρότασις, ἔκϑεσις and διορισμός merely concern terminological preferences, L will be considered prior to correction.)

[πρότασις:] omnis pyramis, quae habet triangulam sedem, diuiditur] These words correspond neatly to the phrase πᾶσα πυραμὶς τρίγωνον ἔχουσα βάσιν διαιρεῖται, which is read unanimously in the Greek manuscripts.

tam in duas pyramidas aequales] This portion shows that the source text of L had the reading of MS b: εἴς τε δύο πυραμίδας ἴσας. The conjunction τε is rendered as tam in Latin.Footnote ³⁰ MS F also exhibits this reading, but τε has been added by a second hand.

triangulas sedes habentes] This phrase evidently corresponds to τριγώνους ἐχούσας βάσεις. As to its position, the Greek manuscripts are divided; it follows immediately after either ἀλλήλαις (MSS BFVq) or τῇ ὅλῃ. In the πρότασις of MS b, however, the phrase is absent altogether. In L it has been given a somewhat odd place; as ac similes si<bi> most likely corresponds to καὶ ὁμοίας ἀλλήλαις (see on ac similes si<bi> below), triangulas sedes habentes interrupts, as it were, the expected sequence aequales ac similes si<bi> (cf. ἴσας τε καὶ ὁμοίας ἀλλήλαις). But a similar interruption seems to occur also in the διορισμός; see on ipse below.

Now, as there is an attested variation in the Greek manuscripts regarding the placement of τριγώνους ἐχούσας βάσεις, in my view it is more probable that L inherited its placement of the phrase from the source text than that this feature had been accomplished by the Latin translator (or at least a Latin scribe); see further on ac similes si<bi> below.

ac similes si<bi>] Undoubtedly the words ac similes translate καὶ ὁμοίας. But as the Greek tradition exhibits two possible καὶ ὁμοίας, one must first ask: to which καὶ ὁμοίας do the words ac similes correspond? The Greek manuscripts read as follows: MS P: ἴσας τε καὶ ὁμοίας ἀλλήλας [sic] καὶ τῇ ὅλῃ. MSS B¹FVq: ἴσας ἀλλήλαις … καὶ ὁμοίας τῇ ὅλῃ. MS b reads: ἴσας ἀλλήλαις καὶ ὁμοίας τῇ ὅλῃ.

According to Geymonat's edition, the words ac similes are followed by si<bi>, which was (partially) suggested by V. De Marco; in his commentary, Geymonat also indicates that si<bi> should correspond to ἀλλήλαις. I. Cazzaniga, on the other hand, conjectured <totius>, which should correspond to τῇ ὅλῃ.Footnote ³¹

As Geymonat reports in his edition, he was unable to read a single letter of the word that follows similes. Consequently De Marco's suggestion was evidently based on the reading of Mai, who also, as it appears, was uncertain as to what should follow similes. For on one occasion in his transcript Mai reports that he read si est; on another he records only si, which he subsequently deletes altogether. However, Mai's surrender of his reading si may not necessarily have been prompted by difficulty in reading si, but by the notion that a si-clause would be completely out of place. If we assume that Mai was right in reading si and what followed si at least appeared to be est, it can be argued that the emendation si<bi> has palaeographical merit. In obscured capital script the letters BI could well be misread as EST, if one considers that the vertical stroke of the letter B could have been read as the letter E, the curved stroke of the letter B as S, and the vertical stroke of the letter I as T. As for Cazzaniga's conjecture, on the other hand, it does not seem to tally with Mai's reading at all; and why the genitive? Would not <toti> be a better match for τῇ ὅλῃ?

In his commentary, Geymonat compares tam sibi <aequa>lia <q>uam paral<lelo>gramma at fol. 331v col. 1, lines 17–19 (= p. 70.20–1 Heiberg ἴσα τε καὶ παραλληλόγραμμα), where sibi (governed by <aequa>lia) is to be understood in the sense of ἀλλήλαις, although none of the Greek manuscripts exhibits any corresponding word.Footnote ³²

In my view, De Marco's si<bi> is convincing; also, it most likely corresponds to ἀλλήλαις. This strongly suggests that the phrase ac similes si<bi> corresponds to καὶ ὁμοίας ἀλλήλαις, not to καὶ ὁμοίας τῇ ὅλῃ. If this is right, in turn, L does not exhibit any reading that corresponds to καὶ (ὁμοίας) τῇ ὅλῃ; either the translator (or a scribe) omitted it by mistake or it was missing already in the source text. Moreover, granted that the source text had the phrase τριγώνους ἐχούσας βάσεις at the place where it stands in L, the source text probably did not read τε after ἴσας, as there is no word corresponding to τε in L at this position.

In conclusion, I suggest that the source text of aequales triangulas sedes habentes ac similes si<bi> probably read ἴσας τριγώνους ἐχούσας βάσεις καὶ ὁμοίας ἀλλήλαις; whether this was followed by καὶ (ὁμοίας) τῇ ὅλῃ or not, cannot be determined.

adque in duas sectiones aequales] These words correspond to καὶ εἰς δύο πρίσματα ἴσα, which is the unanimous reading of the Greek manuscripts. Note that in the accessible portions of L atque is always spelled adque.Footnote ³³

scilicet ex arte discedet] This is an awkward addition, unparalleled in the Greek tradition. In his commentary, Geymonat correctly interprets ‘ex arte sc. geometrica’. The verb form discedet he interprets in Italian as ‘ne risulterà’. The whole phrase could thus be translated: ‘that is to say, it will be the result thereof according to the art (of geometry)’. Geymonat also states that discedere perhaps is to be taken in the sense of diuidi, which in my view seems highly unlikely.Footnote ³⁴

As I have not found any parallel in ancient Latin to the expression ex arte discedere, I suggest that discedet is an erroneously written descendet (or perhaps descendit); cf. TLL s.v. descendo 5.1.641.58–64, where it is noted that the verb descendere may appear as discendere or discedere.Footnote ³⁵ In L, moreover, the verb is divided into two lines thus: disce-det; this might also have contributed to the omission of the letter n. More importantly, however, the phrase ex arte descendere is attested in Late Latin and Early Mediaeval Latin, notably in grammarians from the fifth and the sixth centuries, for example Pompeius Maurus, Commentum artis Donati p. 139.2–3 Keil.Footnote ³⁶ In the grammarians, the expression is used in descriptions of compounds, as in Primae expositiones Sergii de prioribus Donati grammatici urbis Romae p. 146.2–5 Hagen:Footnote ³⁷ figurae accidunt duae, simplex et conposita. simplex est, quae natura uenit; conposita, quae ex arte descendit, ut puta: felix naturale est: conpone illud, infelix: clarus praeclarus, uoco reuoco. Granted my suggestion, the phrase scilicet ex arte discedet (= descendet/–it) ‘that is to say, it (will) follow(s) from the art (of geometry)’ seems to be of grammatical descent and added by the translator, or, presumably, an interpolation added by a scribe; cf. on sit pyramis una cuminproprium ex arte producta below.

ac duo recisamenta totius pyramidis, quae forte contigerint, maior<a sunt> ab eo, quod dimidium esse cernitur] Apart from the relative clause quae forte contigerint, which is unparalleled in the Greek tradition and probably either an addition by the translator or an interpolation, this passage exhibits a feature that is paralleled only in MS b: totius pyramidis (= τῆς ὅλης πυραμίδος) follows immediately after recisamenta (= πρίσματα). For MS b reads: καὶ τὰ δύο πρίσματα τῆς ὅλης πυραμίδος μείζονά ἐστιν ἢ τὸ ἥμισυ.Footnote ³⁸

[ἔκϑεσις:] sit pyramis una cuminproprium ex arte producta] The first two words correspond to ἔστω πυραμίς. The following words up to producta, however, are obviously either an addition by the translator, or, presumably, an interpolation. In his commentary, Geymonat doubts Mai's reading una and states that the letter before na appears to be i; he also mentions that Cazzaniga conjectured non inproprium, which he regards as ‘rectius, sed non videtur’.Footnote ³⁹ On the basis of Geymonat's observation, I suggest: in acumen proprium ex arte producta ‘produced according to the art [of geometry] to its characteristic top’, since a capital E could easily be mistaken for an I. Cf. Ps.-Acro on Hor. Carm. 3.30.1 pyramides autem proprie regum Aegyptiorum sepulchra sunt ingenti mole constructa et in acumen educta, in modum flammae sursum surgentis, unde et nomen est inditum; Footnote ⁴⁰ cf. also Isid. Etym. 3.12.6 pyramis est figura, quae in modum ignis ab amplo in acumen consurgit; ignis enim apud Graecos πῦρ appellatur. Footnote ⁴¹ Granted my emendation, the addition has the character of grammatical and encyclopaedic stock material; cf. scilicet ex arte discedet in the πρότασις.

cuius basis <ea>dem quae ΑΒΓ sit triangulum] That these words correspond to the former membrum of the relative clause (ἧς … τρίγωνον) is clear; but to determine the exact wording of the source text seems to be impossible. As regards L, Geymonat had conjectured <qui>dem (= μέν), but adopted De Marco's <ea>dem. Furthermore, Cazzaniga suggested sit <sit> triangulum—which Geymonat regarded as ‘fortasse recte’—obviously because one would expect one sit in the cuius-clause and another in the quae-clause.Footnote ⁴² But if this particular conjecture is right, one sit must have been omitted by a scribe, as L undoubtedly only exhibits one sit here.

In my view, quae ΑΒΓ suggests that the source text had the erroneous reading ἡ ΑΒΓ (instead of τὸ ΑΒΓ), attested as a first-hand reading in MS F. Furthermore, according to most manuscripts of the Greek tradition, the finite verb form in the ἧς-clause is ἐστι; the sole exception, MS b, reads ἔστω (‘let there be’). As it can be argued that ἔστω would be the expected verb form here,Footnote ⁴³ perhaps ἐστι should be considered spurious. And as Euclidean relative clauses which are subordinated to and follow main clauses with ἔστω(σαν) need not necessarily have ἔστω(σαν) repeated, but implied, one could even suggest that the Greek source text read: ἔστω πυραμίς, ἧς βάσις μὲν ἡ ΑΒΓ τρίγωνον. Unfortunately, L does not provide any evidence in this matter; even if sit were to belong to the cuius-clause, it could correspond to either ἐστι or ἔστω in Greek, as the subjunctive would be the expected Latin form in both cases.Footnote ⁴⁴

sitque apex eius ex Δ] This phrase corresponds to the latter membrum of the relative clause [sc. ἧς] κορυφὴ δὲ τὸ Δ σημεῖον, which is read unanimously in the Greek manuscripts. Unlike its Greek model, however, the phrase sitque … Δ does not belong to the cuius-clause, but is coordinated with the main clause sit pyramis (as the pronoun eius [sc. pyramidis] makes clear). This feature should presumably be attributed to the Latin translator.

As I have argued in a previous article, ex is nonsensical and either a corruption or a misreading of ea; in capital script an ill-written A may easily be confused with an X. The noun implied is either nota or littera, which would correspond to σημεῖον.Footnote ⁴⁵

[διορισμός:] dico quia diuiditur] These words correspond to λέγω, ὅτι ἡ ΑΒΓΔ πυραμὶς διαιρεῖται. In this case the translator may very well be responsible for leaving out ἡ ΑΒΓΔ πυραμίς from L, as it is easily understood from the context that the subject of diuiditur is the aforementioned pyramid.

tam in duas pyramidas aequales] As in the πρότασις, the phrase in duas pyramidas aequales is preceded by tam = τε; on this occasion, however, the conjunction τε is read only in MS b.

trian<gulas sedes> <*> tes] As these words should correspond to τριγώνους βάσεις ἐχούσας, attested in the Greek manuscripts except MS b, I suggest that the illegible portion indicated should be restored thus: <haben>tes. Cf. the πρότασις.

ipse ] This word, which only Mai was able to read, seems awkward. I for my part suggest that it is either a misread or a corrupt ipsi (in the dative). If this conjecture is right, ipsi should correspond to ἀλλήλαις, just as si<bi> in the πρότασις, and the dative should depend on aequales (= ἴσας); cf. on ac similes si<bi> above. With this interpretation ipsi appears to be used as a synonym for sibi with reciprocal sense; the singular in lieu of the plural might be explained as an instance of vulgar analogy with the reflexive pronoun, whose declension is indifferent to number. As the word is placed after trian<gulas sedes haben>tes, the source text probably did not read ἴσας ἀλλήλαις τριγώνους βάσεις ἐχούσας, but ἴσας τριγώνους βάσεις ἐχούσας ἀλλήλαις; cf. on triangulas sedes habentes above.

adquae etiam congruas potius <*> aequa<l> <*> sectiones] This last portion of the διορισμός is difficult; it was read only by Mai, and he was uncertain regarding his reading of the word potius. As it is expected to correspond to καὶ ὁμοίας τῇ ὅλῃ καὶ εἰς δύο πρίσματα ἴσα, I tentatively suggest the following restoration: adquae etiam congruas toti <pyramidi> atqua <e in duas aequales> sectiones. First, adquae (scribal error for adque [= atque]) etiam should, I think, be understood as a strengthened καί.Footnote ⁴⁶ Secondly, congruas should here equal ὁμοίας. Thirdly, toti <pyramidi> would correspond neatly to τῇ ὅλῃ [sc. πυραμίδι], whereby Mai's potius could be explained as a misreading of toti py–.Footnote ⁴⁷ Fourthly, whereas Geymonat on the basis of Mai's transcript read aequa<l>, here one expects a translation of καί. As a capital T can be misread as an E, atqua<e> is a possible reading (scribal error once again). Admittedly, however, one would rather have ad–, as the spelling at– is not attested in L; but considering the fact that L is preserved only in fragments, one cannot exclude the possibility that the spelling was inconsistent. Fifthly, as the last word of the διορισμός is sectiones (= πρίσματα), the illegible portion that precedes it could have read in duas aequales.Footnote ⁴⁸

According to Heiberg's text, the διορισμός ends with καὶ τὰ δύο πρίσματα μείζονά ἐστιν ἢ τὸ ἥμισυ τῆς ὅλης πυραμίδος. MS b, however, does not exhibit this sentence here; nor does L.

Now, having concluded the analysis, I summarize the results by presenting the tentative source text of the Latin πρότασις, ἔκϑεσις and διορισμός in its entirety (all deviations from Heiberg's text are set in bold style; uncertain words are put within round brackets):

This text exhibits the following features paralleled only in MS b: 1) Both the πρότασις and the διορισμός read εἴς τε δύο πυραμίδας ἴσας. 2) In the πρότασις the genitive τῆς ὅλης πυραμίδος appears between πρίσματα and μείζονα. 3) The διορισμός ends without the sentence which states that the two prisms are greater than the half of the pyramid.

But not only do the πρότασις and the διορισμός have the phrase τριγώνους ἐχούσας βάσεις (τριγώνους βάσεις ἐχούσας), which is absent in MS b in both passages, it also seems to be placed in a way that is unparalleled in the whole Greek tradition.

2. κατασκευή (construction) and ἀπόδειξις (proof)

The remaining fragments of proposition 3 in L, which belong to the κατασκευή (construction) and the ἀπόδειξις (proof), are preserved less completely than the preceding ones. The fragments are found at fol. 341v col. 1, line 5 – col. 2, line 23. Because of the many lacunae and uncertain letters in these lines, my attempts to identify readings that could be exclusively attributed either to the text of MS b or to the text of the other Greek manuscripts have generally turned out inconclusive. However, in the beginning of the κατασκευή, at fol. 341v col. 1, lines 5–7, one should, as I have suggested earlier, emend and read as follows: diuidantur quidem <in> duo illae quae sunt ΑΒ, ΒΓ, ??, ??, ΔΒ, ΔΓ or rather diuidantur quippe in duo illae quae sunt ΑΒ, ΒΓ, ??, ??, ΔΒ, ΔΓ.Footnote ⁴⁹ The κατασκευή in Heiberg's edition—which without any significant variation represents the text of the majority of the Greek manuscripts—runs thus:

Granted my emendations, the suggested text in L corresponds neatly to τετμήσϑωσαν … δίχα; one may also suggest that the missing two pairs of Greek letters should be supplied from the Greek text: ΓΑ and ΑΔ.

MS b, on the other hand, exhibits a different text:

First, MS b does not read γάρ; secondly, it reads αἱ πλευραὶ τῆς πυραμίδος δίχα in lieu of αἱ ΑΒ, ΒΓ, ΓΑ, ΑΔ, ΔΒ, ΔΓ δίχα. However, as both MS b and the majority of the Greek manuscripts label the corners of the pyramid in question with the same letters (Α, Β, Γ, Δ), αἱ πλευραὶ τῆς πυραμίδος (apparently = ‘the edges of the pyramid’, not ‘the faces of the pyramid’) and αἱ ΑΒ, ΒΓ, ΓΑ, ΑΔ, ΔΒ, ΔΓ refer to the same geometrical objects.Footnote ⁵² Nevertheless, it is most likely so in this particular instance, that the Greek source text of L exhibited the reading of the majority of the Greek manuscripts, not the one attested in MS b.

1. πρότασις (enunciation)

According to Geymonat's edition, the πρότασις of proposition 8 reads as follows:

The sole significant difference between the readings of MS b (p. 401.9–11 Heiberg) and the readings of the other Greek manuscripts (p. 176.16–18 Heiberg) is as follows:

The letters cem ad were read only by Mai, and he was, as noted in Geymonat's edition, uncertain as regards the letter c. But if his reading indeed was correct, one might suggest this conjecture: <inui>cem ad. As the word inuicem could well be involved in a Latin rendering of πρὸς ἀλλήλας, the possibility cannot be excluded that the Greek source text of L exhibited the reading attested solely in MS b; see further section 2 below.

2. ἔκϑεσις (setting-out) and διορισμός (definition/specification)

According to Geymonat's edition, the ἔκϑεσις reads as follows:

According to Heiberg's text, which can be taken to represent the majority of the Greek manuscripts, the ἔκϑεσις runs thus:Footnote ⁵⁶

MS b, on the other hand, exhibits the following text:

First, it is to be noted that the words sint similes pyramides in L correspond perfectly to the reading ἔστωσαν ὅμοιαι πυραμίδες in MS b; the other Greek manuscripts read καὶ ὁμοίως κείμεναι between ὅμοιαι and πυραμίδες. Secondly, L seems to exhibit pieces of the phrase τριγώνους ἔχουσαι βάσεις, which is read only in MS b: <triangulas> and habeant.Footnote ⁵⁹ Thirdly, L has two pairs of Greek letters which most likely designate the two pyramids in question: ΑΒΓΗ and Δ<Ε>घΘ. This feature is found in the ἔκϑεσις of MS b alone. Fourthly, at the end of the passage in L, two pairs of Greek letters are read: Β<Γ> and Εघ; it seems reasonable to suggest that these should somehow correspond to the ending in MS b: ὁμόλογος δὲ ἔστω ἡ ΒΓ τῇ घΗ.

As regards the lettering of the pyramids, there are some further observations to be made. In his edition, Geymonat not only signals that he supplied Ε by conjecture, but also indicates that his reading of the letters Η (in ΑΒΓΗ), Δ and घ (in Δ<Ε>घΘ) was uncertain. Still, if one assumes that Geymonat read and conjectured correctly, the lettering matches that of the majority of the Greek manuscripts, where Η designates the vertex of the pyramid with the base ΑΒΓ, and Θ the vertex of the pyramid with the base ΔΕघ. MS b, on the other hand, labels the pyramids otherwise: ΑΒΓΔ and ΕघΗΘ (Δ and Θ being their respective vertices).

Now, if we proceed to the διορισμός, it reads, according to Geymonat, as follows:

The διορισμός in the majority of the Greek manuscripts (p. 176.21–3 Heiberg) and in MS b (p. 401.18–20 Heiberg) differs only as regards the lettering:

According to Geymonat's edition, the illegible letters in L immediately preceding quia are three in number. Nevertheless, I would venture to suggest that a four-letter word should be supplied: dico.Footnote ⁶¹ Furthermore, one would expect that ΓΘ ΑΘघΟ in L should designate the two pyramids that appeared in the ἔκϑεσις; frustratingly enough, the lettering does not match. In addition, Geymonat does not indicate that he was uncertain as to his reading of ΓΘ ΑΘघΟ. Still, if one assumes either that Geymonat might have misread or that L exhibits a corrupt reading in this case, one could argue—on palaeographical grounds—for the following emendations: <ΑΒ>ΓΗ (Θ being an error or a misreading of Η), ΔΕघΘ (Α and Ο being errors or misreadings of Δ and Θ; Θ being an error or misreading of Ε, which in L always appears in the lunate form: Ⲉ). For it seems palaeographically less likely that the lettering found in MS b, ΑΒΓΔ and ΕघΗΘ, could have ended up as <ΑΒ>ΓΘ and ΑΘघΟ, in which case Δ and Ε, for example, would have been misread as or corrupted into Θ and Α respectively.

As regards the remaining portion of the διορισμός in L, it does not strike one as similar to the wording in the Greek manuscripts. It is, however, closer to the wording of the Greek πρότασις (p. 176.17–18 Heiberg and p. 401.10–11 Heiberg): (πρὸς ἀλλήλας add. MS b) ἐν τριπλασίονι λόγῳ εἰσὶ τῶν ὁμολόγων πλευρῶν ≈ inuicem positae in triplici ratione esse monstrantur eorum uidelicet laterum quae congruente<s> copulantur. For the prepositional phrase in triplici ratione is equivalent to ἐν τριπλασίονι λόγῳ, the words esse monstrantur can be understood as corresponding to εἰσί, and the genitive τῶν ὁμολόγων πλευρῶν is very likely to have been rendered as eorum uidelicet laterum quae congruente<s> copulantur, whereby the Greek article and the following adjective (ὁμολόγων) are translated with the pronoun is and a relative clause. What now remains to be commented upon are the words inuicem positae, which most probably qualify the two pyramids. In my view, they seem to constitute a rather free Latin translation of the phrase πρὸς ἀλλήλας, which is read only in MS b. Moreover, as the διορισμός in L seems to exhibit this structural similarity with the πρότασις of MS b, one might also suggest that this favours the conjecture <inui>cem, which was proposed in the πρότασις of L; see section 1 above.

3. κατασκευή (construction) and ἀπόδειξις (proof)

According to Geymonat's edition, the κατασκευή in L begins thus: conpleantur quippe sectiones illae <*> (fol. 338v col. 1, lines 1–3). The Greek manuscripts read: συμπεπληρώσϑωσαν γὰρ τὰ ΒΔΜΛ, घΘΡΟ στερεά (MS b): συμπεπληρώσϑω γὰρ τὰ ΒΗΜΛ, ΕΘΠΟ στερεὰ παραλληλεπίπεδα (the other Greek manuscripts). The remaining fragment of the κατασκευή, that is, quae ΒΗ ΕΚ de signis ad meritum (lines 3–5), is awkwardly obscure, and I shall not venture to disentangle it here. What I do suggest, however, is that the subsequent portion in L constitutes the beginning of the ἀπόδειξις; it reads, according to Geymonat's edition, thus:

A comparison between this passage and the text of, on the one hand, the majority of the Greek manuscripts and, on the other, MS b shows, in my view, that L here exhibits more structural resemblance with MS b. For MS b reads:

In the majority of the Greek manuscripts, however, the beginning of the ἀπόδειξις runs as follows:Footnote ⁶⁴

Here the portion indicated in bold style exhibits several similarities with the passage in MS b cited above; but it is preceded by the passage καὶ ἐπεὶ ὁμοία … καὶ ἡ ΒΗ πρὸς τὴν ΕΘ, which does not appear at this place either in MS b or in L. Having pointed this out, I shall now go into the details of analysing the beginning of the ἀπόδειξις in L, piece by piece, in comparison to the Greek texts. I shall also argue for the following tentative restoration of the passage (my conjectures in bold style):

doctrina reserauit quia] The subjunction quia could well correspond to ἐπεί (= ‘as’, ‘since’), but here the Latin translator has added doctrina reserauit, which means that the three words should be interpreted: ‘(geometrical) doctrine has revealed/shown that’.

ista et ] The words ista et were read only by Mai. As the reading ista et does not yield any sensible meaning in the context, I suggest that one should read est ut. In faint or ill-written capital script the letter E (in EST) could easily have been misread as I, and the letter U (in UT) could have been misread as AE thus: the first stroke in the letter U is taken to be the second stroke in the presumed letter A, and the second stroke in U is taken to be the main stroke of the letter E (which, as stated several times, would have appeared very similar to I). Granted this conjecture, est ut would correspond nicely to ἐστίν ὡς.

illa quae ΑΒ <*> quam ΒΓ diducit] This portion corresponds structurally to ἡ ΒΓ πρὸς τὴν ΒΑ (MS b) and ἡ ΑΒ πρὸς τὴν ΔΕ (the other Greek manuscripts). By comparing the following illa quae ΔΕ est ad eā (= eam), I suggest that the illegible portion between ΑΒ and quam should be supplied thus: <est ad eā>.Footnote ⁶⁶ And by comparing the following future retinebunt, I propose that diducit is either a misreading of or an error for deducet, which, in turn, should be an erroneously written deducent; cf. discedet for descendet (or perhaps descendit) in the πρότασις of proposition 3, at fol. 341r col. 2, lines 9–10.Footnote ⁶⁷

Now, granted these emendations, the phrase illa quae ΑΒ <est> ‘the [straight line] which is ΑΒ’ will be equivalent to ἡ + two Greek letters, and the phrase <ad eā> quam ΒΓ deduce(n)t ‘to the [straight line] which [the points] Β [and] Γ will extend’ to πρὸς τήν + two Greek letters; see further on ita … retinebunt below.

ita illa quae ΔΕ est ad eā quam Ϲघ retinebunt] This phrase corresponds structurally to οὕτως ἡ घΗ πρὸς τὴν घΕ (MS b) and οὕτως ἡ ΒΓ πρὸς τὴν Εघ (the other Greek manuscripts): ita = οὕτως; illa quae ΔΕ est = ἡ + two Greek letters; and ad eā quam Ϲघ retinebunt = πρὸς τήν + two Greek letters, whereby the Latin phrase is to be interpreted as ‘to the [straight line] which [the points] Ϲ and घ will enclose’.

Now, if one assumes that the letter Ϲ is either a corrupt or misread lunate Ε (cf. the discussion of the διορισμός of this proposition in section 2 above),Footnote ⁶⁸ one can make the following observations regarding the lettering and the mathematical argument in the whole portion est ut illa quae ΑΒ <est ad eā> quam ΒΓ deduce(n)t ita illa quae ΔΕ est ad eā quam Εघ retinebunt. Given that the triangle ΑΒΓ is the base of the one of the two pyramids in L and the triangle ΔΕघ the base of the other, the proportion ΑΒ : ΒΓ = ΔΕ : Εघ is expressed. In MS b, next, the triangles are denoted ΑΒΓ and ΕघΗ, which means that the phrase ἐπεί ἐστιν ὡς ἡ ΒΓ πρὸς τὴν ΒΑ, οὕτως ἡ घΗ πρὸς τὴν घΕ expresses the same proportion, though in the inverse form and with a different lettering of the second triangle: ΒΓ : ΒΑ = घΗ : घΕ. The majority of the Greek manuscripts, however, reads καὶ ἐπεί ἐστιν ὡς ἡ ΑΒ πρὸς τὴν ΔΕ, οὕτως ἡ ΒΓ πρὸς τὴν Εघ, a phrase which, given the triangles being ΑΒΓ and ΔΕघ, expresses the proportion thus: ΑΒ : ΔΕ = ΒΓ : Εघ. Whereas the lettering in L seems to adhere to that found in the majority of the Greek manuscripts, the proportion itself is expressed in L in a form more similar to MS b; for the first ratio is given as one side to another in the one triangle, and the second ratio as one side to another in the other triangle.

ac sunt circa atquaeles angulos occupatae] As reported by Mai, L here has the corrupt reading atquaeles for aequales.Footnote ⁶⁹ One would expect the words ac sunt … occupatae to correspond to the Greek καὶ περὶ ἴσας γωνίας (τὰς ὑπὸ τῶν ΑΒ, ΒΓ, Εघ, घΗ add. MS b) αἱ πλευραὶ ἀνάλογόν εἰσιν; but presumably there is a lacuna in the text of L after occupatae. Whether this lacuna was accomplished by the Latin translator/scribe or was already present in the Greek source text cannot be determined. At least the translation of the words καὶ περὶ ἴσας γωνίας and εἰσίν has been preserved.

simile igitur est illut quod ba<s> <*>] This phrase should correspond to the phrase ὅμοιον ἄρα ἐστὶ τὸ ΒΜ, which is read unanimously in the Greek manuscripts. According to Geymonat's edition the letters ba follow quod (the letter a was read only by Mai). Evidently Geymonat conjectured that the letter s should be supplied after ba, presumably intending a form of the word basis. I for my part think it to be more likely that the two strokes, interpreted by Mai as the capital letter A, belong to a partly obscured Μ. If this is the case, one should read ΒΜ.