The textual witnessesFootnote ⁷

The text, pp. 107.10–108.8 L, is partly or completely preserved in the following known manuscripts:Footnote ⁸

1. J Jena, Thüringer Universitäts- und Landesbibliothek, Prov. f. 156 (sixteenth century):Footnote ⁹ ff. 69v–70v

2. V Vatican City, Biblioteca Apostolica Vaticana, Vat. lat. 3132 (sixteenth century):Footnote ¹⁰ f. 110rv (pp. 107.10–22; 107.24–108.8 L)

   1. V_V Vatican City, Biblioteca Apostolica Vaticana, Barb. lat. 164 (sixteenth century):Footnote ¹¹ f. 14rv

   2. V_P Paris, Bibliothèque Nationale, lat. 7229 (sixteenth century):Footnote ¹² f. 87rv

3. H London, British Library, Add. 47679 (twelfth century):Footnote ¹³ ff. 46v–47r (pp. 107.10–19; 107.23–108.8 L)

4. H_L Leiden, Bibliotheek der Rijksuniversiteit, Scal. 6 (sixteenth–seventeenth century):Footnote ¹⁴ ff. 14v–15r

J and V are two independent copies of the so-called Codex Arcerianus (Wolfenbüttel, Herzog-August-Bibliothek, 36.23 Aug. 2° (late fifth century–early sixth century); in two parts: A and B), the earliest extant manuscript of the CAR, in which our text, originally located in B, has unfortunately been lost.Footnote ¹⁵ The Codex Arcerianus and its copies constitute one class of CAR manuscripts, the so-called ‘AB-recension’. Although both J and V exhibit alterations – for the most part erroneous emendations – of the text of the Codex Arcerianus, J is generally considered to be more faithful to the original than V.Footnote ¹⁶ In J and V, moreover, our text is illustrated with figures. Thulin argued that they stem from a common source and were drawn by a later hand in the margin of B (whence they were copied into J and V). He also suggested that they were, in turn, either inserted into B from some older and more complete manuscript of the so-called ‘PG-recension’ – another class of CAR manuscripts in which we find P (Vatican City, Biblioteca Apostolica Vaticana, Pal. lat. 1564 (ninth century)) and G (Wolfenbüttel, Herzog-August-Bibliothek, 105 Gud. lat. 2° (ninth century)) – or invented by the drawer himself.Footnote ¹⁷ As this study shows, however, the latter alternative can now be dismissed: the figures were not created by the drawer.Footnote ¹⁸

V_V and V_P are copies of V, and they are of no particular critical value.Footnote ¹⁹

For his editing of the text in question, Lachmann used J and V, whereas H was held to be lost. In 1969, however, Folkerts brought H to light and described it.Footnote ²⁰ H belongs to a third class of CAR manuscripts which originate from a compilation for which a manuscript related to the Codex Arcerianus and a manuscript related to P and G were used; hence the name the ‘mixed class’.Footnote ²¹ Although the group of manuscripts to which H belongs suffers from severe displacements and corruptions, one may nevertheless find readings of our text in H which are less corrupt than in J and V. And if our passage was included in the manuscript related to P and G (which we unfortunately do not know), readings of H could represent the ‘PG-recension’. As in J and V, moreover, in H our text is illustrated with figures.

H_L, finally, is a copy of H by the hand of Joseph Justus Scaliger (1540–1609).

In the following section of this paper, Lachmann's text is presented together with a revised and extended critical apparatus, in which the readings of J, V and H and Lachmann's emendations are recorded. The readings of V_V, V_P and H_L have been excluded with one single exception (see the critical apparatus at p. 107.16 L). Abbreviations etc. are usually expanded, unless they are likely to be of critical significance.

The text according to Lachmann with a new critical apparatus

NotesFootnote ²²

This sentence, which begins our text, is obviously some type of heading, indicating the purpose of the geometrical methods subsequently described. In the text that follows three different methods of constructing right angles are presented. In order to draw exact rectangular and rectilinear figures, it is of course necessary to be able to construct right angles. In this respect, hence, the sentence makes good sense.Footnote ²⁴

Perhaps the sentence will seem oddly expressed: what is the reason for pointing out that a rectangular figure shall be contained by straight lines, not simply consist thereof? The answer is that in Euclidean geometry a figure is defined thus: Σχῆμά ἐστι τὸ ὑπό τινος ἤ τινων ὅρων περιεχόμενον (Euc. Elementa 1 def. 14), ‘A figure is that which is contained by any boundary or boundaries.’Footnote ²⁵ Cf. the Latin translation of this definition in the Expositio p. 104.1–2 L: Forma est quae sub aliquo aut aliquibus finibus continetur. The figure square, for example, is defined not as its boundaries, that is, the four straight lines, but as the plane surface contained by them.Footnote ²⁶

It may seem unnecessary to say that straight lines are to contain a rectangular figure. In fact, it is not unnecessary. For from the account of angles in the Expositio pp. 100.5–103.21 L it follows that rectangular figures are not necessarily contained by straight lines; there are different types of right angles depending on the type(s) of the two lines that form the angle: right angle between two straight lines, right angle between two circular lines, right angle between one straight and one circular line. From the Expositio p. 101.13–16 L, for example, we infer that a semicircle is considered a rectangular figure. The semicircle is contained by a circular line (half a circumference) and a straight line; where the lines meet, they form right angles; consequently the two angles of the semicircle are right angles, which renders the figure rectangular.Footnote ²⁷

In his translation Campbell indicates that something is lost before qualemcumque: ‘… to enclose’ etc. Presumably the lack of a finite verb governing comprehendere was his reason. However, neither Lachmann nor Guillaumin was troubled by this. In my view, the transmitted text (MSS JVH (forma in MS J being, of course, a banal error)) is most likely sound. The infinitive can be satisfactorily explained as an example of the well-attested usage of infinitives in headings, especially common in technical literature; see Löfstedt (Reference Löfstedt1936) 189, Hofmann and Szantyr (Reference Hofmann and Szantyr1965) 367, and Svennung (Reference Svennung1935) 418–24. Cf. e.g. the initial enunciation of the first proposition in Euclid's Elements: Ἐπὶ τῆς δοθείσης εὐθείας πεπερασμένης τρίγωνον ἰσόπλευρον συστήσασθαι (Euc. Elementa 1 prop. 1), ‘On a given finite straight line to construct an equilateral triangle.’Footnote ²⁸

According to Guillaumin, the three figures preserved in MS J f. 70r, appended to Lachmann's edition as figure 122, have nothing to do with the context.Footnote ²⁹ It is true that these figures make no sense at the location given to them in MS J: between the words diastemate and circulos (p. 107.14 L). But taken with Qualemcumque … comprehendere, they aptly serve as illustrations of rectangular and rectilinear figures.

The figure at the bottom of MS H f. 46v is most likely a variant of one such figure.Footnote ³⁰ Furthermore, since the figures in MS H are obviously related to those found in MSS JV, and since MS B itself was not used for the original compilation of the ‘mixed class’ (to which MS H belongs), the figures cannot have been invented by the drawer of the figures in MS B; see the section ‘The textual witnesses’ above and the note on quorum (p. 107.15 L) below.

No diagram illustrating rectangular and rectilinear figures has been preserved in MS V.

Figures 1 and 2 are modern illustrations of the construction, to which I shall be referring in the following notes. Figure 1 illustrates the construction if the point A is placed on the given straight line l, whereas Figure 2 displays the construction if the point A is placed beside the given straight line l.

Fig. 1

Fig. 2

Guillaumin translates the passage thus:

Roth Congès' translation runs as follows:

Campbell, finally, renders the passage thus:

As the illustrating figures in Guillaumin and Roth Congès show, they describe only the case when the point A is placed on the given straight line l.Footnote ³¹ Bouma, for his part, offers a different description of the construction, which has been adopted by Campbell in his commentary;Footnote ³² it will be discussed in the excursus ‘Bouma's explanation of Ex data recta linea … in data linea angulos (p. 107.12–16 L)’ below.

The infinitive ducere is probably a so-called infinitivus imperativus, which is not uncommon in technical literature; see e.g. Hofmann and Szantyr (Reference Hofmann and Szantyr1965) 366–7, Löfstedt (Reference Löfstedt1936) 186–90, and Svennung (Reference Svennung1935) 436–38. It is best rendered in English with the imperative: ‘draw’. Another instance of this type of infinitive in our text is eicere; see the note on it below (p. 107.22 L).

The uolueris which MS H has added to ducere is most likely an erroneous scribal emendation; it has been added in order to supply ducere (and comprehendere (p. 107.11 L)) with a governing finite verb.

Guillaumin translates posito signo thus: ‘D'une droite donnée, mener, en un point que l'on a placé, <une droite perpendiculaire à la première, […] > ’ (my emphasis). To be precise, however, the Latin text as preserved does not explicitly state that the straight line to be drawn (the straight line CD in Figs. 1 and 2) will pass through the point referred to in posito signo (the point A in Figs. 1 and 2); but as a consequence of the construction, this will of course be the case.

In its geometrical sense of ‘point’ the Latin word signum corresponds to the Greek σημεῖον; see e.g. Mugler (Reference Mugler1958–9) s.v. σημεῖον and OLD s.v. signum 1b. Cf. e.g. Expositio p. 97.15 L signum est cuius pars nulla est = Euc. Elementa 1 def. 1 σημεῖόν ἐστιν οὗ μέρος οὐθέν; Expositio p. 98.15–16 L […] lineae autem fines signa = Euc. Elementa 1 def. 3 γραμμῆς δὲ πέρατα σημεῖα.

In his translation, Guillaumin supplied this: ‘D'une droite donnée, mener, en un point que l'on a placé, <une droite perpendiculaire à la première, puis,> […].’ As he mentions in a footnote, he was using an earlier draft of Roth Congès' article.Footnote ³³ In Roth Congès' published article, then, the lacuna is filled similarly: ‘Depuis une ligne droite donnée, élever, après avoir posé un jalon, [une ligne droite perpendiculaire à la première] […].’ In my view, Guillaumin and Roth Congès may well be on the right track in principle.

On the one hand, Lachmann's suggested location of the lacuna is plausible; it seems reasonable that posito signo and relato … circino were originally not asyndetically coordinated (MSS JV) but separated by some portion of text. If, on the other hand, the reading et in MS H is genuine, not a banal scribal emendation, the lacuna could be located either thus: Ex data recta linea <*> ducere posito signo et relato in utramque partem circino etc. Or thus: Ex data recta linea ducere <*> posito signo et relato in utramque partem circino etc.

Guillaumin, Roth Congès and Campbell take relato … circino to be an ablative absolute. Guillaumin translates ‘portant le compas de part et d'autre’; Roth Congès translates literally ‘après avoir envoyé le compas de part et d'autre’,Footnote ³⁴ and then interprets ‘ayant tracé des repères au compas de part et d'autre’; finally, Campbell: ‘when the compass has been moved to both sides’.

Roth Congès explains the purpose of the movement of the compass thus: ‘[…] pour déterminer sur la ligne droite les deux centres des cercles à décrire’.Footnote ³⁵ With Roth Congès' explanation, the idea is that by using the compass one point on each side of and at equal distance to the point A is obtained on the given straight line l (see Figs. 1 and 2). In other words, a circle (or at least two circular arcs) is drawn with its centre at point A and with an arbitrarily taken radius AB, so that the circle intersects the straight line l at two points B and B′ (hence, AB = AB′). These two points, B and B′, will then constitute the centres of two circles that are to be drawn next.

Admittedly, the phrase relato … circino is rather compressed. As it describes such an important part of the construction, one would have wished it to be somewhat more elaborate.

As there are no other attestations in ancient Latin of diastema used in its two senses at the same time, I think that this feature, common to the supposed fragment and the Expositio, can be adduced as one piece of evidence favouring the view that the supposed fragment rightly belongs to the Expositio.

Lachmann rightly, I think, corrected the transmitted iunctorum (MSS JV) to punctorum; MS H has iuncturarum, which could be an erroneous scribal emendation of iunctorum. In our text the word punctum also occurs at p. 107.21 L, where Lachmann corrected the transmitted iuncto (MSS JV) to puncto; see the note on ex quolibet puncto … rectam lineam eicere (p. 107.21–2 L) below. The attestations of the word punctum ‘centre’ in the Expositio are: pp. 101.15; 102.1; 102.7; 102.14 L.

In a footnote to her translation Roth Congès tries to defend iunctorum by suggesting that the participle iunctus could be used as a noun in the sense of ‘ouverture de compas’ or ‘écartement des branches’: aequali iunctorum diastemate ‘à une distance égale à celle des (deux) ouvertures de compas’.Footnote ³⁹ But since there is no support whatsoever to be found in ancient Latin literature for this sense of iunctus, and since the attestation of punctorum diastemate in the Expositio p. 102.14 L – quoted in the note above on aequali punctorum diastemate (p. 107.13–14 L) – is critically certain, Lachmann's correction is in my view as right as an emendation can be.

On the meaning of the word circulus, see the note on conexionem (p. 107.15) below.

The erroneous reading quattuor quorum in MS J^a.c. may well have something to do with the fact that two practically identical pairs of intersecting circles are drawn in MS J f. 70r between angulos (p. 107.16 L) and sed (p. 107.16 L); they are reproduced in Lachmann's edition as figure 123. These four circles, which, as Thulin suggests, were drawn in the margin of MS B and then copied into MS J,Footnote ⁴⁰ may have led the scribe of MS J to commit the error.

On the one hand, the two pairs of intersecting circles could be the result of duplication of one original pair, which, if so, probably had already occurred in some ancestor to the drawings of MS B (see below in this note). On the other hand, it is also possible that both pairs are genuine and originally illustrated different stages of the construction, but, having lost their distinguishing details in the process of copying, now appear identical in MS J.

The four circles found in MS H f. 46v are most likely related to the circles found in MS J, but they appear in a more corrupt state. Their existence in MS H proves that the drawer of the figures in MS B did not draw at his own discretion; see the section ‘The textual witnesses’ and the note on Qualemcumque … comprehendere (p. 107.10–11 L) above.

In MS V f. 110r one pair of intersecting circles is found in the margin. As both MS J and MS H have four circles, I think that MS B had two pairs of intersecting circles and only one such pair was copied into MS V.

In Euclid κύκλος ‘circle’ usually means the plane figure, the surface, if you will, which is contained by the circumference, that is, ‘disk’, not the circumference of the circle;Footnote ⁴¹ see e.g. the Euclidean definition of the circle, which also appears translated into Latin in the Expositio:

Sometimes, however, Euclid evidently uses κύκλος in the sense of ‘circumference of a circle’; see e.g. Euc. Elementa 3 prop. 10 Κύκλος κύκλον οὐ τέμνει κατὰ πλείονα σημεῖα ἢ δύο. ‘A circle does not cut a circle at more points than two.’Footnote ⁴⁴

Now, as the antecedent to the genitive quorum which qualifies conexionem is circulos, this suggests that our author is not using the word circulus here in the usual sense of ‘disk’, but in the sense of ‘circumference of a circle’.

MSS JV read: normalis in recta linea angulos (angulus MS J). MS H reads: normationis in recto angulo. Most likely MSS JV exhibit a more genuine text (Lachmann's minor change of normalis to normales seems to be in order, since there are no indications in the Expositio of a regular use of the ending –is in the accusative plural). Even if normationis were changed to normationem or normationes, the reading of MS H remains awkward and obscure. Perhaps normationis in recto angulo is the futile result of scribal emendation after loss of linea: normalis/–es in recta linea angulos > *normalis/–es in recta angulos > normationis in recto angulo. On the adjective normalis, see the next note.

Lachmann's emendation data for the transmitted recta seems to be supported by factura in data linea rectum angulum at p. 107.24 L; but by comparing ex data recta linea at p. 107.12 L one could also suggest that data should be supplied before recta: in <data> recta linea. In addition, the reading recto in MS H may be adduced in support of maintaining recta. Although these conjectures make the meaning clearer to the reader, I tend to think that they are unnecessary; the author may well have been somewhat vague, and for an attentive reader it is evident that the recta linea in question is the given straight line l, not the straight line CD (see Figs. 1 and 2).

Cf. the following passage in the so-called Fragmentum Censorini, where normalis (sc. linea) is given as a Latin equivalent to the Greek κάθετος (sc. εὐθεῖα (sc. γραμμή)) ‘perpendicular (straight line)’ (see Mugler (Reference Mugler1958–9) s.v. κάθετος):

It should finally be noted that, granted the difference between normalis angulus and rectus angulus suggested above, the author of our text obviously did not hesitate to use the wider term rectus angulus, although the stricter normalis angulus could have been used instead; see p. 107.24 L.

Excursus: Bouma's explanation ofEx data recta linea … in data linea angulos(p. 107.12–16):

Evidently Bouma's tentative and highly unconvincing description of the geometrical construction does not take the following portion of text into account: Ex data recta linea ducere posito signo <*> relato in utramque partem circino (p. 107.12–13 L).Footnote ⁴⁸ Consequently he suggests that the straight line to which the perpendicular shall be dropped is first to be constructed, although it is most likely already given (ex data recta linea). This is done by drawing two intersecting circles with equal radii and then by drawing the straight line AB through their intersections (see

Fig. 3

below). As the phrases posito signo and relato in utramque partem circino are neglected, the construction proceeds by drawing one circle with the centre at B and the radius BA and then by extending the straight line AB to C, so that another circle with the centre at C and the radius CB can be drawn. By drawing the straight line DE, finally, the intersection of DE and AC will be at right angles.

Fig. 3

Furthermore, a great drawback is that the constructed right angles will end up at a location difficult to foresee at the beginning of the construction; this renders Bouma's construction impractical, even in theory.

Unfortunately, Bouma's explanation was uncritically adopted by Campbell in his commenting note.Footnote ⁴⁹

Guillaumin, to begin with, translates thus:

As regards the phrase a circumferentiarum parte, Guillaumin states in a footnote: ‘[O]n peut hésiter entre deux traductions: “sans utiliser les circonférences”, ou “à partir de segments de circonférences”.’Footnote ⁵⁰

Roth Congès, then, interprets as follows:

In a footnote to this portion of text, Roth Congès gives the following comment: ‘C'est-à-dire qu'on se contente de tracer aux endroits nécessaires deux arcs de cercle, dont l'intersection marque le point recherché. Ce sont ces traits croisés en forme de χ que l'on appelle “chiasme”.’Footnote ⁵¹

Campbell, finally, renders the passage thus:

I for my part do not profess to have found a definitive solution to this difficult passage; but in the following notes I tentatively suggest that a text at least as good – or bad – as Lachmann's can be obtained without resorting to conjectural criticism:

This text could be translated thus:

Or perhaps more freely:

As stated in the section ‘The textual witnesses’ above, Lachmann used only MSS JV for his edition. Starting from the transmitted readings quoniam (MS J) and qm (= quoniam) (MS V),Footnote ⁵² Lachmann conjectured quo in, whereby a desired in was restored: in rectarum linearum forma. Possibly he imagined this chain of corruption: *q in (abbreviated quo) > qm (MS B?, MS V) > quoniam (MS J). Exactly what sense of quo he had in mind is unclear, however. According to Guillaumin's translation, quo is interpreted as a final subordinating conjunction, whereas in Roth Congès' and Campbell's translations quo is interpreted as a pronominal adverb; see the preceding note on sed quo … ratione utamur (p. 107.16–19 L). Furthermore, Lachmann preferred interueniat (MS V) to interuenit (MS J).

As we now are able to consult MS H, there is reason to reconsider the text. Unfortunately, however, MS H seems to offer a generally more corrupt text of this passage than MSS JV, but it might still provide us with some clues. First, it could be that qm in (= quoniam in) is the genuine reading, not merely a scribal emendation of a previous qm.Footnote ⁵³ Secondly, the awkward normaliter uenit could, I think, speak in favour of reading the indicative interuenit, although, of course, scribal emendation of a previous ueniat cannot be totally excluded. And thirdly, the reading recisa (MS H) seems to support the genuineness of rectis a (MSS JV), as it is far more likely that rectis a has been corrupted to recisa than vice versa.

The intended sense of the verb interuenire is not perfectly clear; Guillaumin renders it with the verb ‘couper’, Roth Congès with ‘recouper’, and Campbell with ‘intersect’. In my opinion, however, one may compare the following passage in the Expositio:

If the verb interuenire is taken in this sense, ‘be combined with’, one may read the passage in question with quoniam in and interuenit thus: sed quoniam in rectarum linearum forma circularis linea non interuenit rectis ‘but since in a rectilinear figure a circular line is not combined with straight lines’. In other words: by definition a rectilinear figure is not contained by any circular line; however, the previously described method of constructing right angles in such a figure involves the drawing of circles (p. 107.12–16 L). Consequently, the clause reminds the reader of the obvious fact that those circular lines are merely auxiliary, not constituent parts of the figure being constructed.

Lachmann printed a circumferentiarum parte with MS J and then adopted his conjecture chiasmi cuiusdam ratione. In my view, however, there seems to be no need for emendation; I suggest that chiasmis eiusdem rationis be read with MS J. At p. 108.2 L, MS J has preserved the genuine reading chiasmis (the ablative is governed by utendum est), whereas MS V has the corrupt chiasinis (in instead of m); see the note on in hanc autem … utendum est (p. 108.1–2 L) below. Hence there is reason to assume that chiasmis is the genuine reading here too, where MS V also exhibits the corrupt chiasinis. The reading eiusdem, then, is supported by all three MSS, and rationis is supported by MSS JH; rationem in MS V is presumably an ungrammatical emendation introduced by the scribe (with the intention of recovering an “object” for utamur). The text of MS H, finally, is likely to be the result of omissions (recisa for rectis a and parte chiasmis omitted) and subsequent scribal emendations (circumferentia linearum for circumferentiarum and norma added).

As Guillaumin, Roth Congès and Campbell have noted, the word chiasmus here denotes a mark of intersection between two line segments in the shape of the Greek letter χ.Footnote ⁵⁴ According to the TLL s.v. chiasmus 3.1007.76 ff., p. 107.18 L and p. 108.2 L are the only attestations of this Latinised Greek word (cf. LSJ s.v. χιασμός) in ancient Latin gromatical literature. They have also observed that the noun decussis is attested in ancient Latin in a similar sense; see TLL s.v. decussis 5:1.248.57–66 and 77–83. Cf. also TLL s.vv. decusso 5:1.249.5 ff.; decussatio 5:1.248.26 ff.; decussatim 5:1.248.13 ff.

The prepositional expression a circumferentiarum parte is somewhat obscure. It seems to go with the following chiasmis, stating that the intersections may be accomplished by drawing only a part, that is, segments, of the circumferences; for it is not necessary to draw the entire circumferences in order to obtain the intersections.

Consequently the whole clause a circumferentiarum parte chiasmis eiusdem rationis utamur could be translated thus: ‘we may use χ–shaped marks from segments of circumferences according to the same method’; or more freely: ‘we may use χ–shaped marks according to the same method (as described previously) which are obtained from segments of circumferences’.

p. 107.20–4Quod si ab … linea rectum angulum:

In this passage another method of dropping a perpendicular is described. It is an application of the so-called theorem of Thales, which states that the diameter of a circle always subtends a right angle at any point on the circumference of the circle.Footnote ⁵⁵Figure 4 is a modern illustration of the construction, to which I shall be referring in the following notes. Let the straight line AB be the given straight line, and the straight line EB be the constructed straight line at right angles to the given straight line. Cf. the figures given by Guillaumin, Roth Congès and Campbell.Footnote ⁵⁶

Fig. 4

MSS JV have preserved a figure which is a simple illustration of this construction (it is reproduced in Lachmann's edition as figure 124): MS J f. 70r, between utamur (p. 107.19 L) and Quod si (p. 107.20 L); MS V f. 110r, at the bottom of the folio, after angulum (p. 107.24 L). The figure in the top right corner of MS H f. 47r may be a corrupt version of this figure.

As Lachmann's text runs, the transmitted qui makes no sense; it cannot refer to puncto either grammatically or factually. Guillaumin tentatively suggested that et diastemate be supplied before qui, so that the relative pronoun would refer to diastemate, and that recta linea be changed to rectae lineae (since ‘[…] il est inutile de préciser qu'un rayon va en ligne droite’): ex quolibet puncto <et diastemate> qui per caput recta<e> linea<e> transeat rectam lineam eicere ‘d'un centre quelconque [C; see Fig. 4] <et avec un rayon [CB]> ’ ‘qui passe par l'extrémité [B] de la ligne droite [AB]’ ‘prolonger une droite [DCE]’.Footnote ⁵⁸ However, as diastema ought to be a neuter noun (see the note on diastemate (p. 107.14 L) above), should qui not also be changed to quod?

Roth Congès, for her part, tried, unconvincingly in my view, to retain iuncto (masc.) thus: ex quolibet iuncto qui per caput recta linea transeat rectam lineam eicere ‘à partir de n'importe quelle ouverture de compas qui passe en ligne droite (CB) par l'extrémité (B), tracer une droite (DCE)’. Or: ex quolibet iuncto qui per caput recta<e> linea<e> transeat rectam lineam eicere ‘à partir de n'importe quelle ouverture de compas (CB) qui passe par l'extrémité de la ligne droite (B), tracer une droite’.Footnote ⁵⁹

Campbell translates Lachmann's text as follows: ‘from the central point (C) (of the circle) extend a straight line which passes through the extremity of the circle (D) in a straight line’. Obviously, Campbell (inadvertently?) translates as if quae were read, not qui. Moreover, he renders per caput as ‘through the extremity of the circle (D)’, which seems awkward; the circumference of a circle, which is a complete circular line, does not have any extremity. Nor is it likely that caput means ‘circumference’, as it is used in the sense of ‘extremity (of a line)’ at p. 107.23 L (see the note on caput (p. 107.21 L) below). In addition, if the centre C is placed at will (ex quolibet puncto; Campbell does not translate quolibet), the point D will not necessarily coincide with the extremity A of the straight line AB (see Fig. 4), although the figure preserved in MSS JV suggests this; see the note on Quod si ab … linea rectum angulum (p. 107.20–4 L) above.

If the transmitted qui be regarded as genuine, one could venture to emend thus: ex quolibet puncto <et circino> qui per caput recta<e> linea<e> transeat rectam lineam eicere ‘from whatever centre (C) and drawing/trace of the compass, which shall pass through the extremity (B) of the straight line (AB), extend a straight line (DCE)’. Cf. the following occurrences of circinus = ‘drawing/trace of the compass’ in the Expositio:

However, in a note to this passage in the Expositio, Guillaumin suggests that the reason why circinus is used instead of circulus is that the traces of the compass are not complete circles, but arcs.Footnote ⁶¹ If so, one may of course object that the restoration ought to be <et circulo> rather than <et circino>:  ex quolibet puncto <et circulo> qui per caput recta<e> linea<e> transeat rectam lineam eicere ‘from whatever centre (C) and circle, which shall pass through the extremity (B) of the straight line (AB), extend a straight line (DCE)’. Moreover, the required sense of the word circulus here ought to be ‘circle’ = ‘circumference of a circle’, not ‘circle’ = ‘disk’; see the note on conexionem (p. 107.15 L) above.

If the transmitted qui be regarded as a corruption, one could, on the other hand, emend by changing qui and supplying some other word(s) meaning ‘circumference’ attested in our text: ex quolibet puncto <et circumferenti linea> quae per caput recta<e> linea<e> transeat rectam lineam eicere or ex quolibet puncto <et circumferentia> quae per caput recta<e> linea<e> transeat rectam lineam eicere ‘from whatever centre (C) and circumference, which shall pass through the extremity (B) of the straight line (AB), extend a straight line (DCE)’.

p. 107.21caput: The noun caput in the sense of ‘extremity (of a line)’ occurs in our text also at p. 107.23 L. For further attestations, see TLL s.v. caput 3:2.411.83 ff.

p. 107.22eicere: The verb eicere is used in the sense of ‘extend’, ‘produce’; see TLL s.v. eicio 5:2.310.68 ff. Most likely it corresponds to the Greek verb ἐκβάλλειν (see Mugler (Reference Mugler1958–9) s.v. ἐκβάλλειν), as in the following passage of the Expositio, which is a translation of the Euclidean definition of parallel straight lines:

The infinitive eicere is a so-called infinitivus imperativus; see the note on the infinitive ducere (p. 107.12 L) above.

I translate as follows: ‘through/to the point (E; see Fig. 4) of which (i.e. the straight line DCE), that is, to the circumference, a straight line (BE) shall pass from the extremity (B) of the straight line (AB) and make a right angle on the given straight line (AB)’.

As my translation shows, the antecedent to cuius is rectam lineam at p. 107.22 L.

Guillaumin, Roth Congès and Campbell seem to have taken signum to be the antecedent to quod. In my view, however, quod est is to be understood rather as id est ‘that is’, clarifying the direction of the straight line BE.

On the terms signum ‘point’ and caput ‘extremity’, see the notes on signo (p. 107.12 L) and caput (107.21 L) above.

In translating this passage I mainly follow Guillaumin's interpretation:Footnote ⁶³ ‘But if the circumference is eliminated for this method, one must use χ-shaped marks.’ In my view, this sentence states that if the method described at p. 107.20–4 L is applied but, for some reason, a complete circumference is not to be used, it suffices to draw arcs at the points D and E (see Fig. 4). At the point D there will be an intersection between an arc and the straight line AB; the intersection will resemble the Greek letter χ. At the point E, if the straight line DCE is produced somewhat beyond the point E, there will be an intersection between an arc and the straight line; and even if the straight line DCE stops at the point E, a mark will be accomplished which can be described as similar to the letter χ.

On the word chiasmus, see the note on a circumferentiarum … utamur (p. 107.18–19 L) above.

MSS JV have preserved a triangle which is most likely intended to illustrate this construction: MS J f. 70r, between angulum (p. 107.24 L) and In hanc (p. 108.1 L), reproduced in Lachmann's edition as figure 125; MS V f. 110v, between disputauimus (p. 108.4 L) and circuli (p. 108.4–5 L). However, in MS J the triangle has been corrupted to an equilateral triangle, and in MS V its proportions are somewhat distorted, so that it resembles an isosceles triangle.

The MSS JV read: nam quod ad extremam lineae normationem pertinet, uulgaris consuetudinis est sex octo et decem. MS H reads: nam quod ad extremae lineae normationem pertinet, uulgaris consuetudinis est VI VIII uel X numeros ponere. On the one hand, the reading of MSS JV may be considered more likely to be genuine than that of MS H. The reading extremam lineae normationem (MSS JV) may be regarded as lectio difficilior in relation to extremae lineae normationem (MS H), as the adjective extremus is more easily understood as qualifying the concrete noun linea than the verbal noun normatio; the reading uel (MS H) is most likely not original (the required sense is ‘and’, not ‘or’, etc., and uel = et is not attested elsewhere in our text or in the Expositio); numeros ponere (MS H), finally, may be considered to be a later addition. On the other hand, it could be argued that extremae lineae normationem (MS H) is better than the corresponding reading of MSS JV and that MS H has indeed preserved an original numeros ponere, which makes sense of the otherwise elliptical main clause.

I for my part am inclined to think that extremae (MS H) should be preferred to extremam (MSS JV) and that numeros ponere is to be regarded as, at least, a plausible emendation to the text of MSS JV, which, in my view, seems too elliptical. Hence I translate thus: ‘Moreover, as regards the dropping of a perpendicular to the extremity of a line, the common procedure is to employ the numbers six, eight and ten.’

See further the notes on extremam lineae normationem (p. 108.2–3 L) and sex octo et decem (p. 108.3–4 L) below.

As stated in the note on nam quod ad extremam … sex octo et decem (p. 108.2–4 L) above, I tend to think that extremae (MS H) should be preferred to extremam (MSS JV). If the reading of MS H is chosen, the expression extremae lineae normatio means ‘the dropping of a perpendicular to the extremity of a line’ (the adjective extremus is taken partitively), which immediately makes perfect sense. If the reading of MSS JV were preferred, the expression extrema lineae normatio would literally mean ‘the extreme dropping of a perpendicular to a line’, which, in turn, ought to convey the same meaning as ‘the dropping of a perpendicular to the extremity of a line’.Footnote ⁶⁵ Although the reading of MSS JV seems to be both difficilior and possible, the reading of MS H is in my opinion simply more to the point and hence more likely to be genuine.

I translate thus: ‘The method with a circle, which we have talked about above, is more in accordance with the art (of geometry); it does not prescribe numbers.’ The method referred to is evidently the one described at pp. 107.20–108.2 L.

In strict Euclidean geometry, measurement of, for example, lengths is not a valid operation in constructions and proofs. As the method of constructing a right angle by using a triangle with sides of six, eight and ten units of length (p. 108.2–4) requires numbers, that is, measuring, it is rightly pointed out that the method without measurement is magis artificialis ‘more in accordance with the art (of geometry)’.

Finally, it should be mentioned that because of erroneous punctuation (full stop after disputauimus) in Guillaumin's and Roth Congès' Latin text the passage has been mistranslated and misinterpreted by Guillaumin, Roth Congès and even Campbell.Footnote ⁶⁶

Except for the unjustified alteration Eucliden for the transmitted Euclidem (MSS JV, Euclẏdem MS H), I here agree with Lachmann's text. The text of MS H, which, I think, suffers from omissions and erroneous scribal emendations, is as far as I can see nonsensical (it does, however, support the reading concurrerint of MS J against the concurrunt of MS V).

The words normam facturas, governed by habemus, are of course to be understood thus: (sc. eas duas lineas) normam facturas (sc. esse).

I translate thus: ‘For we have it in Euclid that at whatever place on the circumference two (straight) lines from the (end)points of the diameter meet, they will make a right angle.’

MS J has preserved two figures which seem to serve as illustrations to this passage: MS J f. 70v, between disputauimus (p. 108.4 L) and circuli (p. 108.4 L), reproduced in Lachmann's edition as figure 126. One of them is correct, the other not. In MS V f. 110v, in the right margin next to the explicit, only the incorrect figure is found. MS H f. 47r, in the middle of the folio, has three erroneous figures, of which the third one from the left is similar to the incorrect figures found in MSS JV. Now, the incorrectness of the figures found in MSS JV seems to originate from misinterpretation of ex signis dimensionis, literally ‘from the points of the diameter’; apparently the author of the figures interpreted ‘from (whichever) points on the diameter’, instead of ‘from the (end)points of the diameter’.

Although several textual and interpretative uncertainties remain, I hope that this paper may at least to some extent contribute to a better understanding of this important piece of geometry in the Roman tongue.