¹ Proclus, , in Primum Euclidis, p. 379Google Scholar. 2–5, attributes this ascription to Eudemus, the Peripatetic. The fact that Aristotle's friend is the source of this claim itself suggests Aristotle's acquaintance with this version.

² Heiberg, pp. 19–20, followed by Heath, 1949, pp. 29–30, 217, and Ross, 1924, ii. 269–70. Cf. Heath, 1926, i. 321. For Bibliographical details see pp. 371–2.

³ In this discussion I assume τ⋯ διαγρ⋯μματα at 1051 a 22 means diagrams (Ross, 1924, ii. 268), and not geometrical proofs of propositions (as in Bonitz, 1849, p. 407), unless the proofs are seen as the constructed figures. Heath (1949, p. 216) supposes it to mean any geometrical proofs or propositions. So it can sometimes, but not always (cf. Bonitz, 1870, p. 178, lines 3–11), and certainly not here, where the discussion is about what propositions must be known and, more important, what geometrical construction is needed. Division here is not the division of the Topics or Plato's Sophist. It is geometrical analysis, as Bonitz (1849, p. 407) indicates. We must understand the διαγρ⋯μματα as being found (1051 a 22), divided (a 23), and led into actuality (a 29–30). For we must understand the objects to be divided (διαιρο⋯ντες) to be of the same sort as that which get divided, namely τ⋯ διαγρ⋯μματα, which must also be τ⋯ δυν⋯μει⋯ ντα (1051 a 29). And as Bonitz (1848, pp. 407–8) says, these are certainly mathematical objects. For a list of parallel passages where there is the point that from the diagram the matter is evident, cf. Heiberg, p. 6.

⁴ δι⋯ τ⋯ δ⋯ο ⋯ρθα⋯ τ⋯ τρ⋯γωνον⋯τι αἱ περ⋯ μ⋯αν στιγμ⋯ν γων⋯αι ἲσαι δ⋯ο ⋯ρθαῖς. εἰ οὖν ⋯ν⋯κτο ⋯ παρ⋯ τ⋯ν πλευρ⋯ν, ἰδ⋯ντι ἂν ἦν εὐθὺς δ⋯λον δι⋯ τ⋯.

⁵ Heath suggests (1921, pp. 144, 340; 1949, p. 63) that this theorem is Pythagorean. This is enticing, but he offers no evidence. When Proclus mentions the theorem (pp. 383–4) he claims that it, with the general theorem of the interior angles of a polygon, is a preliminary for the Timaeus. Even if he has special reason for this, it will not push the theorem into the world of the Pythagoreans.

⁶ Ps.-Alexander, p. 586. 12–15; Averroes, vi. 1214. 1–2.

⁷ Cf. Bouyges' remarks in Averroes v. cxviii–cxix.

⁸ Cf. Schwegler, p. 185. Aquinas, , In Met. ix. L. 10Google Scholar: C 1889, adopts ps.-Alexander's interpretation, and even those who still put in both constructions, e.g. Averroes, vi. 1216. 6–1217. 1, and Niphius, p. 509, take the construction to which Aristotle refers to be the extended line, CD. Albertus Magnus apparently is unclear as to which construction he adopts (cf. Bürke, p. 144). I gather that the less ambiguous, but obviously corrupt ⋯ περ⋯ τ⋯ν πλευρ⋯ν was accepted because it better describes the accepted construction. Hence the texts of Erasmus or Petrus de la Rovière and the common translation with ‘qui circa latus est’, which may also be found, for whatever reason, in the Parma Aquinas (p. 547).

⁹ It could in the appropriate context mean ‘the area applied to a side’. Cf. Mugler's article. Knorr, pp. 196–7, has given further evidence for the expression's indicating a parallel in Archimedes, although it is rare in Euclid. If his overall argument is correct it certainly points to pre-Euclidean usage.

¹⁰ Cf. Bürke, p. 133, who recognizes this as a clear problem.

¹¹ See note 2.

¹² Cf. Apollonius 1. 50, 51, where ⋯ν⋯γειν does not imply direction up.

¹³ This is not the appropriate place to show this. But one locus of lines will lead to the horizon, i.e. behind HK.

¹⁴ It is really of no importance if they do belong to Euclid or to Apollonius, as the tradition referred to by the scholium to definitions 13—15 maintains. For if an Apollonian definition fits this view of ⋯ν⋯γειν then there is no reason to reject the same usage for Aristotle.

¹⁵ Heath, 1949, p. 73. The variations which one may conjure up are perhaps endless. But the number of cases which fit the text are not. So we propose a version with one construction. CE parallel to AB as in Fig. 1. But then we will have to use something like the fact that the interior angles on the same side of two cut parallels equal two right angles. And this version must be rejected, not because it lacks ancient authority, but because we will not then use the fact that the angles about a point are two right angles, even if this fact is used in proof of the parallel theorem (as Euclid I. 28, 29).

¹⁶ Heiberg, p. 21.

¹⁷ Heath, 1921, pp. 339–40; 1926, ii. 63–4; 1949, pp. 73–4.

¹⁸ Ross, 1924, ii. 271; 1949, p. 641.

¹⁹ In Eudemus' account of the squaring of lunules by Hippocrates of Chios, he refers to an analogue of III. 21 and III. 31 (cf. Simplicius, , in Phys, vol. 1, p. 61Google Scholar. 14–18): κα⋯ γων⋯ας ἲσας δ⋯χεται τ⋯ ⋯μοια τμ⋯ματα. α⋯ γο⋯ν τ⋯ν ⋯μικυκλ⋯ων π⋯ντων ⋯ρθα⋯ ε⋯σι, κα⋯ <αἱ> τ⋯ν μειζ⋯νων ⋯λ⋯ττονες ⋯ρθ⋯ν κα⋯ τοσο⋯τωι ὂσωι μεὂζονα ⋯μικυκλὂων τ⋯ τμ⋯ματα, κα⋯ α⋯ τ⋯ν ⋯λαττ⋯νων με⋯ζονες κα⋯ τοσο⋯τωι ὂσωι ⋯λ⋯ττονα τ⋯ τν⋯ματα. This seems to indicate that the theorem of the angle in the semicircle was proved separately from the case where the segments of the circle are greater and less. Moreover the separate cases of the segments were divided into two parts, one giving the relation of the angle size to the right angle, and the second giving its equivalence in terms of its ‘distance’ from the right angle.

²⁰ An. Post. A 1.71 a 19—21: ὂτι μ⋯ν γ⋯ρ πᾱν τρίγωνον ἒχει δυσ⋯ν ⋯ρθαῖς ἲσας, προήιδει ὂτι δ⋯ τὂδε τ⋯ ⋯ν ⋯μικυκλ⋯ωι τρ⋯γων⋯ν ⋯στιν, ἅμα ⋯παγ⋯μενος ⋯γνώρισεν. The proof starts with I. 32 b and the triangle constructed by the angle in the semicircle as givens. The proof then should use both facts.

²¹ δι⋯ τ⋯ ⋯ρθ⋯ ⋯ ⋯ν ⋯μικυκλ⋯ωι; τ⋯νος ὂντος ⋯ρθ⋯; ἕστω δ⋯ ⋯ρθ⋯ ⋯π' ⋯ς Α, ⋯μ⋯σεια δυοῖν ⋯ρθαῖν ⋯φ' ⋯ς Β, ⋯ ⋯μικυκλ⋯ωι ⋯π' ⋯ς Γ. το⋯ δ⋯ τ⋯ Α τ⋯ν ⋯ρθ⋯ν ὑπ⋯ρχειν τ⋯ι Γ τ⋯ι ⋯ν τ⋯ι ⋯μικυκλ⋯ωι α⋯τιον τ⋯ Β. α⋯τη μ⋯ν γ⋯ρ τ⋯ι Α ⋯ση, ⋯ δ⋯ τ⋯ Γ τ⋯ι Β δ⋯ο γ⋯ρ ⋯ρθ⋯ν ⋯μ⋯σεια. το⋯ Β ο⋯ν ὃντος ⋯μ⋯σεος δύο ⋯ρθ⋯ν τ⋯ Α τ⋯ι Γ ὑπ⋯ρχει (το⋯το δ' τ⋯ ⋯ν ⋯μικυκλ⋯ωι ⋯ρθ⋯ν εἶναι).

²² Ross, 1949, p. 691; also cf. Ross, 1924, ii. 271.

²³ Ross, 1924, ii. 271, thinks the right angle a superfluous oversight: ‘it would be natural enough for Aristotle by an oversight to think that the angle could more easily be proved to be right in the symmetrical case in which it is the sum of two half right angles’. This is the sort of error which would be very much unnatural to the author of the discussion on universal proof at An. Post. A 5.

²⁴ Heath, 1949, p. 72.

²⁵ Ps.-Alexander, p. 596, lines 16–20, 23–4.

²⁶ Ibid., p. 596. 34.

²⁷ Heath, 1949, pp. 72–3.

²⁸ This version of the proof is proffered and rejected by Ross, 1924, ii. 271. So far as I know, Dancy, pp. 374–5, is the only scholar to have endorsed it for An. Post. B 11. He does not say how he deals with Met. Θ 9.

²⁹ This, in effect, is what Roman does in his translation of Aquinas. However, he offers n o justification for it, an d it would seem there is none in any of the Latin translations from the Greek. Cf. Aquinas, p. 694. The only justification he can give for his translation is that Aquinas gives the alternative proof in Euclid, without assuming the line drawn to be perpendicular. But his translation, which is supposed to represent Aquinas's text, is not here based on any edition I know of. The Latin of the Arabic translation used by Averroes, which I shall presently discuss, is a reasonable paraphrase and does not resemble Roman's translation (cf. Bürke, p. 68). Aquinas's discussion shows the influence of Averroes. Ironically, the proof Aquinas gives in his commentary on the An. Post. (XVIII, p. 200), which uses the construction of the perpendicular, fits Met. Θ 9, just as the version given here better fits that passage, as if he had forgotten which goes with which.

³⁰ Ross, 1924, i. clxiv.

³¹ Walzer claims that the Arabic text represents an independent tradition.

³² A verroes, p. 1214, Ins. 4–6. The Arabic goes ‘wa kadhaalika lima fii kulli nisfi daa iratin qaa imatun waahidatun ?alaa lmahiiti tasaawit a suuquhaa am Lam tatasaawi idhaa kaana qaa?idatuhaa lqutra ?alaa wusti lmahiiti waqa?at am ?alaa gayrihi innahu bayyana liman nadhara ilayhu bi'inna lahu ma?rifatu dhaalika’. I think the attempt to reconstruct the Greek text is in vain, the first part of the protasis up to ⋯κ μ⋯σου being very free with the text. The Arabic is bizarre also because it gives no construction at all. It just says that the theorem is obvious.

³³ Aristotle, 1886. Christ's recommendation appears in his notes. By removing ⋯ρθ⋯ from the protasis, we are not committed to keeping the figure which we would get with ⋯ρθ⋯. I would like to think this is what Christ intended. If so he has the right construction, but a bad text. Heath (1949, p. 73) notes that he translated from Christ's edition in the first edition of his text of Euclid (1926, first ed. 1908, ii. 63), though he does not adopt Christ's suggested punctuation. He there translates the relevant phrase, ‘the third set up at right angles at its middle point’. Like Eustathius, Heath takes the middle to be the middle of the circumference. In the later work he adopts Ross's interpretation. Except for his preferring δι⋯ τί to Christ's διότι in line 27, his change of heart is over the interpretation, not the text.

³⁴ Ross, 1924, ii. 271.

³⁵ The plethora of δι⋯ τ⋯ in Ross's text displeases Jaeger, although they are supported by the main Greek manuscripts. Jaeger argues in favour of the Latin translation, also favoured by Bekker and Christ, which puts δι⋯τι for δι⋯ τ⋯ after καθ⋯λου; in line 27, and so makes δι⋯ τ⋯ in line 26 go with the following clause. Besides the weak manuscript tradition behind it, Ross had objected that this reading requires a translation such as ‘Why is the angle in a semicircle always a right angle? Because, if …, it is clear to someone who…that…’ The fact that the theorem is evident becomes the explanation of its truth. Jaeger's defence involves the ways of referring to middle expressions at line 24 and An. Post. B 94a 37, 95a 14, to show that the reply to the ‘why’ question is a ‘because’ answer, whether δι⋯τι or ὅτι, followed perhaps by the expression of the middle expression. Of course the middle expression cannot be the construction. The middle expression which δι⋯τι would initiate is referred to later by ⋯κε⋯νο. One would have a sentence which could be filled out, ‘(The angles of a triangle equal two right angles) because, if a certain construction is made, it will be clear to someone who sees the figure and knows that fact’. The structure is, in fact, very close to the previous argument. But if Jaeger is right, to prevent the properties of the auxiliary constructions from becoming the cause of the theorem, ⋯ρθ⋯ must be emended to something like δ⋯ο ⋯ρθαῖς, with the result that we would have a strong hyperbaton. Note that if the hyperbaton is possible here, it is possible without emendation for Ross's text. Given that the alternative reading is better supported, and makes sense, I see no reason to adopt the Vulgate reading. Hence we might prefer one of two other plausible emendations based on the text as it appears in Ross's edition.

³⁶ The only case in Aristotle attested (and doubted) by Bonitz (1870, p. 173 b 38–40) is Politica E II. 1313 a 18; Vahlen emends δ⋯λον to δ⋯λον ὅτι, and Ross to δηλον⋯τι.

³⁷ Ibid. In his capacity as Secretary to the Delegates of the Oxford University Press, C. Cannan made recommendations to Ross and allowed him to use some unpublished emendations, of which, I presume, the emendation under discussion is one. Cf. ibid. i. v.

³⁸ Ps.-Philoponus on the Metaphysics may have read ὅτι ⋯ρθ⋯. At least that is what he understands as what is clear: ἅμα τ⋯ι θε⋯σασθαι τιν⋯ τ⋯ν τ⋯ι ⋯μικυκλ⋯ωι γων⋯αν, δ⋯λη ἂν ἠν αὐτ⋯ι ὅτι ⋯ρθ⋯ ⋯στιν (cod. urb. 49, p. 110v). Unfortunately the author does not seem aware that constructions may be necessary or even what the problem is (cf. Appendix for how Philoponus fails to understand the example).

³⁹ ⋯τι δ' οὒτε κ⋯κλον οἵ⋯ν τε γ⋯γνεσθαι τ⋯ς ἴριδος οὔτε μεῖζον ⋯μικυκλ⋯ωι τμ⋯μα, κα⋯ περ⋯ τ⋯ν ἄλλων τ⋯ν συμβαιν⋯ντων περ⋯ αὐτ⋯ν, ⋯κ τοὐ διαγρ⋯μματος ἒσται θεωρο⋯σι δ⋯λος. Another less likely possible emendation is based on a parallel at Meteor. A 8. 345 a 22: δ⋯λος ⋯μἳν ἃπας ⋯ κ⋯κλος. But cf. note 37, where ps.-Philoponus has something like this.

⁴⁰ This assumes, of course, that Philoponus wrote the commentary on the second book of the An. Post, which bears his name. I consider the argumentation of this part of the commentary so atrocious, however, as to doubt that it is by Philoponus at all. The account of the duplication of the cube in the commentary to Book I is superb (cf. pp. 102–105). Although Philoponus certainly cribbed that account from another source, even to have done so displays much more intelligence and learning than I would want to attribute to the bungling author of the commentary on the second book. The commentary on the second book is also somewhat terser than one might expect of Philoponus.

⁴¹ I should like to thank J. O. Urmson, G. E. L. Owen, W. R. Knorr, G. Striker, M. Burnyeat, A. A. Long, H. Gelber, P. Suppes, and the editors, for their helpful criticisms and advice in our discussions of this paper. I delivered it to the Philosophy Department, Florida State University, after which there was useful discussion.