Sulla then broke in and said: No doubt this position has its plausible aspects; but what tells most strongly on the other side, did our comradeSee 929 B and note a on p. 48 supra. explain that away or did he fail to notice it? What’s that? said Lucius, or do you mean the difficulty with respect to the half-moon? Exactly, said Sulla, for there is some reason in the contention that, since all reflection occurs at equal angles,This expression is intended to have the same sense as πρὸς ἴσας γίγνεσθαι γωνίας ἀνάκλασιν πᾶσαν (930 A s.v.), and both of them mean (pace Raingeard, p. 100, and Kepler in note 28 to his translation) the angle of reflection is always equal to the angle of incidence. cf. [Euclid], aà (= Euclid, , vii, p. 286. 21-22 [Heiberg]) with Olympiodorus, p. 212. 7 = Hero Alexandrinus, , ii. 1, p. 368. 5 (Nix-Schmidt) and [Ptolemy], , ii = Hero Alexandrinus, , ii. 1, p. 320. 12-13 (Nix-Schmidt); and contrast the more precise formulation of Philoponus, p. 27. 34-35. whenever the moon at the half is in mid-heaven the light cannot move earthwards from her but must glance off beyond the earth. The ray that then touches the moon comes from the sun on the horizonKepler in note 19 to his translation points out that this is true only if μεσουρανῇ is in mid-heaven refers not to the meridian but to the great circle at right-angles to the ecliptic. and therefore, being reflected at equal angles, would be produced to the point on the opposite horizon and would not shed its light upon us, or else there would be great distortion and aberration of the angle, which is impossible. Cleomedes, ii. 4. 103 (p. 186. 7-14 [Ziegler]) introduces as σχεδὸν γνώριμον his summary of this argument against the theory that moonlight is merely reflected sunlight. Yes, by Heaven, said Lucius, there was talk of this too; and, looking at Menelaus the mathematician as he spoke, he said: In your presence, my dear Menelaus, I am ashamed to confute a mathematical proposition, the foundation, as it were, on which rests the subject of catoptrics. Yet it must be said that the proposition, all reflection occurs at equal angles, See note e on 929 F supra. is neither self-evident nor an admitted fact.It has been suggested that οὔθ’ ὁμολογούμενον is a direct denial of ὡμολογηένον ἐστι παρὰ πᾶσιν at the beginning of Hero’s demonstration (Schmidt in Hero Alexandrinus, [ed. Nix-Schmidt], ii. 1, p. 314. However that may be, the law is assumed in Proposition XIX of Euclid’s , where it is said to have been stated in the (Euclid, , vii, p. 30. 1-3 [Heiberg]); and a demonstration of it is ascribed to Archimedes (, 7 = Euclid, , vii, p. 348. 17-22 [Heiberg]; cf. Lejeune, , xxxviii [1947], pp. 51 ff.). It is assumed by Aristotle in , iii. 3-5 and possibly also by Plato (cf. Cornford, Platos , pp. 154 f. on , 46 B); cf. also Lucretius, iv. 322-323 and [Aristotle], , 901 B 21-22 and 915 B 30-35. Proposition XIX of Euclids , referred to above, is supposed to be part of the Dioptrics of Euclid which Plutarch cites at , 1093 E (cf. Schmidt, Op. cit. p. 304). It is refuted in the case of convexi.e. cylindrical, not spherical, convex mirrors; cf. xlvi (1951), pp. 142-143 for the construction and meaning of this sentence. mirrors when the point of incidence of the visual ray produces images that are magnified in one respect; and it is refuted by folding mirrors,For such mirrors cf. [Ptolemy], , xii = Hero Alexandrinus, , ii. 1, p. 342. 7 ff. either plane of which, when they have been inclined to each other and have formed an inner angle, exhibits a double image, so that four likenesses of a single object are produced, two reversed on the outer surfaces and two dim ones not reversed in the depth of the mirrors. The reason for the production of these images Plato explains,Plutarch means , 46 B - C, where Plato, however, describes a concave, cylindrical mirror, not a folding plane mirror. Plutarch apparently mistook the words ἔνθεν καὶ ἔνθεν ὕξη λαβoῦσα, by which Plato describes the horizontal curvature of the mirror, to mean that the two planes of a folding mirror were raised to form an angle at the hinge which joined them. for he has said that when the mirror is elevated on both sides the visual rays interchange their reflection because they shift from one side to the other. So, if of the visual rays (some) revert straight to us (from the plane surfaces) while others glance off to the opposite sides of the mirrors and thence return to us again, it is not possible that all reflections occur at equal angles.See note e on 929 F supra. Consequently (some people) take direct issue (with the mathematicians) and maintain that they confute the equality of the angles of incidence and reflection by the very streams of light that flow from the moon upon the earth, for they deem this fact to be much more credible than that theory. Nevertheless, suppose that thisi.e. the theory that the angle of reflection is always equal to the angle of incidence. must be conceded as a favour to geometry, the dearly beloveds3 In the first place, it is likely to occur only in mirrors that have been polished to exact smoothness; but the moon is very uneven and rugged, with the result that the rays from a large body striking against considerable heights which receive reflections and diffusions of light from one another are multifariously reflected and intertwined and the refulgence itself combines with itself, coming to us, as it were, from many mirrors. In the second place, even if we assume that the reflections on the surface of the moon occur at equal angles, it is not impossible that the rays as they travel through such a great interval get fractured and deflectedWith these words Plutarch means to refer to the effects of refraction; cf. , 894 C = Aëtius, iii. 5. 5 (, p. 372. 21-26); Cleomedes, ii. 6. 124-125 (p. 224. 8-28 [Ziegler]); Alexander, p. 143. 7-10. so as to be blurred and to bend their light. Some people even give a geometrical demonstration that the moon sheds many of her beams upon the earth along a line extended from the surface that is bent away from uscf. the argument given by Cleomedes, ii. 4. 103 (pp. 186. 14-188.7 [Ziegler]) and especially: ὅτι δ᾽ ἀπὸ παντὸς τοῦ κύκλου αὐτῆς φωτίζεται ἡ γῆ, γνώριμον. εὐθέως γὰρ ἅμα τῷ τὴν πρώτην ἴτυν ἀνασχεῖν ἐκ τοῦ ὁρίζοντος φωτίζει τὴν γῆν, τούτων τῶν μερῶν αὐτῆς περικλινῶν ὄντων καὶ πρός τὸν οὐρανόν, ἀλλ᾽οὐχί, μὰ Δία, πρὸς τὴν γῆν ὁρώντων For ἡ ἐκκεκλιμένη cf. Hippocrates, 38 (iv, p. 168. 18 [Littrè]).; but I could not construct a geometrical diagram while talking, and talking to many people too.