¹ See Lejeune, A., “Les tables de réfraction de Ptolémée”, Annales de la Société scientifique de Bruxelles, Series I, lx (1946), 93–101Google Scholar; Lejeune, , Recherches sur la catoptrique grecque (Bruxelles, 1957), pp. 153–163Google Scholar; Crombie, A. C., Robert Grosseteste and the Origins of Experimental Science, 1100–1700 (Oxford, 1953), pp. 120–124, 219–225Google Scholar; Schnaase, Leopold, Die Optik Alhazens (Stargard, 1889), pp. 13–15Google Scholar; Eastwood, Bruce S., “Grosseteste's Quantitative Law of Refraction: A Chapter in the History of Non-Experimental Science”, Journal of the History of Ideas, xxviii (1967), 403–414CrossRefGoogle Scholar. A brief but valuable discussion of medieval theories of refraction appeared after the submission of this article in: Sabra, A. I., Theories of Light from Descartes to Newton (London, 1967), chap. 4.Google Scholar

² For example, see Alhazen's On Light, where (in Baarmann's German translation) it is noted: “Und gerade so verhält sich's mit den durchsichtigen Körpern, in welche das Licht eindringt: die Behandlung des ‘Was’ ihrer Durchsichtigkeit gehört zu den Natur- und die Behandlung des ‘Wie’ der Ausbreitung des Lichtes in ihnen zu den mathematischen Wissenschaften.” (Baarman, J., “Abhandlung über das Licht von Ibn al-Haiṯam”, Zeitschrift der deutschen morgenländischen Gesellschaft, xxxvi [1882], 197.)Google Scholar

³ See the references in n. I.

⁴ On Bacon and Pecham see below; for Henry of Hesse, I have used the unpublished text of his Questiones super perspectivam edited by H. L. L. Busard, Questio 14. Cf. Alessio, Franco, “Questioni inedite di ottica di Biagio Pelacani da Parma”, Rivista critica di storia della filosofia, xvi (1961), Questio 9, 207–213.Google Scholar

⁵ On the relationship between Alhazen and his thirteenth-century followers, see my “Alhazen's Theory of Vision and Its Reception in the West”, Isis, lviii (1967), 321–41.Google Scholar

⁶ The widespread influence of Pecham's Perspectiva communis is discussed in the introduction to my forthcoming edition and translation of this treatise, to be published by the University of Wisconsin Press. On the influence of Alhazen and Witelo, see my introduction to the forthcoming reprint of the Risner edition (1572) of their works. On Descartes' use of Witelo's Perspectiva, see below, n. 52.

⁷ L'Optique de Claude Ptolémée, dans la version latine d'aprés l'arabe de l'émir Eugène de Sicile, ed. Lejeune, A. (Louvain, 1956), Bk. 5, sec. 19, pp. 234–235Google Scholar. In this article I have not included the Latin text of quotations from editions as readily available as this one.

⁸ Ibid., sec. 23, pp. 237–238.

⁹ Ibid., sec. 33, p. 244. Ptolemy notes elsewhere, however, that small differences in density do not alter the angles perceptibly; cf. ibid., sec. 12, p. 30.

¹⁰ Narducci, Enrico, “Intorno ad una traduzione italiana, fatta nel secolo decimoquarto, del trattato d'ottica d'Alhazen, matematico del secolo undecimo, e ad altri lavori di questo scienziato”, Bullettino di bibliografia e di storia delle scienze matematiche e fisiche, iv (1871), 27–30Google Scholar. Delambre's argument that Alhazen probably did not know Ptolemy's Optica (Delambre, J. B. J., Histoire de l'astronomie ancienne [Paris, 1817], vol. ii, pp. 411–412Google Scholar) can be ignored, since Narducci quotes documentary evidence to the contrary.

¹¹ Opticae thesaurus Alhazeni Arabis libri septem, ed. Risner, Friedrich (Basel, 1572), Bk. VII, sec. 8, p. 240Google Scholar: “… motu veloci qui non patet sensui propter suam velocitatem.” All citations are to this sole printed edition of Alhazen's Perspectiva; at a few points, however, I have corrected the text by reference to British Museum Royal MS. la.G.VII., and I have revised the punctuation and spelling.

¹² Opt. thes., Bk. II, sec. 21, pp. 37–38Google Scholar. Alhazen argues that light cannot be propagated instantaneously, since it must reach the part of the air nearest the source before it reaches more distant parts. Thus “air receives light successively”, which implies “motion” of the light. “But there is no motion except in time.”

¹³ Ibid., Bk. VII, sec. 8, p. 240: “Omne enim corpus diaphanum, cum lux transit in ipsum, resistit luci aliquantulum, secundum quod habet de grossitie.”

¹⁴ Ibid., p. 241: “… si occurrerit resistenti, necesse est, ut motus eius transmutetur.”

¹⁵ Ibid.: “Et si resistentia fuerit fortis, tune motus ille reflectetur ad contrariam partem; si vero debilis, non reflectetur ad contrariam partem, nec poterit per illam precedere per quam inciperat, sed motus eius mutabitur.”

¹⁶ Ibid.: “Omnium autem motcrum naturaliter, que recte moventur per aliquod corpus passibile, transitus super perpendicularem, que est in superficie corporis in quo est transitus, erit facilior.”

¹⁷ Ibid.: “Si enim aliquis acceperit tabulam subtilem et paxillaverit illam super aliquod foramen amplum et steterit in oppositione tabule et acceperit pilam ferream et eiecerit eam super tabulam fortiter et observaverit ut motus pile sit super lineam perpendicularem super superficiem tabule, tune tabula cedet pile aut frangetur si tabula subtilis fuerit et vis, qua spera movetur, fuerit fortis. Et si steterit in parte obliqua ab oppositione tabule, et in illa eadem distantia in qua prius erat, et eiecerit pilam super tabulam illam eandem in quam prius eiecerat, tune spera labetur de tabula si tabula non fuerit valde subtilis, nec movebitur ad illam partem ad quam primo movebatur sed declinabit ad aliquam partem aliam. “Et similiter, si acceperit ensem et posuerit coram se lignum et percusserit cum ense, ita ut ensis sit perpendicularis super superficiem ligni, tune lignum secabitur magis. Et si ensis fuerit obliquus et percusserit oblique lignum, tune lignum non secabitur omnino, sed forte secabitur in parte aut forte ensis errabit deviando; et quanto magis fuerit ensis obliquus, tanto minus aget in lignum. Et alia multa sunt similia, ex quibus patet quod motus super perpendicularem est fortior et facilior et quod de obliquis motibus, ille qui vicinior est perpendiculari est facilior remotiore.”

¹⁸ The exception to this rule is a perpendicular ray, which is able to continue in its original direction because of its greater strength.

¹⁹ Ibid.: “Accidit ergo ut declinetur ad partem motus in quam facilius movebitur quam in partem in quam movebatur.”

²⁰ Ibid.: “Sed facilior motuum est super perpendicularem, et quod vicinius est perpendiculari est facilius remotiore.”

²¹ See above; cf. Opt. thes., Bk. IV, sec. 18, pp. 112–113Google Scholar, where Alhazen elucidates reflection by means of the analogy of a rebounding sphere. This is not to claim, however, that the reflection of light is an instance of mechanical rebound, i.e. that light is a flow of particles, but only that the reflection of light is in some respects like the rebounding of spheres and can be elucidated by reference to the latter. Alhazen is not always unambiguous in his remarks on this point, but it seems evident that light, in his view, is not corpuscular; consequently, there is some uncertainty on the extent to which the analogy of mechanical rebound sheds light on the cause of refraction. Cf. my “Alhazen's Theory of Vision” (ref. in n. 5), p. 335.

²² Opt. thes., Bk. VII, sec. 8, p. 241Google Scholar: “Et motus in corpore in quod transit, si fuerit obliquus super superficiem illius corporis, componitur ex motu in parte perpendicularis transeuntis in corpus … et ex motu in parte linee que est perpendicularis super perpendicularem que transit in ipsum. Cum ergo lux fuerit mota in corpore diaphano grosso super lineam obliquam, tune transitus eius in ilio corpore diaphano erit per motum compositum ex duobus predictis motibus.”

²³ The Arabic text of Alhazen's Kitāb al-manāẓir, which is now being edited (I understand) by A. I. Sabra of the Warburg Institute, might shed light on the problem. However, Alhazen's Western followers, who had no access to the Arabic text, must have been as perplexed as I am. Shuja's English translation of this portion of Kamāl al-Dīn's commentary on Alhazen's Kitāb al-manāẓir does nothing to clarify the issue and, in fact, seems to introduce additional confusion by virtue of a faulty translation; cf. Shuja, F. M., Cause of Refraction as explained by the Moslem Scientists (Delhi, 1936Google Scholar [pamphlet printed for private circulation]). There is one phrase in Alhazen's argument that suggests the possibility of a Cartesian interpretation: the expression “lux que extenditur super lineam obliquam moveatur super perpendicularem” (Bk. VII, sec. 8, p. 241), could be taken to mean that oblique light is inspired or enhanced (moveatur) in the direction of the perpendicular. However, such an interpretation is by no means required by the text and receives no support from anything else in Alhazen's account of refraction.

²⁴ Lejeune, in his introduction to Ptolemy's Optica (ref. in n. 7), p. 31*, argues that the Optica was unknown in the West until after 1250 and that Bacon was the first important Western medieval writer on optics to use it. Crombie, in his Grosseteste, pp. 116–117Google Scholar, advances arguments favouring Grosseteste's familiarity with the Optica. It appears to me that Lejeune makes the stronger case.

²⁵ De iride, in Die philosophischen Werke des Robert Grosseteste, ed. Baur, Ludwig (Beiträge zur Geschichte der Philosophie des Mittelalters, ix [Münster, 1912]), p. 74Google Scholar. In his De lineis, Grosseteste associates refraction with the density of the two media; cf. ibid., pp. 61–63.

²⁶ On Pecham's work in optics, see my forthcoming edition of his Perspectiva communis (ref. in n. 6); cf. my “Alhazen's Theory of Vision” (ref. in n. 5).

²⁷ Bacon presents his fullest discussion of the cause of refraction in his De multiplicatione specierum (published in The Opus Majus of Roger Bacon, ed. Bridges, J. H. [London, 1900], vol. iiGoogle Scholar), II, chap. 3, pp. 466–472. On Pecham, see Perspectiva communis, props. 1–14 through 1–16 and III–3. Witelo discusses the cause of refraction in his Perspectiva, ed. Risner (bound with the 1572 edition of Alhazen's Perspectiva). Bk. II, sec. 47, pp. 81–83.

²⁸ De mult. spec., II, chap. 3, p. 469.Google Scholar

²⁹ Opus maius, IV, dist. 3, chap, 1, ed. Bridges, , vol. i, p. 120.Google Scholar

³⁰ Perspectiva communis, prop. II–3: “Rectitudo siquidem lucis cognata processui, sed etiam omni operi nature, dirigit et expedit naturam, omnis enim motus tanto est fortior quanto est rectior.” Text and translation are from my forthcoming edition.

³¹ This conclusion is stated in somewhat vague, but still unmistakable, terms by Pecham and Bacon. See Pecham, , Pers. com., props. 1–15 and 11–3Google Scholar, and prop. 1–6 of the revised version; Bacon, , De mult. spec., II, chap. 3, p. 469Google Scholar; Bacon, , Opus maius, IV, dist. 3, chap, 1, vol. i, p. 120.Google Scholar

³² Perspectiva, ed. Risner, , Bk. II, sec. 47, p. 81Google Scholar: “Omnes enim motus naturales, qui fiunt secundum lineas perpendiculares, sunt fortiores, quoniam coadiuvantur virtute universali celesti secundum lineam rectam brevissimam omni subiecto corpori influente.” At a few points I have corrected the text of the Risner edition by reference to Bodleian Library, MS. Ashmolean 424, and Erfurt, MS. Amploniana F. 374, and I have altered spelling and punctuation when necessary.

³³ Ibid., p. 82: “Facilior autem motuum et plus adiutus celesti influentia est super lineam perpendicularem.”

³⁴ Opera hactenus inedita Rogeri Baconi, fasc. xiii (London, 1935), pp. 402–403Google Scholar; cf. Maier, Anneliese, An der Grenze von Scholastik und Naturwissenschaft (Rome, 1952), 177–182Google Scholar; Duhem, Pierre, Le système du monde, vol. viii, (Paris, 1958), 239–248.Google Scholar

³⁵ This discrepancy would not have been as obvious to Witelo as to us, since for him the most common instance of refraction was from air to water, in which the interface is always horizontal, and the vertical and perpendicular coincide; moreover, the apparatus for investigating refraction described by Ptolemy, Alhazen, and Witelo presupposes, in every case, a horizontal interface.

³⁶ De mult. spec., II, chap. 3, p. 470.Google Scholar

³⁷ Ibid.

³⁸ Ibid.

³⁹ Pers. com., prop. 1–15Google Scholar: “Unde ut transitus per medium secundum proportionetur transitui per primum, radius declinat ad perpendicularem erigibilem a puncto casus sui super secundum medium.”

⁴⁰ Ibid.: “Quia igitur facilior est transitus per unum medium quam per reliquum, necesse est quod in secundo medio, magis scilicet distante a luminoso, reperiatur proportionaliter primo in situ, scilicet similis resistentie.” My translation at this point is free.

⁴¹ Ibid.: “Nec intelligendum radium ad situm fortiorem declinare quasi per electionem, immo transitum per medium primum ad sibi proportionalem in secundo impellere …” Emphasis has been supplied.

⁴² Perspectiva, ed. Risner, , Bk. II, sec. 47, p. 82Google Scholar: “…debilitabitur nec ad aliquid perveniet effectus eius.”

⁴³ Ibid.: “Natura autem frustra nihil agit.”

⁴⁴ Ibid.: “Radius vero … non invenit resistentiam quam prius. Et quia formarum proprium est semper se diffundere secundum amplitudinem omnis capacis materie, patet quod radius … non procedit secundum lineam AC, quia sic dispositio dyafanorum corporum secundum resistentiam ad receptionem luminis esset uniformis, quod est contra ypothesim.”

⁴⁵ Ibid.: “… quoniam illa refractio non fit propter resistentiam materie, sed propter victoriam forme agentis super materiam plus dispositam quam prius. Unde forma diffundit se virtute propria ab incepto progressu … et ad partem contrariam ipsius perpendicularis.”

⁴⁶ The theory of the rainbow is a notable exception.

⁴⁷ The most important use of the emanationist metaphysic is Bacon's remark, quoted above, that “superfluous coarseness in the second substance excites the generative power of a species”. Witelo's brief reference to the self-diffusive property of forms is in the same vein. These remarks are by no means insignificant, but they do not alter the fundamentally mechanistic tone of most of Bacon's and Witelo's account.

⁴⁸ Arguing for the strength of motion along the perpendicular, Bacon writes: “Wherefore a man falling perpendicularly from a high place is killed by the fall. But if something should divert him from direct approach as he descends, he is spared insofar as he diverges from perpendicular approach.” (De mult. spec., II, chap. 3, p. 468.)Google Scholar

⁴⁹ I am not arguing that medieval scientists were mechanical philosophers in the seventeenth-century sense, but only that mechanical modes of explanation were not absent from their work.

⁵⁰ Sabra, Abdelhamid I., “Explanation of Optical Reflection and Refraction: Ibn-al-Haytham, Descartes, Newton”, Actes du dixième congrès international d'histoire des sciences (Ithaca, 1962), i, 551–554.Google Scholar

⁵¹ Of course, the medieval approach was reinforced by seventeenth-century mechanism, and an explanation of Descartes' account of refraction must include both factors.

⁵² On Descartes, see La dioptrique, disc, 2 cf. Burke, John G., “Descartes on the Refraction and the Velocity of Light”, American Journal of Physics, xxxiv (1966), 390–400CrossRefGoogle Scholar. Descartes' knowledge of Witelo's work is amply demonstrated. On the first page of a small notebook found in his trunks at his death, Descartes had written: “Vitellio sic numerat angulos refractos.” Below this statement is a copy of Witelo's table of refraction, followed by a brief analysis. In the first draft of a letter to Golius in 1632, Descartes refers to the instrument for measuring refraction “que décrit Vitellion”. Finally, in a letter to Mersenne in 1638, Descartes briefly discusses Witelo's ideas on the cause of refraction. Cf. Oeuvres de Descartes, ed. Adam, Charles and Tannery, Paul (Paris, 1897–1910), vol. xi, p. 646Google Scholar; vol. i, pp. 239, 241; vol. ii, pp. 142–143.

⁵³ Thomas Hariot seems to have been the first.

⁵⁴ Snell's annotations are quoted and discussed in Vollgraff, J. A., “Pierre de la Ramée et Willebrord Snel van Royen”, Janus, xviii (1913), 622–624.Google Scholar