archive for history of exact sciences volume 6 issue 3 1970 [doi 10.2307%2f41133302] bruce s....

Metaphysical Derivations of a Law of Refraction: Damianos and GrossetesteAuthor(s): Bruce S. EastwoodSource: Archive for History of Exact Sciences, Vol. 6, No. 3 (29.V.1970), pp. 224-236Published by: SpringerStable URL: http://www.jstor.org/stable/41133302 .Accessed: 18/06/2014 16:27

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Metaphysical Derivations of a Law of Refraction: Damianos and Grosseteste

Bruce S. Eastwood

Communicated by M. Clagett

In a peculiar sense the optical treatises of Damianos (fourth century A.D.)1 and Robert Grosseteste (ca. 1167 - 1253) represent the same level of achieve- ment in the history of optics. Although separated by nine centuries, Damianos and Grosseteste incorporate in their respective works on optics similar degrees of sophistication in the treatment of reflection and refraction. Both made much of the equality of angles of incidence and reflection, but failed to deal with more complex known phenomena of reflection. For instance, the focal point of a para- bolic mirror, undiscussed by either of our authors, was accurately described by Apollonius2 and Alhazen.3 Likewise for refraction, briefly covered in the works to be analyzed below, there is only elementary treatment. Most intriguing is the quantitative law of refraction which they stated. Damianos and Grosseteste present to the reader the amazingly inaccurate notion that the angle of refraction is exactly half the angle of incidence. Furthermore, each writer finds his rationale for this quantitative law in the principles of economy and uniformity. Appealing to these a priori bases, each seized on the same form, though incorrect, for the law of refraction. The backgrounds and developments of the relevant optical treatises reveal the similar nature of their conceptual frameworks.

1 Dated thus by George Sarton, Introduction to the History of Science (Baltimore : Williams and Wilkins, 1927), I, 354, following Friedrich Hultsch, "Damianos (3)", Pauly s Real-Encyclopdie der classischen Altertumswissenschaft, ed. G. Wissowa (Stutt- gart: Metzlerscher, 1901), IV (8er Hlbbd.), 2055. Hultsch notes that Damianos is best placed in the fourth century, if Heliodorus (his father or master) is older than Theon Alexandrinus (mid fourth century).

2 See, e.g., Thomas L. Heath, "The fragment of Anthemius on burning mirrors and the 'Fragmentm mathematicum Bobiense'," Bibliotheca Mathematica, 3e Folge, VII (1907), 232; Christian Belger, "Ein neues Fragmentm Mathematicum Bobiense," Hermes, XVI (1881), 271-2.

3 For medieval Latin version, translated by Gerard of Cremona or his school, v. J. L. Heiberg & E. Wiedemann, " Ibn al Haitams Schrift ber parabolische Hohl- spiegel," Bibliotheca Mathematica, 3e Folge, X (1909 - 10), 201 - 37; English transi, of Arabic text in H. J. J. Winter & W. 'Arafat, " Ibn al-Haitham on the Paraboloidal Focussing Mirror," Journal of the Royal Asiatic Society of Bengal: Science, 3rd ser., XV (1949), 25-40.


Derivations of a Law of Refraction 225

I. The optical treatise ascribed to Damianos has come to be considered his only

with difficulty. Now known only from Greek manuscripts of the fourteenth century and later,4 the work has appeared under titles assigning it alternately to Damianos and to Heliodoros of Larissa.5 Hultsch notes that manuscripts assigning the work to Damianos contained fourteen chapters (as edited by Schne), while superscriptions indicating Heliodoran authorship were followed by only thirteen chapters. He concludes that we should ascribe the first thirteen chapters, identical under both titles, to Heliodoros, and the fourteenth to Damianos. In this case, Damianos, mentioned as student (or son) of Heliodoros in the title, redacted his master's work and added a final chapter.6 To complicate the issue there appears a much more extended version drawn up by Angel Vergetius in 1657; 7 the material beyond the original fourteen chapters, however, derives primarily from the last chapters of the quadrivial treatise on geometry by Georgius Pachymeres (1242 - ca. 13IO).8 Vergetius' motive for this ad- dition seems to have been ingenuous enough - the additional money paid for copying a longer work.9

The personality of Damianos is cloaked in far greater obscurity than the treatise ascribed to him. It has been maintained that the name "Domninos" was a corruption of "Damianos". This assertion led one scholar to attribute the optical treatise to the writer of an anti-Nichomachan book on arithmetic. How- ever, DoMNiNOS of Larissa, the mathematician, is well-known as an independent personality via the article on him by Suidas, confirmed by Damascius and Marinus.10 On the other hand, nothing is presently certifiable about the life and work of Damianos beyond the present treatise and its approximate date (fourth century).

Damianos had before him an available optical tradition of great richness - Grosseteste was less fortunate - yet seems to have made parsimonious use of it in his own optics. In catoptrics alone there was detailed work by Archi-

4 Damianos, Schrift ber Optik mit Auszgen aus Geminos, ed. Richard Schne (Berlin: Reichsdruckerei, 1897), PP- vi - viii. 5 Ibid., p. 5; Paul Tannery, "Rapport sur une mission en Italie," Mmoires scientifiques (Paris: Gauthier- Villars, 1912), II, 320 - 1; Hultsch, Pauly- Wissowa, iv, 2054-5. 6 Ibid. However, there remains some doubt 01 this among more recent scholars, e.g., in Paul Ver Eecke's ed. and transi, of Euclide, L'Optique et la Catoptrique (Paris: Descle et Brouwer, 1938), p. xlii; cf. Sarton's statement, "The whole question is very obscure." in Introduction, I, 354. 7 Damiani Heliodori de Opticis libri duo, ed. Erasmus Bartholin (Paris: Cramoisy, 1657); v. Tannery, "Rapport", p. 319. 8 Also, there was added a long fragment from Hero's Catoptrics as a termination of the first of the two books in Bartholin's edition. The material from Pachymeres, constituting the second book of Bartholin, is an abridgment of Euclid's Optics, apparently original with Pachymeres, and found as well in an isolated and anonymous state in Paris ms. gr. 2477, which was copied by Vergetius. Tannery, "Rapport", p. 320; Ver Eecke, Optique et Catoptrique, p. xliii. For Pachymeres' treatment of optics, v. Quadrivium de Georges Pachymre, edd. P. Tannery & E. Stephanou (Vatican City: Bibl. Apost. Vat., 1940), pp. 318 - 28. 9 Tannery, "Rapport", pp. 322 - 3. 10 Paul Tannery, "Domninos de Larissa", Mmoires scientifiques (Paris: Gauthier- Villars, 1912), II, 105 - 6.


226 B. S. Eastwood:

MEDEs,11 Apollonius,12 Hero,13 Ptolemy,14 and a Euclidean author;15 Damianos uses little from them. Making explicit reference to the optical works of Hero16 and Ptolemy,17 he seems to have given no serious attention to their treatments of refraction. It is even possible that he did not make first-hand use of them, preferring more general, encyclopedic sources.18 Or he may have ignored or misunderstood the more difficult material such as Ptolemy's discussion of refractive angles;19 he certainly contradicts Ptolemy on this. Damianos thus seems to remain on the most elementary plane in his "hypotheses" on light.

The brief20 treatise by Damianos aims primarily to establish the identity of visual rays with light rays. The extramission theory of vision21 is assumed from

11 Testimony to a treatise by Archimedes on burning mirrors can be found in Joannes Tzetzes, Historiarum variarum chiliades, ed. Theophilus Kiessling (Leip- zig: Vogel, 1826), p. 479; also Apuleius Madaurensis, Opera, ed. R. Helm (Leipzig: Teubner, 1905), II, i, 18 - 9. An excellent survey on the question of an Archimedean Catoptrics is A. Rome, "Notes sur les passages des catoptriques d'Archimde conservs par Thon d'Alexandrie," Annales de la Socit Scientifique de Bruxelles, LU (1932), 30 - 41. The treatise itself has not survived. 12 Supra n. 3. This work of Apollonius is not extant. 13 Hero of Alexandria, Opera omnia, edd. L. Nix & W. Schmidt (Leipzig: Teubner, 1900), II, 1, pp. 303-65. 14 The relevant portions of Ptolemy's Optics in L'Optique de Claude Ptolme dans la version latine d'aprs V arabe de Vmir Eugne de Sicile, ed. Albert Le jeune (Louvain: Bibliothque de l'Universit, 1956). 15 That is, the Catoptrics ascribed to Euclid. While clearly not by Euclid, this work may well be a product of Theon, whose redaction of Euclid's Optics shows inferiority to the original, e.g., in prop. 22, just as the Catoptrics is generally inferior to the Optics. See J. L. Heiberg, Litterar geschichtliche Studien ber Euklid (Leipzig: Teubner, 1882), pp. 7, 150 - 2; Euclid, Opera omnia, edd. J. L. Heiberg & H. Menge (Leipzig: Teubner, 1895), VII, p. I (note comparison of terms for "ray" on p. xlix); Ver Eecke, Optique et Catoptrique, p. xxix. See esp. The arguments of Albert Lejeune, Recherches sur la catoptrique grecque (Brussels: Palais des Acadmies, 1957), pp. 112 - 49; ibid., pp. 145 - 6 argues for an original Catoptrics by Euclid as well. 16 Damianos, Optik, p. 20. Friedrich Hultsch, in a review of Schne's edition of Damianos' Optics ((Berliner Philologische Wochenschrift, XVIII (1898), 1414)) maintains that Hero's Catoptrics contained details on refraction as well. 17 Damianos, Optik, p. 4. 18 The eclectic, encyclopedic collecting of data on a topic was an established trend before Damianos, as is shown by William H. Stahl, Roman Science (Madison: U. of Wisconsin, 1962). 19 Certainly Theon ignored or misunderstood Ptolemy on refraction (Rome, " Catoptriques d'Archimde", pp. 39 - 40). That a major work in optics may be known but not widely understood or used is evidenced by the history of Alhazen's optics in medieval Islam; Alkindi's optics was commonly used instead, and phenomena of refraction continued to be misunderstood despite the availability of Alhazen's works.

20 In Schne's edition, the Greek text, including a separate list of chapter headings, fills less than eleven pages. The first printed edition, on which the editions of the 1 7th and 1 8th centuries were based, requires only eleven pages for parallel columns of the Greek text with Latin translation; v. "Heliodori Larissaei Capita Opticorum," second appendix (unnumbered pages) to La Prospettiva di Euclide ... & ... Specchi, transi. Egnatio Danti (Florence: Juntas, 1 573)- 21 In the history of optics this position was a major tradition in Antiquity and the Middle Ages. A careful and brief survey of this tradition can be found in Edmund Hoppe, Geschichte der Optik (Leipzig: Weber, 1926), pp. 5 - 25; for the Greco-Roman



the first chapter and seems to imply no problems for our author. Instead the complete parallelism of visual and solar rays, whether direct, reflected, or refracted, is shown. Acknowledging Plato,22 he too admits the necessity of an exterior as well as an interior basis for vision. However, Damianos tell us, there are some people whose eyes emit strong enough light to see unaided in the dark; such a person was the emperor Tiberius.23

The general contents of this relatively unknown treatise are worth summarizing at least briefly, the fourteen chapters dealing with the following subjects.

1 . Vision occurs by emanation from the eye. 2. What emanates from us is true light. 3. Visual rays are rectilinear and are propagated in a right-angled cone. 4. Visual rays must emanate exclusively in the form of a cone. 5. The cone of rays must be right-angled, rather than acute or obtuse; this is

supported by both theory and observation. 6. Visual rays pass through invisible pores in the pupil, yet the rays irradiate

every point between the rays. 7- Any object seen is viewed under a right angle or less. If the former, then

the object must be on the diameter of the base of the cone; otherwise, this may or may not be the case.

8. An object seen under a larger angle appears larger because of the greater number of rays striking it.

9. We see best and primarily along the axis of the cone of vision. 10. We are accustomed to seeing forward, and we therefore attempt to see any

object directly ahead, or to interpret it as being ahead. 11. The apex of the visual cone is within the eye, and the pupil encloses the

cone, which in turn intersects one-fourth of the surface of the eye, a sphere. 12. Vision is completed either by direct rays or broken rays. Of broken rays

two sorts occur, reflected and refracted. 13. In both rectilinear projection and reflection our visual rays act identically

as do solar rays. Both visual and solar rays operate instantaneously over great distances. Both visual and solar rays are reflected along the same lines.

14. Both reflection and refraction follow a definite law. This is the law of equal angles, based on the principle of equality. The law is effective for both visual and solar rays.

tradition a useful work is Vasco Ronchi, Histoire de la lumire, transi. J. Taton (Paris: Colin, 1956), ch. 1. The strength of the emission theory was based upon its early linkeage with geometrical optics. Aristotle's Posterior Analytics linked optics directly to geometry. While Eudoxus was unwilling to play the metaphysical game in geometrical astronomy, Euclid seemed to find it useful in geometrical optics. The great success of Euclidean optics made extramission an acceptable view for Galen, Ptolemy, Theon, and, of course, Damianos.

22 Damianos, Optik, p. 20 (ch. 13). 23 Probably on the basis of Suetonius, Tiberius 68, Damianos asserts this in ch. 2 (d. cit., p. 4).


228 B. S. Eastwood :

Clearly, for refraction we must look at Damianos' last chapter almost exclusively. Additional information from the earlier parts will serve, however, to illuminate further the overall framework of the argument for refraction. The whole of chapter 14 runs as follows.

One should recognize that reflection and refraction of our visual rays do not occur by chance, without a definite law. Both occur without exception under equal angles, based on the surface of the object at which our visual rays are reflected or refracted. Hero the mechanician has proven in his catoptrics that two points are connected more closely by reflection at equal angles than by reflection at any other, unequal angles. He has proven, he said, that if Nature does not wish to permit our visual ray to wander about fruitlessly, she will let it break at equal angles. Likewise it can be shown that when our visual ray penetrates an object, thus altering its direction, the refraction occurs at equal angles (ojllolco ei%0rj0eT(U, ori ned r' iaxXoiOt rfj tpeco xfc ^justsqo tzqo Ool niTEAeTVLi yvioL). But hereupon it is clear that solar rays as well are bent at equal angles. For one cannot maintain that the phenomenon occurs according to the principle of equality with visual rays, but according to inequality and chance with solar rays. And we have shown above that in reflection, at least, both solar rays and visual rays bend at equal angles in reflection.24 In the final chapter on refraction, as well as in the early chapters, on direct

vision, Damianos searches out as simple a geometrical situation as possible to account for the phenomena. In the first six chapters we find that he develops not only the notion of vision as a truly illuminating process but the notion of a right-angled cone of visual rays from the eye. The right-angled cone is chosen, we learn in chapter 5,25 because it is the most determinate, most defined, form of cone : Nature, being rational, prefers the definite to the indefinite, for the former is better! In chapter 14, the same sort of metaphysical appeal suggests itself. Equal angles are chosen for refraction as well as reflection, because an inequality of angles (like a non-rectangular cone of rays) would be indefinite and a matter of chance. Some sort of metaphysical principle of simplicity seems to operate here, according to the text. An authoritative basis is claimed in the Catoptrics of Hero ; likewise, a tacit appeal to the Optics of Ptolemy is obvious in chapter 5 on the rectangular cone.26 In both cases Damianos goes beyond his authorities in applying their conclusions to his situations. To reach a quantitative law of refraction he draws on the quantitative law of reflection. Hero showed that

24 Optik, pp. 20 - 2. It should be noted that the translation given here is not verbatim, though strictly in accord with the sense of the original, and that both the Greek text and Schne's German translation (pp. 21 - 3) have been used as basis for this English version. For Italian and Latin versions, see Danti (ed. cit. supra n. 21), n. pp. [f. 4v and ff. 5 v - 6r of the two texts respectively]. 25 Damianos, Optik, pp. 6 - 8.

26 Damianos clearly had in mind Ptolemy, who experimentally determined the rectangular shape (approximately) of the visual cone ; v. d. cit. (n. 1 5 supra), pp. 35 - 6. A full discussion of the optical cone, or pyramid, in Ptolemy can be found in Albert Lejeune, Euclide et Ptolme: deux stades de V optique gomtrique grecque (Louvain: Bibl. de l'Universit, 1948), pp. 42-55-



reflection occurs at equal angles, because this is the shortest and quickest path.27 He, like others,28 makes use of the metaphysical principle of economy to arrive at his conclusion: he insists that Nature does nothing in vain. Taking up where Hero left off, Damianos adds another metaphysical principle, that of uniformity. Without even bothering to justify himself, he assumes (in the passage given above) that reflection and refraction are similar phenomena and subject to essentially similar laws: a uniformity exists between the laws of reflection and refraction.29 If there is an equality of angles in reflection, then there must also be an equality of angles in refraction.30 Damianos goes no further in describing just where these equal angles of refraction are, but the only reasonable assumption we can make is to locate them on either side of the actual path of the ray in the denser medium into which a visual ray passes. The refracted visual ray bisects the angle formed by a perpendicular and the imaginary unrefracted path of the ray in the denser medium.

visual ray before refraction

_ common surface of two media

/ visual ray after refraction

angle a = angle b

Fig. 1

The fruitfulness of the principle of uniformity is not exhausted here by application to the problem of refracting angles. Damianos also wields a methodological principle of uniformity to argue that both visual rays and solar rays follow the ascertained law of refraction, and that both types of ray are therefore of the same essence.31 To argue that visual rays differ in refraction from solar rays

27 Ed. cit. (n. 4 supra), p. 324; note that Hero assumes the path of the rays to be the shortest possible path, and he proves only that equal angles give this shortest path. 28 E.g., Aristotle, De celo I, 4, 271 a. 29 Ptolemy considered reflection and refraction as two cases of the same phenomenon (Lejeune, Euclide et Ptolme, p. 74, n. 4; e.g., Optique de Ptolme, ed. Lejeune, p. 246). Hero (ed. cit. n. 14 supra, p. 322) used "fractio" as a generic term to cover all types of bent rays. In most Roman authors there was no conception of the difference between reflection and refraction, e.g., in Seneca, Cuestiones naturales I, vii, 1 - 2. 30 Without documentary evidence, it has been suggested that the statement of equal angles in refraction is an error of a later copyist, not to be found in Damianos' thought; v. Julius Hirschberg, Geschichte der Augenheilkunde im Alterthum, Graefe- Saemisch Handbuch der gesamten Augenheilkunde, XII (Leipzig: Breitkopf und Hrtel, 1899), p. 171. 31 Simeon Seth, Conspectus rerum naturalium eh. 74 (Anecdota Atheniensia et alia, ed. Armand Delatte, Liege: Vaillant-Carmanne, 1939, II, 73), refers to Ptolemy's


230 B. S. Eastwood :

would be unnecessary, unfounded, and uneconomical in Damianos* view. Thus chapter 14 exemplifies appeal to the principles of economy and uniformity in both the metaphysical and the methodological senses, and the quantitative law of refraction is based solely on metaphysical principles.

Is there any possible alternative to the interpretation we have given to Damianos' law of refraction? The reference to equal angles cannot be to the equal angles formed by the perpendicular to the surface of the refracting medium, for the reference in reflection is clearly to the actual path of the visual ray, and a parallel situation (the actual path of the ray) is intended for refraction. If we assume that the equality of angles refers to the angles of incidence and refraction, the whole point of refraction disappears, as this would give unbent rays. But Hultsch32 has ingeniously suggested that in a peculiar sense this is exactly what Damianos meant. Stating that the same law was undoubtedly in Hero's Cat- optrics in a more detailed form, Hultsch feels that the law envisages refraction twice, through a glass plate (a parallelepiped) at an angle close to the perpendicular. The ray would then leave the plate at the same angle as it entered, giving equal angles of incidence and refraction. This means that Damianos must have recog- nized the existence of reciprocal angles in refraction from air to glass and back to air, for the equality of angles before entering and after leaving the glass would require such a recognition - unless the bending of the ray at an interface be denied altogether. That knowledge of reciprocal angles was available to Damianos is clear from his reference to Ptolemy's Optics. Whether this information on refraction was assimilated by Damianos remains to be seen. To postulate a likely answer we should remember certain things about Ptolemy's discussion of angles of refraction. While he gives refraction tables for air-water, air-glass, and water- glass, no observations are recorded for rays travelling in the reverse direction. Having formulated in his mind a clear conception of the reciprocity of angles in refraction between two media - even making his observations for water-glass on the basis of such knowledge - Ptolemy declined to treat the question system- atically.33 Seemingly, any student of optics should be able to see the reciprocal law in Ptolemy and to work out its consequences. Yet as able a student as Witelo, in the thirteenth century, failed completely to comprehend Ptolemy's meaning, because of a greater concern for following the text as he understood it rather than for conducting experiments.34 The degraded state of optical science by the time of Theon coupled with the simplicist and non-experimental tenor

view of the visual pneuma appertaining to the quintessence. Simplicius, In Aristotelis de celo, ed. J. L. Heiberg (Berlin: Reimer, 1894), p. 20, supports the notion that Ptolemy considered visual and illuminating rays to be quintessential; Simplicius compares Ptolemy's view with those of Plotinus and Xenarchos. See Le jeune, Euclide et Ptolme, pp. 64 - 6. 32 Op. cit. (n. 16 supra), col. 1414. 33 Optique de Ptolme, p. 243, n. 31, and Lejeune, Catoptrique grecque, p. 158 for the general statement of this law. Reasons for Ptolemy's failure to provide reciprocal tables, as well as an enlightening discussion of his whole treatment of angles of refraction, can be found, inter alia, in Lejeune, Catoptrique grecque, pp. 155 - 66. 34 Witelo, ptica, ed. Friedrich Risner (Basel: Episcopius, 1572), p. 412. Witelo's error, patent enough to a modern eye, has been noted many times, e.g., by Emil Wilde, Geschichte der Optik (Berlin: Rcker und Pchler, 1838), I, 80-2.



of Damianos' treatise, strongly suggest that Damianos was unenlightened and unconcerned over so detailed a question as the reciprocity of angles of refraction.

The geometry of his optics can only be described as metaphysical, not mathematical; in the only place where any serious geometry is discussed, he errs, and he errs despite the availability of a correct solution since the time of Archimedes. In chapter 1 1 Damianos notes that the pupil, which encloses the cone of visual rays, comprises one-fourth of the surface of the ocular sphere. This statement seems very definitely to be aimed at determining an exact point of origin for the cone of vision, i.e., in the middle of the eye;35 certainly his penchant for simplicity and exactness extended here as elsewhere. There is posited a right- angled cone, with its apex at the center of a sphere, intersecting a section of the spherical surface equal to r2n. Yet Archimedes demonstrated much earlier that the area of the intersected segment would be 3/5 r2^.36 Nor does Damianos show ignorance of the treatise37 (On the Sphere and the Cylinder), but rather some confusion and perhaps inability in comprehending it. He may have taken the pertinent proposition (I, 34) to refer to surface areas rather than to volumes as it does, for the basic proposition - not its corollary, which Damianos should have used - gives the ratio of 1 : 4 for the volumes of a right angled cone (with height equal to the sphere's radius) and a sphere. M An even simpler possible basis for the error would be extrapolation from a planar (triangle and circle) situation, where Damianos' explicit ratio of 4:4 would be correct, to a solid (cone and sphere) situation. The very unsophisticated nature of such an error is consonant with the whole tone of Damianos' Optics. The absence of any and all geometrical demonstration is notable. Instead, reference and allusion to geometry create an aura of exactness, where the really significant statements are in the realm of metaphysics rather than mathematics. With respect to the nature of light, of reflection, and of refraction, the important statements he makes are those based on the principles of economy and uniformity. Damianos' quantitative law of

35 This appears to be Ptolemy's view (Lejeune, Euclide et Ptolme, p. 55) and is certainly the way Galen, Hunayn ibn Ishq, Alhazen, and most later medieval Europeans understood the rays, whether visual or solar, to form a cone. This view, in turn, influenced ocular anatomy; in order to avoid the problem of refraction, Galen and most of his medieval successors saw the eye as a series of concentric circles (the tunics and humors), so that rays forming a cone with its apex at the common center would not be refracted, because of their perpendicularity to each successive layer. 36 De sphaera et cylindro I, 34 (corollary) plus some elementary Euclidean geometry. Archimedes, Opera omnia, ed. J. L. Heiberg (Leipzig: Teubner, 1910), I, 132; v. The Works of Archimedes, ed. and transi. T. L. Heath (Cambridge: Cambridge U.P., 1897), p. 44. Damianos' error on this point was first noted in Hirschberg, Augenheil- kunde im Alterthum, pp. 169 - 170. 37 Damianos, Optik, p. 4 quotes book 1, postulate 1 of the work (d. cit. n. 34 supra, I, 8; Eng. transi, cit. p. 3). 38 Whether or not Damianos had read this particular proposition is not certain, though probable. A parallel circumstance is Theon's apparent ignorance of Ptolemy's explanation of horizon magnification in the Optics, for Theon makes use of a contra- dictory theory given earlier by Ptolemy in the Almagest (Rome, "Catoptrique d'Archimde", pp. 39 - 40); Theon may have read only part of Ptolemy's Optics, or he may have misunderstood the discussion of refraction, for he certainly weakens parts of Euclid's Optics (v. Ver Eecke, Optique et Catoptrique, pp. xxvii - xxviii).


232 B. S. Eastwood:

refraction is given on the basis of the principle of uniformity primarily, in order to establish a parallel with reflection. Reflection, in turn, and therefore refraction as well, accords with the principle of economy.

II. The non-experimental nature of Damianos' law of refraction and its openly

metaphysical foundations are mirrored in the optical work of Robert Grosseteste in the thirteenth century. To juxtapose and compare the refraction laws of these two persons is not to claim a historical connection between them. In fact, what makes the similarity interesting is the apparent lack of positive connection. There seems to be absolutely no tradition of a Latin version or summary of Damianos' optics, not even references to the work, before the Renaissance. Yet Grosseteste and Damianos share a similar outlook on science. From an outlook inspired by Platonism, each sees reference to authorities and experience as ancillary information, meaningless without metaphysical bases for synthesis.39 While Damia- nos makes only brief reference to these bases in his optics, Grosseteste develops a veritable metaphysics of light, which he immediately applies to the study of physical light. Throughout his optics, metaphysical principles play an important part. In what was probably the last of his significant works in science, Grosse- teste dealt with the problem of explaining the rainbow. In his treatise, De iride (ca. I235),40 while making some use of empirical data, he relies ultimately upon the principles of economy and uniformity in order to design a theory of the rainbow.41 The same principles lie behind his quantitative law of refraction.

Just as Damianos, Grosseteste seems to use only a small amount of the optical work done before his time. But the lack of attention to available works in the case of the earlier writer is replaced by a lack of availability of many works for the later writer. While the optical treatises of such Greek writers as Archimedes and Apollonius seem to have vanished permanently by the thirteenth century, extant contributions by Anthemius42 (fragmentary) and Ptole- my43 were unavailable to Grosseteste. The primary contribution of the Islamic world to this science, the De aspectibus of Alhazen, seems not to have been

39 The development of this point requires a separate paper in itself; such a paper, entitled "Mediaeval Empiricism: the Case of Grosseteste's Optics", appears in Speculum, 43 (1968), 306-21. 40 Text in Die philosophischen Werke des Robert Grosseteste, Bischofs von Lincoln, ed. Ludwig Baur (Mnster: Aschendorff, 1912), pp. 72-8; dated thus by Richard e. Dales, "Robert Grosseteste's Scientific Works," Isis, LI (196I), 402. 41 see Bruce S.Eastwood, "Robert Grosseteste's Theory of the Rainbow: A Chapter in the History of Non-experimental Science," Archives int. hist. des sciences, 19 (1966), 313- 32.

42 Anthemius is nignly regarded oy tne Araos, e.g., m tne ^aim liui&illiuh ui

Alhazen's De speculis comburentibus (trans, late 12th cent.), but Witelo alone (Per- spectiva IX, 39 - 43) among the Latins shows knowledge of Anthemius. v. Heiberg & Wiedemann, " Parabolische Hohlspiegel", p. 219; G. L. Huxley, Anthemius of Traites: A Study in Later Greek Geometry (Cambridge, Mass., 1959), PP- 3, 40 - 2. 43 Ptolemy's Optics appears to have become current m northwestern Europe only after Grosseteste finished his scientific work, viz., from ca. 1250; Lejeune, Optique de Ptolme, p. 31-



available until after Grosseteste's scientific period;44 in any case, it is certainly never used by Grosseteste. The nature of his optics is largely determined by the elementary level of the treatises he clearly has before him. Little beyond the Euclidean Optics and Catoptrics, Hero's Catoptrics (Liber Ptolemei de speculis), and Alkindi's De aspectibus appears to constitute the optical tradition of the first third of the thirteenth century. None of these discusses refraction, and only the simplest geometrical aspects of reflection and direct light are treated. With these plus a few brief references to refraction,45 Grosseteste has a relatively unlimited field for speculation and development of optical laws.

In his law of refraction more than anywhere else Grosseteste shows how speculative, metaphysical principles can be applied in detail to the business of science. In a notable passage of De iride he lays down the law that the angle of refraction equals half the angle of incidence (when passing from a rarer to a denser medium). This law is closely followed by still another optical law, one for the location of an image in refraction, based on the same principle. Three con- secutive paragraphs by Grosseteste illuminate his whole approach towards optics.46 In the first paragraph, containing the law of refraction, he said.

The size of the angular declination of the refracted ray from a straight ingress can be visualized as follows. First we conceive of a ray which passes from the eye through the medium of the air and meets a second transparent body; we extend into the second transparency the straight line along which this ray travels, and then from the point at which the ray meets the transparency we draw into the depth of that transparency a line which is perpendicular to its surface. I say then that the path of the ray in the second transparency is alone a line dividing equally the angle [my italics] which is formed by the imaginary direct extension of the ray upon the surface of the second transparency into that medium.47

The passage clearly describes the case of refraction from a less to a more dense medium. After constructing a straight line passing through both media and per-

44 George Sarton, "The Tradition of the Optics of Ibn al Haitham", Isis, XXXIX (1938), 403 - 6; but cf. Marshall Clagett, Archimedes in the Middle Ages (Madison: U. Wis. Press, 1964), I, 669. The issue is still very unclear. What seems to be the earliest surviving Latin manuscript of this work is in the Crawford Library of the Royal Observatory, Edinburgh, MS. 9 - 11 - 3 (20), dated 10 May 1269; but the colophon suggests working knowledge of the treatise by a "Magister Johannes Londoniensis" for some time previously. 45 Being the only geometrical treatment clearly available, the most important is in a pseudo-Euclidean De speculis (not the Catoptrica attributed by Proclus to Euclid), appearing in Alkindi, Tideus, und Pseudo-Euklid: drei optische Werke, edd. A. A. BjORNBO & S. VoGL (Leipzig: Teubner, 1911), pp. 105 - 6 (prop. 14 only). 46 In more extended form the following analysis may be found in Bruce S. East- wood, "Grosseteste's 'Quantitative' Law of Refraction: A Chapter in the History of Non-experimental Science," /. Hist. Ideas, 28 (1967), 403 - 14, Some new emphases are made in the present paper, while the earlier article deals as well with the following points unmentioned here: (l) the epistemological basis of Grosseteste's optics, (2) scholarly confusions about the existence of a quantitative law of refraction in the optics, and (3) speculation on the qualitative as opposed to the quantitative aspects in the study of light. 47 Baur, Werke, p. 74.


234 B. S. Eastwood:

pendicular to the interface, he points out the direction taken by a ray in the denser medium in relation to the path it would have followed, had it continued directly and unrefracted. That path is indicated by the phrase "dividentia per aequalia angulum." This peculiar terminology has an obvious meaning - that the ray in the denser medium follows a path bisecting the angle formed between the perpendicular to the surface and the imaginary rectilinear continuation of the ray from the rarer medium into the denser. However, there is a less obvious meaning, only partially explicit, in the phrasing. To speak of " division of an angle into equal parts" rather than of halving that angle may seem no more than an alternative way of speaking. Certainly the result for physics is the same, i.e., the optical law that r = %i; but the result for metaphysics is quite different. Had Grosseteste said explicitly that r = 'i, he would simply have been wrong. In stressing the notion of equality, however, like Damianos before him he was appealing to a metaphysical principle of uniformity as a basis for scientific law. Lest the conclusion seem too ingenious an exercise of too little evidence, let us consider the immediately succeeding paragraphs in De iride. Only then can we appreciate the full significance of the appeal to equality.

Reinforcing the first paragraph, the next one lays down the principle upon which the half-angle law is based. The text runs as follows.

That the quantity of the angle is so determined in the refraction of a ray, similar experiences show us, by which we know that the angle of reflection of a ray upon a mirror is equal to the angle of incidence. And it is shown to us by this principle of natural philosophy, that every operation of nature is by the most finite, most ordered, shortest and best means possible.48

Here Grosseteste has stated exactly the same conclusion as Damianos gave in the last chapter of his Optics. Because reflection occurs at equal angles, refraction also must occur at equal angles. Why? Because of the metaphysical principle of economy, both Hero (followed by Damianos) and Grosseteste asserted the necessity of an equality between the angles of incidence and reflection. Finding no essential difference between reflection and refraction, Grosseteste (like Damianos) easily made the transition from the former to the latter. He says openly that refraction and reflection are

" similar experiences." Certainly this in itself was no novel view, as writers since Antiquity had confused the two types of phenomena.49 But Grosseteste proceeds from this methodological principle of uniformity to a metaphysical use of the principle. Not only is it probable to consider refraction and reflection as similar, it is necessary to do so ! Uniformity becomes a principle of Nature here. All phenomena of light must follow natural

48 Ibid., pp. 74-5. 49 The example of the rainbow is te most common, as it was always considered, the result of reflection. Seneca referred to magnification by water as a case of reflection; v. Snque, Questions naturelles, ed. Paul Oltramare (Paris: Belles Lettres, 196I), I, 33. Lejeune, Euclide et Ptolme, p. 76, n. 4 notes that even Ptolemy considered reflection and refraction as two cases of the same phenomenon. The opening line of Pseudo-EucLiD, De speculis, 14 (art. cit. n. 45 supra) suggests that reflection and refraction are essentially similar. The medieval pseudo- Aristotelian "De pro- prietatibus elementorum" describes the passage of light through a dense sphere as reflection (B. N. ms. lat. 478, f. 58r; Vat. ms. lat. 2083, f. 209r- v).



laws in a simple manner, he appears to say. Both reflection and refraction, different aspects of the same species of phenomenon, must accord with the law of equal angles. Equality is the highest degree of order in reflection. Where does this ordered equality appear in refraction? It can no longer be a relationship of incidence to refraction, and so it becomes a bisection of the imaginary angle formed within the second medium.

The uniformity of the two phenomena, reflection and refraction, is argued again and conclusively established in Grosseteste's mind by the law for the location of an image in reflection and refraction. The third paragraph formulates the law as follows.

A thing which is seen through many transparent media does not appear to be as it truly is, but seems to be at the conjunction of the departing rays, extended from the eye along a straight line, and of a line drawn from the viewed object perpendicular to the surface nearer the eye of the second transparent medium. Moreover, this is shown to us through that experience, and similar reasons, by which we know that a thing seen in a mirror appears at the juncture of the extended line of sight and of a line drawn perpendicular to the surface of the mirror.50

Here again the illuminating parallel of reflection and refraction appears. On the basis of the well-known law of image location in reflection, Grosseteste gives the law for image location in refraction. Again he appeals to "similar reasons/' connecting reflection and refraction. In each case the law described could be stated by the following words; the location of the image is defined by the inter- section of a perpendicular to the surface from the viewed object and a direct continuation beyond the surface of the line of vision from the eye to the medium. The phrasing is identical for both reflection and refraction, but the actual geometry is quite different. The verbal similarity, evidence of a metaphysical uniformity principle, is more important to Grosseteste than any diagrammatic difference. Like Damianos he is hardly concerned for detailed geometrical treatment of optical propositions. The sort of geometry used in optics by Ptolemy, Alhazen, and Witelo is of little use to Grosseteste. For him the applicability, rather than the application, of geometry to optics is sufficient. Geometry represents certitude rather than exactitude, and gives respectability to the metaphysical treatment of light. A reading of his De lineis, angulis, et figuris will makes this abundantly clear; one need not go as far as the cosmogony of De luce to find light metaphysics in the work of Grosseteste.51

50 Baur. Werke, P. 7 S. 51 For some discussion of light metaphysics in Grosseteste v. A. C. Crombie,

Robert Grosseteste and the Origins of Experimental Science, 1100 - 1700 (Oxford: Clarendon Press, 1962), chs. 5 - 6; Robert Grosseteste, On Light (De luce), transi. Clare C. Riedl (Milwaukee: Marquette U. P., 1942); Ludwig Baur, Die Philosophie des Robert Grosseteste, Bischofs von Lincoln (f 1253) (Mnster: Aschendorff, 191 7), pp. 76 - 109. None of these makes much of the geometrical metaphysics involved (cf. Baur, Philosophie, pp. 16 - 7, which misses the point of Grosseteste's separation of physics and mathematics) ; brief attention is given to it in my discussion in op. cit., n. 39 supra. 1 7 a Arch. Hist. Exact Sci., Vol. 6


236 B. S. Eastwood : Derivations of a Law of Refraction

Using geometry as the handmaiden of physics and metaphysics, Damianos and Grosseteste develop optical discussions which contain more speculation than mathematics. The metaphysical principle of uniformity, embodied in the expressed notion of equality, is used by each to design a quantitative law of refraction on the basis of reflection. The initial identity of outlook found on this point in two so widely separated authors is witness to the directive power of elementary treatises on geometrical optics combined with a Neoplatonic environ- ment. For these two generic elements are common to Grosseteste and Damianos. Each seems to have made use of simple geometrical optics, overlaid with a Neo- platonic interest in the reality, physical and metaphysical, of mathematical entities.52 On these bases they laid down the earliest extant statements of a quantitative law of refraction.53

52 A highly suggestive work on this theme is Philip Merlan, From Platonism to Neoplatonism, 2nd edition (The Hague: Nijhoff, I960). 53 Though pointedly inaccurate, they were the earliest attempts to formulate a law. Both Ptolemy and Alhazen, knowing much more about refraction and taking experiments seriously, were not so rash as to formulate a law; the failure to calculate in terms of trigonometric function wTas, of course, an obstacle to such a formulation. Notably, those who approached science with a more Neoplatonic bent seemed intent on discovering only regularity in nature without paying much attention to the evidences of such regularity.

Department of Social Sciences Clarkson College of Technology

Potsdam, New York

(Received November 4, 1969)


Article Contentsp. [224]p. 225p. 226p. 227p. 228p. 229p. 230p. 231p. 232p. 233p. 234p. 235p. 236

Issue Table of ContentsArchive for History of Exact Sciences, Vol. 6, No. 3 (29.V.1970), pp. 171-248Front MatterMaxwell and the Modes of Consistent Representation [pp. 171-213]A Reconsideration of Roger Bacon's Theory of Pinhole Images [pp. 214-223]Metaphysical Derivations of a Law of Refraction: Damianos and Grosseteste [pp. 224-236]Rabbi Levi Ben Gershon and the Origins of Mathematical Induction [pp. 237-248]

archive for history of exact sciences volume 6 issue 3 1970 [doi 10.2307%2f41133302] bruce s....

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