SOURCE: "Euclid's Optics in the Medieval Curriculum," in Archives Internationales D'Histoire des Sciences, Vol. 32, 1982, pp. 159-76.

[In the following excerpt, Theisen discusses the impact of Euclid's Optica on Western scholars in the twelfth and thirteenth centuries and maintains that by the thirteenth century, a "firm tradition " of the critical analysis of Euclid's text was established.]

Defending the utility of a liberal education, John Henry Newman stressed the advantages of learning " … to think and to reason and to compare and to discriminate and to analyze …"¹. Newman's words are an apt description of one of the chief aims of the medieval curriculum, the formation of a discriminating, critical mind. Although it is true that medieval students were concerned primarily with studying the texts of great writers like Euclid, Aristotle and Boethius, they were encouraged, indeed generally constrained, to work their way through a text in a critical fashion. The medieval student did not simply hear a lecture without exerting his own mind; a proof of any proposition was not merely presented, elucidated and passively accepted. The proof was often presented as a test, a statement to be challenged and contested.

This insistence on taking a critical approach to texts is apparent in the glosses to the Latin text of Euclid's Optica, Liber de visu [A Book about Seeing]². These glosses not only yield considerable information about the state of medieval optics, but provide insights into medieval educational theory and practice as well.

There is ample evidence that Western scholars eagerly received and pored over the Optica in the twelfth and thirteenth centuries. Attesting to this interest are the three known translations of the Optica, one from the Greek, Liber de visu, and two from the Arabic, Liber de aspectibus [A Book about Sightings] and Liber de radiis visualibus [A Book about Visual Rays]³. Eight versions of De visu have been identified⁴. It is not surprising that the translators regarded the Optica as important; the large number of extant Greek and Arabic manuscripts with this treatise suggests that the translators did not have to search for copies⁵. Indeed, the overwhelming popularity of Euclid's Elements in the twelfth century would have been incentive enough to give his Optica as well a high priority in the eyes of the translators. Undoubtedly, this could account also for the large number of independent versions of De visu produced by Western scholars in this period. Any work ascribed to Euclid deserved, in the eyes of Western scholars, to be examined carefully and subjected to the closest scrutiny. To be sure, the Optica, as a work limited to an elementary treatment of perspective, could not compete as an optical treatise with Alhazen's Perspectiva or Ptolemy's Optica, both being more thorough, extensive and complete than Euclid's work. However, the brevity and conciseness of Euclid's treatise, as well as its thoroughly mathematical character, induced many scholars to prefer De visu as an introduction to optics to the more discursive and philosophical treatments mentioned above.

For example, prior to their own optical studies, the four major figures involved with optics in the thirteenth century became acquainted with Euclid's De visu. The writings of Grosseteste, Bacon, Pecham and Witelo provide sufficient evidence for this acquaintance, though the degree of interest in De visu is difficult to discern in each. Grosseteste⁶, Bacon⁷ and Pecham⁸ indicated they studied De visu, though references to it in their writings are few. On the other hand, Witelo's Perspectiva⁹ incorporated virtually all of De visu; furthermore, there is good reason to believe that he produced one of the anonymous versions of De visu as well¹⁰. Had Euclid's authorship been insufficient by itself to recommend the study of De visu, the endorsement of these major writers secured its reputation among Western scholars.

Consequently, it is not difficult to understand why De visu became one of the standard texts for the teaching of the mathematical arts in the thirteenth century. The glosses to De visu and its many versions reveal precisely how Euclid's text served the medieval teacher of the mathematical arts, a major field of the medieval curriculum, as Pearl Kibre has established: " … the mathematical or quadrivial arts, although no longer in their traditional framework, did continue into the thirteenth century to constitute an integral part of the university arts curriculum"¹¹. This judgment was made specifically of the University of Paris, but this same conclusion, according to James Weisheipl, O.P.¹², can be extended to the curriculum at Oxford and other universities. Although arithmetic, geometry and astronomy were the core of the curriculum in the mathematical arts, it included the study of optics, the science of weights and computus texts as well. That Euclid's De visu was indeed one of the mathematical arts is apparent from an examination of the thirteenth century manuscripts in which it is found. Nearly all of the thirteenth century copies of De visu are bound with one of the versions of the Elements¹³, and in most cases with several of the texts generally used in the curriculum of the mathematical arts: De sphera of Sacrobosco; Arithmetica of Boethius; De speculis, ascribed to Euclid¹⁴; De ponderibus, ascribed to Jordanus Nemorarius; one of the Computus text; De triangulis of Jordanus Nemorarius¹⁵; De quadratura circuit of Archimedes¹⁶. The consistent association of De visu with such texts is not coincidental; it is clear that De visu was considered to be of the genre of these other mathematical works.

De visu, then, along with the rest of the mathematical works, served the purpose of the medieval arts curriculum admirably, i.e. " … to present the students with the best authorities available and to inculcate in them a logical and scientific manner of thinking"¹⁷. It is not difficult to see, from an examination of the glosses, how the medieval teacher employed De visu to achieve this goal. For a number of reasons De visu was ideally suited for the teaching of the mathematical arts: (1) it was a short text and therefore, could be covered in a brief period, (2) as a completely geometrical text it provided a rapid review of the Elements and (3) at the same time it demonstrated to the students that geometry was more than an abstract discipline but had practical value as well. The brevity of a text was an important consideration, as only seven weeks were allotted to the study of the Elements¹⁸, the only geometrical text required pro forma¹⁹. Any other geometrical text almost certainly would have to be covered more quickly. De visu, in all likelihood was not lectured on ordinarie, by a master, but cursorie, by a bachelor, under the supervision of a master. Therefore, although De visu, as a work on perspective, was extremely limited in scope and ignored important topics like the physiology of the eye, binocular vision and refraction, it did have advantages over the long and abstruse works of Alhazen and Ptolemy.

As a means of teaching rigorous, logical thinking, De visu served nearly as well as the Elements. Indeed, in some respects De visu served this end better, because the propositions in De visu have many lacunae in their arguments and it requires not merely a familiarity with the Elements but also a disciplined, alert mind to detect these lacunae and justify the conclusions reached by Euclid. In fact, this may have been the chief reason for including De visu in the curriculum of the mathematical arts—to provide a test of the students' understanding of the Elements. However, there are enough glosses concerning the nature of vision and perspective to establish that De visu was of interest in its own right. Whatever reasons moved the medieval scholars to examine De visu, it is apparent that they did so in a discerning, critical manner, determined, as in so many other instances, to exhaust the text of its meaning.

It is through the glosses that an appreciation can be gained of the medieval approach to textual understanding. Six texts of De visu have been chosen to exhibit this approach, five of which are from the thirteenth century and one from the second half of the twelfth. The six texts with their sigilla used subsequently are:

BE = Berlin, Staatsbibl., MS Lat. Q. 510, fols 63^v−72^v.

BL = London, British Lib., MS Add. 17368, fols 60^r−69^r.

BOA = Oxford, Bodleian Lib., MS Auct. F. 5.28, fols 17(57)^r−24(64)^r.

CC = Oxford, Bodleian Lib., MS Corpus Christi College 251, fols 1^r−7^v.

CUG = Cambridge, Gonville and Caius College Lib., MS 504/271, fols 86^v−93^v.

VE = Venice, Bibl. Nazionale Marciana, MS Zanetti Lat. 332 (Valentinelli XI.6), fols 242^r−251^v.

BE, BOA, CC and VE have extensive marginal and interlinear glosses; BL has few marginal or interlinear glosses but has added a great deal of material to De visu within the text; CUG has rewritten many of the propositions, but has no marginal or interlinear glosses. The glosses of BE, BL, BOA, and CUG are contemporary with their texts, but those of VE and CC are later than their texts. There is no evidence in the glosses as to the identity of the commentators.

These texts are related in the following ways:

a) BE, BOA and VE have numerous common glosses, but VE has only those glosses from BE that are found in BOA, while BOA has some glosses from BE not found in VE. Consequently, it is safe to conclude that VE acquired the glosses of BE from BOA directly or from some manuscript dependent on BOA.

b) BOA and VE have common glosses not found in the other manuscripts; the direction of transmission is probably from BOA to VE, since VE took the glosses of BE from BOA.

c) BOA, CC and VE have common glosses, but CC generally follows VE more closely than BOA; the conclusion that CC received these glosses from VE is probable, based on certain discrepancies and omissions in the glosses. CC has many glosses peculiar to itself. However, the text proper of CC is closer to BOA than VE and must have been transmitted in a different line than the glosses.

d) CC and BE have no glosses in common.

e) Similarities between BL and CUG suggest that the latter depends on the former.

Although in general the glosses demonstrate a critical approach to the text, one must conclude that there was no attempt to subject the entire text to a thorough, systematically complete analysis. For example, none of the commentators justifies, with suitable references to the Elements, every statement of De visu. In most propositions only a single statement, or perhaps two, is isolated for critical examination. In all of the texts the comments become more rare in the last twenty propositions, an indication, as said above, that the text was probably lectured on cursorie rather than ordinarie and that a relatively short period of time was allotted to this. As do modern teachers, the medieval lecturer found the term too short. Nevertheless, a method of textual analysis was inculcated, given the limitation and restriction of time available.

The most frequent type of gloss found in the manuscripts consists of references to the Elements; in many instances these have found their way into the text itself. But the number of such references found in any one manuscript is not large: 48 in BL, 32 in VE, 25 in CUG, 23 in CC, 17 in BOA, 14 in BE. Again, this indicates that time was short and only a selective treatment was possible, since it would require about 150 references to support all of the statements in the sixtyone propositions of De visu. This incompleteness is revealed by the fact that the references in any given proposition do not agree. In Proposition 9, for example, VE refers to Elements IV, 15; V, 4, 8, 12; VI, 1, 4; CC refers to V, 3, 16, 18; VI, 1, 14, 32; and BL refers to V, 10, 17, 18; VI, 1, 32. The lack of agreement among the references is due to the fact that the commentators singled out different parts of the argument for justification. In an occasional instance the references are in error, either because of the carelessness of the commentator or because of scribal inattention²⁰.

Of the six texts studied, BL shows the greatest familiarity with the Elements. This is apparent, not only from the larger number of references made, but from the keener insight shown. BL alone exhibits a certain uneasiness with Euclid's setting up a ratio of a triangle to a circular sector, seeing that the Elements does not, strictly speaking, allow this in Book V, definition 3: "A ratio is a sort of relation in respect of size between two magnitudes of the same kind"²¹. BL's comment is: "Although a proportion is not permitted except between four quantities of the same kind, of the four quantities the ratio of the first to the second is greater than the ratio of the third to the fourth. This argument about alternate substitution is not geometrical, but the conclusion is still valid"²². Recognizing that De visu was not as complete and rigorous as it ought to be, the commentators, generally with success, showed why Euclid's conclusions were nevertheless valid, or, as will be shown, in some instances invalid.

One of the patent deficiencies of De visu is its paucity of discussion on the visual ray theory. Some of the commentators, BE, BOA, CC and VE, attempted to clarify what this theory entailed. Euclid simply stated that the visual rays are contained within a pyramid²³, but the commentators elaborated somewhat on this construct. They explained that, "An object is seen under a solid angle, each dimension under a plane angle²⁴. The pyramid will be completely closed by plane triangles drawn to the base. If the base has many sides, the pyramid will be many sided as well and if the base is round, the pyramid will be round also"²⁵. In an attempt to better understand the nature of visual rays, BE states that they behave in the same manner as light, which, as "a corporeal power that does not apply itself in full force to each point but distributes itself among each part of an object, point by point, is more efficient for seeing when concentrated. In [becoming] diffuse [it also becomes] weaker; as when light [is concentrated] in one point. Although it is able to spread out over an entire surface, it illuminates more when it is concentrated than when it is distributed over the whole [surface]. The power of seeing [behaves] in a similar fashion. Consequently, that is seen more clearly which is seen under many angles collected together. What is seen under many angles divided among many objects [is seen] more weakly. Because of this, a part, which is visually less than the whole, is seen more clearly than the whole. For the power of many angles is concentrated on a part, whereas before the power was divided among the whole"²⁶. In other ways, too, as BE, CC and VE point out, visual rays behave like ordinary light rays, because they can be refracted and reflected²⁷.

Other glosses represent attempts to account philosophi cally for certain limitations of the eye. The explana tion for the eye's inability to perceive the curvature of an object at a distance is that the eye, after all, is a finite agent. Consequently, it does not have an infinite potency which would enable it to detect such curvature when the angle of vision is small²⁸. CC employs a similar argument to account for the inability of the eye to perceive small details of an object at a distance, like the corners of a square. The commentator states that seeing, whether it is an active or passive natural power, is limited and determined in quantity, and consequently also limited as to what it can perceive²⁹. These brief observations concerning visual rays show that they were physical, not merely geometrical entities. No observations of this kind are found in BL, where De visu is of interest primarily because of its geometrical character. As BL probably precedes the other texts³⁰, one can detect a shift from a mathematical to a more physical interest in De visu on the part of the commentators.

It is surprising, in view of the reverence with which Euclid was regarded, that the limitations of his text were so readily exposed. In numerous instances the commentators expressed their disagreement with the enunciation as he gave it or with his argument….

[I]n the thirteenth century a firm tradition of taking a critical approach to the texts was already established. The fact that De visu did not occupy a significant place in the medieval curriculum and yet was subject to careful analysis confirms this conclusion. What was being taught was more than a text and its contents; it was an attitude, a state of mind, an intellectual stance, a mental disposition and a general, universal method. Whether one takes the position that it was precisely such a pursuit that led to the scientific revolution, the value of such an endeavor in itself cannot be denied.

Not all of the glosses, certainly, are of this same critical nature. Many of them are helpful summaries of a proposition or part of an argument as, "The sense of both translations is …,"⁴⁵ or "The tenor of this argument is …,"⁴⁶. Or the reader/student is alerted to a problem in the demonstration, "Be careful of the proof of this conclusion"⁴⁷. There are many instances of glosses added to clarify the language of the text. Sometimes this indicates that a term used by the translator in the twelfth century is no longer in common use, or the meaning of the term has changed, as visus for radii visuales (visual rays)⁴⁸; epipedum for planum⁴⁹; conos for pyramis⁵⁰. Some passages, because of the de verbo ad verbum technique of the translators, need paraphrasing. Consequently the commentator explained that "lines occupy their places commonly when one is placed immediately adjacent to the other"⁵¹. Or a general statement needs to be made more precise, as when De visu speaks of the eye approaching an object, in Proposition 17⁵², the commentator pointed out that this approaching is not to take place along any line whatsoever, but only along a line perpendicular to the object⁵³. There are several very long enunciations and the reader/student might find it difficult to see what exactly is to be proven; he is aided by the word passio placed over the essential parts of the enunciation⁵⁴.

Only one gloss relates Euclid's propositions to actual experience, and even here it is not clear whether the commentator wished to be taken seriously. Euclid describes in Proposition 13 how, "Of those objects of some length placed in the space before the eye, the ones on the right seem to be inclined towards the left, those on the left seem to be inclined towards the right"⁵⁵. The commentator added the note, "This is confirmed by the nature of sight and by experience, as for example with drunken men, with both eyes open. The force of the statement derives from the fact that it says: they seem to be inclined"⁵⁶. It is not surprising that there is virtually no relating De visu to experience. After all, it was regarded primarily as a mathematical text and the study of natural philosophy followed the study of the mathematical arts in the medieval curriculum.

Nevertheless, one commentator did show an unusual practical bent. In three instances BL suggested that measurements could be made with a rod or a rope⁵⁷. He also proposed the use of visual aids in understanding a figure described in the text; six times he recommended that figures described in the text that do not lie in a plane but are three dimensional be constructed out of reeds⁵⁸. In other instances, anticipating King Richard's suggestion to "make dust our paper", BL stated that the figures could be traced in the dust⁵⁹.

The commentators showed an awareness that Euclid on occasion simply discussed a particular situation or a special case and they recognized that more general conditions should be stated, as in Proposition 5,⁶⁰ where Euclid stated that when equal and parallel lines are seen at unequal distances the closer one is seen to be larger. De visu presents one relatively simple arrangement of the lines. VE has drawings of five different possible arrangements of parallel lines and then indicates the general condition that must be observed if the enunciation is to be valid…

Some glosses show that the commentators were carried away by the geometry of a proposition and became oblivious of the physical situation described. In one gloss the commentator simply ignored the fact that De visu was describing a figure in three dimensions and went on to develop an elaborate proof for lines all lying in the same plane⁶⁴. Although CUG almost completely rewrites Proposition 10: "Squares seen at a distance appear round"⁶⁵, the new version does not take into account that visual rays are contained within a pyramid and consequently cannot be parallel. The commentator proceeded in the following way, forgetting that it is impossible for the rays to fall perpendicularly on all four sides of a square:

Let it be understood that the eye is in the air at A from which the rays are led to fall normally on the four sides of a square. Assume that those sides are at the last place in which they could still be seen. If therefore the rays are led from the eyes to the angles of this square, since they are further from the eye than the sides they will not be seen. Therefore the square will not be seen as angular but in such a way that the rays drawn to the angles will be equivalent to the rays drawn to the sides. However, as the eye will be at the center all rays sent forth to the surface of the object seen are equal. Therefore the object seen appears as circular since, "Such a figure, in which all lines…", et cetera is a circle.⁶⁶

These glosses and similar ones where the commentator lost sight of the physical nature of the discussion emphasize what has already been stated, i.e., De visu was regarded primarily as a geometrical work and its experiential implications were of very minor interest. Furthermore, there is no evidence in the glosses that the propositions of De visu held any significance for the commentators with regard to the development of a metaphysics of light⁶⁷. Nevertheless, the glosses leave no doubt that medieval scholars sensed the power of geometry as a key to discussing and understanding certain elements of ordinary visual experience. This appreciation surely contributed to the growing recognition in the thirteenth and fourteenth centuries that mathematics could be a great aid to natural philosophers.

Notes

¹The Idea of the University (New York: Doubleday and Co., Inc., 1959), 182.

²Optica is generally used to refer to the Greek version of Euclid's work on vision; it has been edited by Johan Heiberg and is included in the seventh volume of the monumental Euclidis opera omnia, 8 vols (Leipzig, 1893-1916), of which H. Menge is co-editor. An English translation of the Optica was produced by Harry Edward Burton and published in the Journal of the Optical Society, 35 (1945), 357-372. I have edited the chief Latin version, Liber de visu, published in Mediaeval Studies, 41 (1979), 44-105.

³ My doctoral dissertation, "The Mediaeval Tradition of Euclid's Optics" (Wisconsin, 1972) [Dissertation Abstracts 32A (1972) 5697A], contains editions of the two translations from the Arabic. For a list of the manuscripts with these two translations see David C. Lindberg, A Catalogue of Medieval and Renaissance Optical Manuscripts (Toronto, 1975), 46-47.

⁴ See Lindberg, op. cit., 50-54. However, London, British Library Add. MS 17378, fols 60^r-69^r should be listed as a distinct version of Liber de visu.

⁵ I am indebted to Dr Emilie Savage-Smith of the Gustave E. von Grünebaum Center for Near Eastern Studies, UCLA, for a list of thirty-two Greek manuscripts that contain the Optica. Numerous copies of the Arabic versions have been discovered. Moritz Steinschneider, Die Arabischen Übersetzungen aus dem Griechischen (Graz, 1960), par. 92, lists ten; he adds the Bodleian manuscript, MS Uri 875 to this list in his article, "Euklid bei den Arabern", Zeitschrift für Mathematik und Physik (historisch-literarische Abteilung), 31 (1886), 100, and states that two manuscripts from London and Leiden have Liber de radiis visualibus. Heinrich Suter, "Die Mathematiker und Astronomen der Araber und ihre Werke", Abhandlungen zur Geschichte der Mathematischen Wissenschaft, 10 (1900), 40, adds two manuscripts from the Palatine collection in Florence, MSS 271 and 286. Johan Heiberg, Litterargeschichtlichen Studien über Euklid (Leipzig, 1882), 20, lists nine other manuscripts.

⁶ The evidence for Grosseteste's acquaintance with De visu is slight indeed, but he does indicate that he began his study of optics with Euclid's work. At the beginning of De iride he mentions the three divisions of perspective and then comments that he has only studied the first two, namely, those dealing with vision and mirrors: "Primam partem complet scientia nominata [vocata in some manuscripts] de visu, secundum illa quae vocatur de speculis. Tertia pars apud nos intacta et incognita usque ad tempus hoc permansit". See Ludwig Baur, "Die philosophischen Werke des Robert Grosseteste", Beiträge zur Geschichte der Philosophie des Mittelalters, 9(1912), 73. The terminology of some of the glosses leads me to conjecture that they may be traced to Grosseteste, but direct dependence is hard to establish. The thirteenth century author of the pseudo-Grosseteste Summa philosophiae is another witness to the widespread knowledge of Euclid's optical treatise. In listing the important ancient writers he states: "Euclides qui et geometriam ac perspectivam cum suis partibus ac speciebus edidit". See Baur, 279. An indication of his preference for Euclid is the fact that Ptolemy's Optica is not mentioned, nor is Alhazen.

⁷ See the Opus Maius of Roger Bacon, trans. Robert B. Burke, 2 vols (Philadelphia, 1928; rpt. New York, 1962), 5. dist. 7.1 (p. 468), 5. dist. 2.2 (p. 510), 5. dist. 3.2 (p. 525), 5. dist. 3.2 (p. 530).

⁸ See his Perspectiva communis in David C. Lindberg, John Pecham and the Science of Optics (Madison, Wisc., 1970), part I, prop. 38 (p. 123), prop. 39 (p. 123), prop. 80 (p. 149); part III, prop. 13 (p. 225).

⁹ Witelo's Perspectiva is part of Friedrich Risner's Opticae thesaurus, Alhazeni Arabis libri septem, nunc primum editi. Eiusdem liber de crepusculis et nubium ascensionibus. Item Vitellonis Thuringopoloni libri X (Basel, 1572; rpt., New York, 1972).

¹⁰ See my "Witelo's Recension of Euclid's De visu", Traditio, 33 (1977), 394-402.

¹¹ "The Quadrivium in the Thirteenth-Century Universities (with special reference to Paris)", Arts libéraux et philosophie au moyen âge. Actes du Quatrième Congrès International de Philosophie Médiévale (Montreal, 1969), 191.

¹² "Curriculum of the Faculty of Arts at Oxford in the Early Fourteenth Century", Mediaeval Studies, 26 (1964), 10.

¹³ These texts are in the following manuscripts: Berlin, Staatsbibl., MS Lat. Q. 510, fols 63^v-72^v ; Florence, Bibl. Naz., MS Conv. soppr. J.I. 32, fols 103^r-113^r; Oxford, Bodleian Lib., MS Auct. F. 5.28, fols 17(57)^r-24(64)^r; Oxford, Bodleian Lib., MS Corpus Christi Coll. 251, fols l^r-7^v; Venice, Bibl. Naz. Marciana, MS Zanetti Lat. 332 (Valentinelli XI.6), fols 242^r-25P; Cambridge, Gonville and Caius Coll. Lib., MS 504/271, fols 86^v-93^v; Oxford, Bodleian Lib., MS Corpus Christi Coll. 283, fols 163^r-165^v; Milan, Bibl. Ambrosiana, MS T.91 sup., fols 39^r-49^r; Milan, Bibl. Ambrosiana, MS R. 47 sup., fols 133^r-148^r; Toledo, Archivo y Bibl. Capitulares de la Catedral, MS 98.22, fol. 90^r-^v. The last three versions of De visu are not bound with the Elements, but with other mathematical works.

¹⁴ There were many works known by the title De speculis in the Middle Ages. This particular one, ascribed to Euclid, had the incipit, Visum rectum esse cuius media termines. Cf. Lindberg, Catalogue, 47-50.

¹⁵ For a complete discussion on Jordanus Nemorarius and his works see Joseph E. Brown, "The Science of Weights", in Science in the Middle Ages, ed. David C. Lindberg (Univ. of Chicago Press, 1978), 179-205. For a brief discussion of Boethius' Arithmetica see Michael S. Mahoney's chapter on mathematics in this book and for compotus see Olaf Pedersen's chapter on astronomy in the same volume.

¹⁶ See Marshall Clagett, Archimedes in the Middle Ages, vol. 1 (Madison, Wis., 1964), 1-222.

¹⁷ James A. Weisheipl, op. cit., 167.

¹⁸ Weisheipl, op. cit., 171.

¹⁹ Weisheipl, op. cit., 149, n. 14. Weisheipl discusses the distinction between those books required pro forma and those required praeter formam. The former were set books required for the degree, and were lectured on ordinarie; the latter were not strictly required for the degree and were lectured on cursorie.

²⁰ For example, see CUG, f. 91^r, left-hand column, where the text refers to the Elements III instead of Elements IV for inscribing a triangle in a circle. Also, see BL, f. 68^v, where the text refers to I, 4 twice and it should be I, 5, which discusses isosceles triangles. Could it be that the commentator was relying on memory here?

²¹The Thirteen Books of Euclid's Elements, trans. by Sir Thomas L. Heath, 3 vols (New York: Dover, 1956), II, 114.

²² Fol. 61^v: Sed cum permutatio non habeat fieri nisi inter quatuor quantitates eiusdem proportionis, propositarum autem quantitatum maior est proportio prime ad secundam quam tertie ad quartam. Argumentum istud de permutatim non est geometricum sed tamen sinegetur conclusio. I have encountered the word sinegetur in only one other place, i.e., in one of the versions of the Elements attributed to Adelard of Bath, found in Oxford, Bodleian Library, MS Digby 174. This strengthens my suspicion that BL is related to Adelard or his school.

²³ See Heiberg, op. cit., 3; Burton, op. cit., 357; or Theisen, Liber, 62.

²⁴ BE, f. 63^v: Res videtur sub angulo solido, unaqueque dimensio sub angulo superficiali.

²⁵ BE, f. 63^v; VE, f. 242^r and CC, f. l^r: Nota quod piramis tota erit clausa superficiebus quot et sua basis lineis et si basis eius fuerit multilatera piramidis erit similiter et si basis rotunda sit et piramis similiter.

²⁶ BE, f. 64^r: Virtus corporalis que non se totam tribuit unicuique sed se ad partes rei per partes diffundit efficacior est visui collectione. In sui autem diffusione debilior sicut patet in uno puncto lucis. est diffundere quam cum diffunditur super totam. Similiter est de virtute videndi. Propter illuminat causam.cum se super totam hoc perspicacius angulis videtur in simul collectis. Sub pluribus enim colligitur super partem pluribus per plura divisis debilius. toto et visualiter minus, maiore perspicacius videtur. Sub pluribus enim angulis collecta est virtus super partem que prius divisa fuit super totum. See D.C. Lindberg, "Alkindi's Critique of Euclid's Theory of Vision", Isis, 62 (1971), 474, n. 24. The identity of luminous and visual radiation seems to have been assumed since antiquity.

²⁷ BE, f. 63^v; VE, f. 242^r; CC, f. l^r: Nota quod, cum tripliciter sit videre, scilicet per reflexionem, per fractionem, per extensionem radiorum, hoc solum tertio et ultimo modo intendit hic. "Note that, although vision occurs in three ways, namely by reflection, refraction and extension of the rays, only the third and final way is treated here". This threefold division of optics according to the mode of propagation was common, as Lindberg, Pecham, p. 25, n. 45 points out.

²⁸ BOA, f. 18^r; BE, f. 66^v; VE, f. 244^v: Ex remotione spere patet propositum cum determinati sint anguli in parvitate sub quibus contingit videri, quia finiti non est infinita potentia que possit actu compleri. Rursus maioris et minoris distantie per elongationem a rebus visis excessus propter predictam causam.

²⁹ CC, f. 2^r: Cuius ratio est quod omnium natura constantium est quantitas determinata circa (?) quam operatur et quam non excedit. Visus est constans natura, ergo et cetera. Sicut omnis accio vel passio naturalis est determinata secundum quantitatem. Visus est accio vel passio naturalis, ergo determinatus secundum quantitatem… res non videbitur.

³⁰ Dr T. A. M. Bishop, when he was on the faculty of history at Cambridge University, examined a photo-copy of this manuscript at my request and conjectured that it was written in England between 1150 and 1170. Because of certain similarities between this version and some of Adelard of Bath's writings, I suspect Adelard may be the author of this version of De visu. …

⁴⁵ BL, f. 62^v: Sensus utriusque translationis est …

⁴⁶ BE, f. 69^r; BOA, f. 21(61)^r; VE, f. 248^r: Tenor huius argumenti est…

⁴⁷ CC, f. 3^r: Cave in probatione huius conclusionis.

⁴⁸ BE, f. 63^v; VE, f. 242^r.

⁴⁹ BOA, f. 17(57)^r; BE, f. 65^r; VE, f. 243^v.

⁵⁰ BE, f. 63^v; VE, f. 242^r.

⁵¹ Propositum 48, Heiberg, 96; Burton, 370; Theisen, Liber, 95. See BOA, f. 22(62)^v; VE, f. 249^v; BE, f. 70^v: Communiter dicuntur linee occupare locos suos quando una applicatur alii ita quod nihil cadit medium inter ipsas sive secundum rectam lineam sive oblique et ad angulum.

⁵² Heiberg, 24; Burton, 360; Theisen, Liber, 71.

⁵³ CC, f. 2^v: Et non est intelligendus iste accessus et recessus secundum quamcumque lineam sed secundum perpendicular em respectu maioris magnitudinis.

⁵⁴ See CC, ff. 1^v, 4^v, 5^r, 7^r, 7^v, and VE, ff. 249^r, 249^v

⁵⁵ Heiberg, 20; Burton, 359; Theisen, Liber, 69.

⁵⁶ BOA, f. 17(57)^r; VE, f. 243^v: Hoc ex natura visus et experimento habet notitiam. In ebriis autem hoc habebis duobus oculis videntibus. Constituatur vis de hoc quod dicit: educi videntur. Videntur is underlined in the gloss.

⁵⁷F. 63^r, props 19, 20, 21: per commensurationem virge, per virgam vel filum commensurari potest.

⁵⁸ F. 65^v, prop. 36; f. 66^v, prop. 40; f. 68^r, prop. 45: Alias vero lineas erige in aere cum festucis (f. 65^v).

⁵⁹King Richard the Second, Act III, Scene 2; BL, ff. 66^v, 67^r, prop. 40: Intelligatur OZ erecta in altum et tunc figuretur in piano in pulvere.

⁶⁰ Heiberg, 8; Burton, 358; Theisen, Liber, 65….

⁶⁴ VE, f. 243^r.

⁶⁵ Heiberg, 16; Burton, 359; Theisen, Liber, 68.

⁶⁶ CUG, f. 88^r: Intelligatur in aere oculus A, a quo intelligantur radii educti ortogonaliter ad quatuor latera quadrati et sint illa latera in ultimo loco in quo possunt videri. Si ergo ab eodem oculo ducantur radii ad angulos eiusdem quadrati cum sint remociores anguli lateribus non videbuntur. Ergo videri non potest aliquis angulus. Ergo videtur non secundum angulos sed secundum quod equabuntur radii protensi ad angulos radiis protensis ad latera. Sic autem erit oculus ut centrum et omnes radii emissi ad superficiem visi sunt equales. Ergo visum apparet circulare cum figura talis a qua omnes linee et cetera sit circulus.

⁶⁷ For an introduction to the subject of light metaphysics see Ch. 3 of Lindberg, Science in the Middle Ages; William A. Wallace, "The Philosophical Setting of Medieval Science", 93, 95-96, 98. See also A. C. Crombie, Robert Grosseteste and the Rise of Experimental Science (Oxford: Clarendon Press, 1953), Ch. 6.