SOURCE: "Alkindi's Critique of Euclid's Theory of Vision," in Isis, Vol. 62, No. 214, December, 1971, pp. 469-89.

[In the following essay, Lindberg presents an analysis of Euclid's Optica by Alkindi (d. 873), an early Islamic philosopher. Lindberg states that Alkindi "placed himself firmly on the side of Euclid" in many respects, but that the philosopher disagreed with Euclid on the nature of the "visual cone, " one aspect of the mathematician 's theory of vision.]

Alkindi, undoubtedly the first great philosopher of the Islamic world, was a leader in the endeavor to communicate Greek philosophy to Islam.¹ Not only did he participate in the translating activity of the ninth century, but he also attempted to integrate Greek philosophy with Mu'tazilite theology and thus, in Walzer's phrase, "to naturalise Greek philosophy in the Islamic world."² Alkindi's deep respect for ancient thought is revealed in the preface to one of his works on metaphysics:

It is fitting then to acknowledge the utmost gratitude to all those who have contributed even a little to truth not to speak of all those who have contributed much. If they had not lived, it would have been impossible for us, despite all our zeal, during the whole of our lifetime, to assemble these principles of truth which form the basis of the final inferences of our research. The assembling of all these elements has been effected century by century, in past ages down to our own time…. It is fitting then for us not to be ashamed to acknowledge truth and to assimilate it from whatever source it comes to us, even if it is brought to us by former generations and foreign peoples.³

His own task, as he conceived it, was to communicate, complete, and correct this body of ancient learning. In the same preface he wrote,

It is fitting then to remain faithful to the principle which we have followed in all our works, which is first to record in complete quotations all that the Ancients have said on the subject, secondly to complete what the Ancients have not fully expressed, and this according to the usage of our Arabic language, the customs of our age and our own ability.⁴

It was in this spirit that Alkindi wrote some 260 works on all branches of learning.⁵ Among them was a work on optics, entitled De aspectibus in its Latin translation, which was to exert a strong and continuing influence on Islamic and Western optics throughout the Middle Ages.⁶ In the preface to De aspectibus Alkindi reveals that one of his motives for writing on optics was the desire to correct and communicate to Islamic society the full legacy of ancient learning: "Since we wish to complete the theoretical sciences, to express what the Ancients have given us of them, and to augment what they have begun, … it is necessary for us to speak … concerning differences of appearance [i.e., optics or perspective] according to the measure of our ability."⁷

However, optics fit into Alkindi's philosophical program in yet another way. That is, optics was not simply a branch of Greek learning, on a par with the others, to be communicated to Islam in its turn, for it had a place of special importance in Alkindi's philosophy of nature. The necessity of seeing Alkindi's work in optics against the background of the natural philosophy expressed in his De radiis stellarum (or De radiis stellatis or stellicis) has been emphasized only recently by G. F. Vescovini, who has also provided a very useful analysis of the latter treatise.⁸ The basic theme of De radiis stellarum is the universal activity of nature, exercised through the radiation of power or force. "It is manifest," Alkindi asserts, "that everything in this world, whether it be substance or accident, produces rays in its own manner like a star…. Everything that has actual existence in the world of the elements emits rays This in radiation which every direction, binds the a vast world…."⁹ This radiation binds the world into a vast network in which everything acts upon everything else to produce natural effects. Stars act upon the terrestrial world; magnets, fire, sound, and colors act on objects in their vicinity. Even words conceived by the mind have the ability to radiate power and thus to produce effects outside the mind.¹⁰ This is a natural philosophy that was destined to influence Robert Grosseteste and Roger Bacon and to reappear in their doctrine of the multiplication of species.¹¹

Optics, then, is of special significance because it treats the most fundamental of all natural phenomena—the radiation of power. The laws of radiation are the laws of nature, and optics is consequently a prerequisite to other studies. This view, I believe, lies behind a passage from Alkindi's De temporum mutationibus sive de imbribus to which Vescovini and Lemay have called attention: "Man is not instructed in philosophy until he can divine superior impressions, and he cannot ascend to that knowledge until he has mastered the quadrivial sciences, which are the introduction to philosophy, and they are mathematical."¹²

Euclid's Theory of Vision

When Alkindi turned his attention to the ancient legacy in the field of optics his gaze fell principally on the Optica of Euclid.¹³ Let us consider briefly the contents of this treatise.

Euclid's Optica contains the first systematic exposition of a mathematical theory of vision. Indeed Euclid's approach to vision was so strictly mathematical as to exclude all but the most incidental references to those aspects of the visual process not reducible to geometry—the ontology of visual light and the physiology and psychology of vision. Lejeune comments that Euclid's Optica

… systematically ignores every physical and psychological aspect of the problem of vision. It restricts itself to that which can be expressed geometrically…. Its model is the treatise on pure geometry, and its method that of the Elements: a few postulates all fully necessary, from which follow deductively and with full mathematical rigor a series of theorems of a traditional form.¹⁴

The postulates on which Euclid bases the geometrical theorems of the Optica are seven in number:

Let it be assumed

1. That the rectilinear rays proceeding from the eye diverge indefinitely;

2. That the figure contained by a set of visual rays is a cone of which the vertex is at the eye¹⁵ and the base at the surface of the objects seen;

3. That those things are seen upon which visual rays fall and those things are not seen upon which visual rays do not fall;

4. That things seen under a larger angle appear larger, those under a smaller angle appear smaller, and those under equal angles appear equal;

5. That things seen by higher visual rays appear higher, and things seen by lower visual rays appear lower;

6. That, similarly, things seen by rays further to the right appear further to the right, and things seen by rays further to the left appear further to the left;

7. That things seen under more angles are seen more clearly.¹⁶

The first three postulates define the visual process and cast it into a geometrical mold: rays proceed in straight lines from the eye, the collection of such rays constituting a cone; to be seen, an object must intercept a visual ray. The rectilinearity of the rays, which Euclid assumes in the first postulate, permits the development of a theory of vision along geometrical lines.¹⁷ Given this simple rule governing the propagation of light (as well as the law of reflection, introduced in Prop. 19), it is possible to employ the straight lines of a geometrical diagram to represent visual rays and thus transform optical problems into geometrical problems. The process of geometrization is completed in Postulates 4-6: the apparent size of a visible object is determined by the angular separation between the visual rays that encounter its extremes, and the position of a visible object in space is determined by the location, within the visual cone, of the rays by which it is perceived. Finally, in the seventh postulate Euclid provides an explanation for variations in the clarity of visual perception: objects seen under more angles—that is, encountered by more visual rays and hence seen under more of the angles formed between adjacent visual rays—are seen with greater clarity.

With few exceptions the fifty-eight propositions of the Optica are based upon these seven postulates. The majority of them treat problems of perspective—that is, the appearance of an object as a function of its spatial relationship to the observer. In only one proposition is there any suggestion of the possibility of depth perception (Prop. 57), and, as many commentators have observed, this is probably a later interpolation.¹⁸ Euclid's own view seems rather to be expressed in the fifth proposition, where it is admitted that if equally large objects are observed from different distances, the closer one will appear larger. Euclid thus ignores, by and large, the physical problems associated with the nature of visual rays and their encounter with visible objects, the physiology of sight, and the psychological factors influencing the perception and localization of visible objects; his is a geometrical theory of vision.

However, the claim that Euclid ignored those aspects of vision not reducible to geometry is true only to a first approximation. For, as even Euclid's ancient commentators recognized, the postulates and several propositions of the Optica have inescapable implications vis-à-vis the ontology of visual rays and thus spill over into the physical realm. In the first place it is apparent from such expressions as "proceeding from the eye" and "those things … upon which visual rays fall" that vision is the result of rays issuing from the observer's eye; there is no warrant, so far as I can see, for construing these as awkward metaphors, intended (but failing) to convey purely geometrical truths.¹⁹ The eye is thus the active member in the visual process, reaching out to apprehend its object. Secondly, within the cone of visual rays there are sentient and insentient regions. In the first proposition it is asserted that an object is not visible in its entirety at one time because of spaces between the visual rays, and in the second that any given object can be removed to a distance from which it will no longer be visible because it falls between adjacent visual rays. It is clear that rays having such properties cannot be mere constructions intended to represent the geometry of sight; they must be the physical agents of sight. A third and final non-geometrical claim can be extracted from the seventh postulate and the second proposition, where Euclid offers a physical explanation of what we would now regard as a psycho-physiological phenomenon. The clarity of perception, he asserts, is dependent on the number of angles under which an object is seen—or, to put it in the clearer terms of the second proposition, on the number of visual rays intercepted by the object.

Euclid devotes the remainder of the Optica to problems of visual perspective, avoiding further discussion of physical issues. He thus leaves it to others to resolve the serious problems raised by his conception of discrete rays emanating from the eye. Euclid fails also to reveal the role in vision that he would assign to external light, though in several propositions he refers to the existence of such light, and in Proposition 18 he even has a solar ray terminating at the observer's eye. It may be that he would have agreed with Ptolemy and Damianus—who argued that luminous and visual rays are identical in nature and that luminous rays, as well as visual rays, are efficacious in the visual process²⁰—but this silence prohibits us from reaching any firm conclusion. It was apparently Euclid's intent to formulate a theory of vision restricted to geometry, and the only nongeometrical claims contained therein are those that slipped in along with the conception of visual rays. It was left to Euclid's followers and commentators (among them Alkindi) to enlarge the theory and invest it with additional physical content.²¹

Alkindi's Defense of the Emission Theory

Historians have long recognized that Alkindi's De aspectibus is based upon, or more exactly is a response to, the Optica of Euclid. It is not, however, merely a recension of Euclid's Optica, as several have suggested. Rather, it is a thorough and determined critique of Euclid's theory of vision, an attempt to remove several serious lacunae and to correct Euclid on a number of fundamental points.²² Let us first consider the lacunae.

Euclid had made several very basic assumptions in the Optica without any attempt at justification. In the first postulate he had asserted that the rays issuing from the eye are rectilinear. Alkindi, for whom rectilinear propagation is demonstrable, devotes the first six propositions of De aspectibus to the removal of this lacuna in Euclid's work. Strangely, however, he attempts to demonstrate the rectilinearity, not of visual rays, but of luminous rays—and not, as we shall see, because he intends to deny the existence of visual rays or give priority in the process of vision to luminous rays. Alkindi's adoption of such a mode of argument can be explained by noting that he is here following the preface to Theon's recension of Euclid's Optica;²³ for a logical justification, however, one must surmise that implicit to his argument is belief in the identity of luminous and visual rays, or at least of their modes of propagation.²⁴

In Propositions 1-3 of De aspectibus Alkindi attempts to demonstrate the rectilinear propagation of luminous rays from a consideration of the shadows cast by opaque bodies exposed to luminous bodies: a body equal in size to the luminous body casts a cylindrical shadow, while bodies smaller and larger than the luminous body cast converging and diverging shadows, respectively. The shadows thus conform to straight lines drawn tangent to the illuminating and illuminated bodies, and from this fact the rectilinear propagation of light follows directly.²⁵ A closely related demonstration appears in Proposition 4, where Alkindi points out that the straight line bisecting any of the shadows passes through the centers of both the opaque and the luminous body. Propositions 5 and 6 provide two more demonstrations based on the same principles. If a candle should be so placed that it stands higher than an opaque body and therefore casts a shadow on a horizontal surface, the length of the shadow … is in the same proportion to the height of the obstacle … as the horizontal distance between the candle and the end of the shadow is to the height of the candle … ; this would not be so if [a] line, … representing a ray emanating from the candle and grazing the top of the opaque obstacle, were not straight….

A second undemonstrated assumption made by Euclid takes us to the very heart of the visual process: the first and third postulates of the Optica claim that the rays by which objects are perceived issue not from the object, but from the observer's eye. In defense of this claim Alkindi develops an elaborate series of arguments in Propositions 7-10. He begins with a brief summary of the various alternative theories formulated in antiquity:

Therefore I say that it is impossible that the eye should perceive its sensibles except [1] by their forms travelling to the eye, as many of the ancients have judged, and being impressed in it, or [2] by power proceeding from the eye to sensible things, by which it perceives them, or [3] by these two things occurring simultaneously, or [4] by their forms being stamped and impressed in the air and the air stamping and impressing them in the eye, which [forms] the eye comprehends by its power of perceiving that which air, when light mediates, impresses in it.²⁶

Now if sight were to occur by the first, third, or fourth methods, Alkindi argues, circles situated edgewise before the eye (i.e., with their planes passing directly through the center of the eye) would impress their forms in the eye and would therefore be visible in their full circularity.²⁷ But this is not the case. "On the contrary, when circles and observer are in the same plane, the circles are by no means seen. Therefore it remains that a power proceeds from the observer to the visible objects, by which they are perceived."²⁸ This power proceeds from the eye in straight lines and falls only on the edges of the circles, perceiving them as straight lines or, if they are truly geometrical circles, not perceiving them at all.

It is essential to pause at this point and consider the import of Alkindi's argument. If the argument is to be grasped at all, one must understand what Alkindi means by a form. What he does not mean is a composite impression (as in our conception of an optical image) produced by a large number of individual rays; rather, forms are coherent units, not susceptible of analysis, which are impressed (or would be if there were such things, for Alkindi rejects this theory) in the observer's eye. These are the forms of the atomistic theory of vision. The first theory described by Alkindi among his four alternatives is that of the atomists, and Alkindi is quite correct in regarding atomistic forms as coherent entities.²⁹ Although it is with considerably less justice that Alkindi attributes the same notion of forms to the Platonic and Aristotelian theories (his third and fourth alternatives),³⁰ his conception of vision by intromission of forms is nevertheless clear: if a circle placed edgewise before the eye should be seen by the entrance of its form into the eye, this would not be the result of radiation from each point on the near edge of the circle entering the eye to produce an image; rather, the form of the circle would enter the eye as a unit, and there its spatial orientation would have nothing to do with its perception, for the laws of perspective no longer apply. Indeed, this appears to be the essential point of Alkindi's argument: if the laws of perspective are to be applicable to vision—that is, if the perception of an object is to depend on its spatial orientation—one must hold to the theory of visual rays. Alkindi sees no means by which the intromission theory, which for him is the theory of coherent forms, can be made compatible with the laws of perspective. Despite the fact that Alkindi submitted the luminous body to punctiform analysis elsewhere in De aspectibus, emphasizing that light issues in all directions from every point on the surface of a luminous body,³¹ it was Alhazen (some 150 years later) who first employed this idea as the basis for a successful intromission theory of vision.³² To summarize, then, Alkindi has presented an effective argument against the only version of the intromission theory (or, for that matter, of the combined emission-intromission theory) of which he could conceive—that of coherent forms. That his argument has no force against the later intromission theory of Alhazen is not relevant.

Alkindi has thus placed himself firmly on the side of Euclid in the struggle between the emission and intromission theories of vision. But Alkindi's arsenal contains additional arguments against the intromission theory. Alkindi repeats Aristotle's argument about weak-sighted people who see their own image before them because "the power proceeding from sight, when it cannot penetrate the air because of weakness, is made to return by the air to the body of the observer."³³ He also argues, following Theon of Alexandria, that the structure of a sense organ implies the mode of its functioning. The ears, for example, are hollow in order to collect the air that produces sound. But God made the eye spherical and mobile. It does not, therefore, collect impressions; rather, through its mobility it shifts itself about and selects the object to which it will send its ray.³⁴

A final argument for the emission theory is that only by it can the selectivity of sight be explained. When we read a book, sight must strain to locate a particular letter and perceives it only after an interval of time; it is thus evident that we perceive objects in the visual field in temporal sequence rather than all at once. Moreover, objects situated at the side of the visual field or far from the observer are poorly perceived. Now if sight occurred through an impression made by the form of the visible object in the eye, everything within the visual field would be seen simultaneously, for all would be equally present to the eye. Furthermore, if objects as far off as celestial bodies are visible, surely objects at the distance of a palm or cubit (such as the letters of a book) must impress their forms on the eye all the more clearly and should not have to be sought by the eye. Therefore it must be the case that a visual power issues from the eye, selecting its objects successively, and that this power is weaker the more it diverges from direct opposition to the center of the eye.³⁵

Alkindi's defense of the emission theory also contains a refutation of a single argument to the contrary. "Certain men," Alkindi reports, "seeing that rays carry the colors of bodies away with them and deposit them at their termini, have judged that sense perceives the figures of things only by the passage of those figures to sense. For since figures, according to them, are merely the termini of colors, they have judged that the figures travel to the observer."³⁶ However, if this were the case, colored glass or transparent stones would retain their color only for a short time before it would disappear through radiation. The truth, according to Alkindi, is that the color of a body is not transported by rays; rather, colored bodies transform the air, imparting to it a color similar to their own (while retaining their own color). Therefore the argument to the contrary is of no force, and it remains that vision occurs through the emission of rays from the observer's eyes.

The Nature of Visual Radiation

Thus far Alkindi has been in full agreement with Euclid, attempting only to supply demonstrations that Euclid had omitted. Both have agreed that from the eye rectilinear rays emanate in the form of a cone. However, on the nature of the visual cone Alkindi finds the Euclidean theory untenable. Whereas Euclid had conceived of the cone as a composite of discrete rays, separated by spaces, Alkindi regards any such conception as absurd. The visual cone, as he reveals in a series of arguments, must be conceived as a continuous body of radiation.³⁷

The first phase of Alkindi's attack on the conception of discontinuous visual radiation is based on the assumption that Euclid's visual rays are geometrical lines, having length but no width:

Certain of the ancients have judged that many rays issue from the observer along straight lines, between which there are intervals. From which opinion follows an absurdity, namely that the definition of a line, according to the one who advanced this opinion and also according to others who are subtle in learning, is a magnitude having one dimension, namely length without width, whereas a ray is the impression of luminous bodies in dark bodies, denoted by the name "light" because of the alteration of accidents produced in the bodies receiving the impression. Therefore a ray is both the impression and that in which the impression is. However, the impressing body has three dimensions—length, width, and depth.³⁸

Thus on account of their nature as impressions made by three-dimensional bodies, rays cannot be considered one-dimensional lines. Moreover, if rays were one-dimensional, they would be imperceptible, "for everything that is perceived [either] has width or is conveyed by that which has width. But a line does not have width. Therefore a line is not seen, nor what is conveyed by it…."³⁹

There is a bit of shifty reasoning in this argument, which must be elucidated. Alkindi begins with the one-dimensionality of Euclidean visual rays, which he then opposes with a discussion of the three-dimensionality of luminous rays: the shift to luminous rays is evident not only from Alkindi's reference to the "impression of luminous bodies," but also from his argument that if rays were one-dimensional they would be imperceptible. It is only luminous rays that are perceptible or imperceptible; visual rays perceive or fail to perceive. Therefore it is once again impossible to make sense of Alkindi's argument unless we assume that implicit to it is belief in the identity of luminous and visual radiation. This identity having been supposed, the visibility of luminous rays does indeed speak against the one-dimensionality of visual rays.

However, this argument is followed immediately by another restricted entirely to visual rays. If that which proceeds from sight to perceive objects is an infinity of lines having no width (separated by an infinity of intervals), then these lines terminate in points having no part. Since the rays are in contact with the visible object only in a point, they are capable of perceiving only a point. "But a point is not perceived, since it possesses neither length nor width nor depth; and what lacks length and width and depth is not perceived by sight.⁴⁰ Therefore these lines [i.e., the visual rays] perceive what is not perceived, which is … a very horrendous absurdity."⁴¹ It must therefore be argued to the contrary: since visual rays perceive points (which, since they are perceived, must in reality be small areas), and since rays perceive only that on which they fall, they must have width as well as length.

Alkindi has thus far denied that visual rays are devoid of width. In a further argument, deriving from the nature of the eye and the visual power, he demonstrates that visual rays are not only three-dimensional, but constitute a single continuous radiant cone:

If the parts of the instrument of sight [the eye] are continuous, i.e., of one substance, then the visual⁴² power is in the whole instrument. What then is it that forms the cone into lines, since the instrument impressing it is a single continuous thing, in which there are no intervals, and the [visual] power would not be in certain places and not others? If part of the instrument impressed a ray in the air and part did not, then the power, having two parts, is diverse; for a power of one part produces one effect. But if the power of the instrument is one and its whole consists of one substance [which Alkindi obviously believes to be the case], then it produces one impression in that in which it makes an impression, and not two, one of which would be the line of the ray and other not.⁴³

However, the cause of discontinuous radiation might be in the medium rather than the source. If this were the case, "then the air would consist of lines of two different substances, some of which admit light and some of which do not."⁴⁴ But that air should "consist of various lines, fixed, unmoved, and not overflowing [into one another], and that the extremity of each of those lines should be fixed over a point, which would be sought by the center of the instrument of sight of each observer until it touches it, … is very unseemly, and all who hear of it would laugh."⁴⁵

The continuity of visual radiation can be established not only from the nature of the eye and the medium, but also from the phenomena of sight. Let us suppose, Alkindi asks, that Euclid did not mean to assert that rays are geometrical lines, that what he meant was only that three-dimensional pencils (separated by intervals) move along straight lines or with the rectitude of straight lines. Even this latter view falls into absurdity, for it predicts that we will observe only those portions of the visual field on which visual rays fall; between them will be blank spaces where no ray is present. In short, we will receive a spotted impression of the visual field. But what we actually observe is something quite different: we see clearly only that which is directly opposite the center of the eye, and as objects are increasingly elongated from opposition, they are seen with decreasing clarity. Therefore the opinion that the visual cone is composed of discrete pencils of radiation violates the teachings of sense and "is worthy of much derision."⁴⁶ It is not Euclid himself, however, who is worthy of derision, but rather certain opinions attributed to him. For there is still another possible interpretation of his position that will altogether exonerate him from the charge of error.⁴⁷ Euclid must have meant, Alkindi argues, that although radiation itself is continuous, its overall shape can be defined by an infinity of discrete geometrical lines, because "the boundaries [i.e., the lateral surface] of the cone-shaped figure impressed in the air by the visual power proceed with the rectitude of straight lines separated by intervals."⁴⁸

For Euclid's followers, however, Alkindi has only harsh words. One of them, in the attempt to save Euclid from the absurdity of discrete, one-dimensional rays, has led him into equal or worse absurdity. This interpreter, whom Alkindi does not identify,⁴⁹ maintains that there is a visual cone consisting of rays separated by intervals, but concedes (in order to avoid the absurdity of discrete rays) that only the central ray of the visual cone, the axis, perceives objects. Alkindi's response is that if visual rays (other than the axis) are neither perceived nor able to perceive, there is no way of ascertaining their existence, and it is as absurd to postulate the existence of a visual cone for which there is no evidence as to suppose that perception occurs through discrete, one-dimensional rays. Thus if one wishes to defend the existence of a visual cone, he is obligated to maintain also that perception takes place through all of its rays.

Euclid's followers are guilty of even more. The same people who maintain that only the axis of the visual cone perceives objects also claim that objects nearer the axis are more clearly perceived than objects far from the axis.⁵⁰ "But if places near the center of the circle [i.e., the base of the visual cone] are perceived, then the ray that perceives visible things is not a single line [i.e., the axis]."⁵¹ Therefore they have contradicted themselves, and the absurdity of their teaching is manifest.

Variations in Sensitivity within the Visual Cone

Alkindi has treated the geometry of the visual cone and concluded that the cone is a continuous beam of radiation, sensitive throughout. But Alkindi's polemic against Euclid's followers has led him to acknowledge variations in the sensitivity of different regions of the cone, which must now be explained. Why should objects near the axis of the visual cone be perceived more clearly than objects on the periphery? Not for the reasons offered by those followers of Euclid who "have regarded themselves as acute teachers in a learned discipline before they have been disciples."⁵² They have argued that the axis of the visual cone is the shortest of all rays,⁵³ and therefore that it perceives most strongly, the assumption being that strength of perception varies inversely as the length of the ray.⁵⁴ The trouble with this explanation, Alkindi argues, is that it ignores several obvious phenomena. An object … on the edge of the visual cone … is closer to the eye than another object on the axis … and yet the latter object is seen more clearly than the former. Indeed, a nearby object at the side of the visual field is not seen as clearly as an object in the orb of the fixed stars on the axis of the visual cone. Therefore the factor that determines clarity of vision is obviously not the length of the ray.

The true cause is discerned through the analogous behavior of light and color. Color is not perceived unless illuminated by light, and the stronger the light the clearer the perception of color. Similarly, the stronger the visual ray, the clearer its perception of color.⁵⁵ But what does it mean for a visual ray to be stronger? Wherein lies its strength? The operation of sight is to effect a conversion or transformation in the surrounding medium—though on the nature of the transformation Alkindi does not at this point elaborate.⁵⁶ A strong ray, then, is simply that which produces a perfect or complete transformation, and a weak ray that which produces an imperfect or incomplete transformation. The axial ray produces the most perfect transformation of the medium and therefore perceives its object most clearly; other rays produce transformations that decrease in perfection with the distance of the ray from the central axis. Alkindi's explanation may raise as many questions as it answers—Why are peripheral rays less capable of transforming the medium than central rays?—but he has at least associated clarity of perception with the object's position in the visual field rather than with its distance from the eye.

However, two propositions later (in Prop. 14) Alkindi takes up the problem again, offering on this occasion a geometrical explanation of variations in the strength of visual radiation and hence in the clarity of visual perception. Here again Alkindi regards the analogy of external illumination as instructive. Just as two candles illuminate the same place better than one, so also places "illuminated" by more visual radiation are more clearly seen….

Thus Alkindi traces the apparent strength of axial rays ultimately to the fact that the central region of the visual field is the recipient of more visual radiation (from various points on the surface of the eye) than is any other region; it is not the case, apparently, that the eye emits rays in various directions that actually vary in strength.

We must take note of several of the consequences of this argument. In the first place Alkindi indicates that radiation is emitted in all directions from every point on the surface of the cornea. This reveals that the part of the eye active in vision is the outermost surface; and, indeed, Alkindi says quite plainly in the passage above that "the part [of the eye] to which belongs the power of comprehending the visible object" is "the aforesaid exterior gibbosity of the eye." Alkindi thus disagrees with Euclid and Damianus, who located the apex of the visual cone within the eye, and with Ptolemy, who also placed the apex within the eye, but, more specifically, at the center of curvature of the front surface of the eye so that visual radiation would emerge only in lines perpendicular to the cornea.⁵⁹

Secondly, Alkindi's argument calls into question the very idea of rays as useful entities for describing the physical process of radiation. I am not, of course, referring to the one-dimensional rays of Euclidean optics or even to three-dimensional, discrete pencils of radiation, both of which Alkindi has already explicitly rejected, but rather to the idea that within the cone of continuous radiation there is a unique, physical correspondence between areas of the visual field perceived and portions of the visual cone or the eye that perceive it—for example, that an object on the axis of the visual cone is perceived by radiation issuing from the center of the eye, while objects at the periphery of the visual field are perceived by radiation issuing from the periphery of the eye.⁶⁰ This entire idea, I say, is here called into question, for Alkindi teaches (in this proposition) that every part of the visual field is "illuminated" by radiation from every part of the eye that has straight-line access to it. It follows that visual radiation does not, after all, proceed along the visual rays of Euclidean or Ptolemaic optics; the single visual cone of Euclidean and Ptolemaic optics no longer represents the physical mode of radiation, but only symbolizes the perception of space geometrically. Or to state the same point in somewhat different terms: if we wish the visual cone to represent the physical process of radiation, there will be not one cone, but many, emanating from every point on the surface of the observer's eye.

I do not know how far along this line of argument Alkindi would have been willing to follow. Since it contradicts what appears to be his clear teaching in other places, I suspect that the answer is, not very far. At this point in his argument Alkindi was attempting only to explain how radiation could vary in strength over the visual field, and there is no reason why the uncomfortable consequences of his explanation must have come to his attention.

One more point must be made regarding Alkindi's explanation of variations in the clarity of perception over the visual field. Alkindi bases his claim that rays issue in all directions from every point on the surface of the eye on the analogous behavior of external light. In fact, the proposition immediately preceding the one that we have been discussing is devoted wholly to the radiation of external light. There Alkindi reveals an explicit understanding of the principle that luminous rays issue in all directions from every point on the surface of a luminous body. He remarks in conclusion, "Therefore we have illustrated how each part of the luminous body illuminates that which is opposite it, namely that to which a straight line can be drawn."⁶² Now such a conception had undoubtedly been implicit to optical theory (at least within certain schools of thought) since its inception, but so far as I have been able to discover, Alkindi was the first to state it explicitly.⁶³ This is one of the most fundamental principles of optics, and we must not permit its present familiarity and self-evidence to obscure its importance. In Alhazen's hands it was to become an alternative to the coherent forms of ancient intromission theories, the very foundation of a new theory of vision; nevertheless, for Alkindi its relationship to vision remained simply that of an analogy, helpful in elucidating the emission of visual rays. Ironically, Alkindi has thus formulated the basis of the new conceptual scheme that would eventually supplant his emission theory.

Alkindi's critique of Euclid's theory of vision is contained in the first fourteen propositions of De aspectibus. The remaining ten propositions pertain to the nature of the visual process in only the most indirect and limited way. Proposition 15 contains a long discussion of the speed with which visual rays are propagated, and the conclusion is that they are propagated instantaneously.⁶⁴ Propositions 16-18 establish the law of equal angles for reflection of both luminous and visual rays from plane, concave, and convex surfaces. Propositions 19-21 discuss physical and geometrical features of vision through a mirror. Finally, Propositions 22-24 treat the effect of distance and of angle subtended by the visible object on perception.

Alkindi as a Galenist

Although De aspectibus is principally a geometrical treatise, on several occasions Alkindi has found it necessary to comment briefly on the physical nature of visual rays. In his attack on the idea of discontinuous radiation Alkindi defines a ray as "the impression of luminous bodies in dark bodies, denoted by the name ' light' because of the alteration of accidents produced in the bodies receiving the impression,"⁶⁵ and he argues that "if part of the instrument [i.e., the eye] impressed a ray in the air and part did not, then the power, having two parts, is diverse."⁶⁶ In his analysis of variations in the sensitivity of the visual cone he argues that the effect of sight is "that it converts that which is opposite."⁶⁷ Finally, in discussing reflection, Alkindi refers to the transformation of the medium provoked by sight as a "resolution."⁶⁸ It is apparent from these remarks that in Alkindi's view the ray is not a substantial entity issuing from the eye—nothing is actually transported from the eye to the visible object—but a transformation of the ambient air produced by the visual power of the eye.

What is important about this position is that it takes Alkindi outside the Euclidean tradition and identifies him with the Galenic or Stoic theory of vision.⁶⁹ Hunain ibn Ishāq, Alkindi's exact contemporary and one of the earliest Arabic authors to expound the Galenic theory, claims that "when [the visual spirit] meets the air in the moment in which it goes forth from the pupil, it transforms it immediately it encounters it, and that which arises from the change runs through it [the air] for a very long distance…. So the change in the air caused by the [action of the] visual spirit penetrates the whole air."⁷⁰ It is precisely this position that Alkindi adopts. I do not wish to argue, however, that Alkindi is a full-fledged Galenist in his theory of vision. Far from it! Alkindi retains the basic geometrical framework of Euclid and Ptolemy and avoids the anatomical and physiological orientation of the Galenists. Moreover, it is clear from his favorable references to Euclid that Alkindi regarded himself as a loyal representative of the Euclidean theory of vision. He has simply given the Euclidean theory a Galenic interpretation at a point where Euclid himself had been entirely silent, namely on the physical nature of visual rays. This is in no sense a violation of the Euclidean theory, but only an extension of it along Galenic lines.

Alkindi's Influence

Alkindi's De aspectibus appears to have exerted a strong influence on Islamic optics. According to the biographies of al-Baihāqi and al-Schahrazūrī, it became a popular texbook.⁷¹ Moreover, the strength of the emission theory among Islamic intellectuals testifies in part to Alkindi's influence. Among those who defended it were al-Fārābī (d. 950-951), 'Ubaid Allah ibn Jibrīl ibn Bakhtyashū' (d. 1058), Ibn Hazm (d. 1064), Nāsir al-Dīn al-Tūsī (d. 1274), al-Qarāfī (d. after 1285), and Ahmad ibn Abī Ya'qūbī (14th c.).⁷² Another, Salāh al-Dīn ibn Yūsuf (fl. 1296), wrote a treatise entitled The Light of the Eyes, in which he developed the emission theory of vision at considerable length; however, he then appended Avicenna's attack on the emission theory and took no firm stand himself on the controversy between the two.⁷³ In the West the emission theory had far less success. Although it was known and discussed, as numerous manuscript copies of Euclid's Optica and Alkindi's De aspectibus attest, nobody adopted it (at least in a pure form) as the true theory of vision.⁷⁴

But Alkindi's influence cannot be measured solely in terms of converts to the emission theory of vision. There were many natural philosophers, particularly in the West, who looked to Alkindi for support in their defense of a combined emission-intromission theory. Grosseteste, an early defender of such a combined theory, was in all likelihood familiar with Alkindi's De aspectibus⁷⁵ and probably had Alkindi in mind when he wrote: "However, mathematicians and physicists [by contrast with natural philosophers], whose concern is with those things that are above nature, … maintain that vision is produced by extramission."⁷⁶ Later in the thirteenth century Roger Bacon argued: "Now it must be considered whether the species of vision is required for the act of sight. And it is manifest that a species issues from sight just as from other things; for since accidents and substances inferior to sight can produce their powers, how much better able is sight";⁷⁷ and as one of his authorities for this position Bacon listed Alkindi. Finally, John Pecham also used the authority of Alkindi to support his contention that rays issue from, as well as enter, the eye.⁷⁸

There were also a number of very specific optical theorems or conceptions bestowed by Alkindi on the subsequent optical tradition. I have already mentioned the principle, first stated by Alkindi, that rays issue in all directions from every point on the surface of a luminous body. Alkindi's own demonstration of this principle was reproduced by Roger Bacon and Witelo down to the very lettering on the figure, and the same demonstration and figure appear as late as 1611 in the Optica of Ambrosius Rhodius.⁷⁹ The first four propositions of De aspectibus, on the shadows cast by luminous and opaque bodies of various sizes, reappear in Bacon's De multiplieatione specierum and Witelo's Perspectiva;⁸⁰ so also do Proposition 14 on the greater strength of axial rays (though turned to other purposes by Bacon and Witelo) and Propositions 17-19 on the law of reflection.⁸¹ The propositions in Witelo's Perspectiva and François Aguilon's Optica on the proportion between the magnitude of a luminous body and the strength of its radiation derive from Proposition 22 of De aspectibus⁸² Finally, Alkindi's views on the speed of propagation of light became a standard part of later discussions of the subject.⁸³

Alkindi's direct influence on Western optics was undoubtedly greatest in the thirteenth century. He continued to be cited in the fourteenth century, but in subsequent times direct references to De aspectibus became infrequent.⁸⁴ However, through Witelo and (to a lesser extent) Bacon, who between them reproduced much of De aspectibus, Alkindi had already entered the mainstream of Western optical thought.

Notes

¹ Ya'qūb ibn Ishāq al-Kindī died about 873. The most useful sources on his life and writings are G. Flugel, "Al-Kindî, gennant 'der Philosoph de Araber'. Ein Vorbild seiner Zeit und seines Volkes," Abhandlungen der deutschen morgenländischen Gesellschaft, 1857, 1 (Pt. 2): 1-54; Lucien Leclerc, Histoire de la médecine arabe (Paris: Clermont, 1876), Vol. I, pp. 160-168; Albino Nagy, "Sulle opere di Ja'qūb ben Ishaq al-Kindī," Rendiconti della Reale Accademia dei Lincei, Classe di Scienze Morali, Storiche e Filologiche, 1895, Ser. 5, 4:157-170; Heinrich Suter, "Die Mathematiker und Astronomen der Araber und ihre Werke," Abhandlungen zur Geschichte der mathematischen Wissenschaften, 1900, 70:23-26; Eilhard Wiedemann, "Ueber das Leben von Ibn al Haitam und al Kindî," Jahrbuch für Photographie und Reproduktionstechnik, 1911, 25:6-11; George Sarton, Introduction to the History of Science (Baltimore: Williams & Wilkins, 1927-1948), Vol. I, pp. 559-560; Francis J. Carmody, Arabic Astronomical and Astrological Science in Latin Translation: A Critical Bibliography (Berkeley: Univ. of California Press, 1956), pp. 78-85; Richard Walzer, "New Studies on al-Kindī," Oriens, 1957, 70:203-232 (reprinted in Walzer, Greek into Arabic: Essays on Islamic Philosophy, Oxford: Bruno Cassirer, 1962); Nicholas Rescher, Al-Kind : An Annotated Bibliography (Pittsburgh: Univ. of Pittsburgh Press, 1964).

² "New Studies," p. 203; cf. Walzer's "Arabic Transmission of Greek Thought to Medieval Europe," Bulletin of the John Rylands Library, 1945-1946, 29:176-178. On Alkindi's translating endeavors, see Leclerc, Histoire de la médecine, Vol. I, p. 161; Flugel, "Al-Kindī," p. 6; Sarton, Introduction, Vol. I, p. 559.

³ Quoted by Walzer, "Arabic Transmission," pp. 172-173.

⁴Ibid., p. 175.

⁵ Flūgel, "Al-Kindî," pp. 20-35, lists 265 works attributed to Alkindi in the Fihrist. On the published works, see Rescher's bibliography.

⁶ No copy of the Arabic text has so far been discovered. The Latin text has been published in Axel Anthon Bjornbo and Sebastian Vogl, "Alkindi, Tideus und Pseudo-Euklid. Drei optische Werke," Abh. Gesch. math. Wiss., 1912, 26 (Pt. 3):3-41; see also the review of Bjôrnbo and Vogl's edition by Alexander Birkenmajer, Bibliotheca Mathematica, 1912-1913, Ser. 3,3, 75:273-280. Bjôrnbo has demonstrated (pp. 146-150) that the translator of De aspectibus was Gerard of Cremona. Latin MSS are relatively numerous and have been carefully described by Bjôrnbo and Vogl; Carmody, Astronomical and Astrological Sciences, p. 79, adds an additional MS to the list.

The Fihrist attributes to Alkindi a number of other optical works, several of which are extant: there are three Arabic MSS (two in Berlin and one in Paris) of an extract of Alkindi's Recension of Euclid's Optics; see Eilhard Wiedemann, "Aus al Kindis Optik," Sitzungsberichte der physikalisch-medizinischen Societal in Erlangen, 1907, 59:247-248. Arabic MSS of Alkindi's treatise on the burning mirror (probably No. 112 in FlugePs list) have recently been discovered (and lost again?); a photocopy of one of them has been published by Mohamed Yahia Haschmi, Propagations of Ray: The oldest arabic manuscript about optics (burning-mirror), from Ya 'kub ibn Ishaq al-Kindi (Aleppo, 1967). Rescher, Al-Kindi, p. 47, lists two other works of marginal optical interest, both available in Arabic editions: "On the Reason for the Azure Color that is Seen in the Air towards the Sky …" and "On the Body which Bears Color Naturally from among the Four Elements …"; the former of these has been translated into English in Otto Spies, "Al-Kindi's Treatise on the cause of the Blue Color of the Sky," Journal of the Bombay Branch of the Royal Asiatic Society, 1937, N.S. 75:7-19.

⁷De aspectibus, ed. Bjôrnbo and Vogl, p. 3: "Oportet, postquam optamus complere artes doctrinales, et exponere in eo, quod antiqui prasmiserunt nobis de eis, et augere, quod inceperunt …, ut de diversitatibus aspectus secundum nostrae possibilitatis mensuram … loquamur." Subsequent citations of De aspectibus will be to this edition.

⁸ Graziella Federici Vescovini, Studi sulla prospettiva medievale (Turin: G. Giappichelli, 1965), pp. 44-47; on the same treatise, see Lynn Thorndike, A History of Magic and Experimental Science (New York: Columbia Univ. Press, 1923-1956), Vol. I, pp. 642-646.

⁹ Quoted by Vescovini, Studi, p. 46: "Manifestum est quod res huius mundi, sive sit substantia sive sit accidens, radios facit suo modo ad instar siderum…. Omne quod habet actualem existentiam in mundo elementorum radios emittit in omnem partem, quo totum mundum replent suo modo."

¹⁰ Thorndike, History of Magic, Vol. I, p. 645, summarizes Alkindi's view of the power of words as follows: "Thus by words motion is started, accelerated, or impeded; animal life is generated or destroyed; images are made to appear in mirrors; flames and lightnings are produced; and other feats and illusions are performed…."

¹¹ The dependence of the doctrine of multiplication of species on Alkindi's De radiis has been claimed (rightly, I believe) by Thorndike, ibid., p. 646, and Vescovini, Studi, p. 40. The large number of Latin MSS (16) of De radiis listed by Carmody, Astronomical and Astrological Sciences, p. 82, gives some indication of its Western influence. I do not mean to suggest, however, that Alkindi was the only or the principal source of the doctrine; nor am I able to demonstrate that Grosseteste or Bacon made firsthand use of De radiis.

¹² Vescovini, Studi, p. 48, n. 58; Richard Lemay, Abu Ma 'shar and Latin Aristotelianism in the Twelfth Century (Beirut: American University, 1962), pp. 47-48: "Homo non est imbutus in philosophia, nisi sit usque quo possit divinare cum scientia impressiones superiores, nee ascendit ad illam scientiam nisi post scientias quadriviales que sunt introitus ad philosophiam, et sunt mathematice." A similar view was later expressed by Alfarabi; see Al-Fārābī : Fusūl al-Madanū Aphorisms of the Statesman, ed. and trans. D. M. Dunlop (Cambridge: Cambridge Univ. Press, 1961), p. 73.

¹³ On the authenticity of the Optica as a Euclidean work, see J. L. Heiberg, Litterargeschichtliche Studien über Euklid (Leipzig: B. G. Teubner, 1882), pp. 129-133; cf. H. Weissenborn, "Zur Optik des Eukleides," Philologus, 1885, 45:54-62. As Heiberg has demonstrated, there is in addition to the genuine version a recension of the Optica prepared by Theon of Alexandria. The Catoptrica attributed to Euclid is probably a later recension of a genuine work, perhaps also prepared by Theon; see Heiberg, pp. 148-153, and Euclide, L'Optique et la Catoptrique, trans. Paul Ver Eecke (Paris: Albert Blanchard, 1959), pp. xxviii-xxix.

¹⁴ Albert Lejeune, Euclide et Ptolémée. Deux stades de l'optique géométrique grecque (Louvain: Bibliothèque de l'Université, 1948), p. 172. I am much indebted to Lejeune's perceptive analysis of Euclid's optics.

¹⁵ More literally, "in the eye"….

¹⁶ Morris R. Cohen and I. E. Drabkin, A Source Book in Greek Science (Cambridge, Mass.: Harvard Univ. Press, 1958), p. 257. The full text of the Optica (Greek and Latin) appears in Euclidis opera omnia, ed. J. L. Heiberg and H. Menge (Leipzig: B. G. Teubner, 1883-1916), Vol. VII. Heiberg's Greek text has been translated into French by Ver Eecke, op. cit., and into English by H. E. Burton, "The Optics of Euclid," Journal of the Optical Society of America, 1945, 35: 357-372. For a critical edition and English translation of the principal medieval Latin version of the Optica, see Wilfred R. Theisen, "The Medieval Tradition of Euclid's Optics," unpublished doctoral dissertation (University of Wisconsin, 1972).

¹⁷ Whereas Euclid assumed the rectilinear propagation of light, others attempted to demonstrate it metaphysically (Hero of Alexandria), ideologically (Damianus), or experimentally (Ptolemy, Alhazen, and Witelo). See Lejeune, Euclide et Ptolémée, pp. 37-41; Eilhard Wiedemann, "Zu Ibn al-Haitams Optik," Archiv für die Geschichte der Naturwissenschaften und der Technik, 1910-1911, 3:24-25, 40-41; Witelo, Perspectiva, published as Opticae libri decem, ed. Friedrich Risner (Basel: Episcopii, 1572), Bk. II, theor. 1, pp. 61-63. Alkindi's demonstration of rectilinear propagation is treated below.

¹⁸ See, for example, Ver Eecke's introduction to Euclide, L'Optique et la Catoptrique, p. xxiv; cf. p. 50, n. 1.

¹⁹ Hirschberg and Mach have claimed that Euclid took no position regarding the direction in which the rays proceed; see Julius Hirschberg, "Die Optik der alten Griechen," Zeitschrift für Psychologie und Physiologie der Sinnesorgane, 1898, 76:327; Ernst Mach, The Principles of Physical Optics: An Historical and Philosophical Treatment, trans. J. S. Anderson and A. F. A. Young (London: Methuen, 1926), pp. 8-9.

²⁰ See Lejeune, Euclide et Ptolémée, pp. 28-29, 62-66, 72.

²¹ Among Euclid's Greek followers were Hipparchus, Ptolemy, Damianus, and Theon of Alexandria. Lejeune's Euclide et Ptolémée is by far the best source on this Euclidean tradition; see also the same author's Recherches sur la catoptrique grecque (Bruxelles: Palais des Académies, 1957).

²² It is astonishing how little attention De aspectibus has received from historians of optics. In 1907 Eilhard Wiedemann discussed the first few propositions in his brief article "Aus al Kindîs Optik." Five years later a critical edition of the Latin text, prepared by A. A. Björnbo, was published along with a free paraphrase by Sebastian Vogl in their "Drei optische Werke," pp. 3-70. The only discussion of any significance to appear in the meantime is in Ch. 3 of Vescovini's Studi sulla prospettiva médiévale, but (as in the rest of her book) Vescovini is there chiefly concerned with epistemological developments rather than with optical theory. As I have already pointed out (n. 6, above), Alkindi did prepare a recension of Euclid's Optica, but De aspectibus is not it.

²³ The influence of Theon's preface is obvious elsewhere in Alkindi's De aspectibus as well. Indeed, there is no way to demonstrate that Alkindi had access to Euclid's Optica except in Theon's recension (although it is demonstrable that both versions were available in Arabic), but this in no way affects my position. It remains that Alkindi's argument is directed toward Euclid, and his reliance on others who had also elaborated or criticized the Euclidean theory, though interesting, is immaterial to the claim that De aspectibus is essentially a critique of Euclid's theory of vision.

²⁴ The identity of luminous and visual radiation seems to have been taken for granted throughout antiquity. It was specifically defended by Hero, Damianus, Theon, and apparently Ptolemy; see Lejeune, Euclide et Ptolémée, pp. 62-66.

²⁵ Nothing in Alkindi's account suggests that he recognized the existence of a penumbral shadow or the difficulty of determining precisely where the penumbra ends and the umbra begins. This, I would suggest, indicates that there was nothing experimental about Alkindi's procedure; he is arguing from the presumed behavior of light as represented in his geometrical diagrams rather than from actual observations.

²⁶De aspectibus, Prop. 7, p. 9: "Dico igitur impossibile esse, quin oculus sua recipiat sensibilia pervenientibus et currentibus suorum sensibilium formis ad eum, quemadmodum plures antiquorum extimaverunt, et sigillentur in eo, aut ab eo procédât virtus ad sua sensibilia, cum qua ea recipiat, aut haec duo sint simul, aut eorum formae sint sigillatae in aère et impressae, et aer sigillet eas et imprimat in oculo, quas oculus comprehendit virtute sua receptibili eius, quod aer in eo impressit lumine mediante." I have translated the Latin recipio as "perceive"; in Alkindi's view, perception was a matter of reception.

²⁷ This argument appeared first in Theon's preface to his recension of Euclid's Optica; see Ver Eecke's translation. It was repeated again by Albertus Magnus in the Appendix to Quest. 22 of the Summa de creaturis, Pt. II, in Opera omnia, ed. A. Borgnet (Paris: L. Vives, 1890-1899), Vol. XXXV, p. 215.

²⁸De aspectibus, Prop. 7, p. 9: "Immo cum circuli et aspiciens in una consistunt superficie, circuli nullo modo videntur. Non ergo restat, nisi ut ab aspiciente ad res, quae aspiciuntur, procedat virtus, qua eas recipiat."

²⁹ On the coherence of forms in the atomistic theory of vision, see Cyril Bailey, The Greek Atomists and Epicurus (Oxford: Oxford Univ. Press, 1928), pp. 406-409; see also Epicurus, "Letter to Herodotus," in Diogenes Laertius, Lives of Eminent Philosophers, trans. R. D. Hicks (London: William Heinemann, 1925), Vol. II, pp. 577-579. The coherence of forms in ancient theories of vision has also been pointed out by Vasco Ronchi, Histoire de la lumière, trans. J. Taton (Paris: Armand Colin, 1956), pp. 31, 38; and Optics, the Science of Vision, trans. E. Rosen (New York: New York Univ. Press, 1957), pp. 24-26. However, as I have attempted to demonstrate in my "Alhazen's Theory of Vision and Its Reception in the West," Isis, 1967, 5S/332-341, Ronchi is quite mistaken when he applies this same conception of forms to medieval optics.

³⁰ Plato and Aristotle speak simply of a motion or qualitative change communicated through a medium to the eye. They do not submit the visual field to analysis; i.e., they do not discuss the interrelations among the various parts of the visual field or a visible object, and hence they never raise the question of coherence.

³¹ See discussion below.

³² See my "Alhazen's Theory of Vision," pp. 322-329.

³³De aspectibus, Prop. 8, p. 10: "virtus a visu procedens cum propter debilitatem in aere penetrare non possit, redire earn facit aer ad corpus hominis aspicientis." Cf. Aristotle, Meteorologica, Bk. III, Ch. 4, 373^b 3-10. The same argument is repeated again by Albertus Magnus, Summa de creaturis, Pt. II, Appendix to Quest. 22, in Opera omnia, ed. Borgnet, Vol. XXXV, p. 215.

³⁴De aspectibus, Prop. 10, p. 12. Cf. Theon's preface, pp. 55-56 in Ver Eecke's translation.

³⁵De aspectibus, Prop. 9, pp. 11-12.

³⁶Ibid., Prop. 8, p. 10: "Plures vero homines, quia videbant radios corporum colores secum defferre et apud eorum fines deponere, sensum non recipere rerum figuras, nisi per cursum earum ad ipsum, extimaverunt. Figurae enim apud eos non sunt nisi colorum fines. Quapropter putaverunt, figuras ad aspicientem currere."

³⁷ Alkindi was surely dependent for his conception of a cone of continuous radiation on Ptolemy, whose Optica circulated widely in Islam. On the Islamic tradition of the Optica, see L'Optique de Claude Ptolémée dans la version latine d'après l'arabe de l'émir Eugène de Sicile, ed. Albert Lejeune (Louvain: Bibliothèque de l'Université, 1956), pp. 28-30; on Ptolemy's theory of the visual cone, see Lejeune, Euclide et Ptolémée, pp. 79-83. Stoic influence also appears likely; on the Stoic theory of vision, see S. Sambursky, Physics of the Stoics (London: Routledge and Kegan Paul, 1959), pp. 23-29; Harold Cherniss, "Galen and Posidonius' Theory of Vision," American Journal of Philology, 1933, 54: 154-161; F. Ogereau, Essai sur le système philosophique des Stoïciens (Paris: Félix Alcan, 1885), pp. 91-94; Rudolph Ε. Siegel, Galen on Sense Perception (New York: S. Karger, 1970), pp. 37-40. On Alkindi's familiarity with Stoic ideas, see Vescovini, Studi, pp. 33-43.

³⁸De aspectibus, Prop. 11, pp. 12-13: "Antiquorum autem quidam estimaverunt, quod radii plures egrediunter ab aspiciente secundum rectas lineas, inter quas existunt intervalla. Quern quidem sermonem inconveniens sequitur, quod est, quia lineae diffinitio apud eum, qui hune protulit sermonem, et apud alios, qui in doctrinalibus sunt subtiles, est magnitudo habens dimensionem unam, longitudinem videlicet sine latitudine. Radius vero est impressio corporum luminosorum in corporibus obscuris, a lumine nominis denominata propter conversionem accidentium delatorum in corporibus illam impressionem recipientibus. Impressio igitur cum eo, in quo est impressio, simul est radius. Imprimens autem corpus est corpus tres habens dimensiones, longitudinem et latitudinem et profunditatem."

³⁹Ibid.: "omne enim, quod sentitur, habet latitudinem, aut in eo, quod habet latitudinem, defertur. Linea vero latitudinem non habet. Non ergo videtur linea, neque quod linea defertur…."

⁴⁰ This point had been very clearly stated by Aristotle in De sensu, Ch. 7,449^a 21-22.

⁴¹De aspectibus, Prop. 11, pp. 13-14: "Punctum autem non sentitur, quoniam longitudinem non habet neque latitudinem neque profunditatem. Quod autem longitudine et latitudine et profunditate caret, non sentitur visu. Hae ergo lineae sentiunt, sed quod non sentitur. Et hoc … est magis horrendum inconveniens."

⁴² The Latin term visibilis can mean "visual," i.e., having the capacity to perceive by visual means, as well as "visible."

⁴³Ibid., p. 14: "Si autem etiam partes instrumenti visus sunt continuae, scilicet ex una substantia, tune virtus visibilis est in toto instrumento. Quae ergo est causa, quae fecit pinealem lineas, cum instrumentum earn imprimens sit unum continuum, in quo non sunt intervalla, et virtus non sit in quadam absque alia? Si ergo pars instrumenti imprimit in aere radium, et pars non imprimit, tune virtus duarum partium est diversa. Virtus enim unius unum perficit opus. Quod si virtus instrumenti est una et totum eius ex una existit substantia, ergo unam perficit impressionem in eo, in quo impressionem facit, et non duas impressiones, quarum una sit linea radii et altera non sit radii."

⁴⁴Ibid.: "ergo aer erit lineae in duabus diversis substantiis, quarum quaedam recipiunt lumen, et quaedam non recipiunt…."

⁴⁵Ibid.: "quod aer inundans sit lineae diversae stantes, non motae et non inundantes, et quod extremitas cuiusque illarum linearum sit fixa supra punctum, et punctum illud inquirat centrum instrumenti visus cuiusque aspicientis, donee ipsum contingat, … est valde turpe, et de quo omnis, qui audierit, rideat." I have altered Björnbo's punctuation slightly.

⁴⁶Ibid., p. 15: "valde deridendus est."

⁴⁷ Alkindi thus reminds us once again of his reverence for the philosophers of antiquity. The full passage is worth quoting: "However, it is essential that we not judge evil of the opinion of this man, nor compare it to error, because we know the excellence of his rank in this art and because he himself is one of the causes of our knowledge of good things; but we would think good things of him and convert his opinion to good purpose, since this [possibility] lies open to us" (ibid.).

⁴⁸Ibid.: "figurae pinealis in aere a virtute visibili impressae fines secundum rectitudinem linearum rectarum tendunt, inter quas sunt intervalla."

⁴⁹ Perhaps Damianus; see the following note.

⁵⁰ Damianus argued that the visual cone consists of discrete rays and also that vision through the axis is clearest; see Richard Schöne, Damianos Shrift über Optik (Berlin: Reichsdruckerei, 1897), pp. 3, 9, 11. Although the position ascribed by Alkindi to his unnamed opponent is not identical to this, I cannot find anybody else to whom Alkindi's accusations would apply as well, and I would thus suggest that Damianus was the probable object of Alkindi's attack.

⁵¹De aspectibus, Prop. 11, p. 17: "Quod si loca praeter centrum circuli comprehenduntur, ergo radius comprehendens visibilia non est linea una."

⁵²Ibid., Prop. 14, p. 23: "qui putant, se esse doctores acutos in doctrina, ante quam sint discipuli in disciplina eruditi."

⁵³ This is true, of course, only if the visible object lies in a plane perpendicular to the axis of the cone.

⁵⁴ This argument occurs in Prop. 12.

⁵⁵ Vogl, in his paraphrase of this proposition, reveals a misunderstanding of Alkindi's point: he supposes that Alkindi assigns two causes for clarity of vision, one on the part of the external illumination and the other on the part of visual rays; see Bjôrnbo and Vogl, "Drei optische Werke," p. 54.

⁵⁶ Alkindi limits himself to the following remark: "Therefore if it is known per se by them [i.e., certain of Euclid's followers] and by us and by all learned men what the effect of sight is, that it converts that which is opposite …" (De aspectibus, Prop. 12, p. 19: "Si ergo per se notum est apud eos et apud nos et apud omnes doctrinales, quod effectus visus est, ut convertat id, quod obviat…"). However, later in De aspectibus, in connection with a problem of reflection, Alkindi touches once more on the transformation of the medium, referring to it as a resolution: "A difficulty arises from the claim that air is affected by sight suddenly, for if sight suddenly resolves (resolvit) all of the air opposite the eye, why is that which is reflected [also] resolved, since it is not opposite the eye, but only opposite the mirror?" (Ibid, Prop. 20, p. 34: "Ex hoc autem, quod aer a visu subito pati dicitur, oritur difficultas. Si enim visus subito resolvit totum aerem, cui videns occurrit, quare ergo, quod conversum e t, resolvitur, cum ipsum non obviet videnti, neque occurrat nisi speculo.") The Latin terms resolutio and resolvere denote a relaxation, dilation, dispersion, or separation into component parts; but unfortunately neither Alkindi nor any subsequent medieval commentator reveals how we are to apply this conception to the process of vision. Avicenna confesses his ignorance of the effect of sight on air, according to emission theorists, when he writes: "If only the effect of sight on air were known!" (De anima, Bk. III, Ch. 5, in Opera philosophica, Venice: Bonetus Locatellus, 1508, fol. 13^r: "Utinam sciretur quid efficiat visus in hoc aere!"). See his analysis of the emission theory, of which this quotation is only a minute part.

Nor does an examination of ancient theories of vision shed much light on Alkindi's intention. The extant portions of Ptolemy's Optica, for example, contain no conception of visual or luminous radiation that could be construed as a relaxation or dispersion. Vogl asserts, in a note to his paraphrase of De aspectibus (p. 65, n. 2), that according to Plato light dilates the air, resolving it into its component parts; but, in fact, the affect of visual light on air in Plato's theory is one of fusion rather than of resolution (although Plato does admit that visual light can be dilated or dispersed by smaller particles of external light; see Timaeus,67^d). The same can be said for the Stoic theory, in which a tension is produced in the air by the efflux of visual pneuma….

⁵⁹ On Euclid and Damianus, see Euclide, L'Optique et la Catoptrique, trans. Ver Eecke, p. 57, n. 1; Lejeune, Euclide et Ptolémée, pp. 55-56. On Ptolemy, see Euclide et Ptolémée, pp. 51-55.

⁶⁰ Ptolemy clearly held such a view: because visual rays issue from the center of curvature of the eye, they emerge without refraction and progress to the object of sight, thereby producing a one-to-one correspondence between points in the visual field and points on the surface of the eye; moreover, this correspondence is a matter not merely of mathematics, but of physical process, for it is the ray or energy issuing from a given point on the surface of the eye that perceives directly opposed points in the visual field. See Lejeune, Euclide et Ptolémée, pp. 54-55….

⁶²De aspectibus, Prop. 13, p. 23: "Iam ergo exemplificavimus, qualiter quaeque pars corporis luminosi illuminet, quod ei obviat, scilicet a quo est possible, ut ad ipsum producatur linea."

⁶³ Surely one can perceive faint foreshadowings or implicit applications of this principle in a number of ancient sources; see, e.g., Lejeune, Euclide et Ptolémée, p. 70; Pseudo-Euclid, Catoptrica, Prop. 30, in Euclide L'Optique et la Catoptrique, trans. Ver Eecke, pp. 122-123. But it was Alkindi who first gave the principle clear, explicit, and extended articulation. Moreover, many of those who stated this principle during the next seven centuries followed Alkindi's exposition closely and also reproduced his geometrical diagram….

⁶⁴ Instantaneous propagation would seem necessarily to have implications for the physical nature of visual rays. However, Alkindi has already asserted that radiation is a change of state of the medium, produced by the visual power, and therefore the claim that propagation is instantaneous does not raise such thorny questions as whether a material entity can be at two different places simultaneously. The only point relevant to the nature of the visual rays made in Prop. 15 is that since the conversion or change of state of the medium must occur instantaneously from one end to the other, it is not the case that the visual power affects the part of the medium nearest the eye, which part in turn affects the part adjacent to it, and so on; nor is it the case that the visual power affects the part of the medium nearest the eye, then the next part of the medium, and so on in temporal sequence. It should be noted that since visual rays are propagated instantaneously, they issue from the eye, not in the sense that they are present first at the eye and later at the object of vision, but only in the sense that the eye is the source of the rays and the active member in the visual process.

⁶⁵ See n. 38 above.

⁶⁶ See n. 43 above.

⁶⁷ See n. 56 above.

⁶⁸Ibid.

⁶⁹ It is a pity that we lack the first book of Ptolemy's Optica and hence any firm knowledge of his theory of the nature of visual radiation. However, by Lejeune's reconstruction (Euclide et Ptolémée, pp. 62-71), Ptolemy was not Galenic or Stoic in this regard.

⁷⁰The Book of the Ten Treatises of the Eye ascribed to Hunain ibn Is-hâq (809-877 A.D.) (Cairo: Government Press, 1928), p. 33. For Galen's similar view, see Siegel, Galen on Sense Perception, p. 75.

⁷¹ See Wiedemann, "Ueber das Leben," pp. 6-7; Max Meyerhof, "Die Optik der Araber," Zeitschrift für ophthalmologische Optik, 1920, 5:20.

⁷² On 'Ubaid Allah, see Max Meyerhof, "An Arabic Compendium of Medico-Philosophical Definitions," Isis, 1928, 10:347-348; on the others, see the same author's "Optik der Araber," pp. 21-22, 24-27, 52-54, 89, 90. Several of these authors presented differing views on the theory of sight in different works. Al-Fārābā, although accepting the theory of visual rays in his De scientiis, defended the Aristotelian theory of vision in his The Model State; see Alfarabi, Catálogo de las ciencias, ed. and trans. Ángel González Palencia (Madrid: Maestre, 1932), pp. 44-45, 100, 149-150; Friedrich Dieterici, Der Musterstaat von Al-Fārābī (Leiden: E. J. Brill, 1900), pp. 70-71. Similarly, Nāsir al-Dīn al-Tūsī, while adopting the emission theory in his answer to al-Qazwīnī regarding Avicenna's theory of the influence of heat and cold on the colors of dry and moist bodies, admits in his recension of Euclid's Optica that in fact sight occurs through intromission; see Meyerhof, "Optik der Araber," p. 53; Sarton, Introduction, Vol. II, p. 1009; Eilhard Wiedemann, "Ueber die Enstehung der Farben nach Nâsir al-Dîn al-Tûsî," Jahrb. Photog. Reprod., 1908, 22:88; Wiedemann, "Ueber die Reflexion und Utnbiegung des Lichtes von Nâsir al-Dîn al-Tûsî," Jahrb. Photog. Reprod., 1907, 27:39. I omit Galenists from this list, although they too may represent the influence of Alkindi.

⁷³ See the German translation of Salāh al-Dīn's work in J. Hirschberg, J. Lippert, and E. Mittwoch, Die arabischen Augenärzte, Pt. II (Leipzig: Von Veit, 1905), pp. 207-230.

⁷⁴ There were also numerous questions written in the Latin West during the Middle Ages devoted to the discussion of visual rays. See, e.g., Albertus Magnus, Summa de creaturis, Pt. II, Quest. 22, Appendix ("Utrum visus est per emissionem radiorum") (in which Alkindi is frequently cited), Opera omnia, ed. Borgnet, Vol. XXXV, pp. 215-228; Pseudo-Grosseteste, Summa philosophie, Bk. XIX, Quest. 14 ("Disputatio subtilis utrum videamus intussuscipientes, an extramittentes, an utroque modo"), Quest. 15 ("De positione radiorum visualium"), Quest. 16 ("Quod secundum Platonem et alios visus fiat per emissionem radiorum visualium"), in Die philosophischen Werke des Robert Grosseteste, ed. Ludwig Baur (Beiträge zur Geschichte der Philosophie des Mittelalters, Vol. IX, Münster: Aschendorff, 1912), pp. 499-504; Rudolph of Erfurt, Questiones de anima, Bk. II, Quest. 18 ("Utrum visio fiat per emissionem radiorum"), Cracow, Biblioteka Jagiello ska, MS 753, fols. 32^V-34^r, and MS 635, pp. 306-309; Anonymous, "An visio fiat per extramissionem," Erfurt, Wissenschaftliche Bibliothek, MS Amploniana Q. 369, fol. 4^r (14th c). The emission theory continued to be debated as late as the 17th century; see the Supplementum of Hugo Cavellus to Duns Scotus' Questiones in libros de anima, Disputatio II, sectio 9, dubium 5 ("Quod est organum visus, et an videat per extramissionem?"), in Duns Scotus, Opera omnia, ed. Lucas Wadding (Paris: Laurentius Durand, 1639), Vol. II, pp. 619-620. I owe several of these references to Nicholas Steneck.

⁷⁵ See A. C. Crombie, Robert Grosseteste and the Origins of Experimental Science 1100-1700 (Oxford: Clarendon Press, 1953), p. 117, n. 2.

⁷⁶De iride, in Grosseteste, Philosophischen Werke, ed. Baur, p. 73: "Mathematici vero et physici considerantes ea, quae sunt supra naturam, … dicunt visum fieri extramittendo."

⁷⁷The Opus Majus of Roger Bacon, ed. J. H. Bridges (London: Williams and Norgate, 1900), Vol. II, p. 49: "Nunc considerandum est, an species visus exigatur ad actum videndi. Manifestum est autem, quod species fit a visu sicut ab aliis rebus, quia accidentia et substantie viliores visu possunt facere suas virtutes, multo magis ergo potest visus."

⁷⁸ See David C. Lindberg, John Pecham and the Science of Optics (Madison: Univ. of Wisconsin Press, 1970), pp. 35, 127-129.

⁷⁹ Bacon, De multiplieatione specierum, Pt. II, Ch. 10, bound with Bridges' edition of the Opus maius, Vol. II, pp. 499-500; Witelo, Perspectiva, ed. Risner, Bk. II, theor. 17-19, pp. 67-68; Ambrosius Rhodius, Optica (Wittenberg: Laurentius Seuberlich, 1611), Bk. I, Prop. 1, pp. 16-17.

⁸⁰ Bacon, De multiplicatione specierum, Pt. II, Ch. 9, pp. 494-496; Witelo, Perspectiva, ed. Risner, Bk. II, theor. 26-30, pp. 71-72.

⁸¹ Bacon, De multiplicatione specierum, Pt. V, Ch. 3, Vol. II. pp. 542-543; Witelo, Perspectiva, ed. Risner, Bk. II, theor. 21, p. 69; Bk. V, theor. 20, 23, pp. 200-202. Rhodius also reproduces the material from Prop. 14 of De aspectibus in his Optica, Bk. I, Prop. 13, pp. 59-60.

⁸² Witelo, Perspectiva, ed. Risner, Bk. II, theor. 6, p. 64; Aguilon, Opticorum libri sex (Antwerp: Vidua et Filii Io. Moreti, 1613), Bk. V, lemma 1, p. 363.

⁸³ See Bacon, Opus maius, ed. Bridges, Pt. V, 1, dist. 9, Chs. 3-4, Vol. II, pp. 68-71 ; Bacon, De multiplicatione specierum, Pt. IV, Ch. 3, p. 528.

⁸⁴ In the 14th century Alkindi was cited by William of Ockham, John Buridan, Themon Judaeus, Nicole Oresme, and Albert of Saxony; see Birkenmajer's review of Björnbo and Vogl's "Drei optische Werke," Bibl. Math., 1912-1913, Ser. 3, 75:277. According to Otto Werner, Zur Physik Leonardo da Vincis (Erlangen: Junge, 1910), pp. 31, 110, Leonardo derived some of his optical ideas from Alkindi. I would suggest, moreover, that the figures on fol. 1^r of Leonardo's treatise on the eye (MS D of the Bibliothèque de l'Institut de France) are derived from one of Alkindi's figures (Fig. 4, above); see Donald S. Strong, "Leonardo da Vinci on the Eye: The MS D in the Bibliothèque de l'Institut de France, Paris, translated into English and annotated with a study of Leonardo's theories of optics," unpublished doctoral dissertation (UCLA, 1967). Finally, Alkindi is cited by Friedrich Risner, Opticae iibri quatuor (Kassel: Wilhelm Wesselius, 1606), p. 2.