Dr. Kim H. Veltman

The Optical Tradition

1. Introduction
2. Ancient Philosophy and Optics
3. Ancient Medicine and Optics
4. Euclid and Optics
5. From Illusion to the Correction of Deception
6. Distance as a Requirement for Sight
7. Distance in Alhazen and Witelo
8. Surveying and Optics
9. Astronomy and Optics
10. Changes in the Scope of Optics
11. Summary


1. Introduction

    Much of Ancient optics is not to be found in optical treatises such as those of Euclid or Ptolemy. Although Aristotle classed optics under geometry, it was partly a philosophical problem. Hence it was discussed in Plato's Timaeus, Aristotle's De Anima and Lucretius' De rerum natura. It was also partly a medical problem. Hence its appearance in Galen's On the Usefulness of the Parts. Moreover, optics was linked with surveying and astronomy. These connections between optics and other disciplines are significant because in the course of time they evolved and gradually transformed the whole scope and content of optics itself. For this reason an outline sketch of their role in the optical tradition will be desireable if we wish to understand the context of Leonardo's optical researches.


2. Ancient Philosophy and Optics

    Aristotle considers the relation of philosophy to optics in the Analytica Posteriora Bk.I.13 (79a 10-15):

As optics is related to geometry, so another science is related to optics, namely the theory of the rainbow. Here knowledge of the fact is within the province of the natural philosopher, knowledge of the reasoned fact within that of the optician, either qua optician or qua mathematical optician.1

    In the following paragraph (79a 17-20) Aristotle refers to optics as one of the "sciences that investigate causes."2 Hence, while the natural philosopher records optical phaenomena, the "optician" is expected to explain why these Lucretius were all, in this sense, opticians and in the optical sections of their philosophical texts they concentrate on three causal problems: (1) how and why the process of vision relates to concepts of matter; (2) why the process occurs through extromission or intromission and (3) how and why the veracity of vision, or its absence, is determined.

    Plato's discussion of optics in the Timaeus is quite short. He begins with analogies between the (cold) fire of sight and the fire of light; between the images of sight by day and the images in dreams by night. Plato distinguishes between different kinds of fire and proceeds to explain reflection in a mirror in terms of a combination of internal and external fire. He believes that sight involves accessory causes not true ones but nonetheless praises it. In a later section he considers colours which, he claims, are different kinds of flame composed of various particles. These in turn yield sensations of different colours.3

    Aristotle's discussion in De Anima opens with the question of the objects of sight. This leads to consideration of basic concepts such as the visible, colour, the transparent, light, the medium and the common sensibles of sight.4 In De sensu Aristotle again begins5 with the objects of sight and common sensibles before broaching the nature of vision; possible links between the five senses and the four elements and hypotheses concerning the ratios of colours.

    Theophrastus devotes much more attention to these questions in his work On the Senses.6 In book one he concentrates on the sensory process outlining a school which holds that vision takes place by similarity between the eye and object (Parmenides, Empedocles, and Plato) and then the opposed school which believes that vision occurs by contrast (Anaxagoras, Heraclitus). In his second book, Theophrastus considers the objects of sight, chiefly colour. He outlines possible links between the four elements and the senses, concluding with a discussion of heavy and light qualities. Lucretius in his On the Nature of Things7 prefaces the optical section with a consideration of the existence and character of both subjective and objective images, the rapidity of their formation and their velocity. He then turns to vision and mental pictures, deceptions of sight and the criteria for their correction with comments on the veracity of the senses.

    This link between philosophy and optics is of great importance because it means that cosmological speculations on the structure of matter are bound up with theories of vision. As a result theories of vision remain a subject for intellectual debate rather than careful observation. This leads Galen to attack the Sophists' notions of vision who care "not for truth but only for glory." Nonetheless, the tradition continues and even in the seventeenth century the habit lingers of making lists of conflicting theories of vision in the manner that one lists conflicting philosophical positions.9 In this context, Leonardo's protracted notes on the pros and cons of both extromision and intromission become more comprehensible.


3. Ancient Medicine and Optics

    While Galen is critical of the habits of some "practical physicians who call themselves oculists," he himself clearly represents a practical school also interested in theory. The section on the eye in On the Usefulness of the Parts10 begins with the question why the eyes are where they are and leads to a description of various parts of the eye such as the crystalline lens, the choroid membrane and the iris. Akin to Plato, Galen believes that the essence of the faculty of sight is of the nature of light. He gives examples to show that excessive light hurts the eye, that a lesser light is overcome by a greater one and that one eye increases when the other is closed - all themes that are discussed by various mediaeval authors and by Leonardo who, studied Galen, according to Vasari.

    Following an excursus on the motions of the eye and eyelids, Galen considers geometrical aspects of the visual process itself: the visual rays, the cone of vision, monocular vision, binocular vision, plus an explanation why the crystalline lens is round. It is striking that Galen is very reticent to use geometrical explanations on the grounds that most people pretending to some education not only are ignorant of this but also avoid those who do understand it and are annoyed with them.11 Indeed, Galen insists that it is "only in obedience to the command of a divinity12 that he has used geometry at all. This is important because it suggests that medical, mathematical and philosophical explanations of the eye traditionally existed independently of one another, with no attempt at synthesis. This habit continued throughout the Mediaeval period and helps us to understand how even Leonardo, who aimed at synthesis, could alternately use medical, mathematical and philosophical approaches to optics and often not compare the results.


4. Euclid and Optics

    Euclid's Optics represents a tradition quite different from that of the philosophers and physicians mentioned above. This Euclidean tradition is presumably what Aristotle had in mind when he wrote in the Physica Bk.II.2 (194a 6-11):

Similar evidence is supplied by the more physical of the branches of mathematics, such as optics, harmonics, and astronomy. These are in a way the reverse of geometry. While geometry investigates physical lines but not qua physical, optics investigates mathematical lines, but qua physical not qua mathematical.13

    Although couched in geometrical terms Euclid's Optics deals, however, with what would in our day be termed psychological optics: its prime concern being subjective appearances and optical illusions. Debates whether vision occurs through intromission or extromission do not interest Euclid. It is likely that these debates were the domain strictly of the philosophers and that those writing on mathematical optics, such as Euclid, had other concerns, as we learn from a passage in Hero's Definitions:

Optics does not deal with physical questions and does not study whether given rays flowing out from the eyes go forth to the boundaries of objects or whether images that are detached go forth from corporeal objects [and] enter the eye along a rectilinear path or whether the intervening air is stretched or contracted by the ray-like pneuma from the eye. It is only concerned whether, at each reception (of an image) the right direction of movement or tension is maintained as well as the requirement that the convergence to a point occurs at an angle when objects are seen that are larger or smaller than the eye.14

    Upon reflection it becomes clear that Ancient optics was not one discipline but at least four which tended to appear in various combinations (Chart 1).

Profession/Discipline Type of Optical Problem/Question
1. Natural Philosophers ‘What’ of optical phenomena, illusions
2. Philosophical Opticians ‘How/why’ in terms of (meta-physical and cosmological structure
3. Mathematical Opticians ‘How’ in terms of mathematics
4. Medical Opticians/Oculists ‘How’ in terms of physical structure

Chart 1. List of different disciplines and their respective optical interests.

    There was often interplay between some of these, e.g. Aristotle, Lucretius (1,2); Galen (3,4); Ptolemy (1,2,3,). Alhazen in the eleventh century is among the first to study all four disciplines together. If we examine the structure of Euclid's treatise more closely (see charts 2, 4) we discover that of the 57 theorems there are four that deal with surveying and stand apart from the rest. As Theisen has rightly noted: " Their inclusion in the work is . . . most significant, since these propositions add a quantitative dimension to what is otherwise a purely qualitative work on vision."15 In Euclid's text the surveying propositions appear simply to be interjections without theoretical justification. This changed, and the seeming accidental link between optics and surveying gradually assumed great significance. To understand this will require an excursus on two basic changes within optics generally: one involving the role of illusions; the second, objects of sight. We shall consider each of these in turn.


5. From Illusion to the Correction of Deception

    Plato's famous attacks on vision/optics due to the deceptiveness of sight introduced a tradition that emphasized the fallibility of optics in particular and the senses in general. There is evidence, however, that the Platonists may have been more concerned with the problem of how one gets beyond the deceptions of vision, than with the deceptions as such. Sextus Empiricus in his Against the Logicians claims to "set forth the Academic tradition from Plato down"16 and refers to careful distinctions made in the school of Carneades between different kinds of vision, in terms of three categories.

Quantitative Problem Qualitative Deceptions, Illusions Proposition
Distinctness, visibility , invisibility 1-3
Size 4-8
Position 9
Relative size with movement 10-14
Size, depth, length 18-21
Amount of shape at a given position 22-23
Shape with movement 24
Size relative to position 25-36
Sizes moving relative to one another
or one object moving relative to eye 50-55
Changes in size and illusion of movement 56
Changes in position 5

Chart 2. Survey of quantitative, qualitative themes in Euclid's Optics.

    Logicians claims to "set forth the Academic tradition from Plato down"16 and refers to careful distinctions made in the school of Carneades between different kinds of vision, in terms of three categories. A first category involves things seen that are evidently false: A second category, those which are apparently true.17 This second category is subdivided into three groups: (1) the probable presentation, (2) the probable and irreversible presentation and (3) the presentation that is "at once probable and irreversible and test."18 This final group requires the certification of all the factors in the visual process, namely:

the subject that judges the object and the object that is being judged and the medium through which judgment is effected and distance and interval, place, time, mood, disposition, activity.19

    Hence within the very school famous for its attacks on the deceptiveness of vision/optics emerged a set of criteria for overcoming such deceptions and certifying the veracity of sight. Perhaps such a quest is also implicit in Euclid's analyses of illusions in the Optics. If so the Optics was ultimately a manual for getting beyond deception. In Lucretius' On the Nature of Things this ideal is more apparent. Lucretius acknowledges the existence of deceptions but, nonetheless, refuses to impugn the veracity of sight:

And yet in this we don't at all concede
That eyes be cheated
Tis after all the reasoning of the mind
That must decide, nor can our eyeballs know
The nature of reality. And so
Attack thou not this fault of mind to eyes
Nor lightly think our senses everywhere
Are tottering.20

    In his list of stock deceptions that follows there is the implicit assumption that experience allows one to see through a deception as when he mentions that

to gazers ignorant of the sea
Vessels in port seem, as with broken poops
To lean upon the water, quite agog.21

    Here the illusion is only caused by lack of familiarity. Through experience one can get beyond deception. That the aim of optics is to give an explanation for illusions is stated specifically by Geminus as reported by Proclus: "optics . . . explains the illusory appearances presented by objects at a distance, such as the converging of parallel lines or the rounded appearance of square towers".22 The passage goes on to discuss two further branches of optics: catoptrics, concerned with the reflection of light and scenography, concerned with assuring that drawings of objects will not be seen as disproportionate or shapeless when seen at a distance.

    A similar classification is found in Hero of Alexandria's Definitions. He too describes optics as a discipline implicitly concerned with illusions; catoptrics, concerned with reflection, mirrors, rainbows, shadows and then scenography, concerned with the painting of buildings. Since things are not what they appear, claims Hero, one must not draw things as they are but as they appear.23 Aulus Gellius, in the Attic Nights provides us with yet another source which confirms that the aim of optics was to explain optical illusions.24

    In Ptolemy's Optics the theme of the basic veracity of vision under proper conditions is continued. Ptolemy is very much aware, however, that the eye can be deceived and hence devotes practically the whole of Book two to these questions of deception. 25 But Ptolemy's underlying concern is to know why the deceptions occur. He also wishes to know at precisely what point in the visual process a deception occurs: be it due to the object's distance or position; to manipulation by the actual eye or due to the mind.26

    Ptolemy also cites examples from everyday experience to illustrate his categories of deception. But he then proceeds to introduce experiments designed to determine when, for instance, we see an image as double and when we see it as normal.27 Hereby, the process of getting behind the deceptions becomes testable. In so doing, Ptolemy has expanded the scope of optics. For whereas Euclid had restricted his treatise to the mathematics of subjective visual appearances, Ptolemy goes beyond an actual description of appearances and seeks to identify the conditions in/by which vision can inform us about objective elements of the measured world.

    Nemesius, carries Ptolemy's approach further in spirit if not in detail. Nemesius emphasizes the role of memory and thought in vision and uses this to defend the veracity of vision: "When then we suppose a wax apple to be a real apple, it is not sight that errs, but thought.28 He proceeds to give examples of deceptions owing to a lack of proper conditions such:

as when someone sets out to meet a friend, meets him and walks right past him, because his thoughts are on other matters. But this is not really a failure of sight as much as mind. For sight saw and gave notice, but mind would not attend to the notice given.29

    This idea, which derives from Aristotle's De sensu 30, recurs in developed form in Macrobius' Saturnalia:

The organ of sight would therefore be of little help without the faculty of reason; this is so true that an oar seen in the water appears broken to us and a polygonal tower appears round to us at a distance. But if reason rectifies these errors, the tower becomes angular again and the oar takes on a straight line. It is by these that we redress so many false impressions which have led the Academics to calumniate the senses, since the senses aided by reason should be counted among the most certain of things, albeit a single sense sometimes does not suffice in distinguishing a species.31

    To illustrate this Macrobius returns to the example of the imitation apple that Nemesius had also used, before concluding that the senses owe their efficacy to reason.

    Alhazen, the great tenth century natural philosopher and mathematician, develops these ideas very considerably. In an approach reminiscent of Ptolemy, Alhazen32 gives detailed attention to the requirements for vision and its objects. He is particularly interested in the criteria needed for the certification of what is seen, sometimes describing experiments akin to those described by Ptolemy33, often providing additional vivid examples, which could have been tested. By Alhazen's time optics is clearly devoted to explaining "what is there," to getting beyond deceptions. Whereas Plato had usually assumed that optics is concerned with describing subjective aspects of what the eye sees, Alhazen is convinced that optics must inform us about objective elements of the physical world of Nature. The great challenge of optics is now increasingly: what are the characteristics of the objects seen, in spite of how they appear? Through the contributions of Ptolemy and Alhazen in particular the realm of optics begins to shift from the debates of philosophical theory to testable predictability of experimental demonstration. What had begun as an objection to illusion has now become a commitment to getting beyond deception. As the aims of vision were being redefined, the requirements and objects of vision were being reconsidered also. Particularly interesting in this regard is the concept of distance.


6. Distance as a Requirement for Sight

    Distance as a requirement for vision can be traced back to Aristotle's De anima:

If what has colour is placed in immediate contact with the eye, it cannot be seen . . . . Hence it is indispensable that there be something in between - if there were nothing, so far from seeing with greater distinctness, we should see nothing at all.34

    Here distance means primarily "lack of contact." The Ancients were also concerned that distance should not be excessive, as Carneades35 noted and as Lucretius illustrated by means of a vivid example:

And when from far away we do behold
The squared towers of a city, oft
Rounded they seem, - on this account because
Each distant Angle is perceived obtuse,
Or rather it is not perceived at all;
And perishes the blow nor to our gaze
Arrive tis stroke, since through such length of air
Are borne along the idols that the air
Makes blunt the idol of the angle's point
By numerous collidings.36

    In such passages distance remains a qualitative prerequisite: a compromise between not in contact and not too far away. By late Antiquity this begins to change. Ptolemy lists distance as one of his distinguishing characteristics37 and he emphasizes the importance of a moderate distance between eye and object. To this end he requires that there be a perceptible proportion between the size of the object and the distance involved:

The eye perceives size accurately when the diameters of the base, which is above the object seen, have a perceptible proportion to our distance from the object, which is the case when the rays containing it are disposed at a perceptible angle at the tip of the pyramid.38

    Unlike Euclid, who had relied solely on a concept of angular size, Ptolemy is convinced that visual angles alone are not sufficient to determine the apparent size and distance of objects and points out that other factors such as the position of the object may also play a significant role.39 Ptolemy does not abandon altogether the Euclidean notion of angular size and yet, in his insistence on the importance of the central ray40, he implicitly establishes a relation between the (measured) size of objects and their (level) distance from the eye.

    Nemesius, in the fourth century, is more extreme in emphasizing the importance of distance: he makes it one of the basic requirements of vision: "Sight needs four chief conditions for clear discernment, unimpaired organs, measured motion, moderate distance and the air clear and light."41 Alhazen, in the eleventh century, makes distance one of six prerequisites for vision, along with position of the object, light magnitude, transparency and density or solidity.42 Later in his treatise Alhazen makes distance one of eight prerequisites of vision, adding time and health of vision to his former list.43 He also repeats the Aristotelian notion that distance in the sense of "lack of contact" is a prerequisite for sight.44 Just how much more distance meant to Alhazen than it did to Aristotle becomes clear, however, when we consider the role of distance as an object of sight.

Distance as an Object of Sight

    In the Ross edition of Aristotle's De anima we read that: "The object of sight is the visible and what is visible is (a) colour...."45 Two words in this translation bear closer attention: "colour" and "object." In today's terminology, colour simply connotes black, white, red, blue, etc. Aristotle defines colour quite differently: "Every colour has in it the power to set in movement what is actually transparent: that power constitutes its very nature".46 Hence colour is, for Aristotle, not just a thing that is passively seen: it is an active agent that is vital to the visual process: colour sets the process of vision in action. The term "object" (of sight) is equally problematic. The original Green To c aTOV** literally means "the seen." In other words, Aristotle is saying: the seen is the visible and what is visible is that which sets the process of vision in action, namely colour. Hence colour is essential for Aristotle because it activates the visual process.

    In Aristotle's student, Theophrastus, a shift in interpretation is evident when he writes concerning the things seen: motion, distance and size are visual objects and yet produce no image."47 This idea that distance is among the things seen, and is an object of sight is not mentioned by Ptolemy who lists instead: "body, magnitude, colour, shape, position, movement and rest."48 Whereas, Ptolemy cites seven things seen, Nemesius in the fourth century goes on to mention twenty-one, including distance:

Vision operates along straight lines and in the first place perceives colours. Along with the colour, it recognizes the body so coloured, its size, its shape, relative position and distance away, together with the number of its parts, whether it is in motion or still, whether it is rough or smooth, even or uneven, sharp or blunt; as well as its constitution, whether, say, it is watery or earthy, moist or dry.49

    Nemesius acknowledges the Aristotelian view that the seen is colour but proceeds to qualify this claim: "But hard upon colour follows perception of the body possessing the colour, the position in which the thing seen may chance to be and the space or distance between the person seeing and the object seen."50 Hence body, position and distance are now objects of sight. Each of these he describes, ending with distance:

Sight, on the other hand, can operate also from a distance. And since it receives its characteristic impression across an intervening space, it necessarily follows that sight by itself can recognize the distance of its object, and, likewise, the size of its object, provided that the object can be apprehended in a single glance.51

    If the starting point of Nemesius' approach is clearly Aristotelian, his interpretation of the concepts is basically different. Colour is, for him, no longer something that stimulates the visual process: it is merely something such as black, white, red or blue. Colour's special role in the visual process is thereby lost, and Nemesius is, therefore, led to the obvious conclusion that other things such as distance should be included among the objects of sight. Hereby the way is set for surveying, which measures distance, to assume a central role in optics. That which Nemesius mentions in the fourth century, Alhazen explores in detail in the eleventh.


7. Distance in Alhazen and Witelo

    A study of Ptolemy's Optics prompted Alhazen to write his Doubts on Ptolemy, in which he notes that instead of seven, there are twenty-two objects of sight52, which he then includes in his Optics: light, colour, distance, position, body, figure, magnitude, continuity, discreteness, separateness, number, motion, rest, roughness, lightness, transparency, thickness, shadow, obscurity, beauty, ugliness, similarity.53 In this list, distance comes directly after light and colour. How important this concept is for him becomes clear when we turn to the second book of his great optical treatise: De aspectibus. Here, he begins with a careful distinction between distance, which can be quantitatively measured, and mere "lack of contact," which is qualitative.54 Alhazen proceeds to discuss the mind's role in vision: it is, he claims of great importance in the certification of distance55 (II:24). If there are a continuous number of ordered bodies the mind begins by determining the distance of one that is fairly close and on the basis of this moves onto the next one, thereby certifying distance as it goes along56 (II:25). If the distance be moderate then the position (situs) of the object can also be determined (II:26)57 and its location (locus) can, in turn, be deduced from its position, provided that moderate distances are involved (II:27).58

    How one discriminates between two kinds of position is now mentioned. A direct position is indicated when the distance from the eye to the extremities on either side is equal. An oblique position is indicated whenever the distance from the eye to the two extremities is not equal (II:28).59 Alhazen points out that if the distance of the objects be extreme, then the eye does not certify their position, with the result that even obliquely positioned things seem as if they were facing the viewer (II:29).60 He claims that the various parts and boundaries of the objects seen, as well as the position of the separate objects, all depend on whether the lines leading to the extremities are equal or unequal in distance (II:30).61

    Alhazen now mentions the different requirements for perceiving a body accurately: sometimes the eye alone is sufficient, sometimes it requires judgment. If the body be too far one cannot be certain at all (II:31).62 Beginning with circles, Alhazen discusses how various shapes are perceived (II:32).63 He then studies how perception of a convex surface is determined (II:33)64 and how this differs from a flat one (II:34)65 which leads in turn to the problem of how we perceive a plane surface at a moderate distance. There follows a consideration of how we perceive the size (magnitude) of objects. This, Alhazen admits, is a matter of debate. There are some who think it depends on the visual angle. Others say it depends on a comparison of this visual angle with the actual distance involved. But neither of these explanations will do66, he claims, and he proceeds to show why they are inadequate, including amidst his arguments a convincing test (fig. 1):


Fig. 1 Diagram from Alhazen's optical treatise to refute the Euclidean theorem of visual angles.

Now if the object seen were one cubit away from the eye and it were then moved until it were two cubits away (i.e. ab. is moved to de) then there will be a great difference between the two angles subtended by the two objects at the eye (i.e. /bca. and /dce.) and yet the eye will not apprehend the object two cubits away as being smaller than the object one cubit away. And similarly if it is moved three or four cubits away it will not appear smaller even though the angles at the eye vary immensely.67

    He goes on to compare a direct view from immediately above the square with various oblique views as the distance increases. What is important in Alhazen's description is the way that he relates distance to vision in terms that can readily be tested experimentally. The actual mention of cubits of measurement indicates how close optical theory has come to the problems of surveying practice. Alhazen's conclusion to this particular demonstration is that if vision depended solely on visual angles then we could not see a square shape. In short, he realizes that a strict acceptance of the visual angles theory precludes any possibility of constancy in perceptual images (II:36).68

    Having destroyed the "conventional wisdom" he turns to set out his own ideas. He claims that the size an object appears depends on the size of the surface of the eye affected by the image (in quam pervenit forma) as well as the angle of the optical pyramid (II:37).69 This serves, however, to introduce his main point (II:38) that the real (measured) size of an object depends on a comparison of the base of the triangle with the length of the optical pyramid, which we can illustrate in terms of a simple diagram in which apparent size depends on a comparison of ab. (distance) with cd. (measured size70 (fig. 2):


Fig. 2 Author's reconstruction of Alhazen's claim (II:38) that apparent size depends on a comparison of distance with measured size (ab. with cd.).

    The implications of this claim are profound, for Alhazen has hereby introduced the notion of a simple relation between measured size and level distance into his theory of vision which implies, in turn, that the basic principles of surveying are now at one with those of optics.

    Alhazen goes on to present a rough version of the inverse size law. The eye, he claims, will note how the object seen will tend to get smaller as one goes further away and larger as one gets closer. Indeed, experience will show that to the extent an object seen is removed from the eye, to that extent will the location of its form in the eye diminish and the angle which the object seen subtends at the centre of the eye. Alhazen is not content to leave the matter here. Granted he does not go as far as Leonardo who insisted on demonstrating these principles experimentally but he proceeds, nonetheless, to drive home his theoretical concept by making an important appeal to the principle of occlusion which, albeit long-winded, is worth citing at length as an example of his approach to problems:

... And to the extent the visible object is moved further away, and the eye certifies the quantity of its remoteness, to that extent is it comprehended to be larger, e.g. when someone looks at a distant wall which is a reasonable distance from the eye and the eye certifies the distance of this wall and its size and it certifies the quantity of its length. If the person then places his hand between one eye and the wall while closing the other eye he will then find that his hand will occlude a great portion of the wall and he will comprehend the quantity of his hand in that situation and he will comprehend that the quantity of the wall occluded by the hand is far greater than the quantity of hand and the eye will simultaneously comprehend the limits (verticationes) of the radial lines and it will comprehend the angle which the radial lines contain.The eye will then comprehend, therefore, that the angle which the hand and the wall subtend, is the same angle, and then he will also comprehend that the part of the wall occluded by his hand is far greater than his hand. And since this is so, the discriminating faculty (virtus distinctiva) comprehends in this comprehension that the more distant of two visible objects - at different distances, both subtending equal angles - is of greater size. Then if someone averts his eye while he is in that position and he looks upon another wall more remote than this wall and he positions his hand between his eye and that all he will find that what is occluded of the second wall is greater than what is occluded of the first. And if he then looks at the sky he will find that his hand occludes a half of what appears in the sky or a great portion of it. Nevertheless, the viewer will not doubt that his hand is nothing with respect to what is occluded in the sky according to his sense (of sight). From this experience it will, therefore, be determined that the eye does not comprehend the size of an object seen, unless from a comparison of the size of the thing seen with the quantity of its remoteness in comparison to the angle and not just from a comparison of the angle. And if the comprehension of the quantity of the magnitude were simply as a result of the angle then it would have to be that two objects seen at different remotenesses, subtending the same angle at the centre of sight, would appear equal. And this is not so ...71

    To put it simply, Alhazen has launched an open attack on classical optics. Euclid's fourth assumption had been that "things seen under a larger angle appear larger, under a smaller angle appear smaller and unequal equal angles appear equal."72 Alhazen rejects this. Whereas Euclid had accepted the convenience of angular size at the expense of a direct relation between (measured) size and (level) distance, Alhazen chooses, instead, to relegate visual angles to an ancillary role. Nor does he stop here. All that remains now, he claims, is to explain how the eye comprehends the distance of continuous ordered bodies (corpora ordinata continuata) and how it determines their size clearly:

These bodies which are ordered and continuous with respect to the distance of the objects seen are, for the most part, pieces of land and objects which are seen regularly and are always comprehended by the eye. The comprehension of the sizes of these pieces of land stretched out between the viewer and the objects seen can only come from a comparative measurement of the pieces amongst themselves and comparing the remoter parts of the land with closer ones whose size has been certified.

    Alhazen notes the importance of habit and memory in these experiences and explains in greater detail how this comprehension of distance occurs:

Its origin, the extent of which is clearly determined by the organ of vision, is that which is at the feet. This is because this extent which is at the feet is comprehended by the eye and the distinguishing power and the organ of sight determines it clearly through a measurement of the human body. This is because that land which is at the feet is always measured by man, without attention (intentione), by his feet where he walks over it and by his arm when he extends his hand to it. And all land which is close to man is always measured by way of the human body without attention and the eye comprehends that measurement and perceives the same. And the distinguishing power comprehends that measurement and knows it and determines clearly from it the extents of the parts of the land with the human body.73

    There is, therefore, an automatic process of judging distance which beings with the ground at one's feet:

Thus if a man were standing up straight and he looked at the earth at his feet there would be lengths of radial lines in keeping with the extent of the man's height and the distinguishing power would know for certain that the distance between the organ of sight and that piece of land is the size of a man standing up and the length of the locations of the land continuous with the body of the man are known and these perceived distances in the distinguishing power and their shapes come to rest in the mind (anima).74

    In volume one we analysed Leonardo's careful attention on Ms.A 37r to the appearance of furrows of land with respect to his perspectival studies.75 This, we now realize, had a mediaeval precedent. In Alhazen's view the judgment of distance is not merely a vague one: the eye perceives distance on level ground in terms of standardized measurement:

... similarly it (the distinguishing power) obtains from a cube and a palm and from some measured quantity a determined quantity. Hence if the viewer apprehends some space and wants to know how many cubits are in it, he compares the form obtained by his imagination of that same space and compares it with the form his imagination had obtained of a cubit and he would comprehend the quantity of that space with respect to a cubit through this comparison.76This standard of measurement also aids in the judgment of heights:And similarly the quantities of the heights of objects above the land at some distance (such as walls and mountains) are apprehended by the eye in the way that quantities of pieces of land are apprehended and the distances of these objects are apprehended through a comprehension of their heights.77

    He concludes this mini-chapter (II:39) by re-emphasizing that these principles only apply within moderate distances.78 Alhazen's further comments on distance need not concern us here. What makes the particular passages we have cited so important is the evidence they provide that Alhazen is very much concerned, at least in theory, with relating vision to distance, not in the vague sense of "interval" or "space between," but specifically in terms of precise standardized measurement.

    With Alhazen, what had traditionally been a philosophical discussion on the role of distance in accurate vision, becomes transformed into a more scientific approach, wherein distance has become a quantifiable factor and whereby the aims of optics and surveying become effectively synonymous. The great twelfth century writer on optics, Witelo79, who borrows heavily from Alhazen's ideas, plays a significant role in making these advances of the Arabic tradition available to the Latin West. For example, Witelo gives a long paraphrase of Alhazen's discussion concerning the certification of distance in the case of continuous, ordered bodies which, as he explains, are those placed in a practically straight line and effectively at equal distances from one another as are trees or mountains or high towers and the like80 (IV:10):

Now the body of land which is interposed between these bodies is measured by the eye through the number of feet, since the foot is the minimal measure ordinarily used by men in measuring nearby sections of land and through these nearby sections of land the more distant sections of land are measured by the distinguishing power of the mind on account of the frequency of the comprehension of sections similar to that section of land, the measure of whose parts remains in the mind such that even the mind does not perceive the duration of these parts in itself. Moreover, these measures come to the mind since the quantity of the spaces which are at the feet of men are comprehended by the eye for they are even measured unintentionally through the feet of men as they frequently walk over those spaces as they are also measured with the lengths of arms (brachiorum cf. braccia). The distinguishing faculty comprehends this true measure and from this it certifies the quantities of the parts of the land which are continuous with the body of the man seeing and this remaining in the mind is the principle of measurement of all distances by the estimative (power).81

    By comparison Alhazen seems unnecessarily wordy (cf. II:24): Witelo not only transmits his predecessor's ideas but also clarifies them. Throughout this description it is striking that the process of measuring distance remains theoretical and notwithstanding references to "true measure" there is no evidence that Witelo is committed to the practical testing of his ideas. In encyclopaedic fashion Witelo reproduces the greater part of Euclid's Optics - in book four of his massive work - and thereby repeats the entire visual angles theory as if it were his own. In the midst of this discussion, however, he introduces Alhazen's attack on the Euclidean theory as faithfully as he had stated the defence. Witelo's summary amply communicates the spirit of this approach (IV:27):

Thus the distinguishing power in distinguishing the true quantity of the thing seen, will not consider the angle alone nor the distance alone, since neither of these suffices in themselves but it will consider the angle and remoteness simultaneously. Hence the quantities of the things seen will not be comprehended unless through distinction and comparison: moreover this comparison will be simultaneous and it will be between the base of the radial pyramid (which is the surface of the object seen by Bk.3, theorem 18) and the angle of the pyramid and the quantity of the length of the axis of the pyramid, which is the line of remoteness of the object seen from the eye.82

    Hence Witelo, like Alhazen, rejects the simple equation between size of the visual angle and apparent size, insisting that this be checked through a comparison with distance. At the same time, as heir to the classical division of optics into four disciplines (Chart 1) Witelo is unconcerned with contradictions arising between the various schools. Thus he can report Alhazen's important philosophical optical comments on distance and then proceed a few folios later to cite Euclid's propositions in terms of geometrical optics without attempting to reconcile the two. Moreover, he can discuss the careful computation of distance in measured cubits, without testing the results. His fourteenth century successor Biagio Pelacani da Parma does test the results but still describes them verbally.83 Francesco di Giorgio Martini (Fig. 5) and Leonardo (Fig. 6) illustrate the results visually.

    In volume one we noted a connection between these diagrams and others in Witelo (Fig. 4) and Euclid (Fig. 3). We are now in a position to understand why these connections existed. The general orientation of optics had shifted from an interest in qualitative appearances and illusions to a study of quantitative elements with a view to getting beyond deception. At the same time the concept of distance had been so integrated into optics that theoretical optics and practical surveying now had parallel goals. The quest for veracity of vision, which had begun on a strictly philosophical level in Antiquity could now be pursued on two levels simultaneously: one in terms of theoretical arguments, the other in terms of practical demonstrations.

    As late as the 1390's Biagio Pelacani da Parma can still pursue both these strands in his lectures on optics. In the course of the fifteenth century, however, these practical demonstrations evolve to such an extent that they inspire independent treatises or parts of treatises by authors such as Alberti, Filarete, Francesco di Giorgio Martini, Piero della Francesca and Leonardo. In the minds of the fifteenth century authors the theoretical and practical demonstrations are interdependent. Hence they refer to both as perspectiva, the mediaeval term for optics. Modern historians, to avoid ambiguity, usually refer to the theoretical demonstrations as optical treatises and those with practical demonstrations as treatises on linear perspective.


8. Surveying and Optics

    In Euclid's Optics, the inclusion of four theorems on surveying appeared fortuitous. Our excuses on how distance became a concept central to optical theory explains why this link between optics and surveying becomes ever more important: both disciplines are concerned with the same concept of distance. Hence as early as the second century A.D. we find evidence in Hero of Alexandria84 that surveying is a branch of optics. By the tenth century, Al-Farabi goes much further. Concerning optics (sic) he writes:

This art makes it possible for one to know the measurement of that which is far distant, for example, the height of tall trees and walls, the width of valleys and rivers, the height of mountains and the depth of valleys, rivers... then the distances of the celestial bodies and their measurements.85

    The twelfth century Toledan philosopher and translator, Gundissalinus, effectively copies this passage verbatim in his De divisione philosophiae.86 His work, as Crombie87 has noted, in turn influence Grosseteste and the Oxford school of the thirteenth century.


Figs 3-6 Surveying principles in the optical tradition: Fig. 3, Euclid, Optics Theorem X; Fig. 4, Witelo, Opticae, IV:22; Fig. 5, Francesco di Giorgio Martini, Codice Torinese Salazziano 148, fol. 33; Fig. 6, Leonardo, CA 36vb.

    Indeed this ideal of distance common to optics and surveying explains why the two disciplines so often appear together in mediaeval manuscripts as, for example, in Grazia de Castellani's De visu, 88 which includes various propositions on surveying. Moreover, this common ideal of distance explains why the practical demonstrations of optics that evolved into the independent science of linear perspective, should have had their start in practical surveying treatises. In short we now have a reason for the links between perspective and surveying described in volume one.


9. Astronomy and Optics

    Aristotle, in the Analytica Posteriora, specifically claims that optics is linked with geometry whereas the data of observation, as he calls them, are linked with astronomy.89 Nonetheless, there also appear to have been direct links between optics and astronomy. Certain propositions in Euclid's Optics may have been written to account for astronomical phenomena.90 Plato in his Timaeus is unequivocal about this link between optics and astronomy:

For I reckon that the supreme benefit for which sight is responsible is that not a word of all that we have said about the universe could have been said if we had not seen stars and sun and heaven. As it is, the sight of day and night, the months and returning years, the equinoxes and solstices, has caused us the invention of number, given us the notion of time, and made us inquire into the nature of the universe .... Let us rather say that the cause and purpose of God's invention and gift to us of sight was that we should see the revolutions of intelligence in the heavens and use their untroubled course to guide the troubled revolutions in our own understanding.91

This connection would explain why Ptolemy, the greatest writer on optics in Antiquity should also have been the author of the Almagest. St. Anatolius in his Fragments of the Books of Arithmetic provides further evidence of this connection when he writes of the mathematician that:

he ought to be cognisant of the course of the stars and their velocity, and their magnitudes, and forms and distances. And besides, he ought to investigate their dispositions to vision, examining into the cause, why they are not seen as of the same form and of the same size from every distance, retaining, indeed, as we know them to do, their dispositions relative to each other, but producing, at the same time, deceptive appearances both in respect of order and position.92



Chart 3 . Summary of different systems of classifying optics and astronomy

    As the mediaeval period progressed the Aristotelian scheme of classification was gradually replaced by others in which optics and astronomy were more closely linked (Chart 3). Among them was that of A1-Farabi who, as we have just cited, claimed optics was concerned with "the distances of the celestial bodies and their measurements."93 These connections between optics and astronomy are reflected in the optical treatises themselves. In the case of Alhazen's great treatise it becomes customary to add a short treatise De crepusculis as an appendix.94 In Witelo's optical compendium astronomical considerations are incorporated within the test itself. In Pecham's treatise astronomy plays a greater role. This interplay between optics and astronomy also explains why Biagio Pelacani da Parma should have been studying Ptolemy's Planisphere in a course on optics.95 It also explains why Dante, in his Convivio (II-III-6), speaking of the moving heavens should write that:

The position of these is manifest and determined by an art, which is called optics (perspettiva) and by arithmetic and geometry, is sensibly and reasonably seen, and by other sense experiences.96

    In this context, Leonardo's preoccupation with optics as a means of understanding phenomena in the heavens can be seen as a natural outgrowth of a tradition. And Kepler's decision, a century later, to focus his optical studies on astronomy, becomes a logical next step. Optics which had begun as a philosophical problem, had now acquired a problem solving function.


10. Changes in the Scope of Optics

    This interplay between optics and disciplines such as philosophy, medicine, surveying and astronomy, dramatically expanded the scope of optics itself as becomes evident from even a cursory survey of the contents of the chief treatises. Euclid, as we have seen, devotes his Optics primarily to deceptions of vision (Chart 4). In a separate work he deals with catoptrics. Ptolemy's Optics, by contrast, is divided into five books. His first book, now lost, dealt with vision and light, how they were imparted, how they were comparable and how they differed. His second book considers those things properly perceptible by means of sight, which topic leads him to pay considerable attention to illusions. In book three, Ptolemy examines basic problems of plane and convex mirrors. In his fourth book, he concentrates on concave mirrors, and his fifth, on refraction.

    Alhazen's great optical treatise, in the version that became famous in the West, has seven books. Book one97 begins with the claim that light has an effect on the eye and then considers colours, composition of the eye, quality of vision, the use of sight and the prerequisites for healthy vision. Book two opens with a description of how vision takes place, examines the twenty-two objects of sight and the diversity of things seen by the eye. In book three he considers deceptions of sight and the various causes thereof. In a fourth book, Alhazen discusses problems of reflection in general, before turning, in book five, to discuss the position of images in plane, convex, concave and cylindrical mirrors. In book six he is concerned with explaining errors brought about be reflection in concave and convex mirrors. In book seven, he deals with refraction, its properties, and the illusions it occasions. Often appended to Alhazen's treatise is a short work entitled De crepusculis, which considers optical phenomena relating to the sun, shadow and clouds.

    Witelo, in the thirteenth century effectively compiles an encyclopaedia of optical knowledge at the time. In a first book he concentrates on mathematical principles gleaned from Euclid, Archimedes, Apollonius, etc.98 In his second book he examines the projection of light and shade. In book three Witelo considers simple vision, disposition of the organ of sight, conditions of sight, and properties of the first two objects of sight: light and colour. In book four he considers the remaining twenty objects of sight and the deceptions that these involved. Here he integrates virtually the whole of Euclid's Optics.



Chart 4. Summary of themes of the other optical treatises from Euclid to Kepler. Evident to a diversification of topics and increasing attention to astronomy.

    In book five Witelo turns to problems of reflection common to all mirrors and the properties of plane mirrors.99 In book six he examines convex, columnar and pyramidal mirrors. These latter types he discusses further in book seven. In book eight, Witelo studies concave mirrors. In his ninth book, he returns to consider some special properties of columnar and pyramidal (conic) mirrors and parabolic burning mirrors. In his tenth book, Witelo concentrates on refraction, which leads him, in the final section, to consider optics with respect to astronomy.

    Pecham's Perspectiva communis100 is most probably an abridged version of Witelo's compendium. Pecham's work is divided into three books and opens with a consideration of light and its properties of propagation. The eye and the visual process are discussed, then visual perception, the conditions for sight and the objects of sight as well as deceptions and the problem of stars on the horizon. Book two turns to investigate mirrors: their nature, differences between them, the manner and positions of reflection, how one locates images, as well as various errors of reflection, ending with a note on the twinkling of stars. In book three, Pecham opens with a discussion of refraction and then devotes no less than eleven propositions to astronomical and meteorological questions ranging from twinkling stars to rainbows.

    Kepler, in his Ad Vitellionem Paralipomena quibus Astronomiee pars optica traditur 101 (1604) focusses on this astronomical dimension of optics. His work is divided into eleven chapters and opens with a discourse on the nature of light, which leads to a consideration of the shape of light (chapter two), a study of mirrors and the position of their images (chapter three), the measure of refraction (chapter four), and in turn to the visual process in chapter five where he makes his distinction between the imago and the pictura. The remaining six chapters are devoted to astronomy, beginning with a consideration of the light of the stars, then the shadow of the earth (chapter seven), the shadow of the moon and diurnal shadows (chapter eight); parallax (chapter nine), the optical basis of the movement of the stars (chapter ten) and how one can determine the diameters of the sun and moon (chapter eleven).


11. Summary

    On the one hand this continuity of the optical tradition during the 1900 years from Euclid to Kepler kelps explain many parallels which we shall find in Leonardo's optical writings. On the other hand, within this continuity the whole nature of the approach had changed. What had begun as an interest in qualitative, subjective aspects of vision has become transformed into a concern for a quantitative, objective physics of light and shade. As our brief survey of the historical context has shown the theoretical roots of this shift were gradually established in the course of the Mediaeval period. Leonardo's optical researches provide us, in turn, with an important chapter in the story of how that theoretical shift became a practical one, in short how a tradition of speculative metaphysical questions was translated into a set of problems in physics.

Last Update: July 2, 1999