Part Three, Chapter 3: Appearance and Illusion

1. Introduction 2. Comparisons with Euclid's Optics 3. Effects of Context and Background

4. Conclusions

1. Introduction

As was noted in the introduction, interest in deceptions and illusions of vision stands in a long tradition which can be traced back to Euclid's Optics.¹ It is not certain whether Leonardo actually studied Euclid directly, but there are sufficient parallels between the two authors to warrant a detailed comparison. An outline of the Optics will be given. The order of the treatise will be followed and its theorems confronted with passages from Leonardo's notebooks. In the latter part of this chapter attention will be given to other illusions studied by Leonardo which are not found in Euclid's Optics, namely, those involving effects of context and background.

2. Comparisons with Euclid’s Optics

The Optics opens with seven definitions pertaining to rectilinear vision, the visual angle and the assumption that apparent size is strictly a function of angular size. In his early theorems (2-9) Euclid explores the consequences of this concept of visual angles for given two-dimensional lines at given distances. He then makes a comparative study of two-dimensional lines at various distances (10-17) and explores the practical implications of this for surveying (18-21). Following this opening section on visual angles in connection with straight lines and rectilinear surfaces, Euclid considers circular forms, first two dimensional (22), then three-dimensional including sphere (23-27), cylinder (28-29) and cone (30-33). Next he compares various parts of circular sphaes seen simultaneously (34-36) and various viewpoints along a circular path (37-38).

In the third section of his treatise Euclid explores two variables which Leonardo studied more systematically: first two dimensional cases with a fixed eye and a moving object (39-40), then those with a fixed object and a moving eye (41-49) and finally cases with a fixed eye comparing different positions of a moving object (50-56). In the last propositions he compares concave, convex and cubic objects (57-58). The general structure of Euclid's Optics thus bears comparison with Leonardo's treatise on linear perspective in the Manuscript A which also begins with straight lines and surfaces, proceeds to circular forms and ends with a three-dimensional cube. A detailed comparison of Euclid's treatise and Leonardo's optical writings reveals more striking parallels.

Definitions One-Three

Euclid's first definition concerns the rectilinear propagation of light. As noted earlier (see above pp. ) this is also the first premise of Leonardo's optical theory. Euclid's second definition pertains to the cone of vision. Leonardo, for reasons we have noted (see above pp. ) calls it a pyramid. Euclid's third definition, that only those objects are seen upon which visual rays fall, is implicit in Leonardo's physics of sight.

Definition Four

Euclid's fourth definition involves three parts:

On a number of occasions Leonardo appears to accept this definition uncritically. On CA237ra (c. 1500), for instance, he paraphrases the first two parts thereof: "That object is said to be greater which comes to the eye with a greater angle and hence the lesser one /i.e. object/ will see the lesser angle." He then illustrates the last part (fig. 1179) adding the caption: "Hence a appears equal to b and hence a grain of millet near the eye occupies a city remote from this eye." The third part of the definition interests him more. On CA353vb (fig. 1184 cf. figs. 1183, 1185-1186, 1485-1487) he notes that "the line cd appears the size of ab." Implicit here is the claim that angles determine apparent size exclusively, an idea which recurs on C27r (fig. 1180, 1490):

Perspective

Immediately following is a comment about the actual sizes of such objects subtended by a same visual angle (fig. 1181, cf. fig. 1182):

Perspective

On A8v (1492) he repeats the Euclidean claim that angles alone determine apparent size:

Perspective

On CA214vb (c. 1497-1500) and again on CA221vc (c. 1500) he writes: "Equal objects, equally distant from the eye are judged to be of equal distance by this eye." He restates this idea on Forst II 5r (c. 1505): "Equal things, equally distant from the eye are judged to be of equal size by this eye." On CA208vb (c. 1513), he claims (fig. 1189 cf. figs. 1191-1193):

He returns to this claim once more on E30r (1513-1514):

Nothwithstanding this series of passages in which he appears simply to accept Euclid's claims, there are others which confirm him to be critical. On C27v (1490), for instance, he points to the role of linear perspective in distinguishing between apparent angular size and measured size:

Perspective

Perspective adds where judgment is lacking int hings which diminish.

This introduces an experiment which we have analysed elsewhere (vol 1, Appendix II B 1). Following this experiment he repeats the idea that angular size determines apparent size:

Perspective

In the context of the accompanying passage, however, it becomes clear that Leonardo no longer accepts at face value the equation of angles and apparent size:

Perspective

By 1508 his rejection of the Euclidean equation is clearly stated on CA190vb (fig. 1197):

One reason for this rejection is revealed by a passage on F40r-39v (fig. 1196, cf. figs. 1194-1195) where he used his concept of images being everywhere ("all in all") in the eye, to undermine the importance of angles subtended at the eye:

Concept

On F29r (fig. 1198) he also mentions the role of eyelids in determining that objects subtending an equal angle, nonetheless, appear different sizes. Such considerations explain why on F37r (fig. 1198), he should speak simply of angular size, without mentioning apparent size:

The angle abd is less than the angle bcd.

The problem continues to trouble him. On CU521 (1508-1510) he notes that linear perspective on its own does not provide sufficient for the perception of distance:

Later, on CU488 (fig. 1200, TPL481, 1510-1515) he broaches a further problem of perception involving visual angles:

Precept

There is another difference to be noted. Euclid considered only visual pyramids having their apex in the eye. Medieval optical writers such as Alhazen (IV.3) and Pecham (I.5) had considered two sets of pyramids, one at the eye and the other at the object. Leonardo also illustrates such pyramids in both directions on A37r and CA131vb (figs. 1202-1202), a theme which he discusses at length on G53v (1510-1515):

Definition Five

Euclid's fifth definition points out that objects seen under higher rays appear higher and conversely: "that those which are seen under lower rays appear lower." Leonardo paraphrases the second part of this definition on CU526 (TPL476a, 1510-1515): "And if it be situated under the eye, the closest to the eye will appear lower."

Definitions Six and Seven

There appears to be no evidence that Leonardo either copied or restated the ideas in definitions six and seven of the Optics.

Theorem 1

There is also no evidence that Leonardo copied theorem one.

Theorems 2 and 3

Euclid's second theorem claims that nearer objects are seen more distinctly and his third theorem adds that at a distance objects are eventually no longer seen. These ideas, as noted elsewhere (see above vol. 1, Part III.2) constitute the starting point of Leonardo's perspective of disappearance of form.

Theorem 4

In his fourth theorem Euclid states that: "among equal lengths finding themselves along a straight line, those which are seen at a greater distance appear smaller." Leonardo expresses a similar idea on CA353vb (fig. , c. 1485-1487): "Among objects of equal size that (which) will show itself of lesser size which is more remote from the eye." This he paraphrases on H249r (January 1494): "Objects hear the eye appear larger than the distant ones" and again on Forst II, 15v (c. 1495) under the heading (fig. 1203)

Perspective

A similar idea is found in a note on BM Arundel 101r (1490-1495), probably not in Leonardo's hand: "Among objects of equal size that which appears to the eye through a thinner pyramid...will be more distant from this eye." On CA214vb (c. 1497-1500) he redrafts this idea: "Among objects of equal size, that (which will) show itself as smaller." This he crosses out and then produces further versions:

He reformulates this idea on CA 1 terv (i.e. CA9v new): "Among objects of equal size that which is more distant from the eye will show itself of lesser appearance.

Theorem 5

In his fifth theorem Euclid claims that: "equal sizes unequally distant appear unequal and that which is situated closer to the eye always appear larger" (fig. 1204 cf. fig. 1205). Leonardo drafts a similar idea on CA214vb (1506-1508 or 1500):

Equal objects, which are remote...the eye by unequal distances, must appear of unequal size.

These drafts he crosses out and produces a revised version: "Equal objects will demonstrate themselves to be unequal when they are remote from the eye at various distances." On CA221vc (c. 1500) there is a version which is again closer to the drafts: "Equal objects in being remote from the eye at various distances, must appear of unequal size." The second part of Euclid's fifth theorem is found on CA225re(c. 1500) in a passage headed:

Perspective

Another paraphrase occurs on I49/1/v (1497) but here the theorem is confronted by an experience from his studies on linear perspective:

Theorem 6

Euclid's sixth theorem claims that "parallel lengths seen at a distance appear unequal." Leonardo does not state this idea directly in the extant notes. Nonetheless, Euclid's diagram (fig. 1206) bears comparison with a figure on CA120rd (fig. 1207) accompanying which Leonardo writes: “All the species of objects positioned opposite the eye concur through radiant rays to the surface of such an eye, which /rays/ are cut at the surface of such an eye under equal angles.” On TPL520 (1508-1510) he draws analogous diagram (fig. 1208cf. fig. 1209) now adding that:

Witelo, in his thirteenth century version of Euclid's theorem (IV. 21), had used it to make a further claim, namely, that "the parts of parallel lines as they are further from the eye appear almost to concur, yet they are never seen to converge" /to a point/. Leonardo expresses the same idea on CA120rd (fig. 1210): "The eye, between two parallel lines, will never see it at any distance so great that these lines converge to a point." In a late precept on TPL476a (1510-1515), however, he claims the converse, namely, that "lateral parallels converge to a point."

Theorem 7

Leonardo does not appear to have studied this theorem.

Theorem 8

Euclid's eighth theorem notes that "equal and parallel sizes, unequally distant from the eye are not seen in /direct or inverse/ proportion to their distances," his point being that angular size, to which he subjects apparent size does not vary in a simple direct or inverse proportion with changing distance. Leonardo appears to be equivocal concerning the relationship between angular and apparent size. He tends to follow Euclid's assumption that apparent size is determined by visual angles but, nonetheless, there are occasions, as on A9v (fig. 1211, cf. figs. 1212-1214, 1492) when he appears to deny the consequence which Euclid stressed and seems to assert that visual angles do vary inversely with distance:

There is an explanation for these apparently contradictory claims. In his studies of linear perspective Leonardo had demonstrated that projected size varies inversely with distance. In this context he had spoken of the visual pyramid in a way reminiscent of Euclid's visual cone, but with an essential twist. Leonardo thinks of the pyramid in terms of its various cross-sections or its projections, rather than in terms of its angles. Moreover, he tends to equate these projected sizes - which can be measured - with the apparent size of an object.

And as a result he can pay lip service to the traditional visual angles theory but in practice equate projected and apparent size and hence insists on a proportional relation, on A8v (1492) for instance, which Euclid would have denied:

On the diminution of things through various distances.

The perspectival implications of this and similar passages have been considered elsewehre (see vol. 1 part I.3).

Proposition 9

Euclid's ninth proposition notes that "rectangular sizes, seen at a distance, appear rounded." This deception of sight had interested Aristotle² and remained a theme of discussion throughout the Mediaeval period.³ Leonardo is also interested in this problem. On A9v (1492) he draws a figure to compare the rays coming from a square and round object respectively (fig. 1226). On A92v (BN 2038 12v) he mentions that

On BM Arundel 112r he introduces "a test whether the square body makes itself like the square aperture in the rays of the sun, which loses its angles" (see above pp. ), which he then illustrates (fig. 1216, cf. figs. 1219-1222). As early as 1490, on C8r, he had noted a related phenomenon with respect to candles:

On light

He offers an explanation for this candle phenomenon on BM115v (c. 1492):

He mentions the phenomenon of the candle again on H91 /43/v (1494): "That luminous body of a long shape will show itself of a rounder shape which is...more distant from the eye." Some fourteen years later on F64r (1504) he pursues the theme in a passage entitled:

Why every luminous body of a long shape appears round over a long distance.

Already in the 1490's he had also explores this transferal from a square to a round shape in terms of percussion. On Forst III 26v (fig. 1225 cf. III 63v, II 54r, fig. 1224), for instance, he notes:

On BM188r (fig. 1223, c. 1510) this percussion problem is pursued in a draft:

The perceptual problem continues to trouble him, hwoever. Hence on CA243ra (c. 1513) he asks:

Here the questions are not answered, but on G26v (CU956, figs. 1217-1217, 1510-1515) he returns to the general problem in connection with trees:

This specific example leads him to a more general conclusion:

Accompanying this he draws a diagram (fig. 1217) reminiscent of Euclid's (fig. 1215). On G53v (1510-1515) Leonardo returns once more to this example of a man:

Whereas Euclid had reduced the phenomenon to a simple geometrical obstraction, Leonardo uses concrete examples and seeks to identify stages in the transformation from the original shape to its rounded appearance.

Theorem 10

In his tenth theorem Euclid makes a quantitative claim that "the more distant parts of planes situated below the eye appear more elevated." Leonardo explores this phenomenon more systematically on CA36vb (c. 1480), TPL936a (1510-1515), CA351va (1500-1505) and elsewhere (see vol. 1, part I.2, 3).

Theorem 11

Euclid's eleventh theorem deals with the converse, namely, that "the more distant parts of planes situated above the eye appear lower" (fig. 1228 cf. fig. 1229). On K121/41/r (post 1505) Leonardo considers this phenomenon in terms of a flying bird (fig. 1233):

On the verso of K121 /41/v he also considers the reverse situation, not mentioned by Euclid:

This he illustrates (fig. 1234) and explains (fig. 1232):

Having explored how movement along a horizontal plane is seen on a vertical plane, he considers how movement along a vertical plane is seen on a horizontal one on K123/43/r (fig. 1231):

Theorem 12

In his twelfth teorem Euclid notes (fig. 1235) that: "Among sizes having their length in front, those which are to the right appear to pass to the left and those that are to the left /appear/ to pass to the right." Leonardo broaches this problem on K120 /40/v (after 1505):

He adds a diagram (fig. 1236) and a concrete example:

Theorem 13

In Theorem thirteen of the Optics Euclid states: "Among equal sizes positioned below the same eye those which are more distant appear more elevated." Leonardo states this idea on A11r (1492) under the heading:

Perspective

He restates this on CA225re (c. 1500?) again under the heading of:

Perspective

On TPL476a (1510-1515) he makes the point anew: "And if they /i.e. the objects.../ are situated below the eye, the nearest to this eye will appear lowest."

Theorem 14

In theorem 14 Euclid consides the converse cae (fig. 1237): "Among equal sizes placed above the eye, those which are more distant appear lower." This Leonardo also considers on A11r (1492) under the heading of:

Perspective

On CA225re (c. 1500) he restates this:

Perspective

On CU526 (TPL476a, 1510-1515) the idea recurs: "Among objects of equal height which are situated above the eye, that which is remote from the eye will appear lower." On CU494 (fig. 1238, TPL480, 1510-1515) he gives a concrete example:

Among things of equal height that which is more distant than the eye will appear lower.

Euclid's theorem's 13 and 14 are also reflected in a passage on A10v (1492):

Theorems 15-17

There is no evidence that Leonardo dealt with these theorems in the extant notebooks.

Theorem 18

This theorem deals with the measurement of an unknown height. Leonardo deals with this problem in a similar fashion on A6r (1492) and on Forst I 48v (1505).

Theorem 19

Here Euclid solves the same problem using a mirror. Leonardo also uses a mirror in solving this problem on BM Arundel 44v (1505-1508).

Theorem 20

In this theorem Euclid measures depth. Leonardo does not deal with this question.

Theorem 21

This theorem deals with measuring the length of objects at a distance. Leonardo also treats this question on Ca148vb (1487-1490), A96r (BN 2038 16r) (1492) and on CA122vb (post 1515).

Theorem 22

In theorem 22 Euclid states that "if one positions the arc of a circle in the same place where the eye is, the arc of the circle appears to be a straight line." Leonardo drafts this problem on BM Arundel 101*v (1490-1495): "that curved line will appear to be straight of which the extremities, along with its centre are found along a perpendicular line." On this same folio he also makes drafts concerning a related phenomenon:

the interval that is between...

These drafts he crosses out and turns to a related problem:

This idea he pursues on BM100r:

On the same folio he also writes another paraphrase of Euclid's theorem 22: "That curved line will appear to the eye to be straight of which all the parts occupy all the parts of a straight line."

Proposition 23

In this theorem Euclid claims that

Leonardo makes a similar claimon CA251ra (fig. 1240, c. 1490):

Fond as he is of the analogy between eye and light source (see above pp. ) he considers an equivalent situation involving light on CA250va (c. 1490):

On CA216ra (c. 1493) he implicitly makes this claim once more in terms of vision: "To the extent that a spherical body is of greater size, it will show a smaller quantity of itself to the eye, the eye being without movement."

Theorem 24

In theorem twenty-four Euclid notes that: "if the eye approaches a sphere, the part seen will be smaller but will appear to be seen as larger." Leonardo makes a similar claim with respect to light on CA250va (c. 1490):

On Ca251ra (c. 1490) he states it differently: "The more distant is the light, the more it sees of the sphere and the less shade there is on the wall mo." He makes a parallel claim with respect to the eye on CA233rd (1490): "The more the spherical body removes itself from the eye, the more it sees." This phrase he repeats under the heading of "perspective" on A10v (1505-1508) where he refers to the "growth of the spherical body and its diminution at various distances." On 216ra (c. 1493) he expresses this basic idea somewhat differently (fig. ): "That spherical body which is of greater size than the eye, the more that it approaches this eye, the less that it will show of itself. On CA174vb (1517-1518) he puts it differently again: "Less quantity is seen of that spherical body which is closer to the eye that sees it."

Theorem 25

In this theorem Euclid claims (fig. ) that in the case of: "A sphere being seen by two eyes, if the diameter of this sphere is equal to the straight line along which the two eyes are separate from one another, its hemisphere will be seen entirely." Leonardo ?***

Theorem 24 **(check because mentioned above)

In theorem twenty-four Euclid notes that: "if the eye approaches a sphere, the part seen will be smaller but will appear to be seen as larger." Leonardo makes a similar claim on CA233rd (1490): "The more the spherical body removes itself from the eye, the more it sees." This phrase he repeats under the heading of "perspective" on A10v (fig. 1249, 1492). On CA216ra (fig. 1243, cf. 1241-1242, 1244-1245, c. 1493) he reformulates the idea: "That spherical body which is of greater size than the eye, the more that it approaches this eye, the less it will show itself." The problem is broached again on BM139r (1505-1508) where he refers to "the growth of the spherical body and its diminution at various distances." On BM199v (fig. 1251, 1508-1510) the idea is restated anew: "the more the greater body removes itself from the lesser, the less one sees it." On CA174vb (1517-1518) he puts it differently still: "Less quantity is seen of that spherical body which is closer to the eye that sees it." Parallel with this area a series of claims with respect to light as on CA250va (c. 1490):

On CA251ra (c. 1490) he considers a case (fig. 1247): "The more distant is the light, the more it sees of the sphere and the less shade there is on the wall mo." Nearby, he mentions a second case (fig. 1246): "To the extent that the luminous point sees less of a spherical body, to that extent does it produce more shadow on the wall bc." A third case follows (fig. 1257, cf. figs. 1252-1256):

These parallels between light and sight are again implicit on a later note on CA112va (fig. , 1505-1508):

Theorem 25

In this theorem Euclid claims that in the case of: "A sphere being seen by two eyes, if the diameter of this sphere is equal to the straight line along which the two eyes are separate from one another, its hemisphere will be seen entirely." Leonardo illustrates this case on CA216ra (fig. , c. 1493) adding the note: "That spherical body which is seen by a single eye will appear of lesser size at an equal distance than if it were seen by two eyes." On K124/44/v he considers the equivalent phenomenon with respect to monocular vision (fig. ):

Theorem 26

Here Euclid observes that "if the distance between the eyes is greater than the diameter of the sphere, one will see more than the hemisphere of the sphere." Leonardo puts this differently on CA175ra (c. 1493-1494): "No opaque body of a spherical shape which is seen by 2 eyes will appear to these /eyes to be/ of perfect rotundity" (see below pp. ). On D4r (fig. 1260, 1508) he adds:

He makes a similar point with respect to monocular vision on CA120vd (figs. 1258-1259, c. 1504 or later c. 1506-1507):

The eye seeing an object less than it sees it as larger than it is.

He pursues this idea on K125/45/r (fig. , post 1504):

Leonardo is also interested in other perceptual problems relating to such objects (see below pp. ).

Theorem 27

In theorem twenty-seven Euclid states that: "if the separation between the eyes is less than the diameter of the sphere, one will see less than the hemisphere." Leonardo considers an equivalent phenomenon with respect to monocular vision on K124/44/v) (fig. ):

Theorems 28-33

The next six theorems in Euclid's treatise deal with the appearance of cylinders and cones from different distances. Leonardo does not discuss these in his extant notes.

Theorems 34-39

These theorems deal with changing appearances of circular objects or circular paths of the eye as it moves around objects. These problems are again not discussed by Leonardo.

Theorem 40

Here Euclid describes a situation in which objects moving relative to a fixed eye sometimes appear larger, soemtimes smaller. Leonardo broaches this problem on H133/10r/(v) (1494): "the objects seen by a same eye appear at one time to be large at another time to be small."

Theorems 41-44

In these theorems Euclid gives further examples of changing appearances as the eye moves relative to given objects. These are not discussed by Leoanrdo int he extant notes.

Theorem 45

Here Euclid notes that there exists: "a common place from which unequal sizes appear equal." Leonardo expresses a similar idea on CA221vc (c. 1500): "unequal objects, on account of various distances from the eye, appear equal."

Theorems 46-50

Theorems 46-49 pursue the theme of changing appearances depending on specific viewpoints. Theorem 50 deals with comparative motion of objects. Leonardo does not discuss these int he extant notes.

Theorem 51

In this theorem Euclid continues with the theme of relative motions (fig. 1261):

Leonardo offers a concrete example of this phenomenon on H89/41/r (c. 1494):

Motion

On BM227r (fig. 1262, 1505-1508) he returns to this theme:

A related idea is expressed on CA207ra (1508-1510):

Theorem 52

Here Euclid notes (fig. 1264) that:

Leonardo considers a related situation on K122/42/r (post 1505, fig. 1265):

On K122/42/v he restates this idea more clearly (fig. 1266):

Theorem 53

Here Euclid notes that "if a size seen approaches the eye this seen size appears to increase." Leonardo does not discuss this phenomenon as such although it is implicit in his perspectival writings (see vol. 1, p. ).

Theorem 54

Continuing with the theme of relative movement Euclid notes that "among sizes carried at equal speed, those which are more distant appear to be carried more slowly." This claim recurs in Ptolemy⁴ and Witelo⁵. Leonardo is particularly interested in this phenomenon and discusses it on A9r (1492) under the heading:

Perspective of motion

He restates this claim on CA225re (c. 1500) under the heading of: "Motion. Among motions of equal speed that will show itself as slower which is more distant fromt he eye." On TPL231a (1505-1510) he offers a concrete example:

He pursues this theme on K124/44/r-123/43/v (post 1504) beginning, as usual, with a heading and general proposition:

This he illustrates (fig. 1273) and explains:ssssss

Above this passage on K123/43/v are further notes relating to appare relative motion:

Beneath this he draws a diagram (fig. 1274) and adds a rather cryptic note:

On BM134v (1505-1508) he makes a preliminary effort to express this perspective of movement in quantitative terms:

Hence that will appear swifter which is closer to the eye.

This he again illustrates with a specific example (fig. 1268) and explanation:

He pursues this theme on CU810 (TPl791b) under the heading (fig. 1270):

Prospettiva commune

He again illustrates this general claim with a specific example (fig. 1270):

In this example it is striking how Leonardo applies the interposed plane principle of linear perspective to perceptual problems of perspective of movement. A note on H28v (1494) may be related to this perceptual problem (fig. 1271): "The eye cannot judge where the object that is high should descend."

Theorems 55-58

In his extant note Leonardo does not consider the final four theorems of Euclid's treatise.

3. Effects of Sound and Contrast

Other aspects of Leonardo's interests in visual deceptions do not come out of the Euclidean tradition. Euclid was concerned with rules for the vision of single objects, and at most three objects, and always out of context. Leonardo, by contrast, wishes to determine effects of context and background on perception.

There were, of course, some classical precedents. Pseudo-Aristotle in De coloribus⁶ mentioned the problem, as did Ptolemy in his Optics⁷ and Galen in De usu partium⁸ broached it mroe explicitly:

Mediaeval authors such as Alhazen⁹ also alluded to the role played by background. Leonardo goes considerably further. He is aware that background affects the size, brightness, shadow, colour and relief of an object. Each of these will be considered in turn.

3.1 Size

By 1490 Leonardo is aware that a dark object seen against a light background appears smaller than it is. On C8v, for instance, he mentions that:

He elaborates on this idea on C24r (1490):

In a note on CA230vb (1485-1487) he expresses this idea in more radical terms: "An object positioned between the eye and a bright object diminishes its size by half." He is equally interested in the converse phenomenon, namely, that bright objects appear larger against a dark background. Hence, on C12r (1490-1491), he notes:

He reformulates this idea on CA126vb (1490-1492): “Among luminous bodies of equal size, distance and brightness that one will show itself of greater size which is surrounded by a darker background.” On CA126rb (1490-1492) he considers both luminous bodies against both a light and a dark background:

On A1r (fig. 1276, 1492) he describes another example of a dark object decreasing against a light background.

Botticelli had depicted (fig. 1275) a similar situation in his Adoration of the Kings (London, National Gallery, c. 1475) but without the joining of the shadows. Leonardo returns to this problem on F31r (1508) (fig. 1278):

On BM97r (c. 1508) he illustrates an opposite, how bright objects tend to merge when the dark background is limited. The phenomenon of a white object appearing larger against a dark background interests him considerably. On A79, for instance, he notes:

On CU186 (TPL258a, c. 1492) he expresses a similar idea:

He drafts two further passages concerning this problem on I18r (1498):

Any dark object seen against a bright background will show itself as smaller than it is.

On I17v, opposite, he reformulates these ideas in more general terms:

He discusses these phenomena at length on Mad II 23v (1503-1504), beginning with a general statement:

To illustrate the first of these claims he returns to his example of a glowing piece of iron:

He now uses the same example to illustrate the converse:

In brackets he adds an explanation for this phenomenon:

As a further illustration of how dark objects against a bright background appear smaller, he describes an experiment:

To the right of this passage he draws a labelled diagram (fig. 1281), two drafts of other diagrams (figs. 1279-1280), and a brief caption:

On Mad II 27v (1503-1505) he bewgins to redraw the diagram (fig. 1282), abandons the attempt and draws a more polished version on II 28v (fig. 1283) now using different letters. In the left-hand margins he begins an explanation:

Beneath the diagram he begins another phrase: "the eye which does not see the object too"..., then stops and leaves the rest of the folio blank. Some four years pass before he takes up the theme anew. On CA124ra (c. 1508), for instance, he drafts two propositions:

These passages he crosses out and writes afresh on F22r (1508) beginning with the first proposition: "That part of a dark object of uniform thickness will show itself as thinner which is seen against a more luminous background." Beneath this he draws a labelled diagram (fig. 1284) which he then explains:

He now reformulates the second of the propositions drafted on CA124ra drafted on CA124ra (c. 1508): “The part of the luminous body of uniform thickness and splendour appears to be thicker which is seen against a darker background and if this luminous body is glowing.” Below this he draws two diagrams (figs. 1285-1286) which he does not explain. Once more he has stopped short. But on F37r, he returns to the theme of the glowing rod under the heading:

This time he stops short for want of space. But on CU540 (TPL457, 1510-1515) he pursues this theme under the heading:

Why parallel towers appear narrower at ther base than at their summit in fog.

He pursues the problem on CU457 (TPL445, 1510-1515), opening once again with a general statement:

To illustrate this he gives three examples beginning with a case from botany:

His second example is the, by now familiar, tower:

His third example involves a woman in black with a white hat:

More than twenty years earlier he had expressed a similar idea on CA320vb (see above p. ). Background can affect not only the perceived size of objects, but also the perceived size of distance between objects as Leonardo notes on L77v (fig. 1287, 1501):

3.2 Brightness

Closely related to these notes on size is a series showing the effects of background on the brightness of objects. Here too Leonardo considers both light and dark backgrounds. On C3r (1490-1491), for instance, he notes: "that luminous body appears less bright which is surrounded by a more luminous background," and on C54, he considers the converse: "that luminous body appears brighter which is surrounded by darker shadows." He explores this relationship between background and brightness at greater length on A113r (fig. 1288, BN 2038 32r; CU750, fig. 1289, 1492):

He notes that the converse is equally true:

These effects of background on brightness are mentioned again in two draft passages on BM Arundel 100v (c. 1490-1495), in a hand probably not Leonardo's:

The problem is touched upon again on Forst III 87v (c. 1493):

The luminous or illuminated body bordering on the shade cuts as much as it touches.

On CU145 (TPL233, 1505-1510) he takes up the problem anew under the heading (fig. 1291):

Of the backgrounds of depicted things.

On CU753 (TPL628, 1508) the problem is broached again:

That shadows must always participate in the colour of the umbrous body.

He adds a second reason which involves changing size of the pupil (see below pp. ). During this period 1505-1510 he develops a particular interest in the effect of background on brightness of reflected light and shade. On CU164 (TPL167), for instance, he broaches this question in general terms:

Where a reflection will be seen more.

On CU165 (TPL163), he pursues this theme under the heading:

Where reflections are most perceptible.

This idea he reformulates as a precept on CU166 (TPL160, 1505-1510):

Where the reflections of lights are of greater or lesser brightness.

Related to these are two other passages concerning background and lustre. A first on CU777 (TPL771, 1508-1510) is entitled:

Of the lustres of umbrous bodies.

This leads directly to a second passage on CU778 (TPL772):

How lustre is more powerful in a black background than in any other background.

In this period 1508-1510 he also consolidates these principles in terms of ready precepts. On CU748 (TPL659), for instance, he notes:

On lights

This he reformulates on CU724 9TPL698): "And this will show itself at equal distances of sharper boundaries which will be seen in a background more disform from itself in brightness or obscurity." On CU854 (TPL650) this principle reappears as a fourth proposition under the heading:

Where the lights deceive the judgment of the painter.

In 1492, he had considered a man's shadow framed by a window. In the period 1505-1508 he restates this experience in more abstract terms on CU752 (TPL757), under the heading:

His various experiments with light and shade in camera obscuras may well have been intended to explain this phenomenon (see above pp. ). On CU760 (TPL694, 1508-1510) he broaches the problem afresh:

He mentions a related idea on E17v (1513-1514): "The eye placed in the illuminated air sees shadows inside the windows of illuminated habitations." On CU669 (TPL719, 1508-1510) he pursues the problem:

Of the brightness of derived light.

To illustrate this he cites a specific example (fig. 926):

He goes on to explain why the brightness is most excellent at r:

Having solved the problem under ordinary conditions (see also above pp. and below pp. ) he considers an underwater situation on CU546 (TPL506, 1510-1515):

To illustrate this he cites a concrete example (fig. 1292):

An explanation why this is so follows:

3.3 Light and Shade

Parallel with these observations on how background affects brightness and darkness, are further passages on contrasts of light and shade (cf. A113r, BN 2038 32r above). Early drafts on this theme, probably in another hand, occur in the Codex Arundel. On BM Arundel 103r, for instance, it is claimsed: "that (that) boundary of derived shade is darker which is surrounded by more air of derived light." On BM Arundel 100*v, it is noted: "that reflection in a body will be more evident which will terminate in a place of greater darkness" and on BM Arundel 101r, there is a third draft: "The straight boundaries of bodies appear twisted which terminate partly in dark places and partly in luminous ones." This idea Leonardo restates on C1r (1490-1491): "The straight boundaries of bodies appear twisted which terminate in a dark place interrupted by the percussion of luminous rays." A passage on CU207 (TPL197, 1505-1510) confirms that these interests in contrasting light and shade are related to his study of contrasting colours:

What colour will make a shadow blacker?

In a passage on Triv. 10v (1487-1490) he considers where contrasting shadows are greatest (328-329):

On CU857 (TPL814, 1508-1510) he pursues this theme on curved surfaces:

Of light

On TPL647 (1508-1510) he describes the nature of light and shade on a curved body with a first proposition under the heading:

Of the size of shadows and primitive lights.

A second proposition follows on CU852 (TPL648, 1508-1510):

Of the greater or lesser obscurity of shadows.

A third proposition follows:

Read in sequence such passage illustrate how Leonardo develops his ideas: he begins with rough drafts, proceeds to concrete demonstrations, reformulates them as pithy rules and finally as numbered propositions. He pursues this problem on CU784 (TPL693, 1508-1510) with the question:

He reformulates the question on CU742 (TPL605, 1508-1510):

Which background will render shadows darker?

He illustrates this with a concrete example (fig. 392):

He considers this problem of contrasts with respect to derived shade on CU712 (TPL602, 1508-1510):

This he again illustrates with an example (fig. 598):

On CU746 (TPL637, 1508-1510) the theme of contrasting shadows is broached afresh under the heading:

Of the shadows made in the umbrous parts of opaque bodies

These ideas lead, on CU ** (TPL553, 1508-1510), to a simple inverse rule:

This he restates almost verbatim on E32v (cf. ) and then develops on E32r into a series of rules:

As was shown elsewhere (see vol. one, part three, 3), Leonardo is very much aware of the consequences of these rules for his painting practice. A late passage on G12v (1510-1515), reflects this awareness clearly:

On the lights among shadows.

3.4 Colour

Aristotle in his Meteorologica had noted that:

Ptolemy, in his Optics,¹⁰ touched on similar phenomena, as did his Mediaeval successors Alhazen¹¹ and Witelo¹². Leonardo's concern with painting practice leads him to study more closely the effects of background on various colours, as is shown in an early note on A84r (BN 2038 4r, 1492) entitled:

On Painting

He is convinced that a colour seen against a background of the opposite colour is most desireable as noted on CU186 (TPL258a, c. 1492):

On Colours

On CU181 (TPL258c, c. 1492) he adds that such contrasting colours are better comprehended:

He restates this intensifying effect of contrasting colour on CU246 (TPL260a, c. 1492):

On CU459 (TPL491, c. 1492) he expresses this idea as a rule:

Precept C

On CA397rb (1497-1499) he begins to formulate a note: "Who looks at the black object on a white background" and then stops short. In the period 1505-1510 he considers these questions afresh on CU151 (TPL204) headed:

He considers this intensifying effect of contrasting colours again on CU154 (TPL231):

On the nature of the colours of backgrounds on which white borders.

On CU184 (TPL238c, 1505-1510) he gives another example of the effects of contrasting colours in a passage entitled:

On the nature of comparisons

He therefore urges that contrasting backgrounds are more appropriate, as on CU148 (TPL229, 1505-1510):

Of the backgrounds that are more appropriate for shadows and lights.

While he generally recommends that white should always border on dark and conversely, on CU150 (TPL230, 1505-1510) he considers a situation where this is not the case:

How one should act when white terminates on white or dark on dark.

On TPL 206 (1505-1510) he suggests that a colour seen against a background of the same colour appears more beautiful:

What part of a same colour will show itself as more beautiful in /a/ painting

This idea he restates on TPl217c (1505-1510):

What part of the surface of bodies will show itself of a more beautiful colour?

Meanwhile, his experiments with camera obscuras had made him aware that the usual rules of contrast do not always hold. On CA195va(1508-1510) for example, he explains:

Nonetheless such cases remain the exception and he continues to favour situations where contrasting colours are positioned opposite one another. On CU183 (TPL190a, 1505-1510), for instance, he expresses this in terms of a rule:

A second rule follows:

He restates these precepts on CU145 (TPL232, 1505-1510):

On the boundaries of objects

On CU153 (TPL252, 1508-1510), he returns to these ideas:

Backgrounds

He reformulates this on CU751 (TPL769, 1508-1510):

In the late period he returns to this theme on CA184vc (1516-1517) in a passage entitled:

Of colours

Other passages concerning this theme have been cited elsewhere (see vol. one, part three.1).

3.5 Relief

As early as 1490 Leonardo had recognized that background and context play an essential role in the perception of objects. Hence on C23r, he claims:

Optics

To illustrate this he cites a concrete example:

Such experiences lead him to recognize that lighting and background play an important role in determining the apparent relief of objects. On A2r (fig. 322, 1492), for instance, he notes that:

While background can obscure relief, it can also serve to heighten the relief of objects. This possibility particularly attracts Leonardo as a painter and, as has been shown elsewhere (see vol. 1, part III.4) he devotes a number of passages to this problem. This creation of relief through contrasting light and shade he terms chiaroscuro which becomes for him, "the first intention of the painter" (TPL412, c. 1492) and eventually the "most fundamental part of painting" (G23v, TPL482, c. 1510-1515).

4. Conclusions

A detailed comparison of Euclid's Optics with Leonardo's writings reveals a number of close parallels. Even so, Leonardo does not cite the Optics explicitly and 31 of its 58 theorems are not discussed in his extant notebooks (see (Chart 26). It therefore remains an open question whether he studied the Optics directly or via some mediaeval source. In any case, the scope of Leonardo's interests in illusions goes well beyond that of Euclid's Optics insomuch as he studies effects of context and background on the perception of size, brightness, shadow, colour and relief in objects. Particularly striking are the experimental demonstrations, which he develops in this connection.

What also emerges from this analysis is that Leonardo's interest in deceptions of vision developed in the 1490's and are, therefore, not to be associated primarily with his late writings after 1510. Indeed, far from representing a late new development in his thought, this concern with illusions confirms his early acquaintance with problems central to the optical tradition.

Notes

1. Euclid's Optics ↩︎
2. Aristotle ↩︎
3. Euclid's ninth ↩︎
4. Ptolemy ↩︎
5. Witelo ↩︎
6. Pseudo-Aristotle - De coloribus ↩︎
7. Ptolemy - Optics ↩︎
8. Galen - De usu partium ↩︎
9. Alhazen ↩︎
10. Ptolemy, in his Optics ↩︎
11. Alhazen ↩︎
12. Witelo ↩︎