SUMS

Dr. Kim H. Veltman

Appearance and Illusion


1. Introduction
2. Comparisons with Euclid's Optics
3. Effects of Sound and Contrast
        3.1 Size
        3.2 Brightness
        3.3 Light and Shade
        3.4 Colour
        3.5 Relief
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:

And that sizes seen under a greater angle appear larger, whereas those seen under a lesser angle appear smaller and that those which are seen under equal angles appear equal.

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Figs. 1179-1182: Visual angles and apparent size. Fig. 1179, CA237ra; figs. 1180-1182, C27r.

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Figs. 1183-1186: Visual angles and apparent size. Figs. 1183-1185, CA353vb; fig. 1186, CA157vb.

    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

The two objects seen within the aforesaid visual pyramids which are not less or do not exceed these lines, even though there be a great interval between them, nonetheless, this distance can never be seen or recognized by this eye.

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

Perspective

To the extent that the distance is greater among the aforesaid bodies included whithin the pyramidal visual lines, the more disform (in size) these need to be formed.

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Figs. 1187-1193: Whether apparent size is determined strictly by visual angles. Fig. 1187, CA23ra; fig. 1188, C27v; fig. 1189, CA208vb; fig. 1190, CA23va; fig. 1191, CA208vb; fig. 1192, C27v; fig. 1193, CA208vb.

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

Perspective

The little object from nearby and the large /one/ from afar, being seen under an equal angle, appear of equal size.

    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):

All the species of those visible objects that come to the eye with and angle of the same size will show themselves to be equal among themselves...even though they be of unequal sizes.

You see the proof that the space bd and the space hi, considerably smaller, come to the eye through the same angle.

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

and by /proposition/ one of perspective: equal objects placed at equal distances will show themselves as equal among one another.

    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.

The eye can never be a true judge...in fixing with truth the extent to which a nearby quantity below is similar to another which is, with its summit, at the same height of the eye, except by means of the interposed plane the master and guide of perspective.

    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

Why all the objects which with their extremities touch the visual lines, even if they are of various sizes at various distances, nonetheless, they all appear of a same size.

    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

That eye which is at the same distance and quite close to this other eye, which sees the objects of various sizes at various distances, they do not appear to it to be judged by a same size. Nonetheless they are neither less or greater than the straight lines of the visual pyramids.

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

The flea and the man can come to the eye and enter into this under equal angles and through this the judgment is not deceived /into thinking/ that the man does not appear larger than this flea. The cause is demanded.

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Figs. 1194-1197: Demonstrations against the Euclidean equation of visual angles and apparent size. Figs. 1194-1195, F95r; fig. 1196, F40r; fig. 1197, CA190vb.

    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

That eye will be less luminous which will be illuminated by a lesser light, even though each part of this place participates in such a light.

By this concept and by the proposition above which I needed to formulate first, we are certain that the illuminating object ng makes a greater impression in the cornea (luce) nmabin the site ab than it would being removed at the distance hl, because the impression of this hl comes to be on the cornea at co which is much less than the first impression ab and for this /reason/ the power of the brightness is diminished, but not the quantity, through the said concept, since the base and the space of the cornea (luce) is as much touched by the image co as by the greater image ab whence the brightness of the rays around the image illumine everywhere within this cornea in the same way as the rays of the large image and no difference is noted by the sense except in degree of brightness, diminished in power but not in quantity.

    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:

Among objects of equal distance the lesser sends a smaller angle to the eye and the larger a larger.
The angle abd is less than the angle bcd.

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Figs. 1198-1199: Discussions of visual angles on F37r and F29r.

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Fig. 1200: Discrepancies between angular and apparent size on CU488.

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Figs. 1201-1202: Double pyramids on A37r and CA131vb.

    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:

Through linear perspective the eye, without moving, will never have cognition of the distance that there is between the object which interposes itself in front of it and another object without the perspective of colours.

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

Precept

Why the painted object, even though it comes to the eye through the same size of angle as that which is more remote than it, it does not appear as remote as that of the natural remoteness. Let us say that I paint on the interposed plane bc an object which has to appear a mile distant and then I place at its side one which has the true distance of a mile, which two things are arranged in such a way that the interposed plane ac intersects the pyramids with equal size, nonetheless, /seen/ with two eyes they iwll never appear of equal distance.

    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):

In distances perspective will adopt two contrary pyramids of which one has an angle at the eye and the base remote towards the horizon. The second has the base at the eye and the angle /of its apex/ at the horizon. But the first extends to the universal, embracing in it all the quantities of bodies placed opposite the eye as would be a large landscape seen by a narrow aperture because a greater number of things are seen by this aperture to the extend that these are more remote from the eye and thus the base at the horizon is generated and the angle /of the apex/ at the eye as was said above.

The second pyramid extends to a particular which shows itself as that much larger or smaller to the extend that one removes it from the eye and this second in persepctive originates from the first.

 

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.

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Fig. 1203:Illustration of Euclid's fourth theorem on Forst.II 15v.

 

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

Among things of equal size that which is more distant from the eye will show itself of a smaller form.

    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:

Among objects of equal colour and size, that iwll diminish its figure more, which is more remote from the eye which sees it.

All objects of equal colour and magnitude will show themselves of...equal size which are equally remote (from the eye) which sees it.

But if equal objects...have unequal distances...which include themselves between the eye...and themselves, that one iwll show itself as more diminished which is situated further from the eye.

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Figs. 1204-1505: Euclid's fifth theorem in the Optics and Leonardo's CA208vb.

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.
Equal objects, in being remote from the eye at various 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

And among those objects of equal size that will show itself of a greater size...which is situated closer to the eye.

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

The more that an object approaches the eye, the more that it shows itself through a greater angle and the similitude of the object does the opposite, because to the extent that it is measured closer to the eye, to that extent is it shown to be of a smaller size.

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Figs. 1206-1210: Euclid's theorem 6 and Leonardo's equivalents on CA120rd, TPl520, CU492 and CA120rd.

 

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:

Of things of equal size situated at various distances the more remote will be seen under a lesser angle. bd is equal to ce. But ce comes to the eye through an angle that is less than bd to the extent that it is more remote from the point p as the angle cae shows with respect to the angle bad.

    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.

Figs. 1211-1214: Concerning visual angles and distance. Fig. 1211, A9v; fig. 1212, CA319rb; figs. 1213-1214, BM125v.

 

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:

Those bodies of equal size situated in various places are seen through various pyramids which will be that much narrower to the extent that they are further from their cause.

    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:

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Figs. 1215-1218: Euclid's theorem 9 and Leonardo's equivalents. Fig. 1216, BM112r; figs. 1217-1218, G26v (CU956).

On the diminution of things through various distances.

The 2nd object which is as far from the first as the first is from the eye, will appear half as small as the first, even if they are of equal size in themselves.

    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

a dark body seen from further back will appear to you as a minimal, dark, round body. It appears round because the distance so much diminishes particular members that only the larger mass appears.

    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

The shape of a luminous body, even if it participates of the length /i.e. is long/, over a long distance will appear /in the shape/ of a round body.

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Figs. 1219-1222: Demonstrations on BM112r concerning square objects appearing round at a distance.

This is proved in the volume of a candle which, even though it be long, appears round over a long distance and this same can appear to the stars which, even though they are horn shaped like the moon, the long distance will make them appear round.

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

That eye which looks at a lighted candle will appear to see around its light, even though it be long, a round brightness.

The reason is that, /when/ the surface of the eye receives the image of the light, this light brightens and illuminates that aperture of the eye filled with transparent humour. The similitude of the light being in the middle of the surface of that aperture, the power of the imprensiva which sees this light, sees it surrounded by the rotundity of this illuminated aperture. And if it sees other circles around this light, they will originate from the circular parts most distant fromthis aperture, which they retain from the transparent /humour/ which they incorporate from this light.

    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.

It is never a perfect circle but it happens as in the case of a die of lead which, when beaten, is severely flattened /and/ which makes itself into a circular shape, so too does this light over a long distance acquire so much size through all the changes that /although/ the acquisition be equal, the first sum of primary light remains as nothing with respect to this acquisition and hence the uniform acquisition makes it appear round.

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Figs. 1223-1227: On square objects becoming and appearing round. Fig. 1223, BM188r; fig. 1224, Forst.III 63v; fig. 1225, Forst.III 26v; fig. 1226, A9v; fig. 1227, Forst.II 54r.

    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:

That which is said to be /the/ centre is an indivisible part and one should sooner imagine it to be round than any other form. Hence the first part which surrounds the circle is divisible however it may be. Being struck in a square it enlarges to a circle.

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

The cube of extendible material, which will be spread by percussion. If this percussion is uniform it will remove its extremities and from a square it will make itself round.

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

Why over a long distance a long bright object makes itself round with respect to us and /yet/ the horns of the moon do not observe this rule and although the light nearby observes the demonstration of this point?

    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:

Of trees, the boundaries of which show plants remote from the air which makes their background.

The boundaries which the branches of trees have with the illuminated air, to the extent that they are more remote, the more they make themselves into a shape tending to the spherical and to the extent that they are closer, they demonstrate themselves to be remote from such sphericity, as a tree at a at first, through being close to the eye shows the true shape of its branches, which diminishes as at b and is entirely lost /at/ c, where not even the branches of this plant are seen, but the whole plant is recognised with effort.

    This specific example leads him to a more general conclusion:

Every dark body, which is of any shape one wishes, appears spherical over a long distance and this occurs because, if it is a square object the angles are lost after a very short distance and soon after it loses more of its lesser sides which remain and thus, before the whole is lost, the parts are lost, through being less that the whole, as a man in such a situation loses first the legs, arms and the head before the bust, then he loses the extremities of length before those of width and when this occurs, if these were equal only the angles /would/ remain, but they do nto remain and the/object/ is round.

    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:

Of every shape placed in the far distance, the definition (notitia) of the smallest parts is lost first and in the end the maximal parts are reserved, deprived of the definition of the extremities...there remaining an oval or spherical figure of confused bodies.

    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):

And if a bird flies along the line of equality, separating it from the eye (and) it will demonstrate itself in its degrees of motion to acquire degrees of lowness.

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

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Figs. 1228-1231: Euclid's Optics theorem 11, Theon's Optics theorem 11 and Leonardo's equivalents on K123/43/r.

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Figs. 1232-1234: Variants on Euclid's theorem 11 on K121/41/v, K121/41/r and 121/41/v.

When a bird flies along the line (equidistant to the sphere of) of equality (and) it appears that, the more it comes closer to the eye, the more it rises.

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

Let gh be the line of equality; let the bird be g which moves from c to s and let the eye be n. I say that the image of a bird, which rises in height in the pupil with every degree of motion, that it appears to the eye that such a bird rises.

    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):

To the extent that a thing which descends from a higher site, to that extent will it appear at the beginning of its motion that it has to descend closer to the eye which sees it, which the object which descends from a low place does not do.

That which is stated arises from the background of the mobile /object/, which background is the sky on which this mobile /object/ borders, which background, to the extent that the mobile /object/ is lower, the more the eye sees it in a more distant background as when the eye p sees the mobile object, it appears to the eye p to have that much more zenith, being in e than in h, such that falling from d it appears that it must fall closer than /when/ falling from a.

 

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):

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Figs. 1235-1236: Euclid's Optics theorem 12 and Leonardo's equivalent on K120/40/v.

If the eye is in the middle of the course of 2 canals which run to their end along parallel courses,...it will appear that these run towards one another.

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

That which is said occurs because the image of the horses which are impressed upon the eye move towards the centre of the surface of the eye.

 

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

No second object can be that much lower than the first, that to the eye standing above it, the second does not appear higher.

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

Perspective

Among things of equal lowness which are positioned below the eye, that will show itself as higher which is positioned more distantly.

    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."

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Figs. 1237-1239: Euclid's Optics theorem 14 and Leonardo's equivalents on CU494 and CU482.

 

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

And that second object will never be that much higher than the first that, /to/ the eye standing below, the second does not appear below the first.

    On CA225re (c. 1500) he restates this:

Perspective

/With/ such objects of equal height, which are situated above the eye, the one that is more distant will show itself to be lower.

    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.

You see that the first cloud, even though it is lower than the second, appears higher than it, as the intersection on the interposed plane shows you of the pyramid of the low, first cloud at na, and the higher second cloud at nm, below an. This occurs to the extent that you appear to see a dark cloud higher than a light cloud through the rays of the sun in the East or in the West.

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

All those plane the extremities of which are joined by perpendicular lines causing right angles, it is necessary that if they are of equal size, the more that they rise to the eye the less that they are seen and the lower that they are the more one sees their true size.

 

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:

Each of the extremities of these 2 separate bodies, one behind the other, are surrounded by a single common line.

if two...bodies /are/ one behind the other separately their boundaries will be surrounded by a single and common line.

    the interval that is between...

Those bodies, the extremities of which are the boundaries born of a single line appear attached together and no interval or any distance between them will ever be recognized.

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

That angle of which its lines are seen by a perpendicular line appears entirely a single straight and continuous...

    This idea he pursues on BM100r:

The angle will not be judged to be other than a straight line which occupies a straight line at the eye.

That angle will not be judged to be anything other than a straight line which occupies a strictly straight line at the eye.

    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."

(figure)

Fig. 1240: CA251ra.

 

Proposition 23

    In this theorem Euclid claims that

the sphere seen in whatever manner it is, under a single eye, always appears smaller than the hemisphere and the part seen by the sphere appears to be the circumference of a circle.

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

From a single point a ball can never be seen except half or less depending on whether one is less close. The prove of the above: to the extent that the point c sees of the mirror, that amount of the mirror sees it and this occurs at point f. Hence point c, /and/ the pyramidal lines parting from it, make their terminus and base at the point/s/ a /and/ b...can neither see nor be seen by the greater quantity of the spherical body and this said pyramid makes its centre at the centre of the mirror and thus the point f is made, making its base at the points /d/ and /e/.

    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):

At any distance it is impossible that a luminous body can illuminate half an opaque sphere which is shown because the angles adb and bcf cannot be equal on the central line bn.

    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."

(figure)

Figs. 1241-1248: Variations on Euclid's Optics 24. Figs. 1241-1245, CA216rab; figs. 1246-1247, CA251ra; fig. 1248, CA250va.

 

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 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):

To the extent that the luminous body is closer to the umbrous body the less it will illuminate if this luminous body is less than the umbrous body.

(figure)

Figs. 1252-1257: Parallels between light and sight with respect to Euclid's theorem 24. Figs. 1252-1253, CA144vb; fig. 1254, C17r; figs. 1255-1256, CA80va; fig. 1257,CA251ra.

    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):

And likewise conversely, to the extent that the light is larger than the object opposite, the more it will see and the less will be the shadow on the wall pq.

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

It is impossible that the eye sees more of a spherical body which is larger than it, than the sun sees and illuminates.

 

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. ):

If the spherical body is equal to the pupil that sees it, even if it were at an infinite variety of distances, given that it could go forth and that the eye can discern it, it will never be seen either more or less than half.

And this occurs because its diameter...with its ectremities always terminate between equal angles between visual lines which are parallel.

 

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:

If the /two/ eyes together see a spherical object of a diameter less than the space interposed between the pupils of the eyes they will see this spherical body beyond the diameter and the more so as this /object/ is closer to these eyes and consequently such a spherical object appears less than it is to the central lines of visual power of the eyes.

(figure)

Figs. 1258-1260: Variants on Euclid's theorem 26. Figs. 1258-1259, CA129vd; fig. 1260, D4r.

    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 a spherical body less than its pupil will see it more and beyond the diameter that is in this diameter.

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):

No spherical body less than the pupil will ever be seen by a single pupil of which more than half is not seen, it being at any distance you wish. And so much more than half /will it be seen/ to the extent /that/ it is closer and so much less to the extent /that/ it is more remote from the eye.

    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. ):

But if the pupil is less than the spherical object positioned in front of it, this /eye/ will never, at any distance, be able to see half and it will see less the closer it is and more, the further it is.

 

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):

(figure)

Figs. 1261-1263: Euclid's Optics theorem 51 and Leonardo's equivalents on BM227r and 120/110/r.

several sizes being carried at unequal speeds, if the eye is also carried along ont he same side with them, those which are carried at the same speed as the eye seem to stop; those which are slower appear to be carried in the contrary direction and those which are more rapid appear to be carried ahead.

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

Motion

That part of the clouds which is nearer to the eye will appear to be faster than that which is higher and for this /reason/ the one will often appear in contrary motion to the other.

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

When two objects move in a same direction, of which the one is more than the other, the less swift will appear to move in contrary movement, relative to the movement of the more swift, it being without other comparison of an immobile object. And this is seen in clouds which move like the sun, or the moon towards the west or like a boat on the high sea with a horison of air and water.

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

If the air moves and the sky stands firm the summit of the air will leave the ray of the comet turned towards the east and if the air stands firm and the sky moves the ray of the comet /turns/ to the west.

(figure)

Figs. 1264-1266: Euclid's Optics theorem 52 and Leonardo's equivalents on K122/42/r and K122/42/v.

 

Theorem 52

    Here Euclid notes (fig. 1264) that:

if one size which is not carried along shows itself among sizes that are carried along, the size that is not carried along appears to be carried in the opposite direction.

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

If an object which moves its position stays firm with its sight on a star, all objects seen along lines not central appear to it to be swift and to flee in a direction contyrary to that of the eye.

Let us say that an eye b stands firm with its sight on a star d, and bodily moves from b to a and it appears to the eye that its non-central lines, having exchanged the species of the object so often, (and) it appears to them that it is moved in a contrary movement to that of the eye from n to c.

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

When the eye moves...its position standing firm..., to a nearby object, it appears to it that remote objects are most swift and that the first is without motion and that the star moves along the line of eye.

Let us say that the eye a stands firm with its visual power on the object c and that it moves itself bodily from a to b, staying firm with its sight on Uc, that the star d, seen by the non-central lines of the eye will appear most swift to it during the time that the eye goes from a to b and the star appears to have moved the whole part of the sky from d/to/e.

(figure)

Figs. 1267-1268: Euclid theorem 54(1) and Leonardo's variant on BM134v.

Figs. 1269-1271: Euclid's theorem 54(2) and Leonardo's variants on CU810 and H28v.

 

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 Witelo.5 Leonardo is particularly interested in this phenomenon and discusses it on A9r (1492) under the heading:

Perspective of motion

If two objects of equal movement are at various distances from you, they appear of various movements and to the extent that the first enters into the second, to that extent iwll the second object appear slower than the first.

    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:

and the snow from nearby appears swift and from afar /it appears/ slow. And the near snow appears of a continuous quantity in the fashion of white cords and the remote /snow/ appears dis-continuous.

(figure)

Figs. 1272-1274: Euclid's Optics theorem 54)3) and Leonardo's variants on K124/44/r and K123/43/v.

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

Among objects of equal motion, that appears swifter which is (remoter) closer and that which is remoter is that much slower /in appearance/.

Since everything which moves is seen in its background where it terminates and /since/ the remote object is equal motion /with respect/ to the nearer will occupy less background than this close one in the same time /period/, for this reason/ the near object/ occupying a greater space of the background appears that much swifter to the extent that the background which it encompasses is greater.

    This he illustrates (fig. 1273) and explains:

Let a be the eye and let e be the first mobile /object/ and let b be the second. d /he means b/ moves to c in the same time that e moves to f. Dc/i.e. bc/ appears slow because it occupies only the space dc/i.e. bc/ and ef appears to have been made extremely swift in occupying the entire space from d/to/g, which space from d to g has such a proportion with the space dc, as that which is close to the eye is more so than d.

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

If the proportion of the motion of 2 mobiles will be the same as their distance from the eye taken in the same direction the movements of these mobile /objects/ will always appear equal even if they were of near infinite variety.

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

When the pyramid has the semi-diameter of its base /equal to/ 3/4 of its hypotenuse then no object will stop on this hypotenuse and even if this hypotenuse is longer it iwll sustain everything.

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

Among objects of equal movement that iwll appear of slower movement which will be remoter from the eye.
Hence that will appear swifter which is closer to the eye.

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

You see the motion /from/ b /to/ c which is made at the same time as the motion from d /to/ e as far as the eye /is concerned/. The motion nm, which is near the eye, is also made in a time equal to the motion from d /to/ e. Hence because it is close to this eye and 4/5 of the motion /from/ d /to/ e, it will appear 4/5 more swift than this motion from d /to/ e.

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

Prospettiva commune

Among things of equal motion, that will appear slower which is more distant from the eye. Let it be that in equal times there are made equal lengths of motions at various distances, which are, from a to f and from g to k and likewise from l to m. I state that such is the proportion of swiftness to swiftness and from length of motion to length of motion, as there is from distance to distance of the thing seen which moves, to the eye which sees it.

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

And hence let lm be in triple proportion of distance from the eye o with /respect to/ the distance af from this /eye at/ o. I say that the motion lm appears in swiftness and in length to be one third of the motion from a to f made in the same time and motion. This is proved because at the distance af, from the eye o, it is demonstrated that lm is only moved through the space cd when a is moved to f and hence it is found that the space cd enters 3 times into the space af. Therefore this space afa is triple the space cd and because the one motion and the other are made during the same time, the motion af appears three times more swift than the motion cd, which is that which was to be proved.

    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.

(chart)

Chart 26. Parallels between Euclid's Optics and Leonardo's Notebooks.

    There were, of course, some classical precedents. Pseudo-Aristotle in De coloribus 6 mentioned the problem, as did Ptolemy in his Optics7 and Galen in De usu partium8 broached it mroe explicitly:

Each object seen appears not alone or isolated but always accompanied by something else, because the visual rays surrounding it fall sometimes on objects beyond the body at which one is looking and sometimes on objects near it.

    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:

Among bodies of equal size and length and of equal shape and darkness, that will appear smaller which is surrounded by a more luminous background.

He elaborates on this idea on C24r (1490):

Among bodies of equal quality, which are equally distant from the eye, that will appear of a smaller shape which is surrounded by a more luminous background.

Every conspicuous body is surrounded by light and shade. That body of equal rotundity which is surrounded by light and shade appears to have one part that much larger than the other, to the extent that one is illuminated more than the other.

    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:

Among things of equal size and brightness of background and length, that object which is of a brighter surface will appear of a larger shape. /A piece of/ iron of equal size, half /of which is/ glowing is the proof, because that which is glowing appears larger than the rest.

    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:

That luminous body will show itself of lesser size which is surrounded by a more luminous background and that luminous body will show itself as larger which confines with a darker background as is shown at night in the heights of buildings when there is a flash behind them such that it immediately appears that the flashing diminishes the building in its height. And from this arises that these buildings appear larger when there is fog or at night than in purified and illuminated air.

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

(figure)

Figs. 1275-1277: Detail from Botticelli's Adoration (London, National Gallery) showing a man in a gateway; an equivalent figure on A1r and a geometrical version on BM97r.

If the window ab sends the sun into a house..., this sun will increase the size of the window and will decrease the shadow of the man in such a way that when the said man brings his decreased shadow close/r/ to that which the true size of the window bears, he will see above him the contact of the shadows lost and confused by the power of the light, close up and not let the solar rays pass, and the shadow made by the man on this contact /has/ the effect precisely as is drawn above.10

    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):

The luminous aperture seen by a dark place, even if it is of uniform size will nonetheless appear to contract strongly near some object interposed between the eye and such an aperture.

That which is stated is proved by the 7th of this which demonstrates that the boundaries of any object interposed between the eye will never be seen clearly but...confusedly through the air which darkens near these boundaries, /and/ which darkness, the closer it comes to these boundaries, the more powerful it is.

    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:

(figure)

Fig. 1278: Demonstration on F31r concerning contrast effects.

That thing which is seen in dark and turbulent air, being white, appears a larger shape than it is. This happens because a bright object increases against a dark background.

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

That thing which is seen in dark and turbulent air, being white, appears larger in size than it is. This happens because, as I have said above, the bright object increases in a dark background, for the reasons assigned before.

    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.

Any bright object (which will be seen against a background of) will show itself to be of a larger size which is seen against a background of a darker colour.

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

That object of uniform size and colour which will be seen against a background of disuniform colour will show itself of disuniform size.

And /if/ an object of uniform size and of various colours be seen against a background of uniform colour, this object will show itself of various sizes.

And to the extent that the colours of the background or of the object against the background will be of a greater variety of colours, the sizes will appear more varied even though the objects seen against the background are of the same size.

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

Bright objects against a dark background show themselves larger than they are and dark objects against a bright background diminish in size.

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

The first part /of that which is/ proposed appears clearly in a rod of iron of uniform thickness, of which one part of its length is glowing. Then it will be shown that its former equality /in thickness/ will have become very disform, because where it is glowing it appears considerably thicker than where the iron remains dark.

    He now uses the same example to illustrate the converse:

The second part of that which was said above is shown by the aforementioned glowing iron when there is placed between it and the eye the /part/ of the iron /rod/ that is not glowing with a line intersecting to the glowing /part/. Then the /part of the/ iron /rod/ which is not glowing will show itself of that much lesser size with respect to the glowing iron, than in any part of its length, even if it be of equal thickness.

    In brackets he adds an explanation for this phenomenon:

And this occurs because the spirits /which are/ spread through the visual power, being overcome by the superfluous light, cause all the pores to contract throughout the entire cornea (luce ), or the parts in every part of this cornea. Hence wehre such a contraction is generated there the things are seen with less power and size than otherwise.

(figure)

Figs. 1279-1280: Preliminary experiments concerning contrast effects on Mad. II 23v

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

The dark object seen against a luminous background will appear much less than if it were seen against a background darker than it.

Let the experience be made by placing a rule or rod of iron of equal width on a /piece of/ white paper, illuminated by the sun, in such a way that half of the rod stands against this illuminated /piece of/ paper and half beyond this paper, on the pavement where the sun does not percuss. Then one will see such a rod /as being/ of various thicknesses, namely, that that part which is in front of the illuminated paper is thin and that which is above the shadow will be thick.

    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:

A /and/ b are the eyes; h is the sun which shines on the pavement of a home on the paper nm and en is the dark pavement, that is, without paper or sun; cd is the rod, pd is the part in the sun and n is the part in shadow.

    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:

The dark object, seen against a luminous background will show itself of lesser size than if it were in front of a background less obscure than it.

(figure)

Figs. 1281-1283: Experiments concerning background and apparent size on Mad.Ii 23v, 27v and 28v.

(figure)

Figs. 1284-1286: Effects of background on apparent size on F22r.

    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:

That part of the (opaque) dark object of uniform thickness and colour will show itself as thinner which will be seen against a more luminous background.

And the part of the luminous body of uniform thickness and brightness will appear thicker and will be seen against a darker background.

    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:

E is the given body, dark of itself, and of uniform thickness; ab and cd are dark backgrounds, the one more so than the other, bc is a luminous background as if it were a place percussed by the sun through an aperture in a camera obscura.

I say that the object...eg appears thicker in ef than in gh, because ef has a darker background than this gh...and the part fg also appears thinner in being seen by the eye o in the background bc which is clear.

    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:

That luminous body will show itself of a smaller size at an equal distance which will lose more brightness.

The rod of iron glowing along one part of its length will show this, (it) being in a dark place, which /rod/, even if it be of a uniform thickness will show itself /to be/ considerably thicker in the glowing part and the more so to the extent that it is hotter. The reason why follows.

    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.

In fog parallel towers in the long distance show themselves thinner at their base than at their summit because the fog which acts as a background is denser and whiter below than above. Whence by the 3rd of this, which states: 'the dark object placed in a white background diminishes its size to the eye' and the converse which states 'the white object placed in a dark background shows itself larger than in a bright background,' the base of the dark tower, having as its background the whiteness of the low and dense...fog, it follows that this fog increases in evidence around the lower boundaries of such a tower and diminishes them in distance, which such a fog cannot do in the upper boundaries of the tower, where the fog is thinner.

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

Among objects of equal obscurity, size, shape and distance from the eye that one will show itself as less which is seen against a backgrouind of greater brightness and whiteness.

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

This the sun seen behind plants without leaves teaches, that all its ramifications, which are found facing the solar body are so much diminished that they remain invisible; a rod interposed between the eye and a solar body does the same.

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

Whether parallel objects placed in front of one, when seen through fog, have to show themselves thicker at their summit than at the base. This is proved by the 9th, which states: the fog or thick air penetrated by solar rays will show itself that much whiter as it is lower.

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

Objects seen from afar are disproportionate and this arises since the brighter part sends its image to the eye with a more vigorous ray than does its dark part. And /hence/ I see a woman dressed in black with a white hat on her head which shows itself twice as large as the size of her shoulders which are dressed in black.

(figure)

Fig. 1287: Effects of background on apparent distance on L77v.

    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):

When I used to be at sea equidistant from the beach and the mountain, that /i.e. the distance/ of the beach showed itself as being much longer than that of the mountain.

 

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):

How bodies accompanied by shadow and light always vary their boundaries in colour and light /depending/ on that thing which borders with its surface.

If you see a body, part of which borders and edges on a dark background, the part of this light which appears brighter is that which borders with a dark /one/ at d. And if this said illuminated part borders on a bright background, the boundary of this illuminated background will appear less bright than before and its maximal brightness will appear between the boundary of the background at f and the shadow.

(figure)

Figs. 1288-1290: Background effects on A113r, CU750 and CU645.

    He notes that the converse is equally true:

And the same happens to the shadow, insomuch that the boundary of that part of the umbrous body which borders on a bright area at l will appear of much greater darkness than the rest. And if the said shadow borders on a dark background, the border of the shadow will appear lighter than before and its maximal obscurity will be between the boundary and the light at the point o.

    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:

That luminous body will appear of greater brightness which is surrounded by greater darkness.

When the luminous body is surrounded by a greater darkness then it will appear of greater brightness.

    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.

As much will be lacking from the extremities of the shadow of bodies, as is touched by the illuminated or luminous background.

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

Of the backgrounds of depicted things.

Of the greatest dignity is the discussion of backgrounds in which opaque bodies invested with shadow and light are situated, because for these it is convenient to have the illuminated parts in dark backgrounds and the obscure parts in bright backgrounds as is partly demonstrated in the margin.

(figure)

Fig. 1291: Demonstration concerning light, shade and background on CU145.

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

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

No object appears of its natural brightness, because the sites in which these are seen, render it that much more or less white to the eye, to the extent that such a site is more or less dark. And this is taught by the moon, which by day shows itself in the sky /as being/ of little brightness and at night which such splendour, that it sends from itself the image of the sun and of the day /light/ with its expelling of shadows. And this arises from two things. And the first is the standard of comparison which has in itself the nature of showing things that much more perfectly in the species of their colours to the extent that they are more disform.

    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.

In a reflection of the same shape, size and power, that part will show itself more or less powerful which will border on a background that is either more or less dark.

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

Where reflections are most perceptible.

That reflection will be the more evident which is seen in a background of greater darkness, and will be less evident which is seen in a brighter background. And this occurs because /with/ things of varying darkness placed in contrast, the less dark makes that which is darker appear tenebrous and /with/ things of varying brightness, placed in contrast, the whitest makes the other appear less bright than it is.

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

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

The reflections of lights are of a greater or lesser evidence to the extent that these are seen in backgrounds of greater or lesser darkness.

And this occurs because if the background is darker than the reflection, than this reflection will be very evident on account of the great difference that these colours have among themselves. But if the reflection be seen in a background that is brighter than it, then this reflection will show itself to be dark with respect to the whiteness on which it borders and thus such a reflection will be imperceptible.

    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.

Of the lustres of bodies of equal smoothness that will have more difference from its background which is generated in a blacker surface and this arises because the lustres are generated on polished surfaces which are practically of the nature of mirrors. And because every mirror render to the eye that which it receives from objects, hence every mirror which has the sun as its object renders this sun a same colour and the sun will appear more powerful in a dark background that in a bright background.

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

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

Among lustres of equal power that shows itself of a more excellent brightness which is in a darker background. This is the same as /the/ above, but it is different /in/ that that speaks of the difference which this /lustre/ has from its background and this of the difference which a lustre has in a black background and of the lustre generated in other backgrounds.

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

On lights

That light will show itself as brighter which approaches further toward darkness and will appear less bright which is closer to other luminous parts of the object.

    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.

Among lights of equal clarity, that appears more powerful which will be less and which is surrounded by a darker background.

    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:

Why the illuminateed background appears brighter around the derivative shade standing in a house than in the countryside.

The bright background which surrounds the derivative shade is brighter nearer this shadow than in a more remote part. And this occurs when such a background receives light from a window and this does not happen in the country side and why this occurs etc. will be defined in its place in the book of light and shade.

    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:

The object seen inside a habitation illuminated by particular and high light from some window will show a great difference between the light and its shadows and maximally if the habitation bne large or dark.

    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.

The more excellent brightness of derived light is where it sees the entire luminous body with half of its right or left umbrous background.

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

This is to be proved and let the luminous body be bc and let its umbrous background to the right and left be dc and ab. And let the umbrous body less than the luminous body be nm and the wall ps is where the umbrous and luminous species are impressed.

I say therefore that on this wall ps at the point r will be a more excellent brightness of the light than in any other part of this pavement.

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

This manifests itself because at r the entire luminous body bc is seen with half of the dark background ad, namely ucd, as the rectilinear concourse of the umbrous pyramid cdr shows and the luminous pyramid ubcr. Therefore at r one sees as great a quantity of the dark background cd to the extent that the luminous body ubc /is large/. But the part s sees the umbrous body ab and also sees the umbrous body cd which two spaces are equivalent to double that of the luminous body bc.

But to the extent to which you move from us to r, the more you will lose of the darkness ab. Therefore from us to r the pavement sr will continually brighten. Again if you move from r to o, you will see continually less of the luminous body and for this /reason/ the pavement ro becomes darker the more one approaches o.

And through such a discussion we have proved that r is the brightest part of the pavement oc.

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

On the bright and dark images which impress themselves on the umbrous and luminous places positioned between the surfaces and the bottom of clear waters.

When the images of dark and luminous bodies impress themselves on the dark and illuminated parts of bodies interposed between the surface and the bottom of waters, then the umbrous parts of these bodies will make themselves darker which are covered by umbrous images and their lumionous parts iwll do the same. But if, over the umbrous and luminous parts, luminous images are imprinted, then the illuminated parts of the aforesaid bodies will make themselves of greater brightness and their shadows will lose their great darkness and such bodies will show themselves as being of less relief than the bodies percussed by dark images. And this occurs because, as was said, the umbrous images augment the shade of umbrous bodies which, even if they are seen by the sun, which penetrates the surface of the water, and making them very different from the lights of these bodies, adding shade to the darkness of the dark image which is mirrored in the surface (pelle) of water and thus such a shadow of these bodies is augmented, making them darker.

And even if such an image tinges with itself the illuminated parts of such submerged bodies they are not lacking the brightness which the percussion of the solar rays gives them, even if they are somewhat altered by this dark image, it hurts little, because there is so much support that it gives to the umbrous parts that the submerged bodies have sufficiently more relief than those which are altered by the luminous image. Which, even if their illuminated parts are lightened like the umbrous ones, the alteration of these umbrous parts are of so much brightness that such submerged bodies will show little relief in such a site.

(figure)

Fig. 1292: Underwater background effects on CU546.

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

Let it be that the sea (pelago) nmtu /has/ jars or herbs or other umbrous bodies at the bottom of the brightness of the waters which takes its light(s) from the solar rays that emerge from the sun d and that one part of a jar has over it the dark image which is mirrored in the surface of such water and which, in another part of the jar, has on it the image of the air bcsm. I say that the jar covered by the dark image will be more visible than the jar which is covered by the brightness of the bright image.

    An explanation why this is so follows:

An the reason is that the part percussed by the dark image is more visible than that which is percussed by the illuminated image because the visual power is overcome and hurt by the part of the water illuminated by the air, which is mirrored in it and thus such a visual power is augmented by the darkened part of this water. And in this case the pupil of the eye is not of uniform power, because from the one side it is hurt by too much light and from the other /side it is/ augmented by darkness.

Therefore that which was said does not arise except from causes which are remote from such waters and such images, because such a thing only originates in the eye which is hurt by the brightness of the air and augmented on the other side by the dark image.

 

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?

That shadow will participate more in black which is generated in a whiter surface and this originates because white is not numbered among the colours and is receptive of every colour and its surface participates more intensely in the colours of its objects than any other surface of any other colour and maximally by its direct contrary, which is black or other dark colours from which white is more remote by nature and for this /reason/ there appears and there is a great difference between its principle shadows and its principle lights.

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

The eye which finds itself in the middle, between the shadow and the lights surrounding the umbrous bodies will see in these bodies the greatest shadows that are in these when looking at it under equal angles, i.e., of the visual /angle/ of incidence.

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

Of light

That light will be of greater quantity which is generated on a body of lesser curvature, such a light being produced by a same cause...

    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.

First. The dilation and contraction of shadows of rather the greater or lesser size of the shadows and lights on opaque bodies will be found in the greater or lesser curvature of the parts of the bodies where they are generated.

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

Of the greater or lesser obscurity of shadows.

Second. The greater or lesser darkness of the shadows is generated in the more curved parts of the members and the less obscure will be found in parts that are flatter (piu larghe).

    A third proposition follows:

Where the shadows deceive the judgment, what gives an indication of their greater or lesser obscurity?

Third. Among shadows of equal obscurity, that will show itself as less obscure which will be surrounded by a light of a lesser power as are the shadows which are generated among reflected lights. Hence you, o painter, should be mindful not to deceive yourself in varying such shadow.

    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:

What colour of a body will make a shadow that is more different from the light, that is, which will be darker?

That body will have its umbrous parts more remote from brightness with respect to its illuminated parts which is of a colour closer to white.

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

Which background will render shadows darker?

Among shadows of equal darkness that one iwll show itself darker which is generated in a background of greater whiteness. It follows that that appears less obscure which is in a darker background.

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

This is proved within a same shadow because its extreme part which on the one side borders with a white background appears most dark and on the other side, where it borders on itself, it appears of little darkness. And let the shadow of the object bd be made on dc, which appears blacker at nc because it borders on a white background ce, than in nd which borders with the dark background nc.

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

How derived shadow, being surrounded in all or in part by an illuminated background is darker than the primitive.

Derived shadow wh ich is entirely or in part surrounded by a luminous background will always be darker than the primitive shadow which is on a plane surface.

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

Let the light be a and let the object which retains the primitive shadow be bc and let the panel de be that which receives the derived shadow in the part nm and its remainder dn and me, remains illuminated by a. And the light dn reflects in the primitive shadow bc and the light me does the same. Hence, the derived /shadow/ nm, not seeing the light a remains dark and the primitive is illuminated by the illuminated background which surrounds the derived. And hence the derived is darker than the primitive /shadow/.

    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

The shadows made in the shadows of umbrous bodies do not have to be of that evidence as are those which are made in the luminous parts of the same bodies, nor do they have to be generated by primitive light, but by derived /light/.

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

That shadow will show itself as darker which is surrounded by a more splendid whiteness and, by the contrary, it will be less evident where it is generated in a darker background.

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

That shadow will show itself as darker which is in a whiter background. The boundaries of that derived shade will be better noted which are closer to the primitive shadow. The primitive shadow will have the boundaries of its impressions better noted which are cut under angles which are more equal on its panel.

That part of a same shadow iwll show itself as darker which will have darker objects opposite it. And that part will show itself as less dark which is seen by a brighter object. And tha tobject which is larger will be brighter and that dark object which is of a grater quantity will render darker the derived shadow in the site of its percussion.

    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.

When you draw some object remember then when you compare the power of the lights of its illuminated parts, that the eye is often deceived, estimating as brigther that which is less bright and the cause originates through the comparison(s) of the parts which border with them, because if you have two parts of unequal brightness and if the less bright borders on dark parts and the brighter borders with brighter parts such as the sky or similar bright objects then that which is less bright, or if you wish to say, lucid, will appear as more lucid and the more lucid will appear darker.

 

3.4 Colour

    Aristotle in his Meteorologica had noted that:

in woven and embroidered stuffs the appearance of colours is profoundly affected by their juxtaposition with one another (purple, for instance, appears different on white and on black wool) and also by differences in illumination.11

    Ptolemy, in his Optics,12 touched on similar phenomena, as did his Mediaeval successors Alhazen13 and Witelo.14 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

The various comparisons of different qualities of light and shade often make uncertain and confused the painter who wishes to imitate and counterfeit the things he sees. The reason is this: if you see a white hat bordering on a black one, it is certain that that part of this white hat which borders on the black will appear much white than that which borders on a greater whiteness. And the reason for this is proven in my perspective.

    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

Among colours of equal perfection that will show itself of greater excellence which is seen in the company of a colour that is directly opposite.

Directly contrary is pallid with red, and black with white, even though neither the one nor the other is a colour, azure and golden yellow, green and red.

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

Every colour is better comprehended in its contrary than in its like as in the case of the dark in the bright and the bright in the dark.

The white which borders on the dark causes that at these boundaries the dark appears blacker and the white appears whiter.

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

That part of a white /object/ will appear whiter which is closer to the confines of black and likewise /that/ will appear less white which is further from this dark.

that part of the black appears darker which is closer to the white and likewise /that part/ will appear less dark which is further from this white.

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

Precept C

Among things equally dark and equally distant, that object will show itself to be darker which borders on a whiter background.

    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:

On the colours which are shown to vary from their essence through the comparison of their backgrounds.

No boundary of uniform colour will show itself to be equal if it does not terminate in a background of a colour similar to itself.

This is seen manifestly when black terminates with white and white with black, each colour appears more noble in the confines of its contrary than it does at its centre.

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

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

A white object will show itself /as/ whiter which is seen against a darker background and will show itself darker which is in a whiter background and this the gleaming (fioccare) of snow has taught. When we see it against the background of the air it appears dark and when we see it against a background of some open window, through which one sees the darkness of the shadow of this house, then this snow iwll show itself as very white.

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

On the nature of comparisons

Black vestments make the skin of human faces appear whiter than they are and white vestments make the skin appear dark and yellow vestments make them appear coloured and red vestments show them pale.

    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.

Among backgrounds that are appropriate for illuminated or shaded boundaries of any colour, those will be more distinct from one another which are more varied, that is, that a dark colour should not terminate in another dark colour, but one that is very different, that is, white or participating of white and similarly a white colour should never terminate on a white background, but, as far as possible, in a dark /background/ or /one/ partaking of dark.

    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.

When the colour of a white object comes to terminate in a white object, then the whites will either he white or not and if they are equal then that which is closer will become somewhat dark/er/ at the boundary which it makes with this white and if this background is less white than the colour that borders in it then the bordering colour will stand out of itself from the /colour/ different from it without other help from a dark background.

    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

Here one wishes to note what part of a same colour will show itself as more beautiful in Nature, whether it is that which has lustre or that which has the light or that of the middling shade or that which is dark or indeed that which is transparent.

Here one must determine which colour it is of which it is asked, because different colours have their beauty in different parts of themselves. And this shows that black has its beauty in the shadows and white in the light and azure and green and tan in the middling shade and yellow and red in the light and gold in reflection and lacquer in the middling shadow.

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

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

The surface of that opaque body will show itself of a more perfect colour which can have as a near object a colour similar to it.

    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:

Why black bordering on white does not show itself as blacker than where it borders on black, nor does white show itself as whiter in bordering on black than on white, as do the species which pass through an aperture or the boundary of some obstacle.

    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:

Now note that if you wish to make an excellent darkness make it through a comparison with an excellent whiteness and likewise you make an excellent whiteness with maximal darkness and a pale colour will make red appear a more fiery rose and this rule will be more distinct in its proper place.

    A second rule follows:

There remains a second rule which does not try to make the colours in themselves of the most supreme beauty that they naturally are, but that company /of another colour/ renders grateful the one and the other, as does green to red and red to green such that the one renders the other graceful reciprocally as does green with blue and here is a second rule, generating from ungraceful company, as blue with yellow which whitens or with white and the like which will be discussed in their /proper/ place.

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

On the boundaries of objects

Among objects of equal brightness that will show itself of lesser brightness which will be seen in a background of greater brightness and that will appear whiter which borders on a space which is darker.

And the skin will appear paler in a red background and the pale /colour/ will appear reddish, being seen in a yellow background and similarly the colours will be judged to be what they are on the basis of the backgrounds which surround them.

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

Backgrounds

On the backgrounds of figures, that is, the bright in the dark and the dark in the white background.

Of white with black or black with white, the one appears more powerful through the other and likewise the contrary, the one always shows itself more powerful.

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

Why the boundaries of opaque bodies sometimes show themselves as brighter or darker than they are.

The boundaries of shaded bodies show themselves /as/ brighter or darker than they are to the point that the background which surrounds them is darker or brighter than the colour of that body which borders on them.

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

Of colours

Of colours of equal brightness that will show itself brighter which is of a darker background. And black will show itself as darker than it is in a background of greater brightness.

And red will show itself more fiery than it is in a background that is more yellow and thus will do all the colours surrounded by their directly contrary 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

No evident body can be well comprehended and judged by human eyes except through the variety of the background where the extremities of these bodies terminate and border and no body, as far as the lines of its extremities /alone are concerned/.

    To illustrate this he cites a concrete example:

The moon, even though it be very distant from the body of the sun, when through eclipses it finds itself /interposed/ between the sun and our eyes, because this moon borders on the sun, it appears to human eyes to be conjoined and attached to this sun.

    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:

The luminous body which is seen along the line of incidence of the light will not show any luminous part of itself to the eye.

For example, let the umbrous body be a. Let c be the light. Let cm and cn be the luminous lines of incidence, that is, those lines which transfer the light to the object a.

Let the eye be at the point b. I say that since the light c sees the entire part mn, that the relief that is there will be fully illuminated. Hence the eye positioned at c will not be able to see shadow and light, and not seeing this every part will appear of one colour, whence the differences between the eminent and globulent parts do not appear.

    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.


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Last Update: July 9, 1999