SUMS

Dr. Kim H. Veltman

Seven Books on Light and Shade


1. Introduction
2. Book One Light and Shade
3. Book Two Primary Shade
4. Book Three Derived Shade
5. Book Four Derived Shade and Interposed Objects
6. Book Five Derived Shade and Reflected Light
7. Book Six Reflected Colour
8. Book Seven Reflected Colour and Distance
9. (Book Eight) Movement of Shadows
10. Conclusions

 

1. Introduction

    A careful study of Leonardo's notes on light and shade reveals a much more systematic approach than is at first apparent. He himself considers various ways of arranging these notes. On BM171r (c.1490), for instance, he writes: "You must first describe the theory and then the practice. First you will describe the shade and light of dense bodies and then of transparent bodies." Much more extensive is his outline on CA250ra (c.1490). This begins with a draft:

Proemium
Having treated of the nature of shadows and their percussion, I shall now deal with the places...in which these shadows are touched and of their curvature or obliquity or straightness or of whatever quality...I could find.

    This he crosses out and begins anew:

Shade is a privation of light
It seeming to me that shade is of the greatest necessity in perspective, because without these opaque bodies and cubes are only poorly recognized unless they terminate in a background of a different colour from this body, and this I propose...in the first proposition of shade, and I state it in this form, how every human body is surrounded...and superficially clothed with shade and light and on this I build the first book. Besides this, these shadows are of various qualities of darkness, because they are...abandoned by various quantities of rays..., and these I call original shadows, because they are the first shadows that invest bodies where they are attached. And on this I shall build the 2nd book,... From these original shadows there result umbrous rays which go spreading out through the air...and they are of as many qualities as are the varieties of original shadows from which they derive and for this [reason] I call these shadows, derivative shadows, because they are born from other shadows, and on this I shall make the 3rd book. Again these derivative shadows, in their percussions, make as many various...effects as are various the locations, where they percuss and of this I shall make the fourth book. And because,...the percussion of the derivative shade is always surrounded by the percussion of luminous rays which, through a reflected concourse, bouncing back towards their source, find the original shadow and mixes itself and converts itself into this, altering its nature somewhat, and on this I shall build the fifth book. Besides this I shall make the sixth book, in which will be treated the various and many diversifications of reflected rays [that are] bouncing back, which alter the original with as many various colours, as are the various locations, whence these reflected luminous rays derive. Moreover, I shall make the seventh division concerning the various distances which exist between the percussion of the reflected ray and the place whence they are born, and the equally various similitudes of colours which this [ray] brings in the percussion of the opaque body.

    The themes of these seven books are summarized in Chart 8 and have been used as a starting point for a reconstruction of various chapters these books might have been contained. In this reconstruction, an attempt has been made to indicate not only the chapters that he foresaw in 1490, but also the modifications resulting from sub-sequent researches (Chart 9).

Books Themes
1 Light and Shade
2 Primary Shade
3 Derived Shade
4 Derived Shade and Interposed Objects
5 Derived Shade and Reflected Light
6 Reflected Colour
7 Reflected Colour and Distance

Chart 8: Survey of Themes of the Seven Books on Light and Shade drafted on CA250ra (c.1490).

 

BOOK ONE: LIGHT AND SHADE
1.   Punctiform Propagation
2.   Central Lines
3.   Opposing Theories

 

BOOK TWO: PRIMARY SHADE
1.   Definitions
2.   Degrees

 

BOOK THREE: DERIVED SHADE
1.   Kinds of Light
2.   Light Source: (a) Equal to Opaque Body
  (b) Smaller than Opaque Body
(c) Larger than Opaque Body
(d) Comparative Sizes
3.   Object: (a) Comparative Distances
(b) Comparative Sizes*
(c) Comparative Sizes and Distances*
4.   Eye: (a) Comparative Positions

 

BOOK FOUR: DERIVED SHADE AND INTERPOSED OBJECTS
1.   Introduction
2.   Degree of Light+
3.   Angle and Intensity of Light*
4.   Angle and Intensity of Shade
5.   Position and Intensity of Light Source
6.   Size/Shape of Light Source/Object
7.   Position/Shape of Projection Plane
8.   How/Why One Light Source and One Object Produce Two Shadows+
9.   Compound Shade: (a) Preparatory Studies
(b) Multiple Lights and Objects
(c) Columns*
(d) Experiments with St. Andrew's Cross*
10.   Conclusions

 

BOOK FIVE: DERIVED SHADE AND REFLECTED LIGHT
1.   Introduction and Basic Propositions
2.   Lustre
3.   Elementary Demonstrations
4.   Interposed Rods
5.   Interposed Walls
6.   Theoretical Demonstrations                 *

 

BOOK SIX: REFLECTED COLOUR
1.   Introduction
2.   Mirrors                                               *
3.   Water                                                 *
4.   White Objects                       *
5.   Faces                                                  *
6.   Landscape and Verdure                       *
7.   Yellow, Azure and Green                     *
8.   Walls                                                   *
9.   Light and Pigments                  *
10.   Further Demonstrations                      *
11.   Precepts                                            *
12.   General Statements               *
13.   Conclusions

 

BOOK SEVEN: REFLECTED COLOUR AND DISTANCE
1.    Introduction
2.    Demonstrations                                    *
3.    General Statements                 *
4.    Links with Perspective                          *

Chart 9: Reconstruction of chapters of the Seven Books on Light and Shade outlined by Leonardo (CA250ra, 1490). Chapters marked with an + indicate themes which he considered in 1490 but subsequently develops. Chapters marked with an * indicate themes that he does not mention in 1490 but discusses at a later stage, especially after 1505.

    Leonardo's researches lead him to consider alternative schemes of organization. On CU841 (TPL673, c.1490-1495), for instance, he considers a fourfold scheme, later adding a fifth, in a passage headed:

Of the four things which one needs to consider primarily in light and shade.

The principal parts which one needs to consider in painting are four, namely, quality, quantity, site and shape. By quality is understood which shade or part of shade is more or less dark. Quantity, that is, how large such a shade is with respect to others nearby; site, that is, in what way things need to be situated and to what part of a member that it attaches itself. Shape, that is, what is the shape of this shadow, that is to say, whether it is triangular or participates of the round or the square, etc.

The aspect of the shadow is also to be numbered along with the parts of shade, that is, whether the shadow is long, [and] to see in which aspect the sum of such a length is directed, whether the shadow of a brow is directed towards the ear, whether the lower shadow of the eye socket is directed towards the nostril of the nose and likewise with similar encounters of the various aspects to situate these shadows. Hence the aspect is to be placed before the site.

    On CU843 (TPL739, 1508-1510) he outlines a further book that he intends to write on the subject:

On the master shadow, which stands between the incident and reflected light.

Note the true shape which the master shadow has, which interposes itself between the reflected and the incident light. Such a shadow is not intersected, nor does it have a boundary, except with the object...to which it attaches itself. And its sides are of various distances from its centre and of various boundaries of this incident and reflected light, such that it sometimes shows clear boundaries and sometimes imperceptible boundaries, sometimes it bends from its rectitude, sometimes it observes rectitude, sometimes the boundaries are at unequal distances from the middle of the principal shadow.
And about this you will compose a book.

 

Books Themes
1. On the usefulness of shadows
2. On the motion of shadows
3. On the shape of shadows
4. On quality
5. On quantity
6. On boundaries
7. On simple shade
8. On compound shade
9. On decompounded shade
10. On darkness
11. On light
12. On light penetrating through apertures of different shapes
13. On light passing through various numbers of apertures
14. On the composition of multiple luminous rays
15. Whether it is possible that rays which penetrate one another depart from a same luminous body
16. Whether parallel rays can [come] from a single light and penetrate through some apertures

Chart 10: List of themes concerning light and shade outlined on CA277va (c.1513-1514). The book numbers have been added by the author.

    On CA277va (c.1513-1514) he composes a completely new list of sixteen chapter headings (see chart 10). At about the same time, on W19076r (K/P 167r, c.1513) he reminds himself to: "Reserve for the last [book] on shade, the figures1 that appear in the study of Gerard the miniaturist at San Marco in Florence." On this same folio Leonardo proposes to include themes relating to light and shade in his treatise of painting:

Do the rainbow in the last book on painting, but first do the book on the colours originating from he mixture of other colours such that, through these colours of painters, you can produce the generation of colours of the rainbow.2

    Hence he himself remains undecided about the final arrangement of his notes on light and shade. Any attempt at a reconstruction of his intended treatise must therefore remain tentative. Our concern is to understand the chief themes that preoccupy him and gain insight into the systematic aspects of his approach. As will be shown, the scheme of seven books outlined on CA250ra (c.1492) lends itself admirably to these concerns. In addition we shall examine his notes on the movement of shadows. Almost all these demonstrations involve light and shade in the open air. In a subsequent chapter we shall show that these demonstrations are parallelled by others involving camera obscuras which theme will lead in turn to the physiology of vision.

 

Book One: Light and Shade

Every opaque body is surrounded and its whole surface is enveloped in shadow and light on this I shall build the first book (CA250ra).

    The purpose of Leonardo's first book on light and shade is to show that every body has its surface covered with luminous and umbrous rays. Had he actually managed to write it, this book might well have had three chapters beginning with a first on punctiform propagation, a second on the role of central lines and a third to deal with opposing theories.

 

Book One - 1. Punctiform Propagation

    We have already analysed the earliest of Leonardo's extant notes on W19147v (K/P 22v, figs. 176-177, c.1489-1490) to demonstrate that light originates in a point and that its punctiform propagation spreads everywhere, or, as he puts it, "all in all and all in every part" (see above p. ).

    These early demonstrations lead him to conclude (W19147v, K/P 22v, 1489-1490):

It will appear clear to experimentors that every luminous body has in itself a hidden centre from which and to which arrive all the lines generated by a luminous surface and from there they return or leap back outwards and unless they are impeded they are dispersed by an equal distance through the air.

(figure)

Figs. 176-182: On the properties of a large light source in front of a small object. Figs. 176-177; W19147v (K/P 22v); fig. 178, CA262v; figs. 178-183, A97r.

    On A97r (BN 2038, 17r, 1492) he returns to this theme:

On light which operates in all its quantity in a single luminous centre.

If a large sphere illumines another sphere much less than it, it would be fitting that if the luminous rays parted from the surface of the light, that the lesser light would be surrounded and illuminated by more than half. If it were so then the shadow, the further it is from its cause, would be smaller until the end. But experience shows the contrary, because the lights of candles which are long and narrow, when they illuminate a spherical body, the shadow on the wall which ought to be round would be long and low.

    He illustrates this with a concrete example (fig. 183 cf. 182):

For this reason, let us suppose that ab, and cd are the height and length of a light. If its surfaces are to function then ab will illuminate that much more than one half of the spherical body to the extent that pn is from ys and the shadow on its wall will appear much smaller than it is on the spherical body.

The surfaces of the width of the light, at cd, will illuminate the spherical body at py, that is, in the middle. It being thus, the shadow will go wide like that of the wall. Hence, the shadow on the ball will be wide and low and like this will be that of the wall which, since it is demonstrated by experience that it is of a round form and always larger than its cause, it is convenient to abandon both of the above demonstrations and to confess that the centre of every light is the cause of shadow.

(figure)

Figs.184-186: Demonstrations how a large light source in front of a small object produces expanding shade. Fig. 184, A97r; fig. 185, A109v; fig. 186, CU628.

    To accompany this passage he draws a preparatory sketch (fig. 182) which he develops (fig. 183) and labels "example." He illustrates the case of a small light source in front of a large opaque body (fig. 180) but crosses this out. He also illustrates how a large light source in front of a small object would theoretically produce a shadow converging towards f (fig. 179 cf. 176). Beneath this diagram he writes "proposition." Next he draws another diagram beneath which he writes "conclusion" (fig. 181, cf. 177). This demonstrates how such a light actually produce an expanding shadow. Immediately following he describes an experiment to verify this (fig. 184):

And the experiment is made as follows: let ab be the wall, cd the ball or a line and let e be the light. Measure how much [the distance] is from the light to the wall and from the line to the light. Then measure the shadow and make two lines which are equal to the distance from the wall to the light, and as large as the shadow, and in these lines observe whether the length of the line cd exceeds or is smaller than these lines.

    On A109v (BN 2038 29v, TPL 615, 1492), he returns to this problem, now taking for granted his demonstrations on A97r:

How separate shadow is never similar in size to its source. If, as experience confirms, luminous rays are caused by a single point and they go increasing and spreading through the air in a circular course around this point. The further they go, the more they expand; and the thing positioned between a light and a wall is always carried by greater shadow because the rays which strike it, joined to the wall in their concourse, make it larger.

(figure)

Figs. 187-189: Concerning the link between punctiform propagation and shade. Fig. 187, CA204ra; fig. 188, CA349vd; fig. 189, CA345rb.

    He pursues this theme on CA204ra (1490-1495) in a passage entitled:

The operation of light with its centre.

If it were the entire light that caused shadows behind the object placed opposite this, it would hold that a body which is much smaller than a light would produce a pyramidal shadow behind it. And since experience does not show this, it must be that it is the centre of this light which performs this function.

    He now draws a diagram (fig. 187) followed by a:

Proof.

ab is the size of the light of a window which gives light to a stick positioned at its foot. From ac and to ad is where the window gives its light entirely. In ce, one cannot see that part of the window which is between lb and similarly df does not see am and for this reason in these 2 places the light begins to become exhausted.

    On CA349vd (1490-1495), he restates what he had claimed to be the basic idea of his first book, and beneath it adds a series of basic claims and definitions (fig. 188).

No opaque body is seen which is not covered by an umbrous and illuminated surface.

(figure)

Figs. 190-200: Preliminary demonstrations of punctiform propagation. Figs. 190-193, CA144va; figs. 194-200, CA179rc.

(figure)

Figs. 201-203: Demonstrations of punctiform propagation confirming that objects have light and shade everywhere on their surfaces. Fig. 201, CA353vb; fig. 202, CA353rb; fig. 203, W19147v (K/P 22v).

The air and every transparent body makes a transit of its species (which) from the objects to the eye [in the case] of those objects that are found either in front of it or above it.

Derived light is surrounded by primitive shade.

Derived shade will be surrounded by derived light.

Derived light is surrounded entirely or in part by primitive or derived shadows.
Through its similitudes every opaque body is all in all and all in every part of the transparent [air] that surrounds it.

    On CA345rb (fig. 189, c.1508) he again alludes to the principle of punctiform propagation: "All the objects seen by a single point are seen again by the same point." Elsewhere he makes a number of sketches to illustrate how objects have light and shade everywhere. Some of these are rough (figs. 190-200). Others are more carefully drawn (figs. 201-203).

 

Book One -  2. Central Line

    Corollary to the principle of punctiform propagation is the idea that the central line plays a determinant role in shadow projection (see below pp. ). On C3v (1490-1491), for instance, Leonardo notes that "in all the propositions that I shall make, it is understood that the middle which finds itself between bodies will be equal." On CA187va (c.1490-1491) he restates this idea more forcefully: "No shadow can imprint the true form of the umbrous body on a wall if the centre of the light is not equidistant from the extremities of this body." The nature of the central line is again considered on BM Arundel 170v (c.1492):

The centre of the length of any shadow always directs itself to the centre of the luminous body....

It is necessary that every shadow regards the centre of its light source with its own centre.

(figure)

Figs. 204-205: Elementary drawings showing the central line. Fig. 204, C17r; fig. 205, BM171r.

    Illustrations of this central line occur on C17r (fig. 204, 1490-1491) and BM171r (fig. 205, c.1492) with further notes on TPl528a (1508-1510), TPL478ab (1510-1515) and CA241vd (1513-1514).

 

Book One - 3. Opposing Theories

    Implicit in the claim that objects have light and shade everywhere is the idea that there can be no object which does not project shade. Hence, on A102r (BN 2038 22r, figs. , 1492) Leonardo takes to task the opinions of some that a triangle [i.e. a pyramid] does not produce any shadow on a wall:

There have been some mathematicians who have firmly held that a triangle which has its base positioned towards the light does not make any shadow on a wall, which thing they prove saying as follows. No spherical object less than a light source can reach as far as the middle with its shadow. Radiant lines are rectilinear. Hence let us suppose that the light is gh, and the triangle is lmn and the wall is ik. They say that the light g sees the face of the triangle ln and the part iq of the wall. And likewise h sees the face lm as g does and in addition it seems mn and the wall pk and if all this wall is seen by the light gh, it follows that the triangle is without shadow and it cannot happen that it does not have shadow. Which thing appears credible in this case if the triangle mpg were...seen by 2 lights, g [and] h. But ip is equal to qk and each is seen by a single light. That is, ip cannot be seen by hg; k will never be seen by g. Hence pq will be twice as bright as the two visible spaces which bear shadow.

(figure)

Figs. 206-210: Demonstrations to refute the claim that some objects are without shadow. Figs. 206-207, A102r; fig. 208, A103v; figs. 209-210, CA204ra.

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Figs. 211-213: Primitive and derived light and shade. Fig. 211, BM171r; fig. 212, CA116rb; fig. 213, CU585 (TPL570).

    This passage helps to explain an otherwise enigmatic note on A103v (BN 2038 23v, fig. 208, 1492): "How 2 lights, which have placed in the middle between them a body of two pyramidal sides with pyramidal bases, leave it without shadow." In the context of the earlier passage this is clearly a statement of the adversary's position. Who this adversary is, becomes evident from a further note on CA204ra (1492) in which Leonardo again launches into a demonstration without explaining the context (figs. 209-210):

Let ab be the luminous window. De produces the shadow gh. Ef produces the shadow ik. The triangle mkg is entirely luminous. Hence, the science of Marliani is false.

Such a demonstration proving that an opaque object necessarily produces shade would presumably have come at the end of Leonardo's first book on light and shade.

 

Book Two: Primary Shade

Shadows have in themselves various degrees of darkness, because they are caused by the absence of a variable amount of the luminous rays; and these I call primary shadows because they are first and inseparable from the body to which they belong. And on this I shall build the second book (CA250ra).

Primary Shade

    Book two of Leonardo's treatise, devoted to primary shade and its various degrees, would probably have opened with a chapter on basic definitions such as those on BM171r (fig. 211, 1492), CA116rb (fig. 212, c.1500) and CU585 (TPL570, fig. 213, c.1505-1508) analysed earlier (pp. ).

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Figs. 214-215: Three degrees of primary shadow on C17r and CU754 (diagram also used in CU796).

    The main part of book two would have been devoted to various degrees of primary shade, a topic on which there are at least six extant passages. The earliest of these, on C17r, (1490-1491), opens with a general statement:

That part of primitive and derivative shade will be less dark which is more distant from its centre.

This occurs because the more the shadow removes itself from its centre, the more it is seen by a greater quantity of luminous rays and everyone knows that where there is more light there is less shade.

    This general statement is followed by a specific example showing three degrees of primary shade (fig. 214):

The triangle dgr does not see anything of the light as and likewise the part of the umbrous body which is enclosed in this triangle. The triangle frk [i.e. frt] and also cri are seen by the light; am and ns and will be shadows that are brighter and more like the part of the ball which enclosed it in their angles.

The triangles bhi and etk are brighter and their external boundaries are the limit of the shadow and likewise of that part of a ball which encloses it at the points of the angles because each is seen by half the light oa and sa.

    Nearly two decades later he again considers three degrees of primary shade on CU754 (TPL631, fig. 215, 1508-1510):

On shadows and which are those primitive ones which will be darker on an object.

Primitive shadows will be darker which are generated on the surface of a denser body and conversely, [they will be] brighter on the surface of less dense bodies. This is manifest because the species of those objects which tinge objects opposite them with their colours will impress themselves with greater vigour which find denser or more polished surfaces on these bodies.

(figure)

Figs. 216-217: Four degrees of primitive shadow on A92v and CU756.

    These leads to a concrete example (fig. 215):

This is proved. And let the dense object be rs interposed between the luminous object nm and the umbrous object op. HEnce, by the seventh of the ninth, which states: the surface of every body participates in the colour of its object, we shall claim that the part bvar of this body will be illuminated because its object nm is luminous and similarly we shall state that the part opposite dcs is umbrous, because its object is dark. And thus our proposition is concluded.

    Meanwhile he had been studying cases with more degrees of primary shade. On A93v (BN 2038 13v, fig. 216; CU756, fig. 217, 1492) he considers four degrees of shade:

That part of the umbrous body is less luminous which is seen by a lesser quantity of light.

The part m of the body is the first degree of light because the window ad sees them all along the line af; n is the second degree because the light bd sees it along the line be; o is the third degree because the light cd sees it along the line ch; p is the penultimate [degree] because cd sees it along the line dv [and] o is the final degree because no part of the window sees it.

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Figs. 218-219: Five degrees of primary shadow on A94r and CU624.

    Directly opposite on A94r (BN 2038 14r, 1492) he considers a case with five degrees of shade (figs. 218=219) at greater length:

Every light that falls on umbrous bodies among equal angles holds the first degree of brightness and that [body] is darker which receives less equal angles, and light and shade function through pyramids. The angle c holds the first degree of brightness because it sees all the window ab and the entire horizon of the sky mx. The angle d is little different from c because the angles which place it in the middle are not so different in proportion as are the others below and there is lacking to it only that part of the horizon that is between y and x. Although it receives as much from the opposite side, nonetheless, its line is of little power because its angle is less than its companion. Angles e [and] c are of less light because the light ms and the light vx see them less, and the angles e [and] i are fairly difform.

The angle k and the angle f are each positioned in the middle by angles very different from one another and hence have little light because at k only the light pt is seen and at f only [the light] tq is seen. Og is the ultimate degree of light because it sees no part of the light of the horizon and these are the lines which once again recompose a pyramid similar to the pyramid c, which pyramid l finds itself in the first degree of shade because it again falls between equal angles and these angles direct themselves and regard themselves along a straight line which passes through the centre of the umbrous body and has its apex at the centre of the light. The luminous species multiplied at the boundaries of the window at the points a [and] b produce a brightness which surrounds the derived shade created by the umbrous body at the locations 4 and 6 and the dark species are multiplies at o [and] g and end at 7 and 8.

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Figs. 220-221: Seven degree of primary shadow on C14r and C21v.

    Elsewhere on C14r (1490-1491) he makes a detailed drawing (fig. 220) showing seven degrees of primary shade with the brief caption: "The boundaries of umbrous bodies, because they are struck by different qualities of luminous pyramids, are surrounded by different qualities of light and shade." On C21v (1490-1491) he pursues this theme now carefully numbering the various degrees of light and shade (fig. 221), again adding the caption: "That part of the luminous body which is struck by a greater luminous angle will be more illuminated than any other." He returns to this problem on Forst II, 5r (c.1495-1497): " The shaded and illuminated parts of opaque bodies will be in the same proportion of brightness and darkness as are those of their objects."

    A few years later on CA199va (c.1500) he claims that the number of degrees of light and shade is infinite:

Even though practitioners put four kinds of brightness in imitating a same colour in all darkened things, [such as] trees, meadows, hair, beards, and skin, that is, first a dark fundament and 2nd, a spot that participates of the form of the parts; 3rd, a part that is clearer and brighter; 4th, lights which have their shapes clearer than other parts. But to me it appears that this variety is infinite on a continuous quantity which is divisible to infinity.

(Figure)

Fig. 222: Demonstration on ca199va that the degrees of shade are infinite.

    He supports this claim with a demonstration (fig. 222):

And I prove it as follows: let ag be a continuous quantity; let d be a light which illuminates it. I say by the 4th which states that; that part of an illuminated body will be more luminous which comes closer to the cause which illuminates it. Hence g is darker than c to the extent that dg is longer than the line dc and by the conclusion that such degrees of clarity or if you wish to say obscurity are not only 4, but can imagined as being infinite, because cd is a continuous quantity and every continuous quantity is divisible to infinity. Hence the varieties of lengths are infinite, which the lines have that extend from the luminous body to the illuminated body and such is the proportion of the lights as is that of the lengths of the lines among them, which extend from the centre of the luminous body to the parts of this illuminated object.

    On TPL810 (1505-1510) he takes for granted this infinite variation and in the following years makes further passing references to degrees of light and shade (TPL672, 634, 683, 1508-1510). Later, on E15r (1513-1514), he restates his earlier rule:

Painting; among bodies of varying obscurity deprived of a single light such will be the proportion between their shadows as is the proportion between their natural obscurity and the same is to be understood of their lights.

    In these later notes, he does not, however, improve on the diagrams (figs. 220-221) made in 1490-1491.

 

Book Three: Derived Shade

From these primary shadows there result certain shaded rays which are diffused through the atmosphere, and these vary in character according to that of the primary shadows whence they are derived. I shall therefore call these shadows derived shadows because they are produced by other shadows; and the third book will treat of these (CA250ra).

    In the case of derived shade, which was to have formed the third book on light and shade, the contrast between Leonardo's early ideas and his later studies in more marked. This third book would probably have opened with a chapter on the three traditional kinds of light, and led to a discussion of various conditions in each of the three variables: light source, object and eye (Chart 9).

 

Book Three - 1. Kinds of Light

    Already in the third century B.C. Aristarchus of Samos3 had made a distinction between three basic kinds of shade: (1) parallel, when light source and opaque object are equal; (2) diverging, when the light source is smaller than the opaque object and (3) converging, when the light source is larger than the opaque object. Aristarchus appears to have been well known to Renaissance humanist circles.4 This distinction had, moreover, been transmitted indirectly through mediaeval commentators such as Witelo.5 By 1490 Leonardo is familiar with this distinction and he illustrates it at least ten times in the next three decades (figs. 223-232). Perhaps the earliest example is on C7v (1490-1491) where roughly drawn sketches (fig. 223) are accompanied by a clear text:

Shade and Light

The shapes of shade are three because if the material, which produces the shade is equal to the light, the shade is similar to a column and it has no end.

If the material is greater than the light, its shade is equal to an inverted or contrary pyramid and its length is without end. But if the material is less than the light, the shade is equal to a pyramid and is finite as is shown in the eclipses of the moon.

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Figs. 223-228: Leonardo's illustrations of Aristarchus' three types of shade. Fig. 223, C7v ; fig. 224, C15v; fig. 225, C347ra; fig. 226, CU615; fig. 227, CU619; fig. 228, CU624.

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Figs. 229-232: Further examples of these basic types of shade (constant, expanding, diminishing). Fig. 229, CA236ra; fig. 230, W12669v; fig. 231, E32v; fig. 232, CU617.

    He illustrates these three kinds more carefully on C15v (fig. 224, 1490-1491), this time without text. On CA347ra (1490-1495) he draws a related series (fig. 225) with a text which he crosses out. By 1508 yhe is no longer certain about the kind of rays propagated by the sun, and hence on F77v (1508) he notes (fig. 233):

If every part of the sun sends its rays to all the surrounding objects what is that part which sends its simulacrum to the waters, that is, is it a columnar ray, or a straight pyramidal or an inverted pyramidal [ray], that is, the columnar is abcd, the truncated pyramid is acfg, the straight pyramid is ace, the inverted pyramid is fgh. Decide which carries the simulacrum of the sun to the water.

    No clear decision ensues, however. When he returns to this theme on CU615 (TPL574, fig. 226, 1508-1510) he merely changes the order of presentation of the three traditional kinds of shadow asking:

Of how many shapes is derived shade?

The shapes of derived shade are three and the first is pyramidal born of an umbrous body less than the luminous body; the second is parallel born of an umbrous body equal to a luminous one; the third is infinitely expanding, and the columnar [kind] is infinite and the pyramidal [kind] is infinite also because beyond the first pyramid it makes an intersection and generates opposite the finite pyramid, an infinite pyramid, finding infinite space.

(figure)

Fig. 233: Possible paths of rays propagated by the sun on F77v.

    On CU619 (TPL588, fig. 227, 1508-1510) he pursues this theme:

Of the three various shapes of derivative shade.

The varieties of derivative shade are three, of which one is large in its origin, and the more it is removed from such a beginning, the more it contracts.

The second observes an infinite length with the same width as at its origin. The third is that which, in every degree of distance behind the width of its origin, acquires a degree of width.

    This passage is directly followed by another (CU620, TP589, 1508-1510), headed:

The variety of each of the said three [kinds of] derivative shade.

The derivative shade originating from an umbrous body less than the body which illuminates it, will be pyramidal and will be shorter to the extent that it is closer to the luminous body. But the parallel [kind] does not vary in such a case. But the expanding will be larger the closer it is to its luminous source.

    Related to this is a further passage on CU625 (TPL591, 1508-1510):

That derived shadows are of three kinds.

Derived shadows are of three kinds, that is, either its intersection on the wall where it percusses is greater than its base, or it will be less than this base or it will be equal. And if it is greater, it is a sign that the light which illumines the umbrous body is less than this body and if it is less, the light is larger than the body and if it is equal, the light is equal to this body.

    On CU624 (TPL601, fig. 228, 1508-1510) the theme is pursued

In how many ways does the quantity of the percussion of shade vary with primitive shade.

Shade, or rather the percussion of shade varies in three ways by the three kinds mentioned above, that is, converging, diverging and parallel. The diverging has a greater percussion than its primitive shade. The parallel always has its percussion equal to its primitive shade. The converging makes two sorts of percussion, namely, one [which is] converging, the other diverging. But the converging always has the percussion of its shade less than the primitive shade and its diverging part does the contrary.

    On CA236ra (fdig. 229, 1508-1510) he considers a dynamic version of these three variables:

To the extent that an umbrous body smaller than a luminous body is closer to this luminous body, it will be illuminated by a smaller quantity of such a luminous body. The converse follows.

To the extent that an umbrous body larger than a luminous body approaches the luminous body it will be illuminated by a greater quantity of this luminous body.

But if the umbrous body is equal to the luminous body, then the quantity that one sees behind it, will never vary in any variety of distance.

    On CA195va (c.1510) he drafts another version which he later crosses out: " Why a light makes pyramidal shade after the umbrous body. Shadows are of 3 kinds of shapes of which the first is pyramidal, the 2nd parallel and the 3rd...a semi-pyramidal intersection." That same year he makes a quick sketch (fig. 230) of these three kinds of shadow on W12669v (c.1510). On E31r (TPL595, 1513-1515) he takes up the theme anew:

On simple derived shade.

Simple derived shade is of three kinds, that is, the one finite in length and two infinite. The finite is pyramidal and of the infinite [kinds] one is columnar and the other is expanding and all three are rectilinear. But the converging shade, that is, the pyramidal [one] originates from an umbrous body less than the luminous body and the columnar originates from an umbrous body equal to the luminous body and the expanding from an umbrous body greater than the luminous source.

    On E32r (fig. 232, CU617, fig. 233, TPL590, 1513-1514), he pursues this theme:

On shade.

Derived shadows are of three kinds of which the one is expanding, the other columnar and the third converging towards the site of the intersection of its sides which, after this intersection are infinitely long, or rectilinear. And if you said that such a shadow was terminated at the angle of conjunction of its sides and does not pass beyond, this is denied, because in the first of the above [mentioned] shadows, I have proven that that thing is entirely bounded of which no part exceeds its boundaries. The opposite of this is seen in such a shadow [however], because along with such a derivative shadow, the shape of two umbrous pyramids manifestly arise, which are conjoined in their angles.

    An adversary's arguments are now considered:

Now, according to the adversary, if the first umbrous pyramid is the limit of the derived shade with its angle whence it originates, then the second umbrous pyramid, claims the adversary, is caused by the angle and not by the umbrous body.

    And these arguments are promptly dismissed:

And this is denied with the help of the 2nd of this which states that shade is an accident created by umbrous bodies positioned between the site of this shade and the luminous body, and by this it is declared that the shade is not generated by the angle of derivative shade, but solely by the umbrous body, etc...

If the spherical lumbrous body is illuminated by a luminous body [that is] long in shape, the shadow produced by the longest part of this luminous body will be of boundaries less well known than that which is generated by the width of the same light. And this is proved by the previous [proposition] which states that the shade created by a creator luminous body, has boundaries which are less clear and conversely, that which is illuminated by a smaller light source has boundaries which are more distinct.

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    Figs. 234-240: Cases where the light source is equal to the opaque body. Figs. 234-237, CA144r; fig. 238, H76[28]v; fig. 239, CU610; fig. 240, CU605.

    Read in the context of his previous notes on the subject, this passage on E32r is of particular interest, because it reveals the extent to which he transforms traditional ideas. What had begun as a passing comment of Aristarchus has now become a much more complex argument. Meanwhile he had been doing further study concerning the particular role played by each of the three variables in this process: namely, light source, object and eye, each of which is effectively a chapter in itself.

 

Book Three - 2. Light Source

    With respect to light sources he considers instances where they are (a) equal to, (b) smaller and (c) larger than an opaque body as well as comparative cases. We shall consider each of these in turn.

 

Book Three - 2a. Light Source Equal to Opaque Body

    Leonardo's earliest examples of this situation are found on CA144ra (figs. 234-237, c.1492) in the form of rough sketches without text. On H76[28]v (1493-1494) he draws a clearer diagram (fig. 238) beneath which he writes two drafts:

Derived shade is never similar to the body from which it originates unless the light is the [same] shape and size as the umbrous body.

Derived shade cannot be similar in shape to primitive [shade] unless it percusses between equal angles.

    He returns to this theme on CU610 (TPL724, 1508-1510):

What is that umbrous body, which does not increase or decrease its umbrous or luminous parts at any distance or proximity of the body which illuminates it?

When the umbrous and luminous body are both of equal size, then no distance or vcinity, which interposes itself between them will have the power of diminishing or increasing their umbrous or luminous sides.

    He illustrates this with a concrete example (fig. 239):

Let nm be the umbrous body which, when taken from the site cd closer to the luminous body ab, the quantity of its shadow is neither increased nor decreased. And this happens because the luminous rays that surround it are parallel in themselves.

    He pursues this theme on CU605 (TPL696b, 1508-1510):

Which luminous body is that which will never see more than half of the umbrous sphere?

When the umbrous sphere is illuminated by a luminous sphere equal in size to this umbrous one, then the umbrous and luminous part of this umbrous body will be equal.

    Again he provides a concrete example (fig. 240):

Let abcd be the spherical umbrous body equal to the luminous sphere ef. I say that the umbrous part abc of the umbrous sphere is equal to the luminous part abd and this is proved as follows: the parallels es and ft are contingent on the front of the diameter ab, that is, the diameter of the umbrous sphere, which diameter passes through the centre of this sphere.

Being divided in the said diameter, it will be divided equally and the one part will be entirely umbrous and the other part entirely luminous.

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Figs. 241-244: Expanding shade. Fig. 241, Triv. 11v; fig. 242,BM170v; fig. 243, C21r; fig. 244, CU601 (TPL638).

    He returns to this situation once more on CU614 (TPL567c, 1508-1510):

Which shade makes its light equal to the umbrous body in the shape of its shadows?

If the umbrous body is equal to the luminous body, then the simple shade will be parallel and infinite in length. But the compound shade and light will be of a pyramidal angle with respect to the luminous body.

 

Book Three - 2b. Light Source Smaller than Opaque Body

    Leonardo is equally interested in cases where the light source is smaller than the umbrous body. Perhaps the earliest example is that found on Triv. 11v (1487-1490) in the context of diminishing intensity of shade (fig. 241). "To the extent that ab enters cb,: he claims "to that extent will ab be darker than cd." He returns to this situation on BM Arundel 170v (fig. 242 cf. fig. 243, c. 1492) now claiming: "The light smaller than the umbrous body makes shadows bounded in this body and produces little mixed shade and sees less than half of it." This idea he develops in a later note on CU601 (fig. 244, TPL638, 1508-1510):

    Which body produces a greater quantity of shade:

That body will be vested with a greater quantity of shade, which is illuminated by a smaller luminous body. Let abcd be the umbrous body, g the small luminous source, which illuminates only the part abc of this umbrous body, whence the umbrous part adc remains much greater than the luminous part abc.

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Figs. 245-247: Contracting shade. Fig. 245, C2v; fig. 246, CA160ra; fig. 247, Mad I 6v.

    He returns to this situation once more on CU638 (TPL568, 1508-1510):

Which shade does the umbrous body larger than the luminous source make?

If the umbrous body is larger than its luminous source, its simple derived shade will have its sides converging to the potential angle beyond the luminous body and the angles of the compound light and shade will regard the entire luminous body.

    Hence on at least four occasions he is content merely to repeat Aristarchus' assumption that a light source larger than an object produces converging shade. This is the more striking because, as we have seen, he had designed his own experiments to demonstrate the contrary (e.g. figs. 176-181).

 

Book Three - 2c. Light Source Larger than Opaque Body

    Ever since Aristarchus it had been assumed that a light source larger than an umbrous body produces a converging pyramidal shadow. Leonardo illustrates this situation on C2v (fig. 245, 1490-1491) adding the caption:

If the umbrous and luminous body are of spherical rotundity, the base of the luminous pyramid will have such a proportion with its body, as the base of the umbrous pyramid will have with its umbrous body.

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Figs. 248-258: Contracting shade. Figs. 248-249, BM170v; figs. 250-252, CA199ra; figs. 253-256, CA112va; fig. 257, W19152v (K/P 118v); fig. 258, CA243ra.

    On Ca160ra (fig. 246, 1490-1491) he draws a similar diagram, this time merely noting that this applies: "where the shade is less than the light." On BM170v (c.1492) he provides two diagrams (figs. 248-249) without text and on BM103v (1490-1495) he drafts a text without a diagram: "Simple derive shade born of an umbrous body less than the luminous source is of a pyramidal congregation." When he returns to this situation on Mad I 6v (c.1499-1500) he alludes both to its astronomical context and his own demonstrations to the contrary (fig. 247):

If the sun is greater than the earth, this earth makes a pyramidal shade through the air behind it. It being thus, a small ball should make a much shorter shadow beyond it when it is illuminated by the sun, and we see the opposite. But in the place of a pyramid one sees a columnar shade.

    Further illustrations of this astronomical context are found on CA199ra (figs. 249-252, c.1500) and on CA112va (figs. 253-256, 1505-1508). He returns to this theme on CU603 (fig. 259, TPL639, 1508-1510) asking:

Which body takes a greater quantity of light?

That body takes a greater quantity of light which is illuminated by a greater quantity of light. Let abcd be the illuminated body. Ef is that body which illuminates it. I say that since the luminous body is so much larger than the illumined body, that the illumined part bcd will be so much greater than its umbrous part bad and this is proven by the rectilinearity of the luminous rays eg [and] fg.

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Figs. 259-260: Two further demonstrations of contracting shade on CU603 and 606.

(figure)

Figs. 261-264: Effects of pinhole apertures and windows on light and shade. Figs. 261-262, BM171v; fig. 263, C12r; fig. 264, CA230vc.

(figure)

Figs. 265-267: Gradations of light and shade in rooms with windows of different sizes. Fig. 265, B20v; fig. 266, A23r; fig. 267, CU133.

    He pursues this theme in the second part of CU606 (fig. 260, TPL660, 1508-1510):

The greater the amount of light by which a body is illumined, the less will be the quantity of shadow which remains on this body.

A is the luminous body, bc is the umbrous body, b is the part of the body which is illumined, c is that part which remains deprived of light and in this the umbrous part is greater than the luminous [part]. E is the luminous body greater than the umbrous body opposite it, fg is the umbrous body and f is the illumined part and g is the part in shade.

    The accompanying diagram (fig. 260) does not show all the details described. Related diagrams occur on W19152v (K/P 118v, fig. 257, 1508-1510) and CA243ra (fig. 258, 1510-1515).

 

Book Three - 2d. Comparative Sizes of Light Source

    Besides considering particular situations in which a light source is either equal to, smaller, or larger than an opaque body, Leonardo also makes comparative studies of light sources. His work on the camera obscura (figs. 261-262 cf. figs. ) may have prompted him to compare the nature of light and shade produced by different sized windows on B20v (fig. 265, 1490-1491). This approach is implicit in examples on A23r (CU133, TPL103, fig. 266, 1492), C12r (fig. , 1490-1491) and CA230vc (fig. 267, 1497-1500).

(figure)

Figs. 268-273: Sketches on Triv. 29r illustrating what happens when candlelight (figs. 268-269) and skylight (figs. 270-273) pass through a window.

(figure)

Figs. 274-277: Further demonstrations of what happens when small and large light sources pass through windows. Figs. 274-275, Triv. 28v; figs. 276-277, CU616.

    On Triv. 29r (1497-1500) after making four preparatory sketches (figs. 268, 270-272) he compares what happens when candlelight (fig. 269) and skylight (fig. 273) pass through a window, adding the caption: "Primitive and derived shade caused by the light of a candle are larger than when caused by that of the air." The two situations which he here presents separately he combines in a single diagram (figs. 274-275) on Triv. 28v (1487-1490), now adding:

The edges of a window illuminated by two different lights of equal brightness will not send light of equal brightness within a habitation.

If b is a candle and ac is our hemisphere, both illuminate the edges of the window mn but the light b only illuminates fg and the hemisphere ac illuminates as far as de.

    Nearly two decades later he returns to this comparative approach on CU616 (TPL584, 1508-1510) in a passage headed (figs. 276-277):

Of derived shade and where it is greater.

That derived shade will be of greater quantity which is born from a greater quantity of light and also conversely. This is proved: ab, a small light produces derived lights cge and dfh which are small. [Now] take the following figure: nm, the light of the sky, which isuniversal, produces a large derived shadow at rtx and also the space osu, because the part pn of the sky produces this shadow rtx and likewise the space lm , a part of the sky, produces the opposite shadow [at] osu.

    Meanwhile, he had also been exploring the links between the intensity of a light source and the resulting shade, as on C10r (1490-1491): "To the extent that the luminous body is of greater obscurity, to that extent will the shadows produced by the bodies illuminated by it be darker."

(figure)

Figs. 278-282: Effects of size of light source and distance on derived shade. Figs. 278-279, CA144vb; figs. 280-281, CA144ra; fig. 282, C2v.

    This idea he develops on A67 (1492), CU702 (TPL620, 1508-1510) and CU860 (TPL694, 1508-1510). Rough sketches of varying light sources are found on CA144vb (figs. 278-279, 1492). On CA144ra (figs. 280-281, c.1492) he drafts two further figures accompanying which he writes:

To the extent that the diameter of the derived shade is greater than that of the primitive shade, to that extent will the primitive shade be darker...than the derived.

To the extent that a more powerful light strikes dense bodies to that extent will the shadows of these bodies appear darker...and more divided by the light.

 

Book Three - 3. Comparative Distances and Sizes of the Object

    Just as Leonardo is intent on studying the role of the light source, so too is he concerned with analysing how changes in an opaque object affect light and shade. In this respect he considers comparative distances, comparative sizes and the combined effect of the two.

 

Book Three - 3a. Comparative Distances

    On C2v (1490-1491) he considers the effect of distance on the intensity of derived shade (fig. 282):

To the extent that the percussion made by the umbrous concourse on the wall positioned opposite it is more distant from the luminous body and closer to its derivation, it will appear darker and with a more distinct boundary.

    He returns to this idea on TPL599 (1508-1510) in a passage entitled:

Which derived shade will show its boundaries as better known?

That derived shade will show the boundaries of its percussion as betterknown, of which the umbrous body is more distant from the luminous body.

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Figs. 278-281: Diagrams analysing the effects of distance on derived shade. Figs. 283-284, A95v; fig. 285, CU641; fig. 286, CU622.

    He is more interested in the effects of distance on the shape of derived shade. On Triv. 29r (1487-1490), for instance, he makes a preliminary sketch (fig. 290) with the caption: "To the extent that the larger derived shade enters into the smaller, to that extent is the cause of the lesser more luminous than the larger." On A95v (BN 2038 15v, fig. 283, CU641, TPL732, fig. 285, 1492) he analyses this problem in detail:

Every shadow with all its varieties which grows in size with distance more than its cause, has its exterior lines join together between the light and the umbrous body. This proposition appears clear and is confirmed by experience. For if ab is a windowwithout any obstruction, the luminous air that stands to the right at a, is seen to the left at d, and the air that stands to the left, illuminates to the right at the point c, and the said lines intersect at the point m.

Every umbrous body finds itself between 2 pyramids, one dark, and the other luminous. The one is seen and the other not. And this only happens when the light enters through a window.

    He now draws a second diagram (fig. 284 cf. CU622, fig. 286) beneath which he writes:

Recall that ab is the window and that r is the umbrous body. The light on the right at z passes the body on the left side of the umbrous body at g and goes to p. The left light K passes this said body on the right side at i and goes to m and these two lines intersect at c and there produce a pyramid.

Then ab touches the umbrous body at ig and produces its pyramid at fig; f is dark because it can never see the light ab and igc is always luminous because it sees the light.

    Having analysed how the pyramid of derived shade, is produced on A95v, he examines what happens to this pyramid at different distances on A90v (1492), beginning with a general claim: "Those bodies which are closer or further from their original light will produce shorter or longer derived shade." This idea he restates in terms of the size of the light source: "Among bodies equal in size that which is illuminated by a larger light source will have a shorter shadow." These claims are followed by a demonstration (fig. 287):

The above mentioned proposition is confirmed by experiment because the body mn is surrounded by a larger part of the light than the body pq, as is shown above. Let us say that vc ab dx is the sky that produces the original light and that st is a window where the luminous species enter and likewise that mn [and] pq and the umbrous bodies positioned opposite this light; mn will be of lesser derived shade because its original shade is little and its derived light is large because the original light cd is also large. Pq will have more derived shade because its original shade is greater [and] its derived light os less than that of the body mn, because that part of the hemisphere ab which illuminates it is less than the hemisphere cd illuminating the body mn.

    This proposition recurs on CU639 (TPL725, 1492) with a slightly modified diagram (fig. 288). On Mad I 31v (1499-1500) he returns to this theme, again beginning with two general claims:

On shade.

The illuminated parts of bodies of equal size are more luminous when the derived shade is shorter.

On shade.

The illuminated parts of bodies of equal size will have such a proportion in their brightnesses as they have in the lengths of their umbrous pyramids.

    To demonstrate this a concrete example is again provided (fig. ):

The body f will be the half less illuminated than the body e, because the part of the sky which illuminates it is twice as small as that of e, as is demonstrated in [comparing] cd and ab.

    On CU453 (TPL440, 1508-1510) he relates these principles to problems of painting practice:

Painting in a universal light.

In the multitudes of men and animals always accustom yourself to making the parts of their shapes or bodies darker to the extent that they are lower and to the extent that they are closer to the centre of their multitude even though they are in themselves of a uniform colour and this is necessary because a smaller quantity of sky illuminating the bodies is seen in the low[er] spaces interposed between the aforesaid animals than in the upper parts of the same spaces.

    A demonstration follows (fig. 291 cf. fig. 290):

(figure)

Figs. 292-293: Cases of lateral derived shade in rooms, on A95r and CU142.

This is shown by the figure placed in the margin where abcd is placed for the arc of the sky, the universal illuminator of bodies beneath it. N [and] M are the bodies which limit the space strh positioned between them, in which space one clearly sees that the site f, being only illumined by the part of the sky, cd, is illumined by a smaller part of the sky than the site e which is seen by the part of the sky ab which is three times greater than the sky dc.

Hence it will be three times more illuminated in e than in f.

    He is also interested in comparing the derived shade of objects off to the side. This situation is implicit on W12604r (fig. 294, c.1488) where he offers a:

Proof how every part of light makes one point.

Although the balls a, b [and] c have light from one window, nonetheless, if you follow the lines of its shadows you will see that these make an intersection and point at the angle n.

    This idea he pursues on A95r (BN 2038 15r, fig. 292, cf. CU642, TPL 293, 1492):

Every shade made by bodies is directed along the central line to a point made by the intersection of the luminous rays in the middle of the space and...the window. The reasoning presupposed above appears clearly through experience, because if you draw a site with a window to the North which is sf you will see the horizon of the East producing a line which, touching the 2 angles of the window of, will end in d and the horizon of the West will produce its line touching the other 2 angles of the window rs, and it will end in c, and this intersection comes precisely in the middle of the space and the size of the window.

This reasoning will be confirmed even better if you take two sticks as in the place gh you will see the line made by the centre of the real shadow directed towards the centre m and with the horizon nf.

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Figs. 294-296: Sketches concerning lateral derived shade. Figs. 294, W12604r; fig. 295, C8r; fig. 296, BM170v.

    On C8r (1490-1491) he examines in detail the case of shadows off to the side, beginning with a general claim:

Umbrous and luminous rays are of a greater power in their points than in their bases.

Even though the points of luminous pyramids extend to dark sites and those of the umbrous pyramids extend to luminous places, and that among them are luminous ones. One is born of a greater base than the other. Nonetheless, if as a result of their various lengths, these luminous pyramids come to angles of equal size they will have equal light amongst them and umbrous pyramids will do the same.

    A concrete example follows (fig. 295):

As is demonstrated in the intersected pyramids abc and def which, even though they originate from different sizes of base, they are, nonetheless, similar in size and in light.

    He pursues this theme on BM170v (1492) beginning with the phrase: "real shade is longer the more it finds itself," which he then crosses out and writes (fig. 296):

When the light of the air is constrained to illuminae umbrous bodies, if these umbrous bodies are equidistant from the centre of this window, that one which is positioned further off to the side will produce a greater shadow behind it.

    He develops this idea on A91r (BN 2038 11r, CU643, TPL726, 1492):

Those scattered bodies situated in a habitation illuminated by a single window will produce derived shade that is more or less short, depending on whether it is more or less opposite this window.

The reason why umbrous bodies which find themselves situated more directly opposite the middle of the window make shadows which are shorter than those situated in a position off to the side is that they see the window in its proper form and the bodies off to the side see it foreshortened. To the one in the middle the window appears large; and those off to the side see it [as] small.

    As usual a concrete example follows (fig. 297 cf. CU643, fig. 298):

The one in the middle sees the hemisphere as large, that is, [as] ef and those to the side see it [as] small, that is, gr sees ab and likewise mn sees cd.

The body in the middle because it has a greater quantity of light than those to the side is illuminated considerably lower than its centre and therefore its shade is shorter. And to the extent that ab enters ef, to that extent does the pyramid g4 enter into ly precisely.

    This discussion leads directly to a consideration of the centres of derived shade (cf. above ):

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Figs. 297-298: Systematic studies of lateral derived shade on A91r and CU643.

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Figs. 299-301: Effects of distance on the shadows of objects in the open air. Fig. 299, Triv. 22v; fig. 300, W12352v; fig. 301, W12635v.

Every centre of derived shade passes through 6 centres and directs itself with the centre of the original shade and with the centre of the umbrous body and of the derived light and with the middle of the window, and ultimately with the centre of that part of the luminous body made by the celestial hemisphere.

Yh is the centre of the derived shade, lh of the original shade, l is the centre of the umbrous body, lk of the derived shade, v is the centre of the window, and e is the ultimate centre of the original light made by that part of the hemisphere of the sky which illuminates the umbrous body.

    In the left-hand margin he returns to the question of relative lengths of shade produced (fig. 297):

Among the shadows produced by equal bodies and at different distances from the aperture illuminating them, that which is longest, its body will be less luminous, and the one body will be that much more luminous than the other, to the extent that its shade is shorter than the other.

The proportion that nm and vK have with st and vx, such will the shadow 4 have with x[and] y.

    His comparative studies of shadows at different distances extend to objects in the open air. On W12635v (fig. 301, c.1500), for instance, he draws two light sources illuminating an opaque body, and notes: " Whatever proportion that the line bc has with the line fc , such will the obscurity m have with the obscurity n."

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Figs. 302-307: Comparative effects of distance on derived shade. Fig. 302, CA236ra; figs. 303-305, BM100r; figs. 306-307, W19102v, (K/P 198v).

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Figs. 308-309: Demonstrations on CU728 concerning comparative sizes of objects.

    Sketches on Triv. 22v (fig. 299, 1487-1490) and W12352v (fig. 300, c.1494) may well represent preparatory drafts for this diagram. He pursues this theme on CA236ra (fig. 302, 1508-1510) where he claims:

That umbrous body will have its simple derived shade with a larger base and a longer pyramid which is more remote from its luminous body. The first conclusion is tested, and let us say (that the) that the first umbrous body, a is closer to the luminous body cf than the second umbrous body br.

Among bodies equal in size, the more remote will make an umbrous pyramid of a longer shape; the reverse follows, etc.

    Related diagrams occur on BM100r (figs. 303-305, 1490-1495) and W19102v (K/P 198v, figs. 306-307, 1510-1515).

 

Book Three - 3b. Comparative Sizes of Object

    He is also concerned how different sizes of an object affect derived shade, as, for instance, on CU728 (figs. 308-309. TPL666, 1508-1510):

On shadow and light.

That object will have its shade and light of more imperceptible boundaries which is interposed between larger dark and bright objects of continuous quantity.

This is proved and let the object be o which is interposed between the umbrous body nm and the luminous body rs. I say that the umbrous body tinges nearly all the object with its pyramid nam and the pyramid of the luminous body rcs does the same at the opposite end.

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Figs. 310-314: Demonstrations with comparative sizes and distances. Fig. 310, C3v; figs. 311-312, CU607; fig. 313, CU602; fig. 314, CU609.

And that which is proposed is concluded by the 8th of the 5th which states that that part of the sphere will be darker which sees more of the darkness placed opposite. It follows that c is darker than any other part of this sphere.

 

Book Three - 3c. Comparative Sizes and Distances of Object

    A next logical step in complexity would be to make comparative studies involving both different sizes and different distances. This Leonardo explores also. On C3v (fig. 310, 1490-1491), for instance, he considers a case:

When two umbrous pyramids, opposite one another, born of a same body...are such that one is doubly dark than the other and the same shape, then the two lights which are the causes thereof are such that one is double the other in diameter and at double the distance from this umbrous body.

    He returns to this theme of different sizes and distances on CU607 (TPL695, 1508-1510) in a passage headed (figs. 311-312):

Equality of shade in unequal umbrous and luminous bodies of different distances.

It is possible that a same umbrous body takes equal shade from luminous bodies of different sizes.

Fogre is an umbrous body of which the shadow is fgo, generated by the privation of an aspect of the luminous body de at the true distance and of the illuminating body bc at a remote distance.

And this arises because both luminous bodies are equally deprived of an umbrous aspect fog through the rectilinearity of ab [and] pc.

    On W12635v (c. 1500) he considers the effects of two light sources of different sizes and at different distances (figs. 315-316) accompanying which is a draft:

[If] the distance of the umbrous body has this proportion to the lights, the lights of this size will have double their shade.

The proportion that the size of the light f has with the light b, such [a proportion] will the darkness of the shade d have with the shadow f.

    He pursues this problem of comparative sizes and distances on CU602 (TPL722, 1508-1510) asking:

Which body is that which, when it approaches the light, its umbrous part increases?

When a luminous body is less than the body illuminated by it, the shade of the illuminated body will increase to the extent that it is closer to the luminous body.

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Figs. 315-316: Derived shade of light sources of different sizes at different distances on W12635.

    By way of illustration he gives a concrete example (fig. 313):

Let a be the luminous body less than the umbrous body rsgl, which illuminates the entire part rsg included between its luminous rays an and am.

When...by necessity of these rays, the whole of rlg remains umbrous.

Then I bring this umbrous body near the same luminous body and there will be dpeo, which is enclosed by the rectilinearity of the lines ab and ac, and is touched by these rays at the point d and the point e and the line de divides the umbrous [part] from its luminous part, [i.e.] dpe from doe, which umbrous part is necessarily greater than the umbrous [part], rlg, of the more distant body. And all arises from the luminous rays which, being rectilinear, separate themselves more distantly from the centre of such an umbrous body, to the extent that this body is closer to the luminous body.

    Having considered what happens with objects larger than the light source, he examines CU609 (TPL723, 1508-1510) what occurs with objects smaller than the light source:

What is that body which, the more it approaches the light, the more its umbrous part diminishes?

When the luminous body is larger than the body illuminated by it, the shadow of the illuminated body will diminish more the closer it is to this luminous body.

    This claim is again demonstrated (fig. 314):

Let ab be the luminous body larger than the umbrous body xgnh which, as it approaches the light fecd, diminishes its shadow because when it stand close to the body which illumines it, it is embraced further beyond its centre with luminous rays than when it is more remote.

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Figs. 317: Light source, eye and object on C27v. 318: Light source, object and eye on C27v.

    In these examples Leonardo's systematic play with variables is again apparent: how he alters first distance, then size, then size and distance. As one might almost expect, he proceeds to study the effects of adding a further variable: the eye.

 

Book Three - 4. Comparative Positions of the Eye

    Leonardo recognizes that the amount of shadow seen depends on the eye's position relative to the light source and the opaque body. On C27v (1490-1491), for instance, he considers the configuration: light source, eye, object (fig. 317):

Perspective

The eye which finds itself sending from itself visual pyramids from the same side as the luminous rays, if it is situated in the middle of these rays, it cannot see any shade on the opaque bodies positioned opposite.

    Immediately following he considers the configuration: eye, object, light source (fig. 318):

Perspective

That spherical body which finds itself between the centre of the natural light and the centre of the visual pyramids is seen by the eye as being completely in shade with an equal luminous circle.

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Figs. 319-321: Various positions of light source, eye and object. Figs. 319, 321, C10r; fig. 320, C12v.

    He develops these two basic demonstrations on C10r (1490-1491). Here the diagrams are much more elaborate (figs. 319, 321) and the accompanying texts more precise:

All umbrous bodies, larger than the pupil, interposed between the eye and the luminous body, will show themselves as being in shade.
The eye positioned between the luminous body and the bodies illuminated by this light will see the said bodies without any shade.

    On C12v (1490-1491) he describes a variant of this situation (fig. 320).

The percussion of derivative shade born and caused by a spherical umbrous and luminous body and interrupted by its percussion on different bodies situated at various distances, appears round to the eye which is situated in front of it near the centre of the original light.

    Some two years later he considers in somewhat more detail the configuration: light source, eye and umbrous object on A2r (fig. 322, 1492; cf. CA112va, fig. 324, c.1505-1508 and CU860, TPL694f, 1508-1510):

The umbrous body which is seen along the line of incidence of light, will not show any protruding part of itself to the eye. For example. Let the umbrous body be a. Let the light be c. Cm as well as cn are incident luminous lines, that is, lines which transfer light to the body a. The eye is at the point b. I say that [since] the light c sees the entire part mn, that those things which are in relief will be entirely illuminated. Hence the eye positioned at c cannot see shade and light. Not seeing this, each part appears to it of one colour. Whence the differences of the protruding and globulous parts do not appear.

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Figs. 322-325: Further variations of eye, object and light source. Fig. 322, A2r; fig. 323, M79v; fig. 324, CA112va;fig. 325, M80r; fig. 326, BM171r; fig. 327, M79v.

    At about the same time he considers the configuration: eye, opaque body, light source on BM171r (fig. 326, c.1492): "The umbrous body situated between a light and the eye will never show a luminous part of itself unless the eye sees all the original light." When he returns to this theme some eight years later on 80r (1499-1500) he is explicit about his methodical approach (fig. 325):

On Painting

Of all the things seen, one has to consider 3 things, that is, the position of the eye that sees, the position of the thing seen and the position of the light that illuminates such a body.

    Having illustrated each of these (figs. 323, 325, 327), he concludes on the folio opposite (M79v): "These show once the eye between the light and the body; 2nd, the light between the eye and the body; 3rd the body between the eye and the light." These passages may well have been drafts for his later statement on K105[25](v) (1506-1507):

On Painting

The aspects of shadows and lights with the eye are 3, of which one is when the eye and the light are seen on the same side of a body; 2nd is when the eye is in front of the object and the light is behind this object; 3rd is that in which the eye is in front of the object and the light, and on the side in such a way that the line which extends from the object to the eye and from this object to the light, when joined together, will be rectangular.

    The third alternative here mentioned is one he had considered as early as 1487-1490 on Triv. 10v (figs. 328-329):

The eye which finds itself between the shadow and the surrounding lights of shaded objects will see in these bodies the deepest shadows that are to be encountered with it, that is, under equal visual angles of incidence.

    He alludes to it again on C27r (fig. 330, 1490-1491) under the heading of:

Perspective

That eye which finds itself between the light and shade surrounding the opaque bodies will see the shadows divided from the luminous side passing transversally through the centre of this body.

    When he returns to this situation nearly two decades later on CU147 (fig. 331, TPL251, 1508-1510) he relates it directly to effects of relief in painting (cf. vol. 1:Pt.3 below and pp. ):

Of things positioned on a bright background and why such a use is useful in painting.

When an umbrous body borders on a background [that is] of a bright colour and illuminated, then by necessity it will appear to stand out in relief and separate from this background.

That which is said happens because bodies with curved surfaces by necessity make themselves umbrous on the side opposite to which they are percussed by luminous rays, since that place is deprived of such rays, for which reason it varies a great deal from its background, and the part of that illuminated body never terminates in that illuminated background with its first [degree of] brightness. Hence between the background and the first [degree of] light of the body there is interposed a background of the body which is darker than [either] the background or than the light of the respective body.

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Figs. 328-331: Cases in which an object is half in light and half in shade. Figs. 328-329, Triv. 10v; fig. 330, C27r; fig. 331, CU147.

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Figs. 332-337: Variants where the eye is positioned obliquely relative to the light source and opaque body. Figs. 332-334, CA144vb; fig. 335, M80r; fig. 336, CA120vd; fig. 337, BM113v.

    He also considers a further variant in which the eye is obliquely positioned relative to the light source and the opaque body. Rough sketches of this alternative appear without text on CA144vb (figs. 332-334, c.1492). On M80r (fig. 335, c.1499-1500) he returns to this variant adding a brief caption: "b is the eye, a is the thing seen, c is the light." He draws further examples of this on Ca120vd (fig. 336, c.1500) and BM113v (fig. 337, c.1510), which as will be shown (see below pp. ) had a certain importance in his astronomical studies. He pursues this theme of various positions of the eye in a series of notes in the Treatise of Painting as on CU645 (fig. 338, TPL685, 1508-1510):

On the middle included between the light and the principle shade.

Middle shade shows itself as being of greater quantity to the extent that the eye which sees it is more opposite the centre of its size. Middle shade is said to be that which tinges the surfaces of umbrous bodies behind the principal shade and is contained inside the reflection and it is darker or brighter to the extent that it is closer or further from this principal shade.
Let mn be a darker shadow. The remainder always becomes brighter towards the point m and the rest of the figure does not apply to this proposition but it will serve for the succeeding one.

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Figs. 338-340: Effects of positions of the eye on derives shade. Fig. 338, CU645; fig. 339, CU647; fig. 340, CU650. On CU647 (fig. 339, TPL687, 1508-1510) he asks:

What is that site where one never sees shade on umbrous spherical bodies?

The eye that is situated between the reflected pyramid of the species illuminated by umbrous bodies will never see any umbrous part of that body.

Let the reflected pyramid of the illuminated species be abc and let the illuminated part of the umbrous body be the part bcd. And let the eye which stands within this pyramid be e, to which all the illuminated species bdc could never converge unless it were seen [on the same side as] the luminous point a, from which no shade is ever seen which it does not destroyimmediately. It therefore follows that e, which only sees the illuminated part odp is more deprived of seeing the boundaries of shade bc, than is a which is further away.

    Having considered a case where the eye is closer to the opaque body than the light source, he asks what happens if the eye is further from the opaque body than the light source on CU650 (TPL688, 1508-1510):

What is that site or indeed that distance around a spherical body which is never deprived of shade?

But when the eye is more distant from the umbrous sphere than the body which illuminates it, then it is impossible to find a site, where the eye is entirely deprived of the umbrous species of such a body.

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Figs. 341-342: How changes in the size of opaque body and light source affect derived shade. Fig. 341, CU648; fig. 342, CU649.

    This general claim is followed, as usual, by a concrete demonstration (fig. 340):

This is proved. Let bnc be the umbrous body. Let bsc be illumined object. Let o be the eye more distant from the umbrous body than the light a, which eye sees all the shade bdce.

And if this eye moves circularly around this body with the same distance, it is impossible that it entirely loses all the aforesaid shade, such that, if through its movement it loses one part of this shade on one side, this is acquired by the other side through the [same] movement.

    Leonardo has explored how various positions and distances of the opaque body, eye and luminous source affect the shade seen. He now adds a further variable: changes in size of the opaque body and light source. On CU648 (fig. 341, TPL734, 1508-1510) he considers cases where the luminous source is either equal in size or larger than the umbrous sphere, under the heading:

What is that light which, even if the eye is further removed from the umbrous sphere than this light, it can never see shade while standing in front of the light?When the luminous body is equal to or larger than the umbrous spherical body, then the eye which is behind this light will never see any part of the shade on the umbrous body, as a result of the difference of said luminous body.

Let cedf be a spherical umbrous body; ab is the luminous source equal to the umbrous body and let cfd be the shade of this spherical body. I say that the eye l which stands behind the light ab at whatever distance one wishes, can never see any part of the shade, through the 7th of the ninth which states: Parallels never converge to a point. Since ac and bd are positioned parallel [to one another] and embrace precisely half of the sphere and [since] the lines nm...converge at the point l, this point can never see half of the sphere at its diameter cd.

    Involved here are problems relating to visual perception (see below pp. ). On CU649 (fig. 342, TPL735, 1508-1510) he considers a case where the luminous source is smaller than the opaque body:

On the eye which, over a long distance will never have the view of the shade on the umbrous body occluded when the luminous source is smaller than the umbrous body.

But when the luminous body is smaller than the umbrous body, there can always be found some distance where the eye can see the shade of this umbrous body.

Let opef be the umbrous body and let the light be ab smaller than this umbrous body by whatever proportion one wishes. I say, that one can never prevent the eye, n, which is behind this light, from seeing some umbrous part of the shade of the spherical umbrous body as the rectilinearity of the lines show.

    Aristarchus' simple distinction between three kinds of light had served as a starting point for Leonardo. But as we have shown he considers variations in the light source, in the object, in the eye and finally in combination, to arrive at a considerably more complex picture of the situation. This picture will become more complex still in his fourth book, when he studies the properties of derived shade on meeting interposed objects.

 

Book Four: Derived Shade and Interposed Objects

Again these derived shadows, where they are intercepted by various objects, produce effects as various as the places where they are cast. And on this I shall make the fourth book (CA250ra).

    What happens when the shadow produced by one body in turn meets another opaque body? This question leads Leonardo to make a series of further detailed studies. Had he managed to compile these systematically in his fourth book on light and shade he would probably have begun with an introductory chapter (1), followed by an excursus on degrees of light (2) and on angles of light (3) which would have led to a consideration of angles of shade (4) and the role played by the position of the light source (5) and size of the light source (6). All this would have been preliminary to his basic concern, namely, consideration of how changes in position and shape of the interposed plans affect shadows (7).

    Experiments had, meanwhile, made him aware that under certain conditions one light source and one opaque body could produce two shadows on an interposed plane. The how and why of this phenomenon would probably have involved a further chapter (8).

(figure)

Figs. 343-346: Basic distinctions between separate and conjoined shade, direct and oblique shade. Figs. 343-344, A102r; figs. 345-346, CU623.

    The shadows produced in cases of compound shade, i.e. when more than one light source and/or more than one opaque body are involved, would have led to at least one further chapter (9, cf. Chart 9 ). Each of these will be considered in turn.

Book Four - 1. Introduction

    By way of introduction to the various possible shapes of shadow Leonardo would probably have begun with a distinction such as he makes on A102r (BN 2038 22r, 1492) between "separate, and conjoined shade" (fig. 343) and "separate, inevident shade" (fig. 344). This bears comparison with his subsequent distinction made on CU623 (figs. 345-346, TPL600, 1508-1510):

In how many principle modes is the percussion of derived shade transformed?

The percussion of derived shade has two varieties, that is, direct and oblique. The direct is always less in quantity than the oblique, which can extend itself to infinity.

This idea of the infinite variations of shadow is pursued on CU859 (TPL809, fig. 347, 1508-1510):

Precept A

Lights and shadows are as various as the variations of the sites where they are found.

F. When the umbrous part is augmented by a dark object, this shade will be darker than at first to the extent that such an augmentation is less clear than the air.

D. The percussion of the derived shade will never be the shape of its original primitive [shade], if the primitive light is not the same shape as the body which makes the shadow.

(figure)

Figs. 347-348: Varieties of shade on CU859 and 588.

    Accompanying this is a diagram (fig. 347) showing shade in an enclosed space. This alternative is contrasted with shade in the open air in two diagrams (fig. 348) illustrating the varieties of primitive shade on CU588 (fig. 348, TPL572, 1508-1510):

In how many ways does primitive shade vary?

Primitive shade varies in two ways of which the first is simple and the second is composed.

Simple is that which regards a dark place and by this such a shade is composed darkness which sees a place illuminated with various colours with the result that such a shade mixes itself with the species of the colours of the objects positioned opposite.

    In the Codex Urbinas this is followed by a passage on the varieties of derived shade (CU759, TPL573, 1508-1510):

What variety does derived shade have?

The varieties of derived shade are of two sorts of which the one is mixed with the air opposite the primitive shade. The other is that which percusses in the object which cuts this derived shade.

    At the end of this introductory chapter he might have considered cases where primitive and derived shade are the same as on C4r (fig. 349, c.1490):

The obscurity produced in the percussion of the umbrous concourse will have conformity with its origin, which is born and terminated between nearby plane surfaces, and of the same quality and in direct opposition.

    He returns to this idea in a sketch without text on Ca144VA (FIG. 350, 1492) and then in greater detail on CU710 (fig. 351, TPL581, 1508-1510), asking:

Whether primitive shade is more powerful than derived shade?

Primitive shade, being simple, will be of equal darkness to simple derived shade. This is proved. And let the simple primitive shade be de and let the simple derived [shade] be fg. I say, by the fourth of this, which states: "darkness is the privation of light," [that] simple shade is therefore that which does not receive any illuminated reflection and for this reason it remains tenebrous as is de which does not see the light a. And the simple derived shade fg also does not see it and hence these shades are of equal obscurity because both the one and other are deprived of light and luminous reflection.

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Figs. 349-351: Cases in which primitive and derived shade are the same. Fig. 349, C4r; fig. 350, CA144ra; fig. 351, CU710.

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Figs. 352-353: Demonstrations relating to degrees of light. Fig. 352, Triv. 3v; fig. 353, Forst. III 58v; fig. 354, W12351r; fig. 355, I33r.

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Figs. 356-366: Concerning the properties of translucent and opaque objects.

Book Four - 2. Degree of Light

    He makes several notes concerning the expansion of light and its varying degrees with distance. These could well have been intended as an introduction to analogous problems in shade. The earliest extant notes on this theme occurs on Triv. 3v (fig. 352, 1487-1490): " If a light be placed in a lanturn that is in the middle, it will enlarge its splendour; at CD its rays will be twice as large at the greater distance FB twice as far away." When he returns to the problem of degrees of light and distance on W12351r (fig. 354, C.1493-1494) he asks: "I ask how and how much one is illuminated more than the other: ab, cd and ef?" To this question he provides a reply nearly a decade later on CA150ra (1500-1503) where he discusses the properties of both translucent and opaque objects, claiming (figs. 356-366):

that part will remain more luminous, which is percussed by a greater sum of luminous rays and...conversely, it will be less luminous which is seen by a lesser quantity of these rays.

...All the parts of the illuminated body which see the entire circle of the luminous body will be as different in clarity among one another as they are closer to the luminous body.

    On CA132vb (c.1508) he provides a more succinct answer: "That part of an illuminated object will be the more luminous which is the closer to the cause of its light," a claim that he repeats nearly verbatim on CU447 (TPL526a, 1508-1510): "That part of an object will be more illuminated which is closer to the luminous object which illuminates it." Related to this question of degrees of light is the problem how these degrees can be multiplied. On A3v (1492), for instance, he notes:

On Lights

Many small lights joined together will be of greater power each in itself than when they were separate. The proof you will see if you take many lights in a straight line and you stand at a certain distance opposite the middle of this line and you note the quality of the light made by these lights and then join them together. You will see [that] the place where you stood will be more luminous that at first....

Again it is known that the stars are of equal light to that of the moon and if it were possible to join them together that they would compose a body much larger than that of the moon, and nonetheless, even if it be a clear night and they are all shining, if the moon is not in our hemisphere, our part of the world remains dark.

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Figs. 367-373: On the multiplication of candlelight. Figs. 367-369,

Ca270va; figs. 370-371, CA270ra; fig. 373, CA260ra.

    He mentions the problem again on Forster III 58v (1490-1493) under the heading (figs. 353):

On the duplication of lights.

If one light makes 4 ounces (and) [then] it appears that 2 of these lights together make 8 ounces.

    He provides a visual demonstration of this principle on CA270va, 270ra and 270va (figs. 367-373, 1508-1510) where he compares the light of smaller candles with larger flames. On W12351r (c.1493-1494) the matter is raised as a question: If one candle consumes itself in one hour, in how much time will 3 candles together consume themselves? This theme he pursues on I33r (fig. 355, 1497), here making explicit the link between his concepts of light and the pyramidal law (cf. vol. 1, pt.2).

Of the luminous rays and the powers of their extremities.

Since the luminous ray is of pyramidal proportion and maximally when the centre is equal it will therefore happen that when 2 rays meet along a straight line parting from equal lights this ray will be doubled throughout and of equal power because where the one has the apex of its pyramid the other has its base as mn shows.

    In addition to such general statements concerning the relation of degrees of light to distance and the pyramidal law, he emphasizes the connection between light intensity and luminous angles.

Book Four - 3. Angle and Intensity of Light

    One of his earliest extant notes on this subject occurs on C12r (1490-1491):

That part of an illuminated wall will be the more luminous which is illuminated by a greater luminous angle. And that place [struck] by said rays will observe the accompanying quality of light less which is shadowed by a greater umbrous angle.

    This idea he restates briefly on C21v (1490-1491):

That part of an umbrous body which is percussed...by a larger luminous ray will be more illuminated than any other.

    On BM103r (1490-1495) is found the draft of another version in a hand probably not Leonardo's: "That pyramid which parts from its base with more unequal and differs angles will be thinner and a more distorted demonstrator of the true size of its base." On the verse of this folio there is another draft in this hand: "If the shade of the umbrous bodies...born of a spherical luminous source falls between equal angles and an unequal centre it will be of various shapes and various [degrees of] obscurity." Leonardo pursues this theme on A85r (BN 2038 5r, 1492):

Painting

That part of an object that receives over it a luminous ray between equal angles will be more luminous than [any] other part of this luminous object.

And that part which is struck by a luminous ray under less equal angles will appear less luminous.

    This idea he repeats more succinctly on A112v (BN 2038 33v, 1492): "That light which strikes under more equal angles is more powerful. Example of the blow." On Mad I 32r (1499-1500) he pursues this theme (fig. 376):

(figure)

Figs. 374-376: Demonstrations concerning intensity angles of light and its intensity. Fig. 374, Mad I 31v; fig. 375, CU671; fig. 376, Mad I 32r.

Lights which close themselves around the axis of the luminous ray and the base of the illuminated object will have that proportion amongst them that the bases of the compound pyramids have.

    On the folio opposite (Mad I 31v, fig. 374, 1499-1500) he quantifies this problem:

Definition

The body f will be the half less illuminated than the body e because the part of the sky which illuminates it is twice as small as is that of c, as is shown in cd and ab.

The proportion that the angle surrounding the illuminated body has to the axis of the illuminated ray...will be such as the quality of the light has.

If the acute angle rcd enters r times into the angle cmn, then cm is 4 times less luminous than cn. Again if the angle dc enters 12 times into the obtuse angle ard and the angle cmn enters 3 times into the obtuse angle mno, the proportion will follow.

    He returns to this theme on CU671 (TPL680, 1508-1510) under the heading:

Of the particular light of the sun or some other luminous body.

That part of the illuminated body will be of greater clarity which is percussed by a luminous ray among more similar angles and least illuminated is that which finds itself among angles that are more difform than these luminous rays.

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Figs. 377-379: Angles and light intensity on CU668.

A specific example demonstrates this claim (fig. 375):

This angle n on the side, which looks at the sun, being percussed by this sun under equal angles will be illuminated with greater power of rays than any other part of this illuminated body and the point c will be less than any other illuminated part since this point is struck by the solar body with angles that are more difform than any other part of the surface where such solar rays extend. And of the two angles, let the greater be dce and the lesser ecf and the equal angles which I have to draw first are ano and bnr which are precisely equal. And for this reason n will be illuminated more than any other part.

    This connection between luminous angles and light intensity is broached afresh on CU668 (TPL718, figs. 377-379, 1508-1510):

In what surfaces is true and equal light found?

That surface will be equally illuminated which is equally remote from the body which illuminates it as [for instance], if from the point a which illuminates the surface bcd, there would be drawn lines equal to this surface. Then by the definition of the circle this surface will be equally illuminated in each of its parts and if this surface were plane, as is demonstrated in the second demonstration efgd, then if the extremities of the surface are equally distant from such lines, the centre h will be the part closest to such a light and will be more illuminated than these extremities, by the extent to which it is closer to its said light e. But if the extremities of such a plane surface are removed from such a light with an unequal distance, as is shown in the third figure iklm, then the closest and the most remote part will have such a proportion in their lights as is that of their distances from the body which illuminates them.

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Figs. 380-381: Angles and light intensity on CU858.

    On CU858 (TPL820, figs. 38-381, 1508-1510) he pursues the question:

On reflected light

To the extent that the illuminated object is less luminous than its illuminating source, to that extent will its reflected part be less luminous than the illuminated part.

That thing will be more illuminated which is closer to the illuminating source.

To the extent that bc enters into ba to that extent will ad be more illuminated than dc.

That wall which is more illuminated appears to have its shadows of greater obscurity.

    On CU675 (TPL694b, 1508-1510) he asks:

What part of a body will be more illuminated by a light of the same quality?

That part of a body which is illuminated by a luminous quality will be of a more intense brightness which is percussed by a greater luminous angle.

    By way of demonstration he offers a specific example (fig. 388):

This is proved. And let the hemisphere be rmc which illuminates the house klof. I say that that part of the house will be more illuminated which is percussed by a greater angle originating from a luminous source of the same quality.

Therefore at f where nfc percusses, there will be a more intense brightness of light than where the angle edc percusses and the proportion of the lights is the same as that of the angles and the proportion of the angles will be the same as is that of their bases nc and ec, of which the larger exceeds the minor in whole by part ne. And likewise at a under the eave of the roof of such a house there will be that much less light than in d to the extent that the base bc of such an angle bac is less than the base ec and thus it always follows proportionately, the light being of a same quality.

And the same which is stated above is confirmed in some object illuminated by our hemisphere and this is manifested in the part of a spherical object under the hemisphere k and f which, at the point b is illuminated by the entire part aec and at the part d by the hemisphere ef and at o by gf and in n by mf and at h by sf and thus you have understood where the first [degree of] light and the first [degree of] shade is in any body.

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Fig. 382: Angles and intensity of light on CU667.

    How these statements concerning light apply also to shade is explored on CU667 (TPL755, 1508-1510) under the title:

Rule for placing the necessary shadows and lights in a figure or some body with sides.

Such is the greater or lesser obscurity of shade, or indeed the greater or lesser brightness of light striking the faces of a body with sides, as is the greater or lesser size of the angle, which is enclosed between the central line of the luminous body, which percusses the centre of the illuminated side, and the surfaces of this illuminated side.

    As usual this is followed by a concrete demonstration (fig. 382):

As [would be the case] if the illuminated body were an octangular column, the front of which is placed here in the margin. And let the central line be ra which extends from the centre of the luminous object r to the centre of the side sc. And again let it be that the central line rd extends itself from the centre of this luminous body to the centre of the side cf. I say that there will be such a proportion between the quality of the light which the side sc receives from this luminous body and that which the second side receives from he second side cf, as there is between the size of the angle bac and the size of the angle edf.

    These principles he summarizes in a late note on CA385vc (1510-1515):

That light is brighter which is of a greater angle.

That shadow is darker which is born of a more acute angle.

Book Four - 4. Angle and Intensity of Shade

    Leonardo's interest in the links between angles and intensities of shade is implicit in an early note on Triv. 28v (c.1487-1490) where he notes that (fig. 383): "to the extent that ab enters cd to that extent an will be darker than cr." On A85r (BN 2038 5r, 1492) he develops this demonstration (fig. 384):

To the extent that the shade made by the object on the wall is less than its cause, to that extent will this object be illuminated by weaker luminous rays.

De is the object [and] dc is the wall. To the extent that de enters fg to that extent will there be more light in fh than in dc. To the extent that the luminous ray is weaker to that extent will it be further from its aperture.

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Figs. 383-384: Links between angles and intensity of shade. Fig. 383, Triv. 28v; fig. 384, A85r.

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Figs. 385-386: Demonstrations concerning angles and light intensity on CU663-664.

(figure)

Figs. 387-390: Abstract and concrete demonstrations of problems of light and shade. Fig. 387, A89v; fig. 388, CU678;

fig. 389, CU675; fig. 390, G12r.

    This link between angles and intensity of shade remains implicit in another note in the same manuscript, on A89v (fig. 387, BN 2038 9v, cf. CU657, TPL555a, 1492):

Painting

Among shadows of equal quality that which is closer to the eye appears of lesser obscurity.

Why is the shade eab in the first degree of obscurity and be[c] in the second [degree] and cd in the third [degree of obscurity]. The reason is that eab does not see any part of the sky. Therefore, no part of the sky sees it, and for this reason it is deprived of original light. Bc sees the part fg of the sky and is illuminated by this. Cd sees the sky hk. Since cdis seen by a greater amount of the sky than is bc it is reasonable that it be more luminous and so on up to a certain distance the wall ad will constantly become brighter until the darkness of the habitation will be overcome by the light of the window.

    This principle he illustrates again on CU675 (TPL694, 1508-1510) analysed above (fig. 389, p. ) and once more on CU678 (TPL694c, 1508-1510) where he claims (fig. 388):

    The shade produced by the sun that remains under the rooves of buildings acquires darkness with every degree of height. He pursues this theme on G12r (c.1515) in the context (fig. 390):

Of universal light illuminating plants.

That part of a plant will show itself as covered with shadow of less obscurity which is more distant from the earth.

This is proved. Let ap be the tree. Let nbc be the illuminated hemisphere. The part below the tree sees the earth pc, that is the part o and it sees a little of the hemisphere in cd. But the part [that is] higher in the concavity a is seen by a greater amount of the hemisphere, that is, bc, and for this reason (and because it does not see the darkness of the earth) it remains more illuminated. But if the tree is covered with leaves as is the laurel, arbutus, box or holm-oak, then it is variegated because even if it does not see the earth and it sees the darkness of the leaves, divided by many shadows which darkness reverberates to the reverse of these leaves and such trees [therefore] have shade that is darker to the extent that they are closer to the middle of the tree.

(figure)

Figs. 391-392: Simple studies of shade on C3v on CU742.

    In each of the six examples above Leonardo has considered various angles of shade produced by eaves of rooves or other overhanging objects. These range from concrete cases to abstract geometrical demonstrations. He is equally systematic in his approach to shade on the ground. At the simplest level he simply depicts a static situation as on C3v (fig. 391, 1490-1491). A next step is to consider the psychological aspects (cf. part 3: 4 below) of this situation as on CU742 (TPL605, 1508-1510) where he asks:

What background will render shadows darker.

Among shadows of equal darkness, that will show itself as darker which is generated in a background of greater brightness. It follows that that part appears less dark which is in a darker background.

    This general claim is, as usual, supported by a specific demonstration (fig. 392):

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

    Having considered the static case he examines a dynamic situation on CU663 (TPL720, 1508-1510) in which the distance, angle and accordingly the shadow changes (fig. 385):

On the remoteness and distance that a man makes in going away from and approaching a same light and on the variety of its shadows.

The shades and lights will vary in shape and quantity in a same body with the variety of the coming closer or going further which a man makes in front of this light.

And this is proved. And let the man be bc who, having lights from a, will produce his shade bcf. Then the man moves from c to e and the light which stays fixed varies the shade in shape and in size, which is the 2nd shade deg.

    In the following passage on CU664 (TPL721, 1508-1510) he examines a related situation. "On the variety of the shade produced by an immobile light generated in bodies that are bent...lower or higher without moving [the position of] their feet." A specific demonstration again follows (fig. 386):

This is proven and let the immobile light be f and let the man not moving from his position by ab who bends to cb. I say that the shadow varies itself infinitely from a to c because the movement is in space. And space is a continuous quantity and consequently divisible to infinity. Therefore shade varies infinitely, that is, from the first shade aob to the second shade bcr and thus our proposition is concluded.

    On CU731 (TPL5778, 1508-1510) his study of dynamic situations continues with a comparison of three cases in a single demonstration (fig. 393):

On derived shade distant from primitive shade.

The boundaries of derived shade will be more confused which are more distant from primitive shade. This is proved. And let ab be the luminous body and cd the primitive shade and ed is the simple derived shade and cg is the confused boundary of the derived shade.

    He returns to this problem of degrees of shade varying with angles and distance on E31v (figs. 394-395, c.1513-1514):

A long and narrow luminous source produces boundaries of derived shade which are more confused than spherical light. And it is this that contradicts the following proposition: that shade will have its boundaries more distant which is closer to the primitive shade, or if you wish to say, to the umbrous body. But the cause of this [contradiction] is the long and narrow shape, ac, of the luminous source.

 

Book Four - 5. Position and Intensity of Light Source

    Meanwhile, Leonardo had been developing model demonstrations in which he explored the role of position and intensity of light source in relation to shade. On C8v (1490-1491), for instance, he begins with a case in which two light sources are of equal intensity (fig. 396):

    That umbrous body which is positioned between two equal lights will make as many shadows as there are lights. Which lights are darker than one another to the extent that the light on the opposite side is closer to this body than the others.

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Figs. 393-395: Angles, distance and shade. Fig. 393, CU731; figs. 394-395, E31v.

(figure)

Figs. 396-397: Experiments concerning intensity of light and shade on C8v.

    Next he considers a case where the lights are not of equal intensity (fig. 397):

That umbrous body which is equidistantly positioned between 2 lights will make two shadows that are darker than one another to the extent that the lights producing the one is larger than the other.

    On C22r (1490-1491) he devises an experiment in four steps to test these factors of position and intensity of light. As a first step he considers a case of two candles of equal intensity with an umbrous body in the centre (fig. 398): "that umbrous body will make 2 derivative shadows of equal darkness which has (in itself) 2 light sources equal in size equidistant from it." As a second step he again takes a case of two candles of equal intensity, but with an umbrous body now off centre (fig. 399): "The proportion of the darkness of the shade ab with the shade bc will be that of the distance of the lights among themselves, that is, of nm to mf."

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Figs. 398-401: Systematic experiments concerning intensity of light and shade on C22r.

    His third step is a case where the two candles are of different intensity and the umbrous body is again in the centre (fig. 400):

If one has positioned an umbrous body equidistantly between 2 lights, it will make two opposite shadows which will differ in their obscurity to the extent that the powers of the 2 light sources...which produce them, differ.

    This general claim is then followed by a specific demonstration (fig. 400): "If the light xv is equal to the light vy the difference of the lights will be such as is that of the sizes." His fourth step is a case where the two candles are of different intensity and the umbrous body is off centre (fig. 401):

But if the large light source is distant from the umbrous body and the small light source is nearby, it is certain that shades can be produced [which are] of the same darkness or brightness.

    Striking here is Leonardo's scientific approach: how he systematically alters one variable while keeping the other constant, thus providing controlled situations. Lacking is a quantitative method of recording his results. Nonetheless, the way is thereby set for the Rumford's photometry experiments. When Leonardo returns to this problem on CA199ra (c.1500) quantitative considerations are alluded to. He begins by drawing a rough diagram in the right-hand margin (fig. 402) beneath which he asks: "Give me the site of the object that produces shades of equal darkness." In the text opposite he drafts a proposition:

Of derived shades opposite, created by a same object (opaque caused by two lights opposite) interposed bet (ween 2 lights) opposite of (various sizes and distances from this object) with various distances between light of various sizes.

    This he crosses out and begins afresh with a general heading: "Of derived shades, opposite [one another], created by a same object,...which is interposed at various distances between lights of different**." He then outlines three specific experiments:

Let there be given two opposed shades around a single object...interposed between 2 lights of double power, and that these shades are...among themselves of equal obscurity. It is asked what proportion have the spaces interposed between these lights and the said object.

You will give 2 shades of a same body interposed between 2 lights of double power and such that with those shades one is treble the darkness of the other.

You will give 2 shades born around a same body, which are of a double darkness in relation to one another and the greater darkness is towards a greater light.

    Whether he actually carried out these specific experiments is uncertain. But these problems are not forgotten. On CU684 (TPL682b, 1508-1510), for instance, he considers how position and intensity of a light source affect the illumination of an umbrous body:

On universal illumination mixed with the particular [illumination] of the sun or other lights.

Without a doubt, that part of the umbrous body which is seen by a lesser quantity of a universal and particular body, will be less illuminated.

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Figs. 402-404: Further experiments involving effects of distance on derived shade. Fig. 402, CA199ra; fig. 403, CU684; fig. 404, CU695.

    To support this general claim he provides a specific example (fig. 403):

This is proved. And let a be the body of the sun positioned in the sky nac. I say that the point o of the umbrous body will be more illuminated by...universal light than the point r, because o sees and is seen by every part of the universal light nam and the point r is only seen by the part of the sky mc.

Moreover, o is seen by the entire quantity of the sun that is facing it, and r does not see any part of this sun.

    A few paragraphs later on CU695 (TPL689, 1508-1510) he returns to this situation from the point of view of intensity of shade (fig. 404), asking:

What light makes the shades of bodies more different from their lights?

That body will make the shades of greater darkness which is illuminated by a light of greater brightness.

The point a is illuminated by the sun and the point b is illuminated by the air illuminated by the sun. And such is the proportion between the illuminated [part] a and the illuminated [part] b, is the proportion which the light of the sun has with that of the air.

    On CU159 (TPL249a, 1508-1510) he provides a catalogue of various possibilities under the heading:

Of the colours of incident and reflected lights.

When two lights [have] an umbrous body put in the middle [between] them, they can only vary in two ways, that is, either they will be equal in power or they will be unequal, that is, speaking of the lights in relation to one another. And if they are equal, they can vary their brightness on the object in two other ways, that is, with equal brightness or with unequal brightness. It [the brightness] will be equal when they [the light sources] are equidistant; unequal [when the light sources are] at unequal distances. [When the light sources are] equidistant they will vary in two other ways, namely, the object will be illuminated by lights equal in brightness and in distance (lights equal in power and in distance from the object opposite).*

The object situated equidistantly between two lights equal in colour and brightness can be illuminated by these two lights in two ways, namely, either equally from every side or unequally. And it will be illuminated equally by these lights when the space which remains between the two lights is of equal colour and darkness or brightness. It will be unequal when these spaces between the two lights are of different darkness.

    These categories become easier to visualize when rendered in tabular form (see chart 11).

(chart)

Chart 11. Categories of Light and Shade described on CU159 (TPL248a, 1508-1510).

(figure)

Figs. 405-406: Elementary cases of derived shade and interposed plans on C9r and C18v.

 

Book Four - 6. Size/Shape of Light Source/Object

    He is also concerned how the size or shape of a light source in relation to an opaque object affects derived shade. On C18v (1490-1491), for instance, he makes a general comment on this theme: "The shapes of shadows often resemble their origin, the umbrous body and often their cause, the luminous source." Directly beneath he draws an introductory example (fig. 406) with the caption:

If the shape and size of the luminous body are equal to that...of the umbrous body, the primitive and derived shade will be of the shape and size of this body, falling between equal angles.

    In the next paragraph he states why this is not always the case:

At a certain distance derived shade will never be the same shape as the umbrous body from which it originates, if the shape of the light of this illuminating body is not the same as the shape of the body of the said illuminated light.

    Again he draws an example (fig. 407) followed by a caption and a restatement of his claim:

A light of a long shape will have the effect that the derived shade, originating from a round body is...wide and low even if it be percussed between equal angles.

It is impossible that the shape of derived shade is the same as that of the luminous body whence it originates unless the light, its cause, is the same shape and size as this umbrous body.

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Figs. 407-408: Further cases of derived shade and interposed plane on C18v and C18r.

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Figs. 409-410: Demonstrations of derived shade and an interposed plane on C12r and C8r.

    Having considered the case of a long light source and a round umbrous body on C18v, he examines, on the recto of the same folio, the case of a round light source and a long umbrous body (fig. 408). He is ever playing with variables. Beneath this drawing he again adds a caption:

    The umbrous percussion originating from a long umbrous body and caused by a round luminous source, at a certain distance is the shape of the umbrous body and at a certain other distance [that] of the luminous source.

    On C12r (fig. 409, 1490-1491) he notes:

Not only the quantity but also the quality of the shade and their boundaries will not show themselves with their true shape in their foreshortening.

Bc being an invisible boundary, by the loss of shade, nonetheless appears visible as a result of foreshortening, not otherwise than ab and dc appear.

    On C8r (1490-1491) he draws an example of a long light source and a round umbrous body (fig. 410), beneath which he asks: "Why in this case does the derived shade show itself as dark in the middle of its height ab and is not discerned at its extremities cd?". In a passage on BM103v (1490-1492) this question of the shape of light source is pursued in a series of drafts:

If a light is of equal proportion to the umbrous body positioned opposite, the derived shade will be the same as the umbrous body if it falls on a plane surface at equal angles.

If a light is of a long shape and the umbrous body is round, the derived shade will be wider than it is high.

Light which falls on a flat place under equal angles...

Of the round aperture and the long light.

...the percussion is long.

    The umbrous and luminous body of spherical shape will produce derived shade of a long shape if this falls on a flat plane at unequal angles.

    He pursues these problems on CA187va (1492) in the context of his camera obscura studies, which constantly run parallel to his light and shade demonstrations (see below pp. ):

No long light will show the true form of the shade separated...from the walls by spherical bodies.

This shadow is long and thin.

No separate shade can stamp on walls the true form of the umbrous body if the centre of the light is not equidistant from the extremities of the body.

    When he returns to these problems on CU632 (TPL607, 1508-1510) he begins with a general claim: "Shade will never have the true similitude of the contour of a body whence it originates even if it be spherical unless the light is of the shape of the umbrous body." Directly following this he lists four specific cases:

If a light is of a long shape which extends upwards, the shades of this illuminated body extend themselves laterally.

If the length of a light is lateral the shade of the luminous body will make itself along its height. And likewise in whatever way the length of the light finds itself, the shade will always have its length intersected cross wise with the length of the light.

If the light is thicker and shorter than the luminous body the percussion of the derived shade is longer and thinner than the primitive shade.

If the length and width of the luminous body is equal to the length and width of the umbrous body, then the percussion of the derived shade will be of the same shape at its boundaries as the primitive shade.

(figure)

Figs. 411-412: Simple cases of derived shadow on an olique plane on C18r and C11v.

    This problem is again broached in the context of his camera obscura studies on CA195va (see below pp. , c.1510): "Why shadow is never similar to the umbrous body if the light is not equal and similar to the umbrous body and it is not stamped over a flat wall between equal angles."

    On E31v (1513-1514) he returns to this theme for the last time in the extant notes:

That luminous body of a long and narrow shape makes the boundaries of derived shade more confused than he spherical light and this is what contradicts the following proposition:

That shade will have its boundaries more noted which is closer to the primitive shade or, if you wish to say, the umbrous body, but the cause of this is the long shape of the luminous body, etc.

Book Four - 7. Position/Shape of Light Source/Object

    Leonardo's expressed aim in book four is to study how shadows vary with the position and shape of the plane on which they are projected. His simplest example of this is a drawing on C18r (1490-1491) beneath which (fig. 411) he writes:

Even though the umbrous and the luminous body are of spherical rotundity and of equal size, nonetheless, its derived shade will not resemble the rotundity of the body whence it originates and will be of a long shape if it falls under unequal angles.

(figure)

Figs. 413-415: Further cases of derived shadows on oblique and irregular planes on CA241vd.

    One step more complex is his drawing on C11v (fig. 412, 1490-1491) where part of the plane is inclined and part is positioned upright, followed by a caption: "Of the derived shade impressed among different qualities of angles, that part which is found between straight angles will hold the first degree of darkness." On CA241vd (1508-1510) he begins with a general claim in draft form, headed:

On the percussion of derived shade.

The percussion of derived shade will never be of the same shape as the umbrous body (if the light does not have the shape) if the luminous body, cause of this shade, (is not similar to ) has not a shape equal to that of this umbrous body and if the umbrous rays...which border with the luminous rays are not equal in length amongst themselves.

    In the right-hand margin beneath this he again draws the shadow of a sphere on a simple inclined plane (fig. 413, cf. fig. 411). Next he draws the shadow of a sphere on a staircase (fig. 414) and finally the shadow of a curved cylinder on another cylindrical surface (fig. 415), adding the caption:

The shadow cd from the umbrous body ab which is itself equidistant [from it], will not show itself as being equal in darkness, through being in a background of various brightnesses.

    This idea he develops in the main text alongside:

Shade will never demonstrate itself...of uniform darkness in the place where it is intersected if such a place is not equidistant from the luminous body.

This is proved by the 7th which states: that shade will demonstrate itself as brighter or darker which is surrounded by a background that is darker or brighter...by the 8th of this: that background...will have its part that much darker or brighter to the extent that it is more remote [from] or closer...to the luminous body and among sites equidistant from the luminous body that will show itself as more illuminated which receives the luminous rays among more equal angles. [It is] always [the case that] shade, impinged on some inequality of sight, will show itself with its true boundaries equal to the umbrous body, if the eye does not position itself where the centre of the luminous body was.

    That shade will show itself as darker which is more remote...from its umbrous body.

    On C13r (1490-1491) he had, meanwhile, drawn a more complex case where a spherical light source strikes a cylindrical object, the shade of which then encounters a spherical body (fig. 416), with the caption: "That part of the umbrous body which is between illuminated bodies is more luminous. The light having been removed it will remain darker." More complex still is the case on C11r (1490-1491) where two spherical light sources have two spherical umbrous objects positioned between them (fig. 418, cf. fig. 417). Here his caption notes: "Often it is possible that there is a derived shadow without original shade."

(figure)

Figs. 419-423: Early illustrations of the phenomenon that a large one light source and a small one object can produce diverging and more than one shadow. Fig. 419, W19147v, K/P 22v; figs. 420-422, K/P 22r; fig. 423 C21v.

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Figs. 424-426: Variations in light source and shade on C1r.

 

Book Four - 8. How/Why One Light Source and One Object Produce Two Shadows

    Leonardo's systematic studies of how the shape/position of the light source, umbrous body and projection plane all affect the shape and quality of shadow make him increasingly aware of a curious phenomenon, namely, how one light source and one opaque body can produce two shadows. In an early diagram on W19147v (K/P 22v, 1489-1490) he shows how a light source larger than an opaque body, nonetheless produces expanding shade (fig. 419). On the recto of this folio he draws three further sketches relating to this theme (figs. 420-422).

    In the Ms. C (1490-1491) he analyses this phenomenon more closely. On C1r, for instance, he draws both a small light source and a longer one to compare their effects (figs. 424-425) noting: "That inferior and superior extremity of the derived shade is less distinct than the lateral one which is caused by the light higher than it is wide." Above this he draws a further example (fig. 426) in which a whole spectrum of shades is produced." A variation on this theme is shown on C21v (1490-1491). Here a central dark shade is surrounded by a larger fainter shade (fig. 423) and beneath it, the caption: "The percussion of derived shade is always surrounded by shade mixed with the illuminated background." Two further examples on C1v (1490-1491) illustrate how different relative sizes of light source and umbrous body can produce two circles of shade which either stand completely separate or overlap (figs. 427-428). In his own words: "The straight boundaries of bodies will appear broken which have their umbrous place rayed by the percussion of luminous rays." On, C2r (1490-1491), the opposite folio, he begins anew with an apparently unrelated statement (cf. below p. ):

The body illuminated by solar rays which has passed through the large ramifications of trees will produce as many shades as there are branches interposed between the sun and it.

    Directly beneath this he draws a further example (fig. ) in which a spherical light source and a spherical umbrous body produce two overlapping circular shadows. He then makes a more complex drawing (fig. ) in which a spherical light source standing in front of a conical pyramid produces three sets of double circular shadows, the lowest of which overlap almost completely, the middle ones less so, while the highest are entirely separate.

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Figs. 427-429: How one light source and one opaque body produce two shadows. Figs. 427-428, C1v; fig. 429, C2r.

(figure)

Figs. 430-431: Three different shadows produced by one light source and a cone on Ca347ra and C2r.

These drawings are followed by a series of notes which he had drafted on CA347ra (fig. , 1490-1495):

CA347ra C2r
The percussion of derived shade originating in a pyramidal umbrous body is of various darkness

The percussion of shades parting
from a pyramidal body is not
similar in shape to its origin.HELLO
The percussion of umbrous bodies
originating from a pyramidal
umbrous body will be of bifurcated

    He pursues this question of the shade produced by conical pyramids on I 38(r) (c.1497) where he draws a schematic diagram (fig. 436), besides which he writes.

(figure)

Figs. 432-437: Sketches of the sun shining through trees and pyramidal forms. Figs. 432-434, I37v; fig. 435, Author's reconstruction of 434; fig. 436, I38r; fig. 437, Author's reconstruction of 436.

Proof how at a certain distance the shadow of the pyramid does to resemble the pyramid whence it originates.

    Below the diagram he adds: "Let ab be the pyramidal umbrous body. Let cd be the part which receives the shade," which text continues in the right-hand margin:

You see the shade n which is joined with its shape on the wall and likewise m, but disproportionate to the continuous proportion. You see the shade e which is joined and d which is joined even worse.

    With the aid of a three-dimensional diagram (fig. 437) Leonardo's intention becomes clear. This applies also to the diagram on the folio opposite (I 37(v), fig. 434 cf. 435 ) where he is considering a curved conical pyramid, beneath which he writes: "Different shadows from pyramids equidistant from their luminous body." This is effectively a heading for the text alongside:

Even if an umbrous body is pyramidal and each of its walls is equidistant from its luminous object, nonetheless, that part of the pyramid that is most smaller than the light which illuminates it, will throw less distant shade on its cause.

    In the upper part of I 37v (1497) he draws a sketch of the sun shining down on a pyramid (fig. 432) and beneath this, a sketch of the sun shining down on a tree, (fig. 433) to which he adds the caption: "The imprint of the shadow of some body of uniform thickness will never be equal to the body whence it originates." His treatment of pyramids and trees on the same page is no coincidence. Indeed his discussions of straight and curved pyramids on I37v -I38r are probably intended as geometrical abstractions to simulate the shape of branches. This would account for his unexpected reference to trees on C2r (see above p. , 1490-1491) while discussing the shade of pyramids.

(figure)

Figs. 438-449: How a large light and small object produce diverging or double shade. Fig. 438, BM171r; fig. 439, BM170v; figs. 440-441, CA353rb; figs. 442-443, CA155re; figs. 444-445, CA144va; figs. 446-448, CA144rb; fig. 449, CA277va.

    In the years that follow the problem of how two shades are produced is often treated as a special case of the phenomenon that a light source larger than an umbrous body nonetheless produces diverging shade. On BM170v, for instance, (fig. 438, 1492 cf. BM171r, fig. 439, 1492) he draws a sketch of two diverging shades accompanying which he notes that "an object larger than the umbrous body sees more than half of it and makes much mixed shade."

    On CA353rb (c.1495) he sketches both the general principle of expansion (fig. 440) and the two divergent shades in particular (fig. 441). On CA155re (1495-1497) he is content merely to sketch the general principle of expansion (figs. 442-443). Meanwhile, his search for an explanation leads him to analyse the phenomenon in terms of geometry. This begins with rough drafts as on CA144va (figs. 44-445, c.1492) where he notes:

The umbrous intersection will occur after which the divided shadows will concur in 2 different concourses...[as] if they derived from 2...divided lights.

Luminous bodies being larger than the umbrous bodies positioned opposite,-...it will happen (that the said body will be more than...than this body being) and the light will operate as if it were...divided and its shadows will divide after their intersection and their concourse will be in different places.

    These drafts continue on CA144rb (figs. 447, 1492; cf. CA222ra, figs. 450-455, 1492; and CA277va, fig. 449, 1508-1510): "The boundaries (of shade made) by the size of the shadows made by a greater...less than it, umbrous bodies will spread out from their centres as if these were born of various qualities of light."

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Figs. 450-460: Further demonstrations of the principle of divergent shadows. Figs. 450-455, CA222ra; figs. 456-457, CA93vb; figs. 458-459, CA258va; fig. 460, CA258ra.

(figure)

Figs. 461-471. Eleven further demonstrations of the divergent shade problem. Figs. 461-465, CA195va; figs. 466-467, CA258va; fig. 468, CA208vb; fig. 469, CA177ve; figs. 470-471, CA195ra.

    More than fifteen years pass before he returns to this problem. On CA258ra (fig. 460, 1508-1510, cf. CA93vb, figs. 456-457, C.1510) he reduces the phenomenon to its geometrical essentials. On the verso of the same folio (fig. 458-459, 466-467) he asks:

How a derived light, even if it be generated by a single light, this light will adopt shades in it as if it were divided into two lights.

The shade generated by a single light is always divided at its bifurcated point, as it if were generated by two lights.

    On CA195ra (figs. 470-471, c.1510, cf. CA177va, fig. 469, 1505-1508), he again asks: "Why a single luminous source makes two shades after a single luminous body. Why a single body illuminated by a single light produces two [shades]." On CA208vb (1508-1510) he draws a nearly identical diagram (fig. 468) to make an astronomical point: "Beneath there is no part which sees the sun entirely." On CA195va (1510) he takes up anew the question of two shadows produced by a single light beginning with a draft:

Why a light makes pyramidal shade after the umbrous body.

Two shades are made by a single light because all...gh sees the entire space abc and from the other side it sees the same space edf. But...

(figure)

Figs. 472-473: One large light source and two shades; two small light sources and two shades on C21r.

    He then begins afresh and gives a thorough explanation (fig. 463, cf. figs. 461-462, 464):

Two shades are made by a single light and a single object when the light is greater than the object and this is caused because...the entire light de illuminates...the space ders. But the space cab...in every degree that it approaches the line cb, always loses sight of this light as the motion of the line fn relative to b shows which, to the extent that n comes closer to b, the more f approaches d above with its other extremity and restricts the space of the light ed...and in a similar way the 2nd shade pno is generated and the triangle Shb is entirely seen by the light de, because half the light,...which is fe, sees with its parts...Sm and the other half of the light df, sees with its parts in nb, but the triangle hSb is seen...by two halves of this light de and hence two halves make one whole.

    In the above examples we have deliberately included some cases involving a camera obscura (see pp. below) in order to give some impression of the connections between various problems in Leonardo's mind.

Book Four - 9. Compound Shade

    He is also concerned with cases of compound derived shade, namely, when more than one light source and/or more than one umbrous body are involved.

Book Four - 9a. Preliminary Studies

    This interest grows partly from his attempts to show how one light source and one object can produce two shadows, as is clear from two diagrams on C21r (figs. 472-473, 1490-1491), alongside which he adds:

If the size of the luminous body surpasses that of the illuminated body, an intersection of shade will occur, beyond which divided shadows will go in two different directions as if they were derived from two different lights.

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Figs. 474-475: Examples of compound light and shade on C4v and C9v.

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Figs. 476-479: One opaque body and one, two and three light sources. Figs. 476-478, C22r; fig. 479, Pecham, Perspectiva communis.

    In the same manuscript he compares different shadows produced by two light sources at different distances. On C4v (fig. 474, 1490-1491), for instance, he notes:

You will find that proportion of darkness between the derived shadows a [and] n as there is between the vicinity of the luminous bodies m [and] b which cause them. And if these luminous bodies are of equal size you will also find such a proportion in the sizes of the percussion of the luminous circles of the shade as is that of the distance of these luminous bodies.

    On C9v (fig. 475, 1490-1491) he draws a related situation in greater detail, this time adding only a brief caption: "The percussion made by umbrous and luminous rays on a same place is mixed and of confused appearance." These preliminary notes lead to more thorough studies.

Book Four - 9b. Multiple Lights and Objects

    Here again his approach involves a systematic play with variables. At the simplest level, on C22r (1490-1491), he draws first one light source and one opaque object (fig. 476); then two light sources and one opaque object (fig. 477) and then three light sources and one opaque object (fig. 478, cf. fig. 479).

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Figs. 480-481: One light source and two opaque bodies on cA144vb and C17r.

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Figs. 482-483: One light source and two opaque bodies in the open air and with an interposed plane on c13r and C14r.

    He also considers the case of one light source and two opaque bodies. In drawings on CA144vb (fig. 480, c.1492, possibly 1490) and on C17r (fig. 481, 1490-1491) he assumes that this will produce two converging pyramidal shadows. He changes his mind, however, and on C13r (1490-1491) demonstrates how intersecting shadows of differing intensities are thereby produced (fig. 482):

It is possible that mixed derived shade, caused by a single light by diverse bodies can intersect and superimpose itself on one another.

Abc is the mixed and intersected derived shade and superimposed on one another since mc is the shade of d and bn is the shade from f and to the extent that abc contains [shade], to that extent one shade superimposes itself on the other.

    After showing what occurs in the open air, on C14r (1490-1491), he examines what happens when this shadow produced by one light source and two opaque objects is intersected by a wall (fig. 483):

Umbrous rays of imperfect and equal obscurity which mix themselves together double the quantity of the darkness.

Reason demands that double equantity produces a double power (And for this reason two imperfect things make a perfect one).

Msn and ktn are the incorporated and imperfect mixed shades and kmstn is the effectively perfect duplicated shade.

    Having studied one light source and two opaque bodies on C13r, 14r, he studies the case of two light sources and one opaque body first in passing on C22r (fig. 477, 1490-1491) and then in more detail in a now partly ruined text on CA230rh (fig. , 1505-1508) entitled:

Why the shadows intersected behind the maximal shade, to the extent that they approach such maximal shade more, the more they lose in darkness.

Let aco be the maximal darkness, codnaobm are the shadows intersected on the maximal shade cao of such a shade in separating...maximal to the extent...they because white...the 2 simple lights and the...which proceed, see the...dark mixed with...surrounded by such a background.

    In the Windsor Corpus this theme is pursued. On W19151v (K/P 118v/b/, 1508-1510) he makes a marginal drawing (fig. 484) alongside which he notes: "Derived shadows will be of equal darkness if they arise from lights of equal power and distance: this is proved." On W19149v (K/P 118v/A/, 1508-1510) he redraws the diagram (fig. 485) this time providing a long explanation of its five degrees of shadow:

The greatest darkness of shadows is the simple derived shadow because it is not seen by either of the lights ab /or/ cd.

The second of lesser darkness is the derived shade efn and this is less dark by half because it is illuminated by a single light, that is, cd. And this is of uniform natural darkness because throughout only one of the two luminous bodies sees it. But it varies with accidental darkness because the more that it is distant from such a light the less it participates in its brightness.

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Figs. 484-485: Two light sources and one opaque body on K/P 118v.

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Figs. 486-487: Two light sources and two opaque bodies on C19r and one light source and three opaque bodies on C13v.

The third /degree of/ darkness is the middle shade. But this is not of uniform natural darkness because the nearer it is to the simple derived shadow, the darker it is and the accidental uniformly difform uniformity is that which corrupts it, that is, the more distant it is from the two luminous sources, the darker it is.

The fourth is the shade krse, and this is so much the darker in natural darkness as it is nearer to ks because it sees less of the light ab but by accidental /shade/ it loses more darkness because that which is closer to the light cd always sees the two lights.

The fifth is of less darkness than each of the others because it always sees the whole of one of the two and the whole or part of the other and this loses more darkness to the extent that it is closer to the two lights and the more so to the extent that it is nearer to the outer side st, because it sees more of the second light ab.

    Meanwhile, he had been exploring other combinations of light sources and opaque bodies. On C19r (1490-1491), he combines two light sources and two opaque bodies (fig. 486) alongside, which he explains: "To the extent that the darkness of two rays of imperfect darkness is different, to that extent the shade resulting from their mixture will differ from its original being." Lower down on the folio he adds two further notes:

(figure)

Figs. 488-490: Three light sources and one opaque body. Figs. 488-489, C6r; fig. 490, CA229rb.

It is impossible that there results a shade of darker quality...from a mixture of 2 perfect shades.

It is possible that there results a perfect shade from a mixture of 2 imperfect shades....

    On C13v (fig. 487, 1490-1491) he considers a case with one light source and three opaque bodies, adding the caption:

It is impossible that simple derived shades originating from different bodies and caused by a single light can ever join or touch one another.

    The converse case of three light sources and one opaque body he illustrates first in two rough sketches on C6r (figs. 488-489, 1490-1491) and then more carefully on CA229rb (fig. 490, 1508-1510, cf. figs. ). Just how systematic is this play of variables becomes clear from Chart 12.

Number of         Number of
Light Sources     Opaque Bodies Codex

1                        1 C22r

2                        1 C22r

3                        1 C22r

1                        1 C21r

1                       2 C13r

1                       3 C13r

Chart 12. Systematic play of variables using light sources and opaque bodies.

    Having considered combinations of one, two and three lights and objects, on C13v (1490-1491), he describes an experiment with four light sources:

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Fig. 491. Four light sources and four objects on C13v; fig. 492. A light source, columns and shadows on F6r.

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Figs. 493-498. Columns casting shadows. Fig. 493, CA347ra; fig. 495, F1v; figs. 496-498, CA236vc; figs. 499-500, CA199va.

If in a room one has positioned 4 light sources and the sky above is completely sifted by the grid which covers it, it will hold much back and sift the grain which, in descending through the air makes the shade evident in the air, standing out and clear as is here shown.

    Beneath this he draws a diagram (fig. 491) which he describes in detail:

That shade is darker which is derived from more diverse umbrous and luminous bodies.

At kfs one sees the lights b, c, d which lacks only the light a, which is a quarter of the number /of the whole/. At fsm one can see only the two lights cd which is the half of all the lights. At mspn one does not see any light whence, not being able to be illuminated it is found to be the first degree of darkness.

Book Four - 9c. Columns

    Related to such experiments are Leonardo's demonstrations involving columns as on F6r (1508) where a light source (fig. 492) in front of a column creates a succession of shadows. His earliest extant illustration on this theme is a rough sketch on CA347ra (fig. 493, 1490-1491) showing a light source in front of a column. Somewhat more developed, but again without text, are three sketches on CA236vc (figs. 496-498, 1508-1510) showing light sources and columns casting shadows.

    On CA199va (fig. 500, cf. fig. 499, c.1500) he draws a column in isolation to illustrate that the degrees of shade on an object can be infinite (see p. above). On F1v (1508) he draws another column (fig. 495) to clarify his claim that the colour of an object is affected by the colours surrounding it. He returns to this theme on E31r (CU621, TPL594, 1513-1514) drawing first a column in isolation (figs. 501, 504) accompanying which he writes:

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Figs. 501-504: Elementary demonstrations with columns. Figs. 501-503, E31r, fig. 504, CU621.

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Figs. 505-507: Preliminary demonstrations with crosses and shadows on C11v, CA229vb and CA37ra.

On pyramidal shade

Pyramidal shade produced by a parallel body will be thinner than the umbrous body to the extent that the simple derived shade is cut at a greater distance from its umbrous body.

    Below this he draws a second column with converging shade (fig. 502) and a third with diverging shade (fig. 503) to illustrate basic types of shade (see pp. above). In themselves these demonstrations with isolated columns are of tangential interest. Their importance lays therein that they form a starting point for a series of experiments involving multiple columns which play an important role in his discussions of simple and compound shade.

Book Four - 9d. Experiments with Crosses and Columns

    The earliest extant note on this theme occurs on C11v (1490-1491) where he draws (fig. 505) a long light source in front of a cruciform opaque object beneath which he adds:

If a long luminous body is the length of a cruciform umbrous body, the simple derived shade of the transverse part of the cross will not be conducive to percussion.

HELLO**

CA37va

    Nearly two decades pass before he returns to this problem on CA229vb (1508-1510). He now draws two separate columns/sticks which overlap one another in cross-form to produce shadows (fig. 506). What interests him in this case, is to show that there is only simple shade and not compound shade where the two shadows intersect. Or as he puts it:

The percussions of simple shadows, even if they are intersected, are not doubled because the shade of the first umbrous body remains impressed in the luminous part of the second umbrous body which intersects with the first one and since with a single light one cannot generate two simple shadows.

    Two compound shadows joined together generate simple shadows. On CA37va (1508-1510) he begins to experiment systematically. He now takes a single obliquely positioned column or stick and demonstrates the shadow that it casts in the presence of one light source (fig. 508). Next he shows that with two light sources this same column produces two shadows (fig. 509) and with three light sources it produces three shadows (fig. 510). In addition he demonstrates how two light sources and two oblique columns can produce a cruciform shadow (fig. 507). This is closely related, in turn, to another demonstration that acquires great significance for him, namely, where two light sources in front of two columns, positioned in the form of a St. Andrew's cross, produce four shadows. Preliminary drafts for this occur on CA37ra (figs. 511-513, 1508-1510).

    On BM248v (1508-1510) these drafts continue. He begins by simply drawing (fig. 514) two points from which emanate four shadows, two of which are marked a and b. In his next version (fig. 515), these two points become the base of a roughly sketched St. Andrew's cross. The four shadows are now marked A, b, a, b respectively. In a third version (fig. 516), he draws the St. Andrew's cross more carefully. The letters a and b are now linked with points representing the two light sources. The sequence of the lettering on the shadows is different however: it is now a a, b b.

    Why should Leonardo be so interested in such problems? Some of the intersections of shadows produced by a St. Andrews cross result in a shadow of double intensity, while others do not. The phenomenon and the reasons for it had been a matter of debate which he hoped to set straight. Hence, having made several drafts without any accompanying text, we find him on CA177rb (1508-1510) redrawing this diagram with two columns in the form of a St. Andrew's cross (fig. 517), beneath which he outlines the problem:

Of simple shade.

Why in the intersections...a /and/ b of the two compound shadows ef /and/ mc, there is generated simple shade, and likewise in ah /sic: eh/ and mg, and such simple shade is not generated in the other two intersections c /and/ d made in the same compound shadows mentioned above.

    In short he is claiming that there is simple shade at the intersections a and b, and compound shade at the intersections c and d. Why this should be so he explains directly following in his:

Reply

Compound shadows are mixtures of bright and dark, and simple /shadows/ are composed of simple darkness. Hence of the two lights n /and/ o, the one sees the compound shadows from the one side and the other sees the compound shadows from the other side. But no light sees the intersections a /and/ b and hence it is simple shade. But in compound shade both see /the/ light source.

    Conscious that this claim is controversial, he introduces the opinion of an adversary:

And here a doubt arises through the adversary, because he states: in the intersection of compound shadows, the two lights causing (...) these shadows are necessarily seen and for this reason such shadows must cancel one another...such that where the two lights do not see, we say that the shade is simple and where a single one of the two lights see, we shall say that such a light is compound, and where the two lights...see, is cancelled shade because...where the two lights see no shade of any kind is generated, but is solely composed of the brightness of the background surrounding the shadows.

    These opinions of the adversary he refutes:

Here it is replied that the above...mentioned /opinion/ of the adversary is true, which only makes mention of that truth which is in his favour, but if he adds the remainder, he will conclude that my proposal is true. And this is that, seeing the two lights, at that intersection such a shadow would be cancelled. This I confess to be true...if the two shadows were not seen in the same place because, where one shadow and one light are seen, compound shadow is generated and where two shadows and two similar lights are seen, this shadow cannot vary in any part of it, the shadows being equal and the lights being equal. And this is proved in the eighth on proportion where it is stated: such a proportion does simple power have with simple resistance, as a double power has with a double resistance.

(figure)

Figs. 511-519: The development of a demonstration involving two light sources, a St. Andrew's cross and four shadows. Figs. 511-513, CA37ra; figs. 514-516, BM248v; fig. 517, CA177rb; figs. 518-519, CA241rc.

    The problem continues to trouble him and on BM243r (1508-1510) he again draws (fig. 526) two light sources a and b. In place of two columns positioned in the form of a St. Andrew's cross, he merely marks the points t and s from which emanate four shadows a, b, s, b intersecting one another at m, n, r and c. This he describes in the text alongside headed:

Definitions

The intersection n is made by the shadow created by the light b because this light b generates the shadow tb and the shadow sb. But the intersection m is made by the light a which generates the shadow sa and the shadow ta. But if you cover the two lights a /and/ b then there are generated the two shadows n /and/ m at the same time and other than this there are generated two other simple shadows, that is r /and/ c, in which...one does not see either of the two luminous sources.

The degrees of darkness that the compound shadows acquire are as many as the number of luminous bodies that see it are less.

    On BM248v (1508-1510) he drafts a three step demonstration of these principles. In the right-hand margin he makes a sketch (fig. 520, cf. fig. 521) which he marks first (prima), showing how light a produces two shadows intersecting at n when light b is extinguished. He then draws another figure (fig. 522 cf. fig. 523) which he marks s (2nd) where light b in turn casts two shadows, while light a is extinguished. In the next figure, (figs. 524-525, cf. figs. 526-527) marked (3rd), he demonstrates how the shadow at g and h is doubled while the shadow at i and k is not.

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Figs. 520-523: Systematic experiments in which the light source on the left, then the light source on the right is extinguished. Figs. 520, 522, BM248v; figs. 521, 523, CA241rc.

    Alongside he writes a draft text:

Why the intersection n, being composed of two compound derived shadows...generates...compound shade and not simple shade as do the other intersections of the compound shade. This occurs by the 2nd of this which states

    A preliminary version of this second proposition follows: "The intersection of the two composed derived shadows originating in the intersections of the umbrous columns will not generate simple shadow which does not acquire any darkness." This is crosses out and restates: "The intersection of derived shadows originating from the intersections of columnar umbrous bodies illuminated by a single luminous source will not generate simple shade." Consideration of a third proposition follows:

And this arises through the 3rd /of the/ 1st which states: the intersections of simple derived shadows never acquires darknesses because...all...of the sums of darkness joined together do not acquire more darkness than a single one because if the many dark sums increase the darkness in their duplication they could not be named sums of darkness but partial darkness /instead/. But if such umbrous intersections are illuminated by a second light source positioned between the eye and the intersected bodies then...such shadows will be compound shadows and will have a uniform darkness int heir intersection as in the remainder.

    The text that follows serves as caption for his third diagram (fig. 525):

By the 1st and the 2nd above, the intersections i /and/ k will not be of double darkness as they are double in quantity but by this 3rd /proposition/ the intersections g /and/ h will be of double darkness and quantity.

    The folio ends with definitions of simple and compound shade (cf. p. above). These drafts on BM248v (1508-1510) serve as a starting point for a more detailed analysis on Ca241rc (1508-1510) which begins with a general description of:

Shade

When the intersection of two umbrous columnar bodies generates its derived shadows, with two luminous sources then it is necessary that four derived shadows are generated,...which shadows are compound and these shadows intersect in four places and of these there are two which compose simple shadow and two remain compound shadow and these 2 simple /shadows/ are generated where the two lights cannot see, and the compound shades are generated where one of the two lights cannot be illuminated. But the intersections...of the compound shade are always generated by a single light source and the simple /shadows/ by two luminous bodies and the right intersection of the compound shade is generated by the left light and the left intersection is generated by the right light. But the two intersections of the simple shadows, both the upper as well as the lower one are generated by the two luminous sources: that is /both/ the right light and the left light.

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Figs. 524-527: Demonstration concerning compound shade. Figs. 524-525, BM248v; fig. 526, BM243r; fig. 527, CA241rc.

    This claim is, as usual, followed by a demonstration (figs. 519, 521, 523):

This is proved.

Let...S be the intersection of the two columnar umbrous bodies af and bL and let the derived shadows, generated by such (shadows) umbrous bodies, be aa and ab generated by the two superior luminous sources a /and/ b,...and the same...2 luminous sources generate the other two derived shadows bb and ba. But each of the shadows which intersect one another do not originate from these two lights.

And this is demonstrated because, removing the light b, the shadow aa remains and the shadow ba intersected at the point x and hence these two shadows, through the removal of such a light, remain simple shadows, being generated by a single luminous source a...and, by the ninth of this, no...quantity of simple shadows generated by more...umbrous bodies,...by means of a single luminous source can generate...intersections among them, if the umbrous bodies, the causes of the shadows do not intersect among themselves.

    The results and implications of these experiments concerning shadows at the outer intersections are now summarized (i.e. v and x in fig. 527):

Therefore we have demonstrated that the 2 lateral intersections do not generate any other darkness of shadow than that which is beyond such an intersection and this originates because the shadow is as diminished at such an intersection...in umbrous bodies as it is beyond this intersection because the shadow of the first umbrous body stamps itself on the back of the second umbrous body, which is in contact at the intersection, and for this reason it does not descend to the pavement...where it intersects the other derived shade. And if you make the cruciform figure without superimposing such an intersection then this cross will be diminished throughout and the derived shade will be diminished in its intersection and for this reason it cannot acquire darkness in its intersection, etc.

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Figs. 528-532: Three light sources in front of a St. Andrew's cross producing six shadows. Figs. 528, 532, CA37va; figs. 529-530, CA177ve; fig. 531, BM243r.

    In the final section of his demonstration he considers the shadows at the two inner intersections (r and t in fig. 527):

But the shadows bb and aa which are double in their intersections r /and/ t, being created by the two luminous sources a /and/ b, in their intersections lose the luminous sources and the remainder only loses one of the two luminous sources.... But the intersections v /and/ x do not lose any luminous source, because each of them always sees one of the luminous sources.

    Having examined cases with one and two light sources the systematic Leonardo cannot resist studying the effects of three light sources positioned in front of a St. Andrew's cross. Preliminary sketches for this occur on CA177ve (figs. 529-530, 1505-1510), BM243r (fig. 531, 1508-1510) and CA37va (figs. 528, 532, 1508-1510). On CA229rb (1508-1510) he goes further. First he shows (fig. 533) how one light source in front of a St. Andrew's cross formation produces two intersecting shadows. Then he shows (fig. 534) how two light sources in front of the same cross produce four intersecting shadows.

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Figs. 533-535: One, two, and three light sources in front of a St. Andrew's cross on CA229rb.

    Finally he shows (fig. 535) how three light sources in front of such a cross produce six intersecting shadows. Alongside this he asks:

If three lights with two columnar umbrous bodies in their intersections produce...six shadows with 9 intersections, why is it that the first two exterior intersections do not double the darkness at q, in their intersection as do the other 7 intersections?

    This he crosses out. In the next column he begins anew, ignoring this complex question and discussing instead a simpler case (fig. 534 cf. figs. 511-519) which he had dealt with previously:

On simple shadow

The light nm...is the cause that the columnar umbrous body os...makes the two shadows of and og and the lights...m /and/ n do the same to the body hp in generating the shadows...ph and PK and for this reason, the simple derived shadows b /and/ c are generated in such intersections b/and/ c and not at the intersections d /and/ e because b /and/ c and not see either one of the two lights but d /and/ e see...these two lights..., that is, one sees the shadow og, and the other sees of which are two compound shadows. Therefore the intersection of such compound shadows sees...or rather is seen by two lights;...but...the two lights do not see this intersection, because such a shadow would be destroyed. But each shadow sees a single light in its entire length and each shadow, joined together with the other will not make simple shadow, because simple is that which does not see...any luminous source and which, in such...an intersection, each in itself only sees its own luminous body and the one can never see the luminous source of the other. It follows that such shadow remains without doubling its darkness and without being destroyed by being seen by the two lights.

    This is clearly a restatement of his answer to an adversary on CA177rb (1508-1510) cited above. An understanding of these passages in turn renders intelligible other diagrams without text. On CA37ra and CA37va (1508-1510), for instance, he makes a series of four sketches (figs. 536-539) which show how two light sources in front of a St. Andrew's cross produce four shadows. In a preliminary sketch on CA37ra (fig. 528, 1508-1510) he shows how three light sources in front of such a cross produce six shadows. This idea he develops in a series of four sketches (figs. 532, 540, 542-543) without text on CA37va (1508-1510).

    He also considers different numbers of light sources in front of a more complex object which he represents simply as four points (cf. fig. 514 where he represents a St. Andrew's cross as two points). On BM243r (1508-1510), for instance, he shows how two light sources (fig. 544) in front of such points produce eight shadows. On CA229rb (1508-1510) he illustrates how three light sources (fig. 545) in front of four points produce twelve shadows. In a further sketch on BM243r (1508-1510) he demonstrates (fig. 546) how four light sources in front of four points produce sixteen shadows.

    On this same folio, BM243r he also considers the configurations of shade produced by three light sources in front of a s St. Andrew's cross coupled with a vertical pole to produce a figure . In one sketch (fig. 547) he marks the three light sources as the points a, b and c, draws in the figure , adds the resulting nine shadows and identifies those which have been produced by light sources a, b or c respectively. In a second sketch he draws (fig. 548) effectively the same situation, with the exception that the three light sources are no longer as far apart. Also on this folio are four preparatory sketches (figs. 549-552) in various degrees of completion.

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Figs. 536-543: Demonstrations of shadows produced by a St. Andrew's cross and either two or three light sources. Fig.536, CA37ra; figs. 537-540, 542-543; fig. 541, CA177va.

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Figs. 544-546: Demonstrations involving three or four light sources. Fig. 544, BM243r; fig. 545, CA229rb; fig. 546, BM243r.

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Figs. 547-552: Shadows produced by three light sources in front of a shape on BM243r.

    The most basic of these (fig. 551) is, in turn, related to a sketch (fig. 553) on CA37ra (1508-1510). Here he shows how three light sources, in front of a figure, produce three shadows at the right-hand column. In a second sketch (fig. 554) he draws this situation again but for purposes of comparison he now shows how the other two columns would each produce two shadows if they had two light sources. In the next sketch (fig. 555) two columns have three light sources and only the column on the far left has two light sources. In a final sketch (fig. 556) all three columns have three light sources and therefore produce nine shadows. In this case, however, the light sources are positioned further off to the right.

Book Four - 10. Conclusion

    Although the extant notes contain no concluding remarks concerning this fourth book, the chief ideas of this section can, nonetheless, be readily summarized. Leonardo has shown that derived shade varies (a) with the light source: its degree, angle, position, size and shape, (b) with the shape of the umbrous body and (c) with the shape of the projection plane on which the derived shade falls.

    His attempts to understand how/why a single light source produces two shadows lead him to systematic studies concerning the properties of compound shade: i.e. situations in which one light source is in combination with one, two or three umbrous bodies and conversely where one umbrous body is in combination with one, two or three light sources.

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Figs. 553-556: Further demonstrations relating to crosses and shadows.

    Equally systematic are his experiments with one, two, three or four light sources in front of columns in the form of a St. Andrew's cross, in order to determine at what points shadow becomes doubled. As will be seen (below pp. ) these experiments are, in turn, paralleled by others involving one, two, three or four pinhole apertures in a camera obscura. The more we penetrate his thought, the more systematic we find his approach to be.

 

Book Five: Derived Shade and Reflected Light

And since all round the derived shadows, where the derived shadows are intercepted, there is always a space where the light falls and by reflected dispersion is thrown back towards its cause, it meets the original shadow and mingles with it and modifies it somewhat in its nature. And on this I shall build the fifth book. (CA250 ra).

    On the basis of this outline on CA250ra a tentative reconstruction of Leonardo's fifth book on light and shade can be suggested. It would probably have opened with basic propositions concerning reflect light and shade (Bk.V: Chapter 1). This might have been followed by notes on lustre (V:2) and elementary demonstrations of reflection (V:3). Separate chapters on reflection involving interposed rods (V:4) and interposed walls (V:5) could have followed. A series of theoretical demonstrations (V:6) might have ended this section. We shall examine each of these in turn.

Book Five - 1. Basic Propositions

    Ludwig's edition of the Treatise of Painting contains two sections, one with seventeen propositions (TPL156-172), the other with eleven propositions (TPL780-790) devoted to problems of reflection in light and shade. Some of these pertain to books six and seven and will be discussed later. Others are of an introductory nature and concern us here (see Chart 13). On CU162 (TPL171, 1508-1510), for instance, Leonardo compares the nature of reflected and direct light in a passage entitled:

On the colours of reflections.

All reflected colours are of less luminosity than direct light and incident light has as much proportion with reflected light, as is that /proportion/ which there is between the luminosity of their causes.

Chart 13: The above provides a summary of the chief themes in the two sections on reflection in the Treatise of Painting (156-172, 780-790). The numbers refer to Ludwig's edition. TPL788 has not been included because it has nothing to do with reflection.

But to return to the promised definition. I say that luminous reflection is not from that part of the body which faces umbrous bodies, as are dark places, meadows of various herbs, green or dry trees which, even if part of each branch facing the original light is vested with the quality of this light, nonetheless, there are so many shadows made by each branch in itself, and so many shadows made by one branch on the other that there results in sum such a darkness that the light is as nothing. Hence objects such as this cannot give any reflected light to bodies /placed/ opposite /them/.

    Those bodies in which reflection is possible he describes briefly on A94v (BN 2038 14v, CU157, TPL156, c.1492):

On Reflection

Reflections are caused by bodies of bright and flat quality and semidense surfaces which, percussed by light, like the bouncing of a ball, repercuss them at the first object.

    On CU167 (TPL158, 1505-1510) he mentions the nature of such reflecting bodies:

On reflections

Reflections participate that much more or less with the object where they are generated, than with the object which generates them, to the extent that the body where they are generated is of a more polished surface than that which generates them.

    A series of five general rules, four of them numbered, on CU172, 170 (TPL168-169, 1505-1510) might also have formed part of this introductory chapter:

On reflections

1. The surface of bodies participate more in the colours of those objects which refelct their similitudes in it at more angles.

2. Of the colours of objects, which reflect their similitudes in the surfaces of the bodies positioned opposite at equal angles, that will be more powerful which will have its ray reflected a shorter length.

3. Among colours of objects that are reflected at equal angles and at equal distance in the surfaces of the bodies positioned opposite, that one will be darker which is a brighter colour.

4. That body reflects its colour more intensely onto the body positioned opposite, which does not have around it other colours than of its species.

Reflection

But that reflection will be of a more confused colour which is generated by various colours of objects.

    In the late period there is a further elementary note on G11v (1510-1515) entitled:

On shade in bodies.

When you draw dark shadows in umbrous bodies always draw the cause of such a darkness and you will do the same with the reflections because the dark shadows originate from dark objections and the reflections from objects of small brightness, that is, of diminished lights, and such is the proportion between the illuminated part and the part brightened by the reflection as there is between the cause of the light of this body and the cause of this reflection.

(figure)

Figs. 557-559: Highlights and lustre on A113r, CU799 and H90/42/v.

    In addition, a series of passages on reflections in relation to backgrounds (TPL160, 163, 167, 172, 780, 786, and 787), to be discussed later (see below pp. ), might have formed part of this opening chapter of book five.

Book Five - 2. Lustre

    Lustre is one of the basic phenomena of reflected light. The Treatise of Painting contains a section of nine passages (TPL771-779) devoted to this theme. There are also other notes scattered throughout the manuscripts. It is likely that these would have served as basis of a second chapter on reflection. On A113r (BN 2038 32r, CU799, TPL746, 1492), for instance, Leonardo writes:

On the highlights which turn and move as the eye seeing this body is moved.

Let us suppose that the said body is this round one drawn here on the side (fig. 557, cf. fig. 558) and that the light is the point a and that part of the illuminated body is bc and that the eye is at the point d. I say that the lustre, because it is all in all and all in a part, that standing at the point d, the lustre will appear at the point c and to the extent that the eye f moves from d to a to that extent will this lustre move from c to n.

    On H90/42/v (1492) he broaches the topic again: "The lights of lights, that is the lustre of some object will not be situated in the middle of the illuminated part. But will make as many mutations as the eye regarding this." Definitions of lustre follow on CU774 (TPL775, 1508-1510) and E31v (CU780, TPL776, 151301514, see above pp. ). On E31v (CU772, TPL777, 1513-1514) he also asks:

    Which bodies are those which have light without lustre?

Opaque bodies which have dense and round surfaces will never generate lustre in any illuminated part of them.

    Immediately following on E31v (CU781, TPL778, 1513-1514) he asks the converse:

Which bodies are those which have lustre and not a luminous part?

Dense opaque bodies with dense surfaces are those which have all the lustre in as many places of the illuminated part as are the sites which can receive the angle of incidence of the light and of the eye, but because such a surface mirrors all the things surrounding it, the illuminated object will not be recognized in that part of the illuminated body.

On CU776, (TPL774, 1508-1510) he makes notes:

On the size of lustres on their terse bodies.

Of lustres generated on spheres equidistant from the eye, that will be of a smaller shape which is generated on a sphere of less size.]

This is seen in the little grains of mercury which are a nearly imperceptible size and their lustres are equal to the size of these grains and this arises because the visual power of the pupil is greater than this little grain and for this reason it surrounds it as was said.

    A diverse series of notes follow on TPL779 (Mad.II 26r, 1503-1504) under the heading:

Of lustre.

Lustre participates more in the colour of the light that illuminates the body which is lustrous than in the colour of this body and this arise in terse surfaces.

The lustre of many umbrous bodies is integrally of the colour of the illuminated body as is that of bronzed gold and silver and other metals and similar bodies.

The lustre of foliage, glass and jewels participate little in the colour of the object where they originate and considerably in the colour of the body that illuminates them.

The lustre made in the depth of dense transparent objects are in the first degree of beauty of such a colour as is seen in a pale red ruby, glass and similar things. This occurs because between the eye and this lustre there is interposed all the natural colour of the transparent body.

The reflected lights of dense and lustrous bodies are of much more beauty than the natural colour of these bodies as is seen in the pleats that occur in gold that is filigreed and in other similar bodies where the one surface reflects on the other positioned opposite it and the other reflects onto it and thus they do successively ad infinitum.

No lustrous and transparent body can show on it the shadows received by any object, as is seen in the shadows of bridges of revers which are never seen except in turbid waters, and in clear /waters/ they do not appear.

    Lustre is found in as many sites as the places where it is seen are various.

The eye and the object standing /still/ without motion, the lustre will move on the object together with the light which causes it. The light and the object standing still, the lustre will move on the object together with the motion of the eye that sees it.

Lustre originates on the polished surfaces of a body that takes more light the more it is dense and polished.

    This section of the Treatise of Painting also contains three further passages (L771-773) concerning lustre in relation to background (see below pp. ). He returns to this theme of lustre briefly in the Manuscript G (1510-1515) where he gives a further definition on G24r (see above pp. ) and on G3v notes:

The shadows that are in the transparent leaves seen from the reverse are directed by this foliage which is transparent from the reverse side along with the luminous part, but the lustre cannot be transparent.

Book Five - 3. Elementary Demonstrations

    A series of basic demonstrations concerning the nature and effects of reflection, scattered through the notebooks, might well have been intended as the basis of a third chapter. In an early passage on C25r (1490-1491), for example, he compares given perceptual effects with those of reflections mirrored in water (fig. 560):

(figure)

Figs. 560-562: Elementary definitions of reflection on C25r, CU212, and CU545.

Shade and Light

If the visual line which sees the shade by the light of the candle is at an equal angle to that...of the shade, it will appear to make shade under the body which causes it like that made by the images of bodies mirrored in the water, which appear to be that much below to the extent that they are above, and thus this shadow does the same such that it appears with its limit to be that much below the plane where it is generated, as the summit of the body which generates it is above this plane, as appears to be on the wall that cb, shade, is as much as ab and that cb stands under ab.

    Reflection in water interests him and becomes the subject of two further propositions. One, on CU212 (fig. 561, TPL227, 1505-1510) is entitled:

Of things mirrored in the water of countrysides and first of the air.

That air alone will be that which gives of itself an image in the surface of the water, which reflects from the surface of the water to the eye at equal angles, that is, that the angle of incidence is equal to the angle of reflection.

    A second passage on reflections in water follows on CU545 (fig. 562, TPL505, 1510-1515) under the heading:

On the shadows made by bridges on their water.

The shadows of bridges will never be seen on their waters if the water does not lose its function of mirroring as a result of turbulence. And this is proved because clear water,...with a lustrous and polished surface, mirrors the bridge in all the places interposed at equal angles between the eye and the bridge and mirrors the air below the bridge where the shade of this bridge is. Which turbid water cannot do because it does not mirror, but it /nonetheless/ receives shade well, as would a dusty street.

    As early as 1490 he had become interested in the properties of light entering a narrow shaft. He makes preliminary sketches in this regard on c12r (1490-1491, fig. 563) and A91v (BN 2038 11v, fig. 564, 1492) without text. This leads to a further diagram on A92r (BN 2038 12r, fig. 565, 1492) again without text which shows light bouncing back and forth down into what is probably a well. Some fifteen years later he illustrates a related situation to make a somewhat different point on CU706, (fig. 566, TPL619, 1508-1510) in a passage entitled:

On derived shade generated in other derived shade.

The derived shade originating from the sun can be made on derived shade generated by the air.

This is proved. And let the shade of the object m which is generated by the air ef be int he space dcb. And let it be that the object n, by means of the sun g, makes the shadow abc and from the remainder of the shdow dme, which in such a site neither sees the air ef nor the sun. Therefore it is double shade because it is generated by the two objects, that is n and m.

    On CU865 (TPL789, c.1510) again uses the example of a well to illustrate a problem of battle painting (fig. 567):

(figure)]

Figs. 566-568: Reflected derived shade in an enclosed space or a well on CU706, 865 and 694.

Of the illumination of the lower parts of bodies which are close together as are men in a battle.

With men and horses engaged in battle, their parts will be that much darker to the extent that they are closer to the earth which sustains them. And this is proved by the walls of wells which are that much darker the deeper they are and this arises because the deepest part of the wells sees and is seen by a smaller part of the luminous air than any other part and the pavements, which are of the same colour as the legs of the aforesaid men and horses, are always more illuminated under equal angles than the other aforesaid legs.

    The reflective nature of light in such a narrow shaft is again discussed on CU694 (TPL587, 1508-1510) in a passage headed (fig. 568):

On the simple shade of prime darkness.

Simple shade is that which cannot be seen by any reflected light but will only be augmented by a shadow opposite.

Let the spherical body be g put in the concavity bcef and let the particular light be a which percusses at b and reflects at d and rebounds with the second reflection onto the spherical body g which, on the one side has simple shade at the angle e, which does not see either the incident light or the light reflected at any degree of reflection. Therefore the shadow of the sphere receives the reflection of the simple shade ue and for this reason it is called simple shade.

    A related situation of light entering a cave is described on Forst I 10r (fig. 569, c.1505):(figure)Figs. 569-570) Further cases of reflection on Forst I 10r and CU160.

Abcd is a cave which is opened on two fronts and has particular light more powerful than the reflected light rs by 2/3. Therefore the particular light will penetrate 2/3 of the cave and the reflected light rs will take the third nd.

    The connection between brightness of reflection and angle of reflection, raised above in connection with a well on CU865 (TPL789) is further explored in a passage on CU160 (TPL161, 1505-1510) entitled:

What part of the reflection will be brighter

That part of a reflection will be more illuminated which receives its light under more equal angles /both/ from the luminous body as well as from its percussion.

    A demonstration follows by way of support (fig. 570):

This is proved. And let the luminous body be n and let ab be the part of the illuminated body which rebounds through all the concavity opposite which is shady. And let it be that the light reflected at e is percussed at equal angles. And below this it will not be reflected at equal angles as the angle cab shows which is more obtuse than the angle eba. But the angles afb, even if it is among angles of less equality than the angle e, this has the base ab which has angles more equal than this angle e, this has the base ab which has angles more equal than this angle c. And hence it is brighter at f than at e. And that which is closer to the thing which illuminates it, will also be brighter, by the sixth which states: that part of the umbrous body is more illuminated which is closer to its luminous body.

 

Book Five - 4. Interposed Rods

    One step more complex are his demonstrations of reflected light and shade involving an interposed rod as on C5r (fig. 571, 1490-1491, cf. fig. ) beneath which he writes: "The more that derived shade approaches its penultimate extremities to that extent will the darkness appear greater." Towards the end of the section that follows he reminds himself: "And this wishes to be at the beginning of the demonstration:"

Let Ab be part of the primitive shade. Let bc be the...primitive light. Let d be the place of intersection. Let the derived shade be fg. Let fe be the derived light.

    This introduces the demonstration proper:

Behind the intersection gz is only seen by the part of the shade;...through the intersection yz takes the shade mn and the shade am directly, whence it has two times as much shade as gz. Through the intersection yx sees no and nma directly whence xy demonstrates itself as having three times as much shade as zg. Through the intersection xf sees ob and it sees onma directly whence we shall say that the shade in between f/and/ x is four times darker than the shade zg because it is seen by four times as much shade.

    On CA357rb (c.1490) he drafts a series of similar sketches (figs. 577-584) accompanying which he writes: "That part of an illuminated place will be more luminous where the rays concur with a greater angle." This idea he develops on CA31vb (c.1495) where he redraws the diagram (fig. 572, cf. fig. 571) claiming:

That place is darker which is seen by a greater sum of umbrous rays.

That place which is percussed by a greater angle of umbrous rays will be darker.
a is twice as dark as b because it arises from a double base at an equal distance.

(figure)

Figs. 571-576: Interposed rods and reflected light. Fig. 571, C5r; fig. 572, C31vb; figs. 573-574, CA144va; figs. 575-576, W12352v.

(figure)

Figs. 577-584: Preliminary sketches involving interposed rods and reflected light on CA357rb.

(figure)

Figs. 585-590: Further cases of interposed rods and reflected light.

Figs. 585-586, CA357rb; fig. 587, CA31vb; fig. 588, CA18rab; figs. 589-590, CA224rb.

    Directly following he demonstrates the converse:

That place will be more luminous which is repercussed by a greater sum of luminous rays.

D is the beginning of the shade df and it tinges a little at c; de is the middle shade df and it tinges more at the percussion b. Df is an entirely umbrous interval and it tinges the place a entirely with itself.

    At the top of this folio is a related phrase: "...is tinted...by the brightness or darkness of the umbrous and luminous bodies placed opposite." Within the next two decades he returns to this diagram, now without text on several occasions: CA144va (figs. 573-574, c.1492); W12352v (figs. 575-576, C.1494); CA18rab (fig. 588, 1500-1504) and CA224rb (figs. 589-590)

    Related to these demonstrations is a more complex series involving the modification of shadow through light that has been reflected backwards. On C5r (1490-1491), for instance, he draws (fig. 593, cf. figs. 591-592) a light source K casting three rays which are then reflected backwards from the ground bf. Above this diagram he notes: "That luminous body will appear...brighter...which is surrounded by darker shadows." Beneath the diagram is a longer passage based on drafts on CA144va (1490, cf. p. 1492):

CA144va C5r
The umbrous and luminous quantity,
even though it is reduced to
/being/ small as a result of
foreshortening, does not diminish
in brightness or darkness.
The width and length of shade or
light, even though their fore-
shortenings appear narrower or
shorter will neither diminish nor
grow in its brightness or darkness.
The length and width of shade and
light even though it appears of less
quantity through foreshortening will
nonetheless not appear diminished in
the quality...of brightness or
darkness.

The width and length of shade or           The width and length of shade and
light, even though they are light,             even though it makes itself
narrower or shorter through...                narrower and shorter through fore-
foreshortenings, nevertheless,                shortening does not diminish or
will neither diminish nor                         augment the quality and quantity
increase the quality and                         of its brightness.
quantity of its brightness
or darkness.

But the function of such light                 The function of shade and light
diminished by foreshortening                diminished by foreshortening will
will be to illuminated or to                      be to shade and...to illuminate the
obscure the counterposed object          body positioned opposite depending
in that quantity and quality on the          quality and quantity that
which appears from that body.             appears in this body.

(figure)

Figs. 591-594: Interposed objects and reflected light and shade. Figs. 591-592, CA144va; fig. 593, C5r; fig. 594, C21r.

    On C21r (1490-1491) he draws a related figure (fig. 594). A spherical light source now casts twelve rays onto the ground which reflect backwards and modify the nature of the shade produced by an opaque body. Accompanying this is the claim:

That part of the derived shade will be darker which is closer to its source.

The /phenomenon in the/ above proposition occurs where a greater luminous angle is joined with a thinner...umbrous body. This luminous body /then/ overcomes it and effectively converts it into its luminous nature. And likewise it is proposed that where the greatest umbrous angle joins itself with the thinnest luminous /body/, the umbrous body will effectively convert the adjacent luminous body into its nature.

    This is followed by a demonstration:

At h the angles of the umbrous and luminous pyramids are joined. The umbrous is mnh. The luminous is oph. The umbrous [pyramid] is the smallest that there is among bcdefgh born of mn and again at h is the largest among the luminous pyramids bcdefgh born at op.

    Which leads him to conclude:

The greatest luminous body always has as its companion the least umbrous angle and similarly you will find that the largest umbrous angle always borders with the smallest umbrous body.

    He considers a more complex situation on C8v (1490-1491) under the heading:

That part of a wall will be darker or more luminous which is obscured or illuminated by a greater dark or luminous angle...

(figure)

Figs. 595-596: Reflected light and shade on C8v and C4r.

    Beneath this he draws (fig. 595) a luminous sphere from which emanate eight rays that are reflected on the wall ae and modify the shadows produced by the interposed plane fg. As he explains:

The above mentioned proposition is clearly proved in this way. Let us say that mq is the luminous body and so fg will be the umbrous body and let ae be the said wall on which the above mentioned angles percuss representing there the nature and quality of their bases. Now a is more luminous...than b because the base of the angle a is larger than that of b and therefore it makes a greater angle...which is amq. And the pyramid bpm is narrower and moe will be thinner /still/ and so on, step by step, the more one approaches e, the more the pyramids are thinner and darker.

    By way of a corollary he claims:

That point of the wall will be of less brightness (d) in which...the size of the umbrous pyramid is greater than the size of the luminous /pyramids).

    This he demonstrates:

At the point a the luminous pyramid will be of as much power as the umbrous /one/ because the base fg is similar to the base rf. And at the point d the luminous pyramid is that much thinner than the umbrous one to the extent that the base sf is less than the base fg.

    As an afterthought he adds:

Divide the above mentioned proposition into two figures, that is, one with umbrous and luminous pyramids and the other with the luminous /ones/.

    Closely related is another diagram on C4r (1490-1491) beneath which (fig. 596) he again begins with a general claim:

The larger the luminous body, the more the course of the umbrous and luminous rays will be mixed together. The effect of the above mentioned proposition occurs because where one finds that there is a greater sum of luminous rays there is greater light and where there is less light, results where the umbrous bodies come to mix together.

    Which leads, as usual, to a demonstration:

M sees and is seen by the entire luminous body ag whence we shall say that among the percussions of luminous rays from m /to/ s, m holds the principle degree of light, n sees af which are 5/6 of the light; o sees ae which are 2/3 of the light; p sees ad which is half the light; q sees ac which is a third, re is seen by ab that is a 1/6 of the light; s sees a the limit of the light and here begins the real and simple shadow.

And this results because the points of the luminous pyramids mnopqrs which are born on the luminous body ag, to the extent that they are narrower, are derived from a lesser base and they make less light in equal distance.

Book Five - 5. Interposed Walls

    The reflections of light and shade produced by objects of various shapes positioned in front of a wall also interest him considerably. On C4v (fig. 599, 1490-1491), for instance, he notes that:

The concourse of shade born and terminated between near and plane surfaces of the same quality and directly opposite will have a darker end than beginning which will terminate in the percussion of luminous rays.

    On CA144va (1490? Pedretti claims C.1492) he sketches what is probably a draft (fig. 597) for the above diagram. A slight variant of this diagram (fig. 598) recurs over fifteen years later on CU712 (TPL602, 1508-1510) where he describes:

How the derived shade being surrounded either entirely or in part by an illuminated background, is darker than the primitive /shade/.

Derived shade which is all or partly surrounded by a luminous body will always be darker than the primitive shade which is on a plane surface.

    This he demonstrates:

Let a be the light and let bc be the object which retains the primitive shade and let the wall de be that which receives the derived shade in the part nm and a remainder of it, dn and me will remain illumined by a. And the light dn is reflected in the primitive light bc and the light me does the same.

Therefore the derived shade nm, not seeing the light a, remains dark and the primitive /shade/ is illuminated by the illuminated background which surrounds the derived shade. And hence the derived shade is darker than the primitive shade.

(figure)

Figs. 597-601: Interposed walls and reflected light and shade. Fig. 597, CA144va; fig. 598, CU712; fig. 599, C4v; fig. 600, CU713; fig. 601, CU715.

(figure)

Figs. 602-605: Spheres, interposed walls and reflected light and shade. Fig. 602, C17r; figs. 603-604, C16v; fig. 605, C20v.

    On CU715 (fig. 601, TPL580, 1508-1510) he considers what happens if the interposed plane is tilted, asking:

What is augmented shade?

Augmented shade is that in which only its derived shade is reflected.

Let a be the luminous body. et bc be the primitive or original shade and dg will be the originated shade.

    What happens when this flat interposed plane is substituted by a spherical object he considers on CU713 (fig. 600, TPL603, 1508-1510):

How the primitive light which is not joined with a flat surface will not be of equal darkness.

This is proved: and let bcd be the primitive shade joined to the object in which is seen the derived shade fg and again one sees its illuminated background ef /and/ gh. I say that such a body will be more illuminated at the extremity b than at the middle d, because at b the primitive light a is seen and the derived light fg is not joined because fbd is the angle of contingence made by the straight line fb and by the curve bd. And all the remainder of such a body is seen by the derived shade fg, more or less according to whether the line fg can be made lower with a triangle with a greater or lesser angle.

    The characteristics of reflected shade produced by a spherical body in front of a wall had concerned him at some length in the Manuscript C. On C17r, for instance, he makes a preliminary drawing (fig. 602), which he then develops on C16v (figs. 603=604, 1490-1491). Between these diagrams on C16v he adds a marginal note:

The rays doubled through intersection in lights and shadows are also of double brightness and obscurity.

The umbrous part of this superior body is brighter in mhn than in tqp because in this part of the two reflected derived lights intersect, that is, ab and dc, as appears in the triangle mno and at tm one will only see ab and not dc.

    In the main body of the text he pursues his ideas on reflected shade in a "proposition" which he had drafted on CA144va (c.1490? Pedretti claims c.1492):

CA144va C16v HELLO
The primitive and derived reflection     The primitive and derived
surrounding dense and spherical           reflected light surrounding
bodies will make that the boundaries   dense and spherical bodies
of primitive shade...of this body will     be the cause that the
are that much better understood         boundaries of the primitive
from the one extremity than from shade are that much more
the other to the extent that the        distinct and bounded with its
primitive light is brighter than               nearby illuminated part, to
the reflected.                                      the extent that the derived
light is brighter than the
derived shade.

    This proposition leads to a "comment":

That is said to be primitive light which illuminates bodies primarily and derived shade is said to be that which rebounds from these bodies in those parts which are distant from this primitive light.

    Which then introduces a (fig. 603):

Demonstration

Let k be the primitive light which illuminates the umbrous body at tp and the places ab/and/ cd. From ab /and/ cd there departs derived shade and it rebounds onto the body opposite at mn and the entire part of the body at h will be more luminous than at q because it is seen by a double light, that is ab and dc. Whence q is not seen except by simple light and therefore it is dark.

    Immediately following he adds another proposition and commentary:

Proposition

That part of primitive shade is more luminous which can see the middles of the derived lights equally.

Commentary

It can clearly be recognized that that part of umbrous bodies seen by a greater quantity of light is more luminous and maximally if that part is illuminated by two lights as is seen in reflected lights which, put in the middle of themselves the derived shadow made through them by the dense bodies opposite.

    This is again followed by a (fig. 604):

Demonstration

Let n be the part of the body which is more luminous than any other in this (body) because it is seen equally by the first 2 powers of the lights positioned opposite it, that is b, greater than the power of the light ac and similarly e is greater than df and all the 2 see (in) the said ndc and similarly af because they are extremities, only less powerful and these see the body in ro and ut, this place being seen by less light. Darker is the part illuminated by these, the entire triangle is seen by double lights of various qualities of brightness.

    This leads to a third: "proposition": again based on a draft on CA144va:

(figure)

Figs. 606-607: Reflected light and shade on C16v and CA144vb.

CA144va C16v
Every luminous body with all of it           Every luminous body with all
and with its part illuminates the               and with part of itself
part and all of the object.                       illuminates the part and all
of the object positioned
opposite it.

    This is followed by a Definition

This proposition is fairly obvious and that which cannot be denied (denied) is that where the entire pupil of the eye looks at this place it does not look at this place (its). Every part of this is a place seen by this pupil /and it/ does the same towards the pupil /i.e. every part of this place also sees the pupil/.

    Which claim is again supported (fig. 606 cf. 607) by a:

Demonstration

Let ac be the luminous body. Let df be the illuminated object which, even though it is composed of infinite points, we shall only make a test of three, that is d, e /and f. Now you see that e is seen by the part b of the luminous body and by all of ac as is demonstrated by the lines ae and ce and by the centric line be and also at the point d, all of ac is seen, and the centre ube and you will find the same at f and the same occurs throughout all the parts of the object df.

    On C20v he develops these diagrams (fig. 605, 1490-1491), this time under the heading:

To the extent that the derived light has less light than the original /light/ to that extent will its pyramids illuminate the place percussed by them.

(figure)

Figs. 606-614: The development of an idea. Figs. 608-609, CA144rb; fig. 610, CA144va; fig. 611, CA144vb; fig. 612, CA144ra; fig. 613, CA144va; fig. 614-616, CA144rb.

    This idea he reformulates beneath the diagram: "Pyramids illuminate the place percussed by them less to the extent that the angles of these are thinner." This is one of the rare cases where a proposition is not followed by a demonstration.

    On CA144vb (c.1490? cf. P.1492) he drafts a related diagram (fig. 606) which he develops on C16v (fig. 607, 1490-1491). Alongside the diagram on CA144vb, he writes "The boundaries of." This he crosses out and begins again. "The boundaries of shade of spherical bodies are." This he crosses out also and begins afresh. "The boundaries of shade on dense and spherical bodies are that much more kn/own.../ and the surroundings than of the reflection." Still unsatisfied he crosses this out too and finally drafts a coherent passage:

The primitive and derived light, surrounding the dense and spherical bodies will have the effect...that the boundaries of the primitive shade...of this body is that much more known from one extremity than from the other to the extent that the primitive light is brighter than the reflected.

    The problem of varying degrees of light continues to trouble him. On CA144ra, rb, va, vb (figs. 604-614, 1490? Pedretti claims 1492) he drafts a further series of diagrams one of which is accompanied by a rough text:

All the lines of shadows are straight, because...among the luminous lines even from r...ts is more than another luminous part...of the half triangle lt because it is seen...no straight and from og and cf one...on which are 2/8 of all the li/ght/...ls is...this same luminous body because it is se/en/...by nf, cg straight and from goec...above, which are again 2/8 of the whole....
Whence we shall say that there is less light...the triangle __x and beyond the shadows 3/8 exactly.

(figure)

Figs. 617-618: Demonstrations of reflect mixed and reflected light and shade on C3r and C4r.

    Beneath this he drafts another section which he later crosses out:

Ts is seen by the whole of no directly and by...ogef inverted, but by the same...is from cf and for this /reason/ it is illuminated more than the other part of the half triangle because it is seen by 5/8 of the light.

S is seen by gn and df.

    The diagram on CA144rb he develops on C3r (fig. 617, 1490-1491) which, as he explains, involves an astronomical phenomenon that had been under discussion in the optical tradition:

That body will appear less bright which is surrounded by a more luminous background.

I have found that those stars which are closer to the horizon appear of a greater size than the others because these see and are seen by a greater sum of the solar body than when they are seen above us. And by seeing more sun they have more light and the body which will be more luminous will show itself as larger as the sun above us demonstrates in fog, which seems greater without fog and in fog diminishes.

    This diagram on C3r is a record of what happens when a luminous body casts light on an opaque body in the open air. On C4v (fig. 618) he adds an interposed wall which causes a complex array of reflected light and shade. Here he develops the ideas he had drafted on cA144rb.

That part of the surface that is percussed by a greater angle by the species of the objects positioned opposite will tinge itself more in the colour of these.

8 below is a greater angle than 4 since its base an is greater than its base en by 4.

This figure below wishes to be terminated by an and 4 and 8.

    Beneath the diagram is a further proposition:

That part of the illuminated body which surrounds the percussion of the shadow is more luminous which is closer to the percussion.

Let 4 be the part of the illuminated object 4, 8 which surrounds...the percussion of shade 9/?/ and 4 and this place 4 is more luminous because it sees a lesser sum of shade than does the place 8 because 4 only sees the shade in and 8 sees and is percussed by the shade ae and by the shade in which is 2 times as dark and this same occurs when you put the air with sun in a place with light and shade.

Book Five - 6. Theoretical Demonstrations

    Among the most fascinating aspects of Leonardo's approach is the way in which he returns to problems at various levels of abstraction. Hence, having provided concrete demonstrations he may well go on to give more abstract geometrical illustrations of the same principle. We have seen, for example, how he discussed the quality of darkness in experimental terms (see above pp. ). On CU689 (fig. 619, TPL641, 150-8-1510) he gives a geometrical demonstration:

Of the quality of darkness of shadows.

The darkness of derived shadows is infinitely variable with so much greater or less power as the distances in which the percussions of the derived shadows are caused are greater or less. Percussions of the derived shadows are caused are greater or less.

(figure)

Figs. 619-621: Theoretical demonstrations of light and shade on CU689, 681 and 682.

This is proved. And let the sun be a which generates the shadow nphi in which enters the light of the air which surrounds the solar rays, that is, eb/to/ rs above and /from/ fc to rs below, and it will brighten this shade which is most dark in the space npo where it sees neither sun nor air nor its extremities b /or/ c.

    On CU723 (fig. 630, TPL783, 1508-1510) he considers the problem in semi-abstract terms:

How reflection is generated in universal lights.

Reflection is generated in bodies illuminated by universal lights when a part of the illuminated body reflects its larger light in that place where a smaller part of the same light sees: As /is the case when/ the sky ef, seeing the place d, and a greater part of the same sky sees h, then the derived light h will reflect in d.

    But of this a separate treatise will be made at the appropriate place.

    Although there is no evidence of the intended treatise itself, he clearly pursued the problem. On CA207ra (1508-1510) he notes in passing "the consummation of shadows by degrees in universal light," and on CU681 (TPL643, 1508-1510) he provides a fully abstract geometrical demonstration of reflection in universal lights (fig. 620) under the heading:

Precept of painting.

In universal lights, shadows occupy little space in the surfaces of their bodies. And this arises because a great sum of the light of our hemisphere tinges as far as the lowest parts of umbrous bodies if it is not impeded with its horizon and maximally if it is suspended from the earth.

Let f be the umbrous body, e the earth, abcd is our hemisphere from which the darkness of the earth vx obscures that much of the umbrous body, as it sees; as the horizon which sees the same parts illuminates the same places and confounds the umbrous species of the aforesaid earth, which will be in a position to make such dark shadows under the object were it not impeded.

    This problem preoccupies him. On CA207ra (1508-1510) he drafts another version:

...part of the bod/y/...

It follows that...of the umbrous body will be more illuminated which sees a greater sum of the body that illuminates it.

    This he develops on CU682 (fig. 621, TPL681, 1508-1510) in a passage entitled:

Of the universal light of the air where the sun does not percuss.

That thing will show itself as more illuminated which is seen by a greater quantity of the luminous body. By that which has been said e will be more illuminated than a because e sees a greater amount of sky, seeing rs which a does not see, it seeing only the sky bcd.

    On CU722 (TPL782, 1508-1510) he pursues this geometrical analysis of conditions in universal light, beginning with a general statement under the heading:

Why reflections are little seen or not at all in universal lights.

The reflections of umbrous bodies are little seen or not at all in universal lights. And this occurs because such a universal light sufficiently surrounds and covers each of these bodies, the surface of which, as has been proved, participates in the colour of its objects.

(figure)

Figs. 622-626: Abstract geometrical demonstrations of reflected light and shade. Fig. 622, CA207ra; fig. 723, CU675; fig. 624, CA207ra; fig. 625, CU690; fig. 626, G3v.

    By way of illustration he carefully describes a geometrical diagram (fig. 632):

As would be /the case/ if the body a were illuminated by its hemisphere gcd and shaded by the earth gfd. Here the surface of such a body is illuminated and shaded by the air and by the earth which stands as its object and it is more or less illuminated or shaded to the extent that it is seen by a lesser or greater amount of a luminous or dark body. As is seen at the point k /which/ is seen by the entire part hci of the hemisphere and is not seen by any part of the darkness of the earth. Therefore it follows that k is more illuminated than a where only the part cd of the hemisphere is seen and such an illumination is corrupted by the darkness of the earth rd, all of which sees and is seen by the point a, as is proved in perspective. And if we wish to speak of the point b, we shall find that this is less illuminated than the point a, such that this b sees half of the hemisphere that a sees; that is, /a/ sees all of cd and b only sees ed, which is half of cd. And it sees all the darkness of the earth which a sees, that is, the earth rd and to this is added the part rf which is darker, because in this, the light of the hemisphere ec is lacking, which is not lacking to the earth rd.

    Which leads him to conclude:

Therefore, for such a reason, this body cannot have reflection, because the reflection of light is after the principal shade of the bodies and here the principal shadow is in the point where such a body is in contact with the plane of the earth, which is why it is entirely deprived of light.

(figure)

Figs. 627-632: Further abstract demonstrations of reflections. Fig. 627, CU676; fig. 628, CU207ra; fig. 629, CU677; fig. 630, CU723; fig. 631, CA207ra; fig. 632, CU722.

    The mode of analysis here used - a geometrical sphere surrounded by a hemisphere -appeals to him and becomes the starting point for no less than ten further demonstrations. On CU687 (TPL736, 1508-1510), for instance, he considers which part of an opaque spherical body will be darker (see above pp. ) under the heading:

Of the shadow of the opaque spherical body positioned in the air.

That part of the opaque spherical body will be darker which is seen by a greater sum of darkness.

    He now describes the geometrical diagram (fig. 641, cf. fig. 640):

Let the dark object be the plane dc and let the luminous hemisphere be dnc, and let the spherical body interposed between the light of the hemisphere and the darkness of the earth be bqop. I say that the part oqp is more obscure than any part of such a spherical body because it alone sees all the sides of the darkness of the earth dc opposite. And every other side of it sees less.

    Directly following is a proof:

This is proved by one of the elements which states: the line produced from the centre of the circle to the angle of contingence will be perpendicular and will fall between two right-angles. It follows that the line which comes from the centre x of the sphere, terminates at sc under right-angles at the point o /and/ sees all the darkness of the earth dc and likewise such an o is seen by this earth. And p opposite does the same for the same reasons.

And likewise q and every part which puts itself between in the space op. But q is of more excellent darkness being in the middle over the earth which o or p is not, being closer to the extremities of such darkness of the earth and begin to see the horizon of this hemisphere and mix themselves with its light.

    As a corollary to this proposition, he considers the case of an opaque body on the ground, first in draft form on CA207ra (1508-1510) and then on CU688 (TPL737, 1508-1510):

CA207ra TPL737 HELLO
On the shadow of the opaque
spherical body positioned over the
earth.
But the shadow of the opaque     But the shadow of the opaque
body...darker than that of the      spherical body positioned in contact
opaque...in the air, because,        with the earth will be of greater
other than receiving the...            darkness than the foregoing, which
earth positioned opposite it,        only sees it as its object.
it also receives...that which
makes it above this earth.

    This claim on CU688 (TPL737) is again supported by a geometrical demonstration (fig. 633):

This is proved and let the opaque spherical body be nms positioned over the earth ac of the point s and the arc abc.

And the same as was said above is confirmed in a body illuminated in our hemisphere and here it manifests itself in the part of the spherical body under the phemisphere k and f which, in the point b, is illuminated by all the part aec and in the part rd by the hemisphere ef and in o by gf and in n by mf and in h by sf. And thus you have understood where the first light and the first shadow in a body is.

    This idea he reformulates in the next passage CU676 (fig. 627, TPL694a, 1508-1510):

That part of an umbrous body will be more luminous which is illuminated by a greater sum of light.

Hence, placing the body abc as umbrous body and dfn as luminous body, that is the illuminated hemisphere, in the part c there is twice as much light as in the part b, and 3/4 more than in a, because c is illuminated by the sky dgfe and b by the sky df which is the half less than de and the part a is only illuminated by the quarter part of de, that is, by gd.

    This passage is followed, in turn, by a geometrical demonstration (CU677, TPL694b, 1508-1510) of another principle that interests him (see below pp. ):

The surface of every body participates in the colour of its object.

Let d be the opaque body. Let an be the luminous body, let ac be a body of a dark colour, let cd be the illuminated plane of the hemisphere afmn. By the aforesaid, r will be more illuminated than o; o than s; s than t and the parts which are facing the dark body, ac, will do the same as will those which are facing the illuminated place cd and from this originate light and shade and reflected light.

    The accompanying diagram (fig. 629) may be based on the draft on CA207ra (fig. 628, 1508-1510) beneath which he notes: "n does not make shadow on the earth." In this series one demonstration builds on the other in the manner of a proposition in Euclidean geometry. Leonardo is set on translating his experimental results into a systematic geometrical language. To this end he makes further drafts on CA207ra (1508-1510) which lead to another series of demonstrations on CU690, 679, 686 (TPL748-750, 1508-1510). On CU690 (TPL748b, 1508-1510) he begins by raising a question which he had already answered elsewhere in concrete terms (see above pp. ): "Which part of the spherical body is less illuminated?". His preliminary answer on CU690 (TPL748b) is again based on drafts on CA207va (1508-1510):

CA207va TPL748b

That umbrous body will have a
lesser...quantity of itself
illuminated which part...is seen
by a smaller...quantity of the
luminous body.
That part of the umbrous body will     That part of the umbrous body
be that much less illuminated               will be less illuminated which
which sees a smaller part of the          is seen by a smaller part of
body which illuminates it.                    the luminous body.

    This is followed by a geometrical demonstration (fig. 625 cf. fig. 624):

This is proved. And let the umbrous body be asqr and let the hemisphere of the luminous source be ncedf. I say that the part a and the part o, being seen by equal arcs bced and cdef, are seen by an equal quality of light and for this reason are equally illuminated by these. But r, seen by a smaller arc edf, receives less light and p which only sees df/which/ is less than edf and for this /reason/ it remains less luminous. And q also remains less luminous which sees only the extremity of the horizon f.

    On CU679 (TPL749, 1508-1510) he poses the converse question: "Which part of the spherical body is more illuminated?". The general claim that follows is again based on an earlier draft:

CA207ra TPL749

And that part of spherical bodies         And that part which is
which is illuminated will be of a            illuminated by spherical
greater brightness, than that                 bodies will be of a more
which is accompanied by a lesser        intense brightness which has
sum of umbrous species.                     a smaller sum of umbrous
                                                           species accompanying it.

    His claim is again supported by a geometrical demonstration:

This is proved. And let fno be the spherical umbrous body and abc is the luminous hemisphere and the plane ac is the darkness of the earth. Therefore I say that the part of the sphere, fn will be of a more intense brightness because it does not see any part of the earth ac and it is, in itself, of equal brightness, being illuminated by the equal arcs of the hemisphere abc, that is the arc are is equal to the arc rbc and the arc gsc and by a proposition (concettione) that states when two things are equal to a 3rd they are equal among themselves. Therefore p, f and n are equal in brightness.

    He returns to the question he had asked on CU690 (TPL748a) in the next proposition in the treatise of painting, i.e. CU686 (TPL656):

Which part of the opaque body is less illuminated?

That part of the opaque spherical body is of darker shadow which is seen by a lesser sum of luminous rays.

    He is conscious that he has already dealt with the problem. What challenges him is the idea of an alternative proof:

Even though this has great resemblance with the 1st above, I will not be content unless I prove it, because this proof is somewhat different.

    A geometrical proof follows (fig. 634, cf.633):

And let the umbrous body be fno and the hemisphere is abc and the darkness of the earth is the line ac. I say firstly that the superior part of the spherical body fpm will be equally illuminated by all the hemisphere abc and likewise I demonstrate it for the three given equal portions, that is are which illuminates the point f and rbs which illuminates p and gsc which illuminates n. Therefore by the 7th of the 9th it is concluded that fpn, the superior part of the spherical body is of equal brightness. Which 7th of the 9th states that all the parts of bodies which are illuminated equidistantly be equal and similar lights will, by necessity always be of equal brightness, which condition occurs at fpn.

    One alternative demonstration is not enough (fig. 636, cf.fig. 635):

There follows a second demonstration.

Let abc be the umbrous spherical body. Let dfe be the illuminating hemisphere. D is the earth which causes shadow. I say by the foregoing /proposition/, that the entire part anb of the sphere is deprived of shadow because it is not seen by the darkness of the earth and all the remainder of the surface of such a sphere is umbrous with more or less darkness, depending on whether a greater or lesser sum of darkness of the earth accompanies a greater or lesser sum of darkness of the hemisphere. Therefore, the point c, which sees a lesser sum of such a hemisphere and a greater sum of earth will be darker than any other part of the shadow, that is, it only sees rd and se of the hemisphere and it sees all the earth de. And the brightest is ab because it only sees the extremities of the earth d, e.

    On CA207ra (1508-1510) he makes further drafts:

That part of a body illuminates.

The illuminated part of a spherical body will be of that much less...shape,...to the extent that it will be seen by a smaller amount of the...luminous body.

That part which...is illuminated by some spherical body,...will be less..., to the extent that it is seen by a lesser sum of the luminous body....

That part which is illuminated by some spherical...will be that much less to the extent that it is seen by a smaller light.

That p/art/ which is illuminated in some spherical body, will be that much less to the extent that the part of the luminous body which sees it will be....

    These drafts, which he crosses out, serve as starting point of his next proposition on CU686 (TPL750, 1508-1510):

That part which is illuminated by some spherical body will be that much smaller to the extent that the part of the luminous body which sees it is smaller.

    On CA207ra (1508-1510) he also drafts a corresponding demonstration (fig. 635):

n has so much...darkness through the object to the extent that it has light and it is shown that ab and ed are equal to bd and do not make shadow on the earth.

(figure)

Figs. 633-651: Abstract geometrical demonstrations of light and shade. Fig. 633, CU688; fig. 634, CU679; fig. 635, CA207ra; fig. 636, CU686; fig. 637, CU686; fig. 638, CA207ra; fig. 639, CU686; fig. 640, CA207ra; fig. 641, CU687.

    This demonstration is developed on CU686 (fig. 736, cf. fig. 635), although the lettering is different:

This is proved. Let ah be the umbrous body. Let cie be our hemisphere. It follows that part a of the umbrous body will be less illuminated, being seen by a smaller part of the umbrous body, that is, by a lesser part of the day of this our hemisphere, as the two parts bc and de show.

    He also claims the converse (fig. 639, cf. fig. 638).

Therefore that part of a spherical body which is illuminated will be of a larger shape which is illuminated by a greater sum of the luminous body.

    This is proved by the converse of the foregoing.

If the minimal light bc and de of our hemisphere illuminate a minimal part of the spherical body ah, the same light will illuminate the maximal part of this spherical body, that is, if bc /and/ df of the following body, that is, if bc /and/ df of the following figure illuminate only the part nmr, the rest of the hemisphere, joined with its part bc /and/ df will illuminate the remainder of the aforementioned spherical body.

Which is why even though bc /and df illuminate nmr it also illuminates the part kn on the side of the spherical body and the other /part/ lr on the side opposite.

    This series ends with a plea in defence of such geometrical demonstrations directed against an adversary:

Here the adversary who does not want such science, says that the practice of drawing natural things suffices. To which it is replied that there is nothing which deceives more than trusting one's judgment without any other reasoning (raggione) as is always proved by experience, the enemy of alchemists, necromancers and other simple spirits (ingiegni)>

    Read in context this oft cited passage is all the more fascinating because it reveals an important link between experience/experiment and geometry in Leonardo's approach. In his conception of science neither practice nor theory is sufficient in itself. Science involves a process of translating particular experience into a universal language of geometry. This is why, when he asks a question, one demonstration is never enough. He needs to provide various demonstrations in order to create bridges between concrete experience and abstract geometry. This is a theme to which we shall return in the eiplogue - see pp.** below.

 

Book Six: Reflected Colour

Besides this in the sixth book I shall investigate the many and various diversities of reflections of these rays which will modify the original /shadow/ by /imparting/ some of the various colours from the different objects whence these reflected rays are derived. (CA250ra)

    That reflected light and shade should influence the colours of surrounding objects was by no means a new idea. (Pseudo-) Aristotle in De Coloribus had, for instance, pointed out:

Lastly we never see a colour in absolute purity: it is always blent, if not with another colour, then with rays of light or with shadows and so it assumes a new tint...This is...why reflections in mirrors resemble the colour of mirrors.6

    This aspect of colours had also been considered by later authors such as Ptolemy7 and Alhazen.8 How Leonardo intended to organize his own scattered notes on this theme is not clear. It is likely, however, that his projected sixth book would have included sections on reflections from 1) mirrors, 2) water, 3) white objects, 4) faces, 5) landscape and verdure, 6) a series of demonstrations show how yellow and azure combined produce green, 7) another set of demonstrations involves walls and lights of different colours, which become a starting point for his parallels, 8) between mixing lights and mixing pigments. There 9) further demonstrations also. Together these form the basis for his 10) precepts and 11) general statements concerning reflected light, shade and colour. Each of these aspects will be considered in turn.

(figure)

Fig. 642: Mirror reflection on Forst III 54r.

Book Six - 1. Mirrors

    In De Coloribus Aristotle had noted that reflections in mirrors resemble the colour of mirrors.9 Ptolemy, in his Optics h ad pointed out that the colour of a mirror affects the colour of things seen.10 This phenomenon had also been mentioned by authors such as Heliodorus of Larissa11, Alhazen12 and Witelo.13 Leonardo's first extant reference to this question is on Forst III 54r (fig. 642, c.1493) under the heading:

Mirror

If the illuminated object is of the size of the illuminating object and of that where this light is reflected, the quality will have such a proportion with the medium light as the second light will have with the first, these bodies being level and white.

    On BM57r (1497-1500) he restates this idea more succinctly:

Whence they say that it transmutes itself in as many natures as the places where it passes are various. And as the mirror transmutes itself into the colour of its objects, so too does this transmute itself into the nature of the place where it passes.

    A slightly different version occurs on BM58v (1505-1508):

And as the mirror transmutes itself into the colour of the objects which pass in front of it, it has nothing in itself, but moves or takes everything and transmutes itself into as many various natures as the places where it passes are various.

    On CU167 (TPL158, 1505-1510) he notes that the phenomenon depends on the degree to which the reflecting surface is polished:

On reflections

Reflections participate that much more or less of the thing where they are generated than of the thing which generates them to the extent that the object where they are generated is of a more polished surface than that which generates them.

    On CU211 (TPL256, 1508-1510) he pursues the question:

Of the colours reflected on the lustres of various colours.

The reflected object always participates of the colour of the body which reflects it. The mirror is tinged in part by the colour reflected by it and participates that much more of the one than the other, to the extent that the object which is mirrored is more or less powerful than the colour of the mirror. And that object appears of a more powerful colour in a mirror which participates more of the colour of this mirror.

    He mentions the phenomenon once more on BM211v (1508-1512): "The image impressed in the mirror participates in the colour of the aforesaid mirror."

Book Six - 2. Water

    He studies the physics of reflections in water in connection with his mirror studies (see below pp. ). In addition to this there are at least four passages where he explores the properties of reflected colour in water. The simplest of these, on CU542 (TPL521, 1505-1510) is headed:

On objects reflected in water.

Of objects reflected in water that will be more similar in colour to the reflected object which is reflected in clearer water.

    On CU5453 (TPL522, 1505-1510) he considers reflection in water:    

On objects reflected in turbulent waters.

Objects reflected in turbulent waters always participate in the colour of that object which renders such water turbulent.

    He considers a more complex situation on CU213 (TPL237, 1505-1510) under the heading:

Of the reflection and colour of the water of the sea seen from various aspects.

The sea with waves does not have universal colour, but he who sees it from firm ground sees it of a dark colour and that much darker to the extent that it is closer to the horizon and he sees some brightness or lustres which move slowly in the manner of a herd of white pigs and he who sees the sea /while/ standing in the high sea/s/, sees it as azure.

And this occurs because from the land the sea appears dark because you see in it the waves which reflect the darkness of the earth and from the high sea/s/ they appear azure because you see in the waves the azure air which these waves reflect.

(figure)

Fig. 643: Reflection from water on CU1007.

    In the late period he considers reflected colours in water once more on CU1007 (TPL943, 1510-1515), this time in connection with painting:

Where the horizon is reflected in the waves.

By the sixth of this the horizon is reflected on the size seen by the horizon and the eye as the horizon f demonstrates, seen by the side of the wave bc and this side is also seen by the eye.
Again you, /o/ painter, who have to draw the inundation of the water, recall that the colour of the water will not be seen by you as being other than bright and dark, whatever the brightness or darkness of the site may be where you are, mixed together with the colour of the other things which are behind you.

Book Six - 3. White Objects

    Leon Battista Alberti, in his On Painting, had described how:

reflected rays carry with themselves the colour they find on the plane. You may have noticed that anyone who walks through a meadow in the sun appears greenish in the face.12

    Leonardo adapts this example and develops it on A100r (BN 2038 20r, 1492) under the heading:

How white bodies must be represented.

If you represent a white body surrounded by much air, because white does not have colour in itself, it is tinged and transmuted in part by the colour which is its object. If you see a woman dressed in white in a countryside, which is seen by the sun her colour will be bright in such a way that she will in part, like the sun, hurt the sight. And that part which is seen by the air or luminous body by the rays of the sun, interwoven and penetrated by it, because the air in itself is azure, that part of the woman seen by this air appears to tend towards azure. If the nearby surface of the earth is covered in meadows and the woman finds herself in that meadow illuminated by the sun and this sun sees all the parts of this which can be seen of the meadow, it will be tinged by the reflected rays in the colour of this meadow and thus it goes transmuting into the colour of nearby luminous and non-luminous objects.

    White objects interest him particularly because he considers them to have no colour of their own (see above p. ) and therefore most apt to adopt the colours of surrounding objects. Hence, on A19v (1492), for instance, he begins with a:

Proposition

Every body without colour is coloured entirely or in part by the colour positioned opposite.

This is seen by experience, because every body that reflects is tinged in the colour which is its object. And that body which is tinged in part, if it is white, then that part which is illuminated by red appears red and by every other luminous or umbrous colour.

    He then gives a second in which he mentions white walls:

Proposition

Every opaque body without colour participates in that colour which it has for object. This happens on a white wall.

    On A20r, the folio opposite, he restates this idea: "Every white and opaque body is tinged in part by the image of the colours that are its object." He mentions this quality of white objects again in the third of a series of drafts on W19141r (K/P99r, 1506-1508):

The surface of every opaque body will participate in the colour of its object.

The surface of the opaque body is tinged by the colour of its object with that much more power to the extent that the rays of the species of these objects strike these bodies between more equal angles.

And the surface of an opaque body is tinged more by the colour of its object to the extent that such a surface is whiter and the colour of its object is more luminous or illuminated.

    On F75r (CU204, TPL247, 1508) he restates the idea, now referring to it as a fourth proposition:

Painting

Since white is not a colour but in power receptive of every colour, when this is in a high landscape all its shadows are azure and this originates by the fourth which states: the surface of every opaque body participates in the colour of its object.

Therefore such a white, being deprived of the light of the sun through the interposition of some object placed between the sun and it, therefore all the white that sees the sun and the participating air remains the colour of the sun and that part which does not see the sun remains umbrous and participating in the colour of the air and if such a white does not see the verdure of the countryside stretching to the horizon, nor the whiteness of such an horizon, without doubt this white would appear to be the simple colour which the air demonstrates itself to be.

    He reformulates this principle on CU206 (TPL196, 1505-1510) again referring to it as a fourth proposition (see Chart 16 ):

Colour of the shadow of white.

The shadow of white seen by the sun and by the air has its shadows tending towards azure. And this occurs because white has no colour in itself but receives some colour and by the 4th of this which states: the surface of every body participates in the colour of its object, it is necessary that that part of the white surface participates in the colour of the air /which is/ its object.

    On CU465 (TPL471, 1508-1510) he develops the principle into two propositions under the heading:

Painting

a. The surface of every opaque body participates in the colour of its object and all the more to the extent that this surface approaches a greater whiteness.

b. The surface of every opaque body participates in the colour of the transparent medium interposed between the eye and this surface and the more so, to the extent that it is denser, and a greater space is interposed between the eye and the said surface.

    He pursues this question of white bodies on CU753 (TPL628, 1508-1510):

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

Nothing appears its natural whiteness because the sites, in which these things 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 of little brightness and at night with such splendour that it renders from itself the image of the sun and by day with its dispelling of shadows.

    To explain this he offers two reasons:

And this arises from two things. And the first is the comparison which has in it the nature of showing things that much more perfect in the species of their colours to the extent that they are more disform. And the second is that the pupil is larger at night than by day as is proved and the larger pupil sees a luminous body of greater quantity of more excellent splendour than the smaller pupil, as is proved by him who looks at the stars through a small aperture made in a piece of cardboard (see below pp. ).

    He returns to the characteristics of white subjects once more on CU785 (TPL704, 1508-1510):

Which object will tinge the white surfaces of opaque bodies more with its similitudes?

That object will tinge the surfaces of white opaque bodies more with its similitudes which is by nature more remote from white. That which demonstrates itself as being the most remote from white is black and this is that by which the surface of a white opaque body is more tinged than by any other colour of other objects.

Book Six - 4. Faces

    What applies to white colours, applies equally to flesh colours, as is clear from a passage on CU174 (TPL162, 1505-1510) headed:

On the colours reflected from the flesh

The reflections from flesh which have the light of other flesh are redder and of a more excellent flesh colour than any other part of the flesh that there is on a man. And this occurs by third of the second book which states: the surface of every opaque body which participates in the colour of its object, is that much greater to the extent that such an object is closer and that much less to the extent that it is more remote and to the extent that it is larger because, being larger, it impedes the species of the surrounding objects which are often of various colours, which corrupt the first species /which are/ closer, in the case of small bodies.

    But it is, nonetheless, possible that the reflection of a nearby small colour tinges more than a

large remote colour by the sixth of perspective which states: large things can be at so great a distance that they appear considerably less than the little ones from nearby (see below pp. )/.

    He outlines the consequences this has for his painting practice on CU175 (TPL170b, 1508-1510) under the heading:

On Reflections

That colour which is closer to the reflection will be more tinged by this reflection and conversely.

Thus you, /O/ painter, need to do in the reflection of the faces of figures /with/ the colour of the parts of vestments which are close to the parts of the flesh /and the more so with/ those which are closer, but not to separate them with too much pronunciation if you need not.

    He discusses the problem of faces and reflected light again on CU798 (TPL644a, 1508-1510) in a passage entitled:

On the shadows which are not accompanied by the illuminated part.

Very rare are those shadows of opaque bodies which are the true shadows of their illuminated part.

This is proved by the 7th of the 4th which states that the surface of every umbrous body participates in the colour of its object. Therefore the illuminated colour of faces, having as its object a black colour will participate in the black shadows and yellow, green and azure will do the same as will every other colour positioned opposite it. And this occurs for the reason that every body sends its similitude through all the surrounding air as is proved in perspective and as is seen by the experience of the sun of which all the objects positioned opposite it participate in its light and reflect this to the other objects as is seen by the moon and the other stars, which reflect to us the light given them by the sun.
And shadows will do the same, because these invest all the things which they strike with their darkness.

    This leads to a more general formulation on CU797 (TPL645, 1508-1510):

Of the light of umbrous bodies which are practically never of the true colour of the
illuminated body.

We can say that it is practically never that the surface of illuminated bodies is the true
colour of this body.

    In the demonstration that follows he again refers to the principle of colour participating as the seventh of the fourth (see Chart 16 ):

That seventh of the fourth states the cause of this and also demonstrates that when a face positioned in a dark place is illuminated on one side by a ray of the air and is, on the other, struck by the ray of a candle, it undoubtedly appears to be of two colours. And before the air sees such a face, the light of the candle will appear its given colour and likewise the intervening air.

    Another demonstration follows which, in turn, relates to his camera obscura experiments (see below pp. ):

If you take a white band and put it in a dark place and you take a light through an aperture, namely, from the sun, from fire and from the air, such a band will be of three colours.

    On CU801 (TPL708, 1508-1510) he discusses the effects of reflected colour on both the clothes and faces of persons:

What the shadows do with the lights in comparison.

Black clothes make persons stand out in greater relief than white clothes and this arises through the 3rd of the 9th which states: the surface of every opaque body participates in the colour of its object. It therefore follows that the parts of the face which see and are seen by black objects show themselves as participating in this black and for this /reason/ the shadows will be dark and there is a great difference between /these and/ the parts of this face which are illuminated.

But white clothes will make the shadows of the face participate in such a whiteness and for this reason the parts of the face will show themselves to you as being of little relief, the bright and the dark having between them little difference from the bright and the dark, it follows that in this case the shadow of the face will not be the true shadow of such skin.

(figure)

Figs. 644-646: Reflected light and colour on A113v, CU199 and CA305va.

In this context Mona Lisa's dark clothes make more sense.

 

Book Six - 5. Landscape and Verdure

    Practical experiences in Nature also serve to demonstrate effects of light and shade on colour. On A113v (BN 2038 32v, CU199, TPL209, 1492), for instance, he cites an example of reflected sunlight in the mountains (fig. 644, cf. fig. 645):

Which part of the colour should reasonably be more beautiful.

If a is the light /and/ b is illuminated along a line from this light, c which cannot see this light sees only the illuminated part, which part let us say that it is red. It being thus, the light which it will throw to the side will be similar to its cause and it will tinge red the /rock/ face uc. And if c is also red you will see that it is much more beautiful than in b and if c were yellow you will see it create a colour between yellow and red.

(figure)

Figs. 647-650: Reflected light and colour. Figs. 647-648, CA144vb; fig. 649, CU203; fig. 650, CU711.

    He considers another case of reflected light and colour in the mountains on CU203 (TPL250, fig. 649, 1509-1510, cf. fig. 647):

Of the colours of shadows

It often happens that the shadows in umbrous bodies are not the same as the colours of the lights: either the shadows are greenish or the lights reddish even though the body is of the same colour.

This happens because the light looks above the object from the east and illuminates the object with the colour of its splendour and from the west there is some other object illuminated by the same light which is of another colour. Which first object where it bounces with its reflected rays towards the east and percusses on the side of the first object facing it and it takes its rays from it and they remain firmly together with their colour and splendour.

I have many times looked at a white object and red lights and bluish shadows. And this occurs in the mountains of snow where the sun is in the west and the horizon shows itself aflame.

A more complex example of reflected light, shade and colour in the mountains occurs on

    CU793 (TPL654i, 1508-1510):

On lights and the shades and colours of these.

No body will ever show itself entirely in its natural colour.

That which is proposed can happen through two different causes of which the first occurs through the interposition of the medium which is included between the object and the eye. The second is when the things which illuminate the said body retain in themselves the quality of some colour.

That part of of a body would show itself of its natural colour which was illuminated by a luminous body without colour and if one did not see any other object than the aforesaid light in such an illuminated object. That this can never happen can be seen unless it were a deep blue colour positioned towards the sky on a very high mountain such that in this place one cannot see another object and the sun, in setting, is covered by low clouds and such that the cloth is of the colour of the air.

But in this case I contradict myself because the red also grows in beauty when the sun, which illuminates it, reddens in the west along with the clouds which are interposed between it. Although in this case one could again accept it for true because if this redness illuminated by the reddening light shows more beauty than elsewhere, it is a sign that the lights of colours other than red take on its natural beauty.

    On CU166 (TPL762, 1508-1510) he cites another case involving objects in the countryside:

On the sites of lights and shadows of things seen in the countryside.

When the eye sees all the parts of the bodies seen by the sun it will see all the bodies without shadow. This is proved by the 9th which states: the surface of every opaque body participates in the colour of its object. Therefore, the sun being the object of all those parts of the surfaces of bodies which see it, these parts of the surfaces will participate in the brightness of the sun which illuminates it. Look at these bodies and /you will see/ that it is impossible that one can see another part of such bodies other than that which is seen by the sun. Therefore you will see neither primitive nor derived shade on any of the aforesaid bodies.

    The reflected light of green meadows which Alberti had mentioned12, interest Leonardo also. On CA305va (c.1508), for instance, he notes (fig. 646): "If ab is green then by reflection nb is also green," and on Mad II 127v (1503-1504; CU225b-225b-226a, TPL767, 1508-1510), he explores the consequences of this phenomenon for his painting practice:

Of the consistency of shadows accompanied by their lights.

In this part you should have great respect for the things surrounding these bodies which you wish to draw by the first of the 4th which proves that the surface of every umbrous body participates in the colour of its object. But you should arrange artfully to make opposite the shadows of green bodies, green things such as green meadows and similar appropriate things such that the shadows participating in the colour of such an object do not come to generate and appear the shadow of a body other than green, because if you put illuminated red opposite a shadow which is green in itself, this shadow will redden and will make a colour of shade which will be most unbecoming and very different from the true green, and that which is said of such a colour is intended for all the others.

    As his studies of Nature continue he becomes more aware of the natural variety of colours, as for instance, on BM114v (c.1510):

The trees of the countryside are of various kinds of green, because some blacken such as firs, pines, cypresses, laurels, box and the like; some tend to yellow as walnuts, pears, vines and young verdure. Some become yellowish with darkness such as chestnuts /and/ holm-oak. Some turn red in autumn as the service-tree, the pomegranate, vine and cherry and some are whitish such as the willow, olive, reeds and the like.

    This awareness leads to further advice concerning painting practice on CU979 (TPL920, 1508-1510):

How to compose the fundament of colours of plants in a painting.

The way of composing the fundaments of colours of plants which border on the air is to make them as you see them at night in little brightness, because you see them equally of a dark colour mixed with the brightness of the air and thus you will see their simple shape clearly, without the impediment of various colours of green, bright or dark.

    From 1508 onwards his interest in Nature focusses on the characteristics of individual plants and leaves. On G28v (CU935, TPL872c, c.1510-1515), for example, he examines reflected light on dark leaves:

The lights of those leaves will be more the colour of the air which is mirrored in them which are of a darker colour. And this is caused because the brightness of the illuminated part /together/ with the dark...composes an azure colour and such brightness arises from the azure of the air which, is reflected in the polished surface of such leaves and augments the azure which the said brightness usually generates with dark things.

(figure)

Fig. 651: Reflected light and colour in leaves on G8v.

    A more detailed analysis of reflected light and colour in leaves follows on G3r-2v (1510-1515):

Even if the leaves of a polished surface are in large part of a same colour on top and their reverse side, it occurs that the part which is seen by the air participates in the colour of this air and it appears to participate in the colour of of this air the more, to the extent that the eye is closer and sees it in more foreshortened form and universally its shadows show themselves as darker on top than on the reverse through the comparison which is made by the lustres which border on such shade.

The reverse of the leaf, even if its colour be the same as that on the top will show itself as being of a more beautiful colour, which colour has a green participating in yellow and this happens when such a folio is interposed between/the eye and the light which illuminates it from the opposite side.

    He pursues this theme on G8v (TPL896, fig. 651, c. 1510-1515):

If the light comes from m and the eye is at n, this eye will see the colour of the leaves ab all participating in the colour of m, that is of the air. And lbc will be seen from behind as transparent with a beautiful green colour participating in yellow.

If m is the luminous body illuminating the leaf s, all the eyes that see the reverse of this leaf will see it as a beautiful green since it is transparent.

    Other examples of reflected light, shade and colour with respect to verdure and landscape have been cited elsewhere (eg. CU782, TPL779 see above p. ; CU936, TPL875; CU980, TPL905; CU961, TPL911 see below pp. ).

Book Six - 6. Yellow, Azure and Green

    His studies of Nature also lead him to study mixtures of colours produced by smoke from chimneys, as he records on CU179 (TPL205, 1505-1510):

Of the changes of transparent colours thrown or mixed on various colours with their different veilings.

When a transparent body is over another colour and varied from it, it composes a mixed colour different than any of the simple ones composing it. This is seen in the smoke issuing from chimneys which, when it meets the black of this chimney, becomes azure and when it rises and meets the azure of the air, it appears grey (berrettini) or reddish and likewise purple (paonazzo) placed on azure makes itself a violet colour and when the azure is set on yellow it will make green and crocus on white makes yellow and white on darkness makes a darkness that is that much more beautiful to the extent that bright and dark are more excellent.

    In this list of examples the combination of azure and yellow to produce green is mentioned in passing. For our purposes this example is of particular interest because it later becomes one of Leonardo's basic demonstrations to show that the surfaces of opaque bodies are tinged by the colours of surrounding objects. On CU790 (fig. 652, TPL701, 1508-1510), for example, he cites it in a passage headed:

(figure)

Figs. 651-654: Experiments with coloured shadows on CU790, CU168, and CU169.

    On the colours of the species of objects which tinge the surfaces of opaque bodies.

Many are the times that the surfaces of opaque bodies in being tinged by the colours of their objects receive colours which are not in these objects.

This is proved. Let cd be the opaque body and let ab be its object which we shall take as being of a yellow colour and the opaque body azure. I say that the entire part of the surface dnc of such an opaque body which is in itself azure will demonstrate itself to be green. And it would do the same if the opaque body were yellow and the object azure. And this occurs because various colours when they are mixed transmute themselves into a third participating of both one and the other and for this /reason/ yellow mixed with azure produces green, which green is a compound of its components which is comprehended clearly by the speculative painter.

    In the above passage he refers to this phenomenon occurring many times. On CU168 (TPL166, 1508-1510), he claims that the phenomenon occurs in almost all cases (fig. 653):

Why the times are very rare when he reflections of colours are the colour of the body to which they attach themselves.

Very rare are the times that reflections are the proper colour of the body to which they are attached. And let the spherical body be dfge and let it be yellow and let the object the back of which reflects its colour, which is azure, be ubc. I say that the part of the spherical body that is percussed by such a reflection will tinge itself in a green colour bc being illuminated by the air and the sun.

    On CU201 (TPL214, 1505-1510) he again cites the example of blue and yellow combining to produce green. This time he refers to the phenomenon as happening with certainty:

On the surface of every umbrous body.

The surface of every opaque body will participate in the colour of its object. Umbrous bodies demonstrate this with certainty, as none of the aforesaid bodies shows its figure or colour if the medium interposed between the luminous body and the illuminated body is not illuminated. Let us therefore say that the opaque body is yellow and the luminous body is azure, I say that the illuminated part will be green, which green is composed of yellow and blue.

    When he next cites this case on CU169 (fig. 654, TPL615, 1508-1510), he refers to it occurring in all cases:

How no reflected colour is simple but is mixed with the species of other colours.

No colour which is reflected on the surface of another body tinges this surface with its proper colour but will be mixed with the concourse of the other reflected colours which rebound in the same place.

As is the case with the yellow colour a which reflects in part of the spherical body coe and in the same place the azure colour b is reflected. I say that through this mixed reflection of yellow and blue, that the percussion of its concourse will tinge the spherical body. If it is white in itself it will be of a green colour, because it has been proven that yellow and azure mixed together compose a most beautiful green.

    On CU196 (TPL248e, 1508-1510) he mentions the need to compare ordinary light with reflected colour:

On Colours

The light of fire tinges everything in yellow. But this does not appear to be correct unless it is compared with things illuminated by the air and this comparison can be seen close to the end of the day or indeed after sunrise and again where, in a dark room, an aperture shines on the object with daylight and again an aperture with candlelight and in such a place the differences will certainly be seen clearly and distinctly.

    Here he is alluding to camera obscura experiments he himself had made (see below p. ). In the paragraph that follows, he again cites the mixing of azure and yellow to produce green:

But without such a comparison their differences will never be known except in the colours which have more similarities but are recognized as white by bright yellow, blue by azure and how to mix together azure and yellow and these compose a beautiful green and if yellow is then mixed with this green it is made more beautiful.

    On W19151v (K/P118v(B), 1508-1510) he notes that this phenomenon of azure and yellow mixing to produce green can also be demonstrated using panes of coloured glass:

If the rays of the sun pass through two panes of glass which are in contact with one another, of which panes the one is azure and the other is yellow, the ray passing through this will not tinge either in azure or in yellow, but in a most beautiful green.

    By now the phenomenon intrigues him the more because he realizes that within the eye blues and yellow together do not produce an impression of green, which he interprets as evidence that images do not interfere at the aperture of the eye (see below p. ). The mixture of azure and yellow to produce green also becomes relevant for his plant studies, as on G28v (CU938, TPL873d, c. 1510):

Of lights of green foliage tending towards yellow.

But the leaves of verdure tending towards yellow in their reflection of the air do not have to make a lustre participating of azure, because everything which appears in a mirror participates in the colour of such a mirror. Therefore the azure of the air, reflecting in the yellow of the foliage appears green, because azure and yellow mixed together compose a most beautiful green. Hence the lustres of bright foliage tending towards the colour yellow will be a greenish-yellow.

    He pursues this theme on CU936 (TPL875, 1508-1510) in a passage headed:

The leaves of plants are commonly of a polished surface as a result of which they partly mirror the colour of the air, which air participates in white when mixed with thin and transparent clouds. The surfaces of these leaves when they are of a dark nature such as those of elms, when not dusty, will render their lustres in a colour participating in azure. And this occurs through the 7th of the 4th which shows: white mixed with dark composes azure.

And such leaves have their lustres that much more azure to the extent that the air which is mirrored in them is more pure and azure. But if such leaves are young, as in the tips of branches in the month of May then they will be green with a participation of yellow. And if their lustres are generated by the azure air which is mirrored in them, then their lustres will be green by the 3rd of this 4th which states: a yellow colour mixed with azure always generates a green colour.

    Which leads him to conclude:

The lustres of all leaves of dense surfaces participate in the colour of the air and to the extent that they are dark leaves, the more they will serve as mirrors and consequently such lustres will participate more in azure.

    Some five years later on CA45va (figs. 655-756, c.1515) he reconsiders the mixture of yellow and blue asking:

Why shadows are made by a luminous body, tinger or surrounder of shadows.

The shadow made by a luminous body and which is yellow sees the shadow as being azure, because there is the shadow of the body a, made on the pavement at b in which it is seen by the azure luminous body. And likewise the shadow made by the luminous body d which which is azure is yellow at the site uc being seen by the yellow luminous body and the background bc surrounding these shadows other than its natural colour, will be tinged by a colour mixed with yellow and blue because it is seen and illuminated by a yellow luminous body and an azure luminous body at the same time.

(figure)

Figs. 655-658: Demonstrations involving the mixture of yellow and blue to produce green. Figs. 655-657, CA45va; fig. 658, CA181ra.

Shadows of various colours depending on the lights seen by them.

    Immediately preceding this passage he notes: "That which makes shade does not see it because shadows are made by the luminous body tinging or surrounding these shadows." This idea he restates directly following the passage: "Shadows of various colours /vary/ depending on the lights seen by them. That light which makes shadow does not see it." On CA181ra (fig. 658, c.1516-1517) he returns once more to his demonstration how azure and yellow mix to produce green, now referring to it as the second proposition:

Painting

The surface of every body participates in the colour of the object.

The colours of illuminated objects impress themselves on the surface of one another in as many sites as are the varieties of the situations of such objects.

O is the illuminated azure object and it alone sees without other company the space bc of the white sphere abcdef and it tinges it with azure colour:...m is the azure object which illuminates the space ab in the company of the azure o and it tinges it in a green colour (by the 2nd of this which proves that azure and yellow make a most beautiful green, etc.).

    He goes on to relate this to his camera obscura experiments (see below p. ) which he intends to include in his book of painting:

And the remainder will be said in the book of painting and in this it will be proved that, making the species and colours of bodies illuminated by the sun enter through a small round hole in a dark place on a flat white wall white in itself etc. But everything will be upside down.

    These eleven demonstrations involving a mixture of azure and yellow to produce to produce green might seem more than sufficient to establish that "the surface of every opaque or umbrous body participates in the colour of its objects." But Leonardo, fascinated and almost obsessed with the phenomenon also uses a series of other demonstrations.

Book Six - 7. Walls

    Among these are a number of experiments involving walls and planes of different colours. On A112v (BN 2038 33v, TPL668e, 1492), for instance, he describes a spherical object positioned on a red plane opposite a green wall:

On shade and light.

Every part of the surface which surrounds bodies is transmuted in part into the colour of that which is positioned as its object.

Example.

If you place a spherical body in the middle of various objects, that is, which from one side is the light of the sun and from the opposite side there is a wall illumined by the sun which is green or another colour /and/ the plane where it is positioned is red. From the two transverse sides it is dark. You will see the natural colour of the said body participate in the colours which are its object. The luminous will be the most powerful. The second will be that of the illuminated wall; the third that of the shadow. There then remains a quantity which participates in the colour of its extremities.

(figure)

Fig. 659: Demonstration of mixture of lights on CU478.

    Some twelve years later, on Mad II 125r (cf. Mad II 26r below pp. ), he records another experience involving green shadows on a white wall on All Saints Day (2 November), 1504 at Piombio:

I have just seen the green shadows made by the cords, mast and lanteen yards on the side of a white wall, the sun going down in the west. And this occurred because the surface of this wall did not tinge itself with the colour of the sun, /but/ tinged itself with the colour of the sea that was its object.

    On CU478 (fig. 659, TPL467, 1508-1510) he derives a more complex play of colours on white walls:

Why towards evening the shadows of bodies generated on a white wall are azure.

The shadows of bodies generated by the redness of the sun near the horizon are always azure. And this arises through the 11th where it is stated: The surface of every body participates with the colour of its object. Therefore the whiteness of the wall being deprived of every colour, is tinged by the colour of its objects, which objects are in this case the sun and the sky, since the sun reddens towards evening and the sky demonstrates azure. And where there is shadow the sun does not see, by the 8th of shadows, which states: No luminous body ever sees the shadow figured by it. And where the sun does not see on such a wall, there it is seen by the sky.

(figure)

Fig. 660: Experiment with mixture of blue and red light, on W19151v (K/P 118v(B)).

Therefore by the said 11th of shade, derived shade will have percussion of an azure colour on the white wall and the background of this shadow seen by the redness of the sun will participate in a red colour.

    He considers the play of blue and red light on a wall at greater length on W19151v (fig. 660, K/P 118v(B), 1508-1510) in a passage headed:

On the colours of simple derived shadows.

The colours of derived shadows always participate in the colours of the bodies which light them up. To prove this, let an opaque body be interposed between the wall sctd and the lights, de, blue and ab, red. I say that de, the blue light, sees all the wall sctd except op which the shadow of the opaque body qr occupies, as the straight lines dqo and erp show. And the same happens to the light ab which sees the whole wall sctd except the place occupied by the shadow /of/ qr as the lines aqn and brm show. Therefore one concludes that the shadow nb sees the blue light de and not being able to see the red light ab, nb remains a blue shadow in a field of red, mixed with blue, because the field sctd sees both lights, but in the shadows it sees only one light. For this reason such a shadow is a half-shadow because if such a shadow was seen by no light at all it would be a maximal shadow, etc. /i.e. darkness/. But the shadow op does not see the blue light because the body qr interposes and prevents it there. Only the red light ab is seen there and this tinges it with the colour red and so this rosy shadow remains in a field of mixed blue and red.

The shadow of qr on op being caused by the blue light de is red and the shadow of this cast by the red light ab is blue at nm. Therefore we shall say that the blue light in this case makes a red derived shadow from the opaque body qr and the red light makes the same opaque body cast a blue derived shadow. But the primary shadow is not of this colour but it is a mixture of red and blue.

    On CU202 (TPL239, 1505-1510), he pursues this theme, beginning with a general claim:

Of the colour of the shadows of some body.

The colour of some body will never be the true or proper shadow if the object which it shades is not of the colour of the body shaded by it.

This he then demonstrates with a case where azure light is reflected from a green wall:

Let us say, for example, that I have a house the walls of which are green. I say that if azure be seen in such a place which is illuminated by the brightness of the azure of the air, that such an illuminated wall will be a most beautiful azure and the shadow will be ugly and not the true shadow of such beauty of azure, because it is corrupted by the green which is reflected in it and it would be worse if such a wall were tan /coloured/.

Book Six - 8. Light and Pyramids

    These demonstrations of reflected light and colour involving walls of different colours become a starting point for his analogies between mixing coloured lights and mixing pigment colours (see above p. on CU469 [TPL433, 1508-1510]):

(figure)

Fig. 661: Concerning the mixing of colours on CU469.

Whether the surface of every opaque object participates in the colour of its object.

    You need to understand that if a white object is put between two walls of which one is white and the other is black, that you will find such a proportion between the umbrous and the luminous part of this body as is that of the said walls and if the wall is of an azure colour it would do the same. Whence, having to paint it, you will do as follows:

Take the black to shade the azure object which is similar to the black or indeed the shadow of the wall which you assume that it has to reflect in your object and you will do thus, wishing to do it with a certain and true science, and you will accustom yourself to doing it in this way.

When you make your walls of whatever colour you want, take a little spoon, a little larger than that of a tea spoon and larger or smaller depending on large or small works in which you have to exercise such operations.

And such a spoon will have its extremities of equal height and with this you will measure the degrees of quantities of colours which you adopt in your mixtures. As would be if in the said walls that you have made the first shade were of 3 degrees of darkness and one degree of brightness, that is 3 level spoons as one does in the measures of grain and if these three spoons were of simple black and one spoon of white, you would have made a composition of a certain quantity without a doubt.

    Immediately following he considers a more complex situation (fig. 661):

Now you have made one wall white and one dark and you have to put an azure object amongst them which object you wish to have the true shades and lights as are fitting for such an azure /object/.

Therefore you put this azure /object/ to one side which you wish to remain without shade and put black on the side. Then take 3 spoons of black and mix them with one spoon of luminous azure and put this with the darkest shade.

Having done this see whether the object is spherical or columnar or square or how it is. And if it is spherical, draw a line from the extremity of the dark wall to the centre of this spherical object and where these lines intersect on the surface of such an object, there the greatest shades terminate at equal angles.

Then it begins to become brighter again as would be at mo which leaves as much of the dark to the extent that this participates in the superior wall ad, which colour will mix with the first shadow of ab with the same distinctness.

    He pursues this analogy between mixing coloured lights and mixing pigment colours on CU869 (TPL756, 1508-1510):

Rule for taking the true brightnesses of lights on the sides of the aforesaid body.

Let there be taken a colour similar to the colour of the body which you wish to imitate and let the colour of the principal light be taken, with which you wish to illuminate this body.

Then if you find that the above mentioned angle is twice the lesser angle then you take a part of the natural colour of the body which you wish to imitate and give it two parts of light which you wish it to receive and you will have placed the light double the lesser light.

Then, in order to make half the light, take only one part of this natural colour of the aforesaid body and add to it only one part of the said light and thus you will have made on a same colour a light which is double the other, because on a quantity of this colour is given a similar quantity of light and to the other quantity two quantities of such a light are given.

And if you wish to measure these quantities of colour exactly, you will have a small spoon with which you can take your equal quantities and when you have taken your colour with it, you level it with a little ruler, as one does with the measures of grain when this grain is sold.

Book Six. 9. Further Demonstrations

    In the meantime, Leonardo has been recording further demonstrations to show the nature of reflected light, shade and colour. On A93v (BN 2038 13v, fig. 216, CU756, TPL728, 1492), for instance, he describes how:

    Every shadow made by an umbrous body less than the original light will send its derived shade tinged with the colour of its origin:

Let the origin of the shade ef be n and let the origin be tinged in its colour by h and let it be o and this is similarly tinged in its colour and likewise the colour of vh is tinged in the colour of p because it originates from it and the shadow of the triangle zky is tinged in the colour of Q because it is derived from it. To the extent that cd enters into ad to that extent is nrs darker than m and all the rest of the background /is/ without shade. Fg is the first degree of light because there all the window ad illuminates and likewise in the umbrous body me is of similar brightness. Aky is a triangle which contains in itself the first degree of shade because in this triangle the light ad does not reach; xh is the 2nd degree of shade because it is only illuminated by a 1/3 of the window, that is, cd; he is the third degree of shade because it sees 2/3 of the window bd. Ef is the ultimate degree of shade because the ultimate degree of light from the window illuminates the place from f.

    A few folios later, on A98v (BN 2038 18v, CU284, TPL146, c.1492), he cites another demonstration involving firelight:

How one should represent a night /scene/.

That thing which is entirely deprived of light is all darkness. The night being in a similar condition and you wish to represent a story, you will make that there is a large fire and that that thing which is closer to that fire is tinged more in its colour because that thing which is closer to its object, participates more in its nature. And making it a red colour, you will make all the things illuminated by it also tend to redden and that those which are further from the said fire can be tinged by the black colour of the night. The figures that are made in front of the fire appear dark in the brightness of this fire because that part of this thing which you see is tinged by the darkness of the night and not by the brightness of the fire and those that are to the side are half dark and half reddening and those which can be seen beyond the boundaries of the flames will be entirely illuminated by the reddening light a against a black background.

(figure)

Fig. 662: Mixing of light and colour on CU766.

    On CU766 (TPL702, 1508-1510) he opens with a general claim under the heading:

On false colour of shadows in opaque bodies.

When an opaque body makes its shadows on the surface of another opaque body, which is illuminated by two various luminous sources, then such a shadow will not show itself to be of the same opaque body, but something else.

    This he illustrates with a demonstration using a candle flame (fig. 662):

This is proved. Let nde be an opaque body and let it be white...and let it be illuminated by the air ab and by the fire cq. Then let there be interposed between the fire and the opaque body, op the shadow of which will cut the surface at dn. Now at this dn the redness of the fire is no longer illuminated but /rather/ the azure of the air. Whence dn participates in azure and nf sees the fire. Therefore the azure shade terminates below the redness of the fire above such an opaque body and above it terminates with the colour of violet, that is, which at de is illuminated by a mixture composed of the azure of the air ab and of the redness of the fire qc which is almost the colour of fire.

And thus we have proved that such a shadow is false, that it is neither the shadow of white nor that of red which surrounds it.

    On CU741 (TPL438b, 1508) he offers a further demonstration involving sunlight:

But a colour will never be seen simply. This is proved by the ninth which states: the surface of every body participates in the colour of its object, even if it is the surface of a transparent body as is air, water and the like. Since the air takes the light of the sun and the darkness from the privation of this sun, therefore it is tinged in as many various colours as are those which are interposed between the eye and them. Since the air does not have in itself a colour any more than the water does, but the humidity which mixes itself with this upper middle region is that which makes it expand and expanding, the solar rays which percuss it, illuminate, it and the air below this said middle region remains tenebrous and since light and dark compose an azure colour it is this azure in which the air tinges itself with this much more or less darkness to the extent that the air is mixed with less or greater humidity.

    Another demonstration on CU796 (TPL633, 1508-1510) is headed simply (fig. 215):

What part of the surface of an umbrous body is it where the colours of objects mix.

Throughout the entire part of the surface of an umbrous body which is seen by the colours of more objects the species of the aforesaid colours will be mixed.

Hence the part abcd of the umbrous body is mixed with light and shade because in such a place it is seen by the light nm and by the darkness op.

    On W19152r (K/P 188r, 1508-1510) he discusses the principle again, this time in connection with his explanation of how rays enter the eye (see below p. ):

Nature of the rays which are composed of the species of bodies and their intersections.

The rectilinearity of the rays which carry through the air the shape and colour of bodies whence they part do not tinge the air of themselves, nor can they tinge one another in the contact of their intersection. But they only tinge the place where they lose their being, because such a place sees and is seen by the origin of these rays and no other thing which surrounds this origin can be seen by that place where that ray being intersected remains destroyed, there leaving the prey carried by it.

And this is proved by the 4th of the colours of bodies where it is stated: the surface of every opaque body participates in the colour of its object. Hence it is concluded that the place which sees and is seen by the origin of such species through the ray carrying the species is tinged by the colour of that object.

    In a demonstration on CU711 (TPL554, fig. 650, cf. fig. 648, 1508-1510) he compares the effects of primitive and derived shade:

Which is darker primitive shade or derived shade?

Primitive shade is always darker than derived shade if it is not corrupted by a reflected light which makes itself a background of the percussion of this derived shade. Let bcde be the luminous body. Let a be the light which causes the primitive shade bec and produces the derived /shade/ bechi. I say that if fh and ig is not reflected, which reflects and corrupts the primitive shade at be with fh and at ce with ig, that this primitive shade will remain darker than the percussion of the derived shade, the one being shade and the other made on a surface of equal darkness of colour or of equal brightness.

    He also examines how the rarity or density of an object affects the reflected colours of objects on CU754 (TPL631, 1508-1510):

Of shadows and which are those primitive ones which will be darker on a body.

Primitive shadows will be darker which are generated on the surfaces of bodies which are denser and conversely, will be brighter on the surfaces of bodies which are rarer. This is clear because the species of those objects which tinge the bodies positioned opposite with their colours will impress themselves with greater vigour which are found on a denser or more polished surface on these bodies.

    This he again demonstrates with a concrete example (fig. 215):

This is proved. And let the dense body be rs interposed between the luminous object nm and the umbrous body op. By the seventh of the ninth which states: the surface of every body participates in the colour of its object, we shall therefore state that the part bar of this body is illuminated, because its object nm is luminous. And in like fashion we shall state that the part opposite dcs is umbrous because its object is dark. And thus our proposition is concluded.

    As will be shown, this interest in reflective properties of rare and dense objects relates to his studies of the moon (see below p. ). On CU755 (TPL632, 1508-1510) he uses the same diagram as that in CU754 (fig. 215) to demonstrate another aspect of the phenomenon.

Which part of the surface of a body is better impressed with the colour of its object.

That part of the surface of a dense body participates more intensely in the colour of its /object/ which is less seen by other objects and other colours.

Let us therefore use the same figure for our proposition and let it be that the surface of the above mentioned body arb is not seen by the darkness op, /then/ it will be entirely deprived of shade and similarly if the surface csd is not seen by the luminous body nm, it will be entirely deprived of light.

    Four further demonstrations on CU677 (TPL694b); CU722 (TPL782, 1508-1510) (see above pp. ) and CU720 (TPL781); CU724 (TPL698, 1508-1510) (see below pp. ), have been cited elsewhere.

Book Six - 10. Precepts

    One result of this volley of demonstrations is a series of pithy precepts and rules. Among the earliest of these is a note on Forst III 74v (1493): "The surface of any umbrous body will participate in the colour of bodies opposite it," which he restates on the adjacent folio, Forst III 75r (1493): "The surface of any opaque body participates in and is tinged by the colour of the bodies positioned opposite it."

On CU794 (TPL655, 1508-1510), he pursues this theme:

On the shadow and lights in objects.

The surface of every umbrous body participates in the colour of its object.

The painter must take great care in situating his things between objects of various powers of light and various illuminated colours, since every body surrounded by these never shows itself fully in its true colour.

    On CU860 (TPL694f, 1508-1510) he lists nine propositions, three of which deal with reflected light, shade and colour:

1. Part of the surface of every body will participate in as many various colours as are those which stand as its object.

7. All illuminated things participate in the colour of the illuminating object.

8. Objects in shade retain the colour of the thing that obscures them.

    These develop into four propositions on CU172 (TPL168, 1505-1510) under the heading:

Of reflections

1st The surfaces of bodies participate more int he colours of those objects which reflect their image in them amidst more equal angles.

2nd On the colours of objects which reflect their images on the surfaces of bodies positioned opposite under equal angles, that will be more powerful which will have its reflected ray of a shorter length.

3rd Among the colours of objects reflected at equal angles and at an equal distance on the surface of bodies positioned opposite, that will be more powerful which will be a brighter colour.

4th That object reflects its colour in the body positioned opposite which does not have around it other colours than of its own species.

    A few of his pithy statements concerning reflected light, shade and colour relate directly to his camera obscura studies. On CA230rb (1505-1508), for instance, he notes: "The surface of every body participates in the colour of its object" (see below p. ). On CA37ra (1508-1510) he refers to a second proposition: "The surface of every opaque body participates in the colour of its object" (see below p. ) and on CA195va (c.1510) he claims: "This is proved by the fourth of this which states: the surface of every opaque body participates in the colour of its object" (see below p. ).He mentions the phenomenon of reflected light, shade and colour once more on W19076r (K/P 167r, c. 1513): "the boundaries of derived shade are surrounded by the colours of the illuminated objects surrounding the luminous body, the cause of this shade," and again in summary form on the same folio: "shade always participates in the colour of its object."

    This he restates on E32v (1513-1514) under the heading: "On shade: The surface of every opaque body participates in the colour of its object." Another restatement occurs on G37r (1510-1515), this time under the heading: "On painting: The colour of the illuminated object participates in the colour of the object illuminating it," which idea he develops in a final passage on G53v (1510-1515):

The surface of every body participates in the colour which illuminates it and in the colour of the air that is interposed between the eye and this body, that is, of the colour of the transparent medium which is interposed between the object and the eye and among colours of a same quality the second will be the same colour as the first and this arises through the multiplication of the colour of the medium interposed between the object and the eye.

Book Six - 11. General Statements

    In addition to the above demonstrations and precepts he makes a series of general statements on the nature of reflected light, shade and colour, as for example, the passage on CU768 (TPL815, 1508-1510) entitled:

Precept

Bodies illumined by various qualities of colours of lights do not have the illuminated parts of their surfaces corresponding to the colours of their shaded parts.

Very rare are the times when the colours of the surfaces of opaque bodies have the necessary colours of the shadows corresponding to the colours of their illuminated parts.

That which is proposed arises because objects which make the shadows on such bodies are not of the natural colour of these bodies, nor of the same natural colour of the illuminator of this body.

    He mentions reflected colours again in passing on CU265 (TPL119a, 1505-1510):

Reflection

But that reflection will be of a more confused colour which is generated by various colours of objects.

    On CA207ra (1508-1510) he makes another comment in passing concerning reflected light:

    "Every reflective body which moves in front of another reflective body which is immobile will infuse itself on this," a theme which he pursues on CU162 (TPL171, 1508-1510):

On the colours of reflections.

All reflected colours are of less luminosity than direct light and the incident light has such a proportion with the reflected light as the luminosities have amongst themselves with respect to their causes.

    On CU758 (TPL608, 1508-1510) he makes another brief note under the heading:

On the boundaries that surround derived shadows in their percussions.

The boundaries of simple derived shadows in their percussions are always surrounded by the colours of illuminated things which send their rays from the same side of the luminous body, which illuminates the umbrous body generator of this shade.

    He pursues this theme of coloured on CU788 (TPL609, 1508-1510):

How every umbrous body generates as many shadows as are the luminous parts that surround it.

Umbrous bodies generate as many sorts of shade around their bases and are of as many colours as are the illuminated colours opposite which surround it, but are that much more powerful the one than the other to the extent that the luminous source opposite it is of greater splendour. And this is taught by various lights positioned around a given umbrous body.

    Which leads to a more radical claim on CU714 (TPL579a, 1508-1510):

Nature or condition of shade.

No shade is without reflection, which reflection increases or weakens it. And that reflection increases it which is born of an object darker than this shadow. And this other reflection weakens it, which is born of an object brighter than this shadow.

    On E17r (1513-1514) there is a further summary note on reflected shade with respect to painting practice:

Painting

If your drawing note how among the shadows there are shadows of an imperceptible darkness and shape and this is proved by the third which states: the globulous surfaces are of as many various darknesses and brightnesses as are the varieties of darknesses and brightnesses which stand as its object.

(chart)

Book Six - 13. Conclusions

    When Leonardo becomes enthusiastic about an idea he repeats it in almost all possible combinations. His "all in all..." passages were a first example of this. His notion that "colour participates," just examined, is another case. He devotes no less than fifty passages to this phrase (see Chart 14) and, in addition twenty-eight others where the concept is expressed more generally (see Chart 15).

    As of 1503-1504 he refers to this idea as a proposition, suggesting that it is to form part of a more coherent treatise. In the course of the next six years he refers to such a proposition no less than thirteen times. What is noteworthy, however, is how the number keeps changing: what begins as a first proposition, becomes the fourth, seventh and ultimately the eleventh proposition (see Chart 16). This re-shuffling gives some impression of the energy with which Leonardo reformulates and reorganizes his ideas. Closely linked to the above passages on reflected light, shade and colour is a further series which involve the distance factor. These he intended to present in his seventh book on light and shade.

Book Seven: Refleced Colour and Distance

will treat of the various distances that may exist between the spot where the reflected rays fall and that where they originate, and the various shades of colour which they will acquire in falling on opaque bodies. (CA250ra)

    The above outline of book seven continues the themes of book six: reflected light, shade and colour, with the addition of a distance factor. Concerning this he again has demonstrations and general statements. Further examples overlap with his studies of perspective of colour and diminution of form. Each of these will be considered in turn.

Book Seven - 1. Demonstrations

    Amongst the earliest of these is a passage on CU858 (TPL820, 1505-1510) under the heading (figs. 381-38):

Of reflected light

To the extent that the illuminated object is less luminous than that which illuminates it, its reflected part will be that much less luminous than the illuminated part.

That object will be more illuminated which is closer to the illuminating source.

To the extent that bc enters ba to that extent will it be more illuminated in ad than in dc.

    He considers the distance factor again on F1v (fig. 495, 1508):

(figure)

Figs. 663-666: Effects of distance on reflected light and colour. Fig. 663, H66/187/r; fig. 664, CU720; ;fig. 665, CU724; fig. 666, CU161.

The surface of every opaque body participates in the colour of its object.

That part of the surface of opaque bodies participates more in the colour of its object which is closer.

LEt ab /aob/ be an opaque body and cd a luminous object and ef an umbrous object. I say that the middle of this opaque body o participates equally in the one and the other object and the part ao is more luminous than the part ob and the closer it is to the luminous body the more it will be illuminated and hence the darker part of this body which which is closer to the umbrous body will make itself darker.

    He expresses a similar idea on CU720 (fig. 664, TPL781, 1508-1510) under the heading:

Where the reflection must be darker

If the light s illuminates the body rhp, primitive light brighter above towards the light than below where this body is positioned on the flat /ground/ by the 4th of this which states: the surface of every body participates in the colour of its object. Therefore the derived shade which stamps itself on the pavement at the site mp rebounds on the side of the umbrous body op and the derived light which tinges such a shade, that is mn, rebounds to or. And this is the cause by such umbrous bodies never have a luminous reflection at boundaries which the umbrous body has with its pavement.

    The accompanying diagram (fig. 664) can be seen as a development of a sketch on H66/18/r (fig. 663, 1494) where he had noted: "The derived shade that borders with the primitive will be darker than this primitive." He develops this theme on CU724 (TPL698, 1508-1510), beginning a general claim:

Of the various darknesses of shadows of bodies imitated in pictures.

The surface of every opaque body participates in the colour of its object and more or less to the extent that the body is closer or further.

    Part one of his demonstration follows (fig. 665):

The first part of this is proved. And let abc be the surface of the opaque body which we shall assume is of a white surface and that the object rs is black and the object nm is also black. And by the 9th of this which proves that every body fills the surrounding air with the species of its colour, and with the similitude of its coloured body.

Therefore, rs, a black body will fill the air which stands in front of it with dark colour which will terminate in gab, part of the opaque object abc, which part will tinge itself in the colour of its object rs and the white colour of the other object nm will whiten the entire part abc of the opaque body.

Therefore, in the opaque body one will find all of ag in a simple participation of /the/ black /object/ urs and in bc in simple white and in ab which is seen by both the white and the black object, there will be a colour composed of white and black, that is, a surface of a mixed colour.

    Part two of the demonstration considers the distance factor:

For the second part of the said proposition it will be much darker at a than at b because a is closer to the black object rs than is b and this is shown by the definition of the circle in geometry as is drawn /alongside, our fig. 665/. And besides this at the angle b, being the smallest angle that there is, as is proved in geometry, at the angle of contingence b, one cannot see other than the extremity of the body rs at the point r and other than this there is joined to b the brightness of the white object nm which, even if it were black, being further away from ub than a from rs, as was proved, b still would never be as dark as is that of a.

    A slightly more complex demonstration, involving two spheres (fig. 661) follows on CU161 (TPL164, 1505-1510) in a passage entitled:

On doubled and trebled reflections.

Doubled reflections have more power than simple reflections and the shadows that interpose themselves between the incident light and these reflections are of little darkness.

Let a be the light source, BC a wall which receives light from this luminous body. Dre and nso are the parts of the two spherical bodies illuminated by direct light. Npm and dhe are the parts of these bodies illuminated by reflections. The reflection dhe is the simple reflection; npm is the doubled reflection. And the simple reflection is said to be that which is only seen by one illuminating body and the doubled /reflection/ is seen by two illuminated bodies. And the simple /reflection/, dhe, is made by the illuminated body Bg; the doubled /reflection/, npm, is composed of the illuminated body BK and the illuminated body dre. And its shadow which is interposed between the incident light n and the reflected light np is of little darkness.

(figure)

Figs. 667-668: Concerning the distance factor in reflected light and colour on CU717 and CU670.

    He again considers the distance factor in reflected light on CU717 (TPL751, 1508-1510):

On the proportion that the luminous parts of bodies have with their reflections.

The illuminated part of incident light will have that proportion to that which is illuminated by reflected light, as is the /proportion of the/ incident light to this reflected light.

    A demonstration follows (fig. 667):

This is proved. Let ab be the incident light which illuminates the sphere cd at cnd and passes with its rays to the object ef and from there it is reflected at cmd. I say that if the light ab has two degrees of power and ef has one, which is half of two, that the reflected light cmd will be one half the light cnd.

    On CU670 (fig. 668, TPL635, 1508-1510) he presents a further demonstration in purely geometrical terms, asking:

Which part of an illuminated surface will be of greater brightness?

That part of an illuminated body will be more luminous which is closer to the object which illuminates it.

This is proved. And let the illuminated part of the object be ucx and let the object which illuminates it be ab. I say that the point c is more illuminated than any other part of such a body because the luminous angle acb which percusses it is greater than any other angle that can be generated on such a surface.

    He pursues the theme on CU818 (TPL786, 1508-1510):

Of the eye which stands in the bright /air/ and looks at a dark place.

In the darkness no second colour is of the same brightness as the first even if they are similar. This is proved by the 4th of this where it is stated: the surface of this body will be more tinged by the transparent medium interposed between the eye and this body, of which the interposed medium is of greater size.

Therefore it remains concluded that a second colour put in a dark transparent medium will have more darkness interposed between it and the eye than the first colour, which which is found closer to the same eye. And such will be the proportion of darkness to darkness of these colours as there is from quantity to quantity of the dark mediums by which they are tinged.

 

Book Seven - 2. General Statements

    In addition to such demonstrations he also makes a number of general statements concerning reflected colours and distance as, for instance, on CU200 (TPL192, 1505-1510) under the heading:

Of the colour of the shade of a colour.

The colour of the shade of some colour always participates in the colour of its object and that much more or less to the extent that this object is closer or further from this shade and to the extent that it is more or less luminous.

On CU171 (TPL216b, 1505-1510) he restates this idea in terms of a question and answer:

What part of a body will tinge itself more in the colour of its object?

The surface of every body will participate more intensely in the colour of that object which is closer.

This occurs because the nearby object occupies a greater multitude of variety of species which, coming to the surface of a body will corrupt the surface of such an object, which it would not do if such a colour were remote and occupying such species, this colour will show its nature more integrally in this opaque body.

    He considers the distance factor again on CU811 (TPL629, 1508-1510), under the heading:

Of white things remote from the eye.

The white thing remote from the eye, the more it is removed the more it loses its whiteness and the more so, to the extent that the sun illuminates it, because it participates in the colour of the sun, mixed with the colour of the air which is interposed between the eye and the white object. Which air, if the sun is in the east, shows itself a turbid red through the vapours, which rise around it. But if the eye turns itself to the east it will only see the shadows of white participating in an azure colour.

    While Leonardo is interested in light and shade for its own sake, as a problem of physics, he is also concerned with its applications to painting. In the period after 1505 this artistic motive becomes explicit in passages such as CU175 (TPL170b, 1505-1510):

On reflection

That colour which is closer to the reflection will tinge this reflection more with itself and conversely.

Hence, /O/ painter, make sure that in the reflections of the faces of figures you use the colours of parts of clothes which are close to the parts of the skin which are nearest. But do not separate them too markedly, if it is not necessary.

    He pursues this artistic interest in reflected light and shade on CU163 (TPL159, 1505-1510):

On the reflections of lights that surround shadows.

The reflections of illuminated parts rebounding in the shadow positioned opposite alleviate their darkness more or less depending on whether they are more or less close or more or less bright. This is taken into consideration by many, and there are many others who avoid it and each /party/ laughs at the other.

But you, in order to avoid the calumnies of both (one and the other, put into operation the one and the other when they are necessary. But make /sure/ that their causes are known/ cf. G11v, that is, that these clearly see the cause of reflections and their colours and equally clearly the cause of the things that do not reflect. And doing this you will neither be entirely scorned nor praised by their various judgments, which persons if they are not entirely ignorant, it will be necessary that they praise you on the whole both one party and the other.

    Leonardo's artistic interest in reflected light, shade and colour also explains why he includes some of his general statements on this topic (eg. G37r, CA181ra) under the heading of "painting" and, in addition, accounts for certain links with his studies of perspective.

Book Seven - 3. Links with Perspective

    Leonardo's studies of perspective of colour and diminution of form have been analysed elsewhere (see Vol. 1, Part 3.1-2). Here, a few examples will serve to draw attention to links between these studies and his interest in reflected light, shade and colour. On CU234 (TPL241b, 1505-1510), for instance, he discusses explicitly a connection between the principle that colour participates and the perspective of colours:

Perspective of Colours

The first colours must be simple and the degrees of their diminution must correspond with the degrees of distance, that is, that the sizes of things participate more in the nature of a point to the extent that they are closer to it i.e. the vanishing point/ and that colours have to participate in the colour of the horizon to the extent that they are closer to it.

    Intimately connected with his colour perspective is an azure rule concerning distant objects which, in turn, relates to his principle that colour participates as on CU814, (TPL630, 1508-1510):

On the shadows of remote things and their colours.

The shadows of remote things participate that much more in azure colour to the extent that they are darker and more remote. And this occurs through the interposition of the brightness of the air which puts itself between the darkness of the umbrous bodies interposed between the sun and the eye which sees it. But if the eye turns opposite the sun, it will not see similar azure.

    This connection between reflected colour and his azure rule applies to verdure, as on G15r (CU980, TPL905, 1510-1515):

On the shadow of verdure.

The shadow of verdure always participates in azure and likewise every shadow of every other thing and it takes more to the extent that it is more distant from the eye and less to the extent that it is closer.

    The same connection also applies to rocks, as on W12414 (c.1511):

the rocks of this mountain will naturally retain a colour tending towards azure and the air which is interposed makes it even more azure and maximally in their shadows...

    And it applies equally to landscapes as on CU961 (TPL911, 1510-1515):

On landscapes.

The umbrous parts of distant landscapes will participate more in an azure colour than the illuminated parts. This is proved by the definition of azure in which the air is tinged, deprived of colour, which /air/ if it did not have darkness over it would remain white because of itself the azure of the air is composed of light and darkness.

    As early as 1492, on A100v (GBN 2038 20v), Leonardo also makes explicit further links between reflected colour and the disappearance of shadow at a distance:

How the shades are lost over a long distance.

Shades are lost over a long distance because the large quantity of the luminous air found between the eye and the thing seen tinges its shadows with this thing in its colour.

    He develops this idea on CU792 (TPL705, 1508-1510) in a passage entitled:

On the accidents of surfaces of bodies.

The surface of every opaque body participates in the colour of its object, which colour will be that much more evident on this surface, to the extent that the surface of such a body is whiter and to the extent that such a colour is closer.

    This leads to a clear connection between his principle of reflected colour and disappearance of form perspective, as on CU870 (TPL690, 1508-1510):

When objects close to one another and small are seen at a long distance, such that the distinctness of their shapes is lost, then a mixture of their species is caused which will participate more in that colour which is invested by a greater amount of such objects.

    Which connection he restates more subtly in passages such as E17r (CU448, TPL472, 1513-1514):

The surface of every opaque body participates in the colour of its object. But with that much greater or lesser impression to the extent that this object is closer or more remote or of greater or less power.

    In the end his principles concerning reflected light, shade and colour become so closely entwined with his perspectival studies that numerous passages apply equally to both domains, which again brings into focus a basic feature of his thought, where one thing literally leads to another.

 

Book Eight: Movement of Shadows

Book Eight - Introduction

    In his list on CA277va (c.12513-1514), Leonardo indicates a number of other books, (ie. chapters in the modern sense), which he intends to write concerning light and shade (see Chart 10). Among these is a proposed chapter on the movement of shadows, for which he writes a series of preparatory notes. On CU658 (TPL582, 1508-1510), for instance, he outlines five basic situations which concern him:

On the motions of shadows.

The motions of shadows are of five natures of which let us say that the first is that which moves the derived shade together with its umbrous body and the light causing this shade remains immobile. And let us say that the second is that of which the shadow and the light moves but the umbrous body is immobile and the third will be that of which the umbrous and the luminous body moves but the luminous body with more slowness than the umbrous body. In the fourth motion of this shade, the luminous body moves with more speed than the umbrous body and in the fifth the motions of the umbrous and luminous body are equal among one another.

And of this will be treated distinctly in its place.

    On CU646 (TPL686, 1508-1510) he outlines a further situation in which the eye moves while the umbrous body and light remain constant. Taken together these six situations provide a probable framework for his intended chapter. Each of them will be considered in turn.

(chart)

Chart 17. Six basic situations concerning movement of shadows based on CU658 (TPL582) and CU646 (TPL686).

(figure)

Figs. 669-691: Demonstrations of movement of shadows, where derived shade and umbrous body move, while light remains immobile on C3v, A110r and CU622

Book Eight - 1. Derived shade and umbrous body move while light constant

    In the early period he considers only the case in which a light source is constant while the umbrous body and derived shade moves, as, for instance, on C3v (fig. 669, 1490-1491):

If the body be moved slowly in front of the luminous body and the percussion of this object is far from this object, the motion of the derived shade will have such a proportion with the primitive shade as the space which is between the object and the light, has with that which there is between the object and the percussion of the shade, in /such a/ way that moving the object slowly the shadow is speedy.

    He continues this theme on A110r (BN 2038 30r, fig. 670, CU703, TPL611, 1492) under the heading:

Of the shade made by a body situated between 2 equal lights.

That body which is found positioned between 2 equal lights will move by itself 2 shadows which direct themselves along a line from the 2 lights and if you remove said body and you bring it closer to the one of the lights, its shadow which will direct itself to the nearest light will be of lesser darkness than that which directs itself to the further light.

    He provides a further illustration of this situation on CA370ra (1497):

Of the shadow which moves with a man.

You will see forms and figures of men or animals which follow these animals and men wherever they flee and such is the motion of the one as of the other. But it appears a wondrous thing from the various sizes in which it transmutes itself.

(figure)

Figs. 672-673: Further cases where derived shade and umbrous body move while light remains immobile on CU659 and 660.

    Of the shadows of the sun and of mirroring in the water at the same time.

    Many times you will see one man becoming 3 (-) and all follow after him, /and/ often the most certain one abandons him.  Approximately a decade later he again considers this situation where the light source is constant while the derived shade and umbrous body move on CU659 (fig. 672, TPL575b, 1508-1510) in a passage headed:

Of the shade which moves with greater velocity than its umbrous body.

It is possible that derived shade is many times more speedy than its primitive shade.

This is proved and let a be the luminous body, let b be the umbrous body which moves from b to c along the line bd and at the same time the derived shade of the body f moves the entire space fe which space can receive the space bc in itself thousands of times.

    He considers the converse of this case on CU660 (fig. 673, TPL576, 1508-1510):

On the derived shade which is muchslower than primitive shade.

It is also possible that the derived shade is much slower than the primitive shade.

This is proved. And let it be that the umbrous body bc moves the whole space ce over the plane ne and that its derived shade be on the opposite plane de. I say that the primitive shade bc will move the whole space bd, while the derived shade does not depart from de.

    On E2v (1513-1514) he drafts a further passage on the problem under the heading:

Of shadow or its movement.

Of 2 umbrous bodies which are one behind the other between the window and the wall with some space interposed,...the shade.... The umbrous body which is close to the side of the wall will be mobile, if the umbrous body near the window is in a transverse motion to this window.

    To illustrate this he gives a concrete example:

This is proved. And let the two umbrous bodies be ab interposed between the window nm and the wall op with some space interposed between them and the same space ab. I say that if the umbrous body moves towards s that the shadow of the umbrous body b which is c will move to...d.

    As will be shown (see below pp. ) this situation overlaps with his camera obscura studies. In Manuscript E 30v (CU662, TPL593, 1513-1514) he pursues this theme under the heading:

On the motion of shadow.

The motion of shadow is always more speedy than the motion of the body that generates it, /if/ the luminous body /is/ immobile.

    This claim is followed by a demonstration (fig. 671):

This is proved and let the luminous body be a and the umbrous body b and the shade d. I say that the umbrous body moves from b to c in the same time that the shadow d moves to e and there is that proportion from speed to speed made at the same time which there is from length of motion to length of motion. Therefore in the proportion made from the umbrous body b to c with the length of motion made from shade d to e, such /a proportion/ will the aforesaid speeds of motion have between them.

(figure)

Figs. 674-675: Further cases where derived shade and umbrous body move while light remains immobile on E30v and G92v.

    Immediately following he examines cases where the light moves (a) as fast as the umbrous body (Situation 5), (b) more quickly than the umbrous body (Situation 3), and (c) more slowly than the umbrous body (Situation 4, see below). The diagram accompanying the main demonstration on E30v (fig. ) is abstract. On G92v (1510-1515) he considers a concrete example based on everyday experience (fig. ):

On the speed of clouds.

The passage of clouds is in itself less rapid than its shade which moves on the earth. This is proved. And let e be the solar body. Let a be the cloud and its shade c. Therefore, moving the cloud from a to b, the shadow will move from c to d. Whence it follows that the shadows which go from the earth to the cloud made by lines concurring towards the centre of the sun, we shall say by the fourth of this that this is true which is proposed. Which fourth states that the intersections equidistant from the angle of the two converging lines will be that much less to the extent that they are closer to the place of convergence. Hence the clouds being without doubt closer to the sun than their shade, the shade will make a greater voyage over the earth than the cloud through the air in the same time.

Book Eight - 2. Derived shade and light move while umbrous body is immobile

    On CU665 (TPL810, 1505-1510) he explores a second situation under the heading:

(figure)

Fig. 676: Movement of shadow on CU665.

Of the illuminated body which turns round without changing position and it receives a same light from various sides and varies infinitely.

The shadows of lights which invest an irregular body in the countryside will be of that many more darknesses and that many more figures to the extent of the variety that this body makes in its turning motion. And it is as much to turn the body around when the light stands firm as to turn the light around an immobile object.

    This he illustrates, as usual, with a concrete demonstration (fig. 676):

This is proved and let en be the immobile body and let the mobile light be b which moves from b to a. I say that when the light is in b the shade of the protrusion d will extend from d to f which, in moving the light from b to a is changed from f to e and thus the said shade is changed in quantity and shape because the place where this is found is not of same shape as the place where it was divided. And such a mutation of shape and quantity is infinitely variable because if all the site which was at first occupied by the shade and is in itself completely various and of continuous quantity and every continuous quantity is divisible to infinity, therefore it is concluded that the quantity of the shadow and its shape is variable to infinity.

Book Eight - 3. Umbrous body moves faster and light moves slower

    He refers to this third situation only in passing on E30v (CU662, TPL593, 1513-1514): "But if the luminous body is slower than the umbrous body then the shade will be more speedy than the umbrous body."

Book Eight - 4. Umbrous body moves slower and light moves faster

    This situation too is referred to only in passing on E30v (CU662, TPL593, 1513-1514): "And if the luminous body is more speedy than the umbrous body, then the speed of the shade will be slower than the speed of the umbrous body."

Book Eight - 5. Umbrous body moves as fast as light

This situation he describes on CU661 (TPL577, 1508-1510) under the heading:

Of the derived shade which is equal to the primitive shade.

The motion of the derived shade will be equal to the motion of the primitive shade when the luminous body, cause of the shade, will be of a motion equal to the motion of the umbrous body or, if you wish to say, of the primitive shade. Otherwise it is impossible because one who walks towards the west from morning to evening will have the shade going in front of the walker in the first part of the day. In the last half of the day the shadow will be much quicker in fleeting behind than the umbrous body in going forward.

    He alludes to this situation again on E30v (CU662, TPL593, 1513-1514):

But if the luminous body is equal in speed to the motion of the umbrous body, then the shade and the umbrous body will be of equal motion relative to one another.

Book Eight - 6. Eye moves while umbrous body and light immobile

    He also considers a case in which the eye moves while the umbrous body and light remain constant on CU646 (TPL686, 1508-1510) in a passage entitled:

(figure)

Fig. 677: Situation where the eye moves while the umbrous body and light are immobile on CU646.

On the site of the eye which sees more or less shadow depending on the motion that it makes around the umbrous body.

The proportions of the quantities that the umbrous and illuminated parts of umbrous bodies have vary as much as the variety of the sites of the eye that sees them.

    A specific example (fig. 677) follows:

This is proved. Let amnu be the umbrous body. Let p be the luminous source which embraces it with its rays pr and ps /and/ illuminates the part uman and the remainder num remains dark.

And let the eye which sees such a body be q which, with its visual rays embraces this umbrous body and sees all of dmo in which glance it sees dm, the illuminated part, considerably less than mo; the umbrous part, as is proved in the pyramid dqo, intersected at kh, equidistant from its base, divided at point i.

And thus will vary similarly in so many ways darkness at the eye which sees it as are the varieties of the sites of the aforesaid eye.

    This factor of the eye's position he again considers on CU770 (TPL676b, 1508-1510) in a passage entitled:

On umbrous bodies which are polished and lustrous.

In umbrous bodies which have polished and lustrous surfaces, which have particular light, they will vary in their shadows and lustres in as many sites as are the mutations of the pupil (luce) of the eye which sees them.

In this case the particular light can be immobile and the eye mobile and also conversely which is the same with respect to the changes of lustres and shadows on the surface of these bodies.

Conclusions

    A comprehensive study of Leonardo's scattered notes on light and shade reveals elements of an unexpectedly systematic approach and confirms that he wrote sufficient material for his projected seven books on light and shade (CA250va, 1490) to permit a reconstruction of their contents. Book One opens with a consideration of the nature of light and shade, its punctiform propagation, the role played by central lines and sets out to dismiss opposing theories. Book Two provides definitions of primary shade and describes its degrees.

    Book Three is devoted to derived shade. It begins with a classification of three basic kinds of shade based on Aristarchus: a first where light source and object are the same size; second, where the light source is larger than the object and third, where the light source is smaller than the object. Derived shade involves three variables (light source, object and eye), each of which Leonardo studies in terms of comparative sizes, distances and positions. While Book Three considers the nature of derived shade in the open, Book Four concentrates on the effects produced when this shade strikes surfaces of different shapes and sizes in various positions and at various distances. This leads to a series of studies on how one light source and one object can produce two shadows. It leads also to an examination of compound shade which, for Leonardo, involves multiple light sources and/or multiple objects.

    In the period after 1501 this culminates in a series of experiments involving interposed columns both in isolation and combined in the manner of a St. Andrew's cross. These experiments again reveal a systematic approach involving one, two, three and four light sources, and are of further interest because they are parallelled by another series of experiments involving one, two, three and four pinhole apertures in a camera obscura (see below p. and pp. ).

    Book Five considers further effects when this derived shade strikes objects and is reflected in such a way that it mixes with the surrounding light. His demonstrations on this theme include interposed rods and walls as well as theoretical situations. In 1490 Leonardo plans to write a sixth book on how reflected and shade alter the colours of surrounding objects, but his extent notes on this theme do not begin until 1492. In that year he begins with demonstrations using white objects, faces and landscapes. In the period after 1505 he explores this theme in more detail with demonstrations involving mirrors, water, the combination of yellow and azure to produce green, and reflections from different coloured walls. This leads to both a series of precepts and general statements. As in the case of Book Six, the ideas for Book Seven were also conceived of in 1490, but not written until later. Here, Leonardo considers the effects of distance on reflected colour, concerning which he has several demonstrations and a number of general statements. Especially in the period after 1505 there are increasing links between these principles of reflected colour at a distance and his studies of perspective of colour and diminution of form.

    In addition to these seven books outlined on CA250va (1490) Leonardo subsequently notes a series of other books that he intends to write on light and shade. The most comprehensive of these lists (CA277va, 1513-1514) mentions sixteen books. Some of these are clearly developments of themes from the earlier seven books (cf. Charts 9 and 10). For instance Book Three, "On the shape of shadows," corresponds to aspects of Book Four, Chapters Six and Seven in the reconstructed earlier version.

    Book Four, "On quality," (cf. the definition thereof on CU841), is very probably a development of Book Four, Chapters Two to Four in the reconstructed version, while Book Five, on CA277va, "On quantity" (cf. again the definition thereof on CU841), is probably a development of aspects of Book Four, Chapters Six and Seven in the earlier version Books Ten and Eleven, "On darkness" and "On light" are probably a development of Book One in the earlier version and of his definitions of darkness and light (see above pp. ). Book One, "On the usefulness of shadows," (CA277va), develops a theme only mentioned in passing in his outline on CA250ra (1490), and is probably based on his eulogies of chiaroscuro (see Vol. 1, Part 3.4, pp. ). Book Two, "On the motion of shadows," involves a theme not mentioned in the earlier outline. But, with the aid of other notes (CU658 and 686) a reconstruction has again been possible (our book eight). In the case of Book Nine, "On decompounded shade," the title is too sketchy to permit a serious attempt at reconstruction.

    The remaining five books mentioned on CA277va (numbers 6, 12, 13, 14 and 16) are related to his camera obscura studies. For instance, Book Six, "On boundaries" is probably based on those camera obscura experiments by means of which he demonstrates the production of a spectrum of boundaries (see below pp. ). Book Twelve, "On light penetrating through apertures of different shapes," involves experiments with apertures in the form of triangles, squares, slits and crosses (see below pp. ). Book Thirteen, "On light passing through various numbers of apertures," is corollary to this and involves a further series of experiments with one to thirty-two apertures (see below pp. ). Book Fourteen, "On the composition of multiple luminous rays" may be based on other demonstrations involving coloured lights passing through a different number of apertures (see below pp. ). Finally Book Sixteen, "Whether parallel rays can come from a single light and penetrate through some apertures," also appears to be based o n camera obscura demonstrations (see below pp. ).

    Hence, notwithstanding some areas of uncertainty, there is a wealth of evidence that Leonardo studied questions of light and shade much more systematically than the surface disorder of the notebooks suggests. While he is initially concerned with these problems as a painter, he gradually develops an interest in the physics of light and shade for its own sake. His analogy between light and sight helps furnish a further motive. since the pupil can be treated as an aperture and the interior of the eye as a camera obscura, the study of light and shade in connection with camera obscuras can offer insight into the nature of the visual process. This leads to further demonstrations and experiments, which are the theme of our next chapter.


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Last Update: July 2, 1999