Part Two, Chapter 2: Seven Books on Light and Shade

Introduction

Book One Light and Shade

Book Two Primary Shade

Book Three Derived Shade

Book Four Derived Shade and Interposed Objects

Book Five Derived Shade and Reflected Light

Book Six Reflected Colour

Book Seven Reflected Colour and Distance

(Book Eight) Movement of Shadows

Conclusions

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

This he crosses out and begins anew:

Shade is a privation of light

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).

Book 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

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 *

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.

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.

And about this you will compose a book.

Book Theme

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

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:

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.

Light and Shade Book One

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):

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

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

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

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):

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.

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

The operation of light with its centre.

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

Proof.

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.

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.

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:

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):

Book Two

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. ).

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:

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

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.

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

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.

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

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:

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

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:

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

Book Three

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 Samos² 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.³ This distinction had, moreover, been transmitted indirectly through mediaeval commentators such as Witelo.⁴ 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

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):

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?

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

Of the three various shapes of derivative shade.

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

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

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

That derived shadows are of three kinds.

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.

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

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.

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

On shade.

An adversary's arguments are now considered:

And these arguments are promptly dismissed:

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):

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

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.

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.

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.

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):

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.

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).

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 is

universal, 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."

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. 3.1. 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 better

known, of which the umbrous body is more distant from the luminous body.

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 window

without 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):

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.

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

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):

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

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

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 ):

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

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."

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. 3. 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.

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.

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?

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

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.

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.

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.

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,

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.

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 destroy

immediately. 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.

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

Let cedf be a spherical umbrous body; ab is the luminous source equal to the umbrous body

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

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).

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.

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.

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.

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.

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

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.

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.

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):

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

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

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):

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

Definition

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.

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

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?

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

On reflected light

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?

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

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.

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

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):

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.

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.

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.

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

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):

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):

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.

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

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.

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

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.”

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):

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):

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:

(Given) Given Two or

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:

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.

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

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?

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

Of the colours of incident and reflected lights.

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

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:

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

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

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:

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:

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. ):

This shadow is long and thin.

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:

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:

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.

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.

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:

This idea he develops in the main text alongside:

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."

Book Four. 8. How/Why One Light Source and One ObjectProduce 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. ):

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.

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 The percussion of umbrous bodies

from a pyramidal body is not originating from a pyramidal

similar in shape to its origin. umbrous body will be of bifurcated

shape and of various darknesses at

its points.

The light which is larger than The light which is greater than the

the point and smaller than the point and less than the base of the

base...of the pyramidal umbrous pyramidal umbrous body placed

body positioned with it, will opposite it will...produce in its

have the effect that the umbrous percussion a shadow of a bifurcating

concourse in its percussion will shape and of various degrees of

cause shade of a bifurcating darkness.

shape and of various degrees of

darkness.

If the luminous body greater

than the umbrous [body].

If the umbrous body, less than If the umbrous body less than the

the luminous, makes two shades luminous source makes two shadows

both the umbrous body [that is] and the umbrous body is similar to

the same size and larger...than the luminous source or the larger

the luminous source produce one makes a single shadow, then it is

[shade]. The pyramidal body fitting that the pyramidal body

which has part of it larger and which has a part which is smaller,

part of it smaller than the a part which is equal and a part

luminous body will make which is larger than the luminous

bifurcating shade. body makes a bifurcated shade.

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.

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:

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:

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.

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:

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.”

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:

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.

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

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:

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:

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).

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):

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):

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:

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:

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:

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

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

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.

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:

On pyramidal shade

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. 10d. 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:

**

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:

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.

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

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

These opinions of the adversary he refutes:

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

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.

Alongside he writes a draft text:

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:

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

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

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

This is proved.

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

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

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.

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

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

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.

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.

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

(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.

He is conscious that reflection is not always possible and on A94v (BN 2038 14v, CU158, TPL157, C.1492) describes these situations under the heading:

Where there can be no luminous reflection

1. Straight vs. 2. Which Objects 3. Surface 4. Universal/ Reflected Light Participates Particular Light

171 156 162

157 164

158 166

168

169

170

781 782/784b

785 783/790

Book V V VI V

5. Role of 6. Role of 7. Reflected

Distance Angle Light/Background

159 161 160

163

168 167

172

789 780

786

787

Book VII V V

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

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

Reflection

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

On shade in bodies.

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.

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?

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

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

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

On the size of lustres on their terse bodies.

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

Of lustre.

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

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:

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):

Shade and Light

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.

On the shadows made by bridges on their water.

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.

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

(figure)]

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

battle.

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.

A related situation of light entering a cave is described on Forst I 10r (fig. 569, c.1505):

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

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

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:"

This introduces the demonstration proper:

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.

Directly following he demonstrates the converse:

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

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.

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.

This is followed by a demonstration:

Which leads him to conclude:

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...

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:

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:

As an afterthought he adds:

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

Which leads, as usual, to a demonstration:

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:

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:

This he demonstrates:

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.

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.

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:

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

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":

Which then introduces a (fig. 603):

Demonstration

Immediately following he adds another proposition and commentary:

Proposition

Commentary

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

Demonstration

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

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

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

Demonstration

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

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:

Beneath this he drafts another section which he later crosses out:

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.

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.

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:

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.

On CU723 (fig. 630, TPL783, 1508-1510) he considers the problem in semi-abstract terms:

How reflection is generated in universal lights.

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.

This problem preoccupies him. On CA207ra (1508-1510) he drafts another version:

...part of the bod/y/...

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.

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.

By way of illustration he carefully describes a geometrical diagram (fig. 632):

Which leads him to conclude:

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.

He now describes the geometrical diagram (fig. 641, cf. fig. 640):

Directly following is a proof:

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

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 idea he reformulates in the next passage CU676 (fig. 627, TPL694a, 1508-1510):

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.

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):

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:

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?

He is conscious that he has already dealt with the problem. What challenges him is the idea of an alternative proof:

A geometrical proof follows (fig. 634, cf.633):

One alternative demonstration is not enough (fig. 636, cf.fig. 635):

There follows a second demonstration.

On CA207ra (1508-1510) he makes further drafts:

That part of a body illuminates.

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):

On CA207ra (1508-1510) he also drafts a corresponding demonstration (fig. 635):

This demonstration is developed on CU686 (fig. 636, cf. fig. 635), although the lettering is different:

He also claims the converse (fig. 639, cf. fig. 638).

This is proved by the converse of the foregoing.

This series ends with a plea in defence of such geometrical demonstrations directed against an adversary:

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

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:

This aspect of colours had also been considered by later authors such as Ptolemy⁵ and Alhazen⁶. 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.

Book Six. 1. Mirrors

In De Coloribus Aristotle had noted that reflections in mirrors resemble the colour of mirrors.⁷ Ptolemy, in his Optics h ad pointed out that the colour of a mirror affects the colour of things seen.⁸ This phenomenon had also been mentioned by authors such as Heliodorus of Larissa⁹, Alhazen¹⁰ and Witelo¹¹. Leonardo's first extant reference to this question is on Forst III 54r (fig. 642, c.1493) under the heading:

Mirror

On BM57r (1497-1500) he restates this idea more succinctly:

A slightly different version occurs on BM58v (1505-1508):

On CU167 (TPL158, 1505-1510) he notes that the phenomenon depends on the degree to which the reflecting surface is polished:

On reflections

On CU211 (TPL256, 1508-1510) he pursues the question:

Of the colours reflected on the lustres of various colours.

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.

On CU5453 (TPL522, 1505-1510) he considers reflection in water:

On objects reflected in turbulent waters.

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.

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.

Book Six. 3. White Objects

Leon Battista Alberti, in his On Painting, had described how:

Leonardo adapts this example and develops it on A100r (BN 2038 20r, 1492) under the heading:

How white bodies must be represented.

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.

He then gives a second in which he mentions white walls:

Proposition

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.

On F75r (CU204, TPL247, 1508) he restates the idea, now referring to it as a fourth proposition:

Painting

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.

On CU465 (TPL471, 1508-1510) he develops the principle into two propositions under the heading:

Painting

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.

To explain this he offers two reasons:

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?

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

But it is, nonetheless, possible that the reflection of a nearby small colour tinges more than a

He outlines the consequences this has for his painting practice on CU175 (TPL170b, 1508-1510) under the heading:

On Reflections

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.

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):

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.

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.

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

CU793 (TPL654i, 1508-1510):

On lights and the shades and colours of these.

No body will ever show itself entirely in its natural colour.

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.

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.

As his studies of Nature continue he becomes more aware of the natural variety of colours, as for instance, on BM114v (c.1510):

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.

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:

A more detailed analysis of reflected light and colour in leaves follows on G3r-2v (1510-1515):

He pursues this theme on G8v (TPL896, fig. 651, c. 1510-1515):

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):

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:

On the colours of the species of objects which tinge the surfaces of opaque bodies.

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):

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.

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.

On CU196 (TPL248e, 1508-1510) he mentions the need to compare ordinary light with reflected colour:

On Colours

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:

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:

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.

He pursues this theme on CU936 (TPL875, 1508-1510) in a passage headed:

Which leads him to conclude:

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.

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.

He goes on to relate this to his camera obscura experiments (see below p. ) which he intends to include in his book of painting:

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.

Example.

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:

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.

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.

On CU202 (TPL239, 1505-1510), he pursues this theme, beginning with a general claim:

Of the colour of the shadows of some body.

This he then demonstrates with a case where azure light is reflected from a green wall:

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]):

Whether the surface of every opaque object participates in the colour of its object.

Immediately following he considers a more complex situation (fig. 661):

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.

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:

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/.

On CU766 (TPL702, 1508-1510) he opens with a general claim under the heading:

On false colour of shadows in opaque bodies.

This he illustrates with a demonstration using a candle flame (fig. 662):

On CU741 (TPL438b, 1508) he offers a further demonstration involving sunlight:

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.

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.

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?

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.

This he again demonstrates with a concrete example (fig. 215):

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.

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.

On CU860 (TPL694f, 1508-1510) he lists nine propositions, three of which deal with reflected light, shade and colour:

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

1^st The surfaces of bodies participate more int he colours of those objects which reflect

their image in them amidst more equal angles.

2^nd 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.

3^rd 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.

4^th 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):

Book Six. 1. 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

He mentions reflected colours again in passing on CU265 (TPL119a, 1505-1510):

Reflection

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.

On CU758 (TPL608, 1508-1510) he makes another brief note under the heading:

On the boundaries that surround derived shadows in their percussions.

He pursues this theme of coloured on CU788 (TPL609, 1508-1510):

Which leads to a more radical claim on CU714 (TPL579a, 1508-1510):

Nature or condition of shade.

On E17r (1513-1514) there is a further summary note on reflected shade with respect to painting practice:

Painting

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

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

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):

The surface of every opaque body participates in the colour of its object.

He expresses a similar idea on CU720 (fig. 664, TPL781, 1508-1510) under the heading:

Where the reflection must be darker

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.

Part one of his demonstration follows (fig. 665):

Part two of the demonstration considers the distance factor:

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.

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.

A demonstration follows (fig. 667):

Which part of an illuminated surface will be of greater brightness?

He pursues the theme on CU818 (TPL786, 1508-1510):

Of the eye which stands in the bright /air/ and looks at a dark place.

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.

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?

He considers the distance factor again on CU811 (TPL629, 1508-1510), under the heading:

Of white things remote from the eye.

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

He pursues this artistic interest in reflected light and shade on CU163 (TPL159, 1505-1510):

On the reflections of lights that surround shadows.

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

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.

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 same connection also applies to rocks, as on W12414 (c.1511):

And it applies equally to landscapes as on CU961 (TPL911, 1510-1515):

On landscapes.

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.

He develops this idea on CU792 (TPL705, 1508-1510) in a passage entitled:

On the accidents of surfaces of bodies.

This leads to a clear connection between his principle of reflected colour and disappearance of form perspective, as on CU870 (TPL690, 1508-1510):

Which connection he restates more subtly in passages such as E17r (CU448, TPL472, 1513-1514):

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

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.

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.

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):

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.

He provides a further illustration of this situation on CA370ra (1497):

Of the shadow which moves with a man.

Of the shadows of the sun and of mirroring in the water at the same time.

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.

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.

On E2v (1513-1514) he drafts a further passage on the problem under the heading:

Of shadow or its movement.

To illustrate this he gives a concrete example:

This is proved. And let the two umbrous bodies be ab interposed between the window nm

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.

This claim is followed by a demonstration (fig. 671):

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.

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:

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

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.

He alludes to this situation again on E30v (CU662, TPL593, 1513-1514):

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:

A specific example (fig. 677) follows:

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 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.

Notes

1. Citation needed ↩︎
2. Aristarchus of Samos ↩︎
3. Aristarchus ↩︎
4. Witelo ↩︎
5. Ptolemy ↩︎
6. Alhazen ↩︎
7. Aristotle - De Coloribus ↩︎
8. Ptolemy - Optics ↩︎
9. Heliodorus of Larissa ↩︎
10. Alhazen ↩︎
11. Witelo ↩︎