Dr. Kim H. Veltman

Seven Books on Light and Shade

1. Introduction

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

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

    This he crosses out and begins anew:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Book One: Light and Shade

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

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

Book One - 1. Punctiform Propagation

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The operation of light with its centre.

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

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

Proof.

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

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

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

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

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

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

Derived light is surrounded by primitive shade.

Derived shade will be surrounded by derived light.

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

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

Book One -  2. Central Line

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

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

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

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

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

Book One - 3. Opposing Theories

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

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

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

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

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

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

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

Book Two: Primary Shade

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

Primary Shade

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Book Three: Derived Shade

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

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

Book Three - 1. Kinds of Light

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

Shade and Light

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

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

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

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

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

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

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

Of how many shapes is derived shade?

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

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

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

Of the three various shapes of derivative shade.

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

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

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

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

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

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

That derived shadows are of three kinds.

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

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

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

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

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

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

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

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

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

On simple derived shade.

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

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

On shade.

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

    An adversary's arguments are now considered:

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

    And these arguments are promptly dismissed:

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

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

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

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

Book Three - 2. Light Source

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    Which body produces a greater quantity of shade:

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

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

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

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

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

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

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

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

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

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

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

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

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

Which body takes a greater quantity of light?

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

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

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

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

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

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

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

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

Book Three - 2d. Comparative Sizes of Light Source

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

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

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

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

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

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

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

Of derived shade and where it is greater.

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

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

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

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

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

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

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

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

Book Three - 3a. Comparative Distances

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

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

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

Which derived shade will show its boundaries as better known?

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

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

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

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

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

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

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

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

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

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

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

On shade.

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

On shade.

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

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

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

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

Painting in a universal light.

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

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

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

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

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

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

Proof how every part of light makes one point.

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

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

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

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

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

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

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

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

    A concrete example follows (fig. 295):

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Book Three - 3b. Comparative Sizes of Object

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

On shadow and light.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    This claim is again demonstrated (fig. 314):

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

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

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

Book Three - 4. Comparative Positions of the Eye

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

Perspective

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

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

Perspective

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

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

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

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

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

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

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

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

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

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

On Painting

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

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

On Painting

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

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

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

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

Perspective

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Book Four: Derived Shade and Interposed Objects

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

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

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

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

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

    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.

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

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

In how many ways does primitive shade vary?

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

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

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

What variety does derived shade have?

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

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

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

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

Whether primitive shade is more powerful than derived shade?

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

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

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

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

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

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

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

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

On Lights

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

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

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

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

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

On the duplication of lights.

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

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

Of the luminous rays and the powers of their extremities.

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

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

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

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

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

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

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

Painting

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

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

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

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

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

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

Definition

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

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

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

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

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

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

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

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

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

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

In what surfaces is true and equal light found?

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

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

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

On reflected light

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

That light is brighter which is of a greater angle.

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

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

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

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

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

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

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

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

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

Painting

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

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

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

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

Of universal light illuminating plants.

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

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

(figure)

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

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