A light which is smaller than the object on which it falls will light up a smaller extent of it in proportion as it is nearer to it, and the converse, as it is farther from it. But when the light is larger than the object illuminated it will light a larger extent of the object in proportion as it is nearer and the converse when they are farther apart.
That portion of an illuminated object which is nearest to the source of light will be the most strongly illuminated.
That portion of the primary shadow will be least dark which is farthest from the edges.
The derived shadow will be darker than the primary shadow where it is contiguous with it.
On the proportion of light and shade (153-157).
That portion of an opaque body will be more in shade or more in light, which is nearer to the dark body, by which it is shaded, or to the light that illuminates it.
Objects seen in light and shade show in greater relief than those which are wholly in light or in shadow.
The shaded and illuminated sides of opaque objects will display the same proportion of light and darkness as their objects [Footnote 6: The meaning of obbietti (objects) is explained in no 153, lines 1-4.--Between the title-line and the next there is, in the original, a small diagram representing a circle described round a square.].
The outlines and form of any part of a body in light and shade are indistinct in the shadows and in the high lights; but in the portions between the light and the shadows they are highly conspicuous.
Among objects in various degrees of shade, when the light proceeds from a single source, there will be the same proportion in their shadows as in the natural diminution of the light and the same must be understood of the degrees of light.
A single and distinct luminous body causes stronger relief in the object than a diffused light; as may be seen by comparing one side of a landscape illuminated by the sun, and one overshadowed by clouds, and so illuminated only by the diffused light of the atmosphere.
THIRD BOOK ON LIGHT AND SHADE.
Definition of derived shadow (158. 159).
Derived shadow cannot exist without primary shadow. This is proved by the first of this which says: Darkness is the total absence of light, and shadow is an alleviation of darkness and of light, and it is more or less dark or light in proportion as the darkness is modified by the light.
Shadow is diminution of light.
Darkness is absence of light.
Shadow is divided into two kinds, of which the first is called primary shadow, the second is derived shadow. The primary shadow is always the basis of the derived shadow.
The edges of the derived shadow are straight lines.
[Footnote: The theory of the ombra dirivativa--a technical expression for which there is no precise English equivalent is elaborately treated by Leonardo. But both text and diagrams (as Pl. IV, 1-3 and Pl. V) must at once convince the student that the distinction he makes between ombra primitiva and ombra dirivativa is not merely justifiable but scientific. Ombra dirivativa is by no means a mere abstract idea. This is easily proved by repeating the experiment made by Leonardo, and by filling with smoke the room in which the existence of the ombra dirivativa is investigated, when the shadow becomes visible. Nor is it difficult to perceive how much of Leonardo's teaching depended on this theory. The recognised, but extremely complicated science of cast shadows--percussione dell' ombre dirivative as Leonardo calls them--is thus rendered more intelligible if not actually simpler, and we must assume this theory as our chief guide through the investigations which follow.]
The darkness of the derived shadow diminishes in proportion as it is remote from the primary shadow.
Different sorts of derived shadows (160-162).
SHADOW AND LIGHT.
The forms of shadows are three: inasmuch as if the solid body which casts the shadow is equal (in size) to the light, the shadow resembles a column without any termination (in length). If the body is larger than the light the shadow resembles a truncated and inverted pyramid, and its length has also no defined termination. But if the body is smaller than the light, the shadow will resemble a pyramid and come to an end, as is seen in eclipses of the moon.
OF SIMPLE DERIVED SHADOWS.
The simple derived shadow is of two kinds: one kind which has its length defined, and two kinds which are undefined; and the defined shadow is pyramidal. Of the two undefined, one is a column and the other spreads out; and all three have rectilinear outlines. But the converging, that is the pyramidal, shadow proceeds from a body that is smaller than the light, and the columnar from a body equal in size to the light, and the spreading shadow from a body larger than the light; &c.
OF COMPOUND DERIVED SHADOWS.
Compound derived shadows are of two kinds; that is columnar and spreading.
Derived shadows are of three kinds of which one is spreading, the second columnar, the third converging to the point where the two sides meet and intersect, and beyond this intersection the sides are infinitely prolonged or straight lines. And if you say, this shadow must terminate at the angle where the sides meet and extend no farther, I deny this, because above in the first on shadow I have proved: that a thing is completely terminated when no portion of it goes beyond its terminating lines. Now here, in this shadow, we see the converse of this, in as much as where this derived shadow originates we obviously have the figures of two pyramids of shadow which meet at their angles. Hence, if, as [my] opponent says, the first pyramid of shadow terminates the derivative shadow at the angle whence it starts, then the second pyramid of shadow--so says the adversary--must be caused by the angle and not from the body in shadow; and this is disproved with the help of the 2nd of this which says: Shadow is a condition produced by a body casting a shadow, and interposed between this shadow and the luminous body. By this it is made clear that the shadow is not produced by the angle of the derived shadow but only by the body casting the shadow; &c. If a spherical solid body is illuminated by a light of elongated form the shadow produced by the longest portion of this light will have less defined outlines than that which is produced by the breadth of the same light. And this is proved by what was said before, which is: That a shadow will have less defined outlines in proportion as the light which causes it is larger, and conversely, the outlines are clearer in proportion as it is smaller.
[Footnote: The two diagrams to this chapter are on Plate IV, No. 1.]
On the relation of derived and primary shadow (163-165).
The derived shadow can never resemble the body from which it proceeds unless the light is of the same form and size as the body causing the shadow.
The derived shadow cannot be of the same form as the primary shadow unless it is intercepted by a plane parallel to it.
HOW A CAST SHADOW CAN NEVER BE OF THE SAME SIZE AS THE BODY THAT CASTS IT.
If the rays of light proceed, as experience shows, from a single point and are diffused in a sphere round this point, radiating and dispersed through the air, the farther they spread the wider they must spread; and an object placed between the light and a wall is always imaged larger in its shadow, because the rays that strike it [Footnote: 7. The following lines are wanting to complete the logical connection.] would, by the time they have reached the wall, have become larger.
Any shadow cast by a body in light and shade is of the same nature and character as that which is inseparable from the body. The centre of the length of a shadow always corresponds to that of the luminous body [Footnote 6: This second statement of the same idea as in the former sentence, but in different words, does not, in the original, come next to the foregoing; sections 172 and 127 are placed between them.]. It is inevitable that every shadow must have its centre in a line with the centre of the light.
On the shape of derived shadows (166-174).
OF THE PYRAMIDAL SHADOW.
The pyramidal shadow produced by a columnar body will be narrower than the body itself in proportion as the simple derived shadow is intersected farther from the body which casts it.
[Footnote 166: Compare the first diagram to No. 161. If we here conceive of the outlines of the pyramid of shadow on the ground as prolonged beyond its apex this gives rise to a second pyramid; this is what is spoken of at the beginning of No. 166.]
The cast shadow will be longest when the light is lowest.
The cast shadow will be shortest when the light is highest.
Both the primary and derived shadow will be larger when caused by the light of a candle than by diffused light. The difference between the larger and smaller shadows will be in inverse proportion to the larger and smaller lights causing them.
[Footnote: In the diagrams A stands for celo (sky), B for cadela (candle).]
ALL BODIES, IN PROPORTION AS THEY ARE NEARER TO, OR FARTHER FROM THE SOURCE OF LIGHT, WILL PRODUCE LONGER OR SHORTER DERIVED SHADOWS.
Among bodies of equal size, that one which is illuminated by the largest light will have the shortest shadow. Experiment confirms this proposition. Thus the body m n is surrounded by a larger amount of light than the body p q, as is shown above. Let us say that v c a b d x is the sky, the source of light, and that s t is a window by which the luminous rays enter, and so m n and p q are bodies in light and shade as exposed to this light; m n will have a small derived shadow, because its original shadow will be small; and the derivative light will be large, again, because the original light c d will be large and p q will have more derived shadow because its original shadow will be larger, and its derived light will be smaller than that of the body m n because that portion of the hemisphere a b which illuminates it is smaller than the hemisphere c d which illuminates the body m n.
[Footnote: The diagram, given on Pl. IV, No. 2, stands in the original between lines 2 and 7, while the text of lines 3 to 6 is written on its left side. In the reproduction of this diagram the letter v at the outer right-hand end has been omitted.]
The shadow m bears the same proportion to the shadow n as the line b c to the line f c.
Of different shadows of equal strength that which is nearest the eye will seem the least strong.
Why is the shadow e a b in the first grade of strength, b c in the second; c d in the third? The reason is that as from e a b the sky is nowhere visible, it gets no light whatever from the sky, and so has no direct [primary] light. b c faces the portion of the sky f g and is illuminated by it. c d faces the sky at h k. c d, being exposed to a larger extent of sky than b c, it is reasonable that it should be more lighted. And thus, up to a certain distance, the wall a d will grow lighter for the reasons here given, until the darkness of the room overpowers the light from the window.
When the light of the atmosphere is restricted [by an opening] and illuminates bodies which cast shadows, these bodies being equally distant from the centre of the window, that which is most obliquely placed will cast the largest shadow beyond it.
These bodies standing apart in a room lighted by a single window will have derivative shadows more or less short according as they are more or less opposite to the window. Among the shadows cast by bodies of equal mass but at unequal distances from the opening by which they are illuminated, that shadow will be the longest of the body which is least in the light. And in proportion as one body is better illuminated than another its shadow will be shorter than another. The proportion n m and e v k bear to r t and v x corresponds with that of the shadow x to 4 and y.
The reason why those bodies which are placed most in front of the middle of the window throw shorter shadows than those obliquely situated is:--That the window appears in its proper form and to the obliquely placed ones it appears foreshortened; to those in the middle, the window shows its full size, to the oblique ones it appears smaller; the one in the middle faces the whole hemisphere that is e f and those on the side have only a strip; that is q r faces a b; and m n faces c d; the body in the middle having a larger quantity of light than those at the sides is lighted from a point much below its centre, and thus the shadow is shorter. And the pyramid g 4 goes into l y exactly as often as a b goes into e f. The axis of every derivative shadow passes through 6 1/2 [Footnote 31: passa per 6 1/2 (passes through 6 1/2). The meaning of these words is probably this: Each of the three axes of the derived shadow intersects the centre (mezzo) of the primary shadow (ombra originale) and, by prolongation upwards crosses six lines.
This is self evident only in the middle diagram; but it is equally true of the side figures if we conceive of the lines 4 f, x n v m, y l k v, and 4 e, as prolonged beyond the semicircle of the horizon.] and is in a straight line with the centre of the primary shadow, with the centre of the body casting it and of the derivative light and with the centre of the window and, finally, with the centre of that portion of the source of light which is the celestial hemisphere, y h is the centre of the derived shade, l h of the primary shadow, l of the body throwing it, l k of the derived light, v is the centre of the window, e is the final centre of the original light afforded by that portion of the hemisphere of the sky which illuminates the solid body.
[Footnote: Compare the diagram on Pl. IV, No. 3. In the original this drawing is placed between lines 3 and 22; the rest, from line 4 to line 21, is written on the left hand margin.]
THE FARTHER THE DERIVED SHADOW IS PROLONGED THE LIGHTER IT BECOMES.
You will find that the proportion of the diameter of the derived shadow to that of the primary shadow will be the same as that between the darkness of the primary shadow and that of the derived shadow.
[Footnote 6: Compare No. 177.] Let a b be the diameter of the primary shadow and c d that of the derived shadow, I say that a b going, as you see, three times into d c, the shadow d c will be three times as light as the shadow a b. [Footnote 8: Compare No. 177.]
If the size of the illuminating body is larger than that of the illuminated body an intersection of shadow will occur, beyond which the shadows will run off in two opposite directions as if they were caused by two separate lights.
On the relative intensity of derived shadows (175-179).
The derived shadow is stronger in proportion as it is nearer to its place of origin.
HOW SHADOWS FADE AWAY AT LONG DISTANCES.
Shadows fade and are lost at long distances because the larger quantity of illuminated air which lies between the eye and the object seen tints the shadow with its own colour.
a b will be darker than c d in proportion as c d is broader than a b.
[Footnote: In the original MS. the word lume (light) is written at the apex of the pyramid.]
It can be proved why the shadow o p c h is darker in proportion as it is nearer to the line p h and is lighter in proportion as it is nearer to the line o c. Let the light a b, be a window, and let the dark wall in which this window is, be b s, that is, one of the sides of the wall.
Then we may say that the line p h is darker than any other part of the space o p c h, because this line faces the whole surface in shadow of [Footnote: In the original the diagram is placed between lines 27 and 28.] the wall b s. The line o c is lighter than the other part of this space o p c h, because this line faces the luminous space a b.
Where the shadow is larger, or smaller, or equal the body which casts it.
[First of the character of divided lights. [Footnote 14: lumi divisi. The text here breaks off abruptly.]
OF THE COMPOUND SHADOW F, R, C, H CAUSED BY A SINGLE LIGHT.
The shadow f r c h is under such conditions as that where it is farthest from its inner side it loses depth in proportion. To prove this:
Let d a, be the light and f n the solid body, and let a e be one of the side walls of the window that is d a. Then I say--according to the 2nd [proposition]: that the surface of any body is affected by the tone of the objects surrounding it,--that the side r c, which faces the dark wall a e must participate of its darkness and, in the same way that the outer surface which faces the light d a participates of the light; thus we get the outlines of the extremes on each side of the centre included between them.]
This is divided into four parts. The first the extremes, which include the compound shadow, secondly the compound shadow between these extremes.
THE ACTION OF THE LIGHT AS FROM ITS CENTRE.
If it were the whole of the light that caused the shadows beyond the bodies placed in front of it, it would follow that any body much smaller than the light would cast a pyramidal shadow; but experience not showing this, it must be the centre of the light that produces this effect.
[Footnote: The diagram belonging to this passage is between lines 4 and 5 in the original. Comp. the reproduction Pl. IV, No. 4. The text and drawing of this chapter have already been published with tolerable accuracy. See M. JORDAN: "Das Malerbuch des Leonardo da Vinci". Leipzig 1873, P. 90.]
Let a b be the width of the light from a window, which falls on a stick set up at one foot from a c [Footnote 6: bastone (stick). The diagram has a sphere in place of a stick.]. And let a d be the space where all the light from the window is visible. At c e that part of the window which is between l b cannot be seen. In the same way a m cannot be seen from d f and therefore in these two portions the light begins to fail.
Shadow as produced by two lights of different size (180. 181).
A body in light and shade placed between two equal lights side by side will cast shadows in proportion to the [amount of] light. And the shadows will be one darker than the other in proportion as one light is nearer to the said body than the other on the opposite side.
A body placed at an equal distance between two lights will cast two shadows, one deeper than the other in proportion, as the light which causes it is brighter than the other.
[Footnote: In the MS. the larger diagram is placed above the first line; the smaller one between l. 4 & 5.]
A light which is smaller than the body it illuminates produces shadows of which the outlines end within [the surface of] the body, and not much compound shadow; and falls on less than half of it. A light which is larger than the body it illuminates, falls on more than half of it, and produces much compound shadow.
The effect of light at different distances.
OF THE SHADOW CAST BY A BODY PLACED BETWEEN 2 EQUAL LIGHTS.
A body placed between 2 equal lights will cast 2 shadows of itself in the direction of the lines of the 2 lights; and if you move this body placing it nearer to one of the lights the shadow cast towards the nearer light will be less deep than that which falls towards the more distant one.
Further complications in the derived shadows (183-187).
The greatest depth of shadow is in the simple derived shadow because it is not lighted by either of the two lights a b, c d.
The next less deep shadow is the derived shadow e f n; and in this the shadow is less by half, because it is illuminated by a single light, that is c d.
This is uniform in natural tone because it is lighted throughout by one only of the two luminous bodies . But it varies with the conditions of shadow, inasmuch as the farther it is away from the light the less it is illuminated by it .
The third degree of depth is the middle shadow [Footnote 15: We gather from what follows that q g r here means ombra media (the middle shadow).]. But this is not uniform in natural tone; because the nearer it gets to the simple derived shadow the deeper it is [Footnote 18: Compare lines 10-13], and it is the uniformly gradual diminution by increase of distance which is what modifies it [Footnote 20: See Footnote 18]: that is to say the depth of a shadow increases in proportion to the distance from the two lights.
The fourth is the shadow k r s and this is all the darker in natural tone in proportion as it is nearer to k s, because it gets less of the light a o, but by the accident [of distance] it is rendered less deep, because it is nearer to the light c d, and thus is always exposed to both lights.
The fifth is less deep in shadow than either of the others because it is always entirely exposed to one of the lights and to the whole or part of the other; and it is less deep in proportion as it is nearer to the two lights, and in proportion as it is turned towards the outer side x t; because it is more exposed to the second light a b.
[Footnote: The diagram to this section is given on Pl. V. To the left is the facsimile of the beginning of the text belonging to it.]
OF SIMPLE SHADOWS.
Why, at the intersections a, b of the two compound shadows e f and m e, is a simple shadow pfoduced as at e h and m g, while no such simple shadow is produced at the other two intersections c d made by the very same compound shadows?
Compound shadow are a mixture of light and shade and simple shadows are simply darkness. Hence, of the two lights n and o, one falls on the compound shadow from one side, and the other on the compound shadow from the other side, but where they intersect no light falls, as at a b; therefore it is a simple shadow. Where there is a compound shadow one light or the other falls; and here a difficulty arises for my adversary since he says that, where the compound shadows intersect, both the lights which produce the shadows must of necessity fall and therefore these shadows ought to be neutralised; inasmuch as the two lights do not fall there, we say that the shadow is a simple one and where only one of the two lights falls, we say the shadow is compound, and where both the lights fall the shadow is neutralised; for where both lights fall, no shadow of any kind is produced, but only a light background limiting the shadow. Here I shall say that what my adversary said was true: but he only mentions such truths as are in his favour; and if we go on to the rest he must conclude that my proposition is true. And that is: That if both lights fell on the point of intersection, the shadows would be neutralised. This I confess to be true if [neither of] the two shadows fell in the same spot; because, where a shadow and a light fall, a compound shadow is produced, and wherever two shadows or two equal lights fall, the shadow cannot vary in any part of it, the shadows and the lights both being equal. And this is proved in the eighth [proposition] on proportion where it is said that if a given quantity has a single unit of force and resistance, a double quantity will have double force and double resistance.
The intersection n is produced by the shadows caused by the light b, because this light b produces the shadow x b, and the shadow s b, but the intersection m is produced by the light a which causes the shadow s a, and the shadow x a.
But if you uncover both the lights a b, then you get the two shadows n m both at once, and besides these, two other, simple shadows are produced at r o where neither of the two lights falls at all. The grades of depth in compound shadows are fewer in proportion as the lights falling on, and crossing them are less numerous.
Why the intersections at n being composed of two compound derived shadows, forms a compound shadow and not a simple one, as happens with other intersections of compound shadows. This occurs, according to the 2nd [diagram] of this [prop.] which says:--The intersection of derived shadows when produced by the intersection of columnar shadows caused by a single light does not produce a simple shadow. And this is the corollary of the 1st [prop.] which says:--The intersection of simple derived shadows never results in a deeper shadow, because the deepest shadows all added together cannot be darker than one by itself. Since, if many deepest shadows increased in depth by their duplication, they could not be called the deepest shadows, but only part-shadows. But if such intersections are illuminated by a second light placed between the eye and the intersecting bodies, then those shadows would become compound shadows and be uniformly dark just as much at the intersection as throughout the rest. In the 1st and 2nd above, the intersections i k will not be doubled in depth as it is doubled in quantity. But in this 3rd, at the intersections g n they will be double in depth and in quantity.
HOW AND WHEN THE SURROUNDINGS IN SHADOW MINGLE THEIR DERIVED SHADOW WITH THE LIGHT DERIVED FROM THE LUMINOUS BODY.
The derived shadow of the dark walls on each side of the bright light of the window are what mingle their various degrees of shade with the light derived from the window; and these various depths of shade modify every portion of the light, except where it is strongest, at c. To prove this let d a be the primary shadow which is turned towards the point e, and darkens it by its derived shadow; as may be seen by the triangle a e d, in which the angle e faces the darkened base d a e; the point v faces the dark shadow a s which is part of a d, and as the whole is greater than a part, e which faces the whole base [of the triangle], will be in deeper shadow than v which only faces part of it. In consequence of the conclusion [shown] in the above diagram, t will be less darkened than v, because the base of the t is part of the base of the v; and in the same way it follows that p is less in shadow than t, because the base of the p is part of the base of the t. And c is the terminal point of the derived shadow and the chief beginning of the highest light.
[Footnote: The diagram on Pl. IV, No. 5 belongs to this passage; but it must be noted that the text explains only the figure on the right-hand side.]
FOURTH BOOK ON LIGHT AND SHADE.
On the shape of the cast shadows (188-191).
The form of the shadow cast by any body of uniform density can never be the same as that of the body producing it. [Footnote: Comp. the drawing on PI. XXVIII, No. 5.]
No cast shadow can produce the true image of the body which casts it on a vertical plane unless the centre of the light is equally distant from all the edges of that body.
If a window a b admits the sunlight into a room, the sunlight will magnify the size of the window and diminish the shadow of a man in such a way as that when the man makes that dim shadow of himself, approach to that which defines the real size of the window, he will see the shadows where they come into contact, dim and confused from the strength of the light, shutting off and not allowing the solar rays to pass; the effect of the shadow of the man cast by this contact will be exactly that figured above.
[Footnote: It is scarcely possible to render the meaning of this sentence with strict accuracy; mainly because the grammatical construction is defective in the most important part--line 4. In the very slight original sketch the shadow touches the upper arch of the window and the correction, here given is perhaps not justified.]
A shadow is never seen as of uniform depth on the surface which intercepts it unless every portion of that surface is equidistant from the luminous body. This is proved by the 7th which says:--The shadow will appear lighter or stronger as it is surrounded by a darker or a lighter background. And by the 8th of this:--The background will be in parts darker or lighter, in proportion as it is farther from or nearer to the luminous body. And:--Of various spots equally distant from the luminous body those will always be in the highest light on which the rays fall at the smallest angles: The outline of the shadow as it falls on inequalities in the surface will be seen with all the contours similar to those of the body that casts it, if the eye is placed just where the centre of the light was.
The shadow will look darkest where it is farthest from the body that casts it. The shadow c d, cast by the body in shadow a b which is equally distant in all parts, is not of equal depth because it is seen on a back ground of varying brightness. [Footnote: Compare the three diagrams on Pl. VI, no 1 which, in the original accompany this section.]
On the outlines of cast shadows (192-195).
The edges of a derived shadow will be most distinct where it is cast nearest to the primary shadow.
As the derived shadow gets more distant from the primary shadow, the more the cast shadow differs from the primary shadow.
OF SHADOWS WHICH NEVER COME TO AN END.
The greater the difference between a light and the body lighted by it, the light being the larger, the more vague will be the outlines of the shadow of that object.
The derived shadow will be most confused towards the edges of its interception by a plane, where it is remotest from the body casting it.
What is the cause which makes the outlines of the shadow vague and confused?
Whether it is possible to give clear and definite outlines to the edges of shadows.
On the relative size of shadows (196. 197).
THE BODY WHICH IS NEAREST TO THE LIGHT CASTS THE LARGEST SHADOW, AND WHY?
If an object placed in front of a single light is very close to it you will see that it casts a very large shadow on the opposite wall, and the farther you remove the object from the light the smaller will the image of the shadow become.