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.]
174
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).
175
ON PAINTING.
The derived shadow is stronger in proportion as it is nearer to its place of origin.
176
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.
177
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.]
178
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.
179
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.]
PROOF.
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).
180
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.]
181
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.
182
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).
183
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