The Notebooks

173. 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.] 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.]