The Notebooks

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. 162. OF SHADOW. 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). 163. 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. 164. 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. 165. 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). 166. 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.] 167. The cast shadow will be longest when the light is lowest. The cast shadow will be shortest when the light is highest. 168. 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).] 169. 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.] 170. The shadow _m_ bears the same proportion to the shadow _n_ as the line _b c_ to the line _f c_. 171. OF PAINTING. 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. 172. 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.