Section 4 of 32 - Table of Contents
131. THE RAYS WHETHER SHADED OR LUMINOUS HAVE GREATER STRENGTH AND EFFECT AT THEIR POINTS THAN AT THEIR SIDES. Although the points of luminous pyramids may extend into shaded places and those of pyramids of shadow into illuminated places, and though among the luminous pyramids one may start from a broader base than another; nevertheless, if by reason of their various length these luminous pyramids acquire angles of equal size their light will be equal; and the case will be the same with the pyramids of shadow; as may be seen in the intersected pyramids _a b c_ and _d e f_, which though their bases differ in size are equal as to breadth and light. [Footnote: 51--55: This supplementary paragraph is indicated as being a continuation of line 45, by two small crosses.] The difference between light and lustre (132--135). 132. Of the difference between light and lustre; and that lustre is not included among colours, but is saturation of whiteness, and derived from the surface of wet bodies; light partakes of the colour of the object which reflects it (to the eye) as gold or silver or the like. 133. OF THE HIGHEST LIGHTS WHICH TURN AND MOVE AS THE EYE MOVES WHICH SEES THE OBJECT. Suppose the body to be the round object figured here and let the light be at the point _a_, and let the illuminated side of the object be _b c_ and the eye at the point _d_: I say that, as lustre is every where and complete in each part, if you stand at the point _d_ the lustre will appear at _c_, and in proportion as the eye moves from _d_ to _a_, the lustre will move from _c_ to _n_. 134. OF PAINTING. Heigh light or lustre on any object is not situated [necessarily] in the middle of an illuminated object, but moves as and where the eye moves in looking at it. 135. OF LIGHT AND LUSTRE. What is the difference between light and the lustre which is seen on the polished surface of opaque bodies? The lights which are produced from the polished surface of opaque bodies will be stationary on stationary objects even if the eye on which they strike moves. But reflected lights will, on those same objects, appear in as many different places on the surface as different positions are taken by the eye. WHAT BODIES HAVE LIGHT UPON THEM WITHOUT LUSTRE? Opaque bodies which have a hard and rough surface never display any lustre in any portion of the side on which the light falls. WHAT BODIES WILL DISPLAY LUSTRE BUT NOT LOOK ILLUMINATED? Those bodies which are opaque and hard with a hard surface reflect light [lustre] from every spot on the illuminated side which is in a position to receive light at the same angle of incidence as they occupy with regard to the eye; but, as the surface mirrors all the surrounding objects, the illuminated [body] is not recognisable in these portions of the illuminated body. 136. The relations of luminous to illuminated bodies. The middle of the light and shade on an object in light and shade is opposite to the middle of the primary light. All light and shadow expresses itself in pyramidal lines. The middle of the shadow on any object must necessarily be opposite the middle of its light, with a direct line passing through the centre of the body. The middle of the light will be at _a_, that of the shadow at _b_. [Again, in bodies shown in light and shade the middle of each must coincide with the centre of the body, and a straight line will pass through both and through that centre.] [Footnote: In the original MS., at the spot marked _a_ of the first diagram Leonardo wrote _primitiuo_, and at the spot marked _c_--_primitiva_ (primary); at the spot marked _b_ he wrote _dirivatiuo_ and at _d deriuatiua_ (derived).] Experiments on the relation of light and shadow within a room (137--140). 137. SHOWS HOW LIGHT FROM ANY SIDE CONVERGES TO ONE POINT. Although the balls _a b c_ are lighted from one window, nevertheless, if you follow the lines of their shadows you will see they intersect at a point forming the angle _n_. [Footnote: The diagram belonging to this passage is slightly sketched on Pl. XXXII; a square with three balls below it. The first three lines of the text belonging to it are written above the sketch and the six others below it.] 138. Every shadow cast by a body has a central line directed to a single point produced by the intersection of luminous lines in the middle of the opening and thickness of the window. The proposition stated above, is plainly seen by experiment. Thus if you draw a place with a window looking northwards, and let this be _s f_, you will see a line starting from the horizon to the east, which, touching the 2 angles of the window _o f_, reaches _d_; and from the horizon on the west another line, touching the other 2 angles _r s_, and ending at _c_; and their intersection falls exactly in the middle of the opening and thickness of the window. Again, you can still better confirm this proof by placing two sticks, as shown at _g h_; and you will see the line drawn from the centre of the shadow directed to the centre _m_ and prolonged to the horizon _n f_. [Footnote: _B_ here stands for _cerchio del' orizonte tramontano_ on the original diagram (the circle of the horizon towards the North); _A_ for _levante_ (East) and _C_ for _ponete_ (West).] 139. Every shadow with all its variations, which becomes larger as its distance from the object is greater, has its external lines intersecting in the middle, between the light and the object. This proposition is very evident and is confirmed by experience. For, if _a b_ is a window without any object interposed, the luminous atmosphere to the right hand at _a_ is seen to the left at _d_. And the atmosphere at the left illuminates on the right at _c_, and the lines intersect at the point _m_. [Footnote: _A_ here stands for _levante_ (East), _B_ for _ponente_ (West).] 140. Every body in light and shade is situated between 2 pyramids one dark and the other luminous, one is visible the other is not. But this only happens when the light enters by a window. Supposing _a b_ to be the window and _r_ the body in light and shade, the light to the right hand _z_ will pass the object to the left and go on to _p_; the light to the left at _k_ will pass to the right of the object at _i_ and go on to _m_ and the two lines will intersect at _c_ and form a pyramid. Then again _a_ _b_ falls on the shaded body at _i_ _g_ and forms a pyramid _f_ _i_ _g_. _f_ will be dark because the light _a_ _b_ can never fall there; _i_ _g_ _c_ will be illuminated because the light falls upon it. Light and shadow with regard to the position of the eye (141--145). 141. Every shaded body that is larger than the pupil and that interposes between the luminous body and the eye will be seen dark. When the eye is placed between the luminous body and the objects illuminated by it, these objects will be seen without any shadow. [Footnote: The diagram which in the original stands above line 1 is given on Plate II, No 2. Then, after a blank space of about eight lines, the diagram Plate II No 3 is placed in the original. There is no explanation of it beyond the one line written under it.] 142. Why the 2 lights one on each side of a body having two pyramidal sides of an obtuse apex leave it devoid of shadow. [Footnote: The sketch illustrating this is on Plate XLI No 1.] 143. A body in shadow situated between the light and the eye can never display its illuminated portion unless the eye can see the whole of the primary light. [Footnote: _A_ stands for _corpo_ (body), _B_ for _lume_ (light).] 144. The eye which looks (at a spot) half way between the shadow and the light which surrounds the body in shadow will see that the deepest shadows on that body will meet the eye at equal angles, that is at the same angle as that of sight. [Footnote: In both these diagrams _A_ stands for _lume_ (light) _B_ for _ombra_ (shadow).] 145. OF THE DIFFERENT LIGHT AND SHADE IN VARIOUS ASPECTS AND OF OBJECTS PLACED IN THEM. If the sun is in the East and you look towards the West you will see every thing in full light and totally without shadow because you see them from the same side as the sun: and if you look towards the South or North you will see all objects in light and shade, because you see both the side towards the sun and the side away from it; and if you look towards the coming of the sun all objects will show you their shaded side, because on that side the sun cannot fall upon them. The law of the incidence of light. 146. The edges of a window which are illuminated by 2 lights of equal degrees of brightness will not reflect light of equal brightness into the chamber within. If _b_ is a candle and _a c_ our hemisphere both will illuminate the edges of the window _m_ _n_, but light _b_ will only illuminate _f g_ and the hemisphere _a_ will light all of _d e_. 147. OF PAINTING. That part of a body which receives the luminous rays at equal angles will be in a higher light than any other part of it. And the part which the luminous rays strike between less equal angles will be less strongly illuminated. SECOND BOOK ON LIGHT AND SHADE. Gradations of strength in the shadows (148. 149). 148. THAT PORTION OF A BODY IN LIGHT AND SHADE WILL BE LEAST LUMINOUS WHICH IS SEEN UNDER THE LEAST AMOUNT OF LIGHT. That part of the object which is marked _m_ is in the highest light because it faces the window _a d_ by the line _a f_; _n_ is in the second grade because the light _b d_ strikes it by the line _b e_; _o_ is in the third grade, as the light falls on it from _c d_ by the line _c h_; _p_ is the lowest light but one as _c d_ falls on it by the line _d v_; _q_ is the deepest shadow for no light falls on it from any part of the window. In proportion as _c d_ goes into _a d_ so will _n r s_ be darker than _m_, and all the rest is space without shadow. [Footnote: The diagram belonging to this chapter is No. 1 on Plate III. The letters _a b e d_ and _r_ are not reproduced in facsimile of the original, but have been replaced by ordinary type in the margin. 5-12. The original text of these lines is reproduced within the diagram.--Compare No 275.] 149. The light which falls on a shaded body at the acutest angle receives the highest light, and the darkest portion is that which receives it at an obtuse angle and both the light and the shadow form pyramids. The angle _c_ receives the highest grade of light because it is directly in front of the window _a b_ and the whole horizon of the sky _m x_. The angle _a_ differs but little from _c_ because the angles which divide it are not so unequal as those below, and only that portion of the horizon is intercepted which lies between _y_ and _x_. Although it gains as much on the other side its line is nevertheless not very strong because one angle is smaller than its fellow. The angles _e i_ will have less light because they do not see much of the light _m s_ and the light _v x_ and their angles are very unequal. Yhe angle _k_ and the angle _f_ are each placed between very unequal angles and therefore have but little light, because at _k_ it has only the light _p t_, and at _f_ only _t q_; _o g_ is the lowest grade of light because this part has no light at all from the sky; and thence come the lines which will reconstruct a pyramid that is the counterpart of the pyramid _c_; and this pyramid _l_ is in the first grade of shadow; for this too is placed between equal angles directly opposite to each other on either side of a straight line which passes through the centre of the body and goes to the centre of the light. The several luminous images cast within the frame of the window at the points _a_ and _b_ make a light which surrounds the derived shadow cast by the solid body at the points 4 and 6. The shaded images increase from _o g_ and end at 7 and 8. [Footnote: The diagram belonging to this chapter is No. 2 on Plate III. In the original it is placed between lines 3 and 4, and in the reproduction these are shown in part. The semi circle above is marked _orizonte_ (horizon). The number 6 at the left hand side, outside the facsimile, is in the place of a figure which has become indistinct in the original.] On the intensity of shadows as dependent on the distance from the light (150-152). 150. The smaller the light that falls upon an object the more shadow it will display. And the light will illuminate a smaller portion of the object in proportion as it is nearer to it; and conversely, a larger extent of it in proportion as it is farther off. 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. 151. That portion of an illuminated object which is nearest to the source of light will be the most strongly illuminated. 152. 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). 153. 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. 154. OF PERSPECTIVE. 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.]. 155. OF PAINTING. 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. 156. OF PAINTING. 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. 157. 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). 158. 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. 159. 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). 160. 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. 161. 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. 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.] 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 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 [10]. 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 [13]. 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.] 184. 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? ANSWER. 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. DEFINITION. 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. 186. 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. 187. 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). 188. 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.] 189. 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. 190. 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.] 191. 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). 192. The edges of a derived shadow will be most distinct where it is cast nearest to the primary shadow.Prev Next All
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