The Notebooks of Leonardo Da Vinci/III
III. Six books on Light and Shade.
_Linear Perspective cannot be immediately followed by either the_ "prospettiva de' perdimenti" _or the_ "prospettiva de' colori" _or the aerial perspective; since these branches of the subject presuppose a knowledge of the principles of Light and Shade. No apology, therefore, is here needed for placing these immediately after Linear Perspective._
_We have various plans suggested by Leonardo for the arrangement of the mass of materials treating of this subject. Among these I have given the preference to a scheme propounded in No._ III, _because, in all probability, we have here a final and definite purpose expressed. Several authors have expressed it as their opinion that the Paris Manuscript_ C _is a complete and finished treatise on Light and Shade. Certainly, the Principles of Light and Shade form by far the larger portion of this MS. which consists of two separate parts; still, the materials are far from being finally arranged. It is also evident that he here investigates the subject from the point of view of the Physicist rather than from that of the Painter._
_The plan of a scheme of arrangement suggested in No._ III _and adopted by me has been strictly adhered to for the first four Books. For the three last, however, few materials have come down to us; and it must be admitted that these three Books would find a far more appropriate place in a work on Physics than in a treatise on Painting. For this reason I have collected in Book V all the chapters on Reflections, and in Book VI I have put together and arranged all the sections of MS._ C _that belong to the book on Painting, so far as they relate to Light and Shade, while the sections of the same MS. which treat of the_ "Prospettiva de' perdimenti" _have, of course, been excluded from the series on Light and Shade._
[Footnote III: This text has already been published with some slight variations in Dozio's pamphlet _Degli scritti e disegni di Leonardo da Vinci_, Milan 1871, pp. 30--31. Dozio did not transcribe it from the original MS. which seems to have remained unknown to him, but from an old copy (MS. H. 227 in the Ambrosian Library).]
You must first explain the theory and then the practice. First you must describe the shadows and lights on opaque objects, and then on transparent bodies.
Scheme of the books on Light and shade.
[Having already treated of the nature of shadows and the way in which they are cast [Footnote 2: _Avendo io tractato._--We may suppose that he here refers to some particular MS., possibly Paris C.], I will now consider the places on which they fall; and their curvature, obliquity, flatness or, in short, any character I may be able to detect in them.]
Shadow is the obstruction of light. Shadows appear to me to be of supreme importance in perspective, because, without them opaque and solid bodies will be ill defined; that which is contained within their outlines and their boundaries themselves will be ill-understood unless they are shown against a background of a different tone from themselves. And therefore in my first proposition concerning shadow I state that every opaque body is surrounded and its whole surface enveloped in shadow and light. And on this proposition I build up the first Book. Besides this, shadows have in themselves various degrees of darkness, because they are caused by the absence of a variable amount of the luminous rays; and these I call Primary shadows because they are the first, and inseparable from the object to which they belong. And on this I will found my second Book. From these primary shadows there result certain shaded rays which are diffused through the atmosphere and these vary in character according to that of the primary shadows whence they are derived. I shall therefore call these shadows Derived shadows because they are produced by other shadows; and the third Book will treat of these. Again these derived shadows, where they are intercepted by various objects, produce effects as various as the places where they are cast and of this I will treat in the fourth Book. And since all round the derived shadows, where the derived shadows are intercepted, there is always a space where the light falls and by reflected dispersion is thrown back towards its cause, it meets the original shadow and mingles with it and modifies it somewhat in its nature; and on this I will compose my fifth Book. Besides this, in the sixth Book I will investigate the many and various diversities of reflections resulting from these rays which will modify the original [shadow] by [imparting] some of the various colours from the different objects whence these reflected rays are derived. Again, the seventh Book will treat of the various distances that may exist between the spot where the reflected rays fall and that where they originate, and the various shades of colour which they will acquire in falling on opaque bodies.
Different principles and plans of treatment (112--116).
First I will treat of light falling through windows which I will call Restricted [Light] and then I will treat of light in the open country, to which I will give the name of diffused Light. Then I will treat of the light of luminous bodies.
The conditions of shadow and light [as seen] by the eye are 3. Of these the first is when the eye and the light are on the same side of the object seen; the 2nd is when the eye is in front of the object and the light is behind it. The 3rd is when the eye is in front of the object and the light is on one side, in such a way as that a line drawn from the object to the eye and one from the object to the light should form a right angle where they meet.
This is another section: that is, of the nature of a reflection (from) an object placed between the eye and the light under various aspects.
As regards all visible objects 3 things must be considered. These are the position of the eye which sees: that of the object seen [with regard] to the light, and the position of the light which illuminates the object, _b_ is the eye, _a_ the object seen, _c_ the light, _a_ is the eye, _b_ the illuminating body, _c_ is the illuminated object.
Let _a_ be the light, _b_ the eye, _c_ the object seen by the eye and in the light. These show, first, the eye between the light and the body; the 2nd, the light between the eye and the body; the 3rd the body between the eye and the light, _a_ is the eye, _b_ the illuminated object, _c_ the light.
OF THE THREE KINDS OF LIGHT THAT ILLUMINATE OPAQUE BODIES.
The first kind of Light which may illuminate opaque bodies is called Direct light--as that of the sun or any other light from a window or flame. The second is Diffused [universal] light, such as we see in cloudy weather or in mist and the like. The 3rd is Subdued light, that is when the sun is entirely below the horizon, either in the evening or morning.
The lights which may illuminate opaque bodies are of 4 kinds. These are: diffused light as that of the atmosphere, within our horizon. And Direct, as that of the sun, or of a window or door or other opening. The third is Reflected light; and there is a 4th which is that which passes through [semi] transparent bodies, as linen or paper or the like, but not transparent like glass, or crystal, or other diaphanous bodies, which produce the same effect as though nothing intervened between the shaded object and the light that falls upon it; and this we will discuss fully in our discourse.
Definition of the nature of shadows (119--122).
WHAT LIGHT AND SHADOW ARE.
Shadow is the absence of light, merely the obstruction of the luminous rays by an opaque body. Shadow is of the nature of darkness. Light [on an object] is of the nature of a luminous body; one conceals and the other reveals. They are always associated and inseparable from all objects. But shadow is a more powerful agent than light, for it can impede and entirely deprive bodies of their light, while light can never entirely expel shadow from a body, that is from an opaque body.
Shadow is the diminution of light by the intervention of an opaque body. Shadow is the counterpart of the luminous rays which are cut off by an opaque body.
This is proved because the shadow cast is the same in shape and size as the luminous rays were which are transformed into a shadow.
Shadow is the diminution alike of light and of darkness, and stands between darkness and light.
A shadow may be infinitely dark, and also of infinite degrees of absence of darkness.
The beginnings and ends of shadow lie between the light and darkness and may be infinitely diminished and infinitely increased. Shadow is the means by which bodies display their form.
The forms of bodies could not be understood in detail but for shadow.
OF THE NATURE OF SHADOW.
Shadow partakes of the nature of universal matter. All such matters are more powerful in their beginning and grow weaker towards the end, I say at the beginning, whatever their form or condition may be and whether visible or invisible. And it is not from small beginnings that they grow to a great size in time; as it might be a great oak which has a feeble beginning from a small acorn. Yet I may say that the oak is most powerful at its beginning, that is where it springs from the earth, which is where it is largest (To return:) Darkness, then, is the strongest degree of shadow and light is its least. Therefore, O Painter, make your shadow darkest close to the object that casts it, and make the end of it fading into light, seeming to have no end.
Of the various kinds of shadows. (123-125).
Darkness is absence of light. Shadow is diminution of light. Primitive shadow is that which is inseparable from a body not in the light. Derived shadow is that which is disengaged from a body in shadow and pervades the air. A cast transparent shadow is that which is surrounded by an illuminated surface. A simple shadow is one which receives no light from the luminous body which causes it. A simple shadow begins within the line which starts from the edge of the luminous body _a b_.
A simple shadow is one where no light at all interferes with it.
A compound shadow is one which is somewhat illuminated by one or more lights.
WHAT IS THE DIFFERENCE BETWEEN A SHADOW THAT IS INSEPARABLE FROM A BODY AND A CAST SHADOW?
An inseparable shadow is that which is never absent from the illuminated body. As, for instance a ball, which so long as it is in the light always has one side in shadow which never leaves it for any movement or change of position in the ball. A separate shadow may be and may not be produced by the body itself. Suppose the ball to be one braccia distant from a wall with a light on the opposite side of it; this light will throw upon the wall exactly as broad a shadow as is to be seen on the side of the ball that is turned towards the wall. That portion of the cast shadow will not be visible when the light is below the ball and the shadow is thrown up towards the sky and finding no obstruction on its way is lost.
HOW THERE ARE 2 KINDS OF LIGHT, ONE SEPARABLE FROM, AND THE OTHER INSEPARABLE FROM BODIES.
Of the various kinds of light (126, 127).
Separate light is that which falls upon the body. Inseparable light is the side of the body that is illuminated by that light. One is called primary, the other derived. And, in the same way there are two kinds of shadow:--One primary and the other derived. The primary is that which is inseparable from the body, the derived is that which proceeds from the body conveying to the surface of the wall the form of the body causing it.
How there are 2 different kinds of light; one being called diffused, the other restricted. The diffused is that which freely illuminates objects. The restricted is that which being admitted through an opening or window illuminates them on that side only.
[Footnote: At the spot marked _A_ in the first diagram Leonardo wrote _lume costretto_ (restricted light). At the spot _B_ on the second diagram he wrote _lume libero_ (diffused light).]
General remarks (128. 129).
Light is the chaser away of darkness. Shade is the obstruction of light. Primary light is that which falls on objects and causes light and shade. And derived lights are those portions of a body which are illuminated by the primary light. A primary shadow is that side of a body on which the light cannot fall.
The general distribution of shadow and light is that sum total of the rays thrown off by a shaded or illuminated body passing through the air without any interference and the spot which intercepts and cuts off the distribution of the dark and light rays.
And the eye can best distinguish the forms of objects when it is placed between the shaded and the illuminated parts.
MEMORANDUM OF THINGS I REQUIRE TO HAVE GRANTED [AS AXIOMS] IN MY EXPLANATION OF PERSPECTIVE.
I ask to have this much granted me--to assert that every ray passing through air of equal density throughout, travels in a straight line from its cause to the object or place it falls upon.
FIRST BOOK ON LIGHT AND SHADE.
On the nature of light (130. 131).
The reason by which we know that a light radiates from a single centre is this: We plainly see that a large light is often much broader than some small object which nevertheless--and although the rays [of the large light] are much more than twice the extent [of the small body]--always has its shadow cast on the nearest surface very visibly. Let _c f_ be a broad light and _n_ be the object in front of it, casting a shadow on the plane, and let _a b_ be the plane. It is clear that it is not the broad light that will cast the shadow _n_ on the plane, but that the light has within it a centre is shown by this experiment. The shadow falls on the plane as is shown at _m o t r_.
[Footnote 13: In the original MS. no explanatory text is placed after this title-line; but a space is left for it and the text beginning at line 15 comes next.] Why, to two [eyes] or in front of two eyes do 3 objects appear as two?
Why, when you estimate the direction of an object with two sights the nearer appears confused. I say that the eye projects an infinite number of lines which mingle or join those reaching it which come to it from the object looked at. And it is only the central and sensible line that can discern and discriminate colours and objects; all the others are false and illusory. And if you place 2 objects at half an arm's length apart if the nearer of the two is close to the eye its form will remain far more confused than that of the second; the reason is that the first is overcome by a greater number of false lines than the second and so is rendered vague.
Light acts in the same manner, for in the effects of its lines (=rays), and particularly in perspective, it much resembles the eye; and its central rays are what cast the true shadow. When the object in front of it is too quickly overcome with dim rays it will cast a broad and disproportionate shadow, ill defined; but when the object which is to cast the shadow and cuts off the rays near to the place where the shadow falls, then the shadow is distinct; and the more so in proportion as the light is far off, because at a long distance the central ray is less overcome by false rays; because the lines from the eye and the solar and other luminous rays passing through the atmosphere are obliged to travel in straight lines. Unless they are deflected by a denser or rarer air, when they will be bent at some point, but so long as the air is free from grossness or moisture they will preserve their direct course, always carrying the image of the object that intercepts them back to their point of origin. And if this is the eye, the intercepting object will be seen by its colour, as well as by form and size. But if the intercepting plane has in it some small perforation opening into a darker chamber--not darker in colour, but by absence of light--you will see the rays enter through this hole and transmitting to the plane beyond all the details of the object they proceed from both as to colour and form; only every thing will be upside down. But the size [of the image] where the lines are reconstructed will be in proportion to the relative distance of the aperture from the plane on which the lines fall [on one hand] and from their origin [on the other]. There they intersect and form 2 pyramids with their point meeting [a common apex] and their bases opposite. Let _a b_ be the point of origin of the lines, _d e_ the first plane, and _c_ the aperture with the intersection of the lines; _f g_ is the inner plane. You will find that _a_ falls upon the inner plane below at _g_, and _b_ which is below will go up to the spot _f_; it will be quite evident to experimenters that every luminous body has in itself a core or centre, from which and to which all the lines radiate which are sent forth by the surface of the luminous body and reflected back to it; or which, having been thrown out and not intercepted, are dispersed in the air.
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).
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.
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_.
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.
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.
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).
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.]
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).]
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).]
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).
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.]
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.]
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).]
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).]
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.
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_.
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).
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.]
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).
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.
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.
WHY A SHADOW LARGER THAN THE BODY THAT PRODUCES IT BECOMES OUT OF PROPORTION.
The disproportion of a shadow which is larger than the body producing it, results from the light being smaller than the body, so that it cannot be at an equal distance from the edges of the body [Footnote 11: H. LUDWIG in his edition of the old copies, in the Vatican library--in which this chapter is included under Nos. 612, 613 and 614 alters this passage as follows: _quella parte ch'e piu propinqua piu cresce che le distanti_, although the Vatican copy agrees with the original MS. in having _distante_ in the former and _propinque_ in the latter place. This supposed amendment seems to me to invert the facts. Supposing for instance, that on Pl. XXXI No. 3. _f_ is the spot where the light is that illuminates the figure there represented, and that the line behind the figure represents a wall on which the shadow of the figure is thrown. It is evident, that in that case the nearest portion, in this case the under part of the thigh, is very little magnified in the shadow, and the remoter parts, for instance the head, are more magnified.]; and the portions which are most remote are made larger than the nearer portions for this reason [Footnote 12: See Footnote 11].
WHY A SHADOW WHICH IS LARGER THAN THE BODY CAUSING IT HAS ILL-DEFINED OUTLINES.
The atmosphere which surrounds a light is almost like light itself for brightness and colour; but the farther off it is the more it loses this resemblance. An object which casts a large shadow and is near to the light, is illuminated both by that light by the luminous atmosphere; hence this diffused light gives the shadow ill-defined edges.
A luminous body which is long and narrow in shape gives more confused outlines to the derived shadow than a spherical light, and this contradicts the proposition next following: A shadow will have its outlines more clearly defined in proportion as it is nearer to the primary shadow or, I should say, the body casting the shadow; [Footnote 14: The lettering refers to the lower diagram, Pl. XLI, No. 5.] the cause of this is the elongated form of the luminous body _a c_, &c. [Footnote 16: See Footnote 14].
Effects on cast shadows by the tone of the back ground.
OF MODIFIED SHADOWS.
Modified shadows are those which are cast on light walls or other illuminated objects.
A shadow looks darkest against a light background. The outlines of a derived shadow will be clearer as they are nearer to the primary shadow. A derived shadow will be most defined in shape where it is intercepted, where the plane intercepts it at the most equal angle.
Those parts of a shadow will appear darkest which have darker objects opposite to them. And they will appear less dark when they face lighter objects. And the larger the light object opposite, the more the shadow will be lightened.
And the larger the surface of the dark object the more it will darken the derived shadow where it is intercepted.
A disputed proposition.
OF THE OPINION OF SOME THAT A TRIANGLE CASTS NO SHADOW ON A PLANE SURFACE.
Certain mathematicians have maintained that a triangle, of which the base is turned to the light, casts no shadow on a plane; and this they prove by saying  that no spherical body smaller than the light can reach the middle with the shadow. The lines of radiant light are straight lines ; therefore, suppose the light to be _g h_ and the triangle _l m n_, and let the plane be _i k_; they say the light _g_ falls on the side of the triangle _l n_, and the portion of the plane _i q_. Thus again _h_ like _g_ falls on the side _l m_, and then on _m n_ and the plane _p k_; and if the whole plane thus faces the lights _g h_, it is evident that the triangle has no shadow; and that which has no shadow can cast none. This, in this case appears credible. But if the triangle _n p g_ were not illuminated by the two lights _g_ and _h_, but by _i p_ and _g_ and _k_ neither side is lighted by more than one single light: that is _i p_ is invisible to _h g_ and _k_ will never be lighted by _g_; hence _p q_ will be twice as light as the two visible portions that are in shadow.
[Footnote: 5--6. This passage is so obscure that it would be rash to offer an explanation. Several words seem to have been omitted.]
On the relative depth of cast shadows (200-202).
A spot is most in the shade when a large number of darkened rays fall upon it. The spot which receives the rays at the widest angle and by darkened rays will be most in the dark; a will be twice as dark as b, because it originates from twice as large a base at an equal distance. A spot is most illuminated when a large number of luminous rays fall upon it. d is the beginning of the shadow _d f_, and tinges _c_ but _a_ little; _d e_ is half of the shadow _d f_ and gives a deeper tone where it is cast at _b_ than at _f_. And the whole shaded space _e_ gives its tone to the spot _a_. [Footnote: The diagram here referred to is on Pl. XLI, No. 2.]
_A n_ will be darker than _c r_ in proportion to the number of times that _a b_ goes into _c d_.
The shadow cast by an object on a plane will be smaller in proportion as that object is lighted by feebler rays. Let _d e_ be the object and _d c_ the plane surface; the number of times that _d e_ will go into _f g_ gives the proportion of light at _f h_ to _d c_. The ray of light will be weaker in proportion to its distance from the hole through which it falls.
FIFTH BOOK ON LIGHT AND SHADE.
Principles of reflection (203. 204).
OF THE WAY IN WHICH THE SHADOWS CAST BY OBJECTS OUGHT TO BE DEFINED.
If the object is the mountain here figured, and the light is at the point _a_, I say that from _b d_ and also from _c f_ there will be no light but from reflected rays. And this results from the fact that rays of light can only act in straight lines; and the same is the case with the secondary or reflected rays.
The edges of the derived shadow are defined by the hues of the illuminated objects surrounding the luminous body which produces the shadow.
Reverberation is caused by bodies of a bright nature with a flat and semi opaque surface which, when the light strikes upon them, throw it back again, like the rebound of a ball, to the former object.
WHERE THERE CAN BE NO REFLECTED LIGHTS.
All dense bodies have their surfaces occupied by various degrees of light and shade. The lights are of two kinds, one called original, the other borrowed. Original light is that which is inherent in the flame of fire or the light of the sun or of the atmosphere. Borrowed light will be reflected light; but to return to the promised definition: I say that this luminous reverberation is not produced by those portions of a body which are turned towards darkened objects, such as shaded spots, fields with grass of various height, woods whether green or bare; in which, though that side of each branch which is turned towards the original light has a share of that light, nevertheless the shadows cast by each branch separately are so numerous, as well as those cast by one branch on the others, that finally so much shadow is the result that the light counts for nothing. Hence objects of this kind cannot throw any reflected light on opposite objects.
Reflection on water (206. 207).
The shadow or object mirrored in water in motion, that is to say in small wavelets, will always be larger than the external object producing it.
It is impossible that an object mirrored on water should correspond in form to the object mirrored, since the centre of the eye is above the surface of the water.
This is made plain in the figure here given, which demonstrates that the eye sees the surface _a b_, and cannot see it at _l f_, and at _r t_; it sees the surface of the image at _r t_, and does not see it in the real object _c d_. Hence it is impossible to see it, as has been said above unless the eye itself is situated on the surface of the water as is shown below .
[Footnote: _A_ stands for _ochio_ [eye], _B_ for _aria_ [air], _C_ for _acqua_ [water], _D_ for _cateto_ [cathetus].--In the original MS. the second diagram is placed below line 13.]
Experiments with the mirror (208-210).
If the illuminated object is of the same size as the luminous body and as that in which the light is reflected, the amount of the reflected light will bear the same proportion to the intermediate light as this second light will bear to the first, if both bodies are smooth and white.
Describe how it is that no object has its limitation in the mirror but in the eye which sees it in the mirror. For if you look at your face in the mirror, the part resembles the whole in as much as the part is everywhere in the mirror, and the whole is in every part of the same mirror; and the same is true of the whole image of any object placed opposite to this mirror, &c.
No man can see the image of another man in a mirror in its proper place with regard to the objects; because every object falls on [the surface of] the mirror at equal angles. And if the one man, who sees the other in the mirror, is not in a direct line with the image he will not see it in the place where it really falls; and if he gets into the line, he covers the other man and puts himself in the place occupied by his image. Let _n o_ be the mirror, _b_ the eye of your friend and _d_ your own eye. Your friend's eye will appear to you at _a_, and to him it will seem that yours is at _c_, and the intersection of the visual rays will occur at _m_, so that either of you touching _m_ will touch the eye of the other man which shall be open. And if you touch the eye of the other man in the mirror it will seem to him that you are touching your own.
Appendix:--On shadows in movement (211. 212).
OF THE SHADOW AND ITS MOTION.
When two bodies casting shadows, and one in front of the other, are between a window and the wall with some space between them, the shadow of the body which is nearest to the plane of the wall will move if the body nearest to the window is put in transverse motion across the window. To prove this let _a_ and _b_ be two bodies placed between the window _n m_ and the plane surface _o p_ with sufficient space between them as shown by the space _a b_. I say that if the body _a_ is moved towards _s_ the shadow of the body _b_ which is at _c_ will move towards _d_.
OF THE MOTION OF SHADOWS.
The motion of a shadow is always more rapid than that of the body which produces it if the light is stationary. To prove this let _a_ be the luminous body, and _b_ the body casting the shadow, and _d_ the shadow. Then I say that in the time while the solid body moves from _b_ to _c_, the shadow _d_ will move to _e_; and this proportion in the rapidity of the movements made in the same space of time, is equal to that in the length of the space moved over. Thus, given the proportion of the space moved over by the body _b_ to _c_, to that moved over by the shadow _d_ to _e_, the proportion in the rapidity of their movements will be the same.
But if the luminous body is also in movement with a velocity equal to that of the solid body, then the shadow and the body that casts it will move with equal speed. And if the luminous body moves more rapidly than the solid body, the motion of the shadow will be slower than that of the body casting it.
But if the luminous body moves more slowly than the solid body, then the shadow will move more rapidly than that body.
SIXTH BOOK ON LIGHT AND SHADE.
The effect of rays passing through holes (213. 214).
If you transmit the rays of the sun through a hole in the shape of a star you will see a beautiful effect of perspective in the spot where the sun's rays fall.
[Footnote: In this and the following chapters of MS. C the order of the original paging has been adhered to, and is shown in parenthesis. Leonardo himself has but rarely worked out the subject of these propositions. The space left for the purpose has occasionally been made use of for quite different matter. Even the numerous diagrams, most of them very delicately sketched, lettered and numbered, which occur on these pages, are hardly ever explained, with the exception of those few which are here given.]
No small hole can so modify the convergence of rays of light as to prevent, at a long distance, the transmission of the true form of the luminous body causing them. It is impossible that rays of light passing through a parallel [slit], should not display the form of the body causing them, since all the effects produced by a luminous body are [in fact] the reflection of that body: The moon, shaped like a boat, if transmitted through a hole is figured in the surface [it falls on] as a boatshaped object. [Footnote 8: In the MS. a blank space is left after this question.] Why the eye sees bodies at a distance, larger than they measure on the vertical plane?.
[Footnote: This chapter, taken from another MS. may, as an exception, be placed here, as it refers to the same subject as the preceding section.]
On gradation of shadows (215. 216).
Although the breadth and length of lights and shadow will be narrower and shorter in foreshortening, the quality and quantity of the light and shade is not increased nor diminished.
The function of shade and light when diminished by foreshortening, will be to give shadow and to illuminate an object opposite, according to the quality and quantity in which they fall on the body.
In proportion as a derived shadow is nearer to its penultimate extremities the deeper it will appear, _g z_ beyond the intersection faces only the part of the shadow [marked] _y z_; this by intersection takes the shadow from _m n_ but by direct line it takes the shadow _a m_ hence it is twice as deep as _g z_. _Y x_, by intersection takes the shadow _n o_, but by direct line the shadow _n m a_, therefore _x y_ is three times as dark as _z g_; _x f_, by intersection faces _o b_ and by direct line _o n m a_, therefore we must say that the shadow between _f x_ will be four times as dark as the shadow _z g_, because it faces four times as much shadow.
Let _a b_ be the side where the primary shadow is, and _b c_ the primary light, _d_ will be the spot where it is intercepted,_f g_ the derived shadow and _f e_ the derived light.
And this must be at the beginning of the explanation.
[Footnote: In the original MS. the text of No. 252 precedes the one given here. In the text of No. 215 there is a blank space of about four lines between the lines 2 and 3. The diagram given on Pl. VI, No. 2 is placed between lines 4 and 5. Between lines 5 and 6 there is another space of about three lines and one line left blank between lines 8 and 9. The reader will find the meaning of the whole passage much clearer if he first reads the final lines 11--13. Compare also line 4 of No. 270.]
On relative proportion of light and shadows (216--221).
That part of the surface of a body on which the images [reflection] from other bodies placed opposite fall at the largest angle will assume their hue most strongly. In the diagram below, 8 is a larger angle than 4, since its base _a n_ is larger than _e n_ the base of 4. This diagram below should end at _a n_ 4 8. That portion of the illuminated surface on which a shadow is cast will be brightest which lies contiguous to the cast shadow. Just as an object which is lighted up by a greater quantity of luminous rays becomes brighter, so one on which a greater quantity of shadow falls, will be darker.
Let 4 be the side of an illuminated surface 4 8, surrounding the cast shadow _g e_ 4. And this spot 4 will be lighter than 8, because less shadow falls on it than on 8. Since 4 faces only the shadow _i n_; and 8 faces and receives the shadow _a e_ as well as _i n_ which makes it twice as dark. And the same thing happens when you put the atmosphere and the sun in the place of shade and light.
 The distribution of shadow, originating in, and limited by, plane surfaces placed near to each other, equal in tone and directly opposite, will be darker at the ends than at the beginning, which will be determined by the incidence of the luminous rays. You will find the same proportion in the depth of the derived shadows _a n_ as in the nearness of the luminous bodies _m b_, which cause them; and if the luminous bodies were of equal size you would still farther find the same proportion in the light cast by the luminous circles and their shadows as in the distance of the said luminous bodies.
[Footnote: The diagram originally placed between lines 3 and 4 is on Pl. VI, No. 3. In the diagram given above line 14 of the original, and here printed in the text, the words _corpo luminoso_ [luminous body] are written in the circle _m_, _luminoso_ in the circle _b_ and _ombroso_ [body in shadow] in the circle _o_.]
THAT PART OF THE REFLECTION WILL BE BRIGHTEST WHERE THE REFLECTED RAYS ARE SHORTEST.
 The darkness occasioned by the casting of combined shadows will be in conformity with its cause, which will originate and terminate between two plane surfaces near together, alike in tone and directly opposite each other.
 In proportion as the source of light is larger, the luminous and shadow rays will be more mixed together. This result is produced because wherever there is a larger quantity of luminous rays, there is most light, but where there are fewer there is least light, consequently the shadow rays come in and mingle with them.
[Footnote: Diagrams are inserted before lines 2 and 4.]
In all the proportions I lay down it must be understood that the medium between the bodies is always the same.  The smaller the luminous body the more distinct will the transmission of the shadows be.
 When of two opposite shadows, produced by the same body, one is twice as dark as the other though similar in form, one of the two lights causing them must have twice the diameter that the other has and be at twice the distance from the opaque body. If the object is lowly moved across the luminous body, and the shadow is intercepted at some distance from the object, there will be the same relative proportion between the motion of the derived shadow and the motion of the primary shadow, as between the distance from the object to the light, and that from the object to the spot where the shadow is intercepted; so that though the object is moved slowly the shadow moves fast.
[Footnote: There are diagrams inserted before lines 2 and 3 but they are not reproduced here. The diagram above line 6 is written upon as follows: at _A lume_ (light), at _B obbietto_ (body), at _C ombra d'obbietto_ (shadow of the object).]
A luminous body will appear less brilliant when surrounded by a bright background.
 I have found that the stars which are nearest to the horizon look larger than the others because light falls upon them from a larger proportion of the solar body than when they are above us; and having more light from the sun they give more light, and the bodies which are most luminous appear the largest. As may be seen by the sun through a mist, and overhead; it appears larger where there is no mist and diminished through mist. No portion of the luminous body is ever visible from any spot within the pyramid of pure derived shadow.
[Footnote: Between lines 1 and 2 there is in the original a large diagram which does not refer to this text. ]
A body on which the solar rays fall between the thin branches of trees far apart will cast but a single shadow.
 If an opaque body and a luminous one are (both) spherical the base of the pyramid of rays will bear the same proportion to the luminous body as the base of the pyramid of shade to the opaque body.
 When the transmitted shadow is intercepted by a plane surface placed opposite to it and farther away from the luminous body than from the object [which casts it] it will appear proportionately darker and the edges more distinct.
[Footnote: The diagram which, in the original, is placed above line 2, is similar to the one, here given on page 73 (section 120).--The diagram here given in the margin stands, in the original, between lines 3 and 4.]
A body illuminated by the solar rays passing between the thick branches of trees will produce as many shadows as there are branches between the sun and itself.
Where the shadow-rays from an opaque pyramidal body are intercepted they will cast a shadow of bifurcate outline and various depth at the points. A light which is broader than the apex but narrower than the base of an opaque pyramidal body placed in front of it, will cause that pyramid to cast a shadow of bifurcate form and various degrees of depth.
If an opaque body, smaller than the light, casts two shadows and if it is the same size or larger, casts but one, it follows that a pyramidal body, of which part is smaller, part equal to, and part larger than, the luminous body, will cast a bifurcate shadow.
[Footnote: Between lines 2 and 3 there are in the original two large diagrams.]