Thus the surface of water does not form part of the water nor does it form part of the air. . . . What is it therefore that divides the air from the water? There must be a common boundary which is neither air nor water but is without substance. . . . A third body interposed between two bodies would prevent their contact and here water and air are in contact without interposition of anything between them. Therefore they are joined together and the air cannot be moved without the water nor the water raised without drawing it through the air. Therefore a surface is the common boundary of two bodies, and it does not form part of either; for if it did it would have divisible bulk. But since the surface is indivisible, nothingness separates these bodies the one from the other.42
The cylinder of a body is columnar in shape and its two opposite ends are two circles enclosed between parallel lines, and through the centre of the cylinder is a straight line passing through the middle of the thickness of the cylinder, ending at the centres of these circles, and called by ancients axis.43
Propositions
Every body is surrounded by an extreme surface.
Every surface is full of infinite points.
Every point makes a ray.
The ray is made up of infinite separating lines.
In each point on any line, there intersect lines proceeding from the points on the surfaces of bodies, and they form pyramids. At the apex of each pyramid there intersect lines proceeding from the whole and from the parts of the bodies, so that from this apex one may see the whole and the parts.
The air that is between bodies is full of the intersections formed by the radiating images of these bodies.
The images of the figures and their colours are transferred from one to the other by a pyramid.
Each body fills the surrounding air with its infinite images by means of these rays.
The image of each point is in the whole and in each part of the line caused by this point.
Each point of the one object is by analogy capable of uniting the whole base of the other.
Each body becomes the base of innumerable and infinite pyramids. One and the same base serves as the cause of innumerable and infinite pyramids turned in various directions and of various degrees of length.
The point of each pyramid has in itself the whole image of its base.
The centre line of each pyramid is full of an infinite number of points of other pyramids.
One pyramid passes through the other without confusion. . . .44
And as the geometrician reduces every area circumscribed by lines to the square and every body to the cube; and arithmetic does likewise with the cubic and square roots, those two sciences do not extend beyond the study of continuous and discontinuous quantities; but they do not deal with the quality of things which constitute the beauty of the works of nature and the ornament of the world.45
(b) Light and Shade and Colour
Among the various studies of natural processes that of light gives most pleasure to those who contemplate it. And among the chief features of mathematics that of the certainty of its demonstrations elevates the minds of the investigators most powerfully. Perspective, therefore, is to be preferred to all the discourses and systems of the schoolmen. In its province the beam of light is explained by methods of demonstration, wherein is found the glory not only of mathematical but also of physical science, adorned as it is with the flowers of both. And, whereas its propositions have been laid down at great length, I shall abridge them with conclusive brevity, with demonstrations drawn either from nature or from mathematics according to the nature of the subject; sometimes deducing the effects from the causes, and at other times the causes from the effects: adding also to my own conclusions some which are not contained in these but which nevertheless may be inferred from them. Even as the Lord who is the Light of all things shall vouchsafe to enlighten me I will treat of light.46
Look at the light and consider its beauty. Blink your eye and look at it again: what you see was not there at first, and what was there is no more.
Who is it who makes it anew if the maker dies continually? 47
Light is the chaser away of darkness. Shade is the obstruction of light.48
The scientific and true principles of painting first determine what is a shaded object, what is direct shadow, what is cast shadow, and what is light, that is to say, darkness, light, colour, body, figure, position, distance, nearness, motion, and rest. These are understood by the mind alone and do not entail manual operation; and they constitute the science of painting which remains in the mind of its contemplators; and from it is then born the actual creation, which is far superior in dignity to the contemplation or science which precedes it.49
In the practice of perspective the same rules apply to light and to the eye.50
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 its outlines and the outlines themselves will be ill understood unless it is shown against a background of a different tone. Therefore, I state as my first proposition concerning shadows that every opaque body is surrounded and its whole surface enveloped in shadow and light. And to this I shall devote the first book.
Moreover these shadows are of varying degrees of darkness, because they have been abandoned by a varying quantity of luminous rays; and these I call primary shadows, because they are the first shadows to form a covering to the bodies concerned. And to this I shall devote the second book.
From these primary shadows there issue certain dark rays, which are diffused through the air and vary in intensity according to the density of the primary shadows from which they are derived; and consequently I shall call these shadows derived shadows, because they have their origin in other shadows. And of this I will make the third book.
Moreover these derived shadows in striking upon anything create as many different effects as there are different places where they strike; and of this I will make the fourth book.
And since where the derived shadow strikes, it is always surrounded by the striking of the luminous rays, it leaps back with these in a reflex stream towards its source and mingles with and becomes changed into it altering thereby somewhat of its nature; and to this I shall devote the fifth book.
In addition to this I will make a sixth book to contain an investigation of the many different varieties of the rebound of the reflected rays, which modify the primary shadow by as many different colours as there are different points from whence these luminous reflected rays proceed.
Furthermore I will make the seventh book treat of the various distances that may exist between the point where each reflected ray strikes and the point whence it proceeds, and of the various different shades of colour which it acquires in striking against opaque bodies.51
As regards all visible objects, three things must be considered. These are the position of the eye which sees, that of the object seen, and the position of the light which illuminates the object. b is the eye, a is the object seen, c is the light. a is the eye, b the illuminating body, c is the illuminated object.52
Of the nature of shadow
Shadow partakes of the nature of universal matter. All such matter is more powerful in its beginning and grows weaker towards the end; I say at the beginning, whatever their form or condition, whether visible or invisible. It is not from small beginnings that it grows to a great size in time, as a great oak from the small acorn. But on the contrary like the oak which is most powerful at its beginning at its stem where it springs from the earth and is largest. Darkness, then, is the strongest degree of shadow and light is its least. Therefore, O Painter, make your shadows darker close to the object that casts it, and make the end of it fading into light, seeming to have no end.53
Shadow is the diminution alike of light and of darkness, and stands between light and darkness.
A shadow may be infinitely dark, and also of infinite degrees of absence of dark
ness.
The beginnings and ends of shadow lie between the light and the darkness and may be infinitely diminished and infinitely increased.54
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.55
What is the difference between light and the lustre [highlight] which appears on the polished surface of opaque bodies? The lights that are on the polished surface of opaque bodies will be stationary even if the eye which sees them moves. But the reflected light on those same objects will appear in as many different places on the surface as different positions are taken by the eye.56
The highlight or lustre on an object is not necessarily situated in the middle of the illuminated part, but moves as the eye moves in looking at it.57
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 bc and the eye at the point d: I say then that as lustre is everywhere and in each part, if you stand at point d the lustre will appear at c, and as the eye moves from d to a, the lustre will move from c to n.58
Of the difference between lustre and light
Lustre [or highlight] does not partake of the colours but is a sensation of white as derived from the surface of wet bodies; light partakes of the colours of the object which reflects it.58
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 illuminated only by the diffused light of the atmosphere.59
In an object in light and shade, the side which faces the light transmits the images of its details more distinctly and immediately to the eye than the side which is in shadow.60
The more brilliant the light of a luminous body, the deeper the shadows cast by the illuminated object.61
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 further they spread the wider they spread; and an object placed between the light and a wall is always imaged larger in its shadow, because the rays that strike it will have spread by the time they have reached the wall.62
The way in which shadows cast by objects should be defined. If the object is the mountain here figured and the light is at the point a, I say that from bd and from cf there will be no light but from reflected rays. And this is because the rays of light can only act in straight lines; and the same is the case with the secondary or reflected rays.58
Where the shadow should be on the face.63
Very great charm of shadow and light is to be found in the faces of those who sit in the doors of dark houses. The eye of the spectator sees that part of the face which is in shadow lost in the darkness of the house, and that part of the face which is lit draws its brilliancy from the splendour of the sky. From this intensification of light and shade the face gains greatly in relief and beauty by showing the subtlest shadows in the light part and the subtlest lights in the dark part.64
The lights which illumine opaque bodies are of four kinds, namely, universal, as that of the atmosphere within our horizon; and particular, like that of the sun or of a window or door or other space; the third kind is the reflected light; and there is also a fourth kind which passes through substances that are semi-transparent to a certain degree like linen, paper, or such-like things; but not those transparent like glass or crystal or other diaphanous bodies where the effect is the same as if nothing was interposed between the body and the light.65
Of the three kinds of lights which illuminate opaque bodies
The first of the lights with which opaque bodies are illumined is called particular, and it is the sun or the light from a window or a flame. The second is called universal and is seen in cloudy weather or in mist or the like. The third is the subdued light when the sun in the evening or the morning is entirely below the horizon.66
The atmosphere is so adapted as to gather up instantaneously and display every image and likeness of whatever body it sees. When the sun appears in the eastern horizon it at once permeates the whole of our hemisphere and fills it with its luminous semblance.
All the surfaces of solid bodies turned towards the sun or towards the atmosphere illumined by the sun, become clothed and dyed by the light of the sun or of the atmosphere.
Every solid body is surrounded and clothed with light and darkness. You will get only a poor perception of the details of a body when the part that you see is all in shadow, or all illumined.
The distance between the eye and the bodies determines how much the part that is illumined increases and that in shadow diminishes.
The shape of a body cannot be accurately perceived when it is bounded by a colour similar to itself, and the eye is between the part in light and that in shadow.67
What portion of a coloured surface ought in reason to be the most intense? If a is light and b illuminated by it in a direct line, then c on which the light cannot strike is light only by reflection from b, which let us say is red. Then the light reflected from this red surface will tinge the surface at c with red. And if c is red also it will appear much more intense than b; and if it were yellow you would see there a colour between red and yellow.58
The surface of an object partakes of the colour of the light which illuminates it, and of the colour of the air that is interposed between the eye and this object, that is to say of the colour of the transparent medium interposed between the object and the eye.68
Colours seen in shadow will reveal more or less of their natural beauty in proportion as they are in fainter or deeper shadow. But colours seen in a luminous space will reveal greater beauty in proportion as the light is more intense.
Colours seen in shadow will display less variety in proportion as the shadows in which they lie are deeper. And evidence of this is to be had by looking from an open space through the doorways of dark churches, where the pictures painted in various colours all appear covered by darkness. Therefore at a considerable distance all shadows of different colours will appear of the same darkness. In an object in light and shade the light side show its true colour.69
Of the nature of contrasts
Black garments make the flesh tints of the representations of human beings whiter than they are, and white garments make the flesh tints dark, and yellow garments make them seem coloured, while red garments show them pale.70
Of the juxtaposition of one colour next to another
so that one sets off the other
If you wish the proximity of one colour to make attractive another which it borders, observe that rule which is seen in the rays of the sun that compose the rainbow, otherwise called the Iris. Its colours are caused by the motion of the rain, because each little drop changes in the course of its descent into each one of the colours of that rainbow, as will be set forth in the proper place.71
Of secondary colours produced by mixing other colours
The simple colours are six, of which the first is white, although some philosophers do not accept white or black in the number of colours, because one is the cause of colours and the other is the absence of them. Yet, because painters cannot do without them, we include them in their number, and say that in this order white is the first among the simple, and yellow is second, green the third, blue is the fourth, red is the fifth, and black is the sixth.
White we shall put down for the light, without which no colour can be seen, yellow for the Earth, green for the Water, blue for the Air, red for Fire, and black for the darknesses which are above the element of Fire; for there is no matter of any density there, which the rays of the sun must penetrate, and, in consequence, illuminate. If you wish briefly to see all the varieties of composed colours, take panes of coloured glass and look through them at all the colours of the country, which are seen beyond them. Then you will see that the colours of things seen beyond the glass
es are all mixed with the colour of the glass and you will see which colour is strengthened or weakened by this mixture. For example, if the glass is yellow in colour, I say that the visual images of objects which pass through that colour to the eye can either be impaired or improved, and the deterioration will happen to blue, black, and white more than to all the others; and the improvement will occur with yellow and green more than with all the others. Thus you will examine with the eye the mixtures of colours, which are infinite in number; and thus you can make a choice of colours for new combinations of mixed and composed colours. You may do the same with two glasses of different colours held before the eye and thus continue experimenting by yourself.72
Make the rainbow in the last book on Painting but first write the book on colours produced by the mixture of the other colours, so that you may be able to prove by those painters’ colours the genesis of the rainbow colours.73
Every object that has no colour in itself is tinged either entirely or in part by the colour set opposite to it. This may be shown for every object which serves as a mirror is tinged with the colour of the thing that is reflected in it. And if the object is white, the portion of it that is illumined by red will appear red; and so with every other colour whether it be light or dark.
Every opaque and colourless object partakes of the colour of what is opposite to it: as happens with a white wall.74
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