Notebooks (24 page)
Authors: Leonardo da Vinci,Irma Anne Richter,Thereza Wells
Tags: #History, #Fiction, #General, #European, #Art, #Renaissance, #Leonardo;, #Leonardo, #da Vinci;, #1452-1519, #Individual artists, #Art Monographs, #Drawing By Individual Artists, #Notebooks; sketchbooks; etc, #Individual Artist, #History - Renaissance, #Renaissance art, #Individual Painters - Renaissance, #Drawing & drawings, #Drawing, #Techniques - Drawing, #Individual Artists - General, #Individual artists; art monographs, #Art & Art Instruction, #Techniques
Only one line of all those that reach the visual power has no intersection and this has no sensible dimensions because it is a mathematical line which originates from a mathematical point which has no dimension.
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Let
ab
be the vertical [picture] plane and
r
the point of the pyramids terminating in the eye and
n
the point of diminution which is always in a straight line opposite the eye and always moves as the eye moves—just as when a rod is moved its shadow moves, precisely as the shadow moves with a body. And each of the two points is the apex of pyramids having common bases at the intervening vertical planes. But, although their bases are equal, their angles are not equal because the point of diminution is the termination of a smaller angle than that of the eye. If you ask me: ‘By what practical experience can you show me these points?’ I reply—so far as concerns the point of diminution which moves with you—when you walk by a ploughed field look at the straight furrows which come down with their ends to the path where you are walking, and you will see that each pair of furrows will look as though they tried to get nearer and meet at the [further] end. . . .
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2. THE SURFACE OF THINGS AND LIGHTab
be the vertical [picture] plane and
r
the point of the pyramids terminating in the eye and
n
the point of diminution which is always in a straight line opposite the eye and always moves as the eye moves—just as when a rod is moved its shadow moves, precisely as the shadow moves with a body. And each of the two points is the apex of pyramids having common bases at the intervening vertical planes. But, although their bases are equal, their angles are not equal because the point of diminution is the termination of a smaller angle than that of the eye. If you ask me: ‘By what practical experience can you show me these points?’ I reply—so far as concerns the point of diminution which moves with you—when you walk by a ploughed field look at the straight furrows which come down with their ends to the path where you are walking, and you will see that each pair of furrows will look as though they tried to get nearer and meet at the [further] end. . . .
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In so far as the art of painting is concerned with the representation of the surfaces of objects it is allied to spatial geometry. The surfaces are conceived as without substance like geometric planes; and the axioms of Euclid’s Elements defining the point, the line, the plane apply to them.
Moreover, the places and colours of objects are revealed by rays of light which radiate from their source in pyramidal formation. At the bases of these pyramids are the objects which the rays strike at different angles as they drive away the surrounding darkness; and through this interplay of light and shadow the objects emerge into view.
Leonardo excelled in modelling by gradations of light and dark, and was a developer of chiaroscuro (the modelling of forms from white to black) and sfumato (the ‘smoked’ effect of blurring edges).
His treatise on painting was imbued with the conception which he had formed of nature. The four simple colours, red, blue, green, and yellow were related to the four elements, fire, air, water, and earth, the way in which objects were enveloped in space was similar to the way in which the elements came in contact with one another.
(a
) The Geometric Foundation
Let no man who is not a mathematician read the Elements of my work.
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The science of painting begins with the point, then comes the line, the plane comes third, and the fourth the body in its vesture of planes. This is as far as the representation of objects goes. For painting does not, as a matter of fact, extend beyond the surface; and it is by its surface that the body of any visible thing is represented.
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A point is that which has no centre. It has neither breadth, length, nor depth. A line is a length produced by the movement of a point, and its extremities are points. It has neither breadth nor depth. A surface is an extension made by the transversal movement of a line, and its extremities are lines. (A surface has no depth.) A body is a quantity formed by the lateral movement of a surface and its boundaries are surfaces. A body is a length, and it has breadth with depth formed by the lateral movement of its surface.
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1. The surface is a limitation of the body. 2. The limitation of the body is no part of that body. 3. That which is not part of any body is a thing of naught. 4. A thing of naught is that which fills no space. The limitation of one body is that which begins another.
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The limiting surface is the beginning of another. The limits of two coterminous bodies are interchangeably the surface of the one and of the other, as water with air. None of the surfaces of bodies are parts of these bodies.
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The boundaries of bodies are the boundaries of their planes, and the boundaries of the planes are lines. Which lines do not form part of the size of the planes, nor of the atmosphere which surrounds these planes; therefore that which is not part of anything is invisible as is proved in geometry.
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The boundary of one thing with another is of the nature of a mathematical line, but not of a drawn line, because the end of one colour is the beginning of another colour—the boundary is a thing invisible.
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Empty space begins where the object ends. Where empty space ends the object begins and where the object ends emptiness begins.
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The point has no centre, but is itself a centre and nothing can be smaller. The point is the minimum. The point is indivisible by the mind. The point has no parts. The point is the end which nothingness and the line have in common. It is neither nothingness nor line, nor does it occupy a space between them. Therefore the end of nothingness and the beginning of the line are in contact with one another, but they are not joined together, for between them, dividing them, is the point. . . .
And from this it follows that many points imagined in continuous contact do not constitute the line and therefore many lines in continuous contact along their sides do not make a surface, nor do many surfaces in continuous contact make a body, because among us bodies are not formed of incorporeal things. . . .
The contact of the liquid with the solid is a surface common to the liquid and the solid. Similarly the contact between a heavier and a lighter liquid is a surface common to them both. The surface does not form part of either—it is merely the common boundary.
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.
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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.
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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. . . .
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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.
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(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.
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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?
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Light is the chaser away of darkness. Shade is the obstruction of light.
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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.
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In the practice of perspective the same rules apply to light and to the eye.
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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.
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