Book One

In all this discussion, I beg you to consider me not as a mathematician but as a painter writing of these things. Mathematicians measure with their minds alone the forms of things separated from all matter. Since we wish the object to be seen, we will use a more sensate wisdom. [7] We will consider our aim accomplished if the reader can understand in any way this admittedly difficult subject--and, so far as I know, a subject never before treated. Therefore, I beg that my words be interpreted solely as those of a painter.

I say, first of all, we ought to know that a point is a figure which cannot be divided into parts. I call a figure here anything located on a plane so the eye can see it. No one would deny that the painter has nothing to do with things that are not visible. [8] The painter is concerned solely with representing what can be seen. These points, if they are joined one to the other in a row, will form a line. With us a line is a figure whose length can be divided but whose width is so fine that it cannot be split. Some lines are called straight, others curved. A straight line is drawn [p. 43] directly from one point to another as an extended point. The curved line is not straight from one point to another but rather looks like a drawn bow. [9] More lines, like threads woven together in a cloth, make a plane. [10] The plane is that certain external part of a body which is known not by its depth but only by its length and breadth and by its quality. Some qualities remain permanently on the plane in such a manner that they cannot be changed without altering the plane itself. Other qualities are such that, due to visual effects, they seem to change to the observer even though the plane remains the same.

Permanent qualities are of two kinds. One is known by the outermost boundary [11] which encloses the plane and may be terminated by one or more lines. Some are circular, others are a curved and a straight line or several straight lines together. The circular is that which encloses a circle. A circle is that form of a plane which an entire line encircles like a garland. If a point is established in the middle, all lines from this point to the garland will be equal. This point in the middle is called the centre. A straight line which covers the point and cuts the circle into two parts is called the diameter among mathematicians, but I prefer to call it the centric line. Let us agree with the mathematicians who say that no line cuts equal angles on the circumference unless it is a straight line which covers the centre.

But let us return to the plane. It is clear that as the movement [12] of the outline is changed the plane changes both name and appearance so that it is now called a triangle, now a quadrangle and now a polygon. The outline is said to be changed if the lines are more or less lengthened or shortened, or better, if the angles are made more acute or more obtuse. It would be well to speak of angles here.

I call angles the certain extremity of a plane made of two lines which cut each other. There are three kinds of angles; right, obtuse, acute. A right angle is one of four made by two straight lines where one cuts the other in such a way that each [p. 44] of the angles is equal to the others. From this it is said that all right angles are equal. The obtuse angle is that which is greater than the right, and that which is lesser is called acute.

Again let us return to the plane. Let us agree that so long as the lines and the angles of the outline do not change, the plane will remain the same. We have then demonstrated a quality which is never separated from the plane.

We have now to treat of other qualities which rest like a skin [13] over all the surface of the plane. These are divided into three sorts. Some planes are flat, others are hollowed out, and others are swollen outward and are spherical. To these a fourth may be added which is composed of any two of the above. The flat plane is that which a straight ruler will touch in every part if drawn over it. The surface of the water is very similar to this. The spherical plane is similar to the exterior of a sphere. We say the sphere is a round body, continuous in every part; any part on the extremity of that body is equidistant from its centre. The hollowed plane is within and under the outermost extremities of the spherical plane as in the interior of an egg shell. The compound plane is in one part flat and in another hollowed or spherical like those on the interior of reeds or on the exterior of columns. [14]

The outline and the surface, [15] then, give their names to the plane but there are two qualities by which the plane is not altered, [although it appears to be]. These take their variations from the changing of place and of light. Let us speak first of place, then of light, and investigate in what manner the qualities of the plane appear to change.

This has to do with the power of sight, for as soon as the observer changes his position these planes appear larger, of a different outline or of a different colour. All of [these qualities] are measured with sight. Let us investigate the reasons for this, beginning with the maxims of philosophers who affirm that the plane is measured by rays that serve the sight--called by them visual rays--which carry the form of the thing seen to the [p. 45] sense. [16] For these same rays extended between the eye and the plane seen come together very quickly by their own force and by a certain marvellous subtlety, penetrating the air and thin and clear objects they strike against something dense and opaque, where they strike with a point and adhere to the mark they make. Among the ancients there was no little dispute whether these rays come from the eye or the plane. This dispute is very difficult and is quite useless for us. It will not be considered. We can imagine those rays to be like the finest hairs of the head, or like a bundle, tightly bound within the eye where the sense of sight has its seat. The rays, gathered together within the eye, are like a stalk; the eye is like a bud which extends its shoots rapidly and in a straight line on the plane opposite. [17]

Among these rays there are differences in strength and function which must be recognized. Some of these rays strike the outline of the plane and measure its quantity. Since they touch the ultimate and extreme parts of the plane, we can call them the extreme or, if you prefer, extrinsic. Other rays which depart from the surface of the plane for the eye fill the pyramid--of which we shall speak more later--with the colours and brilliant lights with which the plane gleams; these are called median rays. Among these visual rays there is one which is called the centric. Where this one touches the plane, it makes equal the right angles all around it. It is called centric for the same reason as the aforementioned centric line. [18]

We have found three different sorts of rays: extreme, median and centric. Now let us investigate how each ray affects the sight. First we shall speak of the extreme, then of the median, finally of the centric.

With the extreme rays quantity is measured. All space on the plane that is between any two paints on the outline is called quantity. The eye measures these quantities with the visual rays as with a pair of compasses. In every plane there are as many quantities as there are spaces between point and point. Height from top to bottom, width from left to right, breadth from near to far and whatever other dimension or measure which is made [p. 46] by sight makes use of the extreme rays. For this reason it is said that vision makes a triangle. The base of [this triangle] is the quantity seen and the sides are those rays which are extended from the quantity to the eye. It is, therefore, very certain that no quantity can be seen without the triangle. The angles in this visual triangle are first, the two paints of the quantity, the third, that which is opposite the base and located within the eye. [19] Nor is this the place to discuss whether vision, as it is called, resides at the juncture of the inner nerve or whether images are formed on the surface of the eye as on a living mirror. The function of the eyes in vision need not be considered in this place. It will be enough in this commentary to demonstrate briefly things that are essential.

Here is a rule: as the angle within the eye becomes more acute, so the quantity seen appears smaller. From this it is clear why a very distant quantity seems to be no larger than a point. Even though this is so, it is possible to find some quantities and planes of which the less is seen when they are closer and more when they are farther away. The proof of this is found in spherical bodies. Therefore, the quantities, through distance, appear either larger or smaller. Anyone who understands what has already been said will understand, I believe, that as the interval is changed the extrinsic rays become median and in the same manner the median extrinsic. He will understand also that where the median rays are made extrinsic that quantity will appear smaller. And the contrary: when the extreme rays are directed within the outline, as the outline is more distant, so much the quantity seen will seem greater. Here I usually give my friends a similar rule: as more rays are used in seeing, so the thing seen appears greater; and the fewer the rays, the smaller.

The extrinsic rays, thus encircling the plane--one touching the other--enclose all the plane like the willow wands of a basket-cage, and make, as is said, this visual pyramid. It is time for me to describe what this pyramid is and how it is constructed by these rays. I will describe it in my own way. [20] The pyramid is a figure of a body from whose base straight lines are [p. 47] drawn upward, terminating in a single point. The base of this pyramid is a plane which is seen. The sides of the pyramid are those rays which I have called extrinsic. The cuspid, that is the point of the pyramid, is located within the eye where the angle of the quantity is. Up to this point we have talked of the extrinsic rays of which this pyramid is constructed. It seems to me that we have demonstrated the varied effects of greater and lesser distances from the eye to the thing seen.

Median rays, that multitude in the pyramid [which lie] within the extrinsic rays, remain to be treated. These behave, in a manner of speaking, like the chameleon, an animal which takes to itself the colours of things near it. Since these rays carry both the colours and lights on the plane from where they touch it up to the eye, they should be found lighted and coloured in a definite way wherever they are broken. The proof of this is that through a great distance they become weakened. I think the reason may be that weighted down with light and colour they pass through the air, which, being humid with a certain heaviness, tires the laden rays. From this we can draw a rule: as the distance becomes greater, so the plane seen appears more hazy. The central ray now remains to be treated. The central ray is that single one which alone strikes the quantity directly, and about which every angle is equal. This ray, the most active and the strongest of all the rays, acts so that no quantity ever appears greater than when struck by it. We could say many things about this ray, but this will be enough--tightly encircled by the other rays, it is the last to abandon the thing seen, from which it merits the name, prince of rays.

I think I have clearly demonstrated that as the distance and the position of the central ray are changed the plane appears altered. Therefore, the distance and the position of the central ray are of greatest importance to the certainty of sight.

There is yet a third thing which makes the plane appear to change. This comes from the reception of light. You see that spherical and concave planes have one part dark and anther [p. 48] bright when receiving light. Even though the distance and position of the centric line are the same, when the light is moved those parts which were first bright now become dark, and those bright which were dark. Where there are more lights, according to their number and strength, you see more spots of light and dark.

This reminds me to speak of both colour and light It seems obvious to me that colours take their variations from light, because all colours put in the shade appear different from what they are in the light. Shade makes colour dark; light, where it strikes, makes colour bright. The philosophers say that nothing can be seen which is not illuminated and coloured. Therefore, they assert that there is a close relationship between light and colour in making each other visible. The importance of this is easily demonstrated for [21] when light is lacking colour is lacking and when light returns the colours return. Therefore, it seems to me that I should speak first of colours; then I shall investigate how they vary under light. [22] Let us omit the debate of philosophers where the original source of colours is investigated, for what help is it for a painter to know in what mixture of rare and dense, warm and dry, cold and moist colour exists? However, I do not despise those philosophers who thus dispute about colours and establish the kinds of colours at seven. White and black [are] the two extremes of colour. Another [is established] between them. Then between each extreme and the middle they place a pair of colours as though undecided about the boundary, because one philosopher allegedly knows more about the extreme than the other. It is enough for the painter to know what the colours are and how to use them in painting. I do not wish to be contradicted by the experts, who, while they follow the philosophers, assert that there are only two colours in nature, white and black, and there are others created from mixtures of these two. As a painter I think thus about colours. From a mixture of colours almost infinite others are created. I speak here as a painter.

Through the mixing of colours infinite other colours are born, but there are only four true colours--as there are four [p. 49] elements--from which more and more other kinds of colours may be thus created. Red is the colour of fire, blue of the air, green of the water, and of the earth grey and ash. [23] Other colours, such as jasper and porphyry, are mixtures of these. Therefore, there are four genera of colours, and these make their species [24] according to the addition of dark or light, black or white. They are thus almost innumerable. We see green fronds lose their greenness little by little until they finally become pale. Similarly, it is not unusual to see a whitish vapour in the air around the horizon which fades out little by little [as one looks towards the zenith]. We see some roses which are quite purple, others are like the cheeks of young girls, [25] others ivory. In the same way the earth [en colour], according to white and black, makes its own species of colours.

Therefore, the mixing of white does not change the genus of colours but forms the species. Black contains a similar force in its mixing to make almost infinite species of colour. In shadows colours are altered. As the shadow deepens the colours empty out, and as the light increases the colours become more open and clear. For this reason the painter ought to be persuaded that white and black are not true colours but are alterations of other colours. The painter will find no thing with which to represent the brightest luster of light but white and in the same manner only black to indicate the shadows. I should like to add that one will never find black and white unless they are [mixed] with one of these four colours.

Here follow my remarks on light. Some lights are from the stars, as from the sun, from the moon and that other beautiful star Venus. [26] Other lights are from fires, but among these there are many differences. The light from the stars makes the shadow equal to the body, but fire makes it greater.

Shadow in which the rays of light are interrupted remains to be treated. The interrupted rays either return from whence they came or are directed elsewhere. They are directed elsewhere, when, touching the surface of the water, they strike the rafters [p. 50] of a house. More can be said about this reflection which has to do with these miracles of painting which many of my friends have seen done by me recently in Rome. [27] It is enough [to say] here that these reflected rays carry with themselves the colour they find on the plane. You may have noticed that anyone who walks through a meadow in the sun appears greenish in the face.

Up to this point we have talked of planes and rays; we have said how a pyramid is made in vision; we have proved the importance of distance and position of the centric ray together with the reception of light. Now, since in a single glance not only one plane but several are seen, we will investigate in what way many conjoined [planes] are seen. Each plane contains in itself its pyramid of colours and lights. Since bodies are covered with planes, all the planes of a body seen at one glance will make a pyramid packed [28] with as many smaller pyramids as there are planes.

Some will say here of what use to the painter is such an investigation? I think every painter, if he wishes to be a great master, ought to understand clearly the similarities and the distinctions [29] of the planes, a thing known to very few. Should you ask some what they are doing when they cover a plane with colours, they will answer everything but what you ask. Therefore, I beg studious painters not to be embarrassed by what I say here. It is never wrong to learn something useful to know from anyone. They should know that they circumscribe the plane with their lines. When they fill the circumscribed places with colours, they should only seek to present the forms of things seen on this plane as if it were of transparent glass. Thus the visual pyramid could pass through it, placed at a definite distance with definite lights and a definite position of centre in space and in a definite place in respect to the observer. Each painter, endowed with his natural instinct, [30] demonstrates this when, in painting this plane, he places himself at a distance as if searching the point and angle of the pyramid from which point he understands the thing painted is best seen. [p. 51]

Where this is a single plane, either a wall or a panel on which the painter attempts to depict several planes comprised in the visual pyramid, it would be useful to cut through this pyramid in some definite place, so the painter would be able to express in painting similar outlines and colours with his lines. He who looks at a picture, done as I have described [above], will see a certain cross-section of a visual pyramid, artificially represented with lines and colours on a certain plane according to a given distance, centre and lights. Now, since we have said that the picture is a cross-section of the pyramid we ought to investigate what importance this cross-section has for us. Since we have these knowns, we now have new principles with which to reason about the plane from which we have said the pyramid issues.

I say that some planes are thrown back on the earth and lie like pavements or the floors of buildings; others are equidistant to these. Some stand propped up on their sides like walls; other planes are collinear to these walls. Planes are equidistant when the distance between one and the other is equal in all its parts. Collinear planes are those which a straight line will touch equally in ever part as in the faces of quadrangular pilasters placed in a row in a portico. [31] These things are to be added to our treatment of the plane, intrinsic and extrinsic and centric rays and the pyramid. Let us add the axiom of the mathematicians where it is proved that if a straight line cuts two sides of a triangle, and if this line which forms a triangle is parallel to a side of the first and greater triangle, certainly this lesser triangle will be proportional to the greater. So much say the mathematicians.

I shall speak in a broader manner to make my statements clearer. It is useful to know what the term proportional means. Proportional triangles are said to be those whose sides and angles contain a ratio to each other. If one side of a triangle is two times as long as its base and the other side three, every single triangle--whether larger or smaller, but having this same [p. 52] relationship to its base--will be proportional to this first, because the ratio which is in every part of the smaller triangle is also the same in the larger. Therefore, all triangles thus composed will be proportional to each other. [32] To understand this better we will use a simile. A small man is proportional to a larger one, because the same proportions between the palm and the foot, the foot and the other parts of the body were in Evander as in Hercules whom Aulus Gellius considered to be the largest of men. [33] There was no difference in the proportions of the bodies of Hercules and Antaeus the giant, for both contained the same ratio and arrangement of hand to forearm, forearm to head and thus through all the members. In the same way a measure is found by which a smaller triangle is equal to a greater--except in size. Here I must insist with the mathematicians, in so far as it pertains to us, that the intercision of any triangle, if it is parallel to the base makes a new triangle proportionate to the larger one. Things which are proportional to each other correspond in every part, but where they are different and the parts do not correspond they are certainly not proportional.

As I have said, the parts of the visual triangle are rays. These will be equal, as to number, in proportionate quantities and unequal in non-proportional, because one of the non-proportional quantities will occupy more or less rays. You see, then, how a lesser triangle can be proportional to a greater, and you have already learned that the visual pyramid is composed of triangles.

Now let us translate our thinking to the pyramid. We should be persuaded that no quantities equidistant to the cross-section can make any alteration in the picture, because they are similar to their proportionates in every equidistant intercision. From this it follows that when the quantity with which the outline is constructed is not changed, there will be no alteration of the same outline in the picture. It is now manifest that every cross-section of the visual pyramid which is equidistant to the plane [p. 53] of the thing seen will be proportional to that observed plane. [34]

We have talked about the plane proportional to the cross-section, that is, equidistant, to the painted plane; but since many planes are found to be non-equidistant, we ought to make a diligent investigation of these in order that our reasoning about the cross-section may be clear. It would be long, difficult and obscure in these cross-sections of triangles and pyramids to follow everything with the rule of mathematics, so let us rather continue speaking as painters. I shall treat most briefly of the non-equidistant quantities. When they are known, we will easily understand the non-equidistant planes.

Some non-equidistant quantities [35] are collinear to the visual rays, others are equidistant to the visual rays. Quantities collinear to the visual rays have no place in the cross-section, because they do not make a triangle nor do they occupy a number of rays. In quantities equidistant to the visual rays, as the angle which is greatest in the triangle is more obtuse at the base, so that quantity will occupy fewer rays and for this reason less space in the cross-section. We have said concerning this that the plane is covered with quantities, but it happens frequently that there are several quantities in a plane equidistant to the cross-section. Quantities so composed will certainly make no alteration in the picture. In such quantities which are truly non-equidistant the greater the angle at the base the greater alteration they will make.