Mighty is geometry; joined with art, resistless.
EURIPIDES
The new currents of thought in the European Renaissance, the search for new truths to replace the discredited ones, the turn to the study of nature to obtain reliable facts, and the revived Greek conviction that the essence of nature’s behavior should be sought in mathematical laws, bore fruit first in the field of art rather than science. While philosophers and scientists sought to unearth basic facts which might somehow be incorporated into their yet to be formulated new scientific method, and while mathematicians were still digesting the Greek works and awaiting inspiration for new themes, the artists, particularly the painters, reacted far more quickly and revolutionized the art of painting.
That the painters turned to mathematics to formulate their new style of painting is a little surprising, but the phenomenon has an explanation. The painters of the fourteenth, fifteenth, and sixteenth centuries were the architects and engineers of their time. They were also the sculptors, inventors, goldsmiths, and stonecutters. They designed and built churches, hospitals, palaces, cloisters, bridges, dams, fortresses, canals, town walls, and weapons. Thus Leonardo da Vinci, in offering his services to Lodovico Sforza, ruler of Milan, promises to serve as engineer, constructor of military works, and designer of war machines, as well as architect, sculptor, and painter. The artist was even expected to predict the motion of cannon balls, by no means a simple problem for the mathematics of those times. In view of these manifold activities the painter necessarily had to be something of a scientist.
Further, the Renaissance painter, unlike the builder of Gothic cathedrals, was influenced by the current doctrines which proclaimed that he learn truths from nature and that the essence of natural phenomena is best expressed through mathematics. Again, in comparison with his predecessors, he had the advantage of gleaning some mathematical knowledge from the newly recovered Greek works that were exciting the Europeans. The Renaissance painters went so far in assimilating this knowledge and in applying mathematics to painting that they produced the first really new mathematics in Europe. In the fifteenth century they were the most accomplished and also the most original mathematicians.
Before we examine just how Renaissance painters employed mathematics and thereby revolutionized the art of painting, let us see what had been going on in this field. Rather early in the medieval period painting became an extensive activity. Kings, princes, and church leaders commissioned works of art to enhance buildings. The system which the medieval painters used until about 1300 was conceptual. Their objective was to portray and embellish the central themes in the Christian drama. Since the intent was to stir up religious feelings rather than to present real scenes, people and objects were drawn in accordance with conventions which had acquired symbolic meaning. Thus people were placed in unnatural, stylized positions; the general impression was one of flatness; and the entire painting had a two-dimensional effect. The backgrounds were usually solid gold to suggest that the action or people existed in some supra-earthly region.
Examples of this style of representation are abundant. A classic example of the late medieval period is found in Simone Martini’s (1285–1344) “Majesty” (Fig. 10–1). Clearly this is no real scene. The background is blue. Despite the assemblage the scene looks flat; the throne especially lacks depth. There is hardly the suggestion of a floor on which the figures stand, and these appear lifeless and unrelated to one another. Moreover, sizes are not important. This painting also illustrates another conceptual device used in medieval painting, known as terraced perspective. To show a group of people arranged in depth, those farther back are placed somewhat above those in front.
Toward the end of the thirteenth century, the painters began to be influenced by the Renaissance. Since the preoccupation with religious themes still existed and paintings, in fact, continued to be commissioned mainly by church officials, the same subjects appear but in more realistic settings. The painters had turned to the observation of nature and saw a real world, physical beings, earth, sea, and air. Their paintings reveal this interest in natural scenes by reflecting their efforts to render space, depth, mass, volume, and other visual effects largely through the use of lines, surfaces, and other geometric forms. To achieve naturalism they also tried to render emotions and to depict drapery folding around parts of the body as drapery actually does. People began to look like real individuals instead of types. Mysticism gradually gave way to realism and art became more and more secular.
Cimabue (ca. 1300), Cavallini (ca. 1250–1330), Duccio (1255–1318), and Giotto (1266–1337) were the leaders of the new movement to inject realism into painting and to incorporate the beauty of nature. Giotto, in particular, is often called the father of modern painting. In the works of the men cited and in those of their immediate successors, we can readily observe the search for an optical system of perspective.
Fig. 10–1.
Simone Martini: Majesty. Pallazzo Communale, Siena.
Duccio’s “Last Supper” (Fig. 10–2) shows what could be a real scene and offers an ambitious attempt at depth. The receding wall and ceiling lines create this effect. Moreover, pairs of lines which are parallel in the actual scene and symmetrically placed with respect to the center are drawn so as to meet on a vertical line through the center of the painting. This scheme is referred to as vertical perspective and was developed further by other painters.
On the whole the picture is not too successful. The table seems to slant toward the front. The objects on the table are too much in the foreground and appear to be on the point of sliding off or toppling over. The table and the room are not seen from the same point of view. The various parts lack proportion. The failure of the painting to depict depth properly causes one to look from side to side instead of into the painting. An interesting feature characteristic of the period is the setting in a partially boxed-in room. The artists were beginning to treat nature, but for the moment limited themselves to scenes which had both interior and exterior components. They were already looking into space and were about to venture into the wide world.
Fig. 10–2. Duccio: Last Supper. Opera del Duomo, Siena.
Fig. 10–3. Giotto: Birth and Naming of St. John the Baptist. Church of Santa Croce, Florence.
Giotto painted with the definite goal of reproducing visual perceptions and spatial relations, and his paintings tend to produce, the effect of photographic copies. His figures possess mass, volume, and vitality, are grouped appealingly, and are interrelated. His “Birth and Naming of St. John the Baptist” (Fig. 10–3) is typical. The partially boxed-in interior is again evident as is the use of lines and surfaces. The side walls are drawn small or foreshortened to suggest depth. The ground plane is a clear surface. Although Giotto’s paintings are not visually correct and although he introduced no new principles, his results are far better than those of his predecessors. He chose homelike scenes, gave human feelings to his figures, and distributed them in space. He catches shades of emotions and expresses them through the features and postures of the bodies. There is no mysticism nor ecstatic piety; “real” angels, Christ, and disciples stand before us. He was aware of the progress he had made and he delighted in showing his skill.
A step forward in the achievement of realism was made by Ambrogio Lorenzetti (fourteenth century). His outdoor panoramas are the best of this period. However, from the standpoint of the significant development which was to follow, his “Presentation in the Temple” (Fig. 10–4) is more worthy of attention. There is a definite foreground or horizontal plane as opposed to the background or vertical plane. The lines on the floor clearly recede and meet in one point. Other pairs of receding parallel lines meet in respective points of a vertical line. Also significant is the gradual decrease (foreshortening) in the size of the floor blocks to suggest distance. But the floor and the rest of the painting are not unified.
These few samples of fourteenth-century Renaissance painting show the increasing efforts to achieve naturalism, real scenes, and three-dimensionality. The innovators were groping for an effective technique but did not succeed. Visualization and sheer artistic skill were not enough.
There was rather little progress in the second half of the fourteenth century because the Black Death seriously disturbed the life of Europe and decimated the population. The fifteenth century witnessed, as we noted in the preceding chapter, a new flood of Greek works to Italy, a new series of translations, and enormous support for artists. The Greek ideals became better known and were discussed enthusiastically in Italy. Secularization was hastened, and the artist acquired a heightened interest in humanity and in the study of nature, and a zeal for science.
Fig. 10–4.
Ambrogio Lorenzetti: Presentation in the Temple. Uffizi, Florence.
To achieve an accurate delineation of actual objects and a system of painting which would yield sound portraiture, painters studied nudes, the body in various postures, anatomy, expression, light, and color. The Madonna and Son were portrayed as human beings suffering human emotions, and Church history was enacted by real people. Religious themes became predominantly a conventional or habitual outlet for the depiction of the real world. Later, instead of humanizing religious themes, the artists turned to glorifying man and nature. The ascetic, mystical, and devotional attitudes were dropped entirely. Still later pagan subjects were adopted. The glory and gladness of nature, the delight in physical existence, the beauty of earth, sea, and air were the new values. Painting became entirely secular.
In their striving for realism the artists went one step further and decided that their function was to imitate nature, to depict what they saw as realistically as they could. Nature was to be the authority for what appeared on canvas, and painting was to be the science of reproducing nature accurately. The objective of painting, says Leonardo da Vinci, is to reproduce nature and the merit of a painting lies in the exactness of the reproduction. Even a purely imagined scene must appear to the spectator as if it existed exactly as pictured. Painting was to be a veridical reproduction of reality.
But how was the reproduction to be achieved? Here, too, the Renaissance artist adopted a Greek ideal. By the fifteenth century he had become thoroughly familiar and imbued with the Greek doctrine that mathematics is the essence of the real world. Hence to penetrate to the real substance of the theme he sought to display on canvas, the Renaissance artist believed that he must reduce it to its mathematical content. To capture the essence of forms, the organization of objects in space, and the structure of space the artist decided that he must find the underlying mathematical laws.
But realistic painting includes more than the mathematical properties of the objects being portrayed. The eye sees the painting, and this must create on the eye the same impression as the scene itself. Also, since vision and the light which carries the scene to the eye are involved, these too must be analyzed. But the study of light also led to mathematics. From Greek times on, as we have already seen in earlier chapters, light had been shown to be subject to mathematical laws. Indeed, the few mathematical laws of light were about the only precise knowledge about the phenomenon which the Greek and Renaissance worlds possessed, because the nature of light itself was a mystery. And so, to study the impress of scene and painting on the eye, the artists were once again led to mathematics.
Thus, although the artists made extensive and intensive physical studies of light and shade, color, the chemistry of pigments, the laws of movement and balance, the eye, anatomy, and the effect of distance on sight, they were chiefly dominated by the new thought that mathematics must be used to achieve realism in painting and that geometry is the key to the solution of this problem. Thereupon they created and perfected a totally new mathematical system of perspective which enabled them to “place reality on their canvases.”
The mathematical system of perspective which the Renaissance painters created and which is known as the system of focused perspective was founded about 1425 by the architect and sculptor Brunelleschi (1377–1446). His ideas were furthered and written down by the architect and painter Leone Battista Alberti (1404–1472). It is not Alberti’s artistic work which entitles him to fame but his technical knowledge. He studied architecture, painting, perspective, and sculpture, wrote several books explaining theoretical matters to artists, and exercised enormous influence. In his Della Pittura (1435) Alberti says that learning is essential to the artist. The arts are learned by reason and method; they are mastered by practice. He says further that the first necessity of a painter is to know geometry and that painting by incorporating and revealing the mathematical structure of nature can even improve on nature.
The mathematical scheme was developed and perfected by Paolo Uccello (1397–1475), Piero della Francesca (1416–1492), and Leonardo da Vinci (1452–1519). The system these men and others created and which Leonardo called the rudder and guide rope of painting has been used since the Renaissance by all artists who seek exact depiction of reality, and is taught in art schools today.
In their study of light, vision, and the representation of objects on canvas, these artists discovered the following facts. Suppose that a person looks at a real scene from a fixed position. Of course, he sees with both eyes, but each eye sees the same scene from a slightly different position. Although in ordinary vision we need both sensations to give us some perception and measure of depth, this perception is really not very good. Experience teaches us how to interpret the combined sensations, as Leonardo points out in his Treatise on Painting. The Renaissance artists decided to concentrate on what one eye sees and to compensate for the deficiency by shading, shadows where pertinent, and by what is known as aerial perspective, that is the gradual diminution of the intensity of colors with distance.*
Let us imagine that lines of light are drawn from one eye to various points on the objects in the scene. This collection of lines is called a projection. Let us imagine next, as did Alberti, Leonardo, and the German artist Albrecht Dürer (1471–1528), that a glass screen is interposed between the eye and the scene itself. Thus when one looks out of a window at a scene outside, the window serves as the glass screen. The lines of the projection will pierce the glass screen, and we may imagine a dot placed on the screen where each line pierces it. The figure formed by these dots on the screen is called a section. The most important fact which the Renaissance artists discovered is that this section makes the same impression on the eye as does the scene itself, for all that the eye sees is light traveling along a straight line from each point on the object to the eye, and if the light emanates from points on the glass screen but travels along the very same lines, it should still create the same impression. Hence this section, which is two-dimensional, is what the artist must place on the canvas to create the correct impression on the eye. Dürer used the word “perspective” because the Latin verb from which it is derived means “to see through.”
Fig. 10–5.
Albrecht Dürer: Designer of the Sitting Man.
Fig. 10–6.
Albrecht Dürer: Designer of the Lying Woman.
Fig. 10–7.
Albrecht Dürer: Designer of the Lute.
Before we investigate just how the painter is to put this section on canvas, let us study the idea of projection and section. Fortunately some woodcuts made by Dürer, who learned the mathematical system of perspective in Italy and then returned to Germany to teach it to his countrymen, are very helpful. The woodcuts are in Dürer’s text Underweysung der Messung mit dem Zyrkel und Rychtscheyed (1525). The first of these, “The Designer of the Sitting Man” (Fig. 10–5), shows an artist looking through a glass screen; he holds his eye at a fixed position, and marks on the screen the point at which a line of light from his eye to some point on the man’s body pierces the screen.
The second woodcut, “The Designer of the Lying Woman” (Fig. 10–6), shows the artist again holding his eye at a fixed position and noting on paper the points where the lines of light from his eye to the woman pierce the screen. To facilitate the process of reproducing the correct location of the dots on the paper, he has divided screen and paper into little squares.
The third woodcut, “The Designer of the Lute” (Fig. 10–7), delineates on the screen the section which the eye would see if it viewed the lute from the point on the wall where the rope is attached.
These woodcuts, then, illustrate what the artists meant by a section on a glass screen. Of course, a section depends upon the position of the glass screen as well as on the position of the observer. But this implies no more than that there can be many different paintings of the same scene. Thus, for example, two paintings can be the same except for size, and size is determined by the distance between glass screen and eye. Two paintings may differ in that one shows a frontal view and the other represents the same scene viewed somewhat from the side. The difference is due to a change in the observer’s position.
Let us accept, then, the principle that the canvas must contain the same section that a glass screen placed between the eye of the painter and the actual scene would contain. Since the artist cannot look through his canvas at the actual scene and may even be painting an imaginary scene, he must have theorems which tell him how to place his objects on the canvas so that the painting will, in effect, contain the section made by a glass screen.
Fig. 10–8.
The image of a line horizontal and parallel to the screen is horizontal.
Suppose then that the eye at E (Fig. 10–8) looks at the horizontal line GH and that GH is parallel to a vertical glass screen. The lines from E to the points of GH lie in one plane, namely the plane determined by the point E and the line GH, for a point and a line determine a plane. This plane will cut the screen in a line, G′H′, because two planes which meet at all meet in a line. It is apparent that the line G′H′ must also be horizontal, but we can prove this fact and so be certain. We can imagine a vertical plane through GH. Since GH is parallel to the screen and the latter is also vertical, the two planes must be parallel. The plane determined by E and GH cuts these parallel planes, and a plane which intersects two parallel planes intersects them in parallel lines. Hence G′H′ is parallel to GH, and since GH is horizontal, so is G′H′. But GH was any horizontal line parallel to the screen. Hence the image on the screen of any horizontal line parallel to the screen or picture plane must be horizontal. Thus in a painting which is to contain what this glass screen contains, the line G′H′ must be drawn horizontally.
We can present practically the same argument to show that the image of any vertical line, which is automatically parallel to the vertical screen, must appear on the screen as a vertical line. Thus all vertical lines must be drawn vertically.
Now let us consider a somewhat more complicated situation. Suppose that AB and CD (Fig. 10–9) are two parallel, horizontal lines in an actual scene. Moreover, assume that these lines are perpendicular to the screen. The eye is at E. If we now imagine that lines go from E to each point of AB, these lines, that is the projection, will lie in one plane for the point E, and the line AB will determine this plane by virtue of the theorem of solid geometry already mentioned. Similarly, E and the line CD determine another plane. The screen cuts the two planes we have just described; The sections must lie on the screen, and our problem is to determine where they should lie.
Fig. 10–9.
The images of two horizontal parallel lines which are perpendicular to the screen meet at a point on the screen.
Of course, the intersection of two planes is a line, and so the section corresponding to AB and that corresponding to CD will be lines, A′B′ and C′D′, respectively. Moreover, as the eye at E looks farther and farther out along the parallels AB and CD, the lines of sight will become more and more horizontal. As the eye follows AB and CD to infinity, so to speak, the lines from E tend to merge into one horizontal line which will be parallel to AB and CD. This line from E will pierce the screen at some point, say O′, and this point corresponds to the imaginary point O where AB and CD seem to meet at infinity. Of course, AB and CD are parallel and do not meet, but it is convenient to think of them as meeting at a point at infinity. Indeed, the eye gets the impression that they do meet. Then the line EO′ will be perpendicular to the screen because it is parallel to AB and CD and these two lines are perpendicular to the screen. The point O′ corresponds to the imagined meeting point at infinity of AB and CD, but because this point does not actually exist, O′ is called the principal vanishing point. It vanishes in the sense that it does not correspond to any actual point on AB or CD, whereas other points on A′B′ or CD′ do correspond to actual points on AB or CD, respectively.
Now the lines AB and CD extend out to infinity to the hypothetical meeting point O; that is, ABO and CDO are lines in the real scene. The sections of these lines, A′B′O′ and C′D′O′, must therefore meet at O′. What we have shown then is that A′B′ and C′D′ must be placed on the screen so that they meet at O′, and O′ is the foot of the perpendicular extending from the eye to the screen. Let us now note that AB and CD are any horizontal lines perpendicular to the screen. Hence all horizontal lines which are perpendicular to the screen must be drawn so as to go through O′, the principal vanishing point, which is the foot of the perpendicular from the eye to the screen.
We may draw another important conclusion from the preceding situation. The distances AC and BD are equal, for they are the distances between parallel lines. However, the corresponding images A′C′ and B′D′ are not equal because the lines A′B′ and C′D′ converge to O′. Moreover, B′D′ will be shorter than A′C′ because it is closer to O′. But B′D′ corresponds to the actual distance BD which is farther from the screen than AC is. Hence lengths which are farther from the screen must be drawn shorter than equal lengths closer to the screen. This fact is often described by the statement that, to obtain proper perspective in a painting, lengths farther away from the observer must be foreshortened.
We shall establish one more theorem about perspective drawing. Let us now suppose that JK (Fig. 10–10) is a horizontal line which makes an angle of 45° with the screen. Assume that the eye at E looks out along the line JK toward infinity. Then the line from the eye to the point at infinity on JK will be parallel to JK. Since JK is horizontal, the new line, EL in Fig. 10–10, will also be horizontal. It will pierce the screen at some point, say D1, and will also make an angle of 45° with the screen. The triangle D1EO′ is a right triangle because EO′ is perpendicular to the screen. In view of the acute angles of 45°, O′D1 = EO′. Then the point D1 is as far from O′ as E is. The projection from E to the various points of JK cuts the screen in some line, J′K′, say. As the eye continues to follow JK toward infinity, the projection cuts the screen in points lying on an extension of J′K′, and we have already established that when the eye looks toward infinity on JK, the projection cuts the screen at D1. Hence J′K′ must go through D1. We now have another important result. The image of any horizontal line which makes an angle of 45° with the screen must go through the point D1 which lies on the screen, on the same level as E, but is as far to the right of the principal vanishing point as E is from the principal vanishing point. The point D1 is called a diagonal vanishing point.
Fig. 10–10.
The image of a horizontal line which makes a 45°-angle with the screen goes through a diagonal vanishing point.
Had we considered instead of JK lines which make an angle of 135° with the screen, we would have found that their images must go through a point D2 which lies as far to the left of O′ as E is from O′. The point D2 is also called a diagonal vanishing point.
We see, then, that the points O′, D1 and D2 correspond to points at infinity in the actual scene. As a matter of fact, all points on the horizontal line D2 O′D1 correspond to points at infinity in the actual scene, and this line is called the vanishing line. It is the image of what one might call the horizon in the actual scene, that is, the points at infinity toward which the eye gazes when it looks in a horizontal direction.
The above theorems hardly begin to illustrate what one must know and apply to draw actual scenes realistically. The treatment of curves is especially difficult. For example, actual circles and spheres cannot, in general, be drawn as circles unless their centers happen to lie on the perpendicular from the eye to the screen. In all other cases, they must be drawn as ellipses or as arcs of parabolas or hyperbolas, depending upon their position relative to the observer. This fact becomes clear if one considers that the lines from the eye to each point on the edge of the circle or sphere, the projection, in other words, form a cone and that the section of this cone on the screen will be one of the conic sections discussed in Chapter 6. We shall not investigate the more complicated theorems because to do so would require a course in the subject and because the detailed theorems are of interest only for the specific purpose of learning to paint realistically. We may have seen enough of the basic principles to appreciate that the problem of painting realistically is handled by the application of a thoroughly mathematical system.
We know that the construction of a painting in accordance with the focused scheme presupposes a definite fixed position of the painter in relation to the scene. To view properly a painting so constructed, the observer should place himself in precisely the position the painter used in planning the painting. Otherwise the observer will get a distorted view. Strictly speaking, paintings in museums should be hung so that the observer can conveniently take that position.
Renaissance painters achieved their goal of devising a mathematical system which permitted the realistic representation of actual scenes and joyously hastened to employ it. Realistic paintings constructed in accordance with the focused scheme of perspective begin to appear about 1430.
The artist who contributed key principles of mathematically determined perspective, including new methods of construction, and who was the best mathematician of his times is Piero della Francesca. This highly intellectual painter with a passion for geometry planned all his works mathematically to the last detail. Each scene to be painted was a mathematical problem. The placement of each figure was calculated to ensure its correctness in relation to other figures and to the painting as a whole. He loved geometrical forms so much that he used them for hats, parts of the body, and other details in his paintings. Piero practically identified painting and perspective. His De pros-pettiva pingendi, a treatise on painting and perspective in which he uses Euclid’s deductive method, presents perspective as a science and provides sample constructions illustrating how perspective problems are to be handled. Though incidental to our purposes, it is worth noting that Piero painted the first Renaissance portraits of real people, the Duke and Duchess of Urbino, Federigo de Montefeltro and his wife Battista Sforza.
There are numerous examples which illustrate Piero’s excellent perspective. His “Flagellation” (Fig. 10–11) is one of the best. As in all of his paintings a geometric framework underlies the design. The principal vanishing point is chosen to be near the figure of Christ. This device of placing the principal vanishing point within the most important area in the painting is deliberate because the eye tends to focus on that vanishing point. All objects are carefully foreshortened; this is especially noticeable in the marble blocks on the floor and in the beams. The immense labor which went into the calculation of these sizes is indicated by a drawing in the book referred to above wherein he explains a similar construction.
Fig. 10–11.
Piero della Francesca: The Flagellation. Ducal Palace, Urbino.
Fig. 10–12.
Piero della Francesca: Architectural View of a City. Kaiser Friedrich Museum, Berlin.
Fig. 10–13.
Leonardo da Vinci: Study for the Adoration of the Magi. Uffizi, Florence.
Piero achieves unity of the various parts by means of the system of perspective. All parts are mathematically tied together to produce this synthesis. Indeed, it was somewhat because of this effect that the Renaissance painters valued the system and were excited about it. The example shown here should be compared with the fourteenth-century works (Section 10–2), where unity is lacking. The entire layout of Piero’s painting is so carefully planned that movement is sacrificed to the unity of design.
To illustrate the power of perspective Piero painted several scenes of cities. His “Architectural View of a City” (Fig. 10–12) gives a striking illusion of depth. These examples of Piero’s paintings show his obsession for perspective and his great technique.
Leonardo da Vinci’s work provides excellent examples of paintings embodying mathematical perspective. Leonardo prepared for painting by deep and extensive studies in anatomy, perspective, geometry, physics, and chemistry. In his Treatise on Painting, a scientific treatise on painting and perspective, Leonardo gives his views. He opens with the statement, “Let no one who is not a mathematician read my works.” Painting, he says, is a science which should be founded on the study of nature and, like all sciences, must also be based on mathematics. He scorns those who think they can ignore theory and by mere practice produce art: “Practice must be founded on sound theory.” Painting, which he regarded as superior to architecture, music, and poetry, is a science because it deals with the geometry of surfaces.
Fig. 10–14.
Leonardo da Vinci: Adoration of the Magi. Uffizi, Florence.
Fig. 10–15.
Leonardo da Vinci: Annunciation. Uffizi, Florence.
The detailed mathematical studies which Leonardo undertook in preparation for his paintings are illustrated by one of several sketches he made for his “Adoration of the Magi” (Fig. 10–13). The painting itself, which was never completed, is shown in Fig. 10–14. His “Last Supper” is another excellent example of mathematical perspective, but is so well known that we shall reproduce instead “The Annunciation” (Fig. 10–15). Although the action takes place in the foreground and the chief figures are far apart, they and the distant scene in the rear are all brought together by the perspective structure.
Raphael (1483–1520) supplies many superb paintings which exhibit excellent perspective. In his “School of Athens” (Fig. 10–16) he boldly tackles an enormous scene encompassing a vast number of people within a magnificent architectural setting. The portrayal of depth, the harmonious organization, coherence, and exactness of proportions achieved despite the difficulty of the undertaking are extraordinary. This picture, especially, shows how perspective unifies a composition and ties figures at the sides to the central theme.
The history of this painting is of interest. Pope Julius II (1443–1513) was impressed with ancient learning and regarded Christianity as the climax of Jewish religious thought and Greek philosophy. He wished to have his idea embodied in paintings and commissioned both Michelangelo and Raphael to develop this theme. Michelangelo treated it in his frescoes on the ceiling of the Sistine Chapel, where he shows the human race led to Christ through a long line of Jewish prophets and pagan sibyls. Raphael executed the same theme in a somewhat different manner. In four frescoes which cover the walls of the Pope’s principal official room, the Camera della Segnatura, he teaches that the human soul is to aspire to God through each of its faculties: reason, the artistic capacity, the sense of order and good government, and the religious spirit. “The School of Athens” glorifies reason and naturally exhibits the people who excelled in the intellectual sphere. Plato and Aristotle are the central figures. Plato points upward to the eternal ideas and Aristotle down to the earth as the field of experience. At Plato’s left is Socrates. In the left foreground Pythagoras writes in a book. The right foreground shows the bald-headed Euclid; Archimedes stoops to demonstrate a theorem; Ptolemy holds up a sphere. All the way to the right is Raphael himself.
Fig. 10–16.
Raphael: School of Athens. Vatican.
Fig. 10–17.
Raphael: Fire in the Borgo. Vatican.
Raphael offers so many examples of excellent perspective that it is difficult to limit oneself to one or two representative samples. His “The Fire in the Borgo” ((Fig. 10–17) shows exquisite depth, perfect handling of figures in various positions, the proper foreshortening, and again the unification of a scene in which many actions take place.
We could offer countless examples illustrating the application of the mathematical system of perspective by the Renaissance masters.* All painters of this period in which western European art reached one of its pinnacles employed it and employed it well. Many, among them Uccello and Piero, were obsessed by it and painted scenes viewed from unusual positions just to solve the mathematical problems involved. The essential difference between the art of the Renaissance and that of the Middle Ages is the introduction of the third dimension, and Renaissance painting is characterized by the importance attached to realism, to the realistic rendering of space, distance, and forms, achieved by means of the mathematical system of perspective. Through it, the process of seeing was rationalized; the extended world was brought under control; and the rational interests of the painters were satisfied.
In the history of culture the accomplishments of the Renaissance artists have broad significance. Their apparent goal was to gaze at nature and to depict what they saw on canvas, but their true, more profound objective was to uncover the very secrets of nature. The Renaissance artist was a scientist, and painting was a science not merely in the sense that it had a highly technical and even mathematical content, but because it was inspired by the ultimate goal of science, understanding nature. Art and science are never separated in the thinking and work of Ghiberti, Alberti and Leonardo, for example. Leonardo’s Paragone, A Comparison of the Arts (Treatise on Fainting) contains a chapter on “Painting and Science” in which he asserts that painting seeks the truths of nature. The artist of that period regarded himself as the servant of science. These men who explored and represented nature with methods peculiar to their art were motivated precisely by the spirit and objectives of the scientists who studied astronomy, light, motion, and other phenomena. They were in fact the forerunners in spirit and goals of the great physical scientists of modern times and they revealed truth in a form which means more to many people than the deep and intricate analyses of modern mathematical physics. That mathematics proved to be the foundation of painting and thereby enabled painting to reveal the structure of nature was no more than fitting, for the Greeks had already shown that mathematics was the essence of design, and later scientists were to confirm this fact in ever more striking fashion.
The works of the Renaissance artists are hung in art museums. They could, with as much justification, be hung in science museums. The lover of Renaissance art is consciously or unconsciously a lover of science and mathematics.
1. Relate the “back to nature” movement of the Renaissance to the development of a mathematical system of perspective.
2. Distinguish between conceptual and optical systems of perspective.
3. Which artists did most to create a mathematical system of perspective?
4. What is the principle of projection and section in the theory of perspective?
5. Draw the rear wall, and the visible portions of the side walls, ceiling, and floor of a room as seen by an observer in the room whose eye is looking directly at the rear wall.
6. Add to the drawing of the preceding exercise a square table, two of whose edges are parallel to the rear wall.
7. Draw a cube positioned in such a way that one edge is closest to you and that neighboring edges make angles of 45° and 135°, respectively, with the canvas. Go as far as you can with the theorems at your disposal.
8. State three theorems of the geometry of perspective drawing.
1. The influence of mathematics on Renaissance painting.
2. Theories of the artists on human proportions. Use the reference to Panofsky’s Meaning in the Visual Arts.
3. Vision and painting. Use the reference to Helmholtz.
BLUNT, ANTHONY: Artistic Theory in Italy, Oxford University Press, London, 1940.
BUNIM, MIRIAM: Space in Medieval Painting and the Forerunners of Perspective, Columbia University Press, New York, 1940.
CLARK, KENNETH: Piero della Francesca, Oxford University Press, New York, 1951.
COLE, REX V.: Perspective, Seeley, Service and Co., Ltd., London, 1927.
COOLIDGE, JULIAN L.: Mathematics of Great Amateurs, Dover Publications, Inc., New York, 1963.
DA VINCI, LEONARDO: Treatise on Painting, Princeton University Press, Princeton, 1956.
FRY, ROGER: Vision and Design, pp. 112–168, Penguin Books Ltd., Baltimore, 1937.
HELMHOLTZ, HERMAN VON: Popular Lectures on Scientific Subjects, pp. 250–286, Dover Publications, Inc., New York, 1962.
IVINS, WM. M., JR.: Art and Geometry, Dover Publications, Inc., New York, 1964.
JOHNSON, MARTIN: Art and Scientific Thought, Part Four, Faber and Faber, Ltd., London, 1944.
KLINE, MORRIS: Mathematics in Western Culture, Chap. 10, Oxford University Press, New York, 1953.
LAWSON, PHILIP J.: Practical Perspective Drawing, McGraw-Hill Book Co., Inc., New York, 1943.
PANOFSKY, ERWIN: “Dürer as a Mathematician,” pp. 603–621 of James R. Newman: The World of Mathematics, Simon and Schuster, New York, 1956.
PANOFSKY, ERWIN: Meaning in the Visual Arts, Chap. 6, Doubleday Anchor Books, New York, 1955.
POPE-HENNESSY, JOHN: The Complete Work of Paolo Uccello, Phaidon Press, London, 1950.
PORTER, A. T.: The Principles of Perspective, University of London Press Ltd., London, 1927.
VASARI, GIORGIO: Lives of the Most Famous Painters, Sculptors and Architects, E. P. Dutton, New York, 1927, and many other editions.
* The difference between a drawing made according to the laws of perspective and a three-dimensional picture is clear when one views a stereoscopic drawing with both eyes through colored glasses.
* The same theme is treated in the author’s Mathematics in Western Culture. Other examples can be found there.
