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PAINTING and PERSPECTIVE 127 the Real World, That the Universe Is Ordered and Explicable Rationally in Terms of Geometry

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PAINTING and PERSPECTIVE 127 the Real World, That the Universe Is Ordered and Explicable Rationally in Terms of Geometry PAINTING AND PERSPECTIVE 127 the real world, that the universe is ordered and explicable rationally in terms of geometry. Hence, like the Greek philosopher, he be­ lieved that to penetrate to the underlying significance, that is, the reality of the theme that he sought to display on canvas, he must reduce it to its mathematical content. Very interesting evidence of x the artist's attempt to discover the mathematical essence of his sub­ ject is found in one of Leonardo's studies in proportion. In it he Painting and Perspective tried to fit the structure of the ideal man to the ideal figures, the flquare and circle (plate VI). The sheer utility of mathematics for accurate description and the The world's the book where the eternal sense philosophy that mathematics is the essence of reality are only two Wrote his own thoughts; the living temple where, Painting his very self, with figures fair of the reasons why the Renaissance artist sought to use mathematics. He filled the whole immense circumference. There was another reason. The artist of the late medieval period and T. CAMPANELLA the Renaissance was, also, the architect and engineer of his day and so was necessarily mathematically inclined. Businessmen, secular princes, and ecclesiastical officials assigned all construction problems During the Middle Ages painting, serving somewhat as the hand­ to the artist. He designed and built churches, hospitals, palaces, maiden of the Church, concentrated on embellishing the thoughts cloisters, bridges, fortresses, dams, canals, town walls, and doctrines of Christianity. Toward the end of this period, the and instru­ ments of warfare. Numerous drawings of such engineering projects painters, along with other thinkers in Europe, began to be interested are in da Vinci's notebooks and he, himself, in offering his services in the natural world. Inspired by the new emphasis on man and the to Lodovico Sforza, ruler of Milan, promised to serve as an engineer, universe about him the Renaissance artist dared to confront nature, to study her deeply and searchingly, and to depict her realistically. constructor of military works, and designer of war machines, as well as architect, sculptor, and painter. The artist was even expected to The painters revived the glory and gladness of an alive world and solve problems involving the motion of cannon balls in artillery fire, reproduced beautiful forms which attested to the delightfulness of a task which in those times called for profound mathematical knowl­ physical existence, the inalienable right to satisfy natural wants, edge. It is nr> exaggeration to state that the Renaissance artist was the and the pleasures afforded by earth, sea, and air. best practicing mathematician and that in the fifteenth century he For several reasons the problem of depicting the real world led was also the most learned and accomplished theoretical mathema­ the Renaissance painters to mathematics. The first reason was one tician. that could be operative in any age in which the artist seeks to paint realistically. Stripped of color and substance the objects that painters The specific problem which engaged the mathematical talents of put on canvas are geometrical bodies located in space. The language the Renaissance painters and with which we shall' be concerned here for dealing with these idealized objects, the properties they possess was that of depicting realistically three-dimensional scenes on canvas. as idealizations, and the exact relationships that describe their rela­ The artists solved this problem by creating a totally new system of tive locations in space are all incorporated in Euclidean geometry. mathematical perspective and consequently refashioned the entire style of painting. The artists need only avail themselves of it. The Renaissance artist turned to mathematics not only because he The various schemes employed throughout the history of painting sought to reproduce nature but also because he was influenced by for organizing subjects on plaster and canvas, that is, the various the revived philosophy of the Greeks. He became thoroughly famil­ systems of perspective, can be divided into two major classes, con­ iar and imbued with the doctrine that mathematics is the essence of ceptual and optical. A conceptual system undertakes to organize the 126 128 MATHEMATICS IN WESTERN CULTURE PAINTING AND PERSPECTIVE 129 persons and objects in accordance with some doctrine or principle dominated. An excellent example, indeed one that is regarded as that has little or nothing to do with the actual appearance of the the flower of medieval painting, is 'The Annunciation' (plate VIII) scene itself. For example, Egyptian painting and relief work were by' Simone Martini (1l~85-1344). The background is gold. There is largely conceptual. The sizes of people were often ordered in rela­ no indication of visual perception. The movement in the painting tion to their importance in the politico-religious hierarchy. Pharaoh is from the angel to the Virgin and then back to the angel. Though was usually the most important person and so was the largest. His there is loveliness of color, surface, and sinuous line, the figures wife would be next in size and his servants even smaller. Profile themselves are unemotional and arouse no emotional response in the views and frontal views were used simultaneously even for different onlooker. The effect of the whole is mosaic-like. Perhaps the only parts of the same figure. In order to indicate a series of people or respect in which this painting makes any advance toward realism is animals one behind the other, the same figure was repeated slightly in its use of a ground plane or floor on which objects and figures rest displaced. Modern painting, as well as most Japanese and Chinese and which is distinct from the gilt background. painting, is also conceptual (plate XXVII). Characteristic Renaissance influences which steered the artists An optical system of perspective, on the other hand, attempts to toward realism and mathematics began to be felt near the end of convey the same impression to the eye as would the scene itself. the thirteenth century, the century in which Aristotle became widely Although Greek and Roman painting was primarily optical, the known by means of translations from the Arabic and the Greek. The influence of Christian mysticism turned artists back to a conceptual painters became aware of the lifelessness and unreality of medieval system, which prevailed throughout the Middle Ages. The early painting and consciously sought to modify it. Efforts toward natural­ Christian and medieval artists were content to paint in symbolic ism appeared in the use of real people as subjects of religious themes, terms, that is, their settings and subjects were intended to illustrate in the deliberate use of straight lines, multiple surfaces. and simple religious themes and induce religious feelings rather than to rep­ forms of geometry. in experiments with unorthodox positions of the resent real people in the actual and present world. The people and figures, in attempts to render emotions, and in the depiction of objects were highly stylized and drawn as though they existed in a drapery falling and folding around parts of bodies as it actually does flat, two-dimensional vacuum. Figures that should be behind one rather than in the flat folds of the conventional medieval style. another were usually alongside or above. Stiff draperies and angular The essential difference between medieval and Renaissance art is attitudes were characteristic. The backgrounds of the paintings were the introduction of the third dimension, that is, the rendering of almost always of a solid color, usually gold, as if to emphasize that space, distance, volume, mass, and visual effects. The incorporation the subjects had no connection with the real world. of three-dimensionality could be achieved only by an optical system The early Christian mosaic 'Abraham with Angels' (plate VII), of representation, and conscious efforts in this direction were made a typical example of the Byzantine influence. illustrates the disin­ by Duccio (1255-1319) and Giotto (1276-1336), at the beginning of tegration of ancient perspective. The background is essentially neu­ the fourteenth century. Several devices appeared in their works that traL The earth, tree, and bushes are artificial and lifeless, the tree are at least worth noticing as stages in the development of a mathe­ being shaped peculiarly to fit the border of the picture. There is no matical system. Duccio's 'Madonna in Majesty" (plate IX) has several foreground or base on which the figures and objects stand. The interesting features. The composition, first of all, is severely simple figures are not related to each other and, of course, spatial relations and symmetrical. The lines of the throne are made to converge in are ignored because measures and sizes were deemed unimportant. pairs and thus suggest depth. The figures on either side of the throne The little unity there is in the picture is supplied by the gold back­ are presumably standing on one level but they are painted one above ground and the color of the objects. the other in several layers. This manner of depicting depth is known Though remnants of an optical system used by the Romans were as terraced perspective, a device very common in the fourteenth cen­ sometimes present in medieval painting. this Byzantine style pre- tury. The drapery is somewhat natural as exemplified by the folds r' 180 MATHEMATICS IN WESTERN CULTURE PAINTING AND PERSPEcnvE 131 over the Madonna's knee. Also there is some feeling for solidity and tended toward a photographic copy. His figures possessed mass, vol­ space and some emotion in the faces. The picture as a whole still ume, and vitality. He chose homelike scenes, distributed his figures contains much of the Byzantine tradition. There is a liberal use of in a balanced arrangement, and grouped them in a manner agree­ gold in the background and in the details.
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