Otto Georg von Simson:
With but a single dimension given, the Gothic architect developed all other magnitudes of his ground plan and elevation by strictly geometrical means, using as modules certain regular polygons, above all the square. The knowledge of this way of determining architectural proportions was considered so essential that it was kept a professional secret by the mediaeval lodges. Only toward the end of the fifteenth century - and of the cathedral age - was it made public by Matthäus Roriczer, the builder of Regensburg Cathedral. He teaches how to take the elevation from the ground plan by means of a single square. From this figure Roriczer derives all proportions of his edifice, in this case a pinnacle, inasmuch as its dimensions are related to one another as are the sides of a sequence of squares, the areas of which diminish (or increase) in geometrical progression. Proportions thus obtained the master considers to be according to true measure.
It was not only the late Gothic architect or the German lodges that made this modular use of the square. Perhaps the most important single piece of evidence regarding the principles of Gothic design is the famous model book by the Picard architect, Villard de Honnecourt, who was active in the second quarter of the thirteenth century. He too teaches how to halve the square for the purpose of determining the true proportions of a building, in this case the ground plan of a cloister.
The canon of proportions did not remained confined to theory. Villard de Honnecourt in another drawing shows its application in the towers of Laon Cathedral. It appears likewise in a number of mediaeval ground plans of Gothic steeples studied by Maria Velte. Here not only the recesses of the different stories, but the dimensions of every single detail, be it the keystone or the width of the walls, hang proportionately together, as do the sides of a series of squares the areas of which increase in geometrical progression. In the famed Church of Our Lady at Trier, as Ernst Gall has recently shown, all proportions are determined by the same formula. Again, the façade of Notre Dame at Paris is composed of a sequence of four squares developed according to true measure. If one compares this façade with the earlier but generally similar façade of Noyon, he is tempted to say that the development of Gothic from its beginning to the classical maturity reached by the mid-thirteenth century is marked by the increasing clarity with which the geometrical principle is realized. Of course, geometrical formulae were also used by pre-Gothic architects and by sculptors and painters as well. Here, however, they seem to have been practical rather than artistic devices, of which the observer usually remains unconscious. Nowhere do they determine the aesthetic appearance as they do in the Gothic system....
The Gothic artist would have overthrown the rule of geometry had he considered it, as most modern artists would, a fetter. It is clear, on the other hand, that he did not use his geometrical canons for purely aesthetic reasons either, since he applied them where they are invisible to the viewer. Thus, all the ribs under the vault of Reims Cathedral circumscribe, according to Viollet-le-Duc, equilateral triangles, a fact no visitor to the church is likely to notice. And even such purely technical data as the width of a wall or buttress were determined by the formula according to true measure.... At least one literary document survives that explains the use of geometry in Gothic architecture: the minutes of the architectural conferences held during 1391 and the following years in Milan....
Two points stand out that are of paramount importance to our present inquiry. First, the reliance on geometric figures - attested by the German architect, Matthäus Roriczer, of the fifteenth century, and the thirteenth-century French architect, Villard - is emphatically confirmed by the Italian document of the intervening century. The question debated at Milan is not whether the cathedral is to be built according to a geometrical formula, but merely whether the figure to be used is to be the square (which had already determined the ground plan) or the equilateral triangle.
The second and even more interesting aspect of the Milan documents is that they suggest the reason for this reliance on geometrical canons. The minutes of one particularly stormy session relate an angry dispute between the French expert, Jean Mignot, and the Italians. Overruled by them on a technical issue, Mignot remarks bitterly that his opponents have set aside the rules of geometry by alleging science to be one thing and art another. Art, however, he concludes, is nothing without science: ars sine scientia nihil est. The terms art and science do not mean what they mean today. Art for Mignot and his contemporaries is the practical know-how gained from experience, science the ability to to account for the reasons that determine sound architectural procedure by rational, and more precisely, geometrical means. In other words, architecture that is scientific and good must invariably be based on geometry; unless he obeys the laws of this discipline, the architect must surely fail. This argument was considered unassailable even by Mignot's opponents. They hasten to affirm that they are in complete agreement as regards this theoretical point and have nothing but contempt for an architect who presumes to ignore the dictates of geometry. And it is taken for granted by both sides that the stability and beauty of an edifice are not distinct values, that they do not obey different laws, but that, on the contrary, both are comprehended in the perfection of geometrical forms....
Jean Mignot's juxtaposition of ars and scientia recalls.... the distinction that occurs almost a millennium before in the most influential aesthetic treatise of the Christian Middle Ages. To this work, to the world view it expounds, and to the tradition it created, we must now turn.[The Gothic Cathedral by Otto Georg von Simson. Harper & Row: New York, 1964]
In the first book of his treatise De musica, St. Augustine defines music as the science of good modulation. Before telling us what good modulation is, he explains why music, properly understood, is a science. He does not deny that music can be produced by instinct or practical skill, just as music can be appreciated by one who just knows what he likes. Such understanding of music, however creative or receptive, is but of a low order, according to Augustine. Vulgar performers and vulgar audiences have such an understanding; even a singing bird has. In fact, there is little difference between man and beast in regard to this kind of musical knowledge, which Augustine contemptuously calls art. The true understanding of music, on the other hand, which knows the laws that are of its very essence, applies them in musical creation, and discovers them in a composition, is what Augustine calls the science of music, and he goes on to explain the nature of this science as mathematical.
The science of good modulation is concerned with the relating of several musical units according to a module, a measure, in such a way that the relation can be expressed in simple arithmetical ratios. The most admirable ratio, according to Augustine, is that of equality or symmetry, the ratio 1:1, since here the union or consonance between the two parts is most intimate. Next in rank are the ratios 1:2, 2:3, and 3:4 - the intervals of the perfect consonances, octave, fifth, and fourth. It is to be noticed that the pre-eminence of these intervals, for Augustine, is not derived from their aesthetic or acoustic qualities. These, rather, are audible echoes of the metaphysical perfection that Pythagorean mysticism ascribes to number, especially to the four numbers of the first tetractys. Without the principate of number, as Augustine calls it, the cosmos would return to chaos. Taking up the Biblical passage thou hast ordered all things in measure and number and weight, the Bishop of Hippo applied Pythagorean and Neoplatonic number mysticism to the interpretation of the Christian universe, thus establishing the cosmology that remained in force until the triumph of Aristotelianism. Augustine shares with Plato both distrust of the world of images and belief in the absolute validity of mathematical relationships. These views form the basis of AugustineÍs philosophy of art. His postulates about the function of the arts in the Christian commonwealth, and even, one may say, their style, left their imprint on Christian art for a thousand years. This influence may be formulated as follows:
1. The principles of good musical modulation and its appreciation that Augustine established in De musica are mathematical principles and therefore apply, in his opinion at least, to the visual arts as they do to music. On the monochord, the musical intervals are marked off by divisions on a string; the arithmetical ratios of the perfect consonances thus appear as the proportions between different parts of a line. And since Augustine deduces the musical value of the perfect consonances from the metaphysical dignity of the ratios on which they are based, it was natural for him to conclude that the beauty of certain visual proportions derives from their being based on the simple ratios of the first tetractys. The place Augustine assigns to geometry among the liberal arts, like the place he assigns to music, is caused by what the Middle Ages called the anagogical function of geometry, that is, its ability to lead the mind from the world of appearances to the contemplation of the divine order. In the second book of his treatise On Order, Augustine describes how reason, in her quest for the blissful contemplation of things divine, turns to music and from music to what lies within the range of vision: beholding earth and heaven, she realizes that only beauty can ever satisfy her, in beauty figures, in figures proportion, and in proportion number.
2. The aesthetic implications are clear. Augustine was nearly as sensitive to architecture as he was to music. They are the only arts he seems to have fully enjoyed; and he recognized them even after his conversion, since he experienced the same transcendental element in both. For him, music and architecture are sisters, since both are children of number; they have equal dignity, inasmuch as architecture mirrors eternal harmony, as music echoes it.
Consistent with this view, Augustine uses architecture, as he does music, to show that number, as apparent in the simpler proportions that are based on the perfect ratios, is the source of all aesthetic perfection. And he uses the architect, as he does the musician, to prove that all artistic creation observes the laws of numbers. The architect, if he is a mere practitioner rather than a scientist of his art, may be unaware of the fact that he is instinctively applying mathematical rules. No beautiful edifice is conceivable, however, unless these rules have been applied and unless their presence is apparent to the observer....
3. For mediaeval art, the greatest significance of this philosophy of beauty lies elsewhere.... While imposing upon artistic creation authoritative regulations that for centuries were more generally and more timidly observed than we often realize, it was precisely this philosophy that invested Christian art with an extraordinary dignity. True beauty, according to Augustine, is anchored in metaphysical reality. Visible and audible harmonies are actually intimations of that ultimate harmony which the blessed will enjoy in the world to come. The place that harmony and proportion came to assume in the art and contemplation of the Christian West is not altogether unlike that which the icon, the sacred image, occupies in the art and thought of the Eastern Church. Here, and under the enduring inspiration of the Greek tradition, the ideal of ultimate beauty remained a visual one; it centered in the image of man. In the West, and under the influence of Augustine, beauty was conceived in musical terms, and even ultimate bliss as the enjoyment of an eternal symphony. And as the icon is thought to partake of the sacred reality it represents so, according to Augustinian aesthetics, the musical consonances in visual proportions created by man partake of a sacred concord that transcends them....
The aesthetic aspects of this philosophy of proportions were taken over from Augustine. He, as well as his pupil, Boethius - for the School of Chartres and the Middle Ages in general the greatest mathematical authority - taught, moreover, how to visualize the perfect consonances in geometrical terms. Boëthius points out that the proportions of double and half, triple and third - those, in other words, that yield the perfect consonances on the monochord - are as readily perceived visually as they are acoustically....
But the Platonists of Chartres expounded not only the aesthetic excellence of these proportions but their technical excellence as well.... In other words, application of the perfect proportions, determined by rigid geometrical means, became a technical necessity as well as an aesthetic postulate if the building was to be stable as well as beautiful.
We now understand why the High Middle Ages defined and practiced architecture as applied geometry; why the experts at Milan paid the same astonishing tribute to this discipline as had Augustine and Boëthius; why the evidence of Gothic architecture itself seems to indicate that all statical problems were actually solved by purely geometrical methods. And we also understand the lofty claim that the great architects of the Gothic period meant to convey in having themselves depicted, compass and measuring rod in hand, as geometricians.... It was with the compass that God himself came to be represented in Gothic art and literature as the Creator who composed the universe according to geometrical laws. It was only by observing these same laws that architecture became a science in Augustine's sense. And in submitting to geometry the mediaeval architect felt that he was imitating the work of his divine master.