Science and the Infinite - Sydney T. Klein (free e books to read online .txt) 📗

Gothic lines. A careful examination of the proportions of the structures themselves, and the character of the decorations found in the finest examples of buildings representing that style, at once shows us that the incentive was the symbolism attached to the mysterious figure called the Vesica Piscis, which appears to be not only the principal feature upon which the whole style rests, but is also employed, as a symbol of the Divine, wherever we have Gothic Architecture, either in painted windows or mural decorations. Every Cathedral has its Vesica Piscis, often of enormous dimensions. Geometry was synonymous with Masonry, and the very foundation of the Science of Geometry, as expounded by Euclid, was his first proposition. Every single problem in the whole of his books necessitates for its construction the use of this one foundation—namely, "how to form an Equilateral Triangle," and this is the Mystical form of "the Knowledge of the Square." This triangle, symbolising the Logos, is therefore not only the beginning of the Science of Geometry, and therefore of Masonry, the Head of all the Sciences, but it is by that triangle that all Geometrical forms, and therefore forms of knowledge, are made, and it became the most mysterious and secret symbol of the Logos, for is it not written by St. John that "In the beginning was the Logos, and by it were all things made"; so the Vesica Piscis, the cradle of the Logos, became the great secret of Masonry, the foundation as we find it upon which Gothic Architecture was evolved, the means by which its wonderful plans were laid down, and the most reverenced figure in Religious Symbolism, as shown by its use in seals, engravings, sculptures, pictures, &c., throughout the Middle Ages.

Let me make this clearer. The more one examines the typical points in the Saxon, Norman, and Gothic styles of Architecture, the more clearly one sees that the Architects of the two former used circles and squares on their tracing-boards, as units for their proportions, in drawing up both ground plans and elevations, with here and there suggestions only of the Equilateral Triangle having been made use of in some of the smaller details; whereas the Gothic Architects seem to have used the Vesica Piscis almost entirely. This explains the reason why true Gothic buildings have always been said to be built mainly on the basis of the Equilateral Triangle; this naturally follows, because the use of the Vesica creates, and therefore necessitates, the appearance of the Equilateral Triangle in every conceivable situation. The following quotation is typical of the leading essay writers on this subject: "The Equilateral Triangle enters largely into, if it does not entirely control, all mediæval proportions, particularly in the ground plans. In Chartres Cathedral the apices of two Equilateral Triangles (vide frontispiece to these Views), whose common base is the internal length of the transept, measured through the two western piers of the intersection, will give the interior length, one apex extending to the east end of the chevet within the aisles, the other to the original termination of the Nave westward, and the present extent of the side aisles in that direction. With slight deviation, most, if not all, the ground plans of the French Cathedrals are measurable in this manner, and their choirs may be so measured almost without exception. Troyes Cathedral is in exact proportion with that of Chartres, and the choirs of Rheims, Beauvais, St. Ouen at Rouen, and others are equally so. Bourges Cathedral, which has no transept, is exactly three Equilateral Triangles in length inside, from the East end of the outer aisle to the Eastern columns supporting the West Towers. Most English Cathedrals appear to have been constructed in their original plans upon similar rules." White's Classical Essay on Architecture compares the Norman with the Gothic, where he says: "In what is usually called the Norman period, the general proportions and outlines of the Churches are reducible to certain rules of setting out by the plain Square. As Architecture progressed the Square gradually disappeared, and the proportion of general outline, as well as of detail, fell in more and more with applications of the Equilateral Triangle, till the art, having arrived at its culminating point, or that which is generally acknowledged to be its period of greatest beauty and perfection in the thirteenth and the beginning of the fourteenth centuries, again began to decline. With this decline the Equilateral Triangle was almost lost sight of, and then a mode of setting out work by diagonal squares was taken up, for such is the basis found exactly applicable to the work of the fifteenth century, since which time mathematical proportions have been generally employed." And after referring to numerous scale drawings of Churches, windows, doors, and arches, he points out that every student of Church architecture must pronounce those of the untraceried and traceried first point to be the most beautiful of all, those of the Norman to be a degree less so, and those of the perpendicular and debased to be far inferior to either, and in that analysis we find that the Equilateral Triangle was used almost exclusively for determining one order (the Gothic), the Square for another (the Norman), and the Square diagonally divided for the other (the debased).

Now let me try to describe the wonderful properties of the Vesica Piscis, so that you may understand the mystery which shrouded it in the minds of those Mediæval builders. The rectangle formed by the length and breadth of this figure, in the simplest form, has several extraordinary properties; it may be cut into three equal parts by straight lines parallel to the shorter side, and these parts will all be precisely and geometrically similar to each other and to the whole figure,—strangely applicable to the symbolism attached at that time to the Trinity in Unity,—and the subdivision may be proceeded with indefinitely without making any change in form. However often the operation is performed, the parts remain identical with the original figure, having all its extraordinary properties, the Equilateral Triangle appearing everywhere, whereas no other rectangle can have this curious property.

It may also be cut into four equal parts by straight lines parallel to its sides, and again each of these parts will be true Vesicas, exactly similar to each other, and to the whole, and of course the Equilateral Triangle is again everywhere.

Again, if two out of the tri-subdivisions mentioned above be taken, the form of these together is exactly similar, geometrically, to half the original figure, and again the Equilateral Triangle is ubiquitous on every base line.

Again, the diagonal of the rectangle is exactly double the length of its shorter side, which characteristic is absolutely unique, and greatly increases its usefulness for plotting out designs; and this property of course holds good for all the rectangles formed by the original figure and for the other species of subdivision. But perhaps its most mysterious property (though not of any practical use) to those who had studied Geometry, and to whom this figure was the symbol of the Divine Trinity in Unity, so dear to them, was the fact that it actually put into their hands the means of trisecting the Right Angle.

Now, the three great problems of antiquity which engaged the attention and wonderment of geometricians throughout the Middle Ages, were "the Squaring of the Circle," "the Duplication of the Cube," and lastly, "the Trisection of an Angle," even Euclid being unable to show how to do it; and yet it will be seen that the diagonal of one of the subsidiary figures in the tri-subdivision, together with the diagonal of the whole figure, actually trisect the angle at the corner of the rectangle. It is true that it only showed them how to trisect one kind of angle, but it was that particular angle which was so dear to them as symbolising their craft, and which was created by the Equilateral Triangle. All these unique properties place the figure far above that of a square for practical work, because even when the diagonal of a square is given, it is impossible to find the exact length of any of its sides or vice versa; whereas in the Vesica rectangle the diagonal is exactly double its shorter side, and upon any length of line which may be taken on the tracing-board as a base for elevation, an Equilateral Triangle will be found whose sides are of course all equal and therefore known, as they are equal to the base, and whose line joining apex to centre of base is a true Plumb line, forming at its foot the perfect right angle, so important in the laying of every stone of a building.

In the volume referred to I have given a skeleton plan upon such a scale of subdivision that a tracing-board, of 5 feet by 8 feet, would be divided up into over one million parts, and, as all these subdivisions are perfect representations of the original Vesica figure with all its properties, the design of the largest building, with the minutest detail, could be drafted with absolute accuracy. There are many other curious properties of this Figure, but they are difficult to explain without diagrams. I will, however, give one more example of its creative power. The problem of describing a Pentagon must have puzzled architects considerably in those early times, but this was again easily accomplished by means of the Vesica. Albrecht Dürer, the great designer and engraver, who lived at the end of the fifteenth century, refers to the Vesica in his works (Dureri Institutune Geometricarum, lib. ii. p. 56) in a way which shows that it was as commonly known in his time as the Circle, Square, and Triangle. His instructions for forming a Pentagon are: "Designa circino invariato tres piscium vesicas" (describe with unchanged compasses three vesicæ piscium). Three similar circles are described with centres at the angles of an Equilateral Triangle, forming the three Vesicæ, by means of which the Pentagon is drawn, and from which also we get a beautiful form of arch very common in the thirteenth century (vide illustrations in Magister Mathesios). This is also the method used in that old manuscript of the fifteenth century named "Geometria deutsch." In this old MS. it is also shown that the easiest method for finding the centre of a circle, however large, or any segment of a circle, is by means of the Vesica Piscis. And just as we see so many Cathedrals of the Middle Ages are stated by antiquarians to have been planned on the Equilateral Triangle, so do we find the Pentagon appearing as the basis of Architectural designs of buildings of a later date, such as Liverpool Castle, Chester Castle, and other similar structures; but the true means by which each were laid down, as in the case of the Equilateral Triangle, was again the Vesica Piscis. A beautiful example of decoration, on the basis of the Vesica, is seen in the tomb of Edward the Confessor in Westminster Abbey.

I will conclude this subject by quoting from the summing up by Prof. Kerrich (Principal Librarian to the University of Cambridge in 1820), in his masterly Essay on Architecture, where he gives the different forms of what he calls the "Mysterious Figure," used in the most noted Gothic buildings: he says, "I would in nowise indulge in conjectures as to the reference these figures might possibly have to the most sacred mysteries of religion; independently of any such allusion, their properties are of themselves sufficiently extraordinary to have struck all who have observed them."

From earliest Christian times the principal doctrine based upon the Mysticism of the Neo-platonists and the Kabalists was what was called the Γνωσις, the Knowledge of the All,