Gothic architecture and mathematics

      Humankind has never thought anything of importance that it did not write in stone.

Victor Hugo

Cathedrals are veritable books of stone, books whose alphabet has been lost over time.  To understand them, one thinks of questioning those who claim to be the custodians of their secrets; to put it briefly, the compagnonnage and freemasonry.
Concerning the former, we shall see that since the Middle Ages they have possessed a body of geometric techniques designed to solve the problems related to the design and construction of a building — the trait. However, in the eighteenth century, this heritage was profoundly revisited and transformed by the descriptive geometry of the mathematician Gaspard Monge.
A certain distance therefore separates the trait of the golden age of builders from that of the Age of Enlightenment. Under the same name, these techniques betray natures arising from almost antinomical ways of thinking. The first was born of a symbolic geometry; it is empirical and sacred. The second is analytical and mathematical.

To return to our question, I observe a fundamental difference between the geometric methods employed by the master builder and the technical trait used by the craftsman. This is not a value judgement, but one of nature. The former conceives the cathedral ex nihilo, from nothing. He works with the concept. The latter carries out work according to his speciality; he works with matter. Yet, as we shall see later on, a common scheme directs the development of Gothic cathedrals. It is an initiation, a specific knowledge entrusted solely to master builders. By definition, it does not form part of the traditional corpus of the compagnonnage.

Our last candidate remains: freemasonry. However, while many Masonic terms and symbols borrow from architecture, nothing attests to any link between this society and the medieval builders. Our questions therefore remain unanswered.

The ideal would be to go directly to the source, to the writings of the ancient master builders. But the tradition that compelled these men to pass on their knowledge orally deprives us of all texts and documents. Only the notebook of Villard de Honnecourt, a master builder of the 13th_e century, has come down to us.

Rich with thirty-three folios — that is, sixty-six pages of drawings — his notebook is divided between architectural sketches and sometimes unusual projects, such as a wheel capable of perpetual motion. While most of the drawings reflect the classic concerns of an architect, floor plans of apses or vault junctions, others seem merely to illustrate a supposed medieval naivety. Men wrestling appear alongside paired lions or summary portraits, all appearing to rely on elementary geometry, an awkward prefiguration of a perspective yet to come.

Extracts from Villard de Honnecourt's notebook - P.D.

As a result, the extraordinary interest represented by the sketches of Villard de Honnecourt was long underestimated. In reality, it partakes of the subtle spirit of the builders' science, that which conceals itself behind the often poorly understood notion of trait. By trait, I mean the body of geometric methods used since the Middle Ages to answer the problems posed by the design and technical realisation of a building.

This science of the art of the trait, the "portraitures" as Villard called it, constitutes the medieval prefiguration of technical drawing, a process defined by Viollet-le-Duc as "an operation of descriptive geometry, a decomposition of the multiple planes that make up the solids to be implemented in construction."
If this definition is to be believed, the art of the trait would be merely a preliminary stage to stereotomy, to the cutting of materials. It would confine itself to describing elements "at execution size," as if a geometric proportion were not, by its very nature, independent of the metric unit. By any measure, this definition is inadequate. It seems to ignore that the trait made it possible to obtain, by purely graphic means, the key elements of a building, such as the distribution of spaces or the height of a nave. For, far better than merely describing, this technique allowed the architect to create. Evidently, Viollet-le-Duc knew only the geometry of Gaspard Monge.

Villard de Honnecourt's notebook magnificently ignores mathematics. As the compagnons like to say: "the trait drives out the number." No crutch is needed for this art, which regards the compass, the ruler and the set square as the only tools capable of apprehending the universe.
The medieval builders came from a society barely discovering the use of zero and laboriously adding in Roman numerals. They had other values, a science and necessities transcending the fevers of mathematics. Their mission was to crown Chartres, Paris or Amiens with enchanting diadems of stone. Tracing on the floor of the lodge, of the "chamber of traits," the floor plan of the edifice, the master builder wielded, more often, and for good reason, the compass rather than the slide rule.

Furthermore, the introduction of algebra and Arabic numerals, and the theories of Euclid, Archimedes and Aristotle, were not disseminated until the end of the 12th_e century, well after the commencement of the first great cathedrals.

No need to belabour the point. Mathematics are foreign to the genesis of cathedrals, but conversely, why refuse their assistance in analysing Gothic layouts? Whether of a numerical or graphic nature, number or magnitude, these methods yield a result that is, in both cases, equivalent. Let us put this assertion to the test by seeking to recover the layout used to establish the elevation of figure 2, which is here that of a Gothic nave.

 


Fig. 2 - Analysis of the height of a nave through geometry

In the spirit of the trait, I shall place the point of a compass at O and open it as far as the springing of the vault at A. If I carry this opening across by an arc of a circle, I obtain the point A'. A perfect square has been revealed. It is thus apparent without further discussion that the elevation of the nave is built upon the diagonal of a square.

This method proves consistent with logic and tradition, for only the ruler and compass are necessary for its construction. It could have been drawn by Villard de Honnecourt himself. On the building site, nothing will be simpler than to retrace this proportion. In medieval symbolism, the elevation represents the projection of the earth toward heaven. It is therefore natural that the base of the square should serve as the module for the height of the nave.

For its part, the mathematical approach will assign the value of one unit to the width of the nave. In relation to this, the measurement of the height will give us the irrational number 1.4142, which is known to be the square root of two, that is, the diagonal of a square with side 1.

This single example allows us to observe the gulf existing between this method and the concerns and techniques of the medieval builders. It is the difference between compass geometry, which visually manipulates proportions, and mathematics, which confines itself to numbers. For all these reasons, to seek to understand Gothic geometric layouts without using the ruler and compass is an error that is both historical and methodological.