Gothic and mathematics

The human race has never thought of anything important unless it was written in stone.

Victor Hugo

Cathedrals are veritable stone books, whose alphabet has been lost over time. To understand them, we need to question those who claim to be the depositories of their secrets; in short, the Compagnonnage and Masonry.
As far as the former are concerned, we'll see that, since the Middle Ages, they have had at their disposal a set of geometrical techniques designed to solve the problems associated with the design and construction of a building: the line. However, in the eighteenth century, this heritage was profoundly revisited and transformed by the descriptive geometry of mathematician Gaspard Monge.
So, there's quite a distance between the line of the Golden Age of Builders and that of the Age of Enlightenment. Under the same name, these techniques betray natures born of almost antinomic ideations. The first is born of 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 processes employed by the master builder and the technical trait used by the workman. This is not a judgment of value, but of nature. The former designs the cathedral ex nihilo, from scratch. He works on the concept. The latter carries out work according to his specialty: he works the material. However, as we'll see later, a common pattern runs through the construction of Gothic cathedrals. It's an initiation, a specific skill entrusted solely to the master builders. By definition, it does not form part of the traditional corpus of Compagnonnage.

Our final candidate is Freemasonry. Although many Masonic terms and symbols are borrowed from architecture, there is no evidence of any link between this society and medieval builders. Our questions therefore remain unanswered.

Ideally, we'd like to go straight to the source, to the writings of the old master builders. But tradition, which required these men to pass on their knowledge orally, deprives us of all texts and documents. Only the notebook of Villard de Honnecourt, a _13th-century master builder, has come down to us.

With thirty-three folios, or sixty-six plates of drawings, his notebook is divided into architectural sketches and sometimes unusual projects, such as a wheel capable of perpetual motion. While most of the drawings reflect the classic preoccupations of an architect, with plans for apses or vault connections, others seem content to illustrate a supposed medieval naiveté. Men in combat rub shoulders with lions or sketchy portraits, all seemingly based on elementary geometry, a clumsy foreshadowing of perspective yet to be born.

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

As a result, the extraordinary interest represented by Villard de Honnecourt's sketches has long been underestimated. In reality, it's part of the subtle spirit of the builders' science, which lies behind the often misunderstood notion of line. By "trait", I mean all the geometric procedures used from the Middle Ages onwards to respond to the problems posed by the design and technical realization of a building.

This science of line art, or "portraiture" as Villard called it, was the medieval forerunner 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 used in construction
According to this definition, the art of line drawing is merely a preliminary step to stereotomy and the cutting of materials. It would be confined to describing elements of "working size", as if a geometric proportion were, in essence, independent of the metric unit. From every point of view, this definition is inadequate. It seems to ignore the fact that, by means of a purely graphic solution, the line could be used to obtain 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 enabled the architect to create. Clearly, Viollet-le-Duc only knew the geometry of Gaspard Monge.

Villard de Honnecourt's notebook is superbly ignorant of mathematics. As the companions like to say: "the line pushes the number". There's no need for a crutch in this art, which considers the compass, ruler and square to be the only tools for understanding the universe.
Medieval builders came from a society that had barely discovered the use of the zero, and laboriously added in Roman numerals. They had other values, a science and necessities that transcended mathematical fevers. Their mission was to crown Chartres, Paris or Amiens with enchanting stone diadems. Tracing the plan of the building on the floor of the lodge, the "chambre aux traits", the master builder wielded the compass more often than the slide rule, and with good reason.

What's more, the introduction of algebra and Arabic numerals, and the theories of Euclid, Archimedes and Aristotle, were not disseminated until the end of the _12th century, well after the first great cathedrals had been built.

There's no need to go on. Mathematics was foreign to the genesis of the cathedrals, but a contrario, why refuse its help in the analysis of Gothic layouts? Whether numerical or graphic, number or magnitude, these methods provide a result that is, in both cases, equivalent. Let's put this assertion to the test by trying to find the layout used to set the elevation of figure 2, in this case that of a Gothic nave.

Fig. 2 - Geometric analysis of nave height

In the spirit of the line, I'm going to place the point of a compass at O and open it up to the birth of the vault at A. If I transfer this distance to an arc of a circle, I obtain point A'. A perfect square has been revealed. It's clear, then, that the nave's elevation is built on the diagonal of a square.

This procedure conforms to logic and tradition, as only a ruler and compass are required for its construction. It could have been designed by Villard de Honnecourt himself. On site, nothing could be simpler than tracing this proportion. In medieval symbolism, elevation represents the projection of the earth towards the sky. So it's only natural that the base of the square should serve as the module for the height of the nave.

For its part, the mathematical approach assigns the value of one unit to the width of the nave. In relation to this, measuring the height will give us the irrational number 1.4142, which we know to be the square root of two, i.e. the diagonal of a square with side 1.

This example alone demonstrates the gulf between this method and the concerns and techniques of medieval builders. This is the difference between compass geometry, which visually manipulates proportions, and mathematics, which is confined to numbers. For all these reasons, attempting to understand Gothic geometric layouts without using the ruler and compass is both a historical and methodological error.