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      There are more things on earth and in heaven, Horatio, than are dreamt of in your philosophy.

Hamlet, William Shakespeare
  

I still have one point to make about the validity of this work. Apart from possible errors on my part, it is possible that future analyses based on strict metric measurement will cast doubt on the legitimacy of a particular layout.

However, it must be understood that if a process is common to a large number of buildings, it cannot be discarded for the simple reason of mathematical precision. Let me explain: in all the examples, the geometric principles are identical. Simple shapes form a polygon, onto which a fraction of the base is sometimes transferred to construct a triangle of proportion 2/3, 2/5, 3/4, 3/5 or 3/4/5.

This process can be found in the profile of the priory at Saint-Leu-d'Esserent, built around an equilateral triangle crowned by a 2/5 transfer, in the profile of Amiens Cathedral, which is made up of a long square topped by a 30° triangle, and the structure of two equilateral triangles and a 1/2 transfer, in Troyes Cathedral, built around two equilateral triangles, and in all the other Gothic monuments I've had the opportunity to study. An investigator would speak of an "orgy of evidence".

This is why a numerical discrepancy is likely to reveal a break in the construction process, a change of master builder, a modification of the project, or one of the many accidents that have dotted the long life of these monuments. For example, many cathedrals have undergone floor repairs, which automatically lead to a change in the height of the vaults. According to Professor Andrew Tallon, the vault at Bourges was thirty centimetres lower than it is today.

I've also noticed that the floor level is not always uniform throughout the cathedral. There can be major differences between the aisles and the central nave, or between the nave and the choir. Beauvais is a perfect illustration of this (see figure 1). So, geometric construction may or may not express itself depending on the location. Let's not forget, however, that the builders' intention was not to showcase these essentially secret layouts. They were guides, aids to design. It is even legitimate to think that, over the years, certain liberties may have been taken with these formal models, with geometric constructions becoming merely indicative.

Fig. 1 - East-west section of eastern Cathedral of Beauvais - Credit: Cyark

Let's also bear in mind that the master builder designed the elevation before finishing the pavements; this last stage involves earthworks and therefore variations in height. Added to this are the margins of error inherent in any medieval building site...

We're sure that this method of measurement was highly accurate ;)

In light of this, it's a good idea to leave formal precision to the background and focus on the Gothic idea, the recurring geometric motifs of these buildings.

That said, there's still one big question. Can the geometric layouts, the segments of the Table and the principle of the three enclosures be successfully applied to all cathedrals? The answer is both simple and complex. As far as layouts are concerned, the answer is clearly positive. However, while elevation layouts and the three-enclosure principle are universal, the principle of segments in the Table is not. Restrained by sexpartite vaults, the first cathedrals kept their layouts simple. In this context, the segment system, which is intended to simplify their design, has no place. By contrast, the barlong-plan cathedrals of the so-called "classical" period feature segments defined according to common procedures. They were born of an operational necessity, a symbolic need. It was important to be able to build simply, with complex rhythms and a common unity.

These secrets did not survive the fall of the Temple in 1307, the 100 Years' War or the Great Plague of 1349, events that signalled the decline of Gothic architecture and the disappearance of its secrets. It was the end of a world. The 14th century saw a definitive end to the enthusiasm and creativity of previous centuries. Faith, the medium of creation, became constraint, and free thought disappeared under formalism.

Gothic art ignored art for art's sake. Its beauty lay in the perfect match between the work and its purpose. In the decades that followed, the technique of craftsmen and sculptors reached new heights. But this very virtuosity, erring on the side of exuberance and excess, stripped the buildings of all unity. This architectural expression needed a guiding line, a moral and technical "authority", to prevent it from drifting off course. This framework had disappeared. We all remember Beauvais Cathedral, whose arrogant spire led to its collapse (1573). The spire had been imposed by the bishop before the nave had been built. In other times, this madness would never have happened. Most architects were lay people, no longer linked to the abbeys or steeped in the mystique of the Trait. They had forgotten the Gothic idea. They couldn't pass on its secrets.

The search for geometric relationships common to the Gothic style was a challenge. The idea was that proportions could be achieved through simple, traditional geometric relationships. And that's exactly what happened here. I've come up with a method, a geometric vocabulary, in fact a complete system of Gothic design that can be summed up in a few figures. First, the circle, the matrix of all regular polygons. Next come triangles, principally the equilateral triangle, the Pythagorean triangle and triangles of proportions 2/3, 2/5 and 3/5. And finally, there's the square. It's astonishing to realize that these figures alone are enough to define the elevation of Gothic cathedrals.

The bases of DNA are four in number, yet the entire universe would not be vast enough to contain the different genes created by their combinations. Thanks to four polygons, we can raise the proportions of all existing cathedrals and those that could have been.
It's easy to see that these figures and the constructions that order them, the triple enclosure system, the division of spaces by three, the double layout system and the construction of the apse, together form the true signature of Gothic cathedrals.

It wasn't easy to rediscover these secrets, but my admiration goes to those who found these magical proportions, the creator or creators of these master plans. They devised a system that was easy to teach and memorize, and which would enable the Gothic style to expand rapidly.
Incidentally, the eternal question of how it was possible to find so many builders capable of implementing this architecture in such a short space of time is answered. A new mystery is solved.

You'll notice that, in this study, I've talked about geometry without focusing for a single second on the units of measurement used by the builders. All too often, these numerical elements give rise to the most questionable arithmological or biblical interpretations. So I'll simply repeat Paul Valéry's phrase: "Pure geometry lives on this ignorance. It doesn't worry about units of measurement, and declares itself true on any scale.