This article presents a geometrical analysis of Bourges Cathedral, based on the application of computer-aided design (CAD) techniques to the results of a recent and highly precise laser survey. This analysis reveals that the cathedral's original designer developed a tightly interlocking and strikingly unified design, in which the five-fold subdivision of the chevet ground plan set proportions that would be vertically extruded into an elevation that can be inscribed both within a square and within a series of progessively smaller equilaterial triangles. These results contribute to an ongoing debate about the use of ‘ad quadratum’ and ‘ad triangulum’ geometries in Gothic architecture, and they provide new evidence for the geometrical coherence of Gothic cathedral design. In methodological terms, meanwhile, this discussion demonstrates the potential of CAD-based geometrical analysis for the study of precisely surveyed medieval buildings.

The sequence of images being analysed can be viewed as supplementary material at:

The following geometrical analysis of Bourges Cathedral can be read as a complement of sorts to the articles that grew out of the 2011 Leiden conference on architectural proportion. For the decade prior to that conference, I had been using computer-aided design (CAD) software to study the proportions of Gothic architectural drawings. As my paper for the conference explains, and as my book

Example (slide 15) of the graphics used for the geometrical analysis of Bourges Cathedral. For the complete set of images, access

I had expected that Bourges would offer a particularly rich subject for geometrical and proportional analysis, for two main reasons. First, the cathedral has an intriguing and very unusual design, since it lacks a transept, and since its five aisles are staggered in height, creating a cross section whose roughly triangular outline is emphasized by the steep slope of its flying buttresses. Second, the cathedral ranks as one of the largest Gothic structures in which the fabric produced in the first building campaign survives substantially unaltered by later modifications and accretions.

My analysis of Bourges has given me a fresh perspective on many of the topics discussed by my fellow medievalists at the Leiden conference, and on my own work as well. Most immediately, of course, my analysis builds on the work of Andrew Tallon, underscoring the value of the exacting survey work that he has undertaken in recent years (

To put this analysis into context, it will be helpful to make a few brief observations about the literature on proportioning figures in Gothic elevation design. Early in the debates on Milan Cathedral, a basic question was whether the elevation of the church should be inscribed in a square or within the lower figure of an equilateral triangle. This distinction certainly mattered for the planning of the building, but it is important to note that ‘ad quadratum’ and ‘ad triangulum’ schemes could be applied in various ways, and even in some respects reconciled with each other, as the case of Bourges will demonstrate. At Milan, moreover, the rejection of the square option did not end the debate about how the triangle scheme should be applied. Ackerman’s article has introduced many students of medieval architecture to the fact that the mathematician Stornaloco recommended a clever modular approximation to the equilateral triangle scheme, which the local builders ultimately rejected in favor of an even lower format based on 3–4–5 right triangles.

The following analysis of the Bourges choir design will demonstrate an important feature of the Gothic design process: namely, the way in which dynamical schemes of geometrical unfolding could develop even within the seemingly rigid frames of basic geometrical figures like the circle, square, and equilateral triangle. So, while these figures were commonly used in the steps of the Gothic design process, Gothic buildings tend to differ widely in the proportions of their respective elements, with little of the convergence to canonical proportions seen in classical design.

Because of the dynamically unfolding nature of the Gothic design process, a fruitful strategy in the analysis of Gothic proportions can be to develop trial sequences of geometrical operations that generate forms matching those seen in the building, or in the associated design drawings. The following analysis of the Bourges choir, like the other such analyses I have undertaken, was developed in that manner. This approach may appear convoluted when compared to simple design strategies like inscribing a building elevation within a square or an equilateral triangle, but the relationship of the building components to those simple schemes only becomes clear when the details are taken into account. These are precisely the qualities that give the buildings their individual flavor. Gothic cathedrals, after all, deserve line-by-line scrutiny as much as literary texts. I thus hope that the following step-by-step analysis of the Bourges choir will help to shed some light on the particular artistic personality of the cathedral’s first designer, as well as on the nature of Gothic design practice as a whole.

Please visit the following link to view the supplementary material:

Supplementary Material

These results (

The standard monograph on the building is Branner (

As noted in Bork (

See note 1 above.

On Saint-Denis, see Bork (

See Ackerman (

For critique of geometrical research, see especially Hecht (

It is interesting, for example, that buildings with very different proportions can be constructed within the frame of an equilateral triangle. Single-aisled buildings like the cathedral of Strasbourg or the church of Saint-Urbain in Troyes will tend to have wide main vessels when designed within this frame, while a double-aisled building like Bourges cathedral will have a comparatively slender main vessel, since this central space takes up a smaller fraction of the building’s total width.

On Reims, Clermont, and Prague, see Bork (