Here is my paradox. All students of Gothic architecture would surely agree that our great churches were laid out using some combination of geometric and arithmetic methods — methods that must leave their traces in the finished edifices. Yet attempts to define the process more closely in any given building, to establish patterns of practice common to many buildings, or to speculate upon the significance of numbers and shapes often result not in consensus or productive scholarly exchange, but rather in rancorous accusations of unacceptable methodology, sloppy measuring, wishful thinking, or skullduggery. Thus, Eric Fernie (1990) opens his ‘Beginner’s Guide to the Study of Architectural Proportions and Systems of Length’ with a rueful reflection on the ‘almost pathological condition once described as pyramidiocy,’ which drives the idiot to explicate the forms and dimensions of the Great Pyramids through reference to geometrically encoded messages.1 After this admirably skeptical introduction, Fernie proposes a useful set of critical working principles: that behind the conception of the great building may lie relatively simple geometric manipulations, often involving a basic proportional relationship like 1:√2. In order to crack the underlying code, Fernie insists, the investigator must actually measure the building and work with the numbers, or with accurate digitally scanned shapes. It is not enough to superimpose thickly limned geometric figures upon small-scale plans or sections. The work most often cited as a negative example is George Lesser’s Gothic Cathedrals and Sacred Geometry with its small plans heavily overlaid with geometric figures (Lesser 1957).2

Interestingly, in the other notable recent wide-ranging essay on medieval architectural design, The Wise Master Builder, Nigel Hiscock takes positions diametrically opposed to Fernie’s, insisting instead that one should work by imposing a limited range of geometric shapes on existing plans since the direct involvement of the investigator in measuring the building will, or so he claims, introduce an unacceptable level of subjectivity (Hiscock 2000; see also Hiscock 2007). Hiscock argues that older studies tended to place too much emphasis on the application of the manipulative mechanism of the square root of two: he suggests that we should concentrate instead upon the three ‘Platonic figures’: square, equilateral triangle and pentagon. Nobody would doubt the importance of these figures in the builder’s design tool box, but Hiscock’s bewildering geometric tangles inscribed upon small plans may leave the reader baffled. Why, the skeptical student might wonder, would the builder adopt such extraordinarily complicated design strategies as these? What advantage would have been gained? How could users or visitors have begun to comprehend the system?

The juxtaposition of Hiscock’s and Fernie’s views serves to illustrate the extent of disagreement in a field where a given building might be ‘explained’ in light of geometric or numerical systems that appear entirely at odds with each other. I think particularly of the conflicting interpretations of the design of St-Etienne of Nevers by Marie-Thérèse Zenner and James Addiss published in the recent volume, Ad Quadratum. Zenner (2002) proposes a series of greater and lesser circles; Addiss (2002) favors a modular system based upon a known foot unit. Given such disagreements in the application of geometric schemes to real buildings, Robert Bork, in the most recent major publication on design in Gothic architecture, The Geometry of Creation, restricts his purview to the plans and drawings made by medieval mason and artisans. These plans can be accurately photographed or scanned, imported into a computer-aided design (CAD) program and subjected to rigorous analysis.3

The extraordinary technological advances of recent decades have provided formidable tools (including the laser scanner and total station as well as computer-assisted drafting) that allow the investigator to avoid one of the pitfalls encountered by the metrological sleuth — the accusation of sloppy measuring. Is it now finally possible to reach ‘scientific’ certainty about the way that medieval buildings were designed? I will argue in the following pages that such ‘certainty’ may still remain elusive and that it would be well to maintain a high level of critical self-consciousness about the basic underlying premises in the search for proportion and measure in Gothic architecture. Having defined four such premises, I shall introduce a case study: an examination of the choir of Beauvais Cathedral based upon the computer-assisted analysis of new laser-scanned data. I will conclude by invoking the notion of ‘plotting’: with the recognition of the multiple meanings of the word ‘plot’ we may begin to find some resolution of the paradox defined at the start of this essay (see Murray forthcoming).

My first premise is this: We may all surely agree that with the means available to the cathedral builders the best way to control the terrain vague was with ropes tightly stretched and pegged on the ground. Right-angled corners could be formed by the application of the Pythagorean triangle and orthogonal correctness assured by equalizing the diagonals. Although this was surely the way that most major building projects were laid out in the Middle Ages, the process left few direct written records.4 Mention of the stretched cord as a means to establish rectitude finds it way into the written sources mainly as a tropological, mnemonic or visionary metaphor (Carruthers 1992; 1993; 1998).5 Optical triangulation devices were probably available to cathedral builders: Villard de Honnecourt, the thirteenth-century ymagier who in the first decades of the thirteenth century witnessed the construction of several Gothic cathedrals and who left us a little book of images (Bibliothèque nationale de France, MS Fr 19093), shows us primitive triangulation devices, and in the archivolts of the central west portal of Chartres Cathedral, the angels carry astrolabes (fols. 20r–v in Barnes 2009: color plates 43–43).6

Second, we can also agree that the design and construction of a Gothic cathedral involved the systematic application of units of measurement of some kind. The range of pre-metric types of foot unit is well documented (see e.g. Zupko 1978; also Machabey 1962). However, we cannot assume that our edifice was necessarily laid out using the local foot unit: The master mason, probably not a local man, might have favored an imported unit of mensuration. The decades around 1200 saw the penetration of royal units of measurement beyond the Île-de-France — in some instances the royal foot of 32.5 centimeters may have displaced the local unit, which often corresponded to a Roman foot of 29.5 cm (Hecht 1979). More than one type of unit might have been in use in the same worksite. This was certainly the case at Beauvais Cathedral, as will be shown shortly.

The picture becomes even more complicated with the recognition that in order to make useful divisions on the stretched cords of the plot, a rigid measuring rod of convenient length would be necessary. To form the side of a square of one hundred feet, for example, a rod of ten feet might be applied ten times. With each application the builders might mark the end of the rod on the fully extended rope and then use that mark to begin another ten-foot stretch. The fewer times the operation is repeated the smaller the risk of error. In other words, there was a distinct advantage to using a rod that was longer rather than shorter. Peter Kidson, in his ‘Metrological Investigation’, explores the range of variation in the multi-foot rod, pole, or perch and the overlap between the methods of the late-Roman agrimensores or land surveyors and the cathedral builders — an overlap that may have helped fix conventions concerning the appropriate length of the perch (Kidson 1990). Stefaan van Liefferinge suggests that control over the building site of Notre-Dame of Paris was established with a grid that enclosed an agrarian unit — an acre — that was subdivided through the application of a ten-foot perch (Van Liefferinge 2006; 2010). In my own work at Amiens I also propose that a ten-foot perch was used to establish a working relationship between a fifty-foot central vessel and thirty-foot aisles.7 The length of the perch might also result from the transformation of the geometry of the Pythagorean triangle into round numbers in order to facilitate the accurate laying out of the building, as demonstrated by Peter Kidson in the article cited above, or it might offer an approximation to a round number in two different kinds of foot unit. In the late Middle Ages the toise, or fathom of six feet was frequently mentioned in the written sources relating to cathedral construction, as we shall see when we discuss Beauvais.

Third, to fix the boundaries and divisions of the roped-out plot which would determine the shape of the building, master builders employed polygonal figures of various kinds: squares (sometimes doubled, tripled or extended geometrically to form rectangles), equilateral triangles, pentagons or octagons, and such units might also have been projected upward to fix heights. Our consensus, however, is lost when it comes to determining what kinds of polygon and how exactly the figure might have been applied to fix plan and elevation: whether to center points or to wall surfaces; whether to the apex of the arch or to the capital? Medieval builders were not too clear on these questions either — we may recall James Ackerman’s exploration of the extraordinary debate over the potential application of triangle and square in the spatial planning of Milan Cathedral (Ackerman 1949). As to the systematic application of such geometric figures, or hypothetical perch units, it seems likely that the rope-stretching exercise might have been applied several times over and for different purposes: at first to control the overall shape of the site at the commencement of foundation digging and, subsequently, as the foundations reached pavement level, to ensure the desired geometric integrity and optical alignment of the piers and wall surfaces.

The fourth and last premise widely shared by students of Gothic design methods was best defined almost half a century ago by François Bucher (‘Design in Gothic’) and Lon Shelby (in his work on Mathes Roriczer): the role of dynamic geometry is understood principally as ‘quadrature’ or rotated squares (Bucher 1968; Shelby 1976). Representation of quadrature, present already in the writings of Vitruvius (1999: 107 and 282, Fig. 110), is found in Villard de Honnecourt’s little book (c. 1230), where an operation based upon inscribed rotated squares is oddly labeled as a means of dividing a stone into two equal parts (fol. 20r in Barnes 2009: color plate 42). The strategy, which introduces irrational numbers (in the ratio 1:√2) to the business of dynamic projection and the appearance of organic growth to the buttress or pinnacle, may be applied to particular details or to the space of an entire building. The two most essential Gothic elements, the pointed or broken arch and the rib vault, bring the potential for dynamic animation as the rotational mechanism of the compasses is transferred to the spatial behavior of the building itself. Gervase of Canterbury, describing the construction of an arch, uses the dynamic verb volvere, to turn with the arc of the compass.8 Once designers had escaped the tyranny of the single center point fixing the height of the arch, proportions became infinitely fungible. Shelby (1976: 47–48) has reminded us of the essential arbitrariness of the application of dynamic geometry.

I have suggested that one way to help resolve the paradox — the credibility gap — defined in the opening paragraphs is for the historian to introduce new technology. Such technology will be especially interesting in the context of a monument that has been carefully studied using the old methods.9 In 1978 I undertook an exhaustive survey of the choir of Beauvais Cathedral using a combination of a fifty-meter steel tape, able to stretch from one end of the choir to the other or from the vault to the pavement, coordinated with a system of quadrangulation, applied through the use of a niveau de chantier (theodolite), and direct triangulation, which is particularly useful for the wedge-shaped bays of the ambulatory and the exterior surfaces and buttresses of the radiating chapels. For pier sections I had recourse to an enormous pair of calipers. For wall thicknesses, armed with plumb bob and tape measure, I scaled a ladder and measured from inner and outer wall surfaces to the window glass, of negligible thickness. The laser scanner was then applied (thanks to Peter Allen and his team and to Andrew Tallon) in two phases: first in 2001–2 and again in 2013.10

Let us start at the west end of the Beauvais choir, where construction began in 1225 (Murray 1989: 60). At a time very close to this, Villard de Honnecourt captured the plan of a Cistercian church designed around squares (une [é]glize desquarié) where the bays of the main vessel are made up of double squares equaling half the crossing square and the aisles are squares equaling one quarter of that central square (fol. 14v in Barnes 2009: color plate 31; see also Hiscock 2004). With such a linear matrix the spatial form of the church can be compressed: conceptualized and held in the head as a mnemonic image. This compressive phase might then lead to expansion as the elements of the building are extended and laid out full scale on the ground, controlled by a grid of stretched ropes. Our search for a comparable mechanism at Beauvais Cathedral is frustrated because only one of the four crossing piers (to the south east) is original: the western crossing piers were built by Martin and Pierre Chambiges in the early sixteenth century, and the northeastern crossing pier was rebuilt after the collapse of the crossing tower in 1573.11

Figure 1 shows the laser-generated plan with inserted bay divisions and dimensions (center to center of the piers and to the interior wall surfaces) of the choir straight bays.12

In Figure 2, having projected the great crossing square, I have divided it into four and placed two such quarter squares on the western bay of the choir to give the main vessel (close to a double square) and one square on each of the transept towers. It should be noted that computer assisted drafting makes it impossible for the author to ‘fudge’ his geometric figures in order to make them fit perfectly between the bay divisions. In Figure 2, I asked my CAD program for a perfect square which I scaled to fit and then duplicated and dragged the identical squares into place, leaving visible the small discrepancies which work their way into the plan as it is worked out on the ground.

Now, what sense can we make of the lateral (north–south) organization of the straight bays of the choir? My old manually measured plan led me to propose three equal spatial corridors corresponding to the width of the main vessel (c. 15.3 m) and each double aisle to just beyond (0.15 m on each side) the exterior surfaces of the aisle walls (Murray 1989: Fig. 18). The new plan allows us to confirm this observation (Fig. 3).

A written description of the Beauvais choir, probably made in the early decades of the fourteenth century, provides a most significant number: the exterior width of the choir is said to be 24 fathoms. Since a fathom equals six feet, we have a total lateral spread of 144 feet divided into three corridors of 48 feet.13 It will be remembered that in the Book of Revelation the Celestial City is represented as 144 cubits wide.14 Closely related to this scheme is the one proposed by Alain Guerreau (1992), who looked at the dimensions between wall surfaces. Here we can find six roughly equal spatial corridors (Fig. 4).15

If the lateral (north–south) divisions of the plan provide broad symmetry, what are we to make of the progressive widening in the three original bays of the main arcade from west to east with the widest bay at the base of the hemicycle (Fig. 1)?16 This is the reverse of the design of neighboring Amiens Cathedral where the widest bay comes next to the crossing. Is it possible that the three irregular bay divisions were simply inserted into a larger matrix designed to make overall sense numerically or geometrically?17

It is with the geometry of the hemicycle that the value of the laser-scanned plan becomes most apparent. In my manual survey of 1978, mimicking the strategy of the builders, I had constructed the configuration of the hemicycle using a great rectangle pegged out with cords stretched on the pavement. Similarly, I pegged out rectangles in each of the radiating chapels, and triangulated measurements taken of the wedge-shaped bays of the ambulatory and around the polygonal shapes of the chapels. Control of the wall thickness then allowed integration of interior and exterior. My manual construction (meticulous as well as laborious) of the hemicycle plan on a 1: 50 scale led me to propose that it was generated from a single center point located about 2.40 meters east of the last transverse bay division, and that the radial geometry was fixed by a regular thirteen-sided polygon (Murray 1989: 15).18 The laser survey has provided striking confirmation of these findings (Fig. 6).

With computer-aided design it was possible to find the centers of the great exterior buttresses and the interior piers and to project lines inwards to find the hidden center point. These lines converged most satisfyingly on a point 2.26 meters east of the base of the hemicycle. It is impossible to escape the conclusion that the accuracy of the concentric composition was ensured through a central peg driven into the ground and the sweep of a cable or rope to fix the position of the hemicycle piers, the divisions of the mouths of the chapels, and the chapel depths. The walls and buttresses of the chapels sit atop a circular exterior plinth struck from the same center point, radius 22.4 meters.19 This plinth, now much restored, helps evacuate rainwater out and away from the foundations. The diameter of this great circle (44.8 m) is slightly less than the total exterior span of the choir straight bays (45.5 m). The CAD program allowed me to generate the thirteen-sided polygon to fix the location of the hemicycle piers and chapel mouths — something which is very difficult to plot using traditional manual methods.20

In planning the spatial forms of the hemicycle the builders were probably driven by three factors: first, the desire for seven radiating chapels of equal depth — quoting the chevet of the Cistercian church at Royaumont (King Louis IX’s favorite) — and thus signaling the close alignment of Bishop Robert de Cressonsac with the king; second, the need to have the width of each chapel match the width of the choir aisles; and, third, a desire for an overall internal shape corresponding to a square (Fig. 5). The reconciliation of these various desires probably lies behind the placement of the hemicycle center point.21

Let us now project plan into elevation and turn to the transverse section of the choir. Our laser scan at the base of the hemicycle where the vault survived the partial collapse of 1284 allows us to study how closely the overall height and width approximate a great square (Fig. 7).

In fact, the height of the keystone of the hemicycle vault, about 46.3 meters above the pavement, somewhat exceeds the total exterior lateral spread of the choir, 45.6 meters. However, the notion of square schematism is reinforced by the fact that the span of the main vessel (15.3 m, center-to-center) projected upwards, will produce the abacus of the capitals of the main arcade and three such squares will reach the main vault.22 The great vertical division into three thus matches the organization of the plan into three corridors of space. The matching of span and height, as well as the number 144, allow the cathedral to embody the attributes of the Celestial City, as witnessed by Saint John the Divine and described in Revelation 21.

The alarmingly steep elevation at Beauvais was not necessarily planned from the start. We might remember that the space frame of two of the cathedrals that formed key points of reference for the builders of Beauvais, Notre-Dame of Paris and Bourges Cathedrals, had both been designed around equilateral triangles (Sandron and Tallon 2013: 31). There is tantalizing evidence that Beauvais, as originally planned, may have shared this configuration. Attached to the outer flying buttress upright on the north side of the westernmost bay of the choir is a now-redundant flyer (Fig. 8).

The form of the crockets set upon its crest suggests a thirteenth-century date, and since work at Beauvais progressed generally from west to east, with the north side ahead of the south, it may be concluded that this buttress is early work. If we project the trajectory of the flyer inwards and upwards it will come very close to intersection with the apex of an equilateral triangle inscribed between the outer surfaces of the lateral walls (Fig. 8). It was from this same base that the square that fixes the height of the existing vault was projected.

What can we conclude from this case study? First, that while the laser-scanned plans and sections coupled with computer-assisted analysis provide invaluable investigative mechanisms and powerful means of representation, they do not ‘solve’ the problem of understanding the application of several different kinds of units of measurement, changing intentions on the parts of the builders, or the scholarly audience’s skepticism about the search for shapes and numbers that appear to emerge, yet not always with sufficient precision to convince us that they necessarily reflect the intentions of the builders. The number 144 emerges particularly clearly at Beauvais, expressed by the external width of the choir, which is very close to 144 royal feet and just a little less than the height of the vaults. The division of this width into three equal parts (48 feet) appears to be a feature of the transverse section where the span of the main arcade fits three times under the high vault and the total width where the main vessel almost equals each of the double aisles.

How should we understand the apparent irregularities of the east-west dimensions and the twisted hemicycle? What we are dealing with at Beauvais is a combination of error and artifice. The error may be easily quantified: as work advanced from west to east, exterior to interior, the builders failed to coordinate the bay divisions anticipated in the great exterior buttresses with the placements of the interior supports, which, at the base of the hemicycle on the north side, have been allowed to slip a full 60 centimeters beyond their due positions.23

This much is susceptible to ‘proof’ through the application of ‘scientific’ methods. But what about artifice? Vitruvius provided the builders of late antiquity and the Middle Ages with invaluable advice on the reconciliation of the demands of symmetry and the ‘flashes of genius’ that allowed the architect to violate the requirements of regularity in response to local considerations and the desire for beauty.24 While we can never be certain about the intention that lay behind the progressive widening of the three original arcade openings of the Beauvais choir, the effect would have been readily available to the eyes of the user or viewer standing in the crossing space — the arrangement served to counter the visual effect of foreshortening, allowing the arcade to retain more of the appearance of regularity than it would have had with equal spacing of the piers.25 The arrangement would have allowed the gaze to penetrate freely into the peripheral spaces, organized in the plan as a double aisle of approximately equal value as the main vessel and in elevation as a spectacular series of descending spatial diaphragms.26 The same kind of optical sense can be found in the fact that the colonnettes attached to the main arcade piers (toward the central vessel) have been recessed into the cylindrical body of the pier, reducing the projection of the colonnette and allowing the gaze to slip more easily past the front surface of the pier. The glimpse of the sublime at Beauvais is not just the result of great height, but in the artful correlation of horizontal expansiveness and the truly awe-inspiring chasm created by the three-to-one elevation of the central vessel.27

While the metrological investigation of great medieval churches may certainly be understood partly as a ‘scientific’ enterprise facilitated through the systematic application of accurate measuring equipment and computer-assisted analysis, we should remind ourselves that cathedral space was and is living space.28 In the creation, application and reception of proportional systems we are dealing with a human phenomenon with an extraordinary range of sociological, anthropological and neurological implications.29 In attempting to correlate and to reconcile the various ways in which meaning is put into the great church by the builders and unscrambled by the users, I have become increasingly preoccupied with the multiple meanings of the ‘plot’ — not just as a piece of land staked out with cords (a cabbage plot, a burial plot, a building plot) but also in a social sense as a conspiracy to ambush the future (Murray 2011). And then, of course, there is a third dimension: ‘plot’ is the story line intended to impel the reader compulsively forward. Common to all three kinds of plot, spatial, social and rhetorical, is the human desire to control, to possess and to represent. We may postulate that humans from earliest times have looked for ways to distinguish a controlled plot of land from the terrain vague ‘out there’. This control is exercised not just in order to secure protection and shelter; there are profound religious and sociological dimensions that appear to transcend cultures.30 In social terms, cathedral building is akin to a kind of conspiracy — an attempt to anticipate and control the future with the appearance of perfection and manifest destiny in the finished edifice — in which the three builders: master mason, churchman and patron or budget provider may be seen as the conspirators (Fig. 9).

The successful working out of a plot may involve technology or a secret weapon of some kind, whether barrels of gunpowder placed beneath the House of Parliament or the mason’s formidable iron compasses and the geometry of the ‘Egyptian triangle’. The plot will always bring the potential for deceit — and it may also bring compelling persuasiveness. The most powerful weapon at the disposal of the master mason as he attempted to sell his vision of the unbuilt church to his prospective employers was the kind of mnemonic device we have encountered in the pages of Villard de Honnecourt in the ‘squared-up’ church. Through such a device even the layperson could commit to memory the complex spaces of a great edifice that could hardly be constructed in a single lifetime.

It is with the third meaning of the word ‘plot’ that I want to end. In storytelling the plot is the mechanism that lends shape and momentum to the narrative. ‘Reading for the plot’ implies the compulsive reaching forward to grasp the dynamics of the story (Brooks 1984). This is what Roland Barthes (1973; Le plaisir du texte) called jouissance. Humans compulsively control the shapes of their artifacts and unscramble them with a similar compulsion — this is not a disease and we are not idiots (Fernie’s pathological ‘Pyramidiocy’). Some, including myself, have confessed to a desire to possess our beloved building by grasping its essential geometric matrix, believing, rightly or wrongly, that we can in some measure enter into the brain of the designer in a process of co-creativity (Murray forthcoming). This is the human condition that Erwin Panofsky (1958) invoked with his notion of ‘mental habits’ and Pierre Bourdieu (1993) with his ‘habitus’, a domain of activity that has more in common with the social than with the physical sciences. In the Middle Ages such activity might have been considered as edification in moral terms or as the quest for salvation, as in the penitential act of measuring the Temple (Carruthers 1998: 232).

The key to the resolution of the paradox outlined in the opening passages of this essay may lie, finally, in the recognition of the unstable combination of science, artifice (including deceitfulness) and desire peculiar to the business of plotting. To stake out the building plot accurately, some understanding of practical geometry was essential (especially the nature of the Pythagorean triangle); and erecting the structure demanded knowledge of the practical working of the principles of physics which lie behind the working of the pulley systems of the great lifting machines. Villard de Honnecourt called such devices engiens (see Godefroy 1884, 3: 171); Gervase of Canterbury applied the word machinas. Both words are synonymous with tricks or ruses — in modern English we might remember the linkage between ‘devices’ and ‘desires,’ and between ‘machine’ and ‘machinate.’ Our enterprise, finally, should lead us not only to equip ourselves with the newest high-tech engiens — the laser scanner and computer-assisted drafting — but also to remain aware of all the other slippery dimensions of the plot. In plotting, science without art is nothing.