To construct a square being exactly equal to a given circle by compass and ruler was one of the three famous geometrical problems of antiquity. A host of circle-squarers have tried their hand on this age-old impossibility, until it was proven futile by Lindemann in 1882. The problem can, however, be represented geometrically, as demonstrated by Dr.-Ing. Helmut Sander in the Nexus Network Journal vol 2 no. 1. I immediadely recognized his diagram as being very close to a diagram that I have found by analyzing two early Norwegian basilica plans, and, that by first sight, I believed might have been used in determining the ratios between some important dimensions. This spurred me to make further investigations, revealing that it was not quite that simple. However, this pursuit revealed some alternative, but related possibilities, including a way of combining root-2 and root-5 - albeit approximately, but close enough to fool a non-mathematician working by small-scale geometry to make a false assumption simliar to Le Corbusier's when he was developing the Modulor (Le Corbusier 1954: p. 229ff.).

On the little island of Selja, immediately south of the Stad promontory, are the ruins of a Benedictine monastery (Figure 1). The abbey church, dedicated to St. Alban, contains remnants of the earliest known stone-built basilica in Norway. This church was built around 1100, maybe a little earlier, and served as the cathedral for the first diocese of western Norway until the see was transferred to Bergen about half a century later. A considerably larger cathedral was then built in Bergen, but the remains of this are too scant to make any assumptions about its design, other than that it must have had an aisled nave.

In the early 12th Century, a new diocese was established south of Bergen. The cathedral of Stavanger is still standing, with its Anglo-Norman nave almost intact (Figure 2). Despite the apparent inaccuracies of its plan layout, some of the main dimensions indicate the use of simple numerical ratios when setting out the plan. However, other dimensions like e.g. the external width, are too far from fitting into this numerical system that it can be ascribed to inaccurate layout.

I therefore looked for ratios generated by geometry, and arrived at a diagram with a square whose sides are divided into 4 equal parts, and a circle cutting sides of the initial square through their outer quarter divisions (Figure 3). The circle diameter is thus is v5 of half the square side. If the circle radius is 1, the square side will be v3,2, which is a rough approximation to vp, and the square area is then somewhat larger than the circle area. This approximation is slightly better than the ratio of 8/9 between square side and circle diameter as stated in the Egyptian Rhind papyrus of c. 1500 BC. Moreover, a square with side equal to the circle generated by this diagram will have approximately the same perimeter as a circle circumscribing the initial square (Watts & Watts 1987, p. 269).

In Stavanger, the circle diameter seems to determine the external width of the nave, while half the square side is the distance between the central axes through the arcade piers at their eastern outset. This distance seems to have been found by dividing a square with sides 1/v2 of the external length of the nave at its central axis by half its diagonal from the square corners (Figure 4). This subdivision of a square, sometimes called "the sacred cut" (Brunés 1967) seems to have been used in building design and town planning in Roman antiquity (Watts & Watts 1987, 1992).

The geometry of the plan also seems to relate to the elevation (Figure 5), as the height from floor level to the roof ridge is very close to 2/v5 of the external width of the nave, or the side of the initial square in the diagram for the approximate squaring of the circle mentioned above, which is twice the distance between the arcade piers' axes. The height of the arcade walls are v2/4 of the external nave length, or half the side of the square from which the pier axes seem to have been determined. The original height of the arcade walls and roof ridge can, however, not be ascertained, as the wall heads and roof were completely rebuilt during the restoration of 1867-69. In addition, the settling of the walls, that at least in some parts must have occurred during the first building campaign, makes it difficult to determine the original base level accurately. The present levels are in accordance with those indicated by the gothic choir gable, and may have been changed after the fire of 1272.

Like most of the early bishops of Norway, the first Stavanger bishop was of the OSB, and the cathedral shows some similarities with the first St. Alban's church at Selja. The ruins there were excavated and meticulously measured in the 1930s, and for comparison, I made a rough analysis of the plan. On the drawing to the scale of 1:100, it seemed that the ratio of internal nave length along its central axis to internal width was root-2, and the pier axes determined by "the sacred cut" from a square with side the internal length (Figure 6). Even more intriguing was that the external length seemed to be ½ v5 of the internal length, or corresponding to the approximate "squaring of the circle" discussed above.

Last summer I had the opportunity to visit Selja and make some measurements. With the present state of the ruins, the length of the nave cannot be determined exactly, but the bases and lower parts of the west front and side walls are fairly well preserved, as are the traces in the west wall of the original arcade walls (Figure 7). Although the walls are clad with ashlar of exquisite craftsmanship, there are pronounced inaccuracies in the plan layout. The nave is 14 cm wider at the west end than in the east, and the north wall is some 30 cm longer than the south wall (internal measures). The wall thick-ness varies between c. 85 and 94 cm. The ratios most close to v2 are found between the inner lengths of the south and east walls (c. 1260 cm : 887 cm ˜ 1,421), and central long axis and west wall (c. 1275 cm : 901 cm ˜ 1.415). Using "the sacred cut" of a square with side 1275 cm gives a distance between the pier axes of 1275(root-2-1) ˜ 528 cm. The actual measure at the west end is 438 cm + (88+89)cm :2 = 526,5 cm. However, with a wall thickness of 85 cm at the north-west corner, the root 5-geometry does not fit the ratio of external to internal length.

A closer approximation to the squaring of the circle can be made by constructing a square with side (5/4)v2 = v3,125 of the circle radius (Figure 8) This accords with 3 1/8 as the value of p, as supposed by the Babylonians. The north wall of the nave of St. Alban's at Selja is c. 1290 cm inside, which by this diagram gives a circle with a diameter 1459,5 cm. Half of the difference is slightly less than 85 cm, which is the thickness of the west wall at the n.w. corner. However, the length of the north wall is the dimension furthest from the root-2 geometry as related to the internal width of the nave, and the thickness of the west wall increases from north to south. Thus, it is impossible to maintain that what appears to be an instance of squaring the circle is more than a mere coincidence.

Back to Stavanger, there is the problem that my assumed geometry relates to the outside faces of the walls, and not the interior dimensions as at Selja. In Stavanger, there is also another possibility for the geometrical determination of the nave width. Instead of the root 5 geometry suggested, it could be defined by the sides of a regular octagon inscribed in the circle inscribed in the square 1/v2 of the external length, thus conforming to a "pure" root 2 geometry (Figure 9). With the dimensions of the nave, the actual difference as generated by the two methods is some 5 cm. On a small scale diagram to, say, 1:100, this is hardly discernible, and even in full scale layout using a cord or a lath on a terrain not completely level, it could be perceived as coinciding if not checked by algebra. Of course one should not speculate upon the ideas of a long dead master builder, but it is tempting to wonder if he might have believed that he believed that there was a connection between root-2 and root-5 here.[1].

Now, there is further the problem of inaccurate plan layout, as the north wall is 90 cm longer than the south wall, with a mean length (central axis) of 31,15 m, which is the measure I have used in the calculations. The width varies between 20,35 m at the east end, 20,40 m at the centre, and 19,90 m at the west end. What is more, the the arcade piers follow rather uneven lines, wavering in and out westwards to meet the west wall almost 60 cm closer than at their ouset in the east. These irregularities could of course be an argument for abandoning any attempt at all to deduce mathemathically based design methods for the building.

I have used the east wall as my point of departure, as the church by evidence of the sculptural detail must have been built in the normal way from the east westwards. The width of the central nave at the eastern wall corresponds very well to the average wall thickness and length along the central axis in ratios of neat numbers. If the wall thickness is (1), the width of the central nave is (8) outside and (6) inside, the length (24) outside and (22) inside. The numbers here set in parenthesis represent relational values; the actual unit (1) measure being slightly less than 130 cm. By using "the sacred cut" on the square with side 1/v2 of the external length (24), the distance between the pier axes would become 7,029… units, which may have been approximated by (7)².[2] By coincidence (?) v5 of this approxi-mation gives an external width of 2035 cm, which at the accuracy level of centimetres is identical to the measure generated by the pure root 2 geometry of an octagon incribed in the circle with diameter 1/v2 of the external length.

The irregularities of the plan, and the especially pronounced slithering of the south arcade pier line should not be expected from a building team capable of erecting the solid and long-lasting edifice with the rather unwieldy stone material they had at hand. It has been suggested that an earlier church may have occupied part of the site, and that the bishop did not permit the demolition of this until the eastern part of the new cathedral could be taken into service.

If this was the case, the master builder would have to set out the plan by measuring his way around the older church. His scheme may have been conceived by geometrical methods, but he would have to employ a measuring rod with all the possibilities for inaccuracies in using this instrument on an uneven terrain. Maybe he was not equally well versant with numerical calculations, and maybe he made shortcuts by taking measures from the wall lines of the old church, which most probably was less than rectangular in plan?

If the unit of ratios cited above is divided by four, it gives a "foot" length of close to 32,5 cm. Could this be the unit deduced from some 12th and 13th C churches in Ile de France of 32,48 cm, that by some scholars is believed to be the pied royale as decided by Charles the Great? The chancel arch of Stavanger Cathedral was, according to measured drawings made before the widening in 1867-69, c. 3,25 m, or 10 such "feet". With an internal width of the central nave of (8)-(2)=(6), or 24 "feet", this would divide the internal east wall by "the sacred cut" approximated as 7+10+7. In the gothic chancel built after the 1272 fire, a series of measures divisible by 32,5 cm or 26 cm (4/5 of 32,5 cm) into whole numbers, can be found.

NOTES

1. Four preserved 12th C. church roofs in the Norwegian region of Trøndelag have a ratio between width and height above the wallheads of root-2 (Nilsen 1999: p. 39). The ratio can be determined in a simple way by dividing the width (W) into four parts, and using the outer quarter division poins as centres for arcs of ¾W starting at the outer ends and meeting at the ridge. On further elaboration of the diagram several of the roof truss joints seem to correspond to significant points in the geometry. The arithmetical "beauty" of this triangle is that, if W is 2, it combines root-2 (height), root-3 (roof slope) and root-4 (=2, or width). This may also have been the pitch of the nave roof of Stavanger Cathedral as originally planned.

2. A numerical approximation of the geometrical ratios might be (17) as the side of the square inscribed in the circle with diameter the external length of the nave (24). The square side division by "the sacred cut" could then be approximated as (5)+(7)+(5)=(17).

REFERENCES

Brunés, T.: The Secrets of Ancient Geometry - and Its Use, Copenhagen 1967.

Le Corbusier: The Modulor, London 1954 (original version, Paris 1951)

Watts, Carol Martin & Donald J.: "Geometrical Ordering of the garden Houses at Ostia", in Journal of the Society of Architectural Historians, vol. XLVI (1987): pp. 265-276

Watts, Carol Martin & Donald J.: "The Role of Monuments in the Geometrical Ordering of the Roman Master Plan of Gerasa", in Journal of the Society of Architectural Historians, vol. LI (1992): pp.306-314

For further reading

Several of the measured drawings from the 1939-41 survey of Stavanger cathedral are published in: Fischer, G.: Domkirken i Stavanger, Oslo 1964, containing an English summary of the text.

For a more thourough discussion in English on the problems concerning the early building history of this cathedral, with references to some related buildings, see:
Hohler, C.: "The Cathedral of St. Swithun at Stavanger in the Twelfth Century", in: Journal of the British Archaeological Association, Vol. XXVII (1964), pp 92-118. A revised version can be found in Universitetets Oldsaksamling, Årbok 1963-64 (Oslo University Archaeological Museum Yearbook), Oslo 1967.

On the possible use of root 2 geometry in a Norwegian mediaeval parish church, see:
Nilsen, D.: "The twelfth Century Church at Værnes, Norway - a Geometrical Speculation", in: Nordisk Arkitekturforskning (Nordic Journal of Architectural Research), Vol. 12, No. 3, 1999.