To construct a square that is exactly equal in area to a given circle by compass and ruler was one of the three famous geometrical problems of antiquity. A host of circle-squarers has tried their hand on this age-old problem, until it was proven futile when Lindemann in 1882 demonstrated that p is a trancendental number.[1] The problem can, however, be represented geometrically, as demonstrated by Dr.-Ing. Helmut Sander in "A geometrical ensemble to generate the squaring of the circle", Nexus Network Journal vol 2 no. 1. I immediately recognized his diagram as being very close to a diagram that I have found by analyzing two early Norwegian basilican- plan cathedrals, and which, at first glance, 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 Ö2 and Ö5 -- albeit approximately, but close enough to fool a non-mathematician working by small-scale geometry into make a false assumption similar to Le Corbusier's when he was developing the Modulor[2]

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 a central nave with aisles.

In the early 12th century, a new diocese was established south of Bergen (Figure 2). The cathedral of Stavanger is still standing, with its Anglo-Norman nave almost intact (Figure 3 and Figure 4).[3] 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, as for example the external width, are systematically too far from fitting into this numerical system that they can be ascribed to inaccurate layout.

I therefore looked for ratios generated by geometry, and arrived at a diagram (Figure 5) with a square DEFG whose sides are divided into 4 equal parts, and a circle cutting the sides of this square through their outer quarter divisions. The circle diameter (BC) is thus Ö5 of AB, which is half the square side. If the circle radius is 1, then square DEFG side will be Ö3.2, which is a rough approximation to Öp, and the square area is then somewhat larger than the circle area. Moreover, a square KLMN with side equal to the diameter of the circle generated by this diagram will have approximately the same perimeter as a circle circumscribing the initial square DEFG.[4]

In Stavanger, such a figure seems to have determined the external width of the nave (Figure 6), as the width equals the diameter of a circle generated in this way from a square with side twice the distance between the axes through the arcade pier centres at their eastern outset. This distance seems to have been found by dividing the sides of a larger square, 1/Ö2 of the external length of the nave at its central axis, by half the diagonal from the square corners (Figure 7 and Figure 8). This subdivision of a square, called "the sacred cut" by Tons Brunés [5], seems to have been used in building design and town planning in Roman antiquity.[6]

The geometry of the plan also seems to relate to the elevation (Figure 9), as the height from floor level to the roof ridge is very close to 2/Ö5 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 Ö2/4 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 wall head level and roof pitch is in accordance with that indicated by the gothic choir gable, but may have been changed after the fire of 1272.

Like most of the early bishops of Norway, the first Stavanger bishop was of the order of Benedictines, 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 [7], 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 Ö2, and the pier axes determined by the "sacred cut" from a square with side the internal length (Figure 10). Even more intriguing was that the external length seemed to be ½ Ö5 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 my own 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 (Figure11). 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 thickness varies between circa 85 cm and 94 cm. The ratios most close to Ö2 are found between the inner lengths of the south and east walls (c. 1260 cm : 887 cm, approximately 1.421), and central long axis and west wall (c. 1275 cm : 901 cm, approximately 1.415). Using the "sacred cut" of a square with side 1275 cm gives a distance between the pier axes of 1275(Ö2-1), approximately 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 northwest corner, the Ö5 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)Ö2 = v3,125 of the circle radius This accords with 3 1/8 as the value of p, which is known as an approximation used by the Babylonians (Figure 12a). [8] 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 northwest corner. However, the length of the north wall is the dimension furthest from the Ö2-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/Ö2 of the external length, thus conforming to a "pure" Ö2 geometry (Figure 13). 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 (Figure 14). 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 there was a connection between Ö2 and Ö5 here, which could be thought to have some esoteric significance.[9]

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 side dorways, and 19.90 m at the west end (Figure 16). 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 held as an argument for abandoning any attempt at all to deduce mathemathically based design methods for the building. However, the erection of a building of this size and quality would not be started without a pre-conceived plan, but even small inaccuracies made when setting out the measures on the site would easily lead to deviations of this order.

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. [10]

If this were the case, the master builder would have to have set out the plan by measuring his way around what was standing in the way of free sight lines. 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 versed in numerical calculations, and maybe he took shortcuts by taking measures from the wall lines of the old church, which most probably was less than rectangular in plan.

I have used the east wall as my point of departure, as the church must have been built in the normal way from the east westwards, as evidenced by the sculptural detail . 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 parentheses 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/Ö2 of the external length (24), the distance between the pier axes would become 7.029… units, which may have been approximated by (7). 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). By coincidence (?), Ö5 of this approximation gives an external width of 2035 cm, which at the accuracy level of centimetres is identical to the measure generated by the pure Ö2 geometry of an octagon incribed in the circle with diameter 1/Ö2 of the external nave length.

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 century churches in Ile de France of 32.48 cm, which some scholars believe to be the pied royale as decided by Charlemagne? 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 (Figure 17).

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 (Figure 18). The interesting relationship between these two "foot" lengths is that they form the hypotenuse and and the longer leg of a Pythagorean triangle with sides 3:4:5. Such a triangle could have been used as a scaling device, and by drawing lines parallel to the sides, all the sides can be divided into 3 or 5 parts, thus yielding a wider range of numerical approximations to geometrically generated measures than the simple method of halving a length and its partitions successively. In the former Bishop's Palace immediately to the south of the cathedral is a chapel contemporary with the choir. No reliable measured drawings have been made of this building, but some measures I made there last year suggest that it is proportioned with the use of a measure unit of c. 23 cm, or half the root two of 32,5 cm (Figure 19).

ACKNOWLEDGMENT
All original measured drawings used belong to the archives of the Norwegian Directorate for Cultural heritage, Oslo.

NOTES
[1] See D. J. Struk, D. J, A Concise History of Mathematics, (New York 1987), p. 180). return to text
[2] Le Corbusier The Modulor, London 1954 (original version, Paris 1951), p. 229ff. return to text
[3] For a discussion in English on the problems concerning the early building history of this cathedral, see C. Hohler, C.: "The Cathedral of St. Swithun at Stavanger in the Twelfth Century", Journal of the British Archaeological Association, vol. XXVII (1964), pp 92-118; C. Hohler, "Remarks on the Early Cathedral of Stavanger and Related Buildings", Universitetets Oldsaksamling, Årbok 1963-64 (Oslo University Archaeological Museum Yearbook), Oslo 1967. return to text
[4] See Donald J. Watts & Carol Martin Watts, "Geometrical Ordering of the Garden Houses at Ostia", in Journal of the Society of Architectural Historians, vol. XLVI (1987), pp. 265-276, esp. p. 269.
return to text
[5] Tons Brunés, The Secrets of Ancient Geometry - and Its Use, Copenhagen, Rhodos, 1967. return to text
[6] See Watts and Watts, "Geometrical Ordering of the Garden Houses at Ostia" (see note 4) and Carol Martin Watts and Donald J. Watts, "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. return to text
[7] Several of the measured drawings from the 1939-41 survey of Stavanger Cathedral are published in Gerhard Fischer, Domkirken i Stavanger, Oslo 1964, which contains a short English summary of the text. return to text
[8] See Struk, A Concise History of Mathematics, p. 29. The Egyptian Rhind papyrus of c. 1500 BC offers an even better approximation to this problem, with the ratio of 8/9 between square side and circle diameter, giving p = approximately 3.1605; see Struk, p. 23 and Figure 12b. return to text
[9] Four preserved 12th c. church roofs in the Norwegian region of Trøndelag have a ratio between width and height above the wallheads of Ö2 (Figure 15); see Dag Nilsen, "The Twelfth Century Church at Værnes, Norway - a Geometrical Speculation", Nordisk Arkitekturforskning (Nordic Journal of Architectural Research), vol. 12, no. 3 (1999), pp 36-46, esp. 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 Ö2 (height), Ö3 (roof slope) and Ö4 (=2, the width). This may also have been the pitch of the nave roof of Stavanger Cathedral as originally planned.
[10] See Hohler, "Remarks on the Early Cathedral of Stavanger and Related Buildings" (see note 3). return to text

FOR FURTHER READING. The following works cited in this article can be ordered from Amazon.com by clicking on the title

• Le Corbusier: The Modulor, London 1954 (original version, Paris 1951)
• D.J. Struk, A Concise History of Mathematics, New York 1987.

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Squaring the Circle: Hobbes on Philosophy and Geometry
Squaring the circle using a special kind of polygon
The Number Pi

ABOUT THE AUTHOR
Dag Nilsen isan architect by profession. He received his diploma NTH (Norwegian Institute of Technology, now NTNU, Trondheim) in 1974, specializing in conservation. Since 1999 he is a member of ICOMOS Norwegian national committee. He is an associate professor (history of architecture and building conservation) at the Faculty of Architecture, Planning and Fine Art, Norwegian University of Technology and Science (NTNU), Trondheim.

Copyright ©2001 Kim Williams