The Geometry of the Arcades
The geometry of the arcades based on equilateral triangles is shown in Figure 6. From the triangle with base d in Figure 4, the length of the bays is (√3/2) x (d/3). It follows that the height to the top of the arcade is 2 x (√3/2) x (d/3) x (√3/2) which evaluates to d/2. By symmetry the springing point of the arch is at a distance d/4 from the floor. A square of side d/4 is shown in blue in Figure 6. The radius of the extrados, or outer curve of the arch is d/4, so its centre is at a corner of the blue square, and it touches the ground, the centre of the pier and the line at the top of the arcade. The thickness of the arch is one sixth of the bay length, so the width of the arch masonry immediately above the capital is one third of the bay length. The outer moulding of the arch terminates at a height of one bay length from the floor. A golden rectangle is shown in yellow, which determines the height of the pier bases. 24 A square shown in violet determines the springing line of the nave capitals.
By way of comparison, the arcade arches in the choir at Wells fit into golden rectangles shown in yellow in Figure 7. The arcade rectangles are in the proportion 5:3. Equilateral triangles based on the floor give the springing line of the arches as at Llandaff, also the top of each rectangle is a tangent to the extrados circle. The extrados curves intersect at a height of one bay length above the floor. The thickness of the arch masonry above the capital is one third of the bay length. The springing line of the capitals is one fifth of the height of the original vault ridge.25
It should be noted that the design of the piers, while evidently based on the geometry of Figures 4 and 6, in practice displays a logic of its own, which can be reconstructed because although the masons delighted in creating irrational lengths by geometrical means, they preferred to make measurements in integers.
From a rectangle of sides 51 and 61 inches, 16½ inches were measured from each corner and chamfers were drawn. Half of a square of diagonal 8 inches was let into each chamfer. Triplet shafts 12 inches wide and 9 inches deep were added to each cardinal face.
It can be seen from the section (Figure 10) that the underside of the ridge timber of the outer roof is four bay lengths from the floor (3,4,5 triangle) and that the springing line of the 14th century presbytery arches is at half the 37' module from the level of the nave floor.
The precision of the geometry of the west bays was relaxed in several ways in the east bays:
i The original determination of the distance between the arcades was abandoned,
ii The pier cores became simple octagons, so the intricacies of the pier design were lost.
iii The arches in the three eastern bays rise to the full height of the part of the elevation devoted to the arcade, but this is at the expense of the circular geometry in Figure 6.
This confirms the building sequence from west to east, as it is easier to relax existing geometry than to introduce rigour into a loose scheme.
The reduction in width of the piers and the merging of the outer mouldings of the arches led to the asymmetry visible in Figures 6 and 9. Because of the tendency to omit capitals, the arch profiles are essentially the same as the pier sections, both in the east and west bays. The junction between the two designs, where it is disguised by a band of foliage and sculpture, is illustrated on the left hand half of the right hand pier in Figure 6, where the contrast with the right hand half of the same pier can be appreciated. See also the frontispiece, extreme left.
By way of comparison, the arcade arches in the choir at Wells fit into golden rectangles shown in yellow in Figure 7. The arcade rectangles are in the proportion 5:3. Equilateral triangles based on the floor give the springing line of the arches as at Llandaff, also the top of each rectangle is a tangent to the extrados circle. The extrados curves intersect at a height of one bay length above the floor. The thickness of the arch masonry above the capital is one third of the bay length. The springing line of the capitals is one fifth of the height of the original vault ridge.25
It should be noted that the design of the piers, while evidently based on the geometry of Figures 4 and 6, in practice displays a logic of its own, which can be reconstructed because although the masons delighted in creating irrational lengths by geometrical means, they preferred to make measurements in integers.
From a rectangle of sides 51 and 61 inches, 16½ inches were measured from each corner and chamfers were drawn. Half of a square of diagonal 8 inches was let into each chamfer. Triplet shafts 12 inches wide and 9 inches deep were added to each cardinal face.
It can be seen from the section (Figure 10) that the underside of the ridge timber of the outer roof is four bay lengths from the floor (3,4,5 triangle) and that the springing line of the 14th century presbytery arches is at half the 37' module from the level of the nave floor.
The precision of the geometry of the west bays was relaxed in several ways in the east bays:
i The original determination of the distance between the arcades was abandoned,
ii The pier cores became simple octagons, so the intricacies of the pier design were lost.
iii The arches in the three eastern bays rise to the full height of the part of the elevation devoted to the arcade, but this is at the expense of the circular geometry in Figure 6.
This confirms the building sequence from west to east, as it is easier to relax existing geometry than to introduce rigour into a loose scheme.
The reduction in width of the piers and the merging of the outer mouldings of the arches led to the asymmetry visible in Figures 6 and 9. Because of the tendency to omit capitals, the arch profiles are essentially the same as the pier sections, both in the east and west bays. The junction between the two designs, where it is disguised by a band of foliage and sculpture, is illustrated on the left hand half of the right hand pier in Figure 6, where the contrast with the right hand half of the same pier can be appreciated. See also the frontispiece, extreme left.