The Geometry of Llandaff Cathedral

THE GEOMETRY OF LLANDAFF CATHEDRAL © Gervais Frykman 2024 1 Contents The Nature of Architectural Geometry .............................................................................. 3 The Norman Cathedral....................................................................................................... 4 Late Norman Enlargement. ................................................................................................ 9 Nave and West Front General .......................................................................................... 13 The Ground Plan of the Nave ........................................................................................... 14 The Geometry of the Arcades and Clerestory .................................................................. 19 The West Front ................................................................................................................ 27 The Lady Chapel and Presbytery ...................................................................................... 41 Decay and Renewal .......................................................................................................... 47 Nineteenth Century Geometry ........................................................................................ 52 Joseph Lord Print ............................................................................................................. 54 The Norman Arch into the Lady Chapel ........................................................................... 56 Interpretation .................................................................................................................. 58 The Geometry of the Chapter House Windows at York Minster ...................................... 62 Notes ............................................................................................................................... 68 Appendix: To construct a pentagon ................................................................................. 73 2 The Nature of Architectural Geometry All great churches in the mediaeval period were designed on a geometrical framework which is unique to each building. The study of the geometry of great churches therefore gives an insight into the creative process. It also reveals beauties which could otherwise be appreciated only unconsciously. The study of lines, figures and proportions is not a substitute for a response such as this written by George Pace after his first visit to Llandaff: "I was overwhelmed with the scale and majesty of the Nave arcades. No illustration I had seen had done the arcade even faint justice".1 The concise treatment of the Lady Chapel in this paper in no way replaces appreciation of its special beauty. Geometrical study is complementary to aesthetic enjoyment, and can only enhance our understanding of these buildings. It can also be of considerable archaeological significance. Architectural Geometry transcended its function in laying out buildings. It became a sophisticated art form, and its finest practitioners could produce geometry of staggering complexity, as in the west front of Llandaff Cathedral. This art is at least as old as the Great Pyramid, with astronomical references which transcended the laying out of fields after the Nile floods. I sit lightly to the term “Sacred Geometry.” It is necessary to start with the building and see what geometry there is without preconceptions. Clearly there was deep intent on the part of the Master Masons. Evidence is given that specific geometrical constructions and procedures were shared across widely dispersed lodges and treasured for centuries. 3 The Norman Cathedral A grid of unit squares can give insight into the layout of a church. There are 54 foot squares in the Nave of York, 42½ foot squares at Bury St Edmunds Abbey, c40 foot squares in Exeter nave, 40 foot squares at Hereford and Tewkesbury, 39 foot squares at Peterborough 2 and Salisbury,3 c37½ foot squares at Tintern, 37 foot squares at Worcester, Bristol and Wells and 33 foot squares at Ely, Beverley and Abbey Dore. The module of Bishop Urban's Norman Cathedral, started on 14th April 1121,4 is 37 feet, which is the distance from the sanctuary arch to the west face of the east wall of the presbytery. A conjectural skeleton of Urban's church consisting of four 37' squares is given on the ground plan (Figure 1). 5 Wide, shallow transepts as indicated here are very unusual. The later transepts and nave at Wells follow this pattern of squares as regards the ground plan, 6 but as there are aisles to both transepts and nave they appear to be long and narrow. The first Norman church at Glastonbury, 7 based apparently on a 33 foot module, had transepts whose projections beyond the aisles were wide shallow spaces each with an apse, similar to the reconstructed transepts at Llandaff. Lanfranc’s transepts at Canterbury had a similar shape. The reconstruction can be made with some confidence. The eastern limb is represented by existing Norman fragments in the presbytery (Figures 22, 24). The spiral staircase which lies somewhat illogically within a wall of the later chapter house is perfectly positioned in the south east corner of the former transept (Figures 1 and 2, and roundel). Its Norman date is confirmed by its construction in stone from Sutton near Southerndown, which is a characteristic of Urban's Cathedral. 8 The north west corner of the north transept can still be seen, inconspicuously incorporated in the exterior of the north aisle wall. 9 4 5 Figure 1 The presumed disposition of the Norman walls on the skeleton of squares, based on this data, is shown in Figure 2. The internal proportions of the presbytery are 1:√2. This was a ratio much used by Norman masons. 10 A square shown in blue whose side is the external east-west width of the transept has diagonal 1½ x 37'. The external width of the transept was therefore (1½ x 37') / √2, or 1½ times the internal width of the presbytery. The transept was not disposed symmetrically over the 37’ square. The west face of the blue square coincides with the west face of the 37’ square. Half the diagonal of the blue square, i.e. ¾ x 37' gives the internal east-west width of the transept. Figure 2 The absence of a third presbytery window to the west of the two that are known (Figures 11 and 22) indicates the former existence of a south transept apse or chapel. Evidence for the equivalent structure leading off the north transept is provided by the measure indicated in yellow in Figure 2, between the line of the north face of the north presbytery arch pier and the north internal face of the chapter house vestibule. This measure is ¾ x 37√2’, the external east west width of the transept. The north face of the north presbytery arch pier is positioned within the area occupied by the east wall of the north transept, and could be the jamb of an opening in that wall. The organ currently prevents examination of the masonry of this face, but the stones in the immediately adjacent east face are coursed with the respond of the Decorated presbytery arcade, and the corner stones are common to both faces, so that the proposed jamb was refaced. The south transept apse was also rebuilt, but for the Norman measure to be present, both faces must be in their original positions. Apart 6 from this detail, the reconstruction of the transept apses in Figure 2 is conjectural. The spiral staircase would originally have opened into the entrance to the apse. Near the north east corner of the north transept some Norman stones can be made out behind later buttresses, but further to the west than expected. There may have been a stair turret in the north transept equivalent to that in the south transept, which was demolished with the east wall when the north presbytery aisle was built. The corner resulting from the laying out of the presbytery aisle somewhat to the south of the 37' line may then have been made good using Norman stones in part. The westward extent of the nave is conjectural. A symmetrical design is consistent with the seal impression on the left 11 but there could have been another 37’ square to the west giving an overall design of five squares, or there may have been another relationship governing the length of the nave. The Norman cathedral had no aisles, as is shown by the position of the remains of the windows in the presbytery. There were two ranges of windows. Bases associated with the upper range were found in the Victorian restoration on the north side of the presbytery at the level of the existing clerestory passage.12 A print by Thornton shows a view of the mediaeval cathedral from the north. What may have been the Norman upper range of windows in the presbytery is shown as an otherwise unknown arrangement of three pairs of alternating large and small windows. The position of the existing fragments of Norman windows in the presbytery shows that it was designed in three bays. The print depicts the cells of the blind arcade on the west front in the same way as the windows. Probably there were three windows on each side in the upper range, separated by four blind arches. There may have been a central tower or turret, and towers on either side of the apse which lay to the east of the existing Norman arch. These are shown on both seal impressions, but have not been demonstrated archaeologically. Bases of the eastern crossing arch were found under the bases of the present sanctuary arch. 13 7 Twin eastern towers may have existed at Old Sarum, and can still be seen at Exeter, though they appear more central now, due to the lengthening of the eastern limb. Small towers adjoin the eastern transepts at Canterbury, though the effect of these is lessened by the subsequent raising of the transept walls. The immediate precedent for the Norman Cathedral appears to be the eastern limb at Hereford, probably designed c 1080-1090 and consecrated by 1115. There was a large central apse opening from a single arch, and smaller apses at the ends of the aisles. Towers stood over the eastern bays of the aisles. The presbytery was laid out in three bays. 14 The designer of the Early English nave at Llandaff probably assumed that the apse and eastern towers would be demolished, for provision was made for towers over the eastern bays of the nave, which would have made the earlier towers superfluous. This intention survived the change of design (end of Section 6), but the towers do not appear to have been built. 8 Late Norman Enlargement. Starting some time about 1180 the south nave aisle wall was built following the line of the Norman transepts and thus clear of the Norman nave. The south door is the principal evidence for this (Figure 3, right). The pattern of separated chevrons on the innermost order is also found on the crossing arches at Hereford and some of the presbytery arcade arches at St David’s. The structure of patterned planes intersecting at right angles in a cylinder of stone is also found in the presbytery of St David’s. The more unusual design of the outer order is found at Hereford on the font, and at St David’s in the decorative band under the eastern lancets and in the nave clerestory. St David’s was started in 1181. Figure 3 The same 37' module was used for the nave aisle walls as in Urban's Cathedral, for the outer face of the south aisle wall is aligned on a 37’ line, and the south door is situated symmetrically in the sixth 18'6" unit measured from the east face of the east wall of the transept (Figure 1). The late Norman design for the north aisle wall would have been similar. The south door is constructed of stone from Dundry near Bristol, as are the existing nave arcades, west front etc.15 The geometry of the opening of this door is that of a square surmounted by a semicircle. These simple shapes can also be seen as elemental, and give the door much of its power. The equivalent rectangle in the north door has sides in the ratio of 6:5 (Figure 3, left). The north door also displays the structure of patterned planes intersecting at right angles, and two pairs intersect in a cylinder of stone. Lovegrove considered that the north door may have been reset, also the outer moulding of dog tooth is characteristic of Early English rather than Norman style. It is constructed of Sutton stone, 9 which is the stone used for Urban's church. It conforms moreover with the geometry of the arcades, which did not exist when the door was first built. It could be the west door of Urban's Cathedral, re-erected here with a new outer moulding of reused Sutton stone when the west front of the Norman nave was demolished. 16 At some time in the late 12th century the presumed south transept apse was replaced by a rectangular chapel with east and south windows, and a vaulted ceiling. The work was contrived so as to allow light from the south east to continue to pass through the western of the two Norman windows of the lower range on the south side of the presbytery. The arch at the west end of the chapel was built into the east wall of the Norman south transept. When the chapter house was built c. 1250 the builders chose the line of the west face of this wall to be the line of the east face of the west wall of the chapter house. With 10 the building of the chapter house the south window of the chapel was blocked, but its apex can still be seen. When the south presbytery aisle was built, an arch was cut into the east wall of the chapel, converting it into the chapter house vestibule. The apex of the east window of the chapel can still be seen above the arch. Externally the corner of the chapel is clearly visible between the chapter house and the presbytery aisle. A buttress was inserted into the corner of the chapel when the chapter house was built. This would have made good visual and structural sense at that time. It took on its present crowded appearance when the aisle was built. Figure 4 The red square in Figure 4 has diagonal of 26’, twice the width of the chapel. The west face is aligned with the centre of the entrance arch. The east face gives the position of the west face of the east wall. The diagonal of 26’ gives the overall length of the chapel, including the east and west walls. The south face of the south wall of the chapel is aligned on a 37’ line, and is not located by the square. Clearly the square could not have been constructed in situ due to existing masonry, so numerical methods were used, or it was enlarged from a drawing using large dividers. 11 Because of the Norman measure of ¾ x 37’ x √2 , indicated in yellow in Figure 2, between the inner faces of the presumed Norman transept apses, the theoretical distance from the south face of the north wall to the south face of the south wall is 37’(1 - 3√2/8) or 17’4½”. Deducting 13’ for the width of the chapel leaves 4’4½” for the thickness of the south wall, which is confirmed by measuring the length of the passage into the chapter house as just that. The similarity of the geometry to that of the Norman presbytery and transepts indicates a master mason who was steeped in Norman tradition. The lack of a boss in the centre of the vault and the profile of the ribs are further archaic features, paralleled in the north nave aisle at Gloucester and the eastern quadripartite vault at Ewenny, though the overall appearance of the chapel is redolent of the developing Early English style that reached maturity at Wells. A date of c.1175 is possible for this chapel, which would mean that it was the first addition to Urban’s cathedral, predating the Norman south nave aisle wall. The triangle of base 11 units, height 7 units and the circle of diameter 3 units are from a diagram by John Mitchell in “The Cosmos in Stone” by Tom Bree. It is similar to a vertical section of the Great Pyramid parallel to an edge. The ratio of 11:7 is a close approximation to π/2. The triangle based on the centre line of the church is used to determine the centre of the chapter houses of Wells, York and Southwell, and the circle is used in various ways to give the dimensions of each chapter house. Well defined pyramid triangles also determine the centres of the chapter houses of Lichfield and Hereford. There is such a triangle at Beverley, based on the north choir arcade. The 3 unit circle at Lichfield gives the backs of the straight walls of the chapter house, and the inner extent of the window tracery At Llandaff the left vertex is aligned with the centre line of two of the piers and the lower vertex gives the position of the south wall, and the centre of it. The right vertex probably referred to architecture no longer extant, and a possible candidate is the centre of the semicircle of the Norman apse. With this information it is tempting to speculate concerning the eastern towers. I have posited circular towers because of the tapering shown in the left hand seal impression and because their demolition left no scars. The inner diameter of the apse is 37/√2, the outer diameter is 37’, the diameter of the towers is 37/2’, the centres of the towers are 37√2’ apart and are aligned on the chord of the apse. The chord is 37/4’ from the east face of the east wall of the presbytery, so the line of the east face of the east wall of the presbytery is tangent to the towers. The right hand seal impression indicates that the towers did not project beyond the transepts. The 3 unit diameter circle has interesting relationships with the window splays, and with the extent of the base of the central column. 12 Nave and West Front General The west towers, west front, arcades, clerestory and presbytery arch, replacing the Norman east crossing arch, were built in a specialised version of Early English style associated with the West Country. The arcades were built from west to east, in order to leave the Norman nave and transepts undisturbed as long as possible. As the 37' module was abandoned in this work, and as the east end of the nave17 is at a considerable distance to the west of the east face of the transept (Figures 1,11), the presumed Norman aisle windows would have been out of step with the new arcades, as the south door still is (Figures 1,11). There are affinities in the Early English work with Wells Cathedral, which was under construction from c. 1175 onwards,18 e.g. in the design of the piers, which consist of a core with straight edges and attached triplet shafts, in the bold chamfering, and in the tendency to omit capitals. The two cathedrals also share important geometrical features. It may be conjectured therefore that the designer of the quire at Wells also designed the Early English work at Llandaff. 13 The Ground Plan of the Nave d Figure 5 Equilateral triangles like that shown in red in Figure 5 relate the inward extremity of the pier bases to the centre line of the opposite arcade. This geometry was also used at Bristol. A square shown in green with side equal to one bay, edge aligned on the centre line of the south arcade gives the position of the glass in the aisle windows. An equilateral triangle shown in blue in Figure 5 gives the length of three bays. It also defines the north-south extent of the piers. Let the length of the base of this triangle be d. Then the height = (√3/2) x d. From the measurements in note 21, d evaluates to 63'1". Triangles like this may be drawn in the nave of Wells 19 and the Choir of Southwell, but in those places they give the length of four bays. 14 Figure 6 A circle diameter d shown in red (Figure 6) passes through certain corners of the pier bases. The tangent shown in yellow defines the western extent of the thickening of the piers that support the towers, also the interior face of the east wall of the Jasper tower. Another circle on the same centre shown in blue, with radius equal to the internal width of the west front (30’6”) or the distance apart of the inner faces of the arcade walls, with a pentagon inscribed, gives the position of the centre lines of the western half of the arcades, and the position of the east-west extremities of the piers. As we have seen, the Early English masons were presented with a south aisle wall with its outer face aligned on a 37' module line. The north aisle wall with its reset Norman door was positioned by the measure d. It bends towards the north in its progress eastward, so that its outer face becomes aligned with a 37’ line. The north arcade is similarly bent. The western half participates in the pentagonal geometry and is parallel with the south arcade. The two eastern piers conform to a 37' module line. The fifth pier from the west is intermediate between the two lines. Its position is justified by an equilateral triangle as shown in yellow (Figure 5). This loosening of the geometry extends to the elevation (page 26). The internal length of the nave from the west front to the sanctuary arch i.e. 150' 2" is another link with Wells, for 150' is the extreme width of the Wells west front including the plinths, also the external width of the transepts. 20 The extra 2" at Llandaff are used to ease the transition from the arcades to the West Front. Theoretically, half of a triplet shaft should be visible at the junction with the west front, but in practice, two shafts of the triplet are present, and the base can be seen to curve inwards before it meets the west front. 15 The geometry may have been developed in the following manner: First the 150' length of the nave was marked out from the centre of the Norman presbytery arch, then the length from the west face of the Norman presbytery arch to the west front was divided unequally into eight to give the bay length. The overall length available for the nave arcades was 146’8”. Assuming spaces of 6” were left at each end of the arcades in respect of the towers, the remainder, divided by 8, gives 18’2½” for the bay length, which matches the calculated value exactly. 21 Measurement of the bay lengths leads to the conclusion that the north arcade was more loosely built than the south arcade. The close adherence of the bay lengths in the south arcade to the theoretical length rules out the possibility that the extra length of the first bay is due to poor measurement. The creation of substantial sub-arches in the bays next to the presbytery and the corresponding thickening of the piers (Figures 1, 11) is evidence that twin towers were intended at this point. The extra 6” allowed for these bays is consistent with the hypothesis. Apparently there was a change of plan for the eastern towers distinct from the change of design in the nave. All four towers would originally have been the same size, with the east walls of the east towers to have been built over the east walls of the Norman transepts, but the reduced eastern towers would have had east walls to the west of this position. An arch to support the east wall of the projected south east tower was inserted with considerable lack of aesthetic and structural sense immediately to the west of the entrance to the chapter house vestibule. 16 Internal and external thickening of the adjacent aisle wall can be associated with this work. When the towers were abandoned, the extra thickness was chamfered off using any available stones. 22 17 The length d, not being an integer, is a derived measure. It may have been obtained as the height of an equilateral triangle with base equal to four bay lengths, and used for the distance between the inner faces of the aisle walls (light blue triangle, figure 5). The centre of the circles in Figure 6 was located using the bay length as determined by measurement. The equilateral triangle base d was set up, not in order to determine the bay length, but to give a measure for the north-south extent of the piers. The tangent in yellow (Figure 6) passes along the upper edge of the chamfer of the pier base. The distance apart of the inner faces of the arcade walls is measured between the tops of the chamfers. The base line of the exterior of the west front (Figure 12) is also aligned, vertically in this case, with the top of the chamfer. It is probable therefore that the geometry of the pier bases relates to the tops of the chamfers. The inner line of the arcade was reflected in the centre line to give the outer line of the arcade, and the intersection of this line with the circle diameter d gave a corner of the upper edge of the chamfer. The other three corners were found by reflecting this point in the north-south and east-west centre lines of the piers. [Detail ad lib: The Norman presbytery arch has wholly disappeared, so its position can only be located by inference. The inner moulding immediately west of the 37’ line is represented by stones of the period of the nave on the north side of the arch, demonstrating that the inner moulding of the arch reproduces mediaeval design. The same is true of the outer moulding. The coursing of the lower part of this moulding on the south side is the same as that of the duplet shaft and the stub of the wall that runs past the back of the Bishop's throne, which are mediaeval features. The line of the outer moulding is the same as that of the west face of the wall above the arch. The line of the east face of the same wall is also taken up in a moulding on the north side of the arch, which too displays mediaeval stones. Some of these stones are common to the east face of the arch and to the half of a triplet shaft visible at the west end of the north presbytery arcade. This shows that the thickness of the wall, which is known to have been altered in the 18th, 19th and 20th Century restorations, is mediaeval at the latest. Geometrically, the east face of the wall is located by the blue transept square in Figure 2 (see also Figure 11: the dotted line to the right of the blue square in Figure 2 does not correspond with any stonework). Moreover, if the wall had been rebuilt or made thinner by repositioning the west face, this would have happened right at the beginning of building operations, causing major disruption to the Norman Cathedral, but we have seen that the new work left the presbytery, transepts and most of the nave undisturbed for as long as possible. The method of building an arch in an existing wall is demonstrated in the presbytery arcades. This procedure was also followed in the south arcade at Brecon, 23 and in the large arches cut into the inner walls of the Norman towers at Exeter. It is probable therefore that the mediaeval presbytery arch was cut into the east wall of the Norman crossing.] 18 The Geometry of the Arcades and Clerestory The geometry of the arcades is shown in Figure 7. From the triangle with base d in Figure 5, 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. 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 quire at Wells fit into golden rectangles shown in yellow in Figure 8. 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 5 and 7, 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. 19 1 bay length 20 Figure 7 66’ Bay Bay/2 44’ 3/5 of total Bay x 5/3 Bay x φ Bay x √2 Bay Bay x √3/2 1/5 of total Figure8 7 Figure 9 21 It would have been a simple matter to make the height of the clerestory equal to d/4. It is short of this by 5½ inches. 26 On the exterior, the top of the parapet is at exactly 1½ times the height of the string course at the base of the clerestory measured from the floor (Figure 9). Figure 9 The designer avoided the simple solution here as he did in rejecting the 37' module for the distance between the arcades. The pentagon in Figure 10 has side equal to half the bay length. This determines the height of the clerestory up to the lower edge of the top course, which adds a further foot to give the total height. The two top vertices of the pentagon and the centre line divide the length of the clerestory bay exactly into quarters (Triangle PQR is isosceles). 22 R T S P Q Figure 10 23 The vertical lines passing through these vertices define the outer edges of the windows, and their corners where the verticals intersect the lower edges of the pentagon. The outer vertices give the positions of the outer columns, and then the other columns are positioned by symmetry. Chords of the pentagon define the tops of the capitals, and the apices of the lancets. The springing line of the capitals is given by the semicircle shown in red, with a radius of half a bay length. The height of the shafts including their bases and capitals is half a bay length. Due to the pentagonal geometry, a golden rectangle is present, shown in yellow. A square, shown in violet, relates the lower edge of the window openings to the lower edge of the sloping sill. The centre lines of all the arches are given by dividing the length of the bay into six. This measure also gives the width of the windows. It follows that QS is one twelfth of a bay length. The width of the stone backing to the shafts is one twenty-fourth of a bay length. 24 25 Figure 11 It can be seen from the section (Figure 11) that the underside of the ridge timber of the outer roof is four bay lengths from the floor (3,4,5 triangle). Half the distance from the east face of the sanctuary arch to the west face of the east wall of the presbytery gives the height of the springing line of the presbytery arcade arches above the nave floor level. 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 original determination of the distance apart of the aisle walls was abandoned. iii The pier cores became simple octagons, so the intricacies of the pier design were lost. iv 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 7. 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 7 and 10. 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 7, where the contrast with the right hand half of the same pier can be appreciated. See also the frontispiece, extreme left. 26 The West Front Figure 12 The extraordinary beauty of the geometry of the west front confirms the artistry and stature of the designer. It also confirms that the round headed Early English doorway was 27 integral to the design of the whole. Such doorways are rare, but can be found in the great transept at Beverley Minster, in the Nine Altars and Refectory at Fountains, and on the exterior of the south nave aisle at Exeter. (If this is Norman, it is very precocious). Exeter, door in south nave aisle. The pentagonal nature of the geometry is disclosed by the angle of the gable, which is 72°, the angle at the centre of a regular pentagon. A further clue lies in the annulets on either side of the doorway, which indicate a pentagonal angle in conjunction with the centre of the base line. The shaft sections above and below the annulets are of equal length, which explains why the capitals are unusually tall in proportion to the length of the shaft. Another device used to disguise the low positioning of the annulets was to form them asymmetrically in their respective stones. 28 In Figure 12, the pentagon round the door has sides of 15'. It is enclosed in a pentagon of side 30', which is the width of the central part of the west front, being 1/5 of 150’, the length of the nave. The top of this pentagon aligns with the bottom of the opening of the upper window. Two equilateral triangles of side 30' are shown. One has its base coinciding with a chord of the large pentagon. Its apex is at the lower edge of the hood mould at the apex of the central lancet, which is the level of the top of the capital of the south stair turret. The second equilateral triangle has its base at this level and its apex at the apex of the gable. The width of the central lancet of the triplet, measured between the edges of the mouldings next to the openings, is equal to the width of the stair turret, which is derived from the bases of the piers of the nave arcade. The distance between the lancets is the same measure, so the inner edges of the outer lancets are defined. Two thirds of the distance from there to the stair turret or edge of the central part of the west front gives the position of the outer edges of these lancets. The remaining one third gives the width of the decorative band round the outer lancets. Naturally the band round the central lancet has the same width. The openings of the lancets are actually a little less than the widths defined above and are defined by the yellow geometry in Figure 14. The effect is a subtle one. What looks like a band of glass quarries round the lancets is in fact stone as regards the vertical and curved edges. The windows as narrowed are shown in Figure 15, with actual quarries of glass round the edges. The position of important mouldings round the outer lancets is given by the upper vertices of the 15’ pentagon in Figure 12. The width of the opening of the upper lancet is one seventh of 30’. A pentagon everted from the 15’pentagon has chord 30’ (Figure 13). The circumcircle of this pentagon can be copied so that the circles touch. The upper circle defines the string course on the gable and the apex of the upper lancet. Its diameter defines the string course at the base of the blind arcade. The upper vertices of the inscribed pentagon define the edges of the gable. The overall width of the pentagon is 30'. Two radii touch the apices of the side lancets. The part of the gable above the line joining the upper vertices can be reflected in that line, and the image of the apex falls at the centre of the pentagon. 29 30 Figure 13 The geometry shown in yellow in Figure 14 is from John Michell as given in “The Cosmos in Stone” by Tom Bree and is used to determine the layout of the 14th century east end of Wells. We have seen the pyramid triangle with base to height ratio of 11:7 in relation to the chapter house (Figure 4). Here its apex gives the horizontal openings of the lancets. Its sides are tangents to important mouldings of the semi-circular arch. The smaller triangle has base to height ratio of 24:5, so it is two 5:12:13 triangles back to back. It is referenced by the annulets of the door. The circle defines the outer vertical openings of the lancets. The ends of the diameter lie on the upper faces of the everted pentagon. A golden rectangle of major side 30’ connects the springing line of the lancets and the string course over the door. The springing line of the lancets is determined by the geometry of the inside of the west front (Figure 15), so the golden rectangle defines the string course. Some of the shafts outlining the lancets on the inside of the west front have fillets below but not above the annulets (Figures 15 and 18). This indicates that the west front may have been built in stages. Figure 14 shows a construction of two half-equilateral triangles which was loosely adhered to in the design of the arcade on the exterior of the west front. This looseness is reminiscent of the treatment of the eastern bays of the nave and confirms the impression that the west front may have been built in stages. 31 Figure 14 32 The illustration below shows that the springing line on the south side is higher than that on the north side. This difference is not perceptible on the inside of the west front (Frontispiece). Nor is it shown on the drawing. The south lancet capitals have evidently been raised after they were built, for the decorative band on the south side of the lancet follows the line of the lancet, but on the north side of the south lancet, where there is a spur on account of the blind lancet which made it difficult to manipulate the stones, the band does not match the lancet. 33 The geometry of the inside of the west front is different, because the ground level is lower, and the width is greater, also the interior treatment of the door and windows is independent of the external treatment (Figure 15). Half the internal width of the west front, measured from the floor, gives the level of the bottom of the string course. The level of the edge of the sloping sill immediately over the string course is that of the springing line of the arcade arches, a distance of d/4 from the floor (Figure 18). The sides of an equilateral triangle based on the floor cut the sill line at points which then determine the position of the strong verticals that divide the lancets of the west window and outline the door. Produced, they define the apices of the outer lancets in conjunction with the centre lines determined on the outside (Figure 16). Another triangle, congruent with the first, is placed so that its base is aligned with the openings at the base of the lancets which were determined by the geometry in yellow in Figure 14. Its apex is at the underside of the apex of the outer moulding of the central lancet. The upper apex of this moulding is at the level of the horizontal face of the large pentagon on the outside. The triangle is confirmed by annulets in the outer lancets. A third triangle, congruent with the other two and inverted rests on the apex of the door and defines the top of the clerestory. Let the side of the triangle be g. A point on the side, a distance g/3 from the door apex defines the shaft, and by analogy, all the other intermediate shafts and was marked with an annulet. A circle radius g/3 centred on this point necessarily passes through the apex of the door and touches the edge of the west front. It also touches the springing line of the arches of the lancets. When reflected in the centre line, a vesica piscis is formed. The diameter of the vesica circles gives the height of the walkway in front of the central lancet from the floor. The points where the lines of the edges of the door opening meet the floor are the centres of each extrados of the door arch. The curve also passes through the junction of the edge of the west front with the floor, and its highest point is its intersection with the opposite side of the equilateral triangle side g based on the floor. A method of determining the centre, to ensure that it should do this, is to strike two sets of arcs from the apex of the door and the corner of the front, and join the two points thus determined with a line. This determines the centre of the extrados where it crosses the base line. The line is shown in green in figure 16. It also meets the edge of the west front at a height of d/4. The centre of the extrados also defines a smaller equilateral triangle. The intersection of a side with the line of the door opening defines the intrados. So let us follow the extrados curve from its origin in the left hand corner. Rising vertically it curves round until it goes through the apex of the small equilateral triangle. This is close to 34 the green vertical and the d/4 line, but they have nothing to do with the geometry of this curve, so they are edited out of Figure 16. Almost at once it passes through the intersection of the left hand vesica circle with the line of the door opening (Figure 15). By this time it is tracing out the extrados. The extrados in the drawing is more curved than the geometry prescribes, but looking at the photograph of the west door in the frontispiece, it may be an error in the drawing. Next it goes through the intersection of both vesica circles and the apex of the triangle that rests on the apex of the arch. Its function as the extrados ends at this point. Its next event is at its apogee where it passes through the intersection of the line of the north opening of the door with the lower large equilateral triangle. It will only do this with this particular width of the door. Declining, its course continues until it meets the opposite edge of the west front at a height of d/4. It cuts the base line at a stair circle diameter from the springing point of its fellow curve (Figure 16). To work out the ratio in which the door opening cuts the base line, let the radius of the extrados be r. Then r is the distance from the right hand end of the diameter of the stair circle to the left hand corner of the west front. Then the length of the vertical tangent of the stair circle is r. Then the remaining part of the base line is of length r/√3 (30° 60° 90° triangle). So the required ratio is r : r/√3, or √3 : 1 The geometry may have been generated in the following manner: a half-equilateral triangle of indeterminate size was set up with a vertex at a bottom corner. The ratios of the vertical, sloping and horizontal sides are 1 : 2 : √3. (Figure 17). A 1 : 1: √2 triangle of indeterminate size was set up on the opposite corner. Where their sloping sides meet, a perpendicular was dropped to the base line, which divides it in the ratio √3 : 1. This is the centre of an extrados with radius such that it goes through a corner of the west front. The generation of the rest is known. So it is the intersection of the extrados curves which locates the vesica piscis and its triangle. To derive the width of the inside of the west front let AB in Figure 16 be r, the radius of the extrados. Then BC = r/√3. So in triangle BCD, r2 - r2/3 = (d/4)2. So 2/3 r2 = (d/4)2. Importing d/4 = 15.77 from the nave floor geometry (Figure 5), r = 19.31. So the width of the inside of the west front is 30.46, or 30’6”. The apex of the inner face of the upper lancet is at a distance d from the floor. The apex of the gable is at a distance of 7d/6 from the floor (Figure 15). 35 7d/6 d d/2 d/4 Figure 15 36 7d/6 d d/2 d/4 d/4 d/4 D Figure 16 A B C 37 d/2 d/4 Figure 17 38 With the quantity of geometrical relationships in the west front, both inside and out, particularly the apparent dual determination of the gable height, the measure d and the height of the clerestory it is interesting to consider the ways in which measures depend on other measures. The measure of d/4 is generated by the geometry of the nave and passed to the west front. The interior west front geometry then generates the 30’6” measure. Then the green equilateral triangle can be drawn in the nave (Figure 6) which gives the radius of the blue circle (Figure 6) which places the blue pentagon which determines the centre lines of the arcades. The west front geometry gives the height of the clerestory which was also determined by the pentagon in figure 10. In Figure 16 using cartesian coordinates, the equation of the right hand extrados curve is (x - r)2 + y2 = r2. Using x = 15.25, r = 19.336, the height of the apex of the door is 18.90. Adding the height of the vesica triangle the total is 45’3½” In Figure 10, triangle PQT, the height of the pentagon is bay/(4 x tan 18°) = 14’0”, so adding d/2 for the arcade, the height to the top of the clerestory is 45’6½” The height of the west front was determined as 7d/6 (Figure 15) bearing in mind that it had to be a little greater than the four bay lengths indicated in Figure 11 (it calculates at 9”), and then the ground level outside the west front was determined by the geometry of Figure 12. There is rising ground to the west of the Cathedral, and some excavation was required in order to create sufficient level space. This is the reason for the steps. The designer could determine how much of the ground he had excavated on the outside. Obviously there were constraints on this, and it seems that here and elsewhere he knew approximately where he wanted the various elements to go, and then worked out a geometrical scheme to fix the positions exactly. They would have had to excavate to the floor level of the nave, then build the lowest courses of the west front, then fill up the ground from the spoil heaps up to the required level. A further sophisticated feature is the way the two outer shafts of the outer sides of the outer lancets on the interior rest on the duplet shaft of the arcade where it meets the west front. Likewise the triplets outlining the central lancet rest, not on the sill, but on the triplet shafts that outline the door. The result is a subtle merging of the arcade and the west front because the duplet shafts are clearly part of the arcade, and just as clearly part of the west front. Also because the plane of the wall above the lancets overhangs the plane of the west front at ground level, there is an interplay between the interior of the west front on these two planes and the main space of the church (Figure 18). Figure 15 shows the roof to be more steeply pitched than the gable. This is to allow room for the parapet, which was a mediaeval feature as shown in the Joseph Lord print. The original roof, however, like the original roofs at Wells, overhung the walls. 39 Figure 18 The approximate line of this roof, aligned with the gable, is shown in Figure 9. This represents a further intricacy in the geometrical design of the west front and the clerestory. Here we may compare the cradling in stone of the shafts in the blind arcade with the similar motif in the Elder Lady Chapel at Bristol, which is attributed to Adam Lock, who was the master mason at Wells up to his death in 1229. 27 The development of this motif can be traced in the treatment of the shafts that flank the triplet of lancets and those in the clerestory. The coherence of the geometry of the arcades, west front and clerestory suggest that the designer of the blind arcade, probably Lock, retained the essential features of the earlier design of the gable. The similarity between the arcades at Wells and Llandaff as regards the geometry is so great that it is likely that one man designed both. Work at Wells started c. 1175, and the Choir design cannot be later than this. The design of Llandaff therefore probably dates from the same period. This means that Llandaff, with Wells and Lincoln, may be one of the earliest Gothic churches in Britain. The difference in style need not imply any considerable break between the construction of the south door and the commencement of the west front, nor is there a reason why a campaign should be halted after the building of an aisle wall. It is attributable rather to a change of masons. It is not known why the masons from the Hereford-St David’s axis were replaced by those from Wells, but it may be because they were called away in 1181 to work at St David’s. 40 The Lady Chapel and Presbytery Figure 19 The Lady Chapel was designed about 100 years after the west front, but its geometry could be read plainly by the master mason. The Lady Chapel is based on the west front module of 30 feet, which is the width from window to window. It is 60 feet long from the Norman wall to the east window (Figure 19). 2½ times 15 feet gives the height of the underside of the vault at the centre line. 3 x 15 feet gives the height of the gable (Figure 20). A slightly smaller measure, from the centre line to the inner face of the window tracery is also significant. Twice this measure from the floor gives the height of the apices of the interior arches of the aisle windows. 2½ times gives the height of the vault sous clef (Figure 20). 41 Figure 20 42 The springing line of the vault is pentagonally determined (Figure 21). Figure 21 43 The designer of the Lady Chapel also created some grand overall proportions. The internal length of the Cathedral, to the west face of the east wall of the Lady Chapel, is 250 feet (Figure 1). The 60 foot length of the Lady Chapel can be measured from here to the centre of the east wall of the presbytery, and from there to the presbytery arch is 40 feet, making a proportion of 3:2. The 150 foot length of the nave relates to the total of 100 feet in the same proportion of 3:2. A line that divides the 60 feet measured from the east window in the same proportion of 3:2 locates the exterior face of the east wall of the north presbytery aisle and the east window of the south presbytery aisle (Figure 19). The east walls of the presbytery aisles belong to the same building campaign as the Lady Chapel, since the buttresses are similar. Geometrical confirmation of this is given by the internal width across the ends of the aisles, which is 60 feet. The internal width of the nave is 63 feet or more. To accommodate this difference the north presbytery aisle wall curves inwards as it approaches the east wall. It is also offset to the south at its junction with the north nave aisle wall. By contrast, the inner face of the south presbytery aisle wall is maintained at a distance of 30’ from the centre line of the presbytery. There was no need to bend this wall as the east wall of the chapel was able to receive the aisle wall at any convenient position, so all the offset on the south side was made at this junction. The south jamb of the arch cut in the east wall of the vestibule is on the same 30’ line, so is the south jamb of the earlier arch at the west end of the vestibule. The width of the presbytery aisles as measured to the windows is half the width of the presbytery measured from the centre lines of the piers (Figure 19). The building history of the presbytery aisles is not easy to determine. The Lady Chapel, which is dated c 1280, together with the end walls of the aisles, could not have existed on their own for 40 years or so, and yet the presbytery aisles seem to be later than the end walls. “The details of this first portion [associated with the Lady Chapel] are by no means clear of Early English; … the window jambs, though under Decorated labels, belong rather to the earlier style. In the south aisle they have a shaft with a broach above the abacus, in the north a continuous roll with a shaft, but no capital. … The next portion embraces the two bays forming the north aisle of the presbytery; here the architecture is decidedly Decorated; the two window-jambs are merely moulded without shafts. In the south aisle we have no work of this portion, probably because of the building added to the vaulted bay being still preserved. The only window here is later.” 28 44 At St David's there are the remains of 13th century passageways that led to the Lady Chapel, which were converted to extensions of the choir aisles in the 14th century. It is possible that a similar arrangement was in place at Llandaff. That would have left the lower range of Norman presbytery windows unencumbered. The string course in the Mathew chapel on the east face of the Norman presbytery wall may be a relic of such passageways. The rubble construction of the aisle walls does not easily lend itself to the recording of architectural traces, but there are discernible differences in the stonework of both aisles at the level of the tops of the corner buttresses. 29 It is also consistent with what may be later window arches resting on earlier jambs. Lovegrove considered that the presbytery aisles must have been built at the same time as the Lady Chapel. In fact they were built in stages, as is shown by the heterogeneous collection of window jambs. 30 Confirmation of this is given by the hagioscope into the Lady Chapel. This is so directed that a celebrant standing behind an altar against the west wall of the Mathew chapel had a view of the Lady Chapel altar. The upper part of this wall still exists, above a 19th century arch, but its former continuation downwards, and its piercing by an opening in line with the hagioscope can be deduced as the altar would not have been placed in the middle of an aisle. Access to the Mathew chapel would have been via a door in this wall or from the Lady Chapel. The exterior of the north presbytery aisle at the junction of the Mathew chapel and the former chapel to the west of it displays “a singular break in the wall, like an enormously wide pilaster sloping backwards and dying into the wall.” 31 The arches leading to the Lady Chapel “by the character of the mouldings, may be of the Middle Decorated style (1320-1350), although the character of these arches is somewhat peculiar, as the bases are Early English and also the foliated caps…while the abaci are Decorated.” 32 Probably therefore the arches were rebuilt with the presbytery aisles, using some existing materials from the arches from the proposed passageways. 33 45 After the presbytery aisles were completed the presbytery north arcade which takes up the pier design concept of a simple core with attached triplet shafts but in Decorated style was inserted into the Norman presbytery wall, and the south arcade was started. Possibly the shifting of the arch to the east compared to the north arcade, leaving insufficient space for Figure 22 the arch mouldings, and the failure to complete the arcade were due to indecision concerning the tomb of St Teilo, seen in the illustration. Also in the 14th century the nave aisles were reconstructed, and probably heightened as Early English roofs were normally steeply pitched. Reticulated windows were inserted now corresponding with the bays. By contrast with the piecemeal building of the presbytery aisles, this was done in a single campaign. 34 In the 15th century Perpendicular tracery was inserted into the windows in the east end of the presbytery aisles, while the Early Decorated jambs were retained. 35 The original north west tower was rebuilt at the charge of Jasper Tudor, but without disturbing the geometry of the central part of the west front. This strong tower may have been built to accommodate the bells following the destruction of the 13th century detached bell tower, probably by Owain Glyndwr. 46 Decay and Renewal The cathedral suffered severe damage in storms in the early eighteenth century. “February 6, 1722-23 [1723 by our current calendar] Wednesday, ten o’clock at night, the main couples of the roof and the south-west tower fell down, and bore with it the timbers of the loft that lay under it, and shattered and bruised a great deal of the tower wall. The other battlements of the north tower, at the east side, were blown down by a storm. November 20th, on Sunday, 1720, they fell on the north side, and beat down 20 feet of it in length to the ground. The storm also threw two pinnacles off the south tower, so that there is but one pinnacle now left. It broke the windows in divers places, and did about £100 damage. On September 3rd, 1723, there fell down 50 feet of the roof at the west end, near the font, and on or about September the 6th the roof of the south aisle fell in, whereupon the choir service was removed to the Lady Chapel, and the west door shut up, and the entrance is now by the south door.” 36 John Wood of Bath constructed an "Italian Temple" in the presbytery and the four eastern bays of the nave, leaving the rest of the nave to decay. The local architect John Pritchard restored the cathedral in the mid 19th century, and it is interesting to try to ascertain what is mediaeval and what is 19th century work, and to gauge the extent to which mediaeval design was recovered in the restoration. 47 J. Buckler In the print of 1820 by J Buckler, the central part of the west front is shown complete. Just enough of the south aisle wall survived to preserve the Norman south door. The aisle windows in the Italian Temple are mediaeval ogee arched reticulated windows, as can be seen more plainly in the print of c.1800 by C Warren. These were used as the model for the reconstruction of the aisle windows in the western part of the nave. In the Buckler print, the part of the arcade arch next to the Italian Temple is missing, presumably due to demolition. Crucially, half a bay of the clerestory is visible, though its parapet is missing. The same half bay of the clerestory is shown as surviving to full height internally (if that is the word) in the engraving of c. 1800 by B Winkles. In addition there are fragments of the clerestory attached to the towers. The Warren print shows that the fragment against the north tower included a window and so must have been a half bay. 48 C. Warren 49 B. Winkles The height of the nave parapet is of 19th century determination therefore (Figure 9). The external height of the presbytery walls is the same as that of the nave parapet, and may also be ascribed to John Pritchard, as may the whole of the clerestory in the presbytery (Figure 11). Here he very properly designed a clerestory such as might have been built in the 14th Century, rather than reconstructing the upper range of Norman windows. He also reconstructed the presbytery arch, and the presbytery aisle windows. Apparently, the bases of the central member of the presbytery arch as well as parts of the other mouldings are mediaeval. They provided clear evidence for the design of the presbytery arch. 50 Half a bay of original clerestory, preferably from the geometrically more exact western bays of the nave, would have been sufficient to permit an accurate reconstruction. The presence of the intricate pentagonal geometry (Figure 10) is evidence that this is precisely what John Pritchard achieved. The gable of the Lady Chapel (Figure 20 left) is 19th Century (Warren shows its predecessor) but the geometry is perfect. A print of 1857 shows a steeply pitched gable, but the corresponding roof was not persisted with as it would have blocked the east window of the presbytery. C. Warren shows the reduced 18th century east window of the Lady Chapel, together with its mediaeval outline. The tracery is Pritchard's, inspired by the side windows of the Lady Chapel, and the chapter house windows at York. Pritchard also built the crown of the Jasper tower. A conscious difference of approach is discernible in the design of the new south west tower and spire. Careful reconstruction gave way to new designing. Because of the spire, the tower had to be square in cross section. Its mediaeval base is rectangular, and the adjustment was made by a series of set backs on the east face of the tower. The turrets are pentagonal. In 1941 a land mine destroyed the roofs of the nave, south aisle and chapter house, and the tracery of the windows in the south aisle, also the sedilia and most of the furnishings. The spire was damaged, and the upper part of it was rebuilt, without the crockets shown in Figure 23. There was no geometrical loss as a result of the explosion. Considerable aesthetic gain was made by George Pace in the reconstruction. He inserted a wooden ceiling over the nave, which involved the resetting of the presbytery arch at a lower level and with smaller radii of curvature. The existence of a mediaeval ceiling is indicated by the ledge immediately under the upper lancet in the west front (Figure 11, Figure 15 right) at a height governed by the large pentagon in Figure 12. He removed the reredos, re-designed the floor levels in the choir and presbytery, and inserted a pulpitum of carefully considered design bearing the Majestas statue by Epstein. 51 Nineteenth Century Geometry S S/3 T G G/2 G T T W J T 5C S T/2 W/3 W J T C 2G Figure 23 52 We have seen two examples of 19th Century geometry in the Nave parapet and Lady chapel gable, both involving simple ratios, i.e. avoiding irrational numbers such as √2, √3 and φ, the golden section. This impression of simplicity is confirmed by the geometry of the Pritchard tower and spire (Figure 23). All measurements appear to be derived from existing measures. T stands for the width of the Pritchard tower. Comparison may be made with Pearson’s geometry at Truro. The nave bays fit into columns of five equal squares. “…Each bay of the arcade is twice as high as long, the springing of the arches being halfway up, and takes two squares; the same goes for the clerestory; while the tribune bays … take a single square each.” 37 53 Joseph Lord Print The print by Josph Lord shows the Cathedral as it was before the 18th Century losses. The print was dedicated to Bishop Adam Ottley of St Davids (1713 – 23). 40 It must be treated with a degree of caution, as it fails to show the buttresses and the stair turret of the Jasper tower, nevertheless it presents some interesting information. The Consistory Court covers and protects the Norman south door. A porch protects the small south door. There are no buttresses to the nave and no door in the presbytery aisle. The roof of the chapter house preceded the low pyramid roof shown in the Buckler print. An external staircase gave access to the school house over the chapter house, and also to a room above the south presbytery aisle. To accommodate this room, the presbytery aisle wall was considerably higher than the nave aisle wall. The roof appears to meet the presbytery wall above the level of the base of the clerestory, and this may explain the astonishing lack of windows. The upper range of Norman windows on the north side of the Presbytery probably survived until the building of the Italian Temple. The clerestory windows in the 8th bay of the nave are wider, lower, and further apart than the others, also there is no vertical break between the nave clerestory and the presbytery wall. The absence of 13th century windows in this bay is consistent with an intention to build towers here. It is possible that these clerestory windows were contemporary with the upper room of the south presbytery aisle. All the parapets are battlemented. The nave roof is of lead. The print gives a view of the 15th Century crown of the Jasper Tower. 54 55 The Norman Arch into the Lady Chapel Figure 24 56 The method of turned squares given by George Lesser 41 and by Colin Dudley 42 has the disadvantage that it very quickly puts a nest of lines over the diagram, few of which directly relate to architectural features of the building, which makes the geometry difficult to follow. It does however have the advantage of being a method used by masons from Norman times through to late Perpendicular. It was a secret method, but nearly given away by a boss in the south Choir aisle at Worcester bearing a design of turned squares, and by the vault under the tower at Tewkesbury, which is a diagram of turned squares carried out on the curved surface of the vault. I resist applying it automatically, for the building should be allowed to speak its geometry without preconceptions, but especially in Norman buildings its use appears to be normal. This method is applied to the Norman arch at the east end of the presbytery. The 19th century floor level shown in the diagram is higher than that in place when the arch was built. The black square is at the level of the top of a chamfer, at the bottom of which was the Norman floor level. It gives the level of the tops of the imposts. The opening under the arch was therefore based on a square, as is that of the south door, but in a different sense. The modern floor level is slightly below the top of the chamfer. The system of turned squares is shown in light blue. The largest square shown has side 18’6”, half of 37’. The circle shown in green has its diameter on the line which joins the intersections of the second right and turned squares, and radius such that the lower sides of the first turned square are tangents. The upper face of the third right square is also a tangent. It gives the width of the arch between the inner extremities of the half-shafts. The circle in dark red has the same radius, and is placed so that the lower face of the fourth right square is a tangent. It gives the intrados, and its centre defines a line above the imposts which is the diameter of all the arch mouldings. This shows why the arch was stilted when there was no reason such as vaulting a rectangular space for doing so. The circle in red has diameter equal to a side of the black square. Concentric with the dark red circle, it also goes through junctions of the third turned square and fourth right square. The circle in mid blue, concentric with the other two defines the extrados, under the decorative disks. The lower face of the second right square is a tangent. The yellow circle goes through the intersections of the green circle and the second right square. There is no guarantee that the nest of squares was in this relationship to the arch, but it is clear that such a nest was used. 57 Interpretation In this section I attempt a partial interpretation of the geometry. The objectivity which has marked the book up to this point is now relaxed. As a Lay Clerk at the cathedral, it was natural to think in musical analogies. Pipes or strings with sounding lengths in the ratio 2:1, other things being equal, give notes one octave apart. A ratio of 3:2 gives a fifth. A ratio of 4:3 gives a fourth etc. In the Renaissance, the analogy with ratios used in architecture was explicit. For Gothic architecture, the analogy does not hold, for pipes in the ratio of √2:1 or √3:2 or φ:1 would produce not only a dissonance, but an intolerable cacophony, whereas we have seen that these ratios were highly prized. But we can extend the analogy and treat these ratios as dissonances. In music it is the handling of dissonance which adds zest. Possibly these ratios can be thought to fulfil a similar purpose in architecture. There was certainly the intention of representing heavenly order and harmony on earth and embodying it. This aspect has a parallel in the great programmes of sculpture and stained glass. They might have had a function in educating illiterate peasants, but their primary purpose was doubtless to anchor the universal in the particular location, and make it present. One row of panels in the great east window of York is not even visible from the ground. There are occasions when different geometrical arguments led to the same result. An example is the height of the clerestory. This showed the superb skill of the designer, but also “locked” the geometry in place. Provided the similarities were sufficiently close, the lady Geometria smiled on the master mason, even if they were not mathematically exact. One lady who attended a presentation of this paper said to me concerning the presence of geometry “That is why these buildings look so right.” Continuing the musical analogy, if the individual ratios are like intervals, the effect of looking at all the geometry at once is that of a mighty geometrical symphony, or like the ratios and combinations of numbers governing section lengths in Tudor masses and other music which build up into a complex numerical structure. 58 The music of the spheres is another concept in the same family. Eternal ruler of the ceaseless round Of circling planets singing on their way… What though in solemn silence all Move round this dark terrestrial ball; What though no real voice nor sound Amid their radiant orbs be found… As with all controversies, there is truth in both positions. There is no microphone that can record the music of the spheres, nor the interval of a minor third in the vicinity of the north door, with its ratio of 6:5. Evidently there are complicated geometrical relationships between the orbits of the planets, and the positions of stars and galaxies, and it is possible to wonder what music bears an analogy with them. An undeniable aspect to architectural geometry is astronomy. This figure by John Mitchell is from The Cosmos in Stone by Tom Bree. It has proved very useful in completing this account of the Geometry of Llandaff. 59 The diameter of the small circle is 3, and that of the red circle 11. These are the respective sizes of the moon and the earth. The pyramid has base 11 and height 7 and is similar to a vertical section through the Great Pyramid, parallel to an edge.. The square then has perimeter of 44. The outer circle has circumference 2πr, or using the approximation of 22/7 for π, the circumference is 44, and the circle is squared. The designers of Llandaff, Wells, York and Southwell were keen to embody this knowledge in their structures. In one sense it is unnecessary to search for a meaning outside the geometry itself. Working with these figures for a long time, one starts to feel that the geometry is a language in its own right, and the intricacy and ingenuity and sheer skill have a meaning in themselves, or that an architect can say things in this language that are expressible in this language alone. We have seen how the designer of the nave “spoke” to the designer of the Lady Chapel at Llandaff. I have had the experience of going backwards and forwards between the elevations of the nave at Llandaff and the quire at Wells, and finding them “saying” similar things, for example, because there was a golden rectangle at Wells and other geometrical features were common to both buildings, I looked to see if there was one at Llandaff. Another “speaking” moment was when I saw how the circle in Figure 12 matched the curve of the door. The language is also capable of sustaining an argument. An example is in the West Font, where each placement depends on its predecessor in a lengthy chain. It was by following this argument that I discovered the golden rectangle in Figure 14. The musical analogy continues here, for one does not need to enquire concerning the extra-musical meaning of a Mozart symphony for example, for the meaning is in the symphony itself, and is communicated in musical language. One of the meanings of the vesica piscis is to join opposing powers and give birth from the union. A deeper meaning is that the two circles represent duality, that which is manifest, and the intersection represents the Unity which is the source of all. The pentagram is an ancient symbol of the Goddess, Venus, Aphrodite, Isis, She has many names. The planet Venus traces out a pentagram as seen from Earth every eight years In some places the pentagram with point downwards is viewed as a symbol of evil. No shape is inherently evil. If you choose to give this meaning to the pentagram round the door, you also see the Unity, symbolised by the circle, shining through every appearance of evil. The same meaning could be seen in the clerestory, but really evil has no place in beauty such as this. 60 One can never say that one has exhausted all the layers of meaning in a great building. To those involved in the design, it may be that numbers possessed properties and meaning in themselves. It is therefore a legitimate line of enquiry to see if there is any consensus in writings on numerology, although the element of subjectivity is necessarily strong here, where it was absent in the demonstration of the pervasive presence of pentagons at Llandaff. Gervais Frykman 2004 2014 2024 61 The Geometry of the Chapter House Windows at York Minster John Britton, 18191819 62 The chapter house windows have been described as “Geometrical at its most handsome.” 1 The excellence of the tracery derives in no small measure from the three large circles in the head of each window. It is not at all straightforward, if it be possible at all, to fit three equal circles into an existing arch. John Pritchard in 1844 inserted tracery based on the York chapter house windows in the re-opened mediaeval east window of the Lady Chapel at Llandaff ,2 but he was constrained to make the upper circle much smaller than the other two. The three circles were probably the starting point of the design, and the arch was fitted round them. The centres from which the arch arcs were struck must, by symmetry, lie on the common tangents of the upper circle and each of the lower circles. They must also lie on the horizontal line through the springing points of the arch, as otherwise the arch would be segmental or horseshoe. Naturally the centres of the three circles lie on the vertices of an equilateral triangle. If the sloping sides of this triangle are produced downwards, the horizontal line through the springing points makes with them another equilateral triangle. Figure 1 H A C G D 63 B If the distance along the common tangent to the centre of the arch arc (AB in Figure 1), is large in relation to the radii of the circles, the springing point of the arch, D, falls within the base of the equilateral triangle CGH. If it is comparatively small, the springing point falls outside the base of the equilateral triangle (Figure 2). Figure 2 H A F E D B C G In the York windows, the springing points are at the lower vertices of the equilateral triangle, so that points C and D in Figure 2 coincide. It is possible to examine the geometry, to see how AB relates to the radii of the circles in order to achieve this coincidence. It is unlikely that we shall recover the thought processes of the designer, but we shall establish the design which he created. In Figure 2, then, let the circles have unit radius, so AH = AE = FE = 1. We obtain expressions for BC and for BD in terms of AB, and then equate them. Triangle ABC is a 30°,60°,90° triangle, so AB:BC = √3:2, so BC = AB x (2/√3) Triangle EAB is right-angled, so by Pythagoras, EB2 = AB2 + 12. 64 FEB is a straight line, as the centres of touching circles and their point of contact lie on as straight line, therefore BF = √(AB2 + 1) + 1. BD = BF, so equating BC and BD, AB x (2/√3) = √(AB2 + 1) + 1. so (2AB - √3)/ √3 = √(AB2 + 1) so, squaring both sides: (4AB2 - 4√3AB + 3)/3 = AB2 + 1 so 4AB2 - 4√3AB = 3AB2 AB cannot be 0, or there would be no arch, so dividing by AB, AB = 4√3. Since BC = AB x (2/√3), BC = 4√3 x (2/√3) = 8 therefore, from triangle ABC, AC = 4, therefore, since AH = 1, CH = HG = GC = 5. The tracery recognises this fivefold length, as there are five lights, equally spaced, though the mullions are of two grades. The head of each light consists of arcs of radius 1 unit, struck from points D,E,F,G,H and I, one unit apart (Figure 3), so they contain equilateral triangles. The tall central lancet has centres A and L (Figure 3), which are also the centres of the arcs of the arch. The radius is 6 units, so the arcs of the tall lancet must necessarily touch the lower circles respectively. 65 The arcs of the outer lancets are struck from centres D,F,G and I, with radius 2 units, so they also contain equilateral triangles, and the apex of each lancet lies on the circumference of one of the lower circles where it is cut by a side of the main equilateral triangle. Figure 3 C D E F G H A and B 66 I J K L The tracery recognises this fivefold length, as there are five lights, equally spaced, though the mullions are of two grades. The head of each light consists of arcs of radius 1 unit, struck from points D,E,F,G,H and I, one unit apart (Figure 3), so they contain equilateral triangles. The tall central lancet has centres A and L (Figure 3), which are also the centres of the arcs of the arch. The radius is 6 units, so the arcs of the tall lancet must necessarily touch the lower circles respectively. The arcs of the outer lancets are struck from centres D,F,G and I, with radius 2 units, so they also contain equilateral triangles, and the apex of each lancet lies on the circumference of one of the lower circles where it is cut by a side of the main equilateral triangle. The manner of fitting a circle in the head of these outer lancets is shown. The arcs of the larger lancet are centred on G and I and have radius 2 units. Arcs of the small lancets are also centred on G and I and have radius 1 unit. The circle therefore must have diameter 1 unit, or radius ½ unit. To locate its centre, set the compasses to 1½ units and strike arcs from G and I to see where they intersect, or from I so that it intersects the centre line. The rest of the design is decoration, though the nine-foils in the large circles deserve special comment. There is no Euclidean construction for a nonagon, so trial and error may have been used. Any error in the compass setting is multiplied by 9 as the circle is stepped round, so it is a simple matter to achieve sufficient accuracy. More sophisticated division of a circle is shown by the gear wheels in the clocks of Wells and Salisbury. 1 Alec Clifton-Taylor, The Cathedrals of England, Thames and Hudson, 1967,1974, p.166 2 J H James, A History and Survey of the Cathedral Church Llandaff, W M Lewis, 1929, p.19. 67 Notes 1. Peter Pace: The Architecture of George Pace: B A Batsford Ltd 1990 2. Frederick Stallard MA and Paul Bush: The Geometric Skeleton of Peterborough Cathedral: Paul Bush 1994 3. Peter Kidson in Salisbury Cathedral - Perspectives on the Architectural History: Royal Commission on the Historical Monuments of England: The Stationery Office 1993, 1996. A 20'6" measure is also important at Salisbury. 4. John H James: A History and Survey of the Cathedral Church, Llandaff: W M Lewis 1929. Lovegrove (Note 9) gives 14th April 1120 5. Illustrations are from John H James op cit, except for the prints which are individually ascribed. 6. Jerry Sampson: Wells Cathedral West Front: Sutton Publishing 1998. 7. C A R Radford in BAA Conference Transactions 1978. 8. F J North: The Stones of Llandaff Cathedral: University of Wales Press 1957. My observation. 9. E W Lovegrove: The Cathedral Church of Llandaff: Western Mail; 1944, also JBAA, 2nd ser., xxxv (1929). The description of the Norman Cathedral and the building sequence are based on this source, also the illustrations of the seal impressions. 10. For its widespread use at Norwich see Eric Fernie: An Architectural History of Norwich Cathedral: Clarendon Press 1993. For its use in Peterborough, see note 2, and for Salisbury and Old Sarum, see note 3. 68 11. The Hereford Chapter Seal of c 1190 gave an image almost identical to this (Ed Aylmer and Tiller: Hereford Cathedral, A History: Hambledon Press 2000, p32) but this does not necessarily mean that the representations were purely abstract. It could underline the essential similarity between the designs of the two cathedrals. 12. E W Lovegrove op. cit. 13. E W Lovegrove op. cit. 14. George Marshall: The Evolution of the Cathedral Church of Hereford: Littlebury & Co: 1950. 15. F J North op. cit. 16. F J North op. cit. 17. The term Nave is used here to refer to the architectural unity of eight bays. 18. Ed L S Colchester: Wells Cathedral - A History: Open Books Publishing Ltd, 1982, 1996. 19. Jerry Sampson Op Cit. The bases of the triangles at Wells and Southwell are measured between the centre lines of the walls in each case. 20. Jerry Sampson op. cit. 69 21. Measured floor to capitals = 15'9½", floor to clerestory base is 31'6½", measured clerestory to the top of the pentagon is 45'11". The top course adds a further 12”. The theoretical length of the bays working backwards from the height of the arcades is 18'2½". Provisional lengths of the bays numbered from the west are as follows. Measurements for the first bay exclude the 2” for the transition to the West Front. North Arcade South Arcade 1st 18’1½” 18’7½” 2nd 18’4” 18'2½" 3rd 18’4” 18’3” 4th 18’1½” 18’2” 5th 18’3½” 18’1” 6th 18'2½" 18’2” The triangles in Figure 7 are confirmed by the above measurements, for 18'2½" x √3 = 31’6.45”. 22. Lovegrove thought that the thickening of the aisle wall was a relic of the Norman transept end. In favour of this view is the use of Sutton stone to chamfer off the extra thickness, together with the presence of the Norman transept end here as deduced from the position of the chapter house stair turret. Against is the apparent continuity of the thickening with the inserted arch, also the dissimilarity between the chamfering and normal window sills and between the vertical finishing and door jambs, also the running into each other of the proposed jamb and sill, also the geometry of Figure 2, which indicates that the thickness of the Norman walls, particularly the extant Norman north wall of the north transept, was the same as that of the south aisle wall excluding the thickening. The single course on the outside, between the chamfer and the string course under the window is also of poor quality, indicating that it may have been inserted after the chamfer, i.e. that the last project for the eastern towers may postdate the Decorated nave aisle windows. 23. The Cathedral Church of St John the Evangelist, Brecon: Royal Commission on the Ancient and Historical Monuments of Wales: Friends of Brecon Cathedral 1994. 70 24. A golden rectangle is such that the ratio of the long side to the short side is the same as the ratio of the sum of the sides to the long side, or φ :1, where φ = (√5+1)/2 25. Geometry from a reconstruction of the original form of the quire by Bilson given in Wells Cathedral – A history (Note 18) 26. Measured to nave capitals x 3 = 47’4½”: Measured to top of clerestory including the top course = 46’11”.See note 21 27. Abbott David of Bristol wrote to the Dean of Wells between 1218 and 1220, asking him to lend his stone-carver “your servant L, to hew out the seven pillars of wisdom’s house, meaning, of course, our chapel of the Blessed Virgin.” (Bristol Cathedral History and Architecture: Ed Rogan: Tempus Books 2000) 28. Freeman’s “Llandaff Cathedral”: W Pickering 1850. 29. “It is by no means impossible but that aisles were added to the presbytery … but at present without disturbing the original Romanesque walls. The evidence on which this supposition rests is the fact that the eastern arch of the vaulted bay is clearly part of the Early English work, and as it must have opened into something, some building must have been added to its eastern face …” (Freeman, op.cit.) The arch may be Decorated, but there could well have been a smaller arch leading into the proposed Lady Chapel passage. 30. The windows were altered in the 18th century, but “even where the windows have given way to Mr Wood’s insertions, enough remained to replace the jambs with perfect accuracy; the tracery alone required to be entirely new.” (Freeman op.cit.) 31. Freeman op. cit. 32. J H James op. cit. 33. Strictly speaking, the foliated caps in the south arch are Early English, or rather Geometrical Decorated, contemporary with the Lady Chapel. The western capital in the north arch, with its naturalistic leaf, is high Decorated, and its eastern fellow may well 71 be. 34. There may have been an element of rivalry with St David’s, for Bishop Gower had the aisle windows replaced with Decorated ones. 35. Freeman op. cit. The tracery in the south window is now 19th century Geometrical Decorated. 36. Cole MS. xxix, fol. 14 b, from J H James, op. cit. 37. Pevsner and Metcalf: The Cathedrals of England: Viking 1985. 38. Peter Kidson op. cit. 39. John James: The Master Masons of Chartres: West Grinstead Publishing 1990, formerly Chartres, the Masons who Built a Legend: Routledge & Kegan Paul 1982. Pentagonal geometry is also to be found at Chartres. 40. Friends of St Davids. The print is from Browne Willis: Survey of Llandaff: 1718 41. George Lesser: Gothic Cathedrals and Sacred Geometry: Alec Tiranti 1957 42. Colin Dudley: Canterbury Cathedral: Aspects of its Sacramental Geometry 43. Dan Pedoe: Geometry and the Liberal Arts: Penguin Books 1976. 72 Appendix: To construct a pentagon The following construction for a regular pentagon is given in Ptolemy’s Alamagest 43 D F A O O E O is the centre of a circle Bisect radius OA at E Construct Radius OD perpendicular to OA Draw arc centre E radius ED to cut AO produced at F ( O cuts AF in the golden section ) DF is the length of the side of the pentagon. 73 74 Other books by Gervais Frykman are Mysticism for Moderns Twelve Past Lives Available on Kindle in e-book and paperback format. He can be contacted on christianityrefreshed160@gmail.com 75