CONCLUSIONS

The method used in the present analysis of the geometry of Peterborough Cathedral has produced an analysis in three dimensions extended throughout the building over four centuries of architectural development, in a totally unified geometry closely related to the forms and dimensions of the fabric.  It maintains complete adherence to the rules of Euclid’s geometry as defined in his Postulates, and as declared in English medieval masons’ documents to have been the basis of their art.

      The geometry presented would appear to be the first to have arrived at such a result,

and the reasons for the comparative success of the method of analysis adopted would seem to me to be as follows:

 1.  The majority of researchers in the field have had little or no experience of geometrical construction.  Most of those who have attempted geometrical analysis have done so in ignorance of the Euclidian postulates and the range of geometrical developments that they offer.  This is largely due to a general misunderstanding of the medieval documents that refer to the importance of Euclid in the geometry of the architects, which has been dismissed as fantasy.  Even historians of mathematics appear to be ignorant of the rules and nature of Euclidian geometrical construction, and this ignorance has militated seriously against the successful geometrical analysis of historical buildings.                   

 

2.  The unawareness of the true nature of Geometrical Progression and its fundamental importance in the function of architectural medieval geometry, having its origins in ad quadratum and ad triangulum constructions, has also seriously inhibited the progress or research in the field.

 

3.  The accumulation of large numbers of measurements in order to search for ratios that would indicated certain geometrical forms, to which much effort has been dedicated, has yet to achieve a full analysis, and I would suggest that it can never uncover a full geometrical analysis, because the range and complexities of ratios that are created by relatively simple Euclidian constructions are far greater than the human brain can deal with, and probably any computer.   In addition, the question of what dimensions to take note of and which to ignore continues to present problems for research by such methods.  Furthermore the unavoidable minor inaccuracies that must exist in all geometrical constructions and in the layout on site of lines, digging of foundations and setting down of footings will always lead to difficulties in geometrical analysis purely from measured ratios.

4.  Theories of proportion can provide endless fascination for researchers with a penchant for arithmetic and algebra, especially theories concerned with the Golden Section, but they have produced no geometry.  The Golden Section is related to the pentagon and is produced in the geometrical construction of the pentagon but pentagonal architectural design is relatively rare in the Middle Ages and mostly fairly late. In any case it is very doubtful if any of the architects drawing pentagons for Lincoln, for instance, had ever heard of a Golden Section. 

     Another theory of proportion that has inhibited research into medieval geometry, particularly at Peterborough, where the theory may well have originated or was revived, is that based on Pythagoras’s Musical Proportions.  Error in this field have emanated from the fact that the portal openings of the West Facade have never been accurately measured until the present investigation, mainly owing to the fact that the lower half of the central opening is buried within the masonry of the Perpendicular porch. The present investigation destroys that theory.    

 

5. Where certain ratios of proportion are obvious, as on the West front at Peterborough, they can lead to an understanding of the basic geometry from which a Euclidian    construction may be deduced, but such deduction depend upon considerable experience of the constructional procedures that are available, and of their possibilities.  The logic that is involved in geometrical analysis is essentially the logic of geometric development, rather than the logic of verbal argument.  It is not unlike the logic that we are aware of when we witness a time-lapsed film of plant growth, or when one hears a Bach fugue, which many can enjoy but few can analyse.

 

6.  The majority of approaches to the problem of medieval church geometry are hampered  by a lack of appreciation of the religious dynamic inspiring the medieval   architect’s vocation, and of the sacramental function of his geometry, which has too often been regarded as no more than a useful tool rather than a high art explored by men of the highest intelligence and great abilities, working in a period of universal religious intensity. Many were lacking in formal education, living in a time of almost universal innumeracy, before the arrival of printing or of Arabic numerals, but many were equal in intelligence and depth of thought and feeling to anyone alive today.  The medieval architect’s geometrical function may well have been inherited from the great scholars of the Church, Lanfranc for example, or Ernulph, whose rather simple geometry they developed into an art of great complexity.  They may or may not have been aware of its ancient origins and purpose, and if they were aware, that awareness may well have withered away as the centuries passed. But in those centuries before the science of statics existed, or the skills of reinforced concrete, how else could one ensure the stability of a mighty House of God, except by reinforcing its stones with the strength of God’s own Geometry?

7.  The great majority of more recent studies have been devoted to church ground-plans which present very little obvious geometry in comparison with that displayed in elevations.

The Present Analysis

The present analysis was based on long experience of geometrical construction, a knowledge of Euclidian construction and a rigorous adhesion to its rules, using only compass and unmarked rule, proceeding from predetermined point to predetermined point.  It was limited to one monument and it was entered into only after a thorough study of that monument’s architectural history according to the most recent professional research. No attempt was made to discover proportions from measure, although the discovery of the 1:Ö2 ratios from the measurement of the portal openings was an unexpected finding.  The evidence provided by geometric work of the medieval period, such as the geometry of masons’ marks and the drawings in the German documents was taken as strong evidence as to the direction to be taken.  The cultural melieu of the medieval architect, as conveyed by such writers as Harvey, his education, his religious beliefs, his values, his responsibilities, his yearly work pattern with five winter months to work at his geometry and his purse of small-scale templates, the deep feeling for his drawing instruments itemized in his will and bequeathed to his good friends and younger colleagues, his love of geometry making - all were considered in the attempt to enter into his mentality as he sat before his drawing board.   This could be regarded as ‘romantic’ by some, but I see it as common sense and rigor, strengthened by twenty years living and working in and around a great cathedral.

  

The method of analysis followed on the above basis has resulted in the following understandings:

      The architectural form of Peterborough Cathedral was designed upon a geometrical structure.  The geometry of that structure is essentially in the ad quadratum mode according to rules set down by Euclid in about 300 BC.  The main alternative form of Euclidean construction, known as ad triangulum, is also found at Peterborough in combination with the ad quadratum mode in the geometry of the early fourteenth century facade and clearly demonstrated in two of the gable windows, c.1230, and also in the plan geometries of the eastern transept and of the  pillars of the Crossing.

       The purpose of the geometrical structure was to sanctify the building by integrating it geometrically with the Divine Geometry with which, in the Pythagorian and Platonic cosmologies, the universe was understood to have been created.  It was believed that thereby the building would have been made receptive to divine grace and wisdom and preserved from the destructive forces of evil.  Although the forms of the building are drawn from the geometry, with which they conform in every detail, the full geometry of the building, like the network of steel within the reinforced concrete of a modern building, is invisible, for it was designed not for human eyes, but essentially for the eyes of Heaven.

      The most important of the ways that the geometry of a sacred building was designed to become integrated with that of the Heavens was through Geometrical Progression, which is fundamental to both the ad quadratum and the ad triangulum geometries, for it is from them that the principle of geometrical progression originates.  By the creation of a geometrical progression of forms the geometrician sets up a geometry that extends outward to the limits of the heavens and inward to the atom.  All sacramental geometry, such as that which informs the architecture of Peterborough Cathedral, is based on this geometrical cosmology.  The architectural designers who worked within this tradition may well have been aware of this fundamental but awe-inspiring concept, though none of them would ever have read one word of Euclid or Plato.

      In creating his geometry it was not an architect’s purpose to determine pleasing proportions.  If the use of the geometry produces proportions pleasing to the human eye this is due to the rigor of the Euclidean geometric process, the integrity of which creates that unity within its variety that is fundamental to the sensation of beauty. The human eye and brain seem fully capable of recognizing a proportional integrity without being aware of its nature.

     The Euclidean rules of geometrical construction allow the architect a great range of possible options within which to exercise his own innate sense of design within the practical and constructional requirements set before him.

  

The Continuity of the Geometry

The integrity of the geometry of the cathedral was maintained through four centuries of architecturally creative development, from c.1100 to c.1500, and this continuity was achieved by means of geometric elements inserted into the fabric by its architects for the express purpose of informing their successors of the nature of their geometry.

 

The Anglo-Norman Abbey Church.  The Geometry of the Plan.

The geometry of the Anglo-Norman building of c.1110 to c.1180 originates from a circle of approximately 153 Norman Feet (45.66m.) radius.  Within this circle an octagram star, formed from two squares, one rotated at 45 degrees, is constructed, and a core octagram is developed from it. From this figure four mutually tangential circles approximately 88 Norman feet (86-87 English feet) in diameter (26.5m.) are generated to form a Greek cross, and this cross provides, by development, the geometry for the whole ground-plan of the monastic part of the church. 

     Each of the four 88 Norman foot circles is then developed by the inscribing of an octagram, a circle within that octagram and a square within that circle. The resulting figure provides by repetition and development the geometry of the whole early church in both plan and elevation.  The basic geometry of the monastic church is completed by the construction of a fifth 88± Norman foot motif in the centre of the cross.   This simple operation leads to the creation of a Greek cross formed by five contiguous squares.

      Finally an octagon with vertical and horizontal sides is inscribed within each of the five 88± Norman-foot circles.  This completes the geometry of the monastic church with the exception of the sanctuary apse.

      The geometry of the people’s church in the nave is created by the addition by geometrical development of two of the 88-Norman-foot diameter motifs.

      The octagons inscribed within each of the 88-Norman-foot circles play an essential role, for they delineate the interior surface of the walls, they divide each of the transepts equally into three chapels, and they situate all the piers of the nave and choir arcades.

     The plan of the sanctuary is developed from a special form of the main geometric motif, and the geometry of that form is indicated in the fabric of the presbytery.

      All the 88-Norman-foot circles of the plan motif are interlocked in such a manner that all the squares within the circles are mutually contiguous, and form a unified Latin cross extending the length of the building.

      Squares circumscribing the 88-Norman-foot circles provide the limits of the whole ground plan, which are shown to lie along the line of the window glass at two thirds of the thickness of the walls.  The glass provides the exact surface separating the sacred interior world of the building from the profane exterior world.  It will be found however that in walls without windows, such as the c.1230 western tower turrets, the geometry would seem to lie at the wall surfaces. The one area in which this seems not to apply totally is in the eastern transepts.  The situation there is complicated by the alteration to the intended east-west dimensions of the transepts with the discarding of the planned western galleries after the foundations had been laid.  Here the geometry would appear to lie along the exterior of the walls, but in the original design they would have simply delineated the thickness of the gallery arcade. This circumstance also certainly explains the discrepancy between the N-S and the E-W dimensions of the crossing.  It is also possible that the continuing existence and use of the old Saxon abbey church created intractable geometrical problems in the transept area.  It will be seen, however, that when the western walls of the transepts were set up in their new position, replacing the gallery arcade, they did conform with elements of the existing geometry. 

     The alternative, and most likely explanation for the geometry of the eastern and western walls of the transepts lying along the exterior of the walls and not along the line of the glass, is that in the original plan, probably by Ernulph, these wall were intended to be unfenestrated.  This would not be unusual in Romanesque churches especially when sufficient light for the virtually unused transepts would come from the many windows of the north and south walls.  The foundation trenches were dug out and the footings put in, but the plans were then changed, perhaps by Jean de Sais, to their present narrower transepts with all walls fenestrated.       

      The square within square or octagram or circle in accordance with Vitruvius’ “correct successive use of compass and rule” results in many 1:Ö2 ratios that may be related to Fernie’s findings in the plan of Norwich cathedral. 

.

The Anglo-Norman Elevations.

      The elevations of the Anglo-Norman church use a geometrical figure identical to the 88-Norman-foot motif of the plan, and from this figure the fenestration of the building is developed, as also is the north-south section of the nave. Again, the motif is developed in two ways, one for the nave and the other for the transepts.  All the geometries of the elevations stand on the level of the cathedral pavement, being centred 44 Norman feet, (43.5 English feet. 13.25m.) above it. This pavement lies at its original early twelfth century level that extends unaltered throughout the cathedral except for the modern steps elevating the altars in the transept chapels and the sanctuary.     

      In drawing up the design of the external walls of the nave and the fenestration, only the middle section of the geometrical motif is used, being repeated for each bay in juxtaposition, slightly but geometrically interlocking with its neighbours.  This system allowed the nave to be extended as required without threatening the integrity of the geometry.   There would have been many experiences of such a need to extend a people’s nave throughout the long history of basilican usage.

       The north-south sectional elevation of the nave, including the roof, even the ceiling, is designed on the same 88-Norman-foot ad quadratum motif as informs the rest of the building, but developed in complexity.

 

The Western Transepts c.1180 - 90

In elevation the extension of the west end of the nave into the western transept c.1180, in the new pointed arch style of transitional Early English Gothic, was developed from the same 88± Norman-foot geometric figure as the rest of the nave, (which converted to 87 English feet), but reduced to its essential elements for the east-west elevation.  In plan an extra unit of the basic motif was added to the nave geometry, and this unit was then developed geometrically to much greater width to include the plan geometry for two new towers.  This increase in width of the façade would appear to exemplify a cultural turning away from the inward-looking character of the early twelfth century church towards an externalization directed at the eyes of the populace rather than to the eyes of Heaven and the enclosed world of the monastic community.

     The geometry of the west interior wall of the western transepts is essentially the same  as that of the eastern nave elevation and of the ground-plan.  The west window of the nave is a late 14th century insertion of similar date to that of the Perpendicular Porch. Nevertheless its geometry is developed from the late twelfth/early thirteenth century geometry of the west wall.       

 

The Portal Façade of c.1235

      The aggrandizement of the western façade proceeded on an even more ambitious scale with the construction c.1235 of the present façade of three great portals and two staircase turrets.  The geometry for this development was again developed from that of the Anglo-Norman church in both plan and elevation, and as in that earlier church the geometries of plan and north-south elevation are essentially identical.

     Although developed geometrically from the earlier geometry of the Romanesque church, the new geometry of the façade is highly original in its final complexity and extent.  The most remarkable new feature is the clearly displayed inclusion of ad triangulum constructions within the ad quadratum framework.  The central ceremonial portal and its gable window is designed in the ad quadratum mode, developing from the octagon star, while the larger side portals and gable windows are determined by an ad triangulum geometry developed from the hexagon star.  The octagon star of the centre portal is undoubtedly related to the octagon star of the Risen Christ, whose cross always proceded the sacred processions through the small portal, while the six-pointed hexagon star is probably related to the lay men and women of the Church, all of whom would have entered throuth the larger side portals.

      The dimensions of the spires upon the staircase towers and upon the small turrets are all developed from the full geometry of the whole facade.  The pinnacles however are mostly modern and do not conform to the medieval geometry.

     The geometry of the gables is developed into the geometry of their windows down to the smallest details. The main lateral proportions of the c.1235 façade are found to be determined not in accordance with a numerical ‘musical ratio’ of 2 to 3, but closely in accordance with the geometrically generated ratio of 1 to Ö2.  Thus the theory of ‘musical’ proportioning at Peterborough is demolished. 

      The hidden geometry of the whole façade is indicated in two ways to those who are practiced in the art of sacramental geometry: first, by the design of the gable windows, which clearly indicate a central geometry in the ad quadratum mode, with the ad triangulum mode informing the northern and southern sectors.  The design of the staircase towers is clearly ad quadratum.  Secondly, the geometry of the façade is indicated by the four false gargoyles whose size and situations indicate nodal points in the whole geometrical scheme.  The lines of the gables intersect at the gargoyles and form equilateral triangles.  They continue down to intersect again and become part of the hexagons in the main intersecting circles north and south.

       The exact conformity of the present gables with the geometry confirms the fact that the essential design and proportions have not been altered from the original.   The lack of exact conformity of the statuary with the main geometry strongly suggests that the statuary scheme may have been an afterthought on the part of the prelate who commissioned the façade, possibly Bishop Grosseteste, and perhaps inspired by arrays of statuary being installed elsewhere. Consequently the architect at Peterborough would have been required to fit the statues into his design in any possible plain area. One only has to imagine the façade without the statues to appreciate the possibility of this assessment. Possibly made elsewhere, the proportions of the statues clearly made it necessary to cut into the lower bouts of the windows’ circular borders.   The resultant design that we see today is thus perhaps more richly decorative than originally conceived. 

      The complete geometry of the facade develops from an originating circle 152.25 English feet in diameter (Fig. 58).

 

The Mid-14th Century Alterations

      The enlarging of the windows of nave and choir carried out between 1290 and 1345, and the heightening of the aisle walls that this operation demanded, were performed in conformity with the geometry of the earlier 12th century windows and walls, but developed geometrically to provide the new dimensions.

     The twelfth century central tower and its replacement by a less lofty tower c.1315 was designed in accordance with a vertical repetition of the basic geometry of the nave elevation, but overlapping the lower geometrical scheme, so that the inscribed squares, exactly as in the ground-plan, are contiguous.

       The turrets added to the central tower in 1780 also conformed exactly to the fourteenth century geometry, from which one is impelled to deduce that they probably presented the height of the fourteenth century wooden octagon that they replaced.  It is also not impossible, even probable, that the height of the wooden octagon itself conformed to the height of the stonework of the originally taller twelfth century tower.   

 

The Perpendicular Porch c.1380   

      The Perpendicular ceremonial porch of c.1380 was designed to a geometrical construction that conformed exactly, in both elevation and plan, to the geometry of the whole 1235 façade, but skillfully developed from it in a most sophisticated and elegant manner. The distance between the inner surface of the forward facing buttresses on either side of the doorway is exactly that of the distance between the geometrical centres of the colonettes on either side of the thirteenth century portal, i.e. 18 feet 4 inches (5.59m.).

     The culminatory central niche, which may well have housed a representation of the Holy Trinity to which the upper chapel is dedicated, is placed exactly in the centre of the whole geometry of the 13th century façade.

The Retrochoir of c.1500

      The plan of the squared retrochoir, built c.1500 around the apse of the Anglo-Norman church, and now known as the New Building, was designed on a geometry very simply developed from that of the sanctuary of c.1120.  The octagonal element of that geometry relates closely to the centres of the three central eastern windows.

        The windows of the east wall of the retrochoir are each designed on a geometrical construction that is developed from one of the four 30.76 foot circles of the basic 87 English-foot motif of the twelfth century church.  It conforms exactly with the basic motif that rests tangent upon the pavement level that extends throughout the building. The highly original, probably unique, group of three triangular merlons surmounting each window suggests a dedication to the Holy Trinity or, less probably, to the three saints Peter, Paul, and Andrew.  At the same time, one can say with certainty that the trio of merlons that surmount each window, and whose geometry is so cleverly developed from the essential geometry of the window, was deliberately created in order to present at the eastern extremity of the cathedral an echo of the great motif of the three triangular gables of the western extremity.  The complete geometry of each window of the New Building is developed from three interlocking circles, of which the central circle is developed ad quadratum, while the outer circles are developed ad triangulum, in a modified imitation of the basic geometry of the western façade of c.1230.  A delightful example of Wastell genius brilliantly achieved.

        The walls of the retrochoir are thinner by a third than the walls of the rest of the cathedral. The glass and the geometry stand halfway through their width, in line with the earlier glass.

 

The geometry of the pillars

 The ground-plan geometries of the pillars of the Nave and the Crossing have been analysed and found to relate in the smallest detail directly to the larger geometry of the whole cathedral plan, being developed from the larger geometry.   It was also found that the geometries of the Crossing pillars involved both Ad Quadratum and Ad Triangulum procedures, and that this combination of geometries also played an essential role in the plans of the transepts, providing the line of the window glass in the north and south walls, and the siting and width of the staircases.                      

     The similarity between this alignment of three circles, the central circle developed Ad Quadratum, the outer circles developed Ad Triangulam, is repeated in the elevation and plan of the c.1230 facade, and in the central east window of c.1500, and would appear to relate to the plan of the great church of Hagia Sophia in Constantinople of  AD 532.  It is also to be found, I believe, in the geometry of Roger van der Weyden’s painting of the Deposition now in the Prado.

 

153 Feet

The figure of 153 feet, 153 being the number of fishes netted after the crucifixion, symbolised in Christendom the whole membership of the Church.  It plays a basic role in the geometry of Peterborough cathedral.  Measured in the Norman feet that remained widely in use until c.1130, it is the radius of the originating circle of the abbey/cathedral’s c.1110 geometry.  Measured in English feet, universal by mid-twelfth century, the diameter of the geometry of the western facade c.1235 is 152.25 feet, which is clearly an intended 153 feet.  It was the basic dimension of Old St. Peter’s cathedral in Rome, (153 Roman feet), and of Hagia Sophia in Constantinople, (153 Byzantine feet). It was also the number of the monastic community for which Lanfranc designed Canterbury Cathedral c.1070.

 

The Continuity

      The geometry of each architectural enterprise over the four hundred years of the building’s history was passed on to future architects by means of geometrical motifs inserted into the fabric in appropriate places, so designed that only one versed and practiced in the art of sacramental geometry would be able to recognize them and interpret their encrypted geometrical purport.  The purpose of this tradition was to ensure that if any alteration or addition to the fabric were ever to be made, the later architects would be able to ensure that their geometry was compatible with the earlier geometry.   Every architect would have been aware of the fact that virtually every building would be altered at some time in its history

          It would seem that one method of indicating the importance of the encrypted motif was to present it in an architectural element that was non-functional, for example: windows that could never be seen from inside the building, and cast no useful light, or gargoyles that spouted no water.  In the Anglo-Norman sanctuary these clues would appear to be presented in a unique form that is unrelated to any other decoration in the building.

      More geometry certainly remains to be discovered at Peterborough Cathedral, particularly in the detail.  It may not exist in the remains of the domestic buildings of the monastery, for they were hardly more sacred than the streets of the town.  The cloister walk was, however, a place for silent private prayers and reading of sacred works and could have had a sacramentally geometric basis.  Very little remains today of the cloister structure, apart from the ground plan, having a proportion which Stallard (1994) has determined to be related to the nave by a Ö2 ratio.

       It is not the aim of this study to pronounce upon the nature of the sacramental geometry that may be present in other religious buildings.   I would nevertheless suggest that the evidence presented at Peterborough leads one to suspect that the canonically Euclidian purity of its geometry, maintained over four centuries, may well be found in other English buildings.  The rigor of its conformity to the Euclidian rules stands in marked contrast to the changes introduced by the Parlers in 15th century Germany.   

      I would suggest that it is unlikely that any other building surpasses Peterborough Cathedral in the remarkable nature of its standing as a superb exemplar of the tradition of sacramental Euclidian geometry and in the insights that it provides into the development of that tradition over four centuries.

 

Related Matters

     These conclusions lead to the consideration of certain related issues, in particular the means by which dimensions in an architect’s drawing could have been scaled up to site dimensions without any form of tape-measure or recourse to attempts to perform geometrical constructions full-scale on the building site. A solution to this problem is that the dimensions would be set out by walking the architect’s two compasses, one small, one large, in succession across his table and across the site according to two arithmetical factors of his decided scale.

       The architect would possess a rule exactly one foot in length, according to the official length laid down at the time.  If the abbot or bishop commissioning the building were to provide a specified essential dimension, say 153 feet from the Crossing to the end of the Sanctuary (as it is at Peterborough), the architect would draw a line one foot long on his parchment, draw from it a circle one foot in radius (Fig 115 & Fig 124), and develop his geometry and ground plan from that circle with the crossing at its centre and the sanctuary wall on its circumference. 

      To scale up his drawing on to the site by a factor of 153 he would use the two factors of 9 and17;  9 probably for walking his small compass across his drawing table, and 17 for walking his “Great compass” on site.

      This method ensured the impossibility of significant errors in measurement, although it was liable to insignificantly small errors.