Chapter 1

tHE TRADITION OF SACRAMENTAL GEOMETRY

This chapter aims to establish the reality of the existence of a special form of geometry that was used in the design of religious buildings and artefacts, particularly in   medieval England, and to present its origins, its general history, character and purpose.

Around 1487, in southern Germany, a distinguished goldsmith, son of an architect, published his Fialenbüchlein, a booklet describing some of the techniques then used by architects in that region in the designing of religious buildings (Shelby, 1977).  As a goldsmith working in a material believed to be of divine origin he would be largely involved in designing and making artefacts such as reliquaries for religious use in the churches. These were often designed in the form of religious buildings.  His creations would be original in design, with experimental ideas for spires, pinnacles, facades etc., which could be taken up by working architects for actual buildings (Fig. 1).  His designs would be created using the same designing skills and within the rules that applied to designs for full-scale church buildings. The architectural importance of the work of medieval goldsmiths is discussed by Bucher (1976).                                                  

     The preface to Hanns Schmuttermayer’s booklet says much about the importance of geometry in medieval church architecture.  

By the Grace of Almighty God (I have written this little book ) in order to answer the petition of many honorable persons for improvement and refinements in the building of holy Christian churches, and for the edification and instruction of our fellowmen and all masters and journeymen who use the high and liberal art of geometry, so that their feeling speculation and imagining can be better subjected after memorization, to the true basis of measured work, and be allowed to take root.  Fundamentally, this art is more freely and truly planted and developed out of the centre of the circle, together with its circumference, correct rules, points and settings out.  I explain these matters not because I wish for my own honour, but more to praise the fame and reputation of the old-timers, our fore-runners, rule-makers and inventors of this high art of building construction which has its original true base in the level, set-square, triangle, dividers and straight-edge, and which is now pursued with precision, subtlety, higher understanding and deeper reckoning. Thus have I, Hanns Schmuttermayer of Nürnberg, correctly shown the technique of such measured work in the square and the round parts of the pinnacle, gablet and buttress – with all of the things belonging to the new and the old techniques, and I have brought it into the most understandable form by means of all its setting out, neither with too little description nor with more words than is necessary.  And I have not discovered such by myself but have received it from many other great and famous masters, such as the Junkers of Prague, Master Ruger and Nicholas of Strasbourg, who for the most part brought this new   technique to light, along with many others (Trans: Shelby 1977, 127).

When Hanns Schmuttermayer wrote his ‘little book’ the ‘Gothic’ style of architecture was dying.  In Germany it continued for some time in what Linda Neagley (1999) has termed ‘disciplined exuberance’, but in other countries it had long been out-dated, in Italy replaced by the Renaissance Classical style, in England modernised into the Perpendicular.  In Italy, Alberti’s famous and influential treatise on the Greek and Roman classical styles had been finished by 1452 and published in 1485.  In Florence, Brunelleschi had begun his Hospital of the Innocents in 1419, and by 1452 Ghiberti had created the bronze east doors of the Baptistery of S.Giovanni, his “Gates of Paradise.” In England the originality and powerful simplicity of the late Gothic of King’s College Chapel in Cambridge, begun in 1446, and very different in style from the contemporaneous buildings of the Italian High Renaissance, nevertheless stood in marked contrast to the elaborately Gothic exuberance of Nüremberg’s Lorenzkirche, of 1493, six hundred miles away.

The preface to the Büchlein leaves one in no doubt that in the Gothic style ‘geometrie’ had for centuries played a major role in the art of religious architectural design.  It was, moreover, no merely useful mason’s tool but a high and liberal art.  In other words it was a creative, inspirational art.  It was not a set either of proportions, or of formulae, but an art that offered its masters a freedom to create works as imaginatively original as those of painters, composers and poets.

On the other hand this art was explored and developed within a set of ancient rules that dated back to its origins, although as we learn from Master Schmuttermayer’s preface, in southern Germany in the fourteenth century, some of those rules had been changed and new methods had been introduced.  This does not mean, however, that the rules had been changed elsewhere, certainly not in England, as will be demonstrated.

 Around 1492, in Regensberg, Germany, a distinguished architect, Mathes Roriczer, inspired, like Schmuttermayer, by the availability of printing presses to attempt to preserve the deteriorating skills of his profession, published his own büchlein.  It was in fact printed on his own press, for in common with many of his fellow architects at that time, he was obliged to depend on a separate source of income, in his case a printing business. His preface begins:

To the Reverend Prince and Lord, the Lord Wilhelm, Bishop of Eystedt, of the family of Reichenau, my very good lord: do I, Mathes Roriczer, at present cathedral architect at Regensburg, signify my obedient humble services to the fore ready and willing:  My very good Lord, - As your princely grace not only was, and now is, an amateur and patron of the free art of Geometry… (Trans: Shelby 1977, 83).

 

Schmuttermayer’s and Roriczer’s booklets, together with other related documents of the period have much to tell us of the manner in which certain elements in the design of churches, mainly small details, were created geometrically in south Germany in the late 15th century. Here we are concerned simply to present the reality of the existence of a form of geometry that was a high and liberal art practised by men of genius in the designing of the great cathedrals and abbey churches of the Middle Ages.  This geometry was not the geometry that one thinks of today as Greek, although it had its roots in the geometry of Pythagoras and Euclid.  Nor was it the geometry of the Renaissance, although some aspects of it continued to echo through early Renaissance religious art and architecture. Medieval geometry involved no definitions, axioms, propositions or proofs.  No words were needed, nor any calculations.  Today it would be known as geometrical drawing, the creation of designs and patterns through the exploitation of a pair of compasses and a straight-edge.  One could learn its rules by watching the geometry being performed.  On the other hand, as will be demonstrated in the design of Peterborough cathedral, the level of sophistication and complexity of which medieval geometry was capable could challenge the resources of architects of the highest intelligence.

In one other important respect the geometry of the medieval architect differed from those of antiquity and of modernity. Its purpose was fundamentally religious.  It was actively theological and a creature of certain beliefs regarding the nature of God and of His universe, and it was employed to sanctify religious buildings and artefacts and thus to magically preserve them from destruction. Its practice, at its highest level, was a meditative openness to the mind of the Almighty, who, if one kept to the rules of His created universe, would guide the hand of the geometer and lead him in the direction that He wished him to go. In Kenneth Clark’s words: ‘One must remember that to medieval man geometry was a divine activity.  God was the great geometer, and this concept inspired the architect’ (Clark.K. 1969, 52).

Von Simson (1956, 34) takes a similar view:

It was with the compass (dividers) that God himself came to be represented as the Creator who composed the universe according to geometrical laws.  It is only by observing these same laws that architecture became a science in Augustine’s sense.  And in submitting to geometry the medieval architect felt that he was imitating the work of his divine master.

 

   The fundamental significance of this final sentence cannot be overstressed, although it has long been overlooked.  In view of the all-pervading religiosity of medieval culture, in which every aspect of a persons’ life, from cradle to grave and beyond, was controlled by some aspect of Christian faith as promulgated by the priesthood, it is impossible to believe that a highly intelligent architect, contemplating the task of designing a Christian building as a House of God, in form as near to the form of Heaven as was humanly imaginable, could avoid seeing himself as a servant of the Creator.  His task was intensely religious, and he would have set himself before his drawing board in awe and humility, as an instrument in God’s hand, a channel, through whom God would design His House.

The identification of geometrical forms with religious belief could be as ancient as human consciousness, for the world of the earliest human beings was flat and had four directions, forward, backward, left and right, as does any mammal’s world today.  But eventually they would have become aware of another world above their heads to which they did not belong and could not inhabit, but which provided the essentials of life, - light, warmth, water and food.  In the hemisphere of the sky they saw a great being, a blindingly brilliant creature that lived in the world above them and brought good things when it was strong.  To keep the creature friendly the men and women on earth gave it praise and admiration.  They danced and sang to it and gave it food, which they would take to the top of the highest hill nearest the dome of the sky and burnt so that its flesh as smoke floated up to their Lord and Master, the Sun.

 This powerful being was perfectly circular, and his world, the sky, appeared hemispherical.  Even drops of rain falling into still water brought perfect rings that spread out larger and larger as if they would encompass the earth. And flakes of snow came in geometrical circularities.  The sky was, for human beings, a hard surface, the inside surface of a sphere or hemisphere, of height unknown, across which the everlasting beings that inhabited it travelled a circular path every day and night.   From such observations the intellectual leaders came to identify the circle and the sphere as forms that belonged to the eternal and all-powerful heavens, while the square belonged to this earthly, changeable world in which every living thing died and rotted into dust.  Their world was divided into four regions; the east where the sun rose, the west where it set, the south where it lived, and the north where it never ventured.  Circular and spherical forms on earth thus became understood by the wise who studied the earth and heavens as providing links between this world and the world above wherein dwelt the great Sol, the eternal lord of the skies upon whom the lives of those living on the flat square earth beneath depended.

All this lies within the field of general knowledge, widely acknowledged, but it needs to be restated here because it demonstrates the essentially religious nature of the origins of geometrical forms and the relationships between them, most importantly the relationship between the square and the circle.  To unify these two forms would be to join Earth and Heaven -- the ephemeral with the eternal.  It was on this understanding that the search for a numerical relationship between the radius of a circle and its perimeter, and later its area, both involving the inexpressible value of p; a search developed. It continued even to the present day, where it has been calculated to the millionth place of decimals.

It will be appreciated that this particular view of Earth and Heaven developed in the northern hemisphere in which most of the world’s landmass lay, inhabited by the great majority of the world’s population, and in the temperate zone, continually under threat from ice from the north, and dependant upon the ever changing vitality and goodwill of the Lord of the Heavens.  In the equatorial tropics where the sun was seldom absent different cosmologies would develop.  Thus it was mainly in the Mediterranean, European and Indo-European nations, and in China and Japan, that the study of geometrical forms became inexorably involved with the need to understand and communicate with the heavenly powers that inhabited the skies.  In Ancient Egypt, throughout its long history, the dominant high god was Re, the sun god, also known as Kheper, and later as Aton, who was always portrayed as wearing the circular sun disc. Sun kings and heroes are central in Indian mythology, as they were in Iran, Greece, Rome and Scandinavia.  In Britain Stonehenge, in the shape of the sun, was undoubtedly built for some form of sun-worship, while in the American civilizations of the Plains Indians and the ancient Mexican and Peruvian religions sun-worship played a central role.  In Japan, another non-tropical country, the sun goddess was held to be the supreme ruler of the world, and her sun disc remains to this day the symbol of the Japanese nation.

Individual commentators such as Sauneron (1960, 177,181) emphasise the importance of sun-worship in ancient Egypt, while W.R. Lethaby, in his Architecture, Mysticism and Myth (1974), cites several authorities on sun worship in China.  On page 47 he quotes Professor Legge: ‘It has ever been accepted as a physical axiom in China that heaven is round and earth is square.[1]… The sovereign of the Chan dynasty…worshipped in a building…112 feet square, and surmounted by a dome, typical of heaven above and earth beneath’.   Lethaby later quotes Professor Beal: ‘Even the coinage, circular with a square hole, is well understood as symbolism for heaven and earth’ (Lethaby 1974, 53).

Lethaby concludes:

The perfect temple should stand…its walls built four-square with the walls of heaven.  And thus they stand the world over, be they Egyptian Buddhist, Mexican, Greek or Christian, with the greatest uniformity and exactitude.  When the world has become round and spherical, the squareness is retained almost universally as a characteristic of the celestial earth (Lethaby 1974, 53).

 

It was in Egypt, around 2500 BC, that a belief in geometry being capable of creating a link between ephemeral life on Earth and eternal life in the sky, found its most primitive but massive expression.  Over the centuries two important aspects of Egypt’s national culture had developed in power.  One was the geometrical or earth-measuring skills of the surveyors, ‘the rope-stretchers of Egypt’ as they were called by Romans, that were needed in the annual re-allocation of land after the Nile floods upon which the life of the nation depended.

The other area of power lay in the theological and ‘magical’ skills of the priesthood who carried, among many functions, the responsibility of facilitating the transportation of their priest-king from earth to heaven where he would live for ever protecting and prospering his nation.  One of the means by which they believed that this great purpose could be achieved was through the heaven-sent gift of geometry. 

Eventually the two realms of thought came together in harmony of function in the great pyramid of Cheops at Giza.  Here the geometry is simple, but powerful in its simplicity (Fig. 2).  The height of the structure at its centre is the radius of a sphere or hemisphere whose circumference on the ground is exactly that of the square perimeter of the pyramid.[2]  Thus on the earth a heavenly circle is created, invisible but understood, with an earthly square of the same circumference. Thus, by squaring the circle, the pyramid geometrically and magically united Heaven and Earth. So the Egyptian priest-magicians believed.  The invisible hemisphere was a microcosm of the heavens, while the square base was a microcosm of the natural world, with its four winds, its four regions and four  corners.[3]

The immensity of the pyramids at Giza bears eternal witness to the faith of the Egyptian priesthood in the supernatural power of geometrical forms, and even today there are highly intelligent persons who, if, say, confined to bed with a broken leg, will have a pyramid of the exact geometrical proportions used at Giza constructed to rest over the leg in order, they believe, to hasten its healing.

It is important in the context of medieval architectural geometry to appreciate that the geometrical link between the geometry of the earth and that of the heavens was the invisible circle on the ground, with its radius the very visible height of the pyramid. 

The invisibility of certain geometrical elements is a characteristic that will be found to continue to be essential to the practice of sacramental geometry throughout its history, and will be shown as such in the architectural design of Peterborough cathedral.  It should not be supposed that this geometry was purely symbolic.  It was part of the world of magic and in Egypt it was firmly believed to have the power to perform the miracle of transporting the human being into eternal life.  Sacramental geometry took its place with all the funerary rites of mummification and entombment that were held to possess the magical power of life over death.

Belief in the magic of geometry continued down the centuries, and is to be found even today in many aspects of New Age culture.  It is also found in folk-lore and fairy-stories, and in the theatre in which, perhaps, children’s toys are brought to life by a magician, who wears a conical cap decorated with stars and moon, a microcosm of the sky.  With his wand he describes a circle on the floor around himself, a magic circle, which is, of course, invisible. But it is certainly there because he has made it, and it will unite him with the miraculous powers of the heavens and protect him from the spirits of evil.

It is in the light of the ancient cosmology that one needs to envisage the culture of medieval western Europe that created the great Christian churches, all of which incorporate a largely invisible but very real geometry, a geometry purposefully created in order to provide, though its supposed supernatural power, divine protection from the destructive powers of the earthly world and the Devil, and to attract the presence within their walls of the wisdom and guidance of the Almighty, creator of all the geometry in the universe.   It is such a potent and purposeful but largely invisible geometry that will be seen to be invested in the architectural design of Peterborough.

 Such a belief in the divine protection from destruction as the function of architectural geometry may well be questioned today, but it would have been very much a part of Christian faith in the medieval world.  The very existence of the vast and powerful monastic organisations developed from the interpretation of the Biblical words to the effect that the world would never be destroyed as long as there were a few men who lived a life of perfection.  Faith in the protective power of geometry existed in parallel with such beliefs.

     The Middle Ages was a world in which belief in the supernatural played a universally active role. Only through an appreciation of the medieval belief in supernatural powers, including the supernatural power of geometry, can we understand that geometry’s true character and function.

 

Pythagoras

 Geometry, from the earliest times, was an essential branch of theology, and the study of geometry was first and foremost a religious activity.  The most famous of all the theologian-geometer-magicians is the man known as Pythagoras. A concise appreciation of his importance is given in Wertheim (1997, 18-22).  Her overview may be summarised as follows: part Greek, part Phoenician, he migrated away from the worldliness of Greek culture into the more mystical culture of Persia.  He was essentially a mystic, studying every kind of religious belief and practice.  According to the legend of his life-story, he migrated to Egypt, where, after long and determined efforts, he was able to explore the culture and beliefs of the priesthood.  He then settled in Babylon where he studied the complications of their arithmetic and astronomy.  Eventually he arrived at a belief in an eternity of reincarnation every 216 years (6x6x6 = 216) and set up a religious community devoted to the study of numbers as a means of discovering the nature of the whole universe and of its Creator.  “All is number” was one of his most well-known sayings, for he was totally convinced that the whole of creation was formed of numbers and of the geometry that created those numbers.  His ‘numbers’ were whole numbers.

Pythagoras gave numbers meaning and character.  Even numbers were held to be female, odd numbers male.  There were square, triangular, rectangular numbers, and certain numbers were ‘perfect’, even divine, beings.  His importance lies in the fact that he removed mathematics from the practical realm related to everyday life to the non-material realm, studying ‘number’ not really for its own sake, but to understand, as he and his followers believed, the nature of the eternal world.

  In this respect his most influential discovery lay in the realm of music.  It was well known that the pitch of a harp-string depended on its length, its tension, and its thickness.  What Pythagoras discovered, by experiment, was that the notes that sounded most pleasantly in harmony were created by string-lengths and tension weights that were related by whole number ratios, i.e. 1 to 2, 2 to 3, and 3 to4. 

Reduce the length of a harp string by a half, retaining its tension, and the pitch of its note rises to the octave.  Reduce the original length by a third and the pitch rises to the Dominant or fifth note of a scale.  Reduce it by a quarter and the pitch rises to the sub-Dominant, the fourth of the scale.  The same rules apply to pipes of different lengths, and they also apply, Pythagoras found, to the tension on harp strings, but in inverse proportion: i.e. increasing the tension produces a rise in pitch. 

 What impressed Pythagoras was the fact that human beings found the four notes produced by whole number ratios to be pleasantly harmonious.[4]  From this he deduced that the souls of human beings responded to this harmony because, he decided, the soul enjoys simple whole number ratios.  Building on this idea he deduced that the whole cosmos had been designed according to musically harmonious numbers and was “a vast musical instrument suffused with mathematical harmonies.”  Pythagoras’s concept of the universe eventually led to a cosmology in which the celestial spheres were created from musically harmonious numbers and in ‘musical’ proportions.  This concept was eventually to have a profound effect on Christian theology and architecture, as recounted by Wertheim (1997, pp. 30-32) among many others.

 

Plato, Aristotle and Euclid

  Pythagoras was an obsessively secretive scholar.  He left no writings, and the members of his sect were vowed to absolute secrecy.  Consequently his ideas had little general effect until Euclid brought some of them to Alexandria in about 300 BC and developed them in his famous system of geometrical logic.  It needs to be understood however that the study of geometry was regarded by Euclid as a purely intellectual pursuit.  It was reported that when a listener asked him what was the practical use of some theorem, he told his slave, “He wants profit from learning.  Give him a penny,  echoing the motto of the Pythagorean brotherhood: “A diagram and a step, not a diagram and a penny.” (Bronowski 1973, 163).

In about 400 BC, well before Euclid arrived on the scene with his intellectual exercises, Plato had developed Pythagoras’s more mystical aspect of geometry and number into a cosmology that saw the world of material experience as a mere imperfect shadow of an eternal and perfect world.

To Plato, mind, not matter, was the fundamental realm of life, and on this basis he developed a philosophy that he applied to every aspect of life.  A thousand years later his belief in a double universe, one perfect and eternal and real, but unseen; the other, the world as we experience it, an imperfect, ephemeral and unreal shadow of the real, was to become the fundamental belief of medieval Christianity. 

A corollary of this belief was an insistence that number and the geometry that created number were elements of the unseen true world, eternal, unchanging and perfect, to be found only as mental concepts.  They were therefore divine in origin, and consequently the study of mathematics was held to provide a channel through which the human mind could enter and interact with the mind of the Deity.

Platonism aroused little enthusiasm among the Greek intelligentsia who took more interest in the philosophy of Aristotle, c.350 BC, who, having studied under Plato, turned away from his mystical perfectionism, to develop a very different philosophy in which sense experience was judged to be the only source of knowledge.  Even Euclid’s works, developed from Pythagorean ideas, are totally free of mysticism, and the Roman Pliny’s early first century Natural History is eminently Aristotelian.

It was not until the third century AD, in Alexandria, where Plotinus founded the school of philosophy known as Neoplatonism, that Platonic–Pythagorean ideas became a powerful field of thought in Mediterranean culture.  This also happened to be the time of the rise of Christianity as a powerful force in the Roman Empire.  Consequently a fierce rivalry developed between what were in many ways very similar cosmologies.  In fact the growth of the Christian faith directed at heavenly perfection paralleled the revival of interest in Platonism by the Neo-platonists.  The difference lay between faith and reasoning, with Roman philosophers, including the scientists and mathematicians, regarding Christian faith as the enemy of reason and knowledge.  The conflict eventually led to the Pagan Revolt and the apostasy of the Emperor Julian, nephew of Constantine, who reverted to paganism and revived pagan worship, fully explored in Momigliano‘s The Conflict between Paganism and Christianity in the Fourth Century (1963).

St Augustine

The situation could have led to the destruction of both pagan and Christian culture and a thousand years of anarchy and barbarism. It was the great philosopher-theologian, Augustine (354-430), Bishop of Hippo, who was able to resolve the potentially disastrous conflict.  Son of a pagan father and Christian mother, he grew up as a pagan and made a prolonged study of Neo-Platonism, before he eventually converted to Christianity.

His great achievement was the development of a Christian theology that could accept Platonism; most importantly through an understanding of mathematical forms as the thoughts of God.  As part of this metaphysical synthesis he found numbers and geometrical forms to be imbued with meaning within the Christian faith. Padovan (1999, 175) relates Augustine’s theology directly to the works of Plato:

For Augustine, Plato’s account of the Creation in the Timaeus was a wonderful anticipation of Christian teaching.  With its doctrine that the ultimate reality is in the immortal world of ideas, and that the sensible world is its imperfect shadow Platonism was for him supreme among philosophies: “There is none who comes nearer to us than the Platonists” (Augustine. The City of God Penguin 1972, 304).

 

Sunderland, in her Symbolic Numbers and Romanesque Church Plans (1959) quotes some of Augustine’s writings:

 

Augustine: ‘If, in reading Genesis, you search the record of the seven days, you will find that there was no evening of the seventh day, which signified that the rest of which it was a type was eternal…hence the eighth day also will have eternal blessedness.  The eighth day, which is the first day of the week, represents to us that original life, not taken away, but made eternal'’(Works of Augustine, Letter LV, Chapter IX, 16 and Chapter X,17.) (Sunderland 1959, 94). This saying, “I have chosen you twelve”, may be understood in this way, that twelve is a sacred number because they were to make known the Trinity throughout the whole world, that is, throughout the four corners of the world.  That is the reason of the three times four (On the Gospel of John, Tract XLIX 8.) (Sunderland.1959, 94).

     

Padovan (1999) provides another typically Augustinian deliberation:

The works of Creation are described as being created in six days….The reason for this is that six is the sum of its parts, that is, of its fractions, the sixth, the half, and the third, for one, two, and three make six….Hence the theory of numbers is not to be lightly regarded, since it is made quite clear, in many passages of the Holy Scriptures, how highly it is to be valued.  It is not for nothing that it was said in praise of God; ’You have ordered all things in measure, number and shape.’

Augustine, The City of God, Penguin 1972 p.465 (Padovan.1999, 176).

 

St.Benedict

Augustine’s many writings were widely copied and became highly influential as the orthodoxy of the Church, particularly among its intelligentsia.  Those who had grown up as Neo-Platonists could now embrace the Christian faith without rejecting their intellectual heritage.  But it was in the following century, when Christian monasticism became a highly successful organisation under the direction of St. Benedict (354-430), that St. Augustine’s Neo-Platonic Christianity made its most profound impact upon the life of the Church.

Escape from the wickedness and temptations of ones social environment by living in solitude in the wilderness, was a path that has been chosen by many throughout the history of mankind.  It grew in popularity particularly in the early Christian period of the late Roman Empire. But isolation was highly precarious, and consequently many ascetics tried to organise themselves into groups sharing domestic life and security, while maintaining as great a degree of solitude and isolation from society at large as might be possible.  Such arrangements seldom met with any success until Benedict’s carefully considered Rule for an orderly way of life, the Life of Perfection, making allowance for human frailties, proved so successful that it was eventually followed by virtually everyone who wished to live a monastic life.  Benedict’s Rule was in fact designed solely for laymen, not for priests. Most of those who entered Benedictine monasteries came from the intellectually active, relatively well-educated members of society, and daily hours of study, essentially of a theological nature, were an essential part of monastic life.  Thus it was through the monastic houses of the Benedictine Order, founded in every far corner of Christendom, that Neo-Platonic Christian theology, the theology of Augustine, having been adopted by St. Benedict, became the basis of learning throughout the whole Benedictine order.  For centuries these monasteries were the only source of learning in Christendom.

Sunderland draws attention to the importance of certain numbers in the Rule of St.Benedict: 

There are four kinds of monks, seventy-two instruments of good works, twelve degrees of humility, seven canonical hours.  ‘As the prophet saith: “Seven times in the day have I given praise to thee.” And we shall observe this sacred number of seven if, at the times of Lauds, Prime, Tierce, Sect, None, Vespers, and Compline, we fulfil the duties of our service (Sunderland. 1959, 97).

 

Boethius

The next important theologian philosopher, Boethius (c.470-524), took great interest in the Pythagorean “musical “ ratios, and taught how to visualise them as geometrical forms.  He even extended into three-dimensional geometry such as the “geometrical harmony” of the cube, which has 6 sides, 8 angles and 12 edges, providing the 6:12, 8:12, 6:8 ratios of the harmonious notes of the musical scale according to Pythagoras.  He also endeavoured to harmonise Neo-Platonism with Aristotelianism, but his works in this field were suppressed on the grounds that his views denied the divinity of Christ, and for his pains he was imprisoned and put to death.  Boethius’ works found more acceptance in the Arab world, but remained unpublished in Latin until the Crusades of the twelfth century brought about the mingling of Christian and Islamic cultures. They were not studied seriously until the thirteenth, when they were expounded and developed by St.Thomas Aquinas, only to be banned by the Church for several centuries. 

However, the possible influence of Aristotelian thought upon medieval architecture is problematical.

An essential element of Augustinian Christian theology was the application of religious meaning to particular numbers and geometric forms. Some of these applications will be found in Mâle (1916, 10-14) and in Lesser(1957, 143-145), although it should be appreciated that all these symbolisms were liable to expansion, development and accretion by individual commentators.

“One” was the singularity of God. “Two” was the duality existent in all creation: light and darkness, good and evil, the New Testament and the Old, the Synagogue and the Church, this life and the Life to Come.  “Three” was the Holy Trinity, and thus became the representation of all things spiritual. (It is of interest that the equilateral triangle is the simplest of all geometrical constructions).  “Four” was the material world with its four corners and four elements.  It also represented the four evangelists, men of this world who, like the four rivers of Paradise, spread the Gospel to the four corners of the earth.  “Five” related to the five virtues, the five wounds of Christ, the five wise and the five foolish virgins, the five talents, the five loaves, and the five books of Moses.  It also symbolised the animal world, with its five senses, created on the fifth day of Creation. “Six” was the human race, created on the sixth day of Creation and thus associated with birth and incarnation, and the completion of the Creation.  It was also the Jewish synagogue, the Old Dispensation, the Church before Christ. “Seven” was the completion of the life-span of men and women; seven decades, (seven times the number of one’s fingers), that brought life to an end.[5]  Seven is also the sum of four, (the number for the earthly and material flesh), and of three, (the number of the soul). With the loss of the soul, the human being returns to the earth, to dust.  Human life consisted of seven ages each associated with the seven virtues and each governed by one of the seven planets.  There are seven sacraments and seven deadly sins (Male 1961, 11).[6]  In medieval geometry seven is usually exemplified by six hexagons surrounding a seventh.  “Eight”, coming immediately after Seven, the life of Man, was the Resurrected Christ, risen from the dead on the eighth day of the Jewish week, when every week began again, also when every musical scale began again. It is therefore the number for Baptism into the New Life and the Promise of Salvation, and the great majority of medieval baptismal fonts and baptisteries were octagonal (Figs. 3, 4).  In music, one discovered a vivid example of number symbolism in Sweelink’s variation’s for harpsichord on a sacred folk-melody,Mein junges leben hat ein end’, clearly related to funeral rites, in which the melody consists of nothing more than the eight notes of the downward scale.

The eight pointed star is frequently found, in several forms, in medieval and early Renaissance paintings relating to the Life of Christ, which in medieval terms meant those events that proclaimed His Divine nature, His Birth, Baptism, Crucifixion and Resurrection (Figs. 7,8,9). 

“Twelve” was the number of the apostles, and it was believed by the Neo-Platonic theologians that it was for profound mathematical reasons that Christ chose twelve as the number of his disciples.  Twelve is the product of three and four, and thus in the Twelve matter is infused with spirit, so that through the Twelve the Truth should be proclaimed to the world and Christ’s Universal Church established (Mâle 1961, 11).  A simple example of the geometrical symbolism of Twelve is to be found in the design of the twelfth-century monastic fishpond at the cathedral priory at Canterbury, which had twelve cusps, one for each of the Fishers of Men (Fig. 5).

The great majority of these numbers were exemplified by the geometric figures that created them.  But there were other special numbers that were not of geometrical origin. 153 was such a number, for it was the number of fish that were recorded in the gospels as being caught by the disciples when the resurrected Jesus advised them to put their net on the other side of the boat.  This number came to represent the whole Church of Christ (Bannister T. 1968).  It has been found to be a basic dimension in several great churches, including Old St. Peter’s in Rome (Bannister. T. 1968) (Fig. 6). Bannister writes:

In the same double square of the Vatican nave, the transverse width of 129.5 Ptolemaic feet converts to 153.24 Roman feet (Fig. 24-1). The number 153 is frequently noted by Early Christian writers as possessing mystical meaning because according to St. John XXI...153.   Multiples of 153 comprise several dimensions of this plan (Bannister T. 1968, 26). 

 

And later:

St. Augustine devoted many sermons elucidating 153.  It is the number of psalms (150) plus the Trinity (3), it is a triangular number, the sum of all the numbers up to and including 17

 

The present  author  has discovered 153 Byzantine feet as an important, in fact basic, dimension in Constantinople’s Hagia Sophia (Fig. 6b).[7]  (The measurements are taken from Mainstone,1988, 240).   It will be found to have a fundamental place in the geometry of Peterborough cathedral.

 

 Another symbolic number was 300. In Sunderland we read:

“Tau (or T) was the symbol for 300 in Greek and to the Christians T symbolized the Cross.  Therefore 300 symbolized the Cross. (Letter of Barnabus, Ante-Nicene Christian Library, IV, 117)” (Sunderland 1959, 99).

 

An example of the interpretation of 300 as a symbol of the Cross is given by Mâle (1961, 12), concerning the occasion when Gideon went forth with 300 companions.

It should not be imagined that mathematical symbolism was no more than an intellectual game with a sanctimonious flavour.   To quote Mâle:

“The middle ages never doubted that numbers were endowed with some occult power.  This doctrine came from the Fathers of the Church who inherited it from Neo-Platonic schools in which the genius of Pythagoras lived again.  It is evident that St. Augustine considered numbers as the thoughts of God” (Mâle 1916, 15).

 

One needs to bear in mind, however, that anyone can, and did, apply their own theological meaning to particular numbers and geometrical forms, and modern interpretations are often at variance with any medieval understanding.  One of the most well-known symbols attributed to Durandus, the deviation of the axis to be found in many churches supposedly symbolising the head of Christ falling to one side on the cross, is in fact the pious invention of his translators, J. Kneal and B.Webb (1843), written into their Preface.[8]  The true explanation of these deviations is given in M.D. Anderson’s The Imagery of British Churches (1955).

Religious number symbolism is to be found in other art forms, such as Bach’s religious works (Bertalot 1981), and the writings of Dante (Mâle 1961, 13), and Chaucer (Jordan, R. Chaucer and the Shape of Creation), while geometrical symbolism is to be found in many medieval paintings.  Sometimes the geometry is overt, either in the design of the framework, or in artefacts portrayed, usually in portrayals of the Madonna and Child, such as the tiles in Fig. 7 or the carpet in Fig. 8.

In Giotto’s portrayal of the Lamentation over the Body of Christ in the Arena Chapel, Padua (Fig. 9), I find unexpected but obvious geometrical symbolism in the border patterns.  The vertical borders aligned with the standing human mourners are constructed of ad triangulum hexagons, symbols of humanity created on the sixth day of Creation, also of the Church before Christ.  The horizontal border aligned with the recumbent body of Christ is made up of ad quadratum eight-pointed stars, symbols of the Divine Christ and His gift of New Life.  Any supposition that these many carefully drawn and painted geometric forms, found in great numbers in Padua and in Assisi, are no more than idle meaningless decorations is surely untenable.  The symbolism is patent.

In Fig.10 the geometry is hidden but suggested.  This portrayal of Christ as the Creator is interesting in that it is developed as eight successive circles, beginning with the circle of the world in creation and completed with the halo of divinity around the head of Christ as the Creator, eight being the number of the New Life in the Resurrected Christ. 

At other times the geometry, as in the Deposition by Roger van der Weyden  (Figs. 11, 12) is complex and covert, although a strong sense of geometricality pervades the composition.   Here, as befits the human race created on the sixth day of Creation, I find the non-divine characters framed within a hexagonal geometry demanded by the proportions of the picture frame, while the figures of the Divine Christ and His Mother are within the octagon geometry demonstrated by the angle of Christ’s body. The geometry also seeks out the five wounds of Christ for emphasis, most notably the horizontal wound in the side and the vertical wound in the right hand, clear examples of ad quadratum geometry.  The most remarkable aspect of this analysis is the geometry of the ladder, for it links the small octagon geometry of the cross with the large geometry of the body of Christ, connecting a particular point in the upper geometry with exactly the same point in the lower geometry.   Thus the deposition from the cross is conveyed geometrically as well as pictorially.[9]

Other examples of canonic Euclidean geometry are to be found in stained glass window designs such as the window at Canterbury (Fig. 13a).  The mosaic pavement at Canterbury is a quite remarkable example (Fig.13b).  Some of the most enlightening geometrical designs are to be found among the 9000 masons’ personal signs studied by the Viennese architect Franz von Rziha, 1000 of which he decoded and published in his Studien uber Steinmerz-zeichen (Rziha 1883), some of which, taken from Ghyka (1977, 122), are shown in Fig. 14.  The interest of these ‘signatures’ is that they could be scribbled like modern signatures, but a mason using such a sign could, if required, prove his ownership by geometric demonstration, which only the true owner could do (Pennick, 1980, 110).  Rhiza in fact found that all the south European signs were developed from four geometrical lattices, each of which belonged to a the masonic lodge of a particular city. A mason would not be required to draw the geometry of his sign but only to define it within the matrix of the lodge (Ghyka 1977 120).  At a time when few but the most senior masons could write their own names these signs were essential.  But furthermore I believe these signs to be of great importance for the subject of this present study, for, as will be shown, they are clear examples of geometrical constructions according to Euclidian rules.  They were designed and used by masons, and thus point the direction in which one must seek to find the geometry of the architecture that some of their owners designed.  Others, such as Jennifer Alexander in her Masons’ marks and stone bonding (Alexander 1996, 219-36), have studied and published many similar English marks, almost all of them geometrical, but discussed only in regard to the organisation of teams of individual stonemasons and their techniques.  The geometrical analysis of these marks remains essentially Rziha’s achievement, even after more than a century, and his brilliant work has never been seriously challenged.  The analysis of any one of his mason’s marks is an extremely difficult task and he appears to be the only person who has ever mastered the skill.  Sadly, the imputations of his discoveries have been overlooked, for it is inconceivable that men who would construct such geometical complexities for use as signatures would not be using similar constructions for their creative work.  There can be little doubt that in regard to complex city lattices, from which individual members were given their signs, the lattice itself would have to have been designed by the master of the lodge.

One important matter displayed by the geometries of the masons’ marks is that every example begins with a circle, and some involve the construction of circles internally, and here we are reminded of Schmuttermayer’s affirmation: “Fundamentally, this art is freely and truly planted and founded on the centre point of the circle, together with the circumference of correctly set point and construction.”  This all underlines the fact that medieval cathedral geometry always began with a circle, and this fact will be clearly exemplified in the geometry of Peterborough cathedral.

There will be those who question the need to begin any geometrical Euclidian construction with a circle, but it will be found impossible to begin any geometrical design except from an originating circle or part of a circle.

Similar geometries to those of the masons’ marks are to be found in musical contexts.  An illuminating study of geometrical number symbolism in renaissance lutes has been made by B.H.Wells (1981) (Fig. 15).  It will be seen that the geometries of their sound-holes are clearly of the same genre as those used by medieval master masons.  Music had always been regarded as a divine gift to mankind, and its performance a means of communicating with the divinities.[10]

The same may be said of theatrical drama, which in Greece certainly was originally religious in function, so that one is not surprised to find the plans of Greek and Roman theatres in Vitruvius designed upon sacramentally geometric figures (Fig.16).  In this area of the theatre, some quite remarkable geometric drawings (Fig. 17) were discovered by Joy Hancox who found them to be related to the original plans for the theatres of Elizabethan London (Hancox 1992).  These would appear to be the first theatres, in Britain at least, if not in Christendom, to have been purposefully designed and built since the fall of the Roman Empire, and in the heady days of the Renaissance in England it is not surprising that the theatre designers and entrepreneurs should seek inspiration and guidance in Vitruvius, illustrated by Palladio, which was by then freely available in print.

The sacramental importance of geometry to architectural design is by no means limited to the Christian world.  As we have seen, it played a vital role in Egyptian religious belief, and in Greek and Roman religious festivals in their theatres.  In Islam we find such buildings as the Dome of the Rock in Jerusalem planned upon an ad quadratum diagram (Fig. 19a), and in India the architect’s plan for the mausolem of the Taj Mahal is seen to have been designed on the same ad Quadratum octagon star motif (Fig. 18).  Islamic buildings also possess a great many tiles designed in ad quadratum geometry that create  carpets of geometry that could extend to encompass the universe (Fig. 19b).

The fundamental nature of sacramental geometry is most clearly expressed in the form of the Mausoleum of the Sassanids at Bukhara c.920 (Fig. 20).  The hemisphere of the heavenly world rests upon the cube of the natural world.   This arrangement is also the essential form of many early Christian Greek churches (Fig. 21), and the great church of Hagia Sophia in Istanbul is essentially a cube supporting an hemisphere (Fig. 22).  To build an hemisphere on an open square of four walls is extremely difficult, for in the corners of the square the dome is not supported.  Only at four points on the circumference is there any support.  In appreciating this fact one is led to realise that the dome on square design was not chosen because it was easy or natural, for it is a most difficult roofing system to build.  There must therefore have been a most compelling non-structural reason for choosing this difficult structure for so many churches, and the reason must lie in its potent symbolism.  One needs to emphasise the magically functional element of the symbolism, for the dome was not just a symbol of Heaven, it was the very form of Heaven as perceived in the sky.  It thus created a geometrical link between the geometry of Heaven and the geometry of the world, and thereby produced a sacred heavenly place on Earth. When one entered that place one entered a part of heaven.  It was sacred.

The problem of supporting the dome in the corners was solved by the construction of pendentives that are in fact segments of a larger hemisphere (Fig. 22).  The interior decoration of domed churches illustrated Biblical events that expressed not only in their subjects, but also in their placing within the architecture, the essential message of the Church.   At Daphni Christ is portrayed typically centrally in the dome (Fig. 23). .   Below him around the edge of the dome the saints or apostles are portrayed, while on the walls of the square church at ground level pictures portray events on earth that foretold the birth of Christ.  But in the four pendentive areas the pictures present the events that were concerned in the union of earth and heaven through such events as the birth of Christ, the Annunciation, the meeting of Mary and Elizabeth, the angels and the shepherds, the Magi at the manger, even the Flight into Egypt which was held to presage the Christ-child’s mission to spread the Word to the world of the Gentiles.  Thus the placing of pictured events within the geometry of hemisphere and cube demonstrates the essential theology of medieval Christianity.  

 

Whole Number Proportioning

There are those who see Number as the alternative to geometry in medieval architecture, and it would seem that there were in fact differences of emphasis at various times and places.  But geometry creates number, and although number can be expressed geometrically, number cannot create geometry.

Both number and geometry can and did exist in perfect harmony in the design of the great churches.  It does seem possible, however that Cistercian architecture, as will be shown, may owe its admired proportions more to numerical whole number ratios than to the irrational ratios created by geometry.  But sacramental geometry, as used in Benedictine churches, was not aimed at beauty, though it often arrived at it, but at harmonisation with the Divine Geometry of God’s universe.

 

Irrational Ratios

Geometry can create simple whole number ratios, but it also creates those strange ratios that express what are known as irrational or silent numbers that cannot be accurately expressed numerically, although they are extremely common and easily constructed in geometry. The most common is the square root of two, created by the diagonal of a square, which today one might take to be 1.414, although even this is only an approximation. In Roman numerals the usual approximation was seven fifths, which in modern decimals is 1.4.  But the Greeks could not accept that anything created by God or the Gods could lie outside the world of whole numbers, (which included fractions drawn from whole numbers).  It must, they believed, exist hidden within the mind if the Creator and the search for its true value must therefore lead one into His mind.[11]  Furthermore, if this “Divine Number” of Ö2 can be accessed simply by drawing the diagonal of a square, then this construction must be part of the geometry of the created cosmos.  Other similar “irrational” numbers, or surds, are the square roots of three and five, all easily produced as geometric ratios, but comprehended only by the Creator of the cosmos. As will be later demonstrated, this is a further reason why the constructions that produce these ratios play such a fundamental role in sacramental geometry.

The most mysterious ratio, however, was that between the diameter and the circumference of any circle.  Known as p, it has already been discussed in relation to the great pyramid of Cheops at Giza.  But p cannot be expressed even as a root, nor as a fraction of a root, or as a ratio between roots, and unlike the square roots of two, three[12] and five, it cannot be created geometrically except by drawing a circle.  Such a number is called ‘transcendental’ and again was regarded as truly Divine.  The same ratio of p is of course involved in the relationship between the radius and the area of a circle.  The search for it, which deeply concerned the Greek mathematicians, involved the construction of regular polygons inscribed within and circumscribed about any circle.  If, for instance, a square were to be inscribed within a circle and another circumscribing the circle, their areas could be calculated and averaged to give an approximation of the area of the circle.  If regular polygons of twelve sides could be drawn instead of squares the approximation would be much closer. Such constructions are closely related to those commonly to be found in examples of medieval sacramental geometry. Taken to the degree of a polygon with an infinite number of sides the concept would eventually lead to the Infinitessimals of Calculus.  

 

The Middle Ages

The essential belief that the world was an imperfect shadow of the real, perfect and eternal heavenly world, and that geometry and number came from that perfect world, created geometrically by God out of primeval chaos, became a central tenet of medieval religious life.  Contact with that world could be achieved only through the divine channels that God displayed to the men and women of this world; in particular music and geometry.

The musical aspect was dealt with through the central function of monastic communities, the Opus Dei, that demanded the unremitting singing of the praises of God in the Choir seven times through every day and night, while the geometrical requirements were invested in the geometry of the building in which .the singing was performed.

Through the centuries from about 500 AD onwards Augustine’s neo-Platonic theology was maintained and preserved in the Benedictine monasteries that were built over the whole of western Christendom.  In Britain, following the Norman Conquest, from 1070 onwards, Benedictine culture underwent an enormous expansion through the number and size of monastic houses that were founded or enlarged.  The explosion was driven by Benedictine zeal, directed by Augustinian theology, powered by Norman organisation and characterised by Anglo-Saxon artistry and individuality, for the many abbey churches and cathedrals built in Britain in the late eleventh century and early twelfth centuries were remarkable for the variety and originality of their architectural character.

At virtually the same time as the Norman Conquest of Britain, the transportation into Europe of many thousands of Saracen prisoners of war took place. They had been taken at the siege and capture of the Moorish city of Barbastro in Spain in 1064.  1500 were sent to France, others to Italy, and 7000 to the city of Constantinople (Harvey 1971, 16). Harvey also points out that in addition to the musicians and artists that were reported as being included in these numbers there must have been many military engineers and craftsmen far more advanced in their technical skills than those north of the Pyrenees.   In England the architect reputed to be King Henry I’s architect was a Saracen prisoner of war, Lalys, who is known to have built Neath Abbey. (Harvey 1971, 16).

Harvey (1971) points out that long before the momentous fall of Barbastro in 1064, throughout the Dark Ages in fact, there had been sporadic and local contact between the cultures of the north Mediterranean and the south-east Mediterranean. In the eighth and ninth centuries the Iconoclastic movement in Constantinople had led to a massive emigration of Greek artists and the religious into Italy.  Then in 1050 the Normans invaded Sicily and southern Italy, which had strong trading links with the Arab world.   Thus, by the time of the Norman Conquest, the culture of the Christian West had absorbed much of the culture of the Near East and as a result gave birth to a new culture that became the Middle Ages, with its new architecture, the ‘Gothic’ (Harvey 1971).  

 

The Gothic

Among the many contributions to the change were such developments as the more accurately cut masonry in larger blocks, the use of which was facilitated by the use of more sophisticated lifting machinery, probably devised by Moorish engineers, plus machines brought back from Jerusalem by returning crusaders after their victorious campaign in c.1100 (Harvey 1971, 24).   The new skills led to the solid thinner walls and to the pointed arch.  It has been shown that a few pointed arches had been built before the Crusades but only after the fall of Jerusalem on July 15 1099 did the pointed arch become general.  The ribbed vault, another basic element of the Gothic style was first seen in England on 4th of September 1104.  As Harvey surmises (1971, 24), it can hardly be coincidence that it happened only five years after the fall of Jerusalem.

The cultural change affected the geometry as well as the design and construction methods involved in church architecture, as will be demonstrated at Peterborough (Figs. 203 to 207), but the rules of the free art of geometry did not change.

 

The Geometry

It would be a mistake to suppose, as many have done, that the geometry that has been described as the very life-blood of the medieval architect (Lesser 1957, 156), was introduced into the design process in order to produce buildings beautiful to the eye, and that this was achieved through special proportions. The purpose of geometry in architecture was to unite with the eternal world of Heaven and thus to preserve the building from disaster. But this does not mean that it had anything to do with the statical structure of the building or with the method of construction.  No-one in the eleventh or twelfth centuries, including master-masons, bishops and abbots, was capable of even the simplest structural calculation, (see Leedy 1980, 23).  Shelby (1971)(2) writes:

Even the more sophisticated treatment of statics by the 16th century Spanish master Rodrigo Gilde Honstanon involves only the simplest mathematical reasoning and relies heavily on empirical evidence and, literally, rule of thumb exercises (238).

 

The permanence of the building was to be achieved by incorporating in its design a geometry that integrated perfectly with the geometry of the universe created by the Almighty.  As the Creator’s balance of forms maintained the stability of the universe, according to Pythagoras and Plato, so according to Augustinian belief, it would also preserve through geometry the stability of the church and protect it from the powers of evil.  Above all it needs to be understood that until the late Middle ages the churches were designed not for the eyes of human beings but for the eyes of Heaven.  If God could see the geometry, invisible to men and women, and it was good, then all would be well.  To build a House of God without His geometry would be vain, sacrilegious and an invitation to disaster.

It is important to appreciate that geometry was used specifically for the design of sacred buildings and artefacts.  Its purpose was to sanctify the building or artefact by making it part of the Heavenly world of the God who was always making geometry.  That does not necessarily mean that it was never used for any other type of building that might need Divine protection.  Roriczer relates it to “measured stone-work,” a term that could apply to some types of building other than churches.  Schmuttermayer uses the term “drawn-out stonework.”  An example of geometry in a secular English building is presented and examined in T.Heslop’s Orford Castle, nostalgia and sophisticated living (1991, 36-58), but it is relatively very simple.  It may well have been used however, in order to promote divine protection of the castle from the forces of evil.

Geometry has been shown to have been used, in combination with numerical measure, in a secular building in Italy c.1340.  Toker (1985) describes the procedure as follows:

We all know…that Renaissance architects did not give up the geometric schemata of their Gothic predecessors.  Alberti and Francesco di Giorgi in particular would begin their drawings in geometry, and only later transfer them into whole number harmonic relationships (n.35)… Only superficially similar to that of the Sansedoni architect, who did not wipe away his geometric schemes as he transformed it into numerical ratios but simply translated it into its nearest equivalent. ...the façade was conceived in the architect’s private language of geometry and executed according to the different language of arithmetic.  Not even the builders knew what the overall controlling scheme was (Toker 1985, 23 Note 35 Idem 161).

Nearly all Renaissance architects used geometric formulae at some point, as designers continue to do today.  In 1421 Brunelleschi used both the old geometric irrational value dimensioning system for the Old Sacristy of San Lorenzo and the arithmetic-modular system for the body of the church (D.Nyberg.‘Brunelleschi and the Use of Proportion in the Pazzi Chapel’ Marsyas VII. 1957. 1-7).

 



[1] Lesser (1957) asserts that the circle also represents the earthly horizon, but this notion is difficult to accept for it has no foundation in scholarship nor in actuality, for a perfectly circular horizon is virtually impossible to experience except from an aircraft or the top-mast of a ship.  

[2] Much has been made of the fact that the Egyptians were quite incapable of calculating the mathematical relationship between the square and the circle involving the p  ratio. But they could easily have arrived at it not by calculation but by putting a cord around a wheel and comparing its length with the wheel’s diameter, or simply by rolling the wheel along a flat surface.  From the diameter and the circumference halved they could obtain have obtained an angle to guide the construction of the pyramid that would have the necessary proportions (p. 16c).

[3]  Leonardo da Vinci’s well-known drawing of a man inscribed within both a circle and a square is closely related to this tradition.  Commissioned by the geometrician Pacioli for his book De Divina Proportione (Venice, 1509), it aimed to demonstrate, through geometry, the human being as both Human and Divine.  The two worlds are related in the drawing not, as often stated, by the Egyptian p ratio but by the f ratio of the Golden Section.  The concept and the geometry are clearly the work of Pacioli, not Leonardo.

[4] They are harmonious only in relation to the key-note and its octave.  The fifth and fourth are mutually inharmonious.  Strangely, the third note of the scale, at 4/5 the length of the string, and only a semitone below the fourth, is very pleasantly harmonious with the key-note, the fifth and the octave, but is never mentioned in literature on the Pythagorean musical ratios

[5] A distinguished surgeon informed the writer that the number seven had a special meaning in hospitals.

[6] The regular seven-sided heptagon is a curious phenomenum.  Hudson (1915) describes it as impossible to construct geometrically.  Roriczer (Shelby 1977) provides a geometrical construction, but it is only an  approximation.  A simpler and  much closer approximation, is given in an Appendix.

[7] In 1510 John Colet, Dean of St. Paul’s, founded St.Paul’s School for up to 153 boys. (Coulton 1961, 11). He stressed the importance of this number.  At Canterbury in 1170 Lanfranc designed his monastery for 150 monks, which, with himself as Abbot and with his Confessors would have made a community of 153.

[8] The true cause of this phenomenum, which can tend either to the north or the south, and which is much more frequent and obvious in more northern latitudes, is related to the rebuilding of choirs separately from their naves, for the new choirs were realigned, not with the modern East, for there were no means of knowing it, but with the rising sun, the direction of which varies from day to day. It rises more northerly in the summer, more southerly in winter, and more so in more northern latitudes (Anderson 1955, 38).  Thus the realignment would often vary from that of the old church.  The directions of the realignments vary, some to the north, some to the south.  The notion that realignments took place on the patronal day has been disproved (Anderson 1955).  At Canterbury Cathedral, two rebuildings took place, the first being Ernulph’s Choir, c.1100, the second the new Trinity Chapel of 1179, both on new foundations and each providing a new alignment in relation to the nave, which was rebuilt c.1400 on the original foundations.  It occurs to me that as building work began then in April, (Knoop & Jones 3)after spring sowing and with improving weather, that was when Easter, the movable Feast that celebrated the Resurrection of Christ occurred.  It would seem a likely occasion for orientating one’s new choir with the rising sun. 

[9] A different analysis is presented in Bouleau (1954) but like virtually all the analyses in that publication, it is totally unconvincing.  Bouleau makes every effort to base his analysis on  pentagons and golden sections, but the number five can only relate in this painting to the five wounds of Christ crucified, yet none of Bouleau’s lines pass anywhere near any of His wounds, one of which, in the

 

side, lies horizontally, while that in his right hand lies at 45 degrees, clear indications of ad quadratum geometry.

[10] “Paintings in the Escorial include a portrayal of David exorcizing Saul by means of music.  According to hermetic belief this kind of exorcism was achieved not by any ‘virtue’ in the sounds themselves but as a result of their connection with the harmony of the celestial spheres, which in turn Fig .8). reflects the harmony of the supercelestial world of God.” (Renee Taylor, 1967.‘Architecture and Magic.’ Essays presented to Rudolf Wittkower, ed. D.Fraser, H.Hubbard, & M.J.Levine. Phaidon. p. 84,

 

 

It was said that one Greek philosopher who proved that such a quantity was inexpressible or irrational was taken out to sea     and drowned, for he was held to be promoting a belief that the Deity was irrational, and such a faith could invite divine wrath.

     [12] A number of constructions have been published, but all prove to be inaccurate.