In 1996 Michael Bispham posted a portion of this essay on the space@intuition.org e-mail list and i was so impressed by it that i asked him to allow me to archive it here. He consented and i am glad to give it space. The illustrations for this article were selected from the "Early Gothic" image archive at Tulane University that accompanies Professor Hugh Lester's courses Period and Style for Designers, I and II. Specific information on the images (place, date of construction, etc.) can be found on my Art and Illustration Credits page.


PLATONIC ATOMISM IN MEDIEVAL DESIGN

By Michael Bispham


TABLE OF CONTENTS

  • Foreword
  • An Introduction to Geometric Design
  • Classical Geometric Design
  • A Specific Example: The Use of Root 3, or Hexagonal Grids
  • Sources and Early Development of 'Gothic' Architecture
  • The Parallel Rise of Intellectual Life: The 12th Century Intellectual Revolution and the Cathedral Schools: Chartres
  • The Commissioners and Builders
  • Medieval Atomism and the Theological Difficulties
  • Ad Triangulum and Ad Quadratum: The Hidden Lattices
  • Conclusions
  • End Note
  • References
  • Reading List
  • FOREWORD

    This is an introduction to some ideas which may, if ultimately found to be valid, offer useful new perspectives on the period of the first widespread exercise in Western intellectual life.

    I will suggest that within the techniques that were developed to accomplish the designs of the great cathedrals, were references to the designers' scientific ideas of the Nature of Creation. My theory suggests that some of the great cathedrals were modelled on the 12th century conception of the form of matter. [1]

    Michael Bispham

    AN INTRODUCTION TO GEOMETRIC DESIGN

    The concept of geometric design will probably be unfamiliar to many readers. There is however a good range of texts available, which explore the geometry of historic buildings and detail the mathematical co-ordination within them. An example of an attempt at analysis might be Jay Hambidge's "The Parthenon and other Greek Temples." Architecture is by no means the only field. Similar, and in my view more precise work, has recently been published in the field of musical instrument design of the Renaissance: Kevin Coates' "Geometry and the Art of Lutherie" is an excellent example. Observations of very precise forms of geometric design have been made in many fields; mostly from the Ancient World and Renaissance period, but also from Medieval work, and particularly from the greatest geometric designs of all time, the Gothic cathedrals.

    Authors frequently observe that mathematical and music theory has been expressed as dimension. Irrational proportions in the form of the whole number roots, roots 2, 3, and root 5+1/2, the so called golden proportion, are often observed. And the designer's wish to echo, or emulate the 'mathematical nature of Creation' is frequently claimed.

    Unfortunately, besides the many well researched books and articles, the area is awash with speculation spilling often into mysticism. 'Sacred Geometry' seems to penetrate every other 'New Age' revelation. There needs to be, as in any ill-defined field, a separation of speculation from valid exploration, before any proper evaluation can begin.

    The subject of geometric design seems, due to these factors, to have been academically neglected. Nevertheless there is in my view, and that of more qualified persons, a real and highly informative subject waiting to be properly explored. There are good grounds for believing that artists and craftsmen did indeed often try to express through their work, something of their beliefs in the mathematical nature of Creation. Many authoritative commentators have drawn attention to the almost continuous high regard given to mathematics and geometry by scholar/philosophers throughout the Medieval period. There are as yet however, gaps in our understanding of how exactly Creation is seen to be ordered.

    For the sake of clarity in a very clouded field, this work addresses only those examples of underlying geometric design that are relevant to the author's hypothesis: that the scientific belief in the atomic nature of matter was expressed within the designs referred to.

    CLASSICAL GEOMETRIC DESIGN

    Most students of the History of Architecture would agree that in those parts of the world that inherited the Classical artistic traditions, there is a persistent and clear trend toward both 'ordered' and 'harmonious' design. In the case of Gothic architecture, a path can be traced which follows the classical Greek traditions through Orthodox and Islamic culture, to its magnificent rebirth as 'Gothic' architecture in the towns around Paris in the early 12th Century. This being so, it will be instructive to look very briefly at certain aspects of ordered design, to try to begin to visualise the very different approach taken by the Ancient builders.

    Much work in the field of geometric design has centred on Classical architecture, and the use of the Classical Orders. The orders focus the attention of both designer and student, not only upon the clear differences of the Doric, Ionic and Corinthian orders, but also upon the approach to design which is conferred by the use of these systems. Firstly the emphasis on proportion. The major dimensions are mostly given in terms of set multiples of a fixed unit, normally the diameter of the columns. The weight of the building is thereby controlled and fixed in both structural and aesthetic terms. Structurally, an architect can operate on the basis that a proportional system which is empirically tested will be applicable to newly designed structures which adhere to the proportional rules of the order. (Note that this applies only to buildings up to a certain size, beyond which a more massive structure is needed.) Aesthetically, an ordered building will harmonize in proportional terms, both within itself, and as part of the wider architectural environment.

    Students of geometric design will note a second feature; the use of geometrically rather than arithmetically defined dimensions. This is an integral feature of ordered architecture, and an understanding of the reasons for it is essential. These will become clearer as the essay progresses, but for the moment it is useful to observe a few points:

    A) A building which is designed geometrically at the small scale design stage can be laid out, and the construction controlled, using exactly the same geometric steps at full scale.

    B) A building which is designed geometrically has at least the potential for an inherent harmony imparted by the very process of taking its proportions from simple geometric figures.

    C) At neither the design nor building stages need anyone have any but the most basic numerical or arithmetic skills. This point perhaps needs a little more emphasis given the near universal reliance today on numerical quantification. In the 500 or so years since the moving point decimal system became known, we have moved to a position from which we can barely conceive quantification without it. The effort to do so is, however, necessary; until comparatively recent times, architecture was a geometrically rather than arithmetically quantified occupation. Try to imagine: You have been asked to design and build a temple in traditional fashion. The first thing you must do is throw away your tape measure! You will have to learn to think in a completely new way, one in which feet, yards, metres play a greatly reduced role. The geometric system you will use however, is an intimate part of a way of thinking, encompassing not only architecture, but the foremost and most basic thinking of contemporary science and metaphysics, with roots in deep antiquity.

    A SPECIFIC EXAMPLE:
    THE USE OF ROOT 3, OR HEXAGONAL GRIDS

    My interest in geometrically ordered design was triggered not by architecture, but by a curious feature of certain paintings. As several well researched books have adequately demonstrated, many paintings, from antiquity to the modern period, can be shown to have been composed upon hidden geometric figures. Despite the fact that there are no contemporary records, recognition that this practice did in fact take place, has in recent years become accepted by many scholars in the various fields of Art History.

    My own observation of hexagonal grids underlying the Trinity with the Virgin and St. John by Masaccio, and The Nativity by Durer, encouraged me to seek an explanation. This led to the observation of a similar scheme which directed the design of Milan Cathedral.

    Called the Ad Triangulum, or 'on the triangle' system, the scheme directs the placing of the elements of architecture by reference to a guiding geometric figure, hexagonal in character. Little is known of Ad Triangulum, nor of its sister, Ad Quadratum, 'on the square'. However, existing buildings as well as contemporary sketches seem to bear out the existence of the systems.

    The discovery of these design systems raises a new set of questions. Was their purpose simultaneously structural, practical, and aesthetic, in the tradition of the Classical orders, and if so, exactly how in each case? And what was the thinking behind the transfer of this practice to the fine arts?

    An early proposal might suggest that the use of such a scheme would co-ordinate all the diverse elements of the design, bringing them to a harmonious unity. Indeed, there is every reason to accept this scenario. It is not however in my opinion, yet complete. A closer look at the systems Ad Triangulum and Ad Quadratum shows a further element common to both systems, and to the fundamental basis of the science of their day.

    I will be rightly accused at this point of muddling my periods; one moment we are discussing 12th century cathedrals, the next 15th and 16th century paintings. I hope, however, to be able to show that the common factor which produced the phenomena, a belief in the concept of Atomism, constant throughout this time.

    SOURCES AND EARLY DEVELOPMENT OF 'GOTHIC' ARCHITECTURE

    The early development of Gothic architecture is known to have taken place in the area roughly described by a fifty mile radius around Paris. It is here that we must look for information about the impetus, purpose, and means by which Gothic architecture took form. This is not, however, an essay on the development of Gothic architecture, and I will restrict myself to brief observations in three main areas.

    * The sources of inspiration and technical knowledge that enabled the construction to take place.

    * The parallel awakening intellectual life centred at this time on Paris and surrounding towns.

    * The role played by the School of Chartres.

    Awareness of what I will cautiously describe as more highly civilised cultures had been taking hold in the prosperous and settled area of Northern France, as well as much of the rest of Europe, over most of the late 11th and early twelfth centuries. This was, at least in part, due to returning crusaders' accounts of vast and splendid cities, high culture and learning in the Byzantine and Islamic lands, visited for the first time by the Northerners. Contact too with the Islamic centres of learning in southern Spain, especially with the schools of Cordoba and Toledo, bought new perspectives to the curious few who undertook the journey. Columns in the vaults of Canterbury Cathedral, for instance, with their distinctive geometric patterns, bear witness to the Eastern influences, with roots in their turn in ancient Greece. Whether these designs were copied from originals in Constantinople or Cordoba perhaps, or were made by craftsman imported from those countries, is unknown. But the influence is unmistakable, and confirms the Old World as the source of inspiration and technical knowledge.

    THE PARALLEL RISE OF INTELLECTUAL LIFE:
    THE 12TH CENTURY INTELLECTUAL REVOLUTION
    AND THE CATHEDRAL SCHOOLS: CHARTRES

    Contemporary with the first experiments in the grand new architecture, was the explosion of intellectual endeavour that has been labelled the Twelfth Century Intellectual Revolution. Centred on what would shortly become the University of Paris, the spirit of enquiry and learning took hold rapidly, and spread to the surrounding towns. It is important to realise that this endeavour was occurring within the Cathedral Schools. The vast buildings we now know as cathedrals did not yet exist; their sites were the administrative centres of the Bishops and Archbishops, and had for several centuries contained schools for the purpose of educating the clergy. Throughout the latter part of the 11th century, a trend toward expanding these schools into centres of learning had begun, as demand for educated professionals grew in the fast expanding bureaucracies of church and states, and in the increasingly specialised demands of commerce. Scholars and teachers of repute were drawn together, and a new age of learning, of letters and of curious enquiry began. The point may be again be emphasised; the coming construction of the great new churches took place in the same places, and at the same time, as the scholars of the Cathedral schools were coming into close contact with the new ways of thinking. The same period saw the establishment and rapid growth of the first universities. New and more complete translations of texts from ancient authors became available on a regular basis, and as their knowledge of the ancient world increased, the 12th century scholars became ever more eager to learn more, both of that world and its ways of thinking, and of its relationship with Biblical history.

    Special attention has to be given the first great experiment in massive Gothic architecture. The cathedral at Chartres was the seed from which Gothic architecture grew; the roots of inspiration and technical know-how were developed there, with help from the speciality of the cathedral school. For two hundred years, Chartres had been the European centre for studies in geometry. Projective geometry, the practical arm of the subject, which enables the angles and real lengths of the pieces of masonry to be calculated, had been mastered and adapted to local needs. 'Drawn out' stonework had arrived, enabling the process of exploring the potential and the limits of compressive architecture to begin.

    THE COMMISSIONERS AND BUILDERS

    Contemporary records show the structure of the commissioning, financing, and construction processes, allowing a reasonable picture of the design process to emerge. The building work was carried out by stonemasons under a master mason, but the details of the design were provided by the clergy; the head theologians supplied the objectives, the major themes, the details of subject, statuary, paintings. The design was arrived at from a combination of technical, theological, and philosophical considerations.

    As massive Gothic architecture developed and spread, it is likely that a certain separation of roles took place. In the early years, and especially at Chartres, it is seems certain that the project was very much a joint affair; theologian/scientist/philosopher/technician were often embodied in one man; together they provided the design. And in the grandiose project, the designers tried hard to express and communicate, as well as their historical, biblical and spiritual knowledge, their technical and scientific knowledge. They sought, for posterity and the glory of God, to embed in stone their most cherished beliefs.

    MEDIEVAL ATOMISM AND THE THEOLOGICAL DIFFICULTIES

    Throughout the Medieval period, many students of Natural Philosophy accepted uncritically the atomistic model of matter. [2] Plato's Timaeus, available, if incompletely, throughout this time, provided a detailed model both of the act of creation, and of the mechanics of matter. It is necessary at this point to take a detailed look at some aspects of the nature of this model.

    In the Timaeus, Plato observes the geometric fact that five and only five regular geometric solids are possible. He goes on to allocate to four of these solids the four theoretical elements: air, Plato proposed was made from particles taking the shape of the octahedron; fire from the tetrahedron; particles of earth took the shape of the cube; to water he ascribed the icosahedron. To the fifth regular solid, the dodecahedron, Plato allocates the structure of the cosmos. He then proceeds, perfectly accurately, to observe that each of the solids can be built in turn from an accumulation of just two triangles. The first is an isosceles triangle, composed of one right angle and two equal sides- the result of dividing a square in half diagonally. The second triangle is the result of dividing an equilateral triangle in half. These are the ultimate atoms from which, according to Plato, all matter is constructed.

    This is how Plato puts it in Timaeus:

    In the first place, then, as is evident to all, fire and earth and water and air are bodies. And every sort of body possesses solidity, and every solid must necessarily be contained in planes; and every plane rectilinear figure is composed of triangles; and all triangles are originally of two kinds, both of which are made up of one right and two acute angles; one of them has at either end of the base the half of a divided right angle, having equal sides, while in the other the right angle is divided into unequal parts, having unequal sides. These, then, proceeding by a combination of probability with demonstration, we assume to be the original elements of fire and the other bodies; but the principles which are prior to these God only knows, and he of men who is the friend of God. And next we have to determine what are the four most beautiful bodies which are unlike one another, and of which some are capable of resolution into one another; for having discovered thus much, we shall know the true origin of earth and fire and of the proportionate and intermediate elements. And then we shall not be willing to allow that there are any distinct kinds of visible bodies fairer than these. Wherefore we must endeavour to construct the four forms of bodies which excel in beauty, and then we shall be able to say that we have sufficiently apprehended their nature. Now of the two triangles, the isosceles has one form only; the scalene or unequal-sided has an infinite number. Of the infinite forms we must select the most beautiful, if we are to proceed in due order, and any one who can point out a more beautiful form than ours for the construction of these bodies, shall carry off the palm, not as an enemy, but as a friend. Now, the one which we maintain to be the most beautiful of all the many triangles (and we need not speak of the others) is that of which the double forms a third triangle which is equilateral; the reason of this would be long to tell; he who disproves what we are saying, and shows that we are mistaken, may claim a friendly victory. Then let us choose two triangles, out of which fire and the other elements have been constructed, one isosceles, the other having the square of the longer side equal to three times the square of the lesser side. Now is the time to explain what was before obscurely said: there was an error in imagining that all the four elements might be generated by and into one another; this, I say, was an erroneous supposition, for there are generated from the triangles which we have selected four kinds- three from the one which has the sides unequal; the fourth alone is framed out of the isosceles triangle. Hence they cannot all be resolved into one another, a great number of small bodies being combined into a few large ones, or the converse. But three of them can be thus resolved and compounded, for they all spring from one, and when the greater bodies are broken up, many small bodies will spring up out of them and take their own proper figures; or, again, when many small bodies are dissolved into their triangles, if they become one, they will form one large mass of another kind. So much for their passage into one another. I have now to speak of their several kinds, and show out of what combinations of numbers each of them was formed. The first will be the simplest and smallest construction, and its element is that triangle which has its hypotenuse twice the lesser side. When two such triangles are joined at the diagonal, and this is repeated three times, and the triangles rest their diagonals and shorter sides on the same point as a centre, a single equilateral triangle is formed out of six triangles; and four equilateral triangles, if put together, make out of every three plane angles one solid angle, being that which is nearest to the most obtuse of plane angles; and out of the combination of these four angles arises the first solid form which distributes into equal and similar parts the whole circle in which it is inscribed. The second species of solid is formed out of the same triangles, which unite as eight equilateral triangles and form one solid angle out of four plane angles, and out of six such angles the second body is completed. And the third body is made up of 120 triangular elements, forming twelve solid angles, each of them included in five plane equilateral triangles, having altogether twenty bases, each of which is an equilateral triangle. The one element [that is, the triangle which has its hypotenuse twice the lesser side] having generated these figures, generated no more; but the isosceles triangle produced the fourth elementary figure, which is compounded of four such triangles, joining their right angles in a centre, and forming one equilateral quadrangle. Six of these united form eight solid angles, each of which is made by the combination of three plane right angles; the figure of the body thus composed is a cube, having six plane quadrangular equilateral bases. There was yet a fifth combination which God used in the delineation of the universe.

    As stated above, until about the middle of the twelfth century, the Platonic model of the geometrically atomic nature of matter was widely accepted, albeit with a history of friction with theologians. From about 1150, however, Atomism began to cause problems. Rome decided its interests were best served by propagating an abridged version of the scientific ideas of Aristotle, who had taken a critical view of Atomism. As the century progressed, and the need for an official view became increasingly desirable, Plato's Atomism was rejected; publication and teaching of Atomism was discouraged. There is much that could be written about the details of this process, and the change of scientific and philosophical direction it engendered. A large corpus of knowledge, informing a complete and self-contained cosmology was, if not destroyed, driven underground. Further, the spirit of free enquiry, and the first attempt to formulate a rational scientific process were stillborn. Dry Scholasticism was not interested in finding new answers to old questions; it sought instead new questions to support the answers it already had. Several writers have observed that the germ of modern science existed in the ideas and methods of the 12th century scholars who called themselves 'moderni.' One could usefully detail the process by which the needs of the Church took precedence over the infant science, and speculate about the likely reaction of scholars ordered to abandon and deny their cherished ancient cosmology. For the purposes of this work however, to say that Atomism was abandoned due to theological difficulties will have to suffice.

    It should be noted that the School of Chartres was one of the outstanding defenders of early science; Thierry of Chartres along with William of Conches for instance stand out as defenders of what were becoming heretical ideas; freedom of enquiry and Atomism. We should note too, that the controversy took place in the very schools attached to the cathedrals; and, given what we know of the role of the senior scholars in commissioning and defining the new buildings, we can observe that the probable supporters of the old cosmology were in a perfect position to ensure that their deep beliefs were timelessly embedded in the stones of the new architecture. [3]

    AD TRIANGULUM AND AD QUADRATUM:
    THE HIDDEN LATTICES

    As noted above, there were two design schemes, Ad Triangulum and Ad Quadratum, which directed the designs of at least some cathedrals. Furthermore, as we have seen, at least one of these schemes seems to have been used in painting design also. We can now show how the two schemes can be directly related in both angular and proportional terms, to the models of atomic structure provided by Plato.

    An Ad Quadratum scheme applied to a three dimensional building will take the form of a regular series of tightly packed, invisible cubes; placed side by side, each layer directly above the one below. [4] The resulting cubic lattice will form an underlying guide to the structure: simple multiples of the side dimension will provide the major dimensions, simple divisions will define and place the smaller elements, the ornamentation etc. In this way the entire building is bought into harmony, 'each smallest detail a harmonious part of a greater whole.' Reductions are also likely to be found in the ratio of root 2, the diagonal of the square.

    Use of the Ad Triangulum scheme will achieve exactly the same objectives in a similar way. The only difference will be that the lattice will be hexagonal, composed of closely packed equilateral triangles, and, and in order to best maintain harmony, proportional reductions are likely to be in the ratio of root 3. [5]

    It is perhaps difficult to visualise the systems, and the effect of their use. (I hope to be able to make graphic illustration available at a later date). It may be useful to note that both schemes result in lattices which match exactly those formed by close-packed spheres. (There are three ways of arranging spheres; Ad Quadratum and Ad Triangulum correspond exactly to two of them).

    In construction terms, these lattices would provide references, datum points, throughout the building. In terms of visual aesthetics, the scheme brings visual harmony. But, and this is the crucial level, in philosophical terms the scheme achieves cosmological harmony; it brings total unity to all facets of the building, aligning the building in perfect harmony with Creation itself, on both atomic and cosmological levels. The Cathedral will fit, with absolute perfection, into the Divine fabric, as revealed by Platonic cosmology. Even the statuary and frescos would fit the Grand Design. The builders achieve an act of creativity which simultaneously dovetails with, and echoes, the act of Creation itself. [6] One can now bring to light the meaning of Pugin's mysterious statement, which I will have to paraphrase as I cannot locate the quotation, that the design should match the fabric of the materials. Many times I have read descriptions of Cathedrals as crystalline in appearance. It now seems possible that they were deliberately designed thus; emulating the architecture of Creation itself, modelled on that ancient magical manifestation of purest matter. The cathedrals are perhaps almost literally, man made jewels.

    CONCLUSIONS

    First to tidy up my loose ends. Masaccio, helped by the architect Brunelleschi, maintained in his seminal work what was by the early 15th century a craft tradition. Likewise Durer, who explored in his woodcuts several geometric schemes. Atomism remained 'underground' until revived by Kepler in his essay "The Snowflake," written in the early part of the 17th century. To those who cry "conspiracy theorist," I can only ask for a more realistic way of looking honestly at the mechanisms by which heretical beliefs are maintained, in the face of great pressure to abandon them.

    There is much detail to be filled in; I have given here only a minimal outline, and have necessarily been guilty of several gross generalisations. Many aspects of this set of ideas need chapter length treatment, in particular details of the exact fate of Atomism. There are' furthermore, several related areas which may offer useful support to the thesis. The important role of music, for instance, to both medieval cosmology and ecclesiastic ritual within the buildings could, for instance, be usefully elaborated. I will finish however with an observation that, regardless of the validity of this work, the cathedrals today receive very little recognition as artifacts of historical scientific and technological achievement. The romantic vision of almost miraculous construction propagated by their caretakers emphasises the undoubted religious enthusiasm of their creators, at the expense of the reality of that creation. Of the synthesis of ways of thinking that produced them, and of the demise of the philosophy that enabled their construction, little is spoken. The cosmology derived from Plato's Timaeus which had, throughout the Medieval period, been the foundation stone of Natural Philosophy, was as responsible for their existence as the religious beliefs that funded and decorated them. At the time, the two impulses were perhaps closer than ever, before or since, to being united as one single belief. [7] In all historical fairness, the buildings ought be viewed as at least as much a monument to early theoretical and cosmological science, as they are currently seen as monuments to religious dedication.

    It would of course be unreasonable to claim any authority whatsoever for this thesis. It is at this stage no more than an intriguing set of ideas which may offer a new perspective to several aspects of history. I can hope only that the theory will be taken seriously, and that anyone in a position to test aspects of the theory will consider doing so.

    END NOTE

    It is now, 17th July 1996, about a year since this essay was written. During that time I have continued my inquiries, as ever on a part time basis. Although I stand by the main thrust, I would, if I were writing now, make some changes; there are several areas that I could update. I will mention here that I have been particularly bolstered by reading "The Byrom Collection" by Joy Hancock.

    Although I feel that Hancock's understanding of much of her material is incomplete, I think it is likely to be a) genuine, and b) very supportive of my theory. (Hancock tries to elucidate the purpose of a collection of ~16th century cards containing intricate geometric designs. Some cards clearly relate to architecture, and among them are geometric studies of the Platonic Solids.)

    I plan in future to look in some detail at aspects of Islamic science and architecture, from whence the traditions were likely introduced. It seems to me that I am coming closer to understanding the intentions of the designers, but there is still a way to go. Any help would of course be most gratefully received. As I said above, I can hope only that the theory will be taken seriously, and that anyone in a position to test aspects of the theory will consider doing so. I'd like to add, that if anyone should wish to fund 'me' to do so, I would of course be extremely happy!

    In order to keep this document as brief as possible, I have included few references and only a shortened reading list, rather than a full bibliography. I will be happy to try to supply more information on request, and to correspond with interested parties.

    Michael Bispham

    REFERENCES

    1
    According to S. Toulmin & J. Goodfield in "The Architecture of Matter": "the medieval conception of matter was broadly 'Platonic geometric atomism." [back]

    2
    Tina Stiefel, "The Intellectual Revolution of Twelfth Century Europe", (St. Martin's Press, 1985), quoting William of Conches (c.1080-c.1154) : "And he [Plato] shows the basic material of the world, that is, the four elements, why four have been created and not fewer, he explains how the world has been constructed from such matter, constructed in such a way that no matter exists beyond it". Also, quoting Thierry of Chartres (c. 1100-1150): "When Moses, for example says 'in the beginning...' what he literally means is is the first thing that God created was actually matter- the basic substratum of things, that include four elements: fire, air, earth and water. Afterwards by a natural process, matter took on distinguishable forms and in these forms the four elements were put in motion." [back]

    3
    Ibid: "Thierry's view of the basic importance of the quadrivium is echoed in the manner of the depiction of the sciences on a portal of the cathedral at Chartres, which was under construction during Thierry's chancellorship." [back]

    4
    See "An Encyclopedia of Architecture," G. Wilt (1847) or "A History of Architecture," G. Wilt, updated by Wyatt Papworth). Wilt supplies a close observation of Gothic geometric design, and notes that the nave of Amiens Cathedral is built upon a cubic lattice, of side 23' 6". [back]

    5
    Caesariano's woodcut of the Ad Triangulum design of Milan Cathedral can be found in "The Cathedral Builders" by Jean Gimpel, and "Sacred Geometry" by Robert Lawlor. [back]

    6
    John James: "The contractors of Chartres" on medieval masons: "Their search for God was in one aspect a search for order. Geometry could express this order more convincingly for them than any other medium except music." [back]

    7
    From "Encyclopedia Britannia": 'Thierry succeeded his brother, the celebrated Platonist Bernard of Chartres, as Chancellor and Archdeacon of Chartres. His unpublished "Heptatechon" contains his cosmology, mainly expounded in his commentry on Genesis, [which] attempts to harmonise scripture with Platonic and other physical or metaphysical doctrines.' [back]

    READING LIST

    Note: For further details on many of these books, including publishers, dates, page counts, ISBNs, and short reviews, see the Sacred Landscape Bibliography

    Alberti: On the Art of Building (translation Cambridge, Mass, 1988)
    Bag, A. K.: Maths in Ancient & Medieval India
    Bartlett, R.: The Making of Europe
    Bailey, C.: The Greek Atomist & Epicurius
    Blair, L.: Rhythms of Vision
    Blunt, A.: Artistic Theory in Italy
    Bouleau, C.: The Painter's Secret Geometry
    Brehier, E.: The history of philosophy vol. 3
    Bridges, J. H.: The Life & Work of Roger Bacon
    Brook, J. H.: Science & Religion: Some Historical Perspectives
    Brown, Peter: The Cult of the Saints
    Brumbaugh, R.: Plato's Mathematical Imagination
    Brunes, T.: The Secrets of Ancient Geometry
    Burckhardt,T.: Sacred Art in East & West
    Bury, J.: A History of Freedom of Thought
    Carpenter, R.: The Architects of the Parthenon
    Cassirer, Ernst: The Individual and the Cosmos in Renaissance Philosophy
    Cassirer, Ernst: The Philosophy of Symbolic Forms
    Charpentier, L.: Chartres Cathedral
    Cordingley, R. A.; Tiranti, A: Normands Parallel of the Orders of Architecture
    Cornford, C.: In Defence of a Preoccupation
    Cornford, F.: Plato's Cosmology
    Coates, K.: Geometry, Proportion and the Art of Lutherie
    Courtenay, William J.: Nominalism and Late Medieaval Religion
    Cramb, Ian: The Art of the Stonemason
    Crane, Lief: Peter Abelard
    Christianson, (may be Cristianson) G.: In the Presence of the Creator
    Critchlow, Kieth: Order in Space: Islamic Patterns, an Analytical and Cosmological Approach
    Crombie, A. C.: Augustine to Gallilao
    Crombie, A. C.: Mediaeval and Early Modern Science
    Crump, C. G. and Jacob, E. F.: The Legacy of the Middle ages
    Christie-Murray, D.: A History of Heresy
    Dampier-Whetham, W. C. D.: A History of Science & its Relations With Philosophy & Religion
    Dantzig, T.: The Bequest of the Greeks
    Doczi, Gyorgy: The Power of Limits
    Eamon, W.: Science and the Secrets of Nature
    Eco Umberto: Art & beauty in the Middle Ages
    Edgar, W.: Pagan Mysteries In the Renaissance
    Edgerton, S. Y.: The Renaissance Rediscovery of Linear Perspective
    Euclid: Elements. Books I - VI, XI and XII
    Evans, Robin: The Projective Cast: Architecture & its Three Geometries
    Fabricus, Johannes: Alchemy, the Medieval Alchemists and....
    Fibbonacci, Leonardo: The Book of Squares
    Fisher, G.: History of Christian Doctrine
    Fletcher, Banister: A History of Architecture (1987 ed. prefered)
    Foster, R.: The Secret Life of Paintings
    Frazer, J. G.: The Golden Bough
    Fulcanelli: Le Mystere Des Cathedrales
    Funkenstein, A.: Theology & the Scientific Imagination
    George, C.: The Legacy of the Middle Ages
    Gillings, R.: Mathematics in the Time of the Pharoes
    Gimpel, J.: The Cathedral Builders
    Gimpel, J.: The Cult of Art
    Gimpel, Jean: The Medieval Machine
    Ghyka, M.: The Geometry of Art & Life
    Gordon, J. E.: Structures
    Grant, E: A source book in medieval science
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    Lesser, G.: Gothic Cathedrals & Sacred Geometry, vols 1 & 2
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    Lund, F.; Ad Quadratum, A Study in the Geometrical Bases of Architecture
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    Mayanander, FIRST NAME: The Wonder Beyond
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    Mitchell, J.: City of Revelation
    Moore, R. I.: The Formation of a Persecuting Society
    Necipoglu, Gulru: The Topkapi Scroll: Geometry and Ornament in Islamic Architecture
    Neugebauer, O.: The Exact Sciences in Antiquity
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    Panofsky, E.: Gothic Architecture and Scholasticism
    Panofsky, E.: The Life & Art of Albrecht Durer
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    Pennick, N.: The mysteries of Kings College Chapel
    Pirenne, H.: Economic and Social History of the middle ages
    Plato: Timaeus
    Pyle, A.: Atomism & its Critics
    Pugin: The True Principles of Pointed Architecture
    Redondi, Pietro Galilao Heretic (translated from the original Galilao Eretico)
    Richard, F.: The Secret Life of Paintings
    Richter, Irma: Rythmic Form in Art
    Robb, N.: Neoplatonism of the Italian Renassance
    Royal Commision on Historic Monuments and Buildings: Salisbury Cathedral: perspectives...
    Ruskin : The Seven Camps of Architecture
    Sandstrom, Sven: Levels of Unreality
    Schneider, Michael: A Beginners Guide to Constructing the Universe: The Mathematical Archetypes of Nature, Art and Science
    Shelby, R.: Gothic Design Technique
    Sorabji, R.: Philoponus & the Rejection of Aristotelian Science
    Southern, R. W.: The Making of the Middle Ages
    Stewart, I.: Fearful Symmetry
    Stiefel, T.: The Intellectual Revolution of Twelfth Century Europe
    Stirling, W.: The Canon
    Taylor, F. S.: The Alchemists Founders of Modern Chemistry
    Thompson, Sir Wentworth D'Arcy: On Growth and Form
    Thorndike, L.: A History of Magic and Experimental science, vol 2
    Toulmin, S.: Architecture of Matter
    Unknown author: The Maths of Plato's Academy
    van Melsen, A. G. M: Van Atomos Naar Atoom
    Vasari, Giorgio: Lives of the Painters
    Vasari, Giorgio: On Technique: an Introduction to the ? Arts and Design
    Victor, S.K.: Practical Geometry in the High Middle Ages (Philadelphia, 1972)
    Vitruvious: The Ten Books on Architecture
    Ward, B.: The Influence of St. Bernard
    Warland, E. G.: Modern Stone Masonry
    Whittaker, T.: The Neo-Platonists
    Williams, E.: Science & the Secrets of Nature
    [Wilson, -]: Wilson's Combined Works, Building, Construction, Carpentry & Joinery, &c.
    Wilt, G.: A History of Architecture, (updated by Wyatt Papworth)
    Wilt, G.: Encyclopedia of Architecture
    Wittkower, R.: Architectural Principles in an Age of Humanism
    White, J.: The Birth & Rebirth of Pictorial Space
    Wood, A.: The Physics of Music
    Wood, R. (translator): Adam de Wodeham's Tractatus de Indivisibilibus
    Yates, F.: Giordano Bruno and the Hermetic Tradition

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