Throughout the nineteenth and twentieth centuries a great deal of intelligence and assiduous investigation was devoted to attempts to discover the true nature of the geometry used by medieval architects. However, it has to be said that very little true geometry has been discovered. In John Harvey’s words:
Discussion has been obscured rather than clarified by a vast literature concerned with a possible symbolism, numerology (a little of which was probably intentional), and arithmetical and algebraic analysis. Almost the whole of this literature must be disregarded in seeking for the empirical means by which the architects reached their remarkable results. (1972, 124).
One needs to add that there have been numerous highly scholarly attempts at establishing philosophical ideas as the bases of architectural design. In fact, however, it has been shown that philosophical thought did not lead architectural development but in fact followed it. Furthermore it has been established that architecture was never mentioned in any philosophical literature.
the other hand, ignoring the literature and looking at the words written by the
medieval masons we find clear indications as to the nature of their
geometry. The English mid-fourteenth
century document known as The
Constitutions of Masonry opens with the words: “Here begin the Constitutions of the art of Geometry according to
even earlier document entitled The Old
Book of Charges gives a similar account of how the geometrical art of
one may think of the accuracy of the history, the fact remains that the
medieval mason clearly understood that the science of Geometry that he used in
architectural design and construction had been received from its inventor,
Euclid. Unfortunately, for the past two
centuries at least, the teaching of
But medieval geometer was not concerned with words, but with the manipulation of his drawing instruments -- the compass and the unmarked straightedge. And when we look at the geometrical designs that remain to us from that era, such as Rixner’s (Fig.36), then at von Rhiza’s analyses of Masons’ marks (Fig.14), then at the geometry of the Dome of the Rock in Jerusalem (Fig. 19b), and of the Arabic tile patterns, even at Mojon’s analyses of German choirs according to the ‘Instructions’ of the medieval German architect Lechler (Fig. 37), we find that they are all constructed according to certain rules. These rules are known as the three Postulates of Euclid.
The rules of geometrical construction according to Euclid
medieval geometry of
in the Middle Ages, as on road and rail today, were always defined by whole
number ratios. An example is sketched,
by a later hand, in Villard de Honnecourt’s folio. It will be found in use importantly at
the geometry used by medieval architects was traditionally ascribed to
‘Let it be granted:
i. That a straight line may be drawn from any point to any other point.
ii. That a terminated line may be produced to any length in a straight line.
iii. That a circle may be described from any centre, at any distance from that centre.
All the constructions used in the first six
books are built up from these three constructions only. The first two tell us what
last postulate tells us what
She also points out that the use of modern instruments such as parallel ruler and set-square, “amount to short cuts in Euclidian constructions without extending the range.” This understanding will be found to be of importance in discussion of the character of architectural geometry in late Gothic.
Another mathematician, E.W. Hobson, in her study: Squaring the Circle. History of the Problem (1913, 7-8), explains the matter of determinate, or predetermined, points:
A new point is determined in Euclidian geometry exclusively in one of the following ways:
Having given four points A,B,C,D, not all incident on the same straight line, then:
Whenever a point P exists which is incident on both (A,B) and on (C,D) that point is regarded as determinate.
1. Whenever a point P exists which is incident both on the straight line (A,B) and on the circle C(D) that point is regarded as determinate.
2. Whenever a point P exists which is incident on both the circles A(B), C(D), that point is regarded as determinate. The cardinal points of any figure determined by a Euclidian construction are always found by means of a finite number of successive applications of some or all of these rules 1, 2, 3.
it will be seen that Euclidian constructions are not necessarily related to any
“In addition to some
Latin, and in England an acquaintance with French as well until the end of the
fourteenth century, we may suppose the more important craftsmen to have a good
knowledge of arithmetic, and of course an outstanding eminence in practical
geometry. The English masons before 1400
described their craft as “according to
is more than ‘lip-service’, and
the Christian context geometry acquired two further rules that do not appear in
But no disposition met with more favour than that controlled by symmetry. Symmetry was regarded as the expression of a mysterious inner harmony. Craftsmen opposed the twelve patriarchs and twelve prophets of the Ancient Law to the twelve apostles of the New. (Male 1961, 9)
The Free and Liberal Art of Architectural Geometry
The art of medieval geometry could be considered as a kind of game played within strict rules, played by one person seeking to enter the mind of God the eternal geometer, and thus to allow God to manifest His will through the medium of the player’s participation in the game. The aim of the game is to design a sacred building or part of a building or any other work of sacred art, by means of the manipulation of rule and compass within the canonic rules as elucidated by Hudson (1915) above and Hobson (1913).
The rules may be expressed more clearly as follows:
(a) Only a pair of compasses (or alternatively a bar compass) and an unmarked straight-edge may be used.
(b) Every exercise of the art must begin with a circle.
(c) Straight lines must join or pass through predetermined (determinate) points.
(d) The intersection of any two lines constitutes a point.
(e) Circles or arcs of circles must be centred on predetermined points and must pass through predetermined points or be tangent to a predetermined line.
(f) No dimensions may be transferred.
(g) Symmetry must be maintained.
The matter of tangency requires some explanation. Tangency occurs only at one predetermined point. Between circles that point is where the line joining their centres crosses their circumferences, while between a circle and a straight line the point of tangency is where a radius of the circle lies at a right-angle to the straight line. There is only one such point, and it is therefore predetermined.
Medieval geometry is easy to construct, provided only that great care and attention to accuracy are maintained. Technically the art is well within the abilities of a ten-year-old child armed with no more than a straightedge, a reliable compass (or should this be a pair of compasses?), a keen eye, and a steady hand. No knowledge of mathematics or of theoretical geometry of any kind is required for the construction of medieval geometry, and a reasonably intelligent person will learn the rules simply by watching it being performed. No words are needed.
question may be raised as to how the medieval stone-mason learnt the
rules. It certainly was not by reading
geometrical construction according to
However, today ignorance of the Euclidian system of construction is widespread and profound, and it is certainly not limited to the scholarly world of architectural history. It is found even in such publications as Heilbron’s Geometry Civilized (1998) and Katz’s The History of Mathematics (1998). In Heilbron (236) one is shown “How to draw a quadrifoil”(Fig. 24a with a description of the process, as follows:
Since the construction has the symmetry of a square, you know that the center of the touching circles must be 90° apart as seen from the center O; since you are trying to find the radius r4 = OA = OB of the loci of the center, you will have to fiddle to draw the circles centred on A and B so as to be tangent to one another and to the circle center O, and also to have their centers on mutually perpendicular lines. The calculation that will relieve you from further fiddling, at least in the case of the quadrifoil, may be accomplished by the obvious step of joining A and B. The setting the radius OG = a you have from right sAOB,
2r4² = AB² = [2(OG – r4)]² = 4(a – r4)²
r4 = a(2 - Ö2) ~ 0.585a.
If you try your cut-and-try drawing, you will find OA = ~ 0.6a. If you are perplexed about the expression for AB in the previous equation, remember that the circles centred A and B are tangent and that, therefore, AB is twice the radius, or 2(a – r4)². (Heilbron 1998, 236)
But there is a very simple Euclidian construction for a quatrefoil (Fig. 24b). Within a circle inscribe a turned square, and in it inscribe a right square. Draw the vertical and horizontal diagonals. The points at which the diagonals intersect the sides of the right square are the centres for the four circles that are then drawn tangent to the turned square and also to each other and to the diagonals of the right square. But no words are really needed. The diagram explains itself. No fiddling, no fudging, no algebra, no need to explain, no perplexing. Only Euclidian determinate rigor.
Katz The History of Mathematics (1998,
61-62) we are given
It is well known that Euclidean constructions are based on the straightedge and compass. Postulates 1 and 2 assert that a straightedge may be used to draw a line between two points or extend a given line, while postulate 3 says that one can use a compass to draw a circle centered at any given point with any given radius...these constructions are all he needed to develop what he considered the basic results, the “elements.” (Katz 1998, 61-62)
is all. The implications of the postulates are not explained or considered as
they are in
ignorance of Euclidian construction remains widespread and the medieval
master’s high art of geometry “according to
Two examples of initial understanding of Euclidian construction are examined in figures 39 to 41. In Fig. 39 it will be seen that Villard’s drawing is seriously wrong. Bucher’s ‘presumed original concept’ is also mistaken and unnecessary. His analysis of the geometry of the actual window is closer to the truth although he admits that ‘I can find no logical reason for the choice of the centers for these four inner circles beyond a conscious wish to create a breach in the geometric modular inscription and thus save the window from a more heavy-handed appearance.’
But there is no need for such strenuous supposition, for the problem he conjures with he has created himself by the unnecessary restriction of his geometry to ‘turned squares’. My purely Euclidian construction (Fig. 40) demonstrates the geometrical rigour of the medieval designer. The circle A drawn tangent to the outer circles provides the centres for the inner circles on its circumference. It also provides the larger of the two central squares (B), which it circumscribes.
The double vesica that is required to provide the four diagonals also provides the diameter of the outer framing circle.
John James’ analyses of vault bosses at
Ad Quadratum and Ad Triangulum
The two simplest kinds of geometrical patterns produced according to the Euclidean postulates are those developed from the square within the circle, (ad quadratum) and those developed from the triangle within the circle (ad triangulum). It is also possible to develop a geometry from the pentagon, and this was sometimes used in the later Middle Ages, but the processes required are sophisticated and even then result only in approximations. The symbolism, moreover, is theologically less fundamental. The pentagon does not appear at Peterborough although it is found with some frequency at Lincoln, particularly in the Chapter House. By far the most commonly used system in medieval art and architecture is the ad quadratum mode.
There will be those who question the need to begin every architectural geometric design with a circle. There are two cogent reasons. The first is that the architect is aiming in his design to bring Heaven down to earth and Heaven’s geometry is circular. The second reason is even more cogent. It is irrefutable. The fact is that it is impossible to construct a true square, or octagon or octagram or hexagon or hexagram without a pre-existing circle. Even an equilateral triangle requires the intersection of two arcs that is simply in essence the intersection of two circles.
The Continuity of the Geometry
One further principle controlling the geometry of any sacred building is yet to be mentioned, and this is the principle of continuity. When any building was extended or altered it was held to be essential that the geometry of the new work must be in conformity geometrically with that of the existing work. If it were not, the building would have no geometrical integrity and would receive no divine protection from the disasters of this world. The existence of this rule can be seen first of all in a non-Christian religious building, the Pantheon of Rome. Built in AD 120-24, its great portal was added c.215. It can be seen (Fig.30) how the geometry of the portal was geometrically developed from that of the existing building. Similarly in Istanbul in the great church of Hagia Sophia (Fig. 31), when the original dome, constructed in 532, collapsed during an earthquake, it was replaced in 563 by a dome with a much higher and more stable profile, but also, as I have shown, to a geometry directly developed from that of the original building. This principle will be discussed at greater length in later pages. Suffice it to say here that it will be found to play a most important role in the geometry of Peterborough Cathedral.
There remains one property of sacramental geometry that has received little attention or understanding in scholarship, although it is of outstanding importance. It is the process known as geometrical progression.
Few mathematicians appear to realise that geometrical progression has its origins, its very being, not in number, but in geometry. As will be demonstrated, geometrical progression played an essential role, a role of outstanding importance, in the sacramental function of architectural geometry in the Middle Ages, for it was through geometrical progression that the geometry of a building was integrated with the geometry of God’s created universe, the achievement of which was the whole purpose of sacramental geometry.
Geometrical progression is created by those constructions that are special to sacramental geometry i.e. square within square within square, and triangle within triangle within triangle, and constructions developed from them with octagons and hexagons, for example Figs. 32, 33, 34. It will be seen that geometry creates geometrical progressions that are often factored by Irrational Ratios such as Ö2, Ö3, Ö6.
Today, geometrical progressions are presented invariably in numerical terms, or occasionally in linear terms. Padovan, in his book Proportion (1999) devotes a good deal of attention to the subject but presents examples only in the form of lines of different lengths, never as geometrical constructions. Heilbron in Geometry Civilized (1998) offers no examples of geometrical geometrical progressions, nor does Katz in his History of Mathematics (1998).
A geometrical progression, (or G.P.), exists when a series of values are consecutively related to one another by a common factor. For example, in a numerical form:
10, 20, 40, 80. In this case the factor is 2. (Only three consecutive values are required to define a geometrical progression).
It is a property of any simple geometric progression that it can extend in either direction, larger or smaller, to an infinite degree, from the atom, the electron, the quark, to the edge of the universe, (if it has an edge), but it can never become a minus quantity or reduce to 0. For example: 1/243 1/81 1/27 1/9 1/3 1 3 9 27 81 243……….
The alternative form of Progression in mathematics is Arithmetical. An arithmetical progression is essentially a series of values that are consecutively related by the addition or subtraction of a common amount, for example: ...10, 15, 20, 25, 30, etc.,
Arithmetical progressions are not created geometrically, but unlike Geometrical progressions they can reduce to 0 and to a minus quality. They would appear to play no part in medieval architecture.
The most important element of any geometrical progression is that once it has been created by three elements it automatically becomes part of a geometry that extends from the infinitely small to the infinitely large, to the dome of Heaven and beyond (Fig.34). It is for this reason that it was installed into the forms of Christian churches in the Middle Ages, and it was for this reason that it was deeply believed by the theologians to have the power to provide security from the destructive forces of the natural world, and to confer sacredness upon the building in which it was installed. It would have been in the light of such beliefs that the theologians reacted so intensely to the non-centrality of Galileo’s view of the Earth’s cosmic geometricality.
The Oral Transmission of the tradition
The tradition of sacramental geometry, its methods and its rules were maintained over many centuries solely by oral transmission and by demonstration. For most of the Middle Ages the vast majority of European peoples, including masons, were illiterate and innumerate. Shelby has paid particular attention to the education of the freemason at that time, (though he also sadly is unaware of the true nature of Euclidean geometry in medieval terms):
But medieval masons founded their craft on geometry, not arithmetic, as explicitly stated in ‘the articles and points of masonry’ which were compiled for the English mason craft c.1400. While the document pays much lip-service to Euclid as the founder of the craft, in fact Euclidean geometry as such was little used by medieval masons...modern students of Gothic architecture should therefore be wary of imputing to medieval architects mathematical skills that would have been beyond their comprehension or their building needs. (Shelby 1971, 238-9).
And again, quite correctly:
The master mason of the medieval period had only the most elementary mathematical knowledge, and no knowledge whatever of geometrical theory (Shelby. 1972, 239).
The master masons were by no means all illiterate, but there is little indication that literacy played a part in the acquisition of the technical knowledge necessary for designing and constructing a building. Whatever knowledge he possessed in the art of building he had learned directly from his master, or from observing the results of the efforts of past masters, or from practical experience of his own successes and failures (Shelby 1964, 388-389).
Before the fifteenth century few would ever have seen a book in their whole lives except perhaps in the hands of a prelate, and all books would have been written in Latin, a foreign language to any English mason. Many monarchs even, though by no means all, could sign documents only with a cross or a seal. Musicians, sculptors, farmers, fishermen, carpenters and cooks did not read books on how to perform theirs crafts. They learnt by verbal advice and by example and trial and correction, in the same way that medieval masons learnt their skills.
The Middle Ages were also characterised by the craft guilds, with their strict rules of entry and levels of proven ability, and above all by strict control of the secrets of the craft skills on which the livelihood of their members, and often of their fellow citizens, depended, a control that depended on those secrets never being written down.
The End of the Tradition
As one has seen from Schmuttermayer’s and Roriczer’s booklets, (and other similar documents), by the late fifteenth century, in south Germany, the situation had changed. Non-qualified persons, unskilled in the traditions of masonry, were entering the field of the professional architect, and were introducing Classical Antique ideas from Italy. In reaction to this development the older architects were attempting to repress those ideas and maintain traditional skills by producing their instructional booklets on the importance and correct use of geometry in any design context.
The booklets were few in number, and limited to readers of German. In the highly unlikely event of one coming into the hands of an Italian or an English architect with free access to the printed copies of Vitruvius and Alberti, it would have been regarded as totally out-of-date and useless, and probably as complete nonsense.
As the Renaissance gained strength and took
control of European culture, bringing the printing industry, grammar schools,
new drawing instruments, machinery, literacy, universities, and increased trade
and travel, the architectural profession became separated from that of the
mason. Books were published, such as
In Italy painters included their own superb designs for buildings as settings for their subjects. Furthermore, great secular projects, such as houses, palaces and civic buildings were re-placing churches as subjects for the architect.
The oral traditions of geometry were no longer applicable to much of the architect’s work. He worked at a distance from the workmen. His designs could be drawn on paper, produced as models and given to others to erect. New measuring devices, most importantly the protractor, were available, while mathematics and Euclid’s geometrical logic could be studied in universities and learned societies. Most importantly, far more craftsmen could read and measure, count and calculate, while the new breed of architects were highly educated and widely cultured and were drawn from other professions.
The old medieval mason-architect became an anachronism. To design a building one now studied such books as Vitruvius and Alberti on Classical architecture, written not by Benedictine theologians but by laymen, historians and architects. Consequently the oral tradition died away into silence, and the old ‘classical’ sacramental geometry of the cathedral designer became a mysterious historical ghost, haunting the halls of modern freemasonry, where very few if any stonemasons are now to be found, but where certain geometrical forms and implements have acquired new meanings within a different kind of religion.
The tradition did not disappear suddenly. Its influence can be seen in the work of certain Renaissance painters, and, as we have seen, in some musical instruments, and in theatre design. In one very fine Rembrandt canvas, now in the Royal Collection at Buckingham Palace, entitled The Shipbuilder and His Wife, the Dutch ship-builder-designer of 1660 sits at his drawing desk with no more than a small brass compass with silver points, and an unmarked rule, apparently only one foot long. Apart from his 17th century ruff he could be any medieval cathedral architect. The scene suggests that the tradition of using Euclid’s special form of geometrical design to protect buildings may have been continued in the design of ships to protect them and their crews from natural disasters. It also suggests that medieval Dutch architects may have moved out of the dying trade of church building into the growing trade of ship building, just as some German architects had been obliged to move into such professions as printing.
· All past references to
 Holbein’s “Ambassadors” painting in the