** THE GEOMETRY
**

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.

On
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 **Euclid**.” *Later
in the document we are told that *Constitutions,* it was taken through St.
Alban to King Athelstan of

An
even earlier document entitled *The Old
Book of Charges *gives a similar account of how the geometrical art of

Whatever
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**

The
medieval geometry of *words* required. It required,
originally, the use of a pair of compasses or ‘dividers’, and an unmarked
straight-edge or rule. There was no squared paper, and there were no measures,
no protractors, no parallel rulers.
Set-squares were of course available, and were certainly used by the
stone-workers, by the architects on site and by carpenters for template making
in the tracing house. But although
set-squares can be used and were used as providing short-cuts in geometrical
construction, they are not reliable enough or easy enough for use in highly
accurate geometrical construction. Most
importantly, Euclid, from whom, as will be shown, all the rules originated,
makes it clear that they are not used in his constructions, but only the compass
and unmarked straight-edge, for he is concerned with geometrical *logic,* which only those instruments can
provide. Why should he provide a
construction for a right-angle, for instance, if a set-square could
suffice? In the later medieval period,
as the Renaissance gradually took effect in the arts, some other instruments
came into use, such as the proportional compass occasionally, and eventually
the divided rule and the protractor, while the compasses were renamed
‘dividers’ as their function changed to the measurement of the proportions of
Classical Greek and Roman masonry (Fig. 261).

Gradients,
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

Although
the geometry used by medieval architects was traditionally ascribed to *The
Genealogical Problems of Medieval Craftsmen. *Undated.)

The
connection with *Elements* in* *the form of his three Postulates.
They are to be found in the Introduction to Hilda Hudson’s *Ruler And Compass*, (1915, 1):

*‘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

The
last postulate tells us what *Euclidean construction* is used for any
construction, whether contained within his works or not, which can be carried
out with

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.*

Thus
it will be seen that Euclidian constructions are not necessarily related to any
of *The
Articles and Points of Masonry*] pays much lip-service to

When

*“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 **Euclid**”, and the poet
Lydgate early in the fifteenth century echoed this with “ by crafft of Ewclyde mason doth his
cure” (Harvey, 1950, 47-8).*

This
is more than ‘lip-service’, and

In
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.

The
question may be raised as to how the medieval stone-mason learnt the
rules. It certainly was not by reading

Nevertheless,
geometrical construction according to *construction. *It follows no
constructional logic as

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 r _{4 }= 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 *

*2r _{4}² = AB²
= [2(OG – r_{4})]² = 4(a – r_{4})² *

*from which *

*r _{4} = a(2 -
*

*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 – r _{4})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.

In
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)*

That
is all. The implications of the postulates are not explained or considered as
they are in

Thus
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.

In
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.

Figs.
25,26,27,28 & 29 illustrate some of the common constructions in
the *ad* *quadratum* and the *ad triangulum* modes.

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.

**Geometric Progression**

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). *

Further:

*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 *.* Anyone with the time, the money and the
education could aim to become the designer of great buildings.

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.[1]

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 17^{th} 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 *Transference
of Dimensions* by supposing that

[1] Holbein’s “Ambassadors” painting in the
National Gallery, *celestial* matters, while the lower shelf
contains* terrestrial* instruments,
including books, music and a lute. Under
the table lies the lute case, black and in shadow, clearly symbolising the
coffin in the nether regions of the *grave*,
which relates to the distorted skull below. (I am indebted to Brian Stewart for
this latter insight). The painting would thus seem to commemorate an important
occasion in the history of Freemasonry, perhaps the founding of a specialised
lodge (restricted to ambassadors?).