**The Geometry of the Thirteenth Century Façade **

**(Figs. 61-75)**

The
most obvious geometrical forms presented by the façade are the three circular windows.
The central window is octagonal in design, while the north and south windows
are hexagonal in design. This arrangement brought to mind the geometry
discovered by this author in the great Hagia Sophia
building in

The
second geometrical element of importance is the arrangement of the three
triangles of the gables, which, enclosing the three circles of the windows,
spread themselves downwards, suggesting the enlargement of the three circles.
Following this idea we arrive at three lower and larger circles overlapping
each other by half their diameter (Fig. 63),
exactly as in the Hagia Sophia plan. Eventually we will arrive at a clear confirmation
of this arrangement in certain hitherto unexplained elements in the fabric, but
this was not appreciated until the analysis of the geometry had been
completed.

In
accordance with the rule that any sacred geometrical construction must begin
with a circle, we create a circle of the width of the complete façade, 156
English feet (47.54m.) in diameter. Various centres for this circle can be
tried, but the only centre that leads forward is the niche at the top of the
Perpendicular porch c.1380, which at first seemed surprising, but, as will be
seen in the consideration of the porch geometry, is exactly as it should be,
for it relates to the fact that the geometry of the c.1380 porch will be found
to derive from the geometry of the whole of the c.1230 façade (See the slide
show 14: The
Geometry of the Perpendicular Porch Elevation).

When this large circle is developed through
inscribed squares (Fig. 62)*,* the second square provides the line of the pavement and the
underside of the string-course 78 feet above. It also provides the central
vertical to each of the three portals. The third inscribed square provides the
centres for three overlapping circles, each of 78 feet (23.77m.) diameter, and
all tangent to the string-course line, the pavement level, and the primary
circle.

It
then appears that the diameters of these three circles are almost exactly three
times the width of the north and south portal openings, i.e. 26 feet. But the division each of the diameters of
these circles into three equal parts must not be done arithmetically, for the
medieval geometers had no measuring rules.
The division must be done geometrically using only unmarked rule and
compasses, and the means by which this is achieved are shown in Fig. 64*.* Centred on point A a
circle is drawn passing through the intersections of the three overlapping
circles. Where it crosses the centre vertical at D is the centre of a smaller
circle that if drawn tangent to the second square within the primary circle has
a diameter one third that of the larger circles. From here it is a simple
matter to develop the two small circles below by swinging the diameter down the
centre line with the compass/dividers.

The
same procedure applied from points B and C divides the north and south circles
into three small circles, and all of these are 26 feet in diameter.

The
rows of 26-ft circles are then extended above the ‘string-course’ line, to
form three circles above that line, and here one encounters one of the
incalculable fortuities that occur in sacramental geometry, for the north and
south circles of this upper row pass exactly tangent to the primary 156 foot
circle.

In
Fig. 65 the hexagrams within the large 78-ft
circles north and south are completed, and also the octagram
within the central large circle, plus another octagram
and a square developed within.

Although
the 26-ft circles fit the width of the north and south portal openings, they do
not immediately provide the 18.4 foot width of the central portal opening. But
26 is 18.4 times the square root of two, 1.414.
If we therefore inscribe the square within the circle we arrive at the
width of the central opening (Fig. 66).

As
prescribed in the chapter concerning the techniques and rules of sacramental
geometry every square should be turned to create an octagram.

Also
in Fig. 66 the square circumscribing the primary 156ft. circle is constructed.

The
octagram constructed within the first octagram in the central gable circle produces a circle
centred on the circular window of the gable and of similar though not identical
diameter Fig. 67. A simple calculation shows that the circle
around the window in the geometry must be 13 feet in diameter, but the diameter
of the actual window is 12 feet. However the geometry is correct and the
development of the 12-foot window from the 13-foot geometry is fully presented
in the following chapter.

The
circles of the north and south portals and gables are similarly inscribed with
their hexagrams in accordance with the design of their windows, and here again
they create 13-foot diameter circles centred on their windows. The development
of the design of the windows from these circles is also presented in the next
chapter.

The
six verticals tangent to the nine small circles need
to be drawn, as also the two horizontals tangent to those same circles (Fig. 66).
At this time the horizontal diameters of all the 26-foot diameter
circles are drawn. It will be seen (Fig. 73)
that three of these horizontals provide the lines of the junctions of the
shafts of the thirty-six detached colonettes
enhancing the three portals, and marked by the ring-shafts holding them in
place.

The geometry of the central arch is
displayed in Fig.68. A circle is drawn
circumscribing the 26ft. square immediately below the string-course. From the
square inscribed within the smaller circle the base line is extended across the
whole design. This gives the level of the terminations of the detached shafts
with their capitals, and also the springing of the arches. The vertical sides
of the next smaller square are extended down to meet the base-line of the
larger square at the two points C and D.
These two points become the centres for the drawing of the arch, the
inner arcs springing from the lower corners X and Y of the 18.4ft square
inscribed within the 26ft circle. They thus create the 18.4ft opening of the
central arch. The outer arcs spring from
the intersections of the larger (36.7ft) circle with the springing level, Z and
E, and create the outer width of the central arch of 32 feet.

The
geometry of the north and south arches is shown in Fig.
69*.* It is developed from an
intriguing combination of hexagon and square both inscribed within the 26ft
circle. The baseline of the square relates to that of the central arch that
defines the springing level of all three arches.

The
intersections of the apex of the inverted triangle with the base of the square,
F and G provide the radial centres for the arcs.

The
outer arcs swing from the apex P of the upright triangle down to the springing
line, N and O, creating a width of 40 feet.

The
inner arcs run from the intersections of the smaller square with the verticals
of the 26ft square up to the centre vertical. This gives an inner width of 26
feet.

The
strictly geometrical design found here in these arches negates the theory of
numerical proportions in arch design that has been advanced hitherto.

**The geometry
of the façade towers (Figs. 70, 71, 72).**

The full width of
the north tower between the centres of the attached shafts is 16 feet 3inches, whereas
that of the south tower is 8 inches less at 15 feet 7 inches. The narrowing of
the southern tower is inexplicable, but could have resulted from a need to
widen the passage between the tower and buildings of the *. *A document of 1245 concerning
the Abbey of Winchcombe quoted by Salzman
(1952, p. 383) may have relevance:

* 15 December. Inquiry whether it would to the
injury of the town or of the Abbey of Winchcombe, or
of anyone else, if the King should allow Master Henry, rector of the church of St.Peter of the same town, to lengthen the church 12 ft. eastwards,and to enlarge an aisle begun on the south side
of the church to the length of 30 ft. and the breadth of 12ft. [The jury] say
that if the chancel were lengthened 12 ft eastwards it would be to the damage
of the Abbot and Abbey, because the Abbot could not have a free way in and out
for his carts and for carrying his timber.*

In view of the
narrowness of the passage at

It will be further
observed that the arrangement of blind arcading and colonnettes
decorating the tower from top to bottom divides the width of the wall into four
equal parts. Both the hexagram and the octagram can
divide a diameter vertically into four, but only the octagram
additionally provide the necessary horizontals. Thus we arrive at an octagram within the 13.25 foot square within an 18.75 foot
circle.

The vertical
space between the verticals of this square and the limit of the north portal,
18 feet 9 inches, is then divided in half, using as starting levels the three
horizontals tangent to the twelve small circles:

Three circles are
drawn tangent to the verticals, and within each circle two

octagrams and a square are progressively
constructed.

It will then be
observed that linking these constructions squares have been created that are
exactly similar to those within the circles.
When these squares are developed in the same way (Fig. 71)*,*
they are found to be centred on the horizontals through the diameters of the
four rows of circles in the main façade.

A final similar construction
is made centred on the horizontal tangent to the tops of the gable circles.

The geometry of
the north tower in relation to the fabric is shown in Fig.72.

Now the centre
verticals of the large north and south circles are extended to meet the primary
circle at X and Y (Fig. 71). A horizontal is drawn between X and Y
crossing the central vertical at Z.
Straight lines are then drawn from X, Y, Z through the intersections of
the 78 foot circles and extended to their horizontal diameters. These lines provide the outlines of the
gables.

Finally a
circle, dotted in Fig. 73*,* is drawn centred on the string-course, radius to the pavement,
and a square drawn around it. This gives
the height of the tower spires.

The complete geometry in
relation to the fabric of the façade is shown in Fig.
73.

**The
geometry of the gable windows**

The geometries of the three gable windows as they are
developed from the general geometry of the whole façade is
provided in Figs. 76-112.

The diameter of
the circle in the general geometry surrounding each window is 13 feet, being
that of the second square within a 26 foot circle. But the diameter of the actual window is 12
feet. The geometrical means by which the difference between the two dimensions
is resolved will be demonstrated in the later analysis.

**The
geometry of the spires**

** **Protractors for the measurement of
angles in degrees did not exist in the Middle
Ages. Angles other than right-angles
were defined numerically as gradients, as they are today on road and rail, and
an example is demonstrated in Villard de Honnecourt’s
notebooks. The gradient of the large
spires upon the staircase towers at

At the same time
all the spires are geometrically developed from the larger geometry of the facade, and their geometry originates from nodal points on
the line of the string-course.

It will be observed
that the angles of the gable profiles do not correspond exactly with the angles
of the geometry. The geometry gives
angles of 60 degrees, while the fabric has angles of 58 degrees. The explanation of this anomaly is that to
produce an angle of 60 degrees one requires, in whole numbers, a gradient of 64
in 111, or 2 to 3.368, which was beyond the limits of medieval expertise. But a simple gradient of 2 to 3, well within
the grasp of any mason, produces an angle of 58 degrees, and that is what they
used. One needs to appreciate that
neither the stone masons nor the master mason had any means of measuring or
checking the angle.

**The
verification of the analysis**

** **It will be observed that the north and south geometries
arrive at the height of the stringcourse. But the geometry of the central arch
achieves a height a
foot below the string-course. In Fig. 44,* *it
will be seen that this unexpected difference in height is in fact presented in
the fabric, where there is a gap of one foot between the string-course and the
upper limit of the central arch. This circumstance proves the veracity of the
geometrical analysis.

A
second powerful proof of the truth of the analysis is provided by the four
false gargoyles. The function of these apparently unique objects has never been
fully explained although Stallard comes very close. They are larger than the usual true
water-spout gargoyles, and they are not waterspouts. Each head is surmounted by
a form of moulded abacus that provides a flat surface about 21 inches below the
string-course and seemingly unrelated to any other feature architectural
feature. They appear in fact to be sited independently of the geometry. And
each gargoyle has a lengthy appendage pointing downwards to nothing in
particular (Fig. 74)*.*

But
when the geometrical analysis is presented superimposed upon the fabric (Fig. 75), it is seen that each of the gargoyles stands
over an apex of a hexagon inscribed within the north or south 78ft circles. In
the case of the two central gargoyles these apexes are also the intersections
of the three 78ft circles. The appendages are now seen to be indicating the
exact points of intersection.

The
flat tops of the gargoyles are then seen to provide a meeting place for the
linear extensions of the gable profiles.

Thus
we realise that in the false gargoyles the architect of the c.1220 façade
provided indications in the fabric that would enable future architects to
deduce his geometry and thus be able to ensure that any alterations or
additions that they might make would conform to his existing geometry.

We
shall find other deliberate indications of the underlying geometries in other parts
of the cathedral. These clues are designed to be recognisable and interpretable
only by those practised in the art of sacramental geometry.