‘Knowledge of numbers should not be despised […] We are instructed in number to avoid confusion. Take away number in all things, and they all perish. Take away computation from the world, and all things are encompassed by blind ignorance; people who are ignorant of the knowledge of reckoning cannot be distinguished from the other animals’. Thus wrote Isidore of Sevilla in the 7th century (Book III, 4). Number clearly mattered. Indeed, the medieval world seems to have taken to heart the words expressed in the Book of Wisdom (11.21): ‘Omnia in mensura et numero et pondere fecisti’. In such a number-obsessed world, one would expect number to play a significant role in the design of medieval architecture, for without number, a building, like everything else, would perish. And indeed, many medieval descriptions of buildings display a great concern with numerical values. After a short introduction into the issues involved, this paper will concentrate on the use of perfect numbers as a design principle in several select examples.
In chapter 14 of the first part of his
The length of three hundred cubits denotes this present age, which extends over
three periods, namely, the period of natural law, the period of the written law,
and the period of grace through which the Holy Church is from the world’s
beginning to its end advancing from this present life towards the future glory.
The fifty cubits breadth denotes all believers everywhere, who are established
under one Head; that is Christ. For fifty is seven times seven that is,
forty-nine, the number that means the total sum of all believers — plus
one, which means Christ, who is the Head of His Church and the goal of our
desires. That is why the ark is gathered to one cubit at the top. The height of
thirty cubits denotes the thirty volumes of the Holy Writ, namely, the
twenty-two of the Old Testament and the eight of the New, wherein is contained
the sum of all the things that God has either done, or else is going to do, for
His Church. The three storeys signify the three ranks of believers that there
are in the Church, whereof the first have commerce with the world, albeit
lawfully, the second are fleeing from it and forgetting it, and the third
already have forgotten it, and they are near to God. The fact that the ark gets
narrower towards the top and wider below means that in the Holy Church there are
more people leading a carnal life than there are persons of a spiritual life, it
being always the rule that the more perfect are proportionately few in number.
The ark narrows to the measure of a single cubit at the top, because Christ, the
head of his Church, who is the Saint of saints, is like to other men in all
respects in nature, but in the uniqueness of His virtue He is above them all.
The hundred years that the ark took to build means the same as a hundred cubits.
For the hundred years signify the period of grace: since the Holy Church, which
began with the beginning of the world, received redemption through the
immolation of the spotless Lamb in the period of grace. (
This was not all Hugh has to say on the dimensions of the ark. He elaborates:
that the length of the ark is six times its width and ten times its height
provides us with an allegorical figure for the human body in which Christ
appeared, for it is itself His body. For the length of a body from crown to heel
is six times its width from one side to the other, and its height moreover is
ten times its thickness through from back to front. So if you measure a
recumbent man when he is lying down quite flat, his length from head to foot is
six times his width across from right to left or left to right, and ten times
his thickness from the ground. Six is the number of times that fifty goes into
three hundred, and there are six periods in the three ages of the world. Again,
three hundred signifies faith in the Trinity, or because of the [Greek] letter
Tau, of which the numerical value is 300, and which still retains the shape of
the cross among the Syrians, it signifies the cross. Fifty denotes the remission
of sins, thirty the measure of the age of the fullness of Christ. (
These two long passages on the measurements of the ark of Noah show that Hugh
considered the numerical values provided for this structure in the Bible to have a
significance elucidating the underlying symbolic meaning of the ark’s
structure. This type of numerological thinking was not unique to Hugh of St.
Victor.
First of all, I will introduce the reader to the most widely used symbolic numbers, after which I will turn to perfect numbers. After a discussion of the way numbers connected to the structure of the universe and God’s plan of Creation, I will discuss a group of closely related 12th-century churches where perfect numbers may have been used to structure the designs.
To the medieval mind, numbers were imbued with symbolic qualities and could be
interpreted in various ways, with a few numbers having fixed meanings (
It is not possible that the Gospels can be either more or fewer in number than
they are. For, since there are four zones of the world in which we live, and
four principal winds, while the church is scattered throughout all the world,
and the ‘pillar and ground’ of the church is the gospel and spirit
of life; it is fitting that she should have four pillars […] He who was
manifested to men, has given us the gospels under four aspects, but bound
together by one spirit […]. For the living creatures are quadriform, and
the Gospel is quadriform. (
More ingenious is the interpretation of the number four given by Hugh of St. Victor
in his
Number itself teaches us the nature of the going out and the return of the soul.
Consider: Three times one makes three; three times three, nine; three times
nine, twenty-seven; and three times twenty-seven, eighty-one. See how in the
fourth multiplication the original ‘one’, or unity, recurs; and you
would see the same thing happen even if you were to carry the multiplication out
towards infinity; always, at every fourth stage of the process, the number
‘one’ recurs. Now the soul’s simple essence is most
appropriately expressed by ‘one’, which itself is also incorporeal.
And the number ‘three’ likewise, because of the ‘one’
which is its constituent link, is fittingly referred to the soul, just as the
number ‘four’, because it has two constituent links and is therefore
divisible, belongs properly speaking to the body. (
In the next chapter he continues:
For the body too they assign its number ‘four’. As the figure
‘one’ fits the soul, so the figure ‘two’ fits the body.
Consider: two times two makes four; two times four, eight; two times eight,
sixteen; and two times sixteen, thirty-two. Here likewise in the fourth
multiplication the same number whence the operation began, namely
‘two’ reappears, and the same thing would undoubtedly happen if one
were to carry the process out to infinity; at each fourth stage of the process,
the number ‘two’ would recur. And this is the number
‘four’ of the body, in which it is given to understand that
everything which is composed of divisibles, or solubles, is itself also
divisible, or dissoluble. (
The number seven is composed of four and three, which medieval authors considered
highly significant. According to St. Augustine, whose ideas had a profound impact on
medieval thinking, four and three are the first even and the first odd number,
Eight could be the number of the Resurrection, of salvation and regeneration (
In his
Of all the numbers with symbolic values, perfect numbers appear to have been valued
more than any other numbers in the Middle Ages. A perfect number is equal to the sum
of its divisors. The first three perfect numbers are 6, 28 and 496. In his
The concept of perfect numbers was first introduced in Euclid’s
[o]f the simple even numbers, some are superabundant, some deficient, like
extremes set over against each other, and some are intermediary between them and
are called perfect. Those which are said to be opposites to one another, the
superabundant and deficient, are distinguished from one another in the relation
of inequality in the directions of the greater and the less[er]; for apart from
these no other form of inequality could be conceived, nor could evil, disease,
disproportion, unseemliness, nor any such thing, save in terms of excess and
deficiency. For in the realm of the greater there arise excesses, overreaching,
and superabundance, and in the less[er] need, deficiency, privation, and lack;
but in that which lies between the greater and the less[er], namely, the equal,
are virtues, wealth, moderation, propriety, beauty, and the like, to which the
aforesaid form of number, the perfect is most akin. (
Nicomachus’
So these numbers, those whose parts are added together exceed the total, are seen
to be similar to someone who is born with many hands more than nature usually
gives, as is the case with a giant who has a hundred hands, or three bodies
joined together, such as the triple-formed Geryon. Or this number is like some
monstrosity of nature which suddenly appears with a multiplicity of limbs. The
other number, whose parts when totalled are less than the size of the entire
number, is like one born with some limb missing, like the ugliness of the
Cyclops’ face. Or the number is like one who is born naturally deficient
in relation to some member, who emerges short of his total fullness. Between
these two kinds of number, as if between two elements unequal and intemperate,
is put a number which holds the middle place between the extremes like one who
seeks virtue. That number is called perfect and it does not extend in a
superfluous progression nor is it reduced in a contracted reduction, but it
maintains the place in the middle; the sum of its parts is not more than the
total nor does it suffer from a lack in comparison with the total, as are 6 and
28 […]. There is in these a great similarity to the virtues and vices.
(
In his
Six is a perfect number, not because God required a protracted time, as if He
could not at once create all things, which then should mark the course of time
by the movements proper to them, but because the perfection of the works was
signified by the number six. For the number six is the first which is made up of
its own [or aliquot] parts, i.e., of its sixth, third, and half, which are
respectively one, two and three, and which make a total of six. (
He then goes on to explain what makes perfect numbers so special:
In this way of looking at a number, those are said to be its parts which exactly
divide it, as a half, a third, a fourth, or a fraction with any denominator
— e.g., four is a part of nine, but not therefore an aliquot part; but one
is, for it is the ninth part; and three is, for it is the third. Yet these two
parts, the ninth and the third, or one and three, are far from making its whole
sum of nine. So again, in the number ten, four is a part, yet does not divide
it; but one is an aliquot part, for it is a tenth; so it has a fifth, which is
two; and a half which is five. But these three parts, a tenth, a fifth and a
half, or one, two and five, added together, do not make ten, but eight. Of the
number twelve, again, the parts added together exceed the whole; for it has a
twelfth, that is, one; a sixth, or two; a fourth, which is three; a third, which
is four; and a half, which is six. But one, two, three, four, and six make up,
not twelve, but more, viz. sixteen. So much I have thought fit to state for the
sake of illustrating the perfection of the number six, which is, as I said, the
first which is made exactly up of its own parts added together; and in this
number of days God finished his work. (
Six was thus the number of creation. No wonder God also used it and its divisors to set out the measurements of the Temple that was to be built in his honour in Jerusalem: ‘And the house which King Solomon built to the Lord was sixty cubits long, twenty cubits wide, and thirty cubits high’, i.e., the proportions used for the temple were 60:20:30, or, 6:2:3. In his writings on the Temple of Solomon, the venerable Bede explained these numbers as follows:
[T]he length of the house is fittingly expressed by the figure of sixty cubits.
For the number six whereby the world was made, conventionally denotes the
perfection of good works. And it is necessary that we should endure the trials
of our sojourn with forbearance so that we may deservedly be able to enter the
promised land of good works when it appears. The breadth is determined by the
number twenty because of the twofold dimension of the same charity wherewith we
love both God and neighbour, the height is determined by the number thirty
because of the belief in the Holy Trinity which is one God in the vision of whom
all our hopes and desires have their fulfilment. So six has to do with the
perfection of the work; two with the love of God and neighbour and three with
the hope of the vision of God. Each number is rightly multiplied by ten because
it is only through faith and the observance of the Decalogue of the law that our
patience gets salutary exercise or our charity burns profitably or our hope is
rapt aloft to yearn for the things of eternity. (
That Bede’s idea was persistent can be shown with reference to Bonaventura, who
in his
The perfect number six even underlies the division of the Bible. According to Hugh of
St. Victor, the whole of Sacred Scripture [1] is contained in two testaments [2],
namely, in the Old and in the New. The books in each Testament are divided into
three groups [3]. The Old Testament contains the Law, the Prophets and
Hagiographers; the New contains the Gospel, the Apostles, and the Fathers [6], i.e.,
1:2:3:6 (
Boethius notes, in what seems an apt conclusion to this section, that there are very
few perfect numbers: ‘You find the perfect numbers rarely, you may enumerate
them more easily, and they are produced in a very regular order. But you find
superfluous or diminished numbers to be many and infinite and not disposed in any
order, but arranged randomly and illogically, not generated from a certain
point’ (
Numbers clearly mattered in the medieval world, which valued the words expressed in
the Book of Wisdom (11:21): ‘You have ordered all things in measure, and
number, and weight’.
prior to all not only because God, the creator of the massive structure of the
world, considered this first discipline as the exemplar of his own thought and
established all things in accord with it; or that through numbers of an assigned
order all things exhibiting the logic of their maker found concord; but
arithmetic is said to be the first for this reason also, because whatever things
are prior in nature, it is to these underlying elements that the posterior
elements can be referred. (
He concludes:
From the beginning, all things that have been created may be seen by the nature
of things to be formed by reason of numbers. Number was the principal exemplar
in the mind of the creator. From it was derived the multiplicity of the four
elements, from it were derived the changes of the seasons, from it the movement
of the stars and the turning of the heavens. (
In the seventh century, Isidore of Seville claimed that ‘we are instructed in
number to avoid confusion. Take away number in all things, and they all perish. Take
away computation from the world, and all things are encompassed by blind ignorance;
people who are ignorant of the knowledge of reckoning cannot be distinguished from
the other animals’ (
[Since] there is no beauty and pleasure without proportion, and proportion is to
be found primarily in numbers; all things must have numerical proportion.
Consequently, number is the principal exemplar in the mind of the Creator and as
such it is the principal vehicle that, in things, leads to wisdom. Since this
device is extremely clear to all and is closest to God, it leads us to Him
through seven stages and it causes us to know Him in all corporeal and sensible
things; and while we learn that things have numerical proportion, we take
pleasure in this numerical proportion and we judge things irrefutably by virtue
of the laws that govern it. (
In a world so preoccupied with number, its significance and its structuring
principle, it would seem logical for number — and perfect and symbolic numbers
in particular — to have played a significant role in the arts of the time and
in the design of medieval architecture. Could it be that number was intentionally
used as a designing principle or played a role in design? Indicative of the use of
number in medieval art is the splendid ‘In Principio’ page at the
beginning of the Book of Genesis in the Bible of St. Hubert
‘In Principio’ page at the beginning of the Book of Genesis in the Bible of St. Hubert of circa 1085. Brussels, Royal Library of Belgium, ms. II. 1639, fol. 6v.
In this Mosan miniature, the figure of Christ is shown in a roundel placed at the
intersection of the overlapping initials IN. Surrounding this bust are four more
roundels with figures personifying the four elements. Each element is identified by
an inscription. Along the frame of each medallion, a Roman numeral attributes a
number to each element. Below and on the sides the divisors of this numeral are
written out in capital letters. The numbers given are: 2 x 2 x 2 = 8 (fire), 2 x 2 x
3 = 12 (air), 2 x 3 x 3 = 18 (water) and 3 x 3 x 3 = 27 (earth). Bober terms this
number theory ‘arithmetic theology’ or ‘arithmology’, a
discipline concerned with the ‘properties, virtues, and powers of numbers and
geometric forms for their bearing on the origins and nature of the universe’
(
There are several buildings in which architectural historians have noted that number
symbolism played an important role in the design.
The word church has two meanings: the one, a material building, wherein the
divine offices are celebrated; the other, a divine spiritual fabric, which is
the collection of the faithful. The Church, that is the people forming it, is
assembled by its ministers, and collected together into one place by ‘Him
who makes men to be of one mind in one house’. For as the material church
is constructed from the joining together of various stones, so is the spiritual
Church by that of various men. (
Like Suger, Durandus furthermore holds that
The piers of the church are the bishops and doctors who especially sustain the
Church of God by their doctrine […]. The bases of the church are the
apostolic bishops who support the frame of the whole church. The capitals of the
piers are the opinions of the bishops and doctors. For as the members are
directed and moved by the head, so are our words and works governed by the mind.
(
In the church of St. Michael in Hildesheim, built by Bishop Bernward (ca.
960–1022) from circa 1000 onwards, number symbolism also played an important
role in the design. This church has a nave of three bays, with four piers, and
twelve columns in total, thus representing the Trinity, four evangelists and twelve
apostles. To give substance to the symbolic meaning of the church as a living
‘Ecclesia’ founded on the apostles and saints, relics of saints were
placed inside the impost blocks above the capitals Schuffels (
The transept elevation of the church of St. Michael in Hildesheim. Photo: author.
By applying the number of creation to each transept arm, the symbolic meaning of the building as the house of God was enhanced. The perfection of the building and what it stood for was underscored by the use of the perfect number six and its divisors.
In 1990, Christopher Wilson (
The choir of the abbey church of St. Rémi in Reims, built under the abbacy of Pierre de Celle (from circa 1162 until 1181, when he became bishop of Chartres) and his successor Simon (1181–1198). Photo: author.
Wilson also states that it is probably not coincidental that the number three is
placed the highest, for this number stands for the Trinity. According to Wilson,
such Trinitarian symbolism was quite common at the time, and he refers to the
12th-century scholar Rupert of Deutz, who held that the tripartite elevation
referred to the Trinity, and to Abbot Suger of Saint-Denis who completed the chevet
of St. Denis in precisely three years and three months.
The choir clerestory of the abbey church of St. Rémi in Reims. Photo: author.
There are thus two systems at work here. The number of arches in each bay (1 + 2 + 3, from bottom to top) is based upon the perfect number 6 and its divisors, thus denoting perfection; at the same time the numbers 3 and 33 have symbolic meanings derived from scripture.
In addition to what Wilson wrote about the glazing programme at St. Rémi, it
should be noted that the 33 windows of the clerestory, which were unfortunately lost
during 19th-century alterations and bombing in the Second World War, showed on its
axis the figure of the enthroned Virgin with child (1 window). She was flanked by
the apostles (12 windows), evangelists (4 windows) and precursors of Christ (16
windows). Below these biblical figures were the saints and bishops of the diocese of
Reims, with St. Rémi taking the position under the Virgin Mary. The central
window of the tribune is still there and shows the Crucifixion (
The idea that number symbolism was used in the choir elevation of St. Rémi is compelling and raises the question of whether there are other examples of early-Gothic architecture where use might have been made of numerical systems based on perfect numbers such as 6 and 28 and their divisors, and on the number 7. In the following paragraphs I would like to introduce three more examples of parts of buildings where this may have been the case.
The choir of the parish and collegiate church of Notre-Dame-en-Vaux in
Châlons-en-Champagne (formerly Châlons-sur-Marne) was rebuilt,
probably from circa 1187, and was ready for use in 1217 (Figs.
The choir of the parish and collegiate church of Notre-Dame-en-Vaux in Châlons-en-Champagne, which was rebuilt, probably from circa 1187 onwards and which was ready for use in 1217. Photo: author.
The choir clerestory of Notre-Dame-en-Vaux in Châlons-en-Champagne. Photo: author.
Dieter Kimpel and Robert Suckale (
The elevation of the Noyon south transept differs from that of every other early
Gothic elevation. It is also unlike that of the choir and north transept of
Noyon Cathedral. As mentioned, the bishop entered the church via the south
transept. Interestingly, the choir has 28 windows in its double clerestory
(Figs.
The four-storeyed south transept of the cathedral of Noyon. Photo: author.
The south transept clerestory of Noyon cathedral. Photo: author.
The perfect number 28 may also underlie the design of Thomas Becket’s
corona at Canterbury Cathedral, which houses the top part of the saint’s
cranium that was struck off during his martyrdom on 29 December 1170.
Twenty-eight, the perfect number of 100, is made up of 4 times 7. The present
corona is a seven-sided rotunda, not counting the extra wide entrance side
(Figs.
The corona of Canterbury Cathedral. Photo: author.
The clerestory of the corona of Canterbury cathedral. Photo: author.
The number seven would have befitted all saints, for it denoted the seven gifts
of the Holy Spirit as well as the seven virtues, but in the case of Thomas
Becket, canonized in 1173, there is more. As Kay Brainerd Slocum shows in her
book,
According to exegetical tradition, the seven eyes represent the manifold
providence of Christ over his Church, or the seven gifts of the spirit of
God. The number seven is also used in scripture as a number of completion
and perfection; as the ‘perfect’ number, it implies a fullness
or completeness of spiritual endowments. […] Thomas’ life was
marked by the seven Tuesdays, and on the seventh, the Translation, he was
‘completed’ so to speak, and arrived at his place of eternal
rest. (
The Canterbury monks clearly considered the number seven to be important and befitting their most famous martyr. It would be entirely fitting, therefore, if it also played an important role in the design of the corona.
As Durandus noted in the 13th century,
everything that pertains to divine worship, the practices and vestments used by
the Church, are full of divine meanings and mysteries. Each and every one of
them, when examined with care and love by an individual who knows how to draw
honey from stones and oil from the hardest of rocks, pours forth a celestial
fragrance. (
The musings of Hugh of St. Victor over the significance of the Biblical measures of
Noah’s Ark are clearly those of an individual who could extract honey from
stones and his interpretation of the ark’s measurements would not have been
common knowledge, and so it is with many other forms of number symbolism. Apart from
interpretations of the number three for the Trinity, four for the four evangelists
and twelve for the apostles, knowledge of number symbolism and perfect numbers would
have been restricted to the intellectual elite. Interestingly, with the choirs of
St. Rémi at Reims and Notre-Dame-en-Vaux at Châlons-en-Champagne, the
south transept of Noyon Cathedral and the Canterbury Corona we are in fact dealing
with spaces intended for a restricted, mainly ecclesiastical audience. If number
symbolism were indeed applied here such an audience could have comprehended and
interpreted the intended symbolism. The writings of the same Durandus, however,
which I have quoted above, should be seen as a caution against reading too much into
a church structure. In his extensive treatise
Hugh of St. Victor (
Hugh of St. Victor (
Hugh’s explication of the ark’s dimensions is to a large extent taken from St. Augustine.
Hugh of St. Victor (
Hugh of St. Victor (
Augustine does not consider one and two to be real numbers. As Elizabeth
Sunderland explains (
St. Augustine (
St. Augustine (
St. Augustine (
Works of St. Augustine, ‘Letter LV’, chapter IX-16 and chapter X-17.
Quoted by Sunderland (
Works of St. Augustine, ‘Reply to Manichaeus’ Fundamental
Epistle’ X-11. Quoted by Sunderland (
Works of St. Augustine, ‘On the Gospel of John’, tract. XLIX-8.
Quoted by Sunderland (
See Sauer (
Hugh of St. Victor (
Isidore of Seville (
Isidore of Seville (
For the number of manuscripts containing Boethius’
Boethius (
St. Augustine (
St. Augustine (
Beda Venerabilis (
Hugh of St. Victor (
St. Augustine (
Boethius (
Book of Wisdom (11:21): ‘Omnia in mensura et numero et pondere fecisti’.
Masi in Boethius (
Boethius (
Boethius (
Isidore of Seville (
Hroswitha of Gandersheim (
Bonaventura (
Brussels, Bibl. Royale. MS II.1639.
For the date of the manuscript, see Smeyers (
Sunderland (
Suger of St. Denis (
The whole idea of the spiritual church as represented in the material church is even more clearly expressed in a 12th-century ivory reliquary from the church of Sayn (now in the Musée Cinquantaire in Brussels), which shows a miniature basilica, made up of ivory plaques, with towers at the west and east ends. Under the arches of the aisles the figures of the twelve apostles are represented. On the front and back more saintly figures are shown under the arches.
Abbot Simon’s epitaph refers to him as a builder of the church: ‘prudens sensu erexit ecclesiam […]’.
Wilson’s idea that number symbolism may have been used in the choir of
St.Rémi is compelling, but whether Pierre de Celle was the initiator of
this use of number symbolism is no more than an assumption. That the abbot was
the leading force behind the building campaign at St. Rémi, furthermore, as
suggested by Otto von Simson, Anne Prache and others, is also no more than an
assumption. As to the first assumption, there is no indication in the extensive
writings of Pierre de Celle, including his surviving sermons, treatises and
letters, of any interest in numerology. The second assumption, that Pierre de
Celle was the main patron of the rebuilding of St. Rémi, is based on the
following three brief references to the rebuilding of the abbey church, the most
extensive one occurring in an undated letter to Prior Inganus of Lapley (letter
144): ‘I am beginning to renew the chevet of our monastery, and with
God’s help the work already begun looks well and promises noble auspices
of the works to follow. I have not forgotten what you once said to me, as it
were in rebuke, that I strove to do other works and not to care for the
monastery. This word, even if it was said in passing, was not heard fleetingly.
For I have laid out a thousand pounds so far, and at the same time have
committed five hundred for a later stage, for the work on the monastery’
(
Unfortunately, Wilson gives no references for the passage concerning Rupert of
Deutz. For Suger’s Trinitarianism, see Suger of St. Denis (
The northwest window is blind, due to the presence of a pre-existing tower in the angle between choir and north transept.
For the dating of the various parts of the building, see Seymour (
Kimpel and Suckale (
There are at least six versions of the Office for the Translation. All of these
texts relate to the life of St. Thomas and his martyrdom, describe the actual
events of the translation and interpret them within a scriptural and prophetic
context, see Slocum (
The passage is taken from Zacharias 3:9.
Slocum (
Durandus,