Marie-Thérèse
ZennerRue des Caves 41800 Fontaine-les-Côteaux, France
Whereas the mathematical content in Vitruvius' work is relatively
easy to discern - because it is explicit in the text - the collection
of Villard is much more difficult to understand, consisting essentially
of drawings which remain obscure except to those initiated in
the same oral tradition prevalent during the thirteenth century.
And yet these drawings can be cracked when studied within the
larger context of applied mathematics - the practical geometry
- from between the first and seventeenth centuries. One perceives,
not surprisingly, that the basic geometric knowledge of the medieval
architect derives ultimately from the
In the abundant and longstanding bibliography on the portfolio,[2] while there
has been occasionally question of practical building geometry,
in terms of an art, there has rarely been question of the mathematical
bases for this geometry, in terms of a science. Indeed, the geometer
Pappus, also of Alexandria (c. 290-c. 350 AD), had long ago advised
that it was impossible to achieve competence in both domains
and, if one were required to work with geometry, the best road
was learning through experience rather than theory.[3] According to Pappus and later, Vitruvius,
one of the only exceptions to this rule was Archimedes of Syracuse.[4] In this
context, Villard was with little doubt a 'working geometer' (as
per the French term But if Villard knew geometry, of what geometry are we speaking?
Historians have long privileged the Greco-Arabic tradition of
translation, established in the eighth century, and the increasingly
important transmission of these ancient texts to Europe, beginning
especially in the second quarter of the twelfth century. In recent
years, research by Wesley M. Stevens and Menso Folkerts has shown,
however, that the corpus of Euclidean plane geometry (books 1-4)
survived largely intact in Latin translations, appearing as early
as the sixth century, principally in works of Boethius and Cassiodorus
Senator ( The geographic location is perhaps relevant, as will be shown, to architectural history as well. There is no graphic documentation for ideas in design and construction during the Romanesque period (eleventh- to early twelfth centuries). The extant corpus of Romanesque monuments is perhaps the most significant group from the pre-modern period missing this type of external evidence. Not surprisingly, therefore, medieval architectural historians have come to rely on comparisons with the only two remaining documents on architectural design, albeit spanning a period of four hundred years, but encompassing the Romanesque period: we have the Plan of St.-Gall (c. 817/19) and the portfolio of Villard (c. 1220/35). Like Corbie, the Abbey of St.-Gall (now in the Germanic-speaking part of Switzerland) was of early Irish foundation, which, according to one school of thought, would suggest a long-standing respect for antique learning. Moreover, the Plan of St.-Gall is contemporaneous with the revival of geometric-gromatic texts at Corbie. Based on the spread of these manuscripts, later additions and commentaries, we may be justified, therefore, in considering an area between northern France and eastern Switzerland, with parts of Belgium and Germany, as a principal zone influenced by this revival. Four hundred years later, Villard too was associated in some
capacity with the Abbey of Corbie. His presumed birthplace, Honnecourt-sur-Escaut,
is situated relatively near Corbie in Picardy (roughly 60 kilometers
east, and 15 kilometers south of Cambrai). A Latin note added
to the manuscript by the so-called Magister II (c. 1250/60) states
that Villard worked on a ground plan (fol. 15r) with a certain
Pierre de Corbie. Other commentaries in the manuscript suggest
a link between Villard's portfolio and the early geometric-gromatic
texts. For example, on folio 18v of the portfolio (Figure 2,
at right), Villard (or his scribe) wrote: It is probable, therefore, that Villard (and Magister II) had access to written texts, having perhaps even studied them at Corbie itself. This hypothesis would necessarily modify the accepted notion that the builder's practical geometry was handed down by means of a strictly oral tradition, and that the oral aspect was propagated because they were all illiterate. It is much more likely that the oral tradition was privileged due to concepts of trade secret and the importance of having information ready-at-hand (that is, memorized) when working in rough conditions on-site. Fortunately for us, Villard's portfolio breached the tradition of corporate secrecy.
Figure
1). If we draw two circles
separated by their common radius (i.e., the base line as in Figure
1), we have created the third point required for constructing
an equilateral triangle, both above and below the given line.In our We suggest, therefore, that it is not at all by coincidence that Villard inscribed the text "here begin the force of lines for drafting" next to the drawing of the two flamingoes. From the two circles with a given finite straight line between their centers as radius, one can construct an equilateral triangle (or hexagon), a square, then a golden-section rectangle and most every other elementary geometric form, moving from point to point with a deductive logic of form, using only compass and straightedge.[11] There is yet another level of interpretation. On the walls of medieval buildings one very often finds a contemporary graffiti in the shape of a six-petalled flower (as seen in the overlay, Figure 2). We suggest that this compass construction is a mnemonic device for recalling the two flamingoes, in other words Euclid's proposition 1.1: firstly, the flower construction is easier to engrave than the forms of two birds; and secondly, whereas the straight lines of an approximately shaped hexagon would be theoretically easier to carve, the flower more directly and precisely recalls the almond-shape created by the overlap of two circles (see Figure 1). The almond-shape is often referred to as the Systematic analysis of those drawings in the Villard portfolio using principles of 'constructive geometry' will permit us to identify additional aspects of Euclidean geometry, which were handed down in the building trade largely through an oral tradition, and subsequently to identify, where possible, their use in architecture and mechanical engineering in the medieval period. The so-called technical leafs (fols. 20r, v) in the portfolio
were added as a palimpsest (i.e., after scraping off the drawings
on the original parchment leaf) by the later hand, Magister II
(c. 1250/60). The subjects of these two folios belong to the
gromatic corpus, a textual and practical tradition of 'measuring
problems' handed down as such at least since the texts by Heron
of Alexandria (c. 10-c. 75 AD). The Euclidean principles at the
basis of these drawings are also identifiable. In a detail at
the bottom of one leaf (fol. 20v,
Figure 4).
One can partly explain this by the fact that the structure works
by laterally distributing forces through a tiered system of blind
arcading in the interior and exterior faces of the nave and aisle
walls; one can partly explain it by the fact that, as we were
able to demonstrate by means of a standard structural analysis,
the stilted quadrant arches in the galleries are the first known
example of functional 'interior' flying buttresses, resulting
in an efficient distribution of remaining transverse forces to
the ground (i.e., along the line of a parabolic arc).[15]We have posited that the architect chose dimensions in such a way as to 'guarantee' in advance the successful distribution of load, in other words, a stable three-dimensional solid. There was no particular experimentation in view of the fact that the archaeological evidence suggests one design, quickly and coherently built, using the finest materials and craftsmanship available at the time. During an analysis of the plan derived from our measured survey, it became clear that three dimensions (or several simple fractions thereof: ½, 1/3, ¼…1/12) were used to determine both the plan and the elevation.[16] Briefly, we propose that the design layout on the ground began by setting out two points of a given finite straight line (Figure 4). This line determines the length of dimension A, and using these two centers draw two circles of dimension C. The former represents the minimum vault height (in the sanctuary and transept arms); the latter the maximum vault height (intrados of the octagonal dome in the crossing). This is not a case of Euclid's proposition 1.1, but our instinctive understanding is that additional overlap of the two initial circles would reinforce the structural stability in a large-scale three-dimensional construction. The rest of the plan design follows on the same principle
(for example, What interests us here is the question whether these three
proportions (A, B and C) have an inherent geometric relationship
and hence, was there a means of predicting a choice of three
"harmonically" proportioned measures? Earlier versions
of answers to this question have appeared in the "Latin
Euclid" and "Three Measures" articles and most
recently in If there is a royal road to geometry it begins with Euclid's proposition 1.1. Its importance in the highly innovative design of an eleventh-century church and in the unique thirteenth-century portfolio of Villard serve as evidence for a long-standing tradition of constructive building geometry, itself best understood in the context of the history of applied mathematics - the geometric, gromatic and mechanical engineering traditions - passed on by either oral or written means in a relatively continuous manner in Europe since Antiquity. Within the larger historical framework of traditions in applied mathematics, the Villard portfolio should be recognized as a key monument of practical geometry. As we have seen, study of the drawings reveals a coded use of Euclid, and we may conclude that Villard (and Magister II) were indeed versed in the mechanical arts, through experience if not through theory.
Pour la Science[ Scientific American, French Edition], dossier no. 37,
Les sciences au Moyen Âge (October 2002/January 2003),
108–9. English translation by the author with substantial
additional material.
return
to text[2]
The first mention of the portfolio (Paris, Bibliothèque
nationale, MS fr. 19093, c. 1220/35) dates to 1666. Refer to
online bibliography maintained by Villard scholar, Carl F. Barnes,
Jr., with continual updates and list of print and online facsimiles,
through links at AVISTA.
[3]
"The science of mechanics…has many important uses in
practical life, and is held by philosophers to be worthy of the
highest esteem, and is zealously studied by mathematicians, because
it takes almost first place in dealing with the nature of the
material elements of the universe. For it deals generally with
the stability and movement of bodies [about their centres of
gravity], and their motions in space…and it contrives to
do this by using theorems appropriate to the subject matter.
The mechanicians of Heron's school say that mechanics can be
divided into a [4]
Ibid, p. 619; Vitruvius, [5]
For bibliography, and summary of its relevance to architecture,
see Marie-Thérèse Zenner, "Imaging a Building:
Latin Euclid and Practical Geometry," in [6]
Zenner, "Latin Euclid" (as in n. 5), p. 234, n.71-72.
These inscriptions are contemporaneous with the production of
the manuscript. [7]
Text produced in Regensburg (MS Munich, Bayerische Staatsbibliothek,
Clm 14836, fols. 45r-52v); see Zenner, "Latin Euclid"
(as in n. 5), p. 234, n. 75. [8]
Roland Bechmann, [9]
Along with bilateral division (hence, symmetry), the concept
of rotation was a key element of medieval practical geometry,
as illustrated in several mnemonic devices on fol. 19v. For an
online facsimile of the Villard portfolio, click
here. [10] An excellent photo of the façade may be found
online by clicking
here. An equally excellent [11] The straightedge is the geometer's ruler, that is,
a ruler without measures to ensure that one works only with proportions.
It is more difficult to obtain the same conditions working in
a CAD environment. [12] [13] An earlier foundation on the site was reputedly
established by the Irish monk, Saint Columbanus, who was credited
as well with the founding of Corbie and St.-Gall. [14] For 3-D modeling of Saint-Etienne in Nevers, taken
as an "ideal" Romanesque church type, see http://perso.wanadoo.fr/fragile/Multimedia/Realisations/NEVERS/NEVERS.html
For a good aerial view, see http://www.amis-saint-jacques-de-compostelle.asso.fr/c/bo08.jpg,
and for interior and exterior detail photographs, see http://www.art-roman.net/nevers.htm.
[15] See Marie-Thérèse Zenner, "Appendix
5: Standard Structural Study," Methods and Meaning of Physical
Analysis in Romanesque Architecture: A Case Study, Saint-Étienne
in Nevers, PhD. dissertation, Bryn Mawr College, 1994 (Ann Arbor,
UMI, 1994, no. 9425215), pp. 370-93, esp. 388-93. [16] Most recently on the Nevers church, see Marie-Thérèse
Zenner, "A Proposal for Constructing the Plan and Elevation
of a Romanesque Church Using Three Measures," in [17 The use of the astrolabe for sighting was discussed
in the text [18] Zenner, "Latin Euclid" (as in n. 5); idem,
"Three Measures" (as in n. 16); idem, "Structural
Stability and the Mathematics of Motion in Medieval Architecture,"
in
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