## Introduction

In his keynote address to the conference ‘Proportional Systems in the History of
Architecture’, hosted by Leiden University, 17–19 March 2011, Howard Burns
posited that the study of proportional systems today lacks ‘ground rules’. In
response, I present the following list of ten principles as a suggested framework for new
discussion of this complex topic.^{1}

## Principle 1: The word ‘proportion’ signifies two unrelated meanings

As I note in the introduction to this special issue of *Architectural
Histories*, ‘Introduction: Two Kinds of Proportion’, the subject of
architectural proportion today is literally defined by ambiguity because the very word
proportion simultaneously signifies two unrelated and in some ways opposite meanings.
Proportion technically denotes a mathematical ratio, which I call
‘proportion-as-ratio’, but in common usage it also connotes a broader meaning
that in 1723 Ephraim Chambers described as ‘a Suitableness of parts, founded on the
good Taste of the Architect’, or, an aesthetic assessment that I call
‘proportion-as-beauty’ (**Le Clerc
1723–1724, 1: 29**).^{2} Since the first
meaning is quantitative and the second qualitative, neither can influence the other
predictably and repeatedly, as the scientific method would require in order to prove a
causal relationship between them. When historians use the word proportion without
qualification, however, they invite their audiences to understand it as a fusion of the
concepts of ratio and beauty, and thus as an implicit assumption that certain proportional
ratios contribute beauty to architecture (see Principle 10). Thus, in my
‘Introduction’ to this special collection, I have proposed that the word
proportion be broken down into its incongruent component meanings that I have termed
‘proportion-as-ratio’ and ‘proportion-as-beauty’, and that one of
these meanings hereafter be specified whenever scholars use this word, either through the
use of the preceding terms or in the context of the discussion.

## Principle 2: Proportional systems must be described with verifiable measurements

In any study of proportional systems that refers to proportional ratios either found in existing buildings or in historical drawings, those proportions-as-ratio must be described accurately, as they really are, based on measurements recorded directly from the object. Those measurements, in turn, must be described in sufficient detail to enable the reader to go to the object and measure between the same points of measurement in order to verify them. When point cloud laser scans are cited, the method by which the data were acquired needs to be made sufficiently transparent to enable the reader to spot potential errors and understand the limitations of the survey. Also, when possible, the digital model or detailed images derived from it need to be made accessible to the reader in order to allow verification of all measurements extracted from it. Descriptions of proportions that are not supported by verifiable measurements can at best be considered subjective interpretations of appearances, which may have their own value but lack the foundation of scholarly documentation. Finally, verifiability requires not only that authors provide measurements, but that readers insist on them and learn how to evaluate them rigorously.

## Principle 3: Proportional systems are never executed exactly as intended

Since irregularity is the normal condition of architecture, some degree of dimensional discrepancy between the proportions the architect intended and the proportions of the built object are unavoidable. Potential structural degradation and measurement error must be considered integral to this problem. Part of the architectural historian’s job, therefore, is to acknowledge these discrepancies and estimate their extents based on internal evidence from the object. For example, my San Lorenzo survey reveals that all six columns in the western three bays (first phase) of the nave differ in height by no more than 1.5 centimeters from one to the next (Fig. 1 shows the nave arcade bay design that is repeated eight times in each nave arcade).

If the masons could achieve this level of precision in duplicating column heights, I
reasoned, they must have been able to execute the various width-to-height proportions that
they intended between each pair of adjacent columns within the same level of precision.
Therefore, in this part of the basilica I considered any proportional discrepancy (i.e.,
between an apparently intended proportion and the building measurements) that exceeded 1.5
centimeters to be abnormally large, meaning that either the architect did not intend that
proportion after all, or that some extenuating historical circumstance that has to be
identified caused the discrepancy. In other buildings, larger discrepancies appear to be
normal. Such estimates need to be made case by case. Thus at San Lorenzo, based on the
analysis described above, I determined that an 11- to 12-centimeter discrepancy between the
column shaft heights and the height of a square + root-2 rectangle proportion that I propose
was intended in each nave arcade bay (Fig. 1, left)
must be the result of construction error. Note that the tops of the column shafts, where
they meet the astragals, rise slightly above the top of the square + root-2 rectangle
overlay in the accurate scale drawing in Figure 1,
though they were probably intended to align perfectly with the top of this overlay (on this
construction error, see **Cohen 2013: 104–111**;
**2008: 33–37**).

## Principle 4: Proportional systems are executed in terms of local units of measure

Proportional systems may contain abstract proportional ratios, but once a building
construction project begins, those ratios become expressed in terms of the local unit of
measure. That unit may be specific to the municipality in which the construction takes
place, or to the construction site itself.^{3} I believe it
is safe to say that every historic building (and perhaps many pre-historic ones) that has
ever resulted from substantial expenditures of labor, resources and capital was built in
accordance with such a unit, regulated by the supervising authority at the time and place of
construction. Available evidence indicates that in historic buildings, measurements
expressed in these units and their subdivisions often add layers of numerical information to
proportional systems, in addition to any abstract ratios that the architect may have
intended.

For example, the basic framework of the San Lorenzo nave arcade bay proportional system
consists of three overlapping geometrical figures: the square, the root-2 rectangle, and the
dual diagon (Fig. 1). These figures embody abstract
ratios that we can describe mathematically as 1:1, 1:√2 and 1:2√2-1,
respectively. The preceding numerals and symbols, however, are modern conventions of
mathematical communication that are not represented as such in the early-15th-century San
Lorenzo proportional system. Only when we convert the measurements from centimeters to
Florentine *braccia* (1 *braccio* = 58.36 cm) do we see that
the architect of this proportional system described these ratios in terms of numbers of
*braccia*.^{4} Significant numbers
expressed as *braccia* in this proportional system include $1{\scriptscriptstyle \frac{2}{3}}$*braccia*, $9{\scriptscriptstyle \frac{2}{3}}$*braccia*, $13{\scriptscriptstyle \frac{2}{3}}$*braccia* and $17{\scriptscriptstyle \frac{2}{3}}$*braccia*. The architect used these numbers to form Boethian number
progressions and numerical approximations of irrational ratios. He thus used numbers of
*braccia* — the local unit of measure — to add numerical layers
of meaning on top of the geometrical layers in this proportional system (Fig. 1; see **Cohen 2013:
53–111**; **2008: 19–37**).

In his paper in this special collection and elsewhere, Stephen Murray notes that the two
tallest Gothic cathedrals, Amiens and Beauvais, express the number 144, a New Testament
dimension of the Celestial City, in their internal heights of the nave vaults above the
floor in terms of their respective local units of measure. These numbers appear to be
integrated into larger programs of iconographically significant numerical and geometrical
relationships in these buildings. Murray’s observation calls attention in a most
poignant way to the important connection between units of measure, iconographical
expression, and scale: In aspiring to this heavenly number symbolism, the Amiens builders
had a somewhat easier structural task before them than their Beauvais brethren, whose
soaring vaults collapsed in 1284, for the local Roman foot used in Amiens was slightly
shorter than the royal foot used at Beauvais, and the physical vault height necessary to
reach 144 local units of measure was consequently shorter by nearly six meters.^{5} In light of this discussion we see that much theoretical
commentary on proportional systems, such as Cesare Cesariano’s plates that show the
Cathedral of Milan’s cross section with geometrical overlays (**Cesariano 1521**), and the treatises of Vitruvius (1st century BC; see **Vitruvius 1914**), Alberti (**1485**), and Vignola (**1562**) that describe
proportional systems in terms of modules, neglect to acknowledge local units of measure, and
thus provide an incomplete picture of how proportional systems worked in practice.^{6}

## Principle 5: Belief-based proportional systems often communicate through
simultaneity^{7}

We have seen that numbers expressed in terms of local units of measure can interact not
only with individual dimensions, or with sets of dimensions associated with geometrical
figures, to add layers of iconographical meaning to an architectural composition (see
Principle 4). Additional layers of numerical meaning can be added through quantities of
repetitious building components. Since numbers of units of measure (such as
*braccia*) and numbers of building components (such as columns or bays) are
fundamentally different from one another, their combinations in proportional systems can be
profoundly interesting.

The portico of Brunelleschi’s Ospedale degli Innocenti, for example, displays a total
of ten columns. Indeed, it does so quite deliberately, for each end terminates with a full
column standing next to the end wall, rather than an engaged column or a pilaster more
typical of the period (Fig. 2). The distances between
the columns measure precisely ten *braccia* on center.^{8} Ten columns, ten *braccia*. You can see the columns and
count them. You cannot see the *braccia*. You can only understand their
quantities conceptually. This confluence of visible and invisible presences of the number 10
is consistent with *simultaneity* in medieval thought, perhaps a companion to
the period’s mystical conception of the visible world as but a reflection of a larger
macrocosmic order (**Crosby 1997: 46–47**).

Examples of this visible/invisible proportional simultaneity abound in the history of
architecture. In the Romanesque basilica of Santi Apostoli in Florence we find two nave
arcades, each containing 6 freestanding columns spaced 6 *braccia* on center
(Fig. 3).^{9}
Furthermore, the total number of columns in this basilica, 12, simultaneously represents the
number of apostles to whom the basilica is dedicated. The Cathedral of Milan floor plan is
16 bays long, and is composed of nave piers placed on a grid of 16 *braccia
milanesi* on center (Fig. 4).^{10} Brunelleschi’s basilica of Santo Spirito has a
nave 9 bays long, 3 transept and apse-like wings each 3 bays long (for a total of 9), and
simultaneously, plinth to plinth distances of 9 *braccia* (**Cohen 2013: 98–99, 105, 146**).^{11} Adding more simultaneity, the numbers 3 and 9 (the latter equivalent
to 3 by 3) symbolize the Trinity, of which the Holy Spirit (Spirito Santo), to which the
basilica is dedicated, is the third point. In the basilica of San Lorenzo, the 7 columns in
each nave arcade correspond to the entablature block frieze reliefs symbolizing the 7-sealed
Book of the Apocalypse (**Cohen 2013: 149**; **Cohen, forthcoming**).

## Principle 6: Proportional systems constitute historical evidence

Proportional systems can facilitate insights into historical ways of thinking, as in my
suggestion above that proportional simultaneity may have sometimes alluded to a larger
macrocosmic order. They can also constitute historical evidence of a more specific nature.
In the basilica of San Lorenzo, for example, I have identified an overall basilica
proportional system that seems to call for deep, approximately square nave chapels rather
than the present chapels that are only half as deep (Figs. 5, 6). This observation is consistent with
Giuliano da Sangallo’s drawing of the San Lorenzo floor plan that shows deep nave
chapels (Fig. 7), and thus may indicate that Sangallo
had inside knowledge that Brunelleschi intended just such chapels.^{12} Whether or not this interpretation of this new proportional evidence
is correct, it led me to search for precedents for this deep nave chapel scheme, and
ultimately to find the late Gothic basilica of Santa Maria del Carmine in Pavia (Fig. 8). Upon visiting this building I not only found notable
floor plan similarities with the basilica of San Lorenzo, as expected, but even more
striking above-ground similarities between the Carmine and Brunelleschi’s basilica of
Santo Spirito (Figs. 9 and 10). These observations leave little doubt that Brunelleschi visited the
Carmine and drew inspiration from it for his work, a significant and unexpected conclusion
to which the use of proportional systems as historical evidence led me (**Cohen 2013: 209–231**; **2009**).

Proportional systems, furthermore, provide sufficiently precise information to permit the
development of provisional taxonomies of proportional systems, and from them, possible
proportional system lineages from one building to the next. For example, in the nave arcade
bays of the basilica of Santa Maria del Fiore, Francesco Talenti appears to have created a
proportional system that includes overlapping rectangular proportions, plinth to plinth
measurement, an accurate whole number approximation of the ratio 1:√2, and a strategic
use of fractions (Figs. 11 and 12). Matteo Dolfini, in turn, appears to have used this proportional
system as a source of proportional raw materials, including specific dimensions and
proportions, for his design of the San Lorenzo nave arcade bay proportional system (**Cohen 2013: 231–242**; **Cohen 2010**). Dolfini died before he could execute his proportional system, but
Brunelleschi appears to have adapted it to his redesign of the basilica we see today (Fig.
1). Brunelleschi, in turn, later used this
proportional system as the starting point for his design of the proportional system of the
Santo Spirito arcade bays, this time making more substantial changes to it (Fig. 13).^{13} Thus we seem
to have a Santa Maria del Fiore–San Lorenzo–Santo Spirito proportional system
lineage that constitutes new historical evidence of previously unrecognized design
influences between these three buildings.

## Principle 7: Belief-based proportional systems have served no practical purposes

Elsewhere I have identified six intended purposes of belief-based proportional systems in
the history of architecture, including structural, aesthetic, iconographical and perhaps
philosophical categories (**Cohen 2013: 25–35, and ch.
6**).^{14} Of these six purposes only structural
stability can be considered practical, and belief-based proportional systems have never
conferred it. The others are all subjective qualities that cannot even be said to have ever
been present in any work of architecture except when someone has believed that they
were.

Modularization, which might seem to be the most logical practical purpose a proportional
system could have served, does not appear to have ever been used during the pre-engineering
period to a sufficient extent to have saved substantial time and money in construction. The
Roman practice of manufacturing column shafts to standard sizes once constituted the
standardization of one building component and therefore did not constitute a proportional
system, though it may have influenced proportional systems. Indeed, large-scale
modularization for the purpose of economization appears to be a phenomenon of mechanical
production associated with the Industrial Revolution. Industrial modularization usually
adopted the simplest possible units based on the prevailing unit of measure — either
the meter or the Imperial foot — and had no need for other units, such as Le
Corbusier’s complicated Modulor, which attempted to impose an arbitrary new module,
even if Le Corbusier developed it to be commensurable with the aforementioned prevailing
units of measure. Consequently the Modulor was never widely adopted.^{15}

## Principle 8: Proportional systems constitute design methods

Pre-engineering, belief-based proportional systems could facilitate an architect’s
creative process by generating forms and dimensional combinations from which he could make
selections. These selections could then be added to the indistinct mix of external design
influences and internal intuitions that fed the process. In the end these selections might
equally likely have been incorporated into the proposed design, or discarded along the
creative path leading to it. One need not agree with or even understand, for example, every
line in the patterns of ‘unfolding geometry’ that Robert Bork (2011) proposes
Gothic architects used in the designs of their complex works to be convinced that Gothic
architecture would not have been Gothic but for the use of such a method.^{16}

The beliefs of architects from the pre-engineering period that proportional systems were
present in already-existing structures, furthermore, encouraged those architects to measure
historical precedents both ancient and recent in order to discover those systems. Such
observations led to many important design developments not necessarily related to
proportional systems. Manetti’s claim, for example, that in their youth Brunelleschi
and Donatello went to Rome to measure ancient monuments in order to understand ‘le
loro proporzioni musicali’ (‘their musical proportions’), whether or not
true, is more significant for its suggestion that the two friends studied ancient Roman
monuments at all, rather than specifically their proportions (**Manetti 1976: 66**). The sketchbook of Villard de Honnecourt likewise
betrays, in at least one proportional diagram sketch, an effort to discover the principles
of French Gothic architectural precedents (**De Honnecourt
1959: 92–93, pl. 41**). Finally, Burns has aptly noted that during the
Renaissance, proportional systems helped to ‘slow down the design process’,
enabling the architect to ensure the design was exactly as he wanted it to be.^{17}

## Principle 9: Proportional systems contribute no aesthetic value to architecture

This principle, which has been treated in detail in my introductory essay to this special
collection of *Architectural Histories*, is included in this list as a point
of discussion, even though consensus as to whether or not it indeed deserves recognition as
a principle may never be reached. It is my hope that the present collection of essays will
inspire new and increasingly informed kinds of discussions of this question.^{18}

## Principle 10: Belief-based proportional systems communicate non-visual narrative content

The sole purpose of belief-based proportional systems has ever been to contribute meaning
to architecture (see Principle 7). Such meaning can behave like a narrative, through a
complex process that necessitates a longer explanation for this principle than the others.
Unlike simple number symbolism, which is usually clearly visible and immediately
comprehensible at least to educated observers (as in the 12 columns of the Santi Apostoli
nave symbolizing the 12 apostles; see Principle 5), proportional systems reveal themselves
over time, because by definition each consists of ‘a set of geometrical, numerical or
arithmetical correspondences’.^{19} A single example
of number symbolism can be part of such a ‘set’, but cannot constitute a
proportional system in itself. The various parts of a proportional system reveal themselves
one by one. Considering the simultaneity of tens in the Ospedale degli Innocenti discussed
above, for example (see Principle 5), the observer is most likely to discover first that
there are 10 columns. Only subsequently, after learning that the columns are spaced 10
*braccia* on center (either by measuring the portico or being told), might
the observer realize that the number of columns is the same as the number of
*braccia* between adjacent columns. The observer could also make these
discoveries in the reverse order. Either way, one discovery constitutes prerequisite
knowledge for the ultimate appreciation of the *simultaneity* of tens in this
proportional system. The observer cannot, however, discover both sets of tens
simultaneously.

More complex proportional systems unfold more slowly, and sometimes in a more prescribed
order. The San Lorenzo nave arcade bay proportional system, for example, took many years to
unfold for me (**Cohen 2013: 53–111**; **2008: 19–37**). First I discovered the overlapping
geometrical figures of square, root-2 rectangle and dual diagon (Fig. 1). Next, by converting my measurements to Florentine
*braccia*, I found that the latter two geometrical figures, which represent
mathematically irrational ratios, are very accurately approximated by the ratios $9{\scriptscriptstyle \frac{2}{3}}$ : $13{\scriptscriptstyle \frac{2}{3}}$ (which is equivalent to 29:41), and $9{\scriptscriptstyle \frac{2}{3}}$ : $17{\scriptscriptstyle \frac{2}{3}}$ (which is equivalent to 29:53), respectively. Lastly I found that these
numbers, together with others in the nave arcade bays that are also expressed in dimensions
of *braccia* ending in the fraction ${\scriptscriptstyle \frac{2}{3}}$ , form part of a Boethian number progression. The significance of each
layer of this proportional system only became evident to me in relation to my understanding
of the preceding layer. Today I require at least fifteen minutes to explain this
proportional system to even the most expert listener, and I must present its various layers
(consisting of geometry, number and arithmetic) one at a time, always beginning with the
geometrical layer, since an understanding of this layer is prerequisite to an understanding
of either of the others. Only after these layers have been unfolded sequentially can they be
appreciated simultaneously, in the mind of the observer.

In light of this temporally unfolding quality, proportional systems can be understood as
narratives, and perhaps plots, not only in the general senses of these terms as
communicators of meaning, but in their specific literary senses as well. A narrative,
according to Steven Cohan and Linda M. Shires, ‘recounts a story, [or] a series of
events in a temporal sequence’ that leads to an ‘outcome’, or,
‘closure’ (**Cohan and Shires 1988: 1, 65**).
This definition can be refined, these authors note, by referring to the novelist E. M.
Forster’s distinctions between the terms *story*, or ‘a narrative
that orders events temporally’, and a *plot*, or ‘a narrative
that orders events causally as well as temporally’ (**Cohan and Shires 1988: 58**).^{20} Thus in a
story, events take place in a particular order for numerous possible reasons, while in a
plot they do so because earlier events influence later events. The British architect John
Soane (1753–1837) recognized the fertile analogy between a similar notion of plot and
architectural experience by comparing the latter to a theatrical performance. According to
Caroline van Eck, Soane understood that ‘looking at a building and watching a play are
similar experiences […] because both [engage] *temporal* arts’.
The experiences of both theatre and architecture, she notes, ‘unfold in the course of
time’, and in both, ‘composition is built on the assumption that the viewer is
able to follow the plot, or grasp the connections on which dramatic or architectural
composition is based’ (**Van Eck 2007: 128**).^{21}

Proportional systems provide similar temporal experiences, though rather than based on
*looking*, as in Soane’s analogy, they are based on
*comprehending*, accomplished through non-visual mental investigations of
the geometrical, numerical and arithmetical relationships expressed in the building
measurements.^{22} These proportional relationships
could be just as laden with cultural associations as the physical objects that Soane uses to
communicate the plot, such as portraits, sarcophagi and sculptures, even though these
relationships are not physical. In the San Lorenzo proportional system, for example,
elements that carry cultural associations include the root-2 rectangle, the number pairs
that closely approximate this ratio (such as 29:41), and the Boethian number progression
identified by the *braccio* dimensions that end in the fraction ${\scriptscriptstyle \frac{2}{3}}$ . The root-2 rectangle attracted much attention in the 15th century because
its incommensurability presented a mathematical and philosophical dilemma. That dilemma,
thinkers as early as Plato realized, could be resolved through accurate whole number
approximations.^{23} Finally, the number progressions I
have termed Boethian carried particularly strong cultural associations because according to
Boethius they formed the basis of arithmetic, the first step in the
*quadrivium*, which led to the study of philosophy, the highest
intellectual pursuit; and this formulation remained a basis for later categorizations of
knowledge.

## Conclusion

If proportional systems are to be understood as narratives or plots, what were their
intended outcomes, and for whom were they intended? Proportional systems could have been
intended as private communications from the architect to himself, as forms of private
contemplation, or more likely, as semi-private communications from the architect and his
patrons to those *cognoscenti* capable of deciphering them. In either case,
they must have been important for the architects to have gone to so much trouble to craft
them. As communications, furthermore, they may have been similar to prayers or incantations,
for they may have facilitated contemplation of spiritual or metaphysical beliefs. Regardless
of the exact character of any particular proportional system, the participants in such
communications believed that proportional systems helped to bring about certain desirable
states in architecture, such as structural stability, *ordine*, and perhaps a
relationship with God and the macrocosm (Cohen, ‘Introduction: Two Kinds of
Proportion’ in this special collection). By imbuing lifeless stone with the agency of
communication, proportional systems thus contributed to making architecture more than merely
a functional art, but a deeply meaningful one. Deciphering these communications that reach
us from across the centuries, in the absence of explicit documentary evidence, remains a
central challenge for architectural historians. A big part of that challenge is to maintain
historical objectivity, and to avoid any temptation to believe the messages.