Cohen, M.A.
Citation
Cohen, M. A. (2011, November 15). Beyond beauty : reexamining architectural proportion in the Basilicas of San Lorenzo and Santo Spirito in Florence. Retrieved from
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One of the underlying assumptions of Chapter 3 of this study is that all the nave arcade bays were intended to be dimensionally identical. They cannot be identical, however, for dimensional
variations from one bay to the next are inevitable. We therefore face the problem of deciding which bays to use in our proportional analysis. The masons may have intended one set of dimensions for all the nave arcade bays, but they gave us sixteen individual sets to choose from. Even after we eliminate the five easternmost bays of the nave (ten individual nave arcade bays) from dimensional consideration due to their notably lower quality of execution compared with the westernmost three bays (see Figure 2-1; and Chapter 2, sections 2.2 and 2.3), we are still left with six individual nave arcade bays to choose from.
The most logical solution might seem to be to take the average dimensions of all six bays. Proportional calculations based on averages, however, are unreliable because averages do not account for
mathematically significant conditions within the data, such as systematic error, or wide dispersion of extremes. The next most logical solution might seem to be to base our proportional analysis on the dimensions of one representative bay selected at random. By this approach, however, we would run the risk of selecting the one bay that least accurately represents the proportions the masons intended; or, conversely, we might tend to choose the one bay that best supports a particular hypothesis of interest to us—and indeed, the very possibility of such a bias would call into question the objectivity of the entire investigation. Finally, we might propose to undertake separate proportional calculations for each nave arcade bay. This approach, however, would execerbate rather than mitigate the problem at hand, for we would then have, in addition to six sets of measurements to choose from, six sets of proportional calculations as well.
Once we resolve the problem of how to calculate the proportions of dimensionally non-identical
nave arcade bays, we must confront the problem of how to interpret these calculations—i.e., how to
correlate the degree of dimensional consistency from one bay to the next, with the degree of precision
with which particular proportions correspond to those measurements. How closely, for example, must
the proportion 1:√2 correspond to the true width-to-height proportions of each nave arcade bay,
according to the points of measurement shown in Figure 2-2, for us to consider that proportion a likely
reflection of the masons’ intentions? Arbitrary evaluation of quantitative data, such as the establishment
of, say, plus-or-minus 5% as an acceptable tolerance level for proportional calculations made from
building measurements, simply because such a figure intuitively seems appropriate, undercuts the mathematical advantages of recording precise measurements at all.
1The best strategy for resolving these problems, albeit an imperfect one, is to turn to the science of statistics. At the simplest level of analysis, statistics can be descriptive. Calculating the “standard deviation,” for example, measures the dispersion of data relative to the mean. The lower the standard deviation, the more closely clustered are the magnitudes of the measurements around the mean. We thus have a quantitative basis for evaluating dimensional variations within a set of repeating dimensions. Our task at San Lorenzo, however, is less one of describing the survey data per se, than of evaluating
inferences we would like to draw from them. Such inferences involve multiple layers of uncertainty, due to the unknown measurement and construction errors embedded in the data. Because of these
uncertainties, our proportional analysis must be expressed in the non-definitive terms of confidence, probability, and ranges of values.
A computer spread sheet program designed for this study gives us a mechanism for
quantitatively testing proportional hypotheses.
2It does so by ruling out proportional values that do not fall within calculated ranges, based on a confidence level that we choose. This spread sheet program takes into account assumed estimates for construction and measurement error, and thus provides the most accurate possible estimates of the nave arcade bay proportions, in light of the bay-by-bay
variations in the arcade measurements that we have observed.
3The use of a spread sheet program such as this in a study of architectural proportion has two notable limitations: First, the numbers of nave arcade bays that we can examine at San Lorenzo—fourteen, if we consider the full nave, or six, if we consider only the earlier nave construction phase—are not truly statistical populations. Ideally, hundreds of repeating elements at a minimum should be analyzed. Second, statistical analysis is best applied to data that is not subject to the whims of human nature. Variations in mass-produced machine tool dimensions, for example, would be more conducive to statistical analysis than six (or fourteen) column heights, which might exhibit variation simply because an otherwise careful mason happened to be having a bad day. Nevertheless, the computerized analysis presented here is useful when treated as one component in a range of documentary and observation-based historical evidence, all of which must be evaluated critically.
Let us consider an example of how the statistical spread sheet works. In Chapter 2, I note that a
dual diagon can be inscribed within each nave arcade bay, measured plinth-to-plinth in width, and from
the floor to the tops of the entablature blocks in height (Figure 2-34). The spread sheet requires that we
enter all the height and width measurements of interest into the “numerator” and “denominator” cells, and that we choose a “confidence level” (Figure 8.4-1).
4It then calculates the standard deviation for each set of width and height measurements per bay, and uses these figures to calculate an upper and lower limit of a “probable proportion range” within which the proportion intended by the masons is assumed to fall.
5The greater the confidence level, the farther apart the limits of the probable proportion range will be. In other words, the more certainty we demand of our findings, the broader will be the range of proportions that we will have to consider. Statisticians typically choose a confidence level of 95%.
Let us first analyze my dual diagon hypothesis based on the measurements from all sixteen nave arcade bays. We enter the plinth-to-plinth distances and total order heights into the appropriate cells and, based on a confidence level of 95%, the spread sheet calculates a confidence interval of 1:1.812 to 1:1.827 (Figure 8.4-1). Since the dual diagon proportion, 1:1.828… (or, 22-1), falls outside this range (albeit just barely), we may state with 95% certainty that the spread sheet has rejected this proportion, and, therefore, that the masons did not intend to use it here. Of course, there is a 5% chance that the results of this calculation are incorrect—i.e., that the masons intended this proportion after all. We cannot eliminate this uncertainty. If we enter a higher confidence level into the spread sheet, say, 99%, the probable proportion range increases to 1.809 to 1.830 (Figure 8.4-2). The dual diagon proportion now falls inside the interval, but so do many others; and due to this greater inclusiveness, the test becomes less effective at screening out the proportions the masons did not intend.
Due to the nature of statistical studies, the probable proportion range can only be used to reject proportions that do not fall within it. It cannot be used as evidence that a particular proportion that falls within it is more likely to have been intended by the masons than one that falls outside of it.
Nevertheless, since the proportions the masons intended will presumably fall within the range, the more proportions that fall outside it, the better, so that we can begin to isolate the intended proportions—
provided, of course, that intentional proportions exist here at all. In short, we need to maximize our certainty (the confidence level), while minimizing the range of possible proportions (the probable proportion range). In this effort, our observations regarding the construction history of the nave arcades provide a significant advantage, as we will now see.
A mediator between the confidence level and the probable proportion range is the standard
deviation. Generally, the lower the standard deviation, the narrower the probable proportion range for a
given confidence level will be. Thus, had the masons built the basilica of San Lorenzo with greater care,
the dimensional irregularities would presumably be fewer and smaller, the standard deviation would be lower, and the probable proportion range would be narrower relative to the confidence level. We would therefore be able to reject a broader range of proportions as candidates for the ones the masons intended, and we would be able to do so with a high degree of confidence. Although we cannot go back in time and implore the masons to be more careful, Cosimo’s apparent demands for speedy completion of work notwithstanding, there is of course another way to significantly reduce the standard deviation: we can limit our data to the measurements of the more carefully-constructed western portion of the nave.
Returning now to our dual diagon test, and eliminating the ten easternmost nave arcade bays from consideration, the standard deviation of the total order height drops substantially, from 2.32 cm.
(Figure 8.4-1) to 0.57 cm. (Figure 8.4-3), and that of the plinth-to-plinth distance, from 4.24 cm. (Figure 8.4-1) to 0.45 cm. (Figure 8.4-3). More importantly, the probable proportion range narrows from
1:1.812–1:1.827 (Figure 8.4-1) to 1:1.824–1:1.827 (Figure 8.4-3). As it turns out, eliminating the measurements of the ten easternmost nave arcade bays does not affect the upper limit of the probable proportion range, which remains at 1:1.827. Thus, with the confidence level maintained at 95%, even these revised spread sheet calculations reject the dual diagon proportion. With the confidence level increased to 99%, however, this proportion falls within the probable proportion range (Figure 8.4-4), as it did when we considered the measurements of all sixteen nave arcade bays at this confidence level (Figure 8.4-2). This time, however, the probable proportion range seems to be quite narrow, at 1:1.823—1: 1.828 (Figure 8.4-4). How are we to interpret these results?
First we must note that we are now dealing with exceedingly small tolerances. If each total order height were to measure just 0.5 cm. (0.05%) taller than it does at present, the dual diagon proportion would fall within the confidence intervals in all four of the preceding tests. Such a small and consistent insufficiency in the heights of the orders could be the result of mortar shrinkage. In each order there are eight mortar joints (nine in the case of Col. 4; see Chapter 2, sections 2.2 and 2.3), and each would have had to shrink by slightly less than one millimeter to account for this shortfall—a reasonable estimate for mortar joint shrinkage in medieval construction.
6Thus, for now let us simply consider the dual diagon hypothesis to be a promising one, and defer final judgement of its merit until we have completed similar spread sheet tests for the other proportions in the nave arcade bay that appear to be related to it both geometrically and historically.
For our next spread sheet test, let us return to our very first proportional observation, the
comparison of the plinth-to-plinth distances in each nave arcade bay, measured first between the nearer
edges of the column plinths, and then between the farther edges. Note that the measurements of the two nave arcade bays nearest the crossing square cannot be included in this test because each terminates with a pilaster on the west side. Thus, the second plinth-to-plinth measurement in question (taken between the farther edges of the column plinths) is not found in these bays. Based on the measurements of the
remaining four nave arcade bays that date to the earlier nave construction phase, we find very low standard deviations of 0.17 cm. and 0.19 cm., and a probable proportion range of 1:1.413—1:1.414 (Figure 8.4-5). This test result could not be more supportive of my root-2 rectangle hypothesis, which calls for a proportion of 1:1.414… (1:√2).
When we next test how closely the height of a root-2 rectangle, inscribed between each pair of adjacent column plinths, corresponds to the heights of the column shafts, however, the results are less encouraging (Figure 8.4-6). The standard deviations of approximately 0.5 cm. indicate that the column shaft heights and intercolumniations were executed with great consistency from one to the next, thus making ambiguous test results unlikely, and the very narrow probable proportion range of 1:1.433—
1:1.436 would seem to definitively exclude the root-2 rectangle proportion of 1:1.414… (1:√2) as an accurate description of the proportions of the nave arcade bays, when measured plinth-to-plinth. Indeed, the discrepancy represents approximately 11-12 cm. of excess column shaft height, which is quite substantial when we observe that the maximum height difference between any two column shafts within the western six nave arcade bays is just 1.5 cm.
7When we next test how closely the height of a square, circumscribed about the farther edges of adjacent column plinths, corresponds to the heights of the column shafts (Figure 80), we obtain similar results. For a true square, the upper and lower ends of the probable proportion range should both be 1. Instead, both are slightly larger, indicating excess column shaft heights of approximately 11-12 cm.
In order to interpret the preceding four test results correctly, we must now clarify one important
distinction, already implicit in this discussion: These tests help us to identify the masons’ intentions, not
necessarily the architect’s intentions. After all, if the masons could make all the column shafts consistent
in height, with a maximum variation of 1.5 cm., we may assume that they could also make each column
shaft conform to the height dimension they intended within the same tolerance. Thus, judging from the
preceding tests, the masons appear to have intended to execute the dual diagon proportion and the 1:√2
relationship between the two plinth-to-plinth dimensions that we examined in the first two tests; but
neither the root-2 rectangle nor the square examined in the last two tests. We need not come to the same
conclusion with regard to the architect’s intentions, however. We have seen that Brunelleschi’s
involvement with the basilica probably ended by early 1429 (modern style), or, nearly two decades before construction of the nave arcades began—ample time for his specifications for the nave arcade proportions to have become corrupted before finding their way into the masons’ hands. The masons may not have intended the overlapping square and root-2 rectangle to align with the tops of the column shafts (Figure 2-2), but did Brunelleschi?
8The three criteria that we established earlier for the purpose of evaluating the likelihood that particular proportions reflect the architect’s intentions, now come crucially into play (see Chapter 2, section 2.1). The first criterion, let us recall, emphasizes the spread sheet test as the fundamental standard for evaluation, but allows for flexibility in the interpretation of this test if a convincing explanation for dimensional discrepancies can be found through historical research. Since the four proportional relationships examined in the preceding tests are closely interrelated, and since two of them conform to the measurements within—or very nearly within—the accuracy level established by the tests, the failure of the other two (the root-2 rectangle and the square) to similarly conform, by a significant 11-12 cm., seems likely to be the result of an error brought about by some historical circumstance, yet to be identified.
In order to continue toward our objective of identifying the proportions Brunelleschi intended for the nave arcades, we will temporarily ignore these 11-12 cm. discrepancies. Doing so will allow us to focus on the second and third of our evaluation criteria, which are concerned with discovering the particular historical contexts within which any set of intentional, rather than coincidental, proportions would likely be situated. We can then revisit the column shaft height discrepancies, and attempt to determine their cause (see Chapter 2, section 2.7).
Before we proceed to these context studies, however, which will explore the subjects of geometry, number, and arithmetic as they relate to our hypothesis, a word about other proportional possibilities is in order. Thus far we have examined only one interpretation of the proportions of the San Lorenzo nave arcade bays, because it is the only one to show serious promise as a likely reflection of the architect’s intentions. This hypothesis consists of three overlapping geometrical figures—the square, the root-2 rectangle, and the dual diagon—all of which can be described in terms of the four proportional relationships tested above. Although other scholars have provided alternative interpretations of the proportions of the San Lorenzo nave arcade bays, none of them meet the three criteria established for this study. Saalman’s proposal, for example, that each bay of the nave arcades conforms to the
proportions of a root-5 rectangle inscribed in the clear (i.e., between adjacent column shafts) in width,
and from floor level to the undersides of the arches in height, fails to satisfy Criterion #1 (see Chapter 2, section 2.1), for it fails to meet the requirements of the spread sheet test by approximately 5-7 cm. It furthermore fails to warrant the exception allowed under Criterion #1, because it neither appears in documentary sources relevant to the early fifteenth century (Criterion #2), nor does it appear to have any significant relationship to other proportions in the basilica (Criterion #3).
9The same may be said of numerous other potential proportional hypotheses, whether based on geometrical or modular
proportions. My methodology, therefore, which combines rigorous observation-based and documentary
analysis, helps to keep our attention focused on the historically most likely proportions, and away from
the unproductive distractions offered by the many potential coincidental proportions that inevitably
compete for our attention.
1
See, for example, the following arbitrary tolerances established by Saalman: “Any suggested proportional relationship must be demonstrable in carefully surveyed measurements of the buildings involved, with a tolerance of no more than 15 cm in large dimensions and no tolerance at all in small dimensions. A small margin of error in laying out or surveying dimensions from 5 to 40 metres cannot be excluded. In dimensions up to 2 metres no tolerance is permissible. Between 2 and 5 metres
discrepancies of perhaps 2 cm may be allowed.” (Howard Saalman, Filippo Brunelleschi: The Buildings, University Park, Pennsylvania, 1993, p. 361).
2
The spread sheet program was designed by James E. Georges of Statistics Unlimited, Inc., Wellesley Hills, Massachusetts. I thank Mr. Georges, as well as Stephen Blyth, Department of Mathematics, Imperial College of Science, Technology & Medicine, London, who generously provided additional advice.
3
The standard deviations give estimates of the combined measurement and construction error. If the measurements are normally distributed, which is one of our assumptions, 95% of all measurements should be within two standard deviations above and two standard deviations below the mean.
4
Since each bay contains one width dimension but two height dimensions, one for each total order height measured at each column, I have entered the average total order height for each bay into the spread sheet. Since the spread sheet can accommodate only one numerator and one denominator, and since only two numerator (height) measurements are involved, in this case my use of the average is mathematically appropriate.
5
The spread sheet formulae are as follows: For the mean of both numerator and denominator:
=AVERAGE(R[-15]C:R[-2]C); for the standard deviation of both numerator and denominator: = STDEV(R[-16]C:R[-3]C); for the lower limit of the probable proportion range: =(-B-SQRT(B^2-
4*A*C_))/2/A; and for the lower limit of the probable proportion range: (-B+SQRT(B^2-4*A*C_))/2/A.
According to James E. Georges (see n. 123, above) these calculations are based on Fieller’s Theorem, as discussed in: E. C. Fieller, “Some Problems in Interval Estimation,” Journal of the Royal Statistical Society 16 (1954): 175-185.
6
John Fitchen, Building Construction Before Mechanization (Cambridge, Massachusetts: MIT Press, 1994): 80.
7
The approximately 11-12 cm. discrepancy has been calculated for the purpose of obtaining a quick,
reasonably accurate estimate, as follows: The mean plinth-to-plinth dimension, 563.883 cm., is
multiplied by the hypothetical ratio of 1:√2 (i.e., 1.414), to obtain a hypothetical column shaft height of 797.331 cm. This figure is then subtracted from the heights of the shortest and tallest column shafts in the nave, respectively, those being: Column Shaft 8, which measures 808.0 cm. high, and Column Shaft 10, which measures 809.5 cm. high (note the 1.5 cm. height difference between them).
8
Note that the question of whether Brunelleschi inherited the the nave arcade bay proportional system from Dolfini and adapted it to his own design, or whether Brunelleschi designed it entirely, is not relevant to this discussion, which treats of the instructions Brunelleschi gave to the masons. Thus, the authorship of the proportional system is not at isssue here.
9
