Designing Orsanmiche: the rule rediscovered.

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Designing Orsanmichele: The Rediscovered Rule maria teresa bartoli

Transcript of Designing Orsanmiche: the rule rediscovered.

Designing Orsanmichele:

The Rediscovered Rule

maria teresa bartoli

37 bartoli Designing Orsanmichele: The Rediscovered Rule

1. Orsanmichele, Florence, east facade on Via dei Calzaioli

Digital composite by the Dottorato in Rilievo e Rappresentazione dell’Architettura e dell’Ambiente, Facoltà di Architettura, Università di Firenze, 1998; courtesy of Emma Mandelli

Orsanmichele is an enigmatic building, not only in its size and design but also in the ambiguous message con-

veyed by its form, which is consistent with no architectural type. The formal elegance of its design clearly identifies the monument as a masterpiece of late Gothic architecture (figs. 1 and 2), but the unusual double-axis layout (fig. 3) continues to disorient visitors, who may not recognize the building as a church even though a church has occupied the ground floor for centuries. In any case, the original function of this space was dif- ferent: it began as a grain market. What is now the museum on the upper two floors (figs. 4 – 6) was once storage space for grain that would be sold at the street-level market; under a later political regime, the upper floors were the repository of the archives of the grand duchy of Tuscany.1

The building was originally designed for several purposes. The first, that of a covered civic market and granary, was social. The second concerned its urban setting. During the fourteenth-century transformation of the city, Orsanmichele became a landmark because of its noteworthy location in relation to the new city walls. Its thirteenth-century miraculous image of the Virgin, which attracted popular devotion, gave a religious overlay to these other functions. In time, the religious function supplanted the market, which was transferred elsewhere. Yet because the market and the storage of grain are them-selves tied to the themes of charity and care of the poor, in concept the building was closely connected to a religious outlook from the start. Architecturally, this unique multipurpose building was conceived as a three-span dou-ble loggia, completely open, without continu-

ous walling on the ground floor. Its arcades were almost forerunners to the pilotis, or ground-level supporting columns, of modern architecture; they allowed access to the cov-ered market from all four directions.2 The grain-storage area was built over this as a tall and compact parallelepiped. The four facades, with opposites paired in identical design, line up exactly with the points of the compass. Over time, the terms loggia and palazzo were both associated with the struc-ture, which was never fully one or the other but had elements of both. The graceful, ele-gant triforium within the arches gave expres-sion to the structure’s transformation into a church. On a map of the city center (fig. 7), we immediately notice Orsanmichele’s position in relation to the oldest existing monument in the city, the Baptistery. The north-south axis of the lantern on the Baptistery’s pyra-midal roof passes right through the loggia. When the longitudinal axis of Orsanmichele, an exact orthogonal to the first axis, is extended toward the city walls, we surpris-ingly find two main components of the urban fabric — the Porta San Frediano to the west and the Porta La Croce to the east — exactly equidistant from Orsanmichele. Therefore, Orsanmichele’s very position is the defining element in its importance, the exact cardinal orientation of its facades and its outstanding height (more than forty-one meters) recalling its erstwhile role as a tower station positioned between two of the most important points in the city walls (fig. 8). Work began on the city gates in 1284, when construction of the first loggia in the open area known as Orto Sammichele and its square was also under way.3 The walls of Florence were designed and plotted out on paper following exact

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2. Orsanmichele, Florence, south facade, rectified photograph

Digital composite by the Dottorato in Rilievo e Rappresentazione dell’Architettura e dell’Ambiente, Facoltà di Architettura, Università di Firenze, 1998; courtesy of Emma Mandelli

39 bartoli Designing Orsanmichele: The Rediscovered Rule

geometrical configurations and were con-structed in accordance with the laws of the Arabic science of optics.4 To make the design geometrically exact and achieve the desired intersection of sightlines, it was necessary to establish stations from which to visually line up preestablished points, using procedures similar to those of navigation. Orto Sam- michele was one of the sites that could be used to construct a tower, perhaps on the site of the first wooden-roofed loggia, to locate the position of the western gate on the other side of the Arno and its exactly correspond-ing point on the eastern side of the city. The loggia-palazzo gave particular importance to the Via dei Calzaioli. Indeed, it preceded the construction of the Palazzo dei Priori and its square as the center of civic power, and it marked — exceptionally for a medieval city — a third pole of urban life, the city’s economic heart, physically positioned between the reli-gious and political centers.5

The enigmatic nature of the design of this Gothic structure has in some way deflected critical attention, which has focused largely on the extraordinary statues adorning the building’s exterior and the interior frescoes.6

3. Orsanmichele, Florence, plan of the ground floor

Author survey and drawing

4. Orsanmichele, Florence, plan of the second floor

Author survey and drawing

5. Orsanmichele, Florence, plan of the top floor

Author survey and drawing

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41 bartoli Designing Orsanmichele: The Rediscovered Rule

6. Orsanmichele, Florence, longitudinal section

Author survey and drawing

7. Detail of a map of Florence showing the center before the nineteenth-century transformations, from Federico Fantozzi, Nuova guida, ovvero, Descrizione storico-artistico-critica della città e contorni di Firenze (Florence, 1842)

This essay is devoted to the building’s design and aims to answer the following questions: What were the methodology and the concep-tual tools with which its designers approached this project? Considering that the complex articulation of the building’s structural com-ponents exhibits an understanding that is remarkable in the context of medieval archi-tecture, how was it regulated? The perfection

of the plan and execution of the structure, without deviation from symmetry, cannot escape observation. The arch openings are the same on all four sides (the only exception being an opening on the northwest, which is narrower because of the placement of a stair-case on the pier at that corner), as are the perfectly uniform piers, all governed by the axis of symmetry.7

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8. Geometry of the fourteenth- century city walls of Florence: (A) Baptistery; (B) Orsanmichele; (C) Palazzo dei Priori (Palazzo Vecchio)

Author drawing

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certainly not unfamiliar: the loggia was based on those of private residences, but doubled to six bays and completely open. The Sala d’Arme on the ground floor of the Palazzo dei Priori (Palazzo Vecchio), the largest en- closed civic space in Florence, consisted of six almost square, cross-vaulted bays with an almost perfect three-to-two ratio of length (45 braccia, or 15 × 3 braccia) to breadth (30 braccia, or 15 × 2 braccia) (figs. 10 and 11). That model for an interior space provided the guiding principle for the ground floor of Orsanmichele, except that the latter is boldly transformed into an open loggia. Super- imposed on this construction is a closed con-tainer, still reflecting the configuration of the urban palace, but compacting the effect by having the same number of windows as openings in the loggia. This is a modification of the classic patrician palace, which has five windows above three loggia openings. Increasing the spans of the bays from 15 to 18 braccia modified the module suggested by the Sala d’Arme. The span measurement

Form and order are the elements of this architecture. My aim here is to give an accounting of both, with the hope of shed-ding light on how design ideas in Gothic architecture were developed. The essay is based on detailed drawings I made between 1996 and 1998 from on-site surveys of the interior, and on an instrumental survey of the facades made by students of the doctoral program in surveying and architectural and environmental drawing of the architecture department of the Università degli Studi di Firenze.8 No measurement of the entire com-plex had been made since the publication of the survey by Riccardo and Enrico Mazzanti in the late nineteenth century; and, as docu-mented in the volumes by Diane Finiello Zervas, other studies have been partial.9

Private family palaces like the Palazzo Davanzati, tall buildings with three-bay log-gias on the street level for mercantile activi-ties, were common in the cityscape (fig. 9).10 So even though the design of Orsanmichele was a new one for a public structure, it was

9. Palazzo Davanzati, Florence, facade, measured drawing

From Emma Mandelli, “Palazzi Rinascimentali,” special issue, Studi e documenti di architettura 16 (1989): 95

10. Sala d’Arme, Palazzo dei Priori (Palazzo Vecchio), Florence, interior

Author photograph

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is the surface area of the building, and the second is the graphic devices used in describ-ing the project. These parameters are not well known to historians because they belong to a category of implicit knowledge that is evidenced indirectly in literary documents and mentioned only very generally in trea-tises because of their technical application.12 Both of these require a slight digression. In the chapter on measuring fields in Prac-tica geometriae (1220), Leonardo Fibonacci discusses methods for calculating area.13 Fibonacci’s unit of measure was the staioro (about 525 square meters), made up of 66 square pertiche, or square perches, measur-ing 6 piedi on a side. In Florence, Fibonacci’s perch is the land surveyors’ canna, 5 braccia in length: the staioro is thus equal to 1,650 square braccia. An analysis made in a survey of the Palazzo dei Priori demonstrates the purposeful reference of its design to precisely calculated measurements of area expressed in whole or common fractions of staiora (fig. 12).14 Even the loggia in the Piazza della Signoria was planned and measured so that it covers an area exactly equivalent to a staioro.15 Orsanmichele, a building planned and managed by the Florentine republic for the city, follows the same design practice. The building’s footprint covers a surface area that corresponds exactly to 1 1⁄3 staiora, or 2,200 square braccia, an easy number to use in making calculations. A famous early fourteenth-century Sienese drawing reproducing fairly closely Giotto’s campanile for the cathedral of Florence (fig. 13) is a good example of how Gothic architects worked out designs.16 The scale for the drawing is 1:48. One oncia (equal to one-twelfth of a braccio in the drawing) cor-responds to one Florentine canna mercantile (4 braccia).17 Thus one-fourth of an oncia stands for one braccio (fig. 13). An oncia equals 20 denari. Thus one denaro (2.43 mm) in the drawing represents 1⁄5 of a braccio, or 11.67 cm. The braccio can also be expressed in soldi (1 soldo = 2.918 cm; 1 braccio = 20 soldi). Each soldo can be further divided into 12 denari, a unit of division that works very

of 18 is a multiple of 6, and multiples of 6 are commonplace in medieval architecture: they were well integrated into the way medi-eval architects approached calculation. At this time in Florence, the system of measure-ment included both division by 5 (note that the Sala d’Arme module of 15 is divisible by 5) and division by 6.11 Thus, 36 (= 18 × 2) and 54 (= 18 × 3) could have been useful starting numbers for planning a rectangular space similar to the one in the Sala d’Arme, but larger. The statics of a series of arches in rectan-gular buildings like this are well known. To distribute stress, it is necessary to vary the sizes of the piers according to their positions: those on the perimeter of the building need to be larger in circumference than those on the interior, and on the perimeter, those at the corners need to be larger than those between. The footprint of the six-squared rectangle needs to be increased by the thickness of additional masonry forming a shoulder to contain the thrust of the large vaults. But what should this thickness be? Two parameters relevant to Gothic archi-tecture now need to be introduced. The first

11. Palazzo dei Priori (Palazzo Vecchio), plan of the ground floor by Arnolfo di Cambio

Drawing from survey directed by the author

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12. Palazzo dei Priori (Palazzo Vecchio), geometric layout of successive additions

Author drawing

8,250 sq. br.

5 staiora

2.5 staiora

2 staiora 3,300 sq. br.

4,125 sq. br.

10041

.25

1.25 staiora2,062 sq. br.

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well with multiples of 6 in architecture. On paper, on a grid broken down into sub- divisions of the braccio, the 36 × 54 braccia of the preestablished rectangular unit of Orsanmichele would be represented by 180 × 270 denari, or 15 × 22.5 soldi.18

The design for Orsanmichele was developed on the basis of these relationships (fig. 14). The predetermined area of 1 1⁄3 staiora (or 2,200 square braccia), if graphed on paper on a grid of 1-denaro units, is 55,000 square denari (fig. 15). To find the measurements for the sides of the building, including the thickness of the additional masonry, around a rectangle measuring 36 × 54 braccia, a simple equation can be set up containing a single unknown: (36 + x) × (54 + x) = 2,200, or (180 + x) × (270 + x) = 55,000. The latter computation yields a value for x of 13.8 denari. Consequently, 1 soldo (12 denari) needs to be added to the breadth and length of the rectangle in the design, only slightly more than an actual braccio, and this yields the thickness of the sides at the height of the piers above the base of the building. The cyma of the actual base in fact meets the required measurement. In the plan of Orsan-michele, the long side measures 282 (= 270 + 12) denari, and the short side 192 (= 180 + 12) denari.19

Now let us attempt to derive the dimen-sions of the architectural divisions of the facades. On the long sides these are formed by three arches. The first step is to establish the arch span. In Florence during the four-teenth and fifteenth centuries, the breadth of the windows of palaces was usually made equal to that of the alternating spaces.20 Thus, in the length L of a facade with three windows, each window opening and each filled area would be L/7 wide. The total of all window openings would then be 3⁄7 L, and the total of all filled areas 4⁄7 L. In order to reasonably broaden the window openings, we can (applying the scholastic principle that Roger Bacon called concordiae violentes, or “forced harmonizations”) invert the assigned values, allowing 4⁄7 L for the open-ings and 3⁄7 L for the filled spaces.21 Using the total length, 4 (282 ÷ 7) ÷ 3 = 53.71, which

13. Drawing of campanile, early fourteenth century, with scaled measurement by the author

Archivio dell’Opera del Duomo, Siena

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14. Plan of Orsanmichele, with a scale in divisions of the Florentine braccio, 1:48

Author drawing

can be rounded up to 54 denari. This mea-surement is exactly the drawn span of each arch and conveniently corresponds to exactly 4.5 soldi (fig. 16). On the short sides, which have two arches and measure a total of 192 denari each, the arches take up 108 (= 54 × 2) denari. We thus have 84 denari, or (84 ÷ 12) = 7 soldi, to

divide between the corner piers and the cen-tral one. A reasonable subdivision is 2 soldi at the center and 2.5 soldi on the corners. Applying these same measurements to the long side, we see that their sum is less than the total: (4.5 × 3) + (2.5 × 2) + (2 × 2) =

22.5 soldi = 270 denari, compared to the 282 denari allotted. But instead of creating a

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1,944 sq br =

48,600 sq den

2,200 sq br =

55,000 sq den

15. Orsanmichele, Florence, showing the length of the sides with and without additional wall thickness (br = braccia, den = denari)

Author diagram

16. Orsanmichele, Florence, solutions for calculating measure-ments of window openings: top, window openings 3⁄7 and wall areas 4⁄7 of the total facade length; bottom, the inverse, with window openings 4⁄7 and wall areas 3⁄7 of the total facade length

Author diagram

17. Orsanmichele, Florence, pro- portional relationships between window openings and wall spaces, final scheme

Author diagram

problem, this disparity provided the solution for the staircase that needed to be inserted for access to the upper floors. The propor-tions of the different elements are now defined (fig. 17). From this point on, the design unfolds with confidence and convic-tion (fig. 18). A scale drawing of the design can also logically solve the height of the structure: the three floors are proportioned according to the progression (7 × 3), (6 × 3), (5 × 3), in which the second term is the arithmetic mean of the other two (fig. 19). The height of the interior fasciae on the upper two levels is such that special proportions are provided for the window rectangle: 5:7 on the middle floor (side and diagonal of the square), and 5:8 (adjacent numbers in a Fibonacci series) for the upper floor (fig. 20). In the early 1950s Erwin Panofsky pub-lished a famous study on Gothic architec-ture in which he argued that architectural design reflected the scholastic mentality, dif-fused throughout the culture by religious preaching and the custom of public debate.22 The principle of starting with an axiom and arriving at a solution through deductive rea-

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18. Orsanmichele, Florence, figure 17 superimposed on figure 14

Author diagram

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19. Orsanmichele, Florence, dia-grammatic plan and elevation showing floor heights measured in half soldi

Author diagram

20. Orsanmichele, Florence, south facade, rectified photograph with overlay showing window propor-tions measured in half soldi

Digital composite by the Dottorato in Rilievo e Rappresentazione dell’Architettura e dell’Ambiente, Facoltà di Architettura, Università di Firenze, 1998; courtesy of Emma Mandelli

soning is certainly apparent in the descrip-tion of architectural design just provided. On the other hand, when the design of Orsanmichele is drawn on paper, results emerge that were perhaps unanticipated by its creators (for example, the nonalignment of the bays and the side arches and the dis-parity between the pairs of bays, which must have caused problems). These, how-ever, were accepted as the functional conse-quences of the irrefutable logic of compu-

tation as it was then understood. The results at Orsanmichele, which open up the prob-lem of working with a completely modular system of design, would provide succeeding generations of builders with the starting point, if not the solutions, for the correction of such irregularities.

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square of the Orto Sanmichele, where grain is sold, and it was paved and bricked, and it was a very useful, beautiful, and rich project.)

4. On the design of the Florentine walls and their rela-tionship to Orsanmichele, see Maria Teresa Bartoli,

“Quadrante geometrico e bastone di Giacobbe, un’ipotesi indiziaria per le mura trecentesche di Firenze,” Dise- gnare, idee, immagini 22 (2001): 39 – 50; Maria Teresa Bartoli, “La torre di Palazzo Vecchio e le mura di Firenze,” Disegnare, idee, immagini 25 (2002): 8 – 19; Maria Teresa Bartoli, “Un laboratorio dell’architettura gotica: Firenze, la città, le mura, il palazzo,” in Città e architettura, le matrici di Arnolfo, ed. Maria Teresa Bar-toli and Stefano Bertocci (Pieve di San Giovanni Battista, San Giovanni Valdarno, 2003), 17 – 53.

5. Villani 1991, vol. 2, book 9, ch. 26, 45 – 46:

Nel detto anno mcclxxxxviii si cominciò a fondare il palagio de’ priori per lo Comune e popolo di Firenze, per le novità cominci-ate tra ’l popolo e’ grandi, che ispesso era la terra in gelosia e in commozione, a la riformazione del priorato . . . e i priori che reggeano il popolo e tutta la repubblica non parea loro essere sicuri ove abitavano innanzi, ch’era ne la casa de’ Cerchi bianchi dietro a la chiesa di San Brocolo. E colà dove puosono il detto palazzo furono anticamente le case degli Uberti, ribelli di Firenze e Ghibellini; e di que’ loro casolari feciono piazza, acciò che mai non si rifacessono. E perché il detto palazzo non si ponesse in sul terreno de’ detti Uberti coloro che·ll’ebbono a far fare il puosono musso, che fu grande difalta a lasciare però di non farlo quadro, e più discostato da la chiesa di San Piero Scheraggio.

(In the said year 1298 construction on the palace of the priors for the Commune and people of Florence began. But because the whole place was in upheaval for all the news among the popu-lace and seigneurs about the reform of the prioriate . . . and the priors who ruled the people and the Republic did not feel safe where they lived before, which was in the house of the Cerchi of the White Guelfs behind the church of San Procolo. And so they founded the said palace where once there were the residences of the Uberti, Florentine rebels and Ghibellines. They made the square out of their houses so that they could never be rebuilt. And so that the said palace would not be placed on the land of the said Uberti, those who had been commissioned to build it, placed it on a slant, which was a great mistake instead of not building it square and farther away from the church of San Piero Scheraggio.)

6. Among the very few studies dealing with the monu-ment’s design is Gert Kreytenberg, “Or San Michel und die florentiner Architectur um 1300,” Mitteilungen des Kunsthistorischen Instituts in Florenz 27, no. 2 (1983): 171 – 192. The study individually identifies the Floren-tine models to which the monument refers and discusses its relationships with them. Elena Fossi, “La loggia di Orsanmichele,” in Bartoli and Bertocci 2003, 65 – 73, deals with the architecture of Orsanmichele.

notes

Translated from the Italian by Joan Marie Reifsnyder

1. The most complete and comprehensive account of the building’s structural vicissitudes is in Diane Finiello Zer-vas, ed., Orsanmichele a Firenze / Orsanmichele Florence, 2 vols., Mirabilia Italiae, vol. 5 (Modena, 1996) (hereaf-ter Zervas 1996a).

2. Gothic architects took full account of the Vitruvian idea of meaningful articulation of the function of build-ings. “In architectura haec duo insunt, quod significatur et quod significat. Significatur proposita res de qua dici-tur, hanc autem significat demonstratio rationibus doc-trinarum explicata.” (In all matters, but particularly in architecture, there are these two points: the thing signi-fied, and that which gives it its significance. That which is signified is the subject of which we may be speaking; and that which gives significance is a demonstration on scientific principles.) Vitruvius, De architectura, ed. Pierre Gros, translation and commentary by Antonio Corso and Elisa Romano, 2 vols. (Turin, 1997), 1:12. English translation: The Ten Books on Architecture, trans. Morris Hicky Morgan (Cambridge, ma, 1914), book 1, chs. 3, 5).

3. Giovanni Villani, Nuova cronica, ed. Giuseppe Porta, 3 vols. (Parma, 1990 – 1991), vol. 1 (1990), book 8, ch. 99, 562:

Nel detto anno, del mese di febbraio, essendo i Fiorentini in buono e pacefico stato, e la città cresciuta di popolo e di grandi borghi, sì ordinarono di crescere il circuito della città, e comin- ciarsi a fondare le nuove porte, ove poi conseguirono le nuove mura, cioè quella di Santa Candida di là di Santo Ambruogio, e quella di San Gallo in sul Mugnone e quella del Prato d’Ognisanti, e quella d’incontro a le Donne che·ssi dicono di Faenza ancora in sul Mugnone; il quale fiumicello di Mugnone alquanto dinanzi era adirizzato, che prima correa avolto per Cafaggio e presso alle seconde cerchie della città, faccendo molesto assai alla città quando crescea, e fecionvi su i ponti dinanzi alle dette porte, e rimase il lavoro di quelle innanzi che fossono a l’arcora, per la novella che venne in Firenze che ‘l prenze Carlo era stato scon-fitto in mare da Ruggieri di Loria e da’ Ciciliani. E in questi tempi si fece per lo Comune di Firenze la loggia sopra la piazza d’Orto Sammichele, ove si vende il grano, e lastricossi e amattonossi intorno, la quale allora fu molto ricca e bella opera e utile.

(In the said year in the month of February, the Florentines being in a good and peaceful state, and the city increased in popula-tion and in suburbs, it was ordered to extend the circumference of the town, and to begin to construct new gates, and by conse-quence new walls. These were those of Santa Candida beyond the church of Sant’Ambrogio, San Gallo on the Mugnone, and Prato d’Ognisanti, and that at the crossroads where there are the nuns called of Faenza, also on the Mugnone. The stream of the Mugnone had been shortly before straightened. Before, it wrapped around the Cafaggio and near the second walls of the city, causing a lot of trouble when the city grew. And they built the bridges before these gates, and there remained the work of the gates which had only gotten as far as the arches when the news came to Florence that Prince Charles was defeated in sea battle by Ruggieri di Loria and the Sicilians. And in these times, for the Commune of Florence a loggia was constructed in the

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17. Multiples of the braccio could come from either base 5 (the land surveyors’ canna, 2.918 meters, which is derived from the Roman pertica of 10 feet, or 2.956 meters), or base 4 (the canna mercantile, 2.33 meters). Both measurement systems play a role in Florentine architecture.

18. Subdivisions of the braccio are thoroughly explained in Bartoli 1994, 84 – 91.

19. The sides of the rectangle of the monument, over the bases of the pillars, are 32.63 and 22.30 m long.

20. This point is further developed and illustrated in the essays cited in note 7.

21. On Bacon, see Erwin Panofsky, Gothic Architecture and Scholasticism (Latrobe, pa, 1951), 67.

22. Panofsky 1951, 20 – 25.

7. Having surveyed and analyzed much of the monu-ment, I published two studies: “La fabbrica di Orsanmi-chele,” in La Lonja un monumento del ii milenio para el iii milenio . . . actas del congreso internacional: Lonjas del Mediterráneo, marzo, 1998, ed. Salvator Lara Ortega (Valencia, 1998), 285 – 294, and “Il modulo progettuale di Orsanmichele a Firenze,” Disegnare, idee, immagini 16 (1998): 9 – 18.

8. I thank the coordinator of the doctoral program, Emma Mandelli, for permission to publish the rectified photographs of the facades (figs. 1, 2, and 20).

9. Riccardo Mazzanti, Enrico Mazzanti, and Torquato Del Lungo, Raccolta delle migliori fabbriche antiche e moderne di Firenze (Florence, 1876), plates 80 – 90; Zer-vas 1996a.

10. On the Palazzo Davanzati, see Emma Mandelli, “Palazzi del rinascimento dal rilievo al confronto,” spe-cial issue of Studi e documenti di architettura 16 (1989): 87 – 98. I thank Professor Mandelli for permission to publish the measured drawing of the facade.

11. The Florentine braccio (58.36 cm) could be divided into 20 soldi or into 12 oncia. On Florentine systems of measurement before the metric system, see Angelo Mar-tini, Manuale di metrologia (Turin, 1883), 206 – 211.

12. That these parameters are technical does not mean that they are not also connected to a relevant symbolic meaning that emerges from their reference in scientific circles. For the impetus given by scientific knowledge to art in Tuscany, see Diane Finiello Zervas, “The Trattato dell’Abaco and Andrea Pisano’s Design for the Floren-tine Baptistery Door,” Renaissance Quarterly 28, no. 4 (1975): 483 – 503.

13. Leonardo Fibonacci, Practica Geometriae, in Scritti di Leonardo Pisano, ed. Baldassarre Boncompagni, 2 vols. (Rome, 1862), 2:2 – 4.

14. Maria Teresa Bartoli, Musso e non quadro, la strana figura di Palazzo Vecchio dal suo rilievo (Florence, 2007), 34 – 42.

15. Including the back wall and the west side, the Log-gia della Signoria measures 26 × 63.75 braccia (up to the exterior edge of the bench-parapet), equal to 1657.5 square braccia.

16. The drawing of Giotto’s campanile, in the Archivio dell’Opera del Duomo in Siena, is one of the few surviv-ing medieval architectural drawings. According to Decio Gioseffi, Giotto architetto (Milan, 1963), 13, 74 – 78, 90, it represents Giotto’s original design for the Campanile of Florence. It has also been studied by Emma Mandelli: see “Lettura di un disegno: La pergamena di Siena,” Studi e documenti di architettura 11 (1983): 99 – 134. See also Maria Teresa Bartoli, “Giuliano da Maiano e la canna ferrata: Il braccio come scalimetro,” in Giuliano e la bottega dei da Maiano: Atti del con-vegno internazionale di studi, Fiesole, 13 – 15 giugno 1991, ed. Daniela Lamberini, Marcello Lotti, and Roberto Lunardi (Florence, 1994), 84 – 86.