1 Frankl, Paul, ‘The Secret of the Medieval Masons,’ Art Bulletin, 27 (1945), 46–60.Google Scholar He expanded his ideas in The Gothic (Princeton, 1960), pp. 145-152.

2 Saalman, Howard, ‘Early Renaissance Theory and Practice in Antonio Filarete's Trattato di Architettura ,’ Art Bulletin, 41 (1959), 89–106.CrossRefGoogle Scholar

3 White, John, ‘Measurement, Design and Carpentry in Duccio's Maesta,’ Parts l and 2, Art Bulletin, 55 (1973), 334–366 Google Scholar; 547-569; ‘Donatello's High Altar in the Santo at Padua,’ Parts 1 and 2, Art Bulletin, 51 (1969), 1-14, 119-141.

4 Baxandall, Michael, Painting and Experience in Fifteenth Century Italy (London, 1972), pp. 82–102.Google Scholar

5 Shelby, Lon R., ‘The Geometrical Knowledge of Medieval Master Masons,’ Speculum, 42 (1972), 395–421.CrossRefGoogle Scholar For the education of English masons, see Shelby, , ‘The Education of Medieval English Master Masons,’ Medieval Studies, 32 (1970), 1–26.CrossRefGoogle Scholar

6 See Baxandall, , Painting and Experience, pp. 86–87 Google Scholar; Sapori, A., ‘La cultura del mercante medievale italiano,’ Studi di storia economica (secoli xiii-xiv-xv), 1 (1955), 66–67 Google Scholar; Fanfani, A., ‘La préparation intellectuelle et professionnelle à l'activité economique in Italie du XV^e au XVI^e siècle,’ Le Moyen Age, 57 (1951), 327–346 Google Scholar; Bee, Christian, Les marchands écrivains à Florence 1375-1434 (Paris, 1967), pp. 383–393.Google Scholar

7 For an excellent introduction to the subject with particular emphasis on the Quattrocento and Cinquecento, see Goldthwaite, Richard A., ‘Schools and Teachers of Commercial Arithmetic in Renaissance Florence,’ Journal of European Economic History, 1 (1972), 418–433.Google Scholar

8 Many of them are contained in a Quattrocento Trattato dell'Abbaco by the Florentine Maestro Benedetto: cf. Arrighi, Gino, ‘Il Codice L.iv.21 della Biblioteca degli Intronati di Siena e La “Bottega dell'Abbaco a Santa Trinita” in Firenze,’ Physis, 7 (1965), 369–400.Google Scholar

9 Goldthwaite, ‘Schools and Teachers,’ p. 429.

10 These manuscripts are listed in Appendix 1 at the end of this article.

11 Bib. Ric. Codice 2404 and Bib. Naz. Magi, xi 74 are unpublished. Bib. Naz. Magl, xi 86 has been published: cf. Arrighi, Gino, Paolo dell'Abbaco, Trattato d'Aritmetica (Pisa, 1964).Google Scholar Arrighi also treats Bib. Naz. Magi, xi 87 briefly in ‘Due Trattati di Paolo Gherardi Matematico Fiorentino. I Codici Magliabechiani ci.xi. nn. 87 e 88 (Prima metà del Trecento) della Biblioteca Nazionale Firenze,’ Atti della Accademia delle Scienze di Torino, Classe di Scienze Morale, Storiche e Filologiche, 101 (1967), 61-82. Calculations in these trattati, and indeed in all those in the three Florentine libraries, are carried out in Arabic numerals. The opening chapters of most trattati provide a table listing Roman numerals and their Arabic equivalents for the reader's benefit.

12 Additional evidence that Florentine goldsmiths of the Trecento had training at a scuola dell'abbaco is offered by Statute 133 of the Arte di For Santa Maria. Written in 1335, it states that all goldsmiths who were members of the Arte would be required to keep a clear and concise book of debits and credits in their shop. It was to be kept according to the legal mercantile rules of the city; failure to comply was punishable by a 100 soldi fine. Cf. Dorini, U., Statuti dell'Arte di Por Santa Maria del Tempo della Repubblica (Florence, 1934-1942), p. 148.Google Scholar At least some training in accounting was offered in the scuola dell'abbaco, since a number of Trecento trattati, including one dated 1315, Bib. Ric. Codice 2252, provide examples of the form used for a debits and credits book.

13 See footnote 24 below.

14 Codice Magi, xi.87, fol. 60; Codice Magi, xi.86, fol. 36^v.

15 Codice Magl, xi.87, fols. 41^v, 44.

16 Cf. Frankl, ‘Secret of the Medieval Masons,’ p. 58.

17 Ibid.

18 Codice Magl. xi, 87, fols, 37, 38^v; Codice Magl, xi.74, fols. 40-41, 42-42^v, 45, 48^v, 51.

19 Goldthwaite, ‘Schools and Teachers,’ p. 428.

20 The documents for Andrea Pisano's door have been published by 1. Falk, , Studien zu Andrea Pisano (Hamburg, 1948).Google Scholar As early as 1322 a plan existed to cover the wooden doors with gilded copper or metal. Presumably two new sets of wooden doors were ordered shortly before the bronze one, since there is a reference to work beginning on ‘le porte di legname’ in a document dated January 13, 1330, nine days before Andrea began work.

21 Falk, , Studien zu Andrea Pisano, pp. 40–58.Google Scholar

22 The height of the door opening is only 1.4 cm under the actual altitude of an equilateral triangle with a side equal to twice the width of the door opening.

23 Codice Magl, xi.86, fol. 22; Bib. Ric. Codice 2161, fol. 61^v; Bib. Laur. Codice Ashburnham 1662, fol. 115^v; Bib. Laur. Codice Plutcus 30.26, fol. 49^v.

24 Andrea used either the Pisan braccio or its Florentine equivalent, the bractio di terra, for the actual dimensions of the door. According to Uzielli, G., Le Misurc Lincari Mediocvale e L'Effigie di Cristo (Florence, 1899), p. 13 Google Scholar, the Florentine braccio di terra originated from the Pisan braccio. The Pisan braccio was derived from the legal Palestinian braccio, which was introduced to the West during the crusades of the twelfth and thirteenth centuries. The Palestinian braccio was adapted as a unit of measure in Pisa during the twelfth century because of its special religious significance: it was used to calculate the length of the body of Christ (3 Palestinian braccia). This is verified by a number of Trecento and Quattrocento guidebooks to the Holy Land, which contain a linear measure identified as 1/16 the length of Christ's body: the measure is usually a subdivision of the Palestinian braccio (55.48 cm). There is a Florentine document dated May 16, 1209, which mentions ‘brachium pisanum’ in connection with work done on tower walls: cf. Santini, P., Documenti dell'antica Costituzione del Comune di Firenze (Florence, 1859), p. 534.Google Scholar

Both the Pisan braccio and the Florentine braccio di terra are slightly smaller than the Florentine braccio di panna (55.12 cm compared with 58.36 cm), which was used mainly as a measure for selling cloth. Piero della Francesca, for example, in his Trattato d'Abbaco, consistently differentiates the braccio di panna—used in problems involving cloth calculations— from the braccio used in geometrical problems involving two or three dimensions, and in surveying problems involving land. Luca Pacioli also retains that distinction in his Summa aritmetica, printed in 1494. The braccio di terra remained in use until 1782, when was abolished in favor of only the braccio di panna. While it is true that the braccio de panna was used by masons and sculptors during the Quattrocento, it is also true that the braccio di terra has received little or no consideration as a second possible unit of measure in the literature concerned with Florentine proportional problems. Like the braccio di panna, the braccio di terra and the Pisan braccio were divided both into 20 soldi and into 12 crazie. See Tavole di Ragguaglio per la Riduzione dei Pesi e Misure che si Usano in Diversi Luoghi del Granducato di Toscana (Florence, 1782), I, 369.

25 There are three technical problems which must be considered in any discussion of the dimensions of Andrea's door. In the first place, due to the slight shrinkage involved in the casting process, the door's actual dimensions are smaller than those of the original wax model would have been (according to Cav. Bruno Bearzi, the normal rate of shrinkage for bronze is between 13 to 16 mm per meter, or 1.3-1.6%). Secondly, we know from the documents that the left wing was cast before the right one. The separate casting of the wings introduces the possibility of differences in their dimensions. In fact, the entire left wing is slightly smaller than the right. Finally, as emphasized in the documents, the right wing was severely miscast, and had to be straightened and patched (both the cracks and patches in the frame are still clearly visible today). This introduces yet another possibility of irregularities in its dimensions. For these reasons, the following discussion has been kept general, while detailed information on the door's dimensions is contained in Appendix II.

26 This includes all the frame moldings, but excludes the inner dentils decorating the rectangular panels.

27 The proportions of some of the door's dimensions can be generated by the Roriczer method, but not according to any logical design procedure. If a Roriczer sequence of inscribed squares is generated from the door height, and the consecutively smaller squares are given numbers beginning with R2, while the spaces between the squares are given numbers beginning with R1A, the lengths correspond to certain door dimensions. For example, R7 (an eighth of the door height) equals the height of the raised rectangular frames, R8 the width of the rectangular panels and the height of the quatrefoil reliefs, R4A the height of the rectangular panels, and R5A the width of the quatrefoil reliefs. R14 equals the lengths of the outer door moldings, and R1OA those of the latticework moldings. By this method, however, Andrea would more or less arbitrarily have selected Roriczer lengths, relating heights of some sections geometrically with widths of others at random. Gone would be all the carefully planned relations among the door's sections, relations clearly apparent in any rational analysis of the door's dimensions, and those stressed in the Trecento Italian's mathematical education.

28 Had Andrea wanted to retain the proportions generated by the equilateral triangle, the back dimensions would have been 108 soldi by 187 soldi (297.7 cm by 515.37 cm).

29 Design and proportional methods used by early Quattrocento Florentine sculptors will be the subject of a forthcoming study.

30 White, John, ‘Carpentery and Design in Duccio's Workshop: The London and Boston Triptychs,’ Journal of the Warburg and Courtauld Institutes, 36 (1973), 92–105 CrossRefGoogle Scholar, contains a good discussion of rational design methods used by Duccio's workshop. All of the measurements in those two works, moreover, are either identical or extremely close to subdivisions of the Sienese braccio (the Sienese braccio equaled 60.105 cm and was divided into 24 once of 2.50 cm; see Tavole di Ragguaglio … , p. 553). The Sienese braccio was clearly not identical with the Florentine braccio di panna, although such an assumption has crept into the literature on the subject.

page 497 note * The starred manuscripts were kindly brought to my attention by Mr. Warren Van Egmond, Department of History and Philosophy of Science, Indiana University.