## Design Method and Mathematics in Francesco di Giorgio’s Trattati

During the 1950s, Rudolf Wittkower encapsulated his understanding of architectural proportion in the Italian Renaissance in the following statement: “. . . two different classes of proportion . . . derived from the Pythagoreo-Platonic world of ideas . . . the Middle Ages favoured Pythagoreo-Platonic geometry, while the Renaissance . . . preferred the numerical, i.e. the arithmetical side of that tradition.” 1 He further contended that the Middle Ages favoured the perfect geometrical fi gures for both the plans and the elevations of buildings. Conversely, Renaissance artists adopted an empirical approach to measurement, in turn associated with a renewed interest in nature. As a result, the irrational proportions which often characterize geometrical fi gures were found to be inadequate. The Renaissance preference lay with integral numbers, or simple fractions. Consequently, “architects of the Renaissance and not of the Middle Ages, fully embraced Vitruvius’ well known module system which was the only way of guaranteeing a rational numerical relationship throughout a whole building.” 2 Commensurability of measurement, Wittkower concluded, was the “nodal point of Renaissance aesthetics.” 3

As Matthew Cohen has recently pointed out, Wittkower was the fi rst scholar to engage in an historical study of Renaissance proportion, providing a “paradigm,” which still underlies modern scholarship. 4 To a certain extent this would explain why architectural historians have traditionally scrutinized Italian Renaissance buildings for the modular system that underlies their design and have placed particular emphasis on the study of the orders. Cohen has also stated explicitly what intuitively appears to be a major shortcoming of Wittkower’s thesis, namely that it consists of an “oversimplifi cation of available evidence.” 5 He has argued, instead, that geometry and numbers were employed equally in the architectural theory and practice of the Renaissance and the Middle Ages. 6

While Wittkower’s argument might be considered dated and overgeneralized, the writings of Francesco di Giorgio Martini invest it with a certain qualifi ed validity. Francesco’s treatise contains the only theoretical statement of a modular system in the fi fteenth century. Unpublished during the Quattrocento, the treatise survives in a number of manuscripts dating from the 1470s to the late 1490s. The manuscripts are considered to fall into two major stages of preparation of the treatise. The fi rst version, Trattati I , is found in the codex Ashburnhamianus 361 in the Biblioteca Laurenziana in Florence and the codex Saluzzianus 148 in the Biblioteca Reale in Turin. The second version of the treatise, Trattati II , is found in the codices S.IV.4 in the Biblioteca Comunale in Siena and Magliabechianus II.I.141 , in the Biblioteca Nazionale in Florence. 7 These reveal successive stages of preparation of the text, illustrating the trajectory of Francesco’s architectural thinking. 8 The discussion of the modular

system is found in codex Magliabechianus II.I.141 , which represents the latest version of Francesco di Giorgio’s text. 9 In this chapter, I will examine the nature of Francesco’s modular system in relation to arithmetic and geometry, as well as its Vitruvian prototype. I will argue that Francesco’s intent is to establish a correspondence between geometrical and numerical methods. In this process, he inventively applies the Vitruvian proportions of columns to the entire design of a building.