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This volume explores the mathematical character of architectural practice in diverse pre- and early modern contexts. It takes an explicitly interdisciplinary approach, which unites scholarship in early modern architecture with recent work in the history of science, in particular, on the role of practice in the “scientific revolution”. As a contribution to architectural history, the volume contextualizes design and construction in terms of contemporary mathematical knowledge, attendant forms of mathematical practice, and relevant social distinctions between the mathematical professions. As a contribution to the history of science, the volume presents a series of micro-historical studies that highlight issues of process, materiality, and knowledge production in specific, situated, practical contexts. Our approach sees the designer’s studio, the stone-yard, the drawing floor, and construction site not merely as places where the architectural object takes shape, but where mathematical knowledge itself is deployed, exchanged, and amplified among various participants in the building process.

Inhaltsverzeichnis

Frontmatter

Introduction

This volume explores the mathematical character of architectural practice in diverse pre- and early modern contexts. It takes an explicitly interdisciplinary approach, which unites scholarship in early modern architecture with recent work in the history of science, in particular, on the role of practice in the scientific revolution. As a contribution to architectural history, the volume contextualizes design and construction in terms of contemporary mathematical knowledge, attendant forms of mathematical practice, and relevant social distinctions between the mathematical professions. As a contribution to the history of science, the volume presents a series of micro-historical studies that highlight issues of process, materiality, and knowledge production in specific, situated, practical contexts. Our approach sees the designer’s studio, the stone-yard, the drawing floor, and construction site not merely as places where the architectural object takes shape, but where mathematical knowledge itself is deployed, exchanged, and amplified among various participants in the building process.
Anthony Gerbino

Foundations

Frontmatter

Proportion and Continuous Variation in Vitruvius’s De Architectura

It is important to balance Vitruvius’s discussion of the architectural orders, centered on temples, with his sections on civil and, in particular, domestic architecture. It is in this domain, the subject of Book 6 (Chapters 3 and 4) of the De Architectura, that the relationships implied by the term symmetria appear explicitly, in both functional and aesthetic terms and without interference from the question of whether the recommended ratios are affected by the transformation of wooden temples to stone ones. Based on a review of his rules for designing atria, the Vitruvian conception of order as genus appears not as a fixed set of ideal relationships laid down once and for all, but as a series of variations in proportion. While certainly not obeying the concept of “function” as developed in the seventeenth century, these variations can nevertheless be shown to follow continuous curves interpolated from sets of derived values. In this respect, the Vitruvian project finds contemporary expression in today’s CAD/CAM software.
Bernard Cache

Mathematics and Material Culture in Italian Renaissance Architecture

Frontmatter

The Palazzo del Podestà in Bologna: Precision and Tolerance in a Building all’Antica

The Palazzo del Podestà in Bologna offers an important case study for some of the quantitative and numeric features of built architecture of the last quarter of the fifteenth century. It shows, in particular, how imperfections in construction and difficult site conditions could hinder the much-desired ideal of geometrical, mathematical, and proportional exactitude that was already well diffused in both the theory and practice of Renaissance architecture. The Palazzo, a project of the early 1470s, can serve as a model for understanding how the idea of a building conceived on the model of the geometric grid—with precisely calculated, exact, and whole measurements—was the strongest prerogative of the well-educated Renaissance architect. Such characteristics imply a knowledge of precise geometrical and mathematical rules, the ability to render meticulous and accurate drawings, and to execute them in built form. It also reflects the capabilities of stonemasons to create architectural elements of great precision.
Francesco Benelli

Practical Mathematics in the Drawings of Baldassarre Peruzzi and Antonio da Sangallo the Younger

Combining technical practice with aesthetic intent, Renaissance architecture was by nature a mathematical art. Although the limitations of surviving documents hinder efforts to discern what Italian Renaissance architects knew of mathematics, where they learned it, and how they applied this knowledge, extant drawings from the period offer one means of addressing these questions. Inscribed numerals and calculations, in particular, abound in the drawings by two leading architects of early sixteenth-century Italy, Baldassarre Peruzzi and Antonio da Sangallo the Younger, suggesting that both attained a high degree of numeracy. Comparing these contemporaries is also revealing since, while each incorporated mathematics as a central element in their architectural practice, their approaches diverge in ways that point to and illuminate significant differences in their background and design methods.
Ann C. Huppert

Geometric Survey and Urban Design: A Project for the Rome of Paul IV (1555–1559)

The slow development of survey technology—from the first statement of its geometric principles in the mid-fifteenth century to its application in the administration of property and the design of urban spaces—spans an arc of almost two centuries. One of the landmarks of this progress is a drawing in the Uffizi collection, catalogued under the number 4180A. It is a large drawing, composed of 10 joined sheets, and measuring 117 cm by 133 at its widest points. It is a project for a large building complex on an urban site. The constituent elements identify it as a cloister: the cruciform space of a church, an atrium, and an arcaded court. A “rota” and “parlatoio”, located between the two latter spaces allow communication between the cloistered religious and lay visitors. This is not an ideal scheme, and it is the survey that makes it specific. The “Piazza del arco di camillo” to the right side and the “piazza di S. Ma(c)uto” at the bottom left, place the project in Rome, on the site occupied today by the late sixteenth-century structures of the Collegio Romano and the seventeenth-century church of Sant’Ignazio. The drawing represents a project for a convent of Franciscan nuns, or Poor Clares, sponsored by the Marchesa Vittoria della Tolfa and was executed in the period 1555–1559, during the pontificate of the marchioness’s uncle, Paul IV Caraffa.
David Friedman

The Baroque Institutional Context

Frontmatter

Architecture and Mathematics in Early Modern Religious Orders

For most of us, a familiar image from Raphael’s School of Athens serves to illustrate our intuitive notions about the links between early modern architecture and mathematics. The artist’s portrait of the great Renaissance architect Bramante as the geometer Euclid recalls the medieval traditions of Gothic architects and master masons using geometry. Moreover, the inclusion of Zoroaster and Ptolemy—identified by celestial and terrestrial globes—in the group huddled around Euclid/Bramante further seems to associate geometry and architecture with astronomy, vaguely echoing the medieval quadrivium of arithmetic, music, geometry, and astronomy. In short, the architect as mathematician (or mathematician as architect) operating within a larger group of quantifiable crafts and sciences seems obvious, and not particular to the early modern world. Yet a closer look at a well-defined culture which produced such individuals illuminates much about the period’s understanding of both architecture and mathematics. The religious orders traditionally associated with the Counter Reformation, such as the Jesuits, Theatines, and Barnabites, provide rich material for investigating the relationship between architecture and mathematics, and they nurtured a specific type of priest-architect.
Susan Klaiber

The Master of Painted Architecture: Andrea Pozzo, S. J. and His Treatise on Perspective

Andrea Pozzo’s illusionistic work is well-known among historians of architecture and art, as is his Perspectiva pictorum et architectorum, published in the last decade of the seventeenth century. Historians have presented the general outline of the treatise and have discussed its valuable descriptions of the author’s own designs. The book’s place, however, in the history of the literature on perspective has received far less attention. This article examines the style and content of the Perspectiva in relation to the broader tradition of perspective writings in Italy. Being a Jesuit played an essential role for Pozzo’s self-understanding; hence it is also natural to ask how common it was for men in holy orders to write on the subject.
Kirsti Andersen

Narratives for the Birth of Structural Mechanics

Frontmatter

Geometry, Mechanics, and Analysis in Architecture

Mathematics, in the form of rules of proportion, was used in architecture from very early times. However, it was not until the mid-seventeenth century that the three disciplines in the title of this paper came to be gradually introduced. Their use was crystallised into modern codes of practice for elastic design, which claim to evaluate the state of a given structure under given loading. In fact, these states cannot be observed: a real structure is subject to unknown and unknowable imperfections, which profoundly alter its behavior. Safe designs may still be made: the way forward is to use so-called plastic methods (or limit design in the US). The now established use of the word “plastic” is misleading and could be replaced more meaningfully by “equilibrium”. The equations of statics, which preceded those of analysis, turn out to be the key to structural design.
Jacques Heyman

Epistemological Obstacles to the Analysis of Structures: Giovanni Bottari’s Aversion to a Mathematical Assessment of Saint-Peter’s Dome (1743)

Visible faults in the dome of Saint Peter’s basilica in Rome had raised fears about the structure’s stability ever since its completion in 1593. The most extensively documented episode of this long history erupted in the early 1740s, a few years after Prospero Lambertini was elected Pope Benedict XIV. The debates over the causes of the cracks, the ensuing scientific analyses, and the adopted solutions are well known, due to the Memorie istoriche della gran cupola del Tempio vaticano, the magisterial treatise published in 1748 by Giovanni Poleni (1685–1761), the mathematician entrusted with the supervision of the restoration work. One of the great points of interest of this episode was the involvement of competing protagonists and factions, including architects, master carpenters, and natural philosophers. Each of these groups benefited from varying degrees of credibility. Beyond the technical issues concerning the dome’s structure, the debate raised important questions about the social and intellectual legitimacy conferred by different forms of expertise.
Pascal Dubourg Glatigny

A Scientific Concept of Beauty in Architecture: Vitruvius Meets Descartes, Galileo, and Newton

The artistic and architectural theories of the seventeenth and eighteenth centuries were strongly marked by a critical comparison between the greatness of the ancients and the inventions of the moderns. In some cases, the parallel assumed the sterile form of a purely academic debate, but in others it gave rise to entirely original reasoning and theoretical elaboration. This is the case, for example, of the critical revision of Vitruvius’s concepts in light of the extraordinary developments in modern science. A number of architects and architectural theorists believed they could reinvigorate architecture by re-elaborating the ancient theory of proportions in light of the recent achievements in optics, mechanics and projective geometry. Three cases are, I believe, particularly eloquent: the oblique architecture of Juan Caramuel de Lobkowitz in relation to Descartes’ philosophical thought, the optical-perceptivity theory of Bernardo Vittone in relation to Newton’s optical discoveries, and the rationalism, or functionalism, of Carlo Lodoli in relation to Galileo’s studies in mechanics.
Filippo Camerota

Architecture and Mathematical Practice in the Enlightenment

Frontmatter

Breathing Room: Calculating an Architecture of Air

How do the findings of science—the precise mathematical measurements and calculations of phenomena that reveal the deep truths of the natural world—get translated into real life, into real practices, and, for the purposes of this paper, into architectural theory and practice? The answer, far from straightforward now in an era of rapid change, was equally difficult at a time when architects were concerned about tradition as much as innovation. By correlating pneumatic research in the 1700s to architectural designs specifically intended to promote a healthier internal air, this paper tries to trace how scientific findings became practical knowledge. The pneumatic research sought to quantify internal volumetric requirements and to outline ideal patterns of air movement in the creation of healthy spaces. Its practical application, however, posed a particular challenge for architects, who typically paid little attention to room occupancy and air flow. How architects dealt with (or ignored) this challenge illuminates the larger historical issue of how innovation is disseminated from initial laboratory-based mathematical findings to later empirically-processed practical changes. This work focuses on prison and hospital design. Those building types were the subject of intense discussion and experimentation, particularly over their air quality. They have, moreover, received considerable historical scrutiny.
Jeanne Kisacky

James “Athenian” Stuart and the Geometry of Setting Out

A characteristic feature of the neoclassical attitude to Greco-Roman architecture that ran from the middle of the eighteenth century to the middle of the nineteenth has long been held to be the minute surveys of ancient buildings that were undertaken and published during that period. Ultimately inspired by Antoine Desgodetz’s Les édifices antiques de Rome (1682), measured surveys of antique buildings across the Mediterranean world became a staple part of architectural and antiquarian study from the 1750s, especially in relation to the growing interest in Greek architecture. The British were especially assiduous in framing these surveying activities as part of a discourse about “truth” (as Robert Wood put it in 1753) and “accuracy”, a term used by James Stuart in the preface to the first volume of The Antiquities of Athens in 1762. However, the process of surveying an existing structure is by no means commensurate with that of setting it out in the first place, since some dimensions are effectively concealed by the fabric of the building itself. Further still, methods appropriate for drawing on the smooth surface of a drawing board may be quite different from those appropriate for the staking out of the plan in the field or the marking of stone by the mason. This situation raises a number of related conundra: How did Stuart take measurements in the field? How did they get translated to published form? What assumptions did he make about Greek setting out, and how did these assumptions color his measurements and his reconstructions?
David Yeomans, Jason M. Kelly, Frank Salmon

Backmatter

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