Ad Quadratum: The Practical Application of Geometry in Medieval Architecture is the first publication in the Association Villard de Honnecourt (AVISTA) series, Studies in the History of Medieval Technology, Science, and Art. The goal of this series is to provide a publication venue for interdisciplinary studies in the fields of medieval art, architecture, science, and technology. The eleven essays included in the inaugural volume, edited by Nancy Y. Wu, address the geometry and systems of measure that were used to determine the design and construction of medieval buildings. United by their focus on the mathematics and metrology underlying medieval building practices, the essays cover a broad chronological span from the Carolingian era to the late fifteenth century. The majority, however, examine the geometry underlying the complex designs of Gothic churches.

Eric Fernie’s excellent introduction to the anthology offers insightful guidelines that any researcher investigating medieval building design should follow. He stresses the importance of accurate measurements made on-site, clearly defining the tolerable margin of error in overall design and construction, and seeking the simplest solution suggested by the evidence. He advises scholars to limit themselves to using documented measuring units, such as the Roman foot (29.57 cm); to check for incommensurate ratios, for example √2:2, that might have determined a plan’s conception; and not to disregard automatically unexpected features in a building’s plan. For the most part, the authors of the papers included in the volume have followed his sensible advice and concur with his observation that more often than not it is basic geometric shapes and the proportions to which they give rise that serve as the basis for medieval building design.

The challenge in accounting for the geometry or ratios that underlie the appearance of a medieval church arises from the paucity of written and graphic sources that might provide a check for the evidence offered by the fabric of a building. In most instances, bricks and mortar alone must suffice to reveal the methods used by a building’s designer. Because almost all medieval churches conform, or at least strive to conform, to regular plans characterized by symmetry, right angles, and rectilinear and semicircular forms, any number of geometrical solutions can be arbitrarily selected to account for their design. The speed, ease, and flexibility afforded by AutoCAD programs, as Wu cautions in her essay, sometimes encourages the impulse to discover endless circles, squares, pentagons, or other geometric forms that may just by chance conform to a plan. Herein lies the value Fernie’s advice to work with measures, ratios, and geometry for which there are documented examples.

The most consequential papers in this volume are those in which the authors have been both conscientious and open about their methods and scrupulous to avoid explanations based on coincidental correspondences between buildings and the design strategies they propose. Texts by James Addiss and Ellen Shortell are good examples. Both authors base their analysis on precisely measured plans, documented systems of measure and geometry, and the way these work in the most significant portions of the buildings they explore.

In his account of the design of the eleventh-century Cluniac priory of St.-Etienne at Nevers, Addiss initially asserts that a modular system conceived in Roman feet according to a three-foot unit was used as the basis for the construction of the church. However, skepticism about his first observation led him to reconsider, leading subsequently to the discovery that the building was in part conceived in terms of a geometrical system based on the Golden Section. Without rejecting his first conclusion, he added to it the idea that the Golden Section was used to plot a simple schema for the overall plan. In her paper, “The Plan of Saint-Quentin: Pentagon and Square in the Genesis of High Gothic Design,” Shortell demonstrates that the chevet and choir of the collegiate church of St.-Quentin were begun in the early 1190s rather than in the early thirteenth century, as other scholars have suggested. This earlier dating places the conception of St.-Quentin’s eastern portion around the same time as those of the more frequently studied cathedrals at Chartres, Soissons, Troyes, and Reims. Using precise measurements of the major portions of the choir and chevet, Shortell discovered that their design was based on the Roman foot, Golden Section proportions, and the manipulation of a square and a pentagon. She shows how the plan, although complex in its underlying geometry, could be set out simply with only a few measurements. Both Shortell and Addiss add depth to their studies of the design factors underlying the appearance individual buildings by placing them in their broader contemporary architectural contexts.

Several of the most intriguing papers in the book use an analysis of design choices as a springboard for more complex issues. For instance, Stephen Murray in his “Reconciling the Feet at Beauvais and Amiens Cathedrals” examines how two units of measure—the royal foot (32.5 cm) and the Picard foot (29.5 cm)—were used simultaneously to plan and construct the cathedrals mentioned in the title of his article. While this simultaneous use of different measures had little consequence at Amiens, Murray convincingly demonstrates that it created a mismatch of forms at Beauvais that ultimately compromised the cathedral’s structure. In hopes of discovering the provenance of the masons who built St. George of the Latins in Famagusta, Cyprus, around the turn of the fourteenth century, Alpy Özdural examines the measuring units used to construct this Gothic building. Although he finds that the Roman pace (74 cm) is the design module for the church, this widely used measure did not provide enough evidence to determine the masons’ homeland. In the course of his investigations, nevertheless, the author learned that the main dimensions of St. George relate to the Pythagorean concept of musical proportions, a rare feature in medieval architecture but frequently used in Renaissance buildings. In his essay on the late Gothic choir of Metz Cathedral, Robert Bork likewise explores measure and geometry as a means to a larger issue. His paper convincingly demonstrates how geometric design gives not just coherence but also visual harmony to a complex plan built over a long period of time.

In their papers Wu and Nigel Hiscock examine how the mathematics of cathedral design reflected contemporary intellectual and philosophical beliefs. Wu, for instance, finds parallels to the design process used in the cathedral of Reims in the philosophy of medieval thinkers such as Boethius and Thomas Aquinas. In a controversial paper, Hiscock accounts for the design of Norwich Cathedral in terms of three figures of Platonic geometry: the equilateral triangle, the square, and the regular pentagon. He claims that designers used these shapes because they were imbued with metaphysical significance. He takes issue with Fernie’s suggestion that the √2:1 ratio served as the basis for the church’s design because this ratio lacks spiritual resonance. While Hiscock’s theory is intriguing, it presupposes a philosophical knowledge on the part of the masons with which it seems difficult to credit them.

In “On the Drawing Board: Plans of the Clermont Cathedral Terrace,” Michael Davis examines one of the rare graphic sources for medieval building design. He explores the full-scale drawings of the window and portal designs that are incised into the platform covering the aisles and chapels of the choir of Notre-Dame, Clermont. Ironically, the tracery patterns of the windows and designs for the doors etched into the terrace were never actually built at Notre-Dame, although they exist in other contemporary churches in the Île-de-France. Davis accounts for this by explaining that the master mason, Jean Deschamps, used the drawings as an early draft in the design process but that he subsequently changed his mind before actually cutting the stone for the building.

Ad Quadratum is a book that will be engaging to scholars who are specifically interested in the mathematics that determine the grandeur of medieval architecture. Its essays are dense and not intended for the general reader. Taken as a volume, they give a clear overview of the type of research being done in the field of medieval metrology and building design. These highly specialized studies, which are largely monographic in focus, should provide a sound foundation for future, more synthetic and comprehensive studies.