Early Sciences
Noam Andrews, Harvard University

Training for the Abstract: Visualizing Geometria in the 16th-Century German Lehrbuch


While the typical geometry course in Renaissance Europe revolved around the teaching of Euclid’s Elements, the 16th-century German Lehrbücher were something altogether different. Including proportionately little explanatory text to accompany geometrical and perspectival prints, these popular textbooks were intended for practical use by mixed-mathematicians/artists for whom knowledge of geometry and perspective had become essential. The innovative use of geometry in the Lehrbücher remains emblematic of a shift in priorities away from a “transcendent” geometry towards a set of “physicalized” geometrical operations inspired by the tactile properties of generating geometrical objects, albeit visualized in two dimensions. The resultant forms of abstraction, certainly an anachronistic yet useful term, substituted the theoretical geometrical exercises and definitions of Euclid (an abstraction in which geometry was divorced from real world application) for a theoretical geometry used to teach perspectival construction (the realistic rendering of “abstract” geometry on paper), making use of novel graphic manipulations of information. The paper explores the epistemological possibilities of geometria, a specifically Renaissance category of knowledgeembodied in two such Lehrbücher, Augustin Hirschvogel’s Geometria (1543) and Heinrich Lautensack’s Des Circkels unnd Richtscheyts (1564), in order to unpack the unique contributions of mixed mathematics to the way that geometry was understood, taught and used in Early Modern Germany.