Conceptual Change in Early Modern Practical Geometries

As recognized by a growing body of literature, most conceptual and methodological shifts in early modern mathematics cannot be accounted for in terms of internal theoretical developments. On the contrary, this literature suggests, social and institutional contexts and the social practices of arithmetic, geometry, and metrology provide inputs that may explain the momentous transformations mathematics went through in early modern Europe. This paper analyses some 16th-century practical geometry books that contain theoretical innovations — as compared to contemporary authoritative editions of Euclid’s Elements. It pays particular attention to old notions (such as ratio and curve) that were newly defined, to new ideas (such as measure) that were introduced as if they were old ones, and to new methods (such as the use of material instruments) that were legitimized in practical geometry books. By paying attention to them the paper aims to document ways in which mathematical innovations “sneaked in” so to speak into the established, ordinary, authoritative body of mathematical results. The emphasis is not only on documenting new concepts and methods connected to social practices, but also on analyzing how practice and practical tools added legitimacy and authority to new concepts and new methods.