The rare traces of constructional procedures in “ practical geometries ” Jens

From a sociological point of view, pre-Modern “non-theoretical geometry” is not adequately described as merely “practical”. The “practical geometry” we find in written treatises is mostly that of “scribal” environments, and aims at calculating lengths, areas or volumes from already performed measurements. As a rule it is not interested in geometrical construction, nor in the making of measurements – the fields, broadly speaking, of master builders/architects and surveyors. The paper discusses two cases – one fairly well-established, another more conjectural – where none the less “scribal” practical geometry does reveal traces of (very simple) geometrical construction. Both of these concern the “long run”, connecting Old Babylonian, classical ancient and late medieval material. A final instance of weak communication between “scribal” and “surveying” geometry is located in thirteenth-fourteenth33-century France. In a customary dichotomy, geometry (like many other fields, mathematical as well as non-mathematical) falls into “theoretical” and “practical”. In full agreement with this, Stephen K. Victor [1979: ix] writes about his Ph.D.-project that My first assumption, and that of most of the people I have spoken to about the topic, was that “practical geometry” must relate somehow to architecture, surveying and city planning, to those areas, in other words, where geometry plays a central role in the exercise of other professions. The study of medieval buildings, fields and towns from extant physical evidence was not a fruitful approach for me, and I have left it to those better trained in the methods of archaeology and art history. Since I was working as a historian of science, I chose to concentrate on the written tradition of treatises called “practical geometry”. The treatises he chose to work on – the Latin late-twelfth–century Artis cuiuslibet consummatio and a vernacular (Picardian) Pratike de geometrie from the late thirteenth century which is largely a translation of the former work – led him to a different view, namely that practical geometry has its greatest importance as a popularization of mathematics. The treatises on practical geometry were a way of teaching some basic principles to those who would not remain in school or university long enough to become philosophers or theologians and would not necessarily exercise a mathematical profession, but who might want, or even need, some mathematics in their everyday lives. The sampling of arithmetic and astronomy in ACC and of commercial arithmetic and metrology in the Pratike argues for the generally pedagogic, rather than scholastic, purpose of the treatises. The development of a vernacular version of ACC is further evidence that the practical geometries sought their homes outside of the universities, perhaps in the bureaucratic and commercial milieus. Nonetheless, as the Introduction shows, the formalized structures of university education had an influence even on the non-scholastic tradition of practical geometry. As the tradition developed, the practical geometries acquired an increasingly theoretical underpinning, to the point where they are sometimes considered works on measurement rather than simply practical geometries. Part of this conclusion depends critically on the choice of a Latin treatise and a vernacular treatise which in the main was derived from it. Other features, however, are shared not only with the Italian vernacular Pratiche di geometria and with Fibonacci’s Pratica geometrie but also with most Arabic, Sanskrit, Chinese, Greek, Babylonian and ancient Egyptian writings on the subject-matter. They deal – not at al or not much with measurement, as Victor states euphemistically, but rather with how to calculate something on the basis of measurements that have already been performed or which are presupposed to have been performed, either on pre-existing 1 An Arabic exception to this rule is Abū’l-Wafa ’s Book on What is Necessary for Artisans in Geometrical Construction [ed., Russian trans. Krasnova 1966].