ABSTRACT
MOTION AND TRANSFORMATIONS IN GEOMETRY From the mid ninth century onwards, mathematicians were readier than
before to make use of geometrical transformations. The works of alFarghānī, of the brothers Banū Mūsā – particularly those of the younger one, al-Ḥasan – and those of Thābit ibn Qurra provide the most striking examples. A century later, geometrical transformations have even acquired a group name: al-naql, as it is written by al-Sijzī.1 A careful reading of Ibn Sahl, al-Qūhī and al-Sijzī, for example, shows that geometers were not concerned solely with studying figures but also with investigating relationships between them. Transformations do, of course, appear before the ninth century: for instance they are used by Archimedes and Apollonius.2 But in the ninth century they are used much more frequently and applied much more widely. There is a noticeable difference between the ancients and the moderns: among the former certain transformations arise in the course of proofs – as can be seen in Archimedes – whereas among the latter a new point of view emerges: transformations are used directly in geometrical investigations. We have had several occasions to draw attention to the emergence of this new attitude, this changed perception of geometrical objects. We have also presented it as one of the consequences of research in geometry becoming more active from the ninth century onwards – but
not an element in causing the revival. Let us now look at the areas covered by the revival.
