  Proclus’ distinctively non-empirical approach towards physics and astronomy also influences his philosophy of mathematics, which is set out in the two prologues to his commentary on the first book of Euclid’s Elements. The first prologue deals with the mathematical sciences in general, while the second prologue focuses on geometry proper. Proclus argues in great detail that the objects of mathematical sciences cannot be derived from sensible particulars by means of abstraction. Because of the imperfect and deficient character of the sensible objects one cannot derive from them objects that are as perfect and as precise as mathematical objects are. Therefore, mathematical objects reside primarily in intellect and secondarily in souls (as logoi). As universal concepts (cf. 3.2) we can grasp mathematical objects by means of recollection (anamnêsis). Since geometrical objects are not universal, but particulars, and since by definition they possess extension, Proclus argues that their place is human imagination (phantasia). Imagination acts as a mirror and provides the mathematical objects which are projected into it by the soul with intelligible matter. By means of the latter geometrical objects gain extension and particularity. As with physics and astronomy, the ultimate aim of geometry is not the study of these extended, material objects. Rather, geometry serves an anagogical task (just as in Plato’s Republic), leading the soul upwards to a study of the true and unextended causes of geometrical objects in the divine mind (In Eucl. 54.14–56.22).