Commentary on the First Book of Euclid's Elements

dianoia discursive reasoning relates to noēsis intellectual intuition. For eikasia imagination/conjecture knows the images of sensible things as they appear in waters and other mirrors, having a sort of final rank among forms; they are truly images of images. And dianoia contemplates the icons of intelligible things, which have descended from the first, simple, and partless forms into multiplicity and division. Because of this, its knowledge is dependent upon other, older hypotheses, whereas noēsis ascends to the unhypothetical principle itself. If, therefore, mathematical subjects possess neither a partless existence separated from all division and variety, nor the kind of existence known by sense-perception that is highly changeable and divisible in every way, it is clear to everyone that they are dianoeta objects of discursive reasoning in their essence. Dianoia stands over them as a criterion, just as sense-perception stands over sensible things and eikasia over the things imagined. Hence, Socrates also determines the knowledge of these to be clearer than doxastic opinion-based apprehension, but dimmer than the primary science. For they possess more of the unfolding and discursive character of noēsis, while they surpass doxa opinion in the permanence and irrefutability of their logoi rational principles. And they obtain their descent from the first science because they start from hypotheses, yet they exist in immaterial forms because they possess a more perfect knowledge than that of sensible things.

We define the criterion of all mathematical subjects according to the mind of Plato in this way,