Commentary on the First Book of Euclid's Elements

97 B.

the more holistic and the more particular principles to each. For this reason, Socrates in the Theaetetus Plato, Theaetetus 197b, mixing jest with earnestness, likens the knowledge within us to birds. He says they fly, some in flocks and others separately from the rest. For the more common and holistic logoi rational principles contain many more particular ones within themselves, while those that grasp the things known by their species stand apart from one another and are unconnected, having set out from differing first principles. Therefore, one science should be placed before the many sciences and mathematical subjects: the one that knows the common things running through all kinds and that provides the principles to all mathematical sciences.

And up to this point, let the teaching concerning it be defined among us. After this, let us contemplate what the criterion of mathematical subjects might be. Let us put forward as our guide for this transmission Plato, who, in the Politeia Republic Refers to the Divided Line analogy in Book VI, divides the things known separately from the knowledges, and assigns the knowledges to the known things in correspondence. For having posited that of existing things, some are intelligible and others sensible, and of the intelligible, some are again "intelligible" purely objects of thought and others "dianoetic" objects of mathematical reasoning; and of the sensible, some are "sensible" and others "imaged"—he sets noēsis intellectual intuition over the intelligible, which are the first of the four kinds; dianoia discursive reasoning over the dianoetic; pistis belief/trust over the sensible; and eikasia imagination/conjecture over the imaged. He declares that the relation of eikasia conjecture to aisthesis sensation is the same as that which...