Commentary on Euclid's Elements

Book

As we have arranged, our discussion concerns principles. There is, therefore, one science that is not simply ranked alongside the many other sciences and mathematical branches; rather, it identifies for us what is common and pervades through all of them, supplying the principles for all the mathematical sciences. Up to this point, our precise teaching regarding it has been as follows. Following this, we must consider how we might subordinate these to the divine nature of mathematics original: theiō tōn mathēmatōn. Proclus views mathematics not just as a tool, but as a divine ladder for the soul's ascent.. We shall set forth that we consider Plato Plato (c. 427–347 BCE), whom Proclus considers the ultimate authority in the "Golden Chain" of philosophers. the leader of this tradition. In the Republic original: politeia. Specifically, Proclus refers to the analogy of the "Divided Line" found at the end of Book VI., he divides the objects of knowledge into one category and the modes of knowing into another, pairing each mode of knowledge with its corresponding object.

Among all beings, he says some are intelligible original: noēta; things understood by the mind alone, such as the Forms. and others are sensible original: aisthēta; physical things perceived by the senses.. Of the intelligible realm, some are again pure intelligibles, while others are objects of understanding original: dianoēta; these are the intermediate mathematical objects.. Of the sensible realm, some are sensible objects themselves, while others are likenesses original: eikasta; shadows or reflections.. To these four states surrounding us, he assigns Intellection original: noēsis; pure, non-discursive insight. to the pure intelligibles, Understanding original: dianoia; the discursive, step-by-step reasoning used in geometry. to the objects of understanding, Belief original: pistis; the conviction we have about physical objects. to sensible objects, and Conjecture original: eikasia; the lowest form of perception, like looking at reflections. to likenesses.

Having set forth this proportion, he shows that Conjecture stands in the same relation to Perception The text uses aisthēsin here to refer to the broader sensory realm or the act of believing in it. as Understanding stands to Intellection. For Conjecture knows the mere images original: eidōla of sensible things—those things that appear through colors and shadows, occupying the lowest rank, like reflections seen in water, which are images that have truly become images. Similarly, Understanding contemplates the images of the intelligibles—those truths that have descended from the primary, simple, and indivisible Forms original: eidōn into multiplicity and divisibility. For this reason, the knowledge belonging to this [Understanding] depends on other, more ancient original: presbyterōn. In this context, "ancient" means higher or logically prior hypotheses. hypotheses.