Commentary on Euclid's Elements

By these demonstrations, do they not reveal things to be more confused?
8 9 For we say, for instance, in a demonstration, that while the isosceles triangle has angles equal to two right angles, and that one might first understand the equilateral triangle scientifically Proclus is distinguishing between "knowing" a specific triangle and "understanding" the universal geometric truth that applies to all such shapes., it is actually the person who has demonstrated the universal principle universal? reason? original: katholou logon and who possesses the knowledge in its simplest form who truly holds science within themselves. And again, if the universal is superior to the particular for the purpose of demonstration, then the demonstrations of the universal are more primary. The principles from which demonstrations arise are older and prior by nature to individual cases; they are the causes of what is being shown.

Therefore, the demonstrative arts are far from being concerned with things that come "after" This refers to the "posterior" nature of physical objects compared to the "prior" nature of mathematical truths in Neoplatonic thought., and these unmixed forms remain far removed from the realm of the senses. Rather, they are grasped by the discursive mind original: dianoia; the capacity of the soul to reason through steps, as opposed to direct sensory perception or pure intellectual intuition. and are more perfect than sensations; they are a cultivation of the very essence of knowledge.

But consider the consequence for those who speak this way Those who believe geometry is merely abstracted from physical matter.: they first make the soul more dishonorable than matter. For if matter—being what it is—receives these forms as things that are better and more wondrous than its own nature, while the soul merely receives a secondary portion, having surpassed its own ignorance only to model within itself things more dishonorable than matter, how does this not portray the soul as weaker and more deficient?

By claiming the soul merely "subtracts" from matter the qualities matter is naturally worthy of, how do they not make the soul inferior? For matter is the "place" of the principles embodied in living things, and the soul is the "place" of the Forms. But while matter holds the secondary, "later" forms, the soul holds the older, prior forms. Matter holds the forms established in nature, but our soul holds the essential forms original: ousiodē eidē; forms that constitute the very being of the soul.. How can it be that these forms are merely "mental concepts" original: kat' epinoian; things that exist only as thoughts rather than having real, objective existence. as they claim? How—