Commentary on Euclid's Elements

Acting from a secondary position, they complete its entire generation and craftmanship original: demiurgian; refers to the creative act of the "Demiurge" or divine craftsman from the mathematical form, and they have established the causes of all things within it. Of these, the seven starting points encompass the origins of the linear, the planar, and the solid Proclus refers here to the seven numbers Plato uses in the Timaeus (1, 2, 3, 4, 8, 9, 27) to describe the soul's mathematical composition.. Regarding all the ratios, the seven have defined a rational subordination original: hypallēlian; an ordered arrangement or logical sequence. of essence within it. Furthermore, the principles of shapes original: schēmatōn are established within it in a creative manner. Of all movements, the first—which contains all causes and sets them in motion—co-exists with the soul; for of all things that are moved, the circle and circular motion are the beginning.

It is for these reasons that the ratios of mathematics are self-moving, filling the souls; and the discursive reason original: dianoia; the faculty of the mind that thinks through things step-by-step, in its circling and unfolding of all this mathematical variety, exists according to knowledge. It will never cease begetting, bringing forth other things through itself, and declaring its own clear ratios. For having embraced all things, and according to its infinite power, it puts forward the beginning of all kinds of theorems original: theōrēmatōn; literally "things to be looked at" or objects of contemplation as its projections.

But indeed, after considering the essence of mathematical ratios,
What is the work and what are the powers of mathematical science, and how far do they extend in their activities?

let us run back to that one science original: epistēmēn of them, which we previously showed would exist, and let us consider what its work is, what its powers are, and how far it extends in its activities. We must declare that the work of all mathematics is intellectual original: dianoētikon; related to the discursive mind, and we must state what its field of action is—whether it is the kind of thing that the Intellect original: nous; the higher, intuitive mind finds permanently established and perfect within itself, or whether it is something innate—