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The philosophical and mathematical commentaries of Proclus on the first book of Euclid's elements (Vol. 1 of 2) cover

The philosophical and mathematical commentaries of Proclus on the first book of Euclid's elements (Vol. 1 of 2)

Chapter 40: DEFINITION VI.
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About This Book

A learned commentary unpacks Euclid’s first book by combining close geometric exposition with Pythagorean and Platonic philosophical reflection. Propositions, definitions, and diagrams are explained alongside arguments that pure geometry, pursued for its own sake, elevates intellectual understanding and serves as a pathway to theological insight rather than merely a tool for practical crafts. The edition features a translator’s preface that addresses corrupt Greek texts and dependence on a Latin rendering for clarity, alongside editorial decisions about terminology, chapter arrangement, paraphrase, and explanatory notes to clarify obscure passages. Recurring themes contrast contemplative ancient wisdom with empirical modern approaches.

DEFINITION VI.

The Extremities of a Superficies are Lines.

From these also, as images, we may understand, that things more simple procure bound and an end to every one of their proximate natures: for soul perfects and determines the operations of nature; and nature the motion of bodies. And prior to these, intellect measures the convolutions of soul; and unity the life of intellect; for that is the measure of all. Just as in these also, a solid is terminated by a superficies; but a superficies by a line; and a line by a point; for that is the boundary of them all. Hence, the line existing uniformly in immaterial forms and impartible reasons, terminates and restrains the various motion of a superficies in its progression, and proximately unites its infinity. But in the images of these, when that which bounds supervenes that which is bounded, it causes, by this means, its limitation and bound. But if it should be enquired how lines are the extremities of every superficies, since they are not the extremes of every finite figure; for the superficies of a sphere is terminated indeed, yet not by lines, but by itself? In answer to this, we must say, that by receiving a superficies so far as it is distant by a twofold interval, we shall find it terminated by lines according to length and breadth. But if we behold a spherical superficies, we must receive it as that which is endued with figure; which possesses another quality, and conjoins the end with the beginning; and loses its two extremities in the comprehensive embraces of one: and this one extremity subsists in capacity only, and not in energy.