The philosophical and mathematical commentaries of Proclus on the first book of Euclid's elements (Vol. 1 of 2) : $b To which are added, A history of the restoration of Platonic theology, by the latter Platonists: And a translation from the Greek of Proclus's Theological elementsProclus
Philosophy
The philosophical and mathematical commentaries of Proclus on the first book of Euclid's elements (Vol. 1 of 2) : $b To which are added, A history of the restoration of Platonic theology, by the latter Platonists: And a translation from the Greek of Proclus's Theological elements
Proclus
Euclid. Elements; Platonists
Every composite receives its bound from that which is simple, and every
thing partible from that which is impartible; and the images of these
openly present themselves in mathematical principles. For when it is
said that a line is terminated by points, it seems manifestly to make
it of itself infinite, because, on account of its proper progression,
it has no extremity. As, therefore, the duad is terminated by unity,
and reduces its own intolerable boldness under bound, when it is
restrained in its comprehensive embrace: so a line also is limited by
the points which it contains. For, since it is similar to the duad,
it participates of a point having the relation of unity, according to
the nature of the duad. Indeed, in imaginative, as well as in sensible
forms, the points themselves terminate the lines in which they reside.
But in immaterial forms, the reason of the impartible point pre-exists
separate and apart; but when proceeding from thence by far the first of
all, by determining itself with interval, moving itself, and flowing
in infinite progression, and imitating the indefinite duad, it is
restrained indeed, by its proper principle, is united by its power,
and on every side seized by its coercive bound. Hence it is, at the
same time, both infinite and finite: infinite, indeed, according to its
progression; but finite according to its participation of a terminating
cause. So that, when it approaches to this cause, it is detained in its
comprehension, and is terminated according to its union. Hence too, in
the images of incorporeal forms, a point is said to terminate a line,
by occupying its beginning and end. Bound, therefore, in immaterials,
is separated from that which is bounded: but here it is twofold; for
it subsists in that which is terminated. And this affords a wonderful
symptom, that forms; indeed, abiding in themselves, precede their
participants according to cause; but when giving themselves up to their
subordinate natures, subsist according to their diversified properties:
since they are multiplied and distributed together with these, and
receive the division of their subjects. Besides, this also must be
previously received concerning a line, that our geometrician uses it in
a threefold acceptation. As terminated on both sides, and finite; as
in the problem[137] which says, Upon a given terminated right line to
construct an equilateral triangle. And as partly infinite and partly
finite; as in the problem which commands us from three right lines,
which are equal to three given right lines, to construct a triangle;
for in the construction of the problem, he says, Let there be placed a
certain right line, on one part finite, but on the other part infinite.
And again, a line is received by Euclid as on both sides infinite; as
in the problem which says, Upon a given infinite right line, from a
given point, which is not in that line, to let fall a perpendicular.
But, besides this, the following doubts, since they are worthy of
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