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
the circumference of the circle are equal, is either within the circle,
or without (for every circle has a pole, from which all the lines drawn
to its circumference are equal), on this account he adds, _of the
points within the figure_, because, here he receives the centre
alone, and not the pole. For he wishes to behold all its properties in
one plane, but the pole is more elevated than the subject plane. Hence,
he necessarily adds, in the end of the definition, that this point,
which is placed within the circle, and to which all right lines drawn
from it to the circumference, are equal, is the centre of the circle.
For there are only two points of this kind, the pole and the centre.
But the former is without, and the other within the plane. Thus, for
instance, if you conceive a perpendicular standing on the centre of a
circle, its superior extremity is the pole: for all lines drawn from
it to the circumference of the circle, are demonstrated to be equal.
And, in like manner, in a cone, the vertex of the whole cone, is the
pole of the circle at the base. And thus far we have determined what a
circle is, and its centre, and what the nature is of its circumference,
and the whole circular figure. Again, therefore, from these, let us
return to the speculation of their exemplars, contemplating in them
the centre, according to one impartible and stable excellence. But the
distances from the centre, according to the progressions which are made
from one, to multitude infinite in capacity. And the circumference of
the circle, according to the regression of the progressions to the
centre, by means of which the multitude of powers are rolled round
their union, and all of them hasten to its comprehension, and desire
to energize about its indivisible embrace. And, as in the circle
itself, all things subsist together, the centre, intervals, and
external circumference; so in these which are its image, one thing
has not an essence pre-existent, and another consequent in time;
but all things are, indeed, together, permanency, progression, and
regression. But these differ from those, because the former subsist
indivisibly, and without any dimension; but the latter with dimension;
and in a divisible manner; the centre existing in one place, the lines
emanating from the centre, in another; and the external circumference
terminating the circle, having a still different situation. But there
all things abide in one: for if you regard that which performs the
office of a centre, you will find it the receptacle of all things. If
the progression distant from the centre, in this, likewise, you will
find all things contained. And, in a similar manner, if you regard
its regression. When, therefore, you are able to perceive all things
subsisting together, and have taken away the defect proceeding from
dimension, and have removed from your inward vision, the position about
which partition subsists, you will find the true circle, advancing to
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