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
and a rhombus, have their opposite sides and angles equal. But in a
rhomboides he only adds this, _that its opposite sides are equal_,
lest he should define it by negations alone, since he neither calls it
equilateral, nor rectangular. For where we want proper appellations,
it is necessary to use such as are common. But we should hear Euclid
shewing that this is common to all parallelograms. But a rhombus
appears to be a quadrangle having its sides moved, and a rhomboides
a moved oblong. Hence, according to sides, these do not differ from
those; but they vary only according to the obtuseness and acuteness
of angles; since the quadrangle and the oblong are rectangular. For
if you conceive a quadrangle or an oblong, having its sides drawn in
such a manner, that while two of its opposite angles are dilated, the
other two are contracted; then the dilated angles will appear obtuse,
and the contracted, acute. And the appellation of rhombus[182] seems
to have been imposed from motion. For if you conceive a quadrangle
moving after the manner of a rhombus, it will appear to you changed in
order, according to its angles: just as if a circle is moved after the
manner of a sling, it will immediately exhibit the appearance of an
ellipsis. But here you may perhaps enquire concerning the quadrangle,
why it has this denomination? and why the appellation of quadrangle
may not be applied to other quadrilateral figures, as the name of
triangle is common to all those which are neither equiangular nor
equilateral, and in like manner of quinquangles or pentagons; for
the geometrician, in these, adds only the particle _an equilateral
triangle_, or a _quinquangle_, _which is equilateral and
equiangular_, as if these could not be otherwise than such as they
are? But when he mentions a quadrangle, he immediately indicates that
it must be equilateral and rectangular. But the reason of this is as
follows: a quadrangle alone has the best space, both according to
its sides and angles. For each of the latter is right, intercepting
a measure of angles, which neither receives intention nor remission.
As it excels, therefore, in both respects, it deservedly obtains a
common appellation. But a triangle, though it may have equal sides,
yet will in this case have all its angles acute, and a quinquangle all
its angles obtuse. Since, therefore, of all quadrilateral figures,
a quadrangle alone is replete with equality of sides, and rectitude
of angles, it was not undeservedly allotted this appellation: for,
to excellent forms, we often dedicate the name of the whole. But it
appeared also to the Pythagoreans, that this property of quadrilateral
figures, principally conveyed an image of a divine essence. For they
particularly signified by this, a pure and immaculate order. Since
rectitude imitates inflexibility, but equality a firm and permanent
power: for motion emanates from inequality, but quiet from equality
itself. The gods, therefore, who are the authors to all things
Public-domain text, read in full here on John Shaqi.
Reviews
Reviews
No reviews yet
Be the first to share your thoughts on this work.
Elsewhere in the archive
Join the Discussion
Join the discussion
Sign in to leave a comment or review.
Sign InorCreate an account