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
comprehended by more than two, there is an infinite procession. For
there are certain figures contained by three and four and succeeding
circumferences. Thus, if three circles touch each other, they will
intercept a certain trilateral space; but if four, one terminated by
four circumferences, and in like manner, by a successive progression.
But of those contained by right lines, some are comprehended by three,
others by four, and others by a multitude of lines. For neither is
space comprehended by two right-lines, nor much more by one right-line.
Hence, every space comprehended by one boundary, or by two, is either
mixt or circular. And it is mixt in a twofold manner, either because
the mixt lines comprehend it, as the space intercepted by the cissoidal
line; or because it is contained by lines dissimilar in species,
as the apsis: since mingling is twofold, either by apposition or
confusion. Every right-lined figure, therefore, is either trilateral,
or quadrilateral, or gradually multilateral; but every trilateral, or
quadrilateral, or multilateral figure, is not right-lined; since so
great a number of sides is also produced from circumferences. And thus
much concerning the division of plane figures. But we have already
asserted[177], that rectitude of progression is both a symbol of motion
and infinity, and that it is peculiar to the generative co-ordinations
of the gods, and to the producers of difference, and to the authors
of mutation and motion. Right-lined figures, therefore, are peculiar
to these gods, who are the principles of the prolific energy of the
whole progression of forms. On which account, generation also, was
principally adorned by these figures, and is allotted its essence from
these, so far as it subsists in continual motion and mutation without
end.
DEFINITION XXIV.
Of three-sided FIGURES: an EQUILATERAL TRIANGLE is
that which has three equal Sides.
DEFINITION XXV.
An ISOSCELES TRIANGLE, is that which has only two Sides
equal.
DEFINITION XXVI.
A SCALENE TRIANGLE, is that which has three unequal
Sides.
DEFINITION XXVII.
A RIGHT-ANGLED TRIANGLE is that which has a Right
Angle.
DEFINITION XXVIII.
An OBTUSE-ANGLED TRIANGLE is that which has an
Obtuse Angle.
DEFINITION XXIX.
An ACUTE-ANGLED TRIANGLE is that which has three
Acute Angles.
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