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 — Plato — John Shaqi
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
Plato · en
But if it be necessary that all assumptions should be demonstrated by
others, and these again by others; either the enquiry must be continued
to infinity, (but infinity can never be absolved), or if, wearied by
the immense process, you at length stop, you must doubtless leave those
propositions unknown, whose demonstration was declined through the
fatigue of investigation. But how can science be derived from unknown
principles? For he who is ignorant of the principles, cannot understand
the conclusions which flow from these as their proper source, unless
from an hypothesis or supposition of their reality.
This argument of the sophists is, indeed, so far true, that he who does
not understand that which is first in the order of demonstration, must
remain ignorant of that which is last:--But in this it fails, that
all knowledge is demonstrative; since this is an assertion no less
ridiculous than to maintain that nothing can be known. For as it is
manifest that some things derive their credit and support from others,
it is equally obvious that many, by their intrinsic excellence, possess
indubitable certainty and truth; and command our immediate assent as
soon as proposed. They inherit, indeed, a higher degree of evidence
than those we assent to by the confirmation of others; and these are
the first principles of demonstration: propositions indisputable,
immediate, and perspicuous by that native lustre they always possess.
By means of these, we advance from proposition to proposition, and
from syllogism to syllogism, till we arrive at the most complicated
and important conclusions. Others, willing to decline this infinite
progression, defend the necessity of a circular or reciprocal
demonstration. But this is nothing more than to build error upon error,
in order to attain the truth; an attempt no less ridiculous than that
of the giants of old. For since, as we shall hereafter accurately
prove, demonstration ought to consist from that which is first, and
most known; and since it is impossible that the same thing should be
to itself both prior and posterior: hence we infer the absurdity of
circular demonstration; or those syllogisms in which the conclusions
are alternately substituted as principles, and the principles as
conclusions. It may, indeed, happen, that the same thing may be both
prior and posterior to the same; but not at one and the same time, nor
according to the same mode of existence. Thus, what is prior in the
order of our conceptions, is posterior in the order of nature; and what
is first in the arrangement of things, is last in the progressions of
human understanding. But demonstration always desires that _first_
which is prior in the order and constitution of nature. But the folly
of such a method will more plainly appear from considering its result:
let us suppose every _a_ is _b_, and every _b_ is _c_; hence we justly
infer, that every _a_ is _c_. In like manner, if we prove that every