Abhandlungen zur Geschichte der Mathematik.

On an allusion in Aristotle to a construction for parallels. 157
easily deduced from the other; and whether construction or demonstration is implied by yqoacpLv is immaterial also, since, even if construction is meant, a construction in geometry is not complete unless
followed by a demonstration that what was required is in fact done.
Let us now compare the construction or demonstration as thus explained with EUCLID'S course of procedure in Book I of the Elements.
In I. 27 he proves that, if a straight line falling on two straight lines
makes the alternate angles equal to one another, the two straight
lines are parallel, and in I. 28 he proves that, if a straight line falling
on two straight lines makes the interior angles on the same side
equal to two right angles, the two straight lines are parallel; lastly
in I. 31 he draws, through a given point A, a parallel to a given straight
line BC by joining A to any point D on BC and then drawing through A
a straight line E A making, with AD, the alternate angle EAD equal to
the alternate angle ADC. That is, if we accept the interpretation of the
passage of ARISTOTLE given by the scholiast and WAITZ, ARISTOTLE must
be supposed to say that the argument in the three propositions of EUCLID
referred to involves a petitio principii.
But is it true that we have here a petitio principii? I think that,
if the question is considered for a moment, it will be clear that there is
no petitio principii whatever involved. EUCLID defines parallel straight
lines as straight lines which, being in the same plane, will never meet
however far they are produced in either direction. Then in I. 27, 28 he
proves that, if the alternate angles are equal, or if the two interior angles
on the same side are equal to two right angles, the two straight lines in
question will never meet however far they are produced; whence, by the
definition, they are parallel. Lastly in I. 31 he draws a straight line
in accordance with the criterion of parallelism furnished by I. 27, and it
follows that the straight line so drawn is parallel to the given straight
line. There is not here even any assumption of a difficult Postulate such
as Postulate 5; and the observation of WAITZ that the equality of the
(alternate) angles cannot be proved except from the fact that parallel
straight lines are taken is certainly not true, because one angle is drawn
equal to the other angle (an operation which I. 23 has taught us to
effect without any reference to parallels), and the equality of the angles
does not depend upon the existence of parallels or upon anything else
than the construction.
Further than this, I think that we may conclude from other passages
of ARISTOTLE himself that he would not have regarded EUCLID'S procedure
as open to the charge of petitio principii. Thus in Anal. post. I 5