Abhandlungen zur Geschichte der Mathematik.

158 T. L. Heath:
(74a 12), where he is showing that a&z6Otltg should be not only xara
7wtvrog but rov'ov 7recov xoV zO4Aov, he mentions as an instance the proof
that "right angles do not meet" (by which of course he means that straight
lines making the two interior angles equal to two right angles will not
meet), observing that it is not enough to prove this in the case where
each of the two angles is right, because this is only one way in which
the two angles can be equal to two right angles, whereas the property
depends on the equality of the sum of the angles to two right angles,
while the individual angles may bear any ratio to one another. Can it
be supposed that ARISTOTLE would have referred to the general proof in
this case as a scientific 6eöt~EIgL if he had regarded it as a petitio
principii?
I think it clear therefore that the interpretation of the scholiast and
WAITZ cannot be right, and that ARISTOTLE must have referred to some
other construction for parallels given by somie conteniporary geometers.
What then was this alternative construction? I think that the conment
of PHILOPONUS on the passage gives a clue. PHILOPONUS does in fact
allude to a construction different from EUCLID'S, and, though the description of it is somewhat vague, it seems possible to make out its
essential features:
(PHILOPONUS f. CXI1) TO av otoro iolo xl oi Ol rTag caoaXjXLovg
yQao;ovEs CTO EV o% x CLZEl~azCL ßovXovcaIt yCo 7acaQaXAovg EvsOIgC
&7tC roi E 3erßly VO XVKiov erayQca[uL JvvaCo6v, ca iaov'ßP'VOVL
6r]iLE8Ov Cog iCEbLt5V 7rTrov 8EQI TO Ei7tEiTe1ov E'IVOV Kat ozlrco ExPaCLovtL r'ag EvEltCag. XabC EiYTai ö rOVO ET'oYoTL* 6 ya& 0
GvycoQov yiyvEuau zrIv 7tcraQdurtov o&6E 1o ';>iLEoOV (5vYXCOQi)sE E8ELVO.
"The same thing is done by those who draw parallels, namely
begging the original question; for they will have it that it is
possible to draw parallel straight lines from the meridian circle,
and they assume a point, so to say, falling on the plane of that
circle, and thus they draw the straight lines. And what was sought
is thereby assumed; for he who does not admit the genesis of the
parallel will not admit the point referred to either."
What is ineant is, I think, somewhat as follows. Given a straight
line and a point through which we have to draw a parallel to it, we
are to suppose that the given straight line is placed in the plane of the
meridian. Then we are told to draw through the given point another
straight line in the plane of the meridian (strictly speaking, it should be
a plane parallel to the plane of the meridian, but the idea is that, compared with the size of the meridian circle, the distance of the given point