A history of Greek mathematics - Wilbourhall.org

GEMINUS 227
Attempt to prove the Parallel-Postulate.
Geminus devoted much attention to the distinction between
postulates and axioms, giving the views of earlier philosophers
and mathematicians (Aristotle, Archimedes, Euclid,
Apollonius, the Stoics) on the subject as well as his own. It
was important in view of the attacks of the Epicureans and
Sceptics on mathematics, for (as Geminus says) it is as futile
to attempt to prove the indemonstrable (as Apollonius did
when he tried to prove the axioms) as it is incorrect to assume
what really requires proof, as Euclid did in the fourth postu-
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late [that all right angles are equal] and in the fifth postulate
[the parallel-postulate] '}
The fifth postulate was the special stumbling-block.
Geminus observed that the converse is actually proved by
Euclid in I. 17; also that it is conclusively proved that an
angle equal to a right angle is not necessarily itself a right
angle (e.g. the angle between the circumferences of two semi-
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circles on two equal straight lines with a common extremity
and at right angles to one another) ; we cannot therefore admit
that the converses are incapable of demonstration. 2
And
we have learned from the very pioneers of this science not to
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have regard to mere plausible imaginings when it is a question
of the reasonings to be included in our geometrical
doctrine. As Aristotle says, it is as justifiable to ask scientific
proofs from a rhetorician as to accept mere plausibilities
from a geometer ... So in this case (that of the parallelpostulate)
the fact that, when the right angles are lessened, the
straight lines converge is true and necessary ; but the statement
that, since they converge more and more as they are
produced, they will sometime meet is plausible but not necessary,
in the absence of some argument showing that this is
true in the case of straight lines. For the fact that some lines
exist which approach indefinitely but yet remain non-secant
(dcrvfi7rTCQToi), although it seems improbable and paradoxical,
is nevertheless true and fully ascertained with reference to
other species of lines [the hyperbola and its asymptote and
the conchoid and its asymptote, as Geminus says elsewhere].
May not then the same thing be possible in the case of
1
Proclus on Eucl. I, pp. 178-82. 4; 183. 14-184. 10.
2 lb., pp. 183. 26-184. 5.
Q 2