GEMINUS 
227 
Attempt to prove the Parallel-Postulate. 
Geminus devoted much attention to the distinction between 
postulates and axioms, giving the views of earlier philoso 
phers 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 
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 
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 
1 we have learned from the very pioneers of this science not to 
have regarofe to mere plausible imaginings when it is a ques 
tion of the reasonings to be included in our geometrical 
doctrine. As Aristotle says, it is as justifiable to ask scien 
tific proofs from a rhetorician as to accept mere plausibilities 
from a geometer... So in this case (that of the parallel- 
postulate) the fact that, when the right angles are lessened, the 
straight lines converge is true and necessary; but the state 
ment that, since they converge more and more as they are 
produced, they will sometime meet is plausible but not neces 
sary, 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 
[aavpiTTcoTOL), 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 End. I, pp. 178-82. 4; 183. 14-184. 10. 
2 lb., pp. 188. 26-184. 5. 
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