PTOLEMY ON THE PARALLEL-POSTULATE 297 
two right angles, it must also make the other pair BFG, FGD 
together greater than two right angles. 
But the latter pair of angles were proved less than two 
right angles: which is impossible. 
Therefore the sum of the ^angles AFG, FGG cannot be 
greater than two right angles. 
(2) Similarly we can show that the sum of the two angles 
AFG, FGG cannot be less than two right angles. 
Therefore Z AFG + Z GGF — two right angles. 
[The fallacy here lies in the inference which I have marked 
by italics. When Ptolemy says that AF, GG are no more 
parallel than FB, GD, he is in effect assuming that through 
any one'point only one*parallel can he drawn to a given straight 
line, which is an equivalent for the very Postulate he is 
endeavouring to prove. The alternative Postulate is known 
as ‘ Playfair’s axiom but it is of ancient origin, since it is 
distinctly enunciated in Proclus’s note on Eucl. I. 31.] 
III. Post. 5 is now deduced, thus. 
Suppose that the straight lines making with a transversal 
angles the sum of which is less than two right angles do not 
meet on the side on which those angles are. 
Then, a fortiori, they will not meet on the other side on 
which are the angles the sum of which is greater than two 
right angles. [This is enforced by a supplementary proposi 
tion showing that, if the lines met on that side, Eucl. I. 16 
would be contradicted.] 
Hence the straight lines cannot meet in either direction : 
they are therefore parallel. 
But in that case the angles made with the transversal are 
equal to two right angles : which contradicts the assumption. 
Therefore the straight lines will meet.