"From any point P drop PC, a perpendicular to any given straight line AB. If D move off indefinitely on the ray CB, the sect will approach as limit PF copunctal with AB at infinity.
Fig. 5.
PD is said to be at P the parallel to AB toward B. PF makes with PC an angle CPF which is called the angle of parallelism for the perpendicular PC. It is less than a right angle by an amount which is the limit of the deficiency of the triangle PCD. On the other side of PC, an equal angle of parallelism gives the parallel P to BA towards AM.[8] Thus at any point there are two parallels to a straight. A straight has, therefore, two separate points at infinity."
"Straights through P which make with PC an angle greater than the angle of parallelism and less than its supplement do not meet the straight AB at all not even at infinity."
The parallel-postulate is stated in the non-Euclidean geometry as follows:
"If a straight line meeting two straight lines make those angles which are inward and upon the same side of it less than two right angles the two straight lines being produced indefinitely will meet each other on this side where the angles are less than two right angles."
It is stated by Manning[9] in the following language:
"If two lines are cut by a third and the sum of the interior angles on the same side of the cutting line is less than two right angles the line will meet on that side when sufficiently produced."