"Two straights perpendicular to a third straight intersect at a point half a straight from the third either way." "A pole is half a straight from its polar." "A polar is the locus of coplanar points half a straight from its pole. Therefore, if the pole of one straight lies on another straight the pole of this second straight is on the first straight." "The cross of two straights is the pole of the join of their poles." "Any two straights inclose a plane figure, a digon." "Two digons are congruent if their angles are equal." "The equidistantial is a circle with center at the poles of its basal straight." A typical postulate based upon the BOLYAI hypothesis of the acute angle is the following: "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. [Illustration: 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." [8] NOTE.--_M_ may be any point on the line _BA_ indefinitely produced. 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."