  The construction introduced above to explain Euclid's Postulate can also be used for elucidating its negation. Draw the straight line a through point P at right angles with the segment PQ. If Euclid's Postulate is denied, there are countless straight lines through Q, coplanar with a, that make acute angles with PQ but never meet a.  Consider the set of real numbers which are the magnitudes of these acute angles. Let the greatest lower bound of this set be μ.  Evidently, μ &gt; 0. There are exactly two straight lines through Q, coplanar with a, that make an angle of size μ with PQ. (See Figure 2.) Call them b1 and b2. Neither b1 nor b2 meets a, but a meets every line through Q that is coplanar with a and makes with PQ an angle less than μ. Gauss, Lobachevsky and Bolyai—unbeknownst to each other—coincided in calling b1 and b2 the parallels to a through Q. μ is called the angle of parallellism for segment PQ. Its size depends on the length of PQ, and decreases as the latter increases.