. . .es referred to in the above postulate as to whether it is equal to, greater or less than two right angles.

The assumption that the angular sum is congruent to a straight angle is called the Euclidean or parabolic hypothesis and is to be distinguished from the[Pg 55] synthetic or hyperbolic hypothesis established by Gauss, Lobachevski and Bolyai and which assumes that the angular sum is less than a straight angle. The elliptic or Cayley-Klein hypothesis assumes that the angular sum is greater than a straight angle. Lobachevski, however, not satisfied with the statement of the parallel-postulate as given by Euclid and which had caused the age-long controversy, substituted for it the following:

"All straight lines which, in a plane, radiate from a given point, can, with respect to any other straight line, in the same plane, be divided into two classes—the intersecting and the non-intersecting. The boundary line of the one and the other class . . . Read More