"If a straight line meet two straight lines so as to make the two interior angles on the same side of it taken together less than two right angles, these straight lines being continually produced, shall at length meet upon that side on which are the angles which are less than two right angles."
On this postulate hang all the "law and the prophets" of the non-Euclidean Geometry. In it are the virtual elements of three possible geometries. Furthermore, it is both the warp and the woof of the loom of present-day metageometrical researches. It is the golden egg laid by the god Seb at the beginning of a new life cycle in psychogenesis. Its progeny are numerous—hyperspaces, sects, straights, digons, equidistantials, polars, planars, coplanars, invariants, quaternions, complex variables, groups and many others. A wonderfully interesting breed, full of meaning and pregnant with the power of final emancipations for the human intellect!
When the conclusions which were systematically formulated as a result of the investigations along the lines of hypotheses which controverted the parallel-postulate were examined it was found that they fell into three main divisions, namely: the synthetic or hyperbolic; the analytic or Riemannian and the elliptic or Cayley-Klein. These divisions or groups are based upon the three possibilities which inhere in the conception taken of the sum of the angles 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 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 is called parallel to the given line."
This is but another way of saying about the same thing that Euclid had declared before, and yet, curiously enough it afforded just the liberty that Lobachevski needed to enable him to elaborate his theory.
For the purposes of this sketch the field of the development of non-Euclidean geometry is divided into three periods to be known as: (1) the formative period in which mathematical thought was being formulated for the new departure; (2) the determinative period during which the mathematical ideas were given direction, purpose and a general tendence; (3) the elaborative period during which the results of the former periods were elaborated into definite kinds of geometries and attempts made at popularizing the hypotheses.
The Formative Period