The necessity of geometrical determinations is merely the necessity which inheres in logical inferences or deductions. These may or may not be valid. Inasmuch as the necessariness of deductions is primarily based upon the conditional certainty of premises and definitions it appears that this quality is in no way peculiar to geometry whether Euclidean or non-Euclidean. In like manner, the universality of geometric judgments may not properly be regarded as a peculiarity of geometry; but is explicable upon the basis of the formal character of the assumptions which underlie it. The chief value, then, of non-Euclidean geometry seems to abide in the fact that it clarifies our understanding as to the complex processes by which it is possible to organize and systematize our spatial experiences for assimilation and use in other branches of knowledge. With the above statement of the case of the non-Euclidean geometry it is now thought permissible to state briefly some of the elements thereof.[7] [7] The science of pure mathematics is perhaps indebted to no one in so great a degree as to GEORGE BRUCE HALSTEAD, formerly of the University of Texas, whose labors in connection with the popular exposition of the non-Euclidean geometry have been most untiring and effectual. Vide _Popular Astronomy_, Vol. VII and VIII, 1900, Dr. G. B. HALSTEAD. Below will be found some of the elements obtained as a consequence of efforts made both at proving and disproving the parallel-postulate of Euclid: "If two points determine a line it is called a straight." "If two straights make with a transversal equal alternate angles they have a common perpendicular." "A piece of a straight is called a sect." "If two equal coplanar sects are erected perpendicular to a straight, if they do not meet, then the sect joining their extremities makes equal angles with them and is bisected by a perpendicular erected midway between their feet." "The sum of the angles of a rectilineal triangle is a straight angle, in the hypothesis of the right (angle); is greater than a straight angle in the hypothesis of the obtuse (angle); is less than a straight angle in the hypothesis of the acute (angle)." "The hypothesis of right is Euclidean; the hypothesis of the acute is BOLYAI-LOBACHEVSKIAN; the hypothesis of obtuse is RIEMANNIAN." "If one straight is parallel to a second the second is parallel to the first." "Parallels continually approach each other." "The perpendiculars erected at the middle point of the sides of a triangle are all parallel, if two are parallel." "If the foot of a perpendicular slides on a straight its extremity describes a curve called an equidistant curve, or an equidistantial." "An equidistantial will slide on its trace." "In the hypothesis of the obtuse a straight is of finite size and returns into itself." "Two straights always intersect."