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J. H. Lambert(1776) first suggested that the hypothesis of the acute angle, in which the sum of the angles of a triangle is always less than two right angles, implies the existence of an absolute measure of distance, which he implicitly defined as the linear constant relating the length of the side of a quadrilateral with three right angles to the magnitude of its acute angle (the first geometrical representation). He also interpreted the constant in terms of the radius of an imaginary sphere (the second representation). In his work on non-Euclidean hyperbolic geometry in the second decade of the nineteenth century, Carl Gauss denoted the constant by the letter k. He expressed its (unknown) magnitude in terms of the area of the triangle of maximum size whose vertices lie at infinitely distant points and whose angle sum is zero (the third representation). Ferdinand Schweikart (1818) expressed the constant in terms of the altitude of a right angled isosceles triangle whose acute angles approach zero as its sides are indefinitely extended (the fourth representation). In his log-spherical geometry based on a sphere with an imaginary radius Franz Taurinus (1826) deduced what has come to be known as the fundamental identity of hyperbolic geometry. Using this identity the constant can be defined as the length of the segment corresponding to an angle of parallelism of approximately 40 23’ 42’’ (the fifth representation). Nikolai Lobachevsky (1829) and János Bolyai (1832) both represented the constant in terms of horocycles, circles of infinite radius peculiar to hyperbolic geometry. They defined the constant as the radial distance separating two concentric horocyclic arcs between the same parallels where the ratio of the outer to the inner arc is e : 1 (the sixth representation). Karl Schwarzschild (1900) equated the linear constant to the radius of curvature of hyperbolic space (the seventh representation). Finally in numerous expository texts on hyperbolic geometry in the last century the constant was defined as the length of a horocycle whose tangent at one extremity is parallel to the axis through the other extremity (the eighth representation).