I believe that the thorough investigation of the conditions for the validity of this theorem will throw a new light upon the idea of distance, as well as upon other elementary ideas, e. g., upon the idea of the plane, and the possibility of its definition by means of the idea of the straight line, the construction and systematic treatment of the geometries here possible seem to me desirable. Flat metrics Desargues's theorem:If two triangles lie on a plane such that the lines connecting corresponding vertices of the triangles meet at one point, then the three points, at which the prolongations of three pairs of corresponding sides of the triangles intersect, lie on one line.The necessary condition for solving Hilbert's fourth problem is the requirement that a metric space that satisfies the axioms of this problem should be Desarguesian, i.e.,: if the space is of dimension 2, then the Desargues's theorem and its inverse should hold; if the space is of dimension greater than 2, then any three points should lie on one plane. For Desarguesian spaces Georg Hamel proved that every solution of Hilbert's fourth problem can be represented in a real projective space or in a convex domain of if one determines the congruence of segments by equality of their lengths in a special metric for which the lines of the projective space are geodesics. Metrics of this type are called flat or projective. Thus, the solution of Hilbert's fourth problem was reduced to the solution of the problem of constructive determination of all complete flat metrics. Hamel solved this problem under the assumption of high regularity of the metric. However, as simple examples show, the class of regular flat metrics is smaller than the class of all flat metrics. The axioms of geometries under consideration imply only a continuity of the metrics. Therefore, to solve Hilbert's fourth problem completely it is necessary to determine constructively all the continuous flat metrics. Prehistory of Hilbert's fourth problem Before 1900, there was known the Cayley–Klein model of Lobachevsky geometry in the unit disk, according to which geodesic lines are chords of the disk and the distance between points is defined as a logarithm of the cross-ratio of a quadruple. For two-dimensional Riemannian metrics, Eugenio Beltrami (1835–1900) proved that flat metrics are the metrics of constant curvature. For multidimensional Riemannian metrics this statement was proved by E. Cartan in 1930. In