The conventional method of a generalized geometry construction, based on deduction of all propositions of the geometry from axioms, appears to be imperfect in the sense, that multivariant geometries cannot be constructed by means of this method. Multivariant geometry is such a geometry, where at the point P there are many vectors PP', PP'',... which are equivalent to the vector QQ' at the point Q, but they are not equivalent between themselves. In the conventional (Euclidean) method the equivalence relation is transitive, whereas in a multivariant geometry the equivalence relation is intransitive, in general. It is a reason, why the multivariant geometries cannot be deduced from a system of axioms. The space-time geometry in microcosm is multivariant. Multivariant geometry is a grainy geometry, i.e. the geometry, which is partly continuous and partly discrete. Multivariance is a mathematical method of the graininess description. The graininess (and multivariance) of the space-time geometry generates a multivariant (quantum) motion of particles in microcosm. Besides, the grainy space-time generates some discrimination mechanism, responsible for discrete parameters (mass, charge, spin) of elementary particles. Dynamics of particles appears to be determined completely by properties of the grainy space-time geometry. The quantum principles appear to be needless.