It is a well-known fact that Euclidean geometry, familiar to all of us since childhood, is not the only geometric system. Widely known, for example, non-Euclidean geometry Lobachevsky. Interest in it is caused, first of all, by the very fact of the “non-uniqueness” of geometry, which sheds light on many features of mathematical science. However, not all "nemathematicians" are likely to know that many non-Euclidean geometries are known. Among them, Lobachevsky’s geometry is quite difficult to understand. For the first acquaintance with non-Euclidean geometries, the simplest geometry would be better suited, which would be, say, no more complicated than Euclidean geometry. With one of these geometries - non-Euclidean geometry associated with the Galilean principle of relativity - the author introduces us in this book.