The first part, which constitutes the content of this volume, is devoted to a detailed exposition of the geometry of Lobachevsky, this main creation, on which the modern doctrine of the foundations of geometry is primarily built, if not on the justification of all mathematics in general. Quite naturally, the desire to publish in our country, where non-Euclidean geometry arose and was deeply developed in the works of N .. and . Lobachevsky, a detailed and serious exposition of hyperbolic geometry, according to which a geometer could thoroughly study it, assimilate about as the classical geometry of Euclid is assimilated. I think it is quite wrong to think that Lobachevsky's geometry is sufficient to understand itself by general acquaintance with one of its interpretations or models. Accordingly, hyperbolic geometry in this work is presented so that the student can learn it and master it to the same extent as he owns classical geometry ;. and for this it is necessary to state it in such order and in such volume as the geometry of Euclid is stated (elementary, analytical, differential) .. This is what I tried to do in the present volume ..