The invention of non-Euclidean geometries is often seen through the optics of Hilbertian formal axiomatic method developed later in the 19th century. <...> However such an anachronistic approach fails to provide a sound reading of Lobachevsky’s geometrical works. <...> Although the modern notion of model of a given theory has a counterpart in Lobachevsky’s writings its role in Lobachevsky’s geometrical theory turns to be very unusual. <...> Lobachevsky doesn’t consider various models of Hyperbolic geometry, as the modern reader would expect, but uses a non-standard model of Euclidean plane (as a particular surface in the Hyperbolic 3-space). <...> Introduction A popular story about the discovery of Non-Euclidean geometries goes like this. <...> HO] 11 Aug 2010 Rodin A., 2015 Did Lobachevsky have a model of his «imaginary geometry»? 35 bachevsky guessed that non-P5 opens a door into a vast unexplored territory rather than leads to the expected dead end. <...> However the issue remained highly speculative until Beltrami in his [2] found some Euclidean models of Lobachevsky’s geometry, which proved that Lobachevsky’s new geometry is consistent (relatively to Euclidean geometry). <...> Finally Hilbert in his [11] put things in order by modernizing Euclidean axiomatic method and clarifying the logical structure of Non-Euclidean geometries. <...> This assumption concerns the very notion of mathematical theory, which dates back to [11] and goes on a par with the above story. <...> The following passage makes it clear how the author’s notion of mathematical theory has a bearing on his interpretation of history: † The formalist viewpoint just stated is a radical departure from the older notion that mathematics asserts “absolute truths”, a notion that was destroyed once and for all by the discovery of Non-Euclidean geometry. <...> The principal aim of this short study is to reconsider the Hilbertian “formalist viewpoint” through looking back at the history of geometry of 19th century, and more specifically at Lobachevsky’s works. <...> But if one allows for the the talk of models in this context one finds something surprising: Lobachevsky didn’t look for models of the geometrical theory known by his name but used a non-standard model of Euclidean plane. <...> Hyperbolic intuition Lobachevsky in his writings presents his main geometrical discovery in various <...>