Did Lobachevsky Have a Model of His “imaginary Geometry”?

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). In this paper I consider this Lobachevsky’s construction, and show how it can be better analyzed within an alternative non-Hilbertian foundational framework, which relates the history of geometry of the 19th century to some recent developments in the field.