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Russian mathematician Lobachevsky (1792-1856) composed Pangeometry the year before he died as a summary and culmination of his life's work in non-Euclidean geometry, now called Lobachevskian geometry. Blind by then, he dictated the treatise in Russian and in French, and for this first complete English version, Papadopoulos (mathematics, U.
Let lines of direction be drawn from my eye to every point of a plane triangle that I see, and let there be interposed between this triangle and my eye a saddle-shaped surface of the kind used to model Lobachevskian geometry. The lines of direction will intercept this surface in a triangle one side of which curves outward and two sides of which curve inward in relation to the eye.
In Euclidean geometry, the three angles of a triangle add up to 360 degrees; in Lobachevskian geometry, they add up to less than 360 degrees; and in Riemannian geometry, they add up to more than 360 degrees.