Essay

Geometrical Researches on the Theory of Parallels

Nikolai Lobachevsky

Develops a self-consistent geometry in which Euclid's parallel postulate fails, giving the first worked demonstration that a non-Euclidean geometry is logically possible.

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FreeIntermediate50 pageslink checked 17 Sept 2026

Read these first

Not a difficulty warning — a route. These are in the library too.

ElementsEuclid, 300 BCEDerives the whole of classical geometry from five postulates, and in doing so invents the axiomatic method every later mathematics is written in.

If this is the wrong level

Too hard? Start hereNon-Euclidean Geometry: A Critical and Historical Study of its DevelopmentRoberto Bonola, 1912Traces two thousand years of failed attempts to prove the parallel postulate and shows that the failures were the discovery: the postulate is independent, so consistent geometries exist without it.

Filed under Geometry in Mathematics.