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Non-Euclidean geometry: motivations to develop it at the times of Gauss?

The motivation of discoverers of non-Euclidean geometry (Gauss, Lobachevski and Bolyai) was their attempts to prove the Fifth postulate of Euclid (to deduce it from the other axioms, or to replace by ...
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How was curvature originally defined and calculated?

From Curvature of Surfaces in 3-Space by Michael Garman & Jessica Bonnie published in Verge 6: The notion of curvature first began with the discovery and refinement of the principles of geometry ...
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