Perrault's watch and Beltrami's pseudosphere: A story without a moral
If a specter haunted 19th century mathematics, it was the specter of the pseudosphere, i.e. the two-dimensional space with constant negative curvature. Already toward the end of the previous century, Johann Heinrich Lambert (1728 1777), in his original investigation about Euclid’s fifth postulate,1 hinted that the angles of a triangle could sum to less than two right angles in the case where the triangle lies on an "imaginary sphere" (imaginäre Kugelfläche) [Lambert 1895, p. 203]. Lambert’s highly speculative remark was suggested by the «analogie entre le cercle et l’hyperbole» he had already exploited in his work on the irrationality of pi [Lambert 1768]; however, it would have been impossible to express that insight in a mathematically more definite form simply because, at the time, essential geometrical notions – first of all, that of curvature – were still to be developed.
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