Abstract
So far as I am concerned Mr. Frankland answers too soon (p. 170), for I am sorry to say I have not read Klein in the meantime. Therefore my reply is provisional. A hint was given of Mr. Frankland's explanation by Mr. Newcomb in a phrase quoted by Mr. Halsted (American journ. of Math., I. iii. 275, paper on the bibliography of hyperspace, &c.): “The first elements of complex functions imply that a line can change direction without passing through infinity or zero.” We do not require even the first elements of complex functions to tell us that we can get to the other side of a point without passing through it, provided we can go round it. But the question was not whether “a line” simply could be thus reversed, but whether it could be so with the geodetic perpendicular in question described in a uniform continuous manifold of two dimensions. Mr. Frankland's explanation expressly takes account of a third dimension. It supposes the moving line to generate a sort of skew helicoid about the fixed line to which it is perpendicular. But how can even initial portions of successive generators be in the same plane, Euclidean or other? This point may seem incidental, but I think it is essential, so I omit further questions.
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MONRO, C. On the Simplest Continuous Manifold of Two Dimensions and of Finite Extent. Nature 22, 218 (1880). https://doi.org/10.1038/022218b0
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DOI: https://doi.org/10.1038/022218b0
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