Abstract
IT could hardly fail to be instructive if Mr. Frankland would explain the following obvious paradox in his theory (NATURE, vol. xv., p. 515) Let two “straight lines” XOX′, LOL′, make an angle XOL other than a right angle, and consider the shortest line PN from a moving point P in LL′ to XX′; from the assumptions, this is a “straight line” perpendicular to XX′. AS P moves from O along OL, it will by and by, according to the theory, be at L′; that is, on the other side of XX′, if our “straight lines” are “of the same shape all along.” Now, to put it algebraically, how does the perpendicular come to change sign? It does not pass through infinity, for the manifolduess is of finite extent: it does not vanish except when P is at O; and though it is conceivable in itself that N should travel to a maximum distance along OX and come back again while P moves on, yet this contradicts our principal assumption, for each perpendicular will then have two points in common with LL′. Is a door of escape to be found through any interpretation of “continuous”? Or, while “there is nothing self-contradictory in the definition,” is there something in it contradictory of the superposition-principle by means of which its consequences are worked out?
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MONRO, C. On the Simplest Continuous Manifoldness of Two Dimensions and of Finite Extent. Nature 15, 547 (1877). https://doi.org/10.1038/015547d0
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DOI: https://doi.org/10.1038/015547d0
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