On the Simplest Continuous Manifoldness of Two Dimensions and of Finite Extent

Abstract

THERE appeared in your pages some three years ago (vol. xv. p. 515) an article of mine “On the Simplest Continuous Manifoldness of Two Dimensions and of Finite Extent.” In a succeeding number a correspondent (Mr. Monro, of Barnet) propounded a query which may be shortly stated as follows:— “How does it happen that the perpendicular on a geodesic from a point moving along another geodesic changes sign without passing through either the value zero (0) or the value infinity (∝)?” The problem here suggested is a peculiarly knotty one. In the case of the Euclidian plane the perpendicular of course changes sign by passing through the value ∝, while in the case of a spherical surface it is equally obvious that the perpendicular passes through zero, since the two geodesies intersect twice. But what are we to say of the strange hybrid surface which formed the subject-matter of my paper? Your correspondent appeared to insinuate that the problem was insoluble, and that the definition of the surface must therefore involve a logical contradiction. For a while I was greatly puzzled by this unforeseen difficulty, but after a little thought came to the conclusion that the perpendicular changes sign by passing through the value l/2√−1, where l is positive and represents the absolute length of a complete geodesic. In other words, I conceived that the sign of the perpendicular changed from + to − by a continuous variation of the real numbers a and b in the complex number a + b√- 1. I conceived a to diminish continuously till, passing through o, it became − -a, while b at the same time increased with simple harmonic motion from o to a maximum, and then decreased from a maximum to o.