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Further, the spherical surface is a non-Euclidean continuum of two dimensions, that is to say, the laws of disposition for the rigid figures lying in it do not agree with those of the Euclidean plane. This can be shown in the following way. Place a paper disc on the spherical surface, and around it in a circle place six more discs, each of which is to be surrounded in turn by six discs, and so on.
In their non-Euclidean geometry the part is always greater than the whole. In their unnatural history the tail always wags the dog. On the plane of politics it is not necessary to press the case against "Ulster" any farther than that. Even majorities have their rights.
This distinguished scientist has expounded his views in a book entitled "Verschwinden und Seine Theorie," which has attracted some attention, "particularly," says one writer, "among the followers of Hegel, and mathematicians who hold to the actual existence of a so- called non-Euclidean space that is to say, of space which has more dimensions than length, breadth, and thickness space in which it would be possible to tie a knot in an endless cord and to turn a rubber ball inside out without 'a solution of its continuity, or in other words, without breaking or cracking it."
But it has gradually appeared, by the increase of non-Euclidean systems, that Geometry throws no more light upon the nature of space than Arithmetic throws upon the population of the United States. Geometry is a whole collection of deductive sciences based on a corresponding collection of sets of axioms. One set of axioms is Euclid's; other equally good sets of axioms lead to other results.
The logical problem has arisen through the suggestions of non-Euclidean geometry. Given a body of geometrical propositions, it is not difficult to find a minimum statement of the axioms from which this body of propositions can be deduced.
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