By Felix Klein

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348-408. 30 LECTURE IV. treated alike. We may here confine ourselves to the case of an hyperboloicl of one sheet with four distinct lines of each set. These lines divide the surface into 16 regions. Shading the alternate regions as in the figure, and regarding the shaded regions as double, the unshaded regions being disregarded, we have a surface consisting of eight separate closed portions hanging together only at the points of intersection of the lines ; and this is a Kummer surface with 16 real double points.

The special case when the original points of contact happen to lie on a circle Fig. 11. being excluded, it can be shown analytically that the continuous curve which is the locus of all the points of contact is not an analytic atrve. The points of contact form a manifoldness that is everywhere dense on the curve (in the sense of G. Cantor), although there are intermediate points between them. At each of the former points there is a determinate tangent, while there is none at the intermediate points.

22 (1883), pp. 249-259. f Vorlesungen über neuere Geometrie, Leipzig, Teubner, 1882. g, Leipzig, Teubner, 1882. ictionen, Math. Annalen, Vol. 29 (1887), pp. 123-140. 46 • LECTURE VI. Pasch's idea of building up the science purely on the basis of the axioms has since been carried still farther by Peano, in his logical calculus. Finally, it must be said that the degree of exactness of the intuition of space may be different in different individuals, perhaps even in different races. It would seem as if a strong naïve space-intuition were an attribute pre-eminently of the Teutonic race, while the critical, purely logical sense is more fully developed in the Latin and Hebrew races.