Springer, 1932 . vii-f-310 pp.+330 figures. R.M. 25.80. The foundation of this extremely interesting book is a course of lectures delivered by Hilbert at the University of Göttingen in the winter of 1920-21. The notes were taken by W. Rosemann and supplemented and edited by S. Cohn-Vossen. In the preface Hilbert points out that in mathematics as in all other scientific research we meet two sorts of tendencies: one towards abstraction, the other towards intuition. The first seeks to work out the

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... logical points of view from the extensive material and to connect it systematically ; the second strives for an intuitional conception and an understanding of relations of content. The first has led to the magnificent systematic theories embodied in algebraic geometry, Riemannian geometry, and topology. Nevertheless, Hilbert maintains the position that intuitional geometry is still of great importance as a superior force of research and for the appreciation of the results of research. The reviewer may be permitted to illuminate this by a personal experience. It is well known that the sextic of genus four, as the intersection of two generically located cubic and quadric surfaces, admits of 120 tritangent-planes. Just as in the case of the 28 double tangents of a general plane quartic, which may all be real, the question arises whether all 120 tritangent-planes may be real.