# Lectures on Mathematics by Felix Klein

By Felix Klein

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The writer wish to recognize his legal responsibility to all his (;Olleagues and associates on the Institute of Mathematical Sciences of latest York collage for his or her stimulation and feedback that have contributed to the writing of this tract. the writer additionally needs to thank Aughtum S. Howard for permission to incorporate effects from her unpublished dissertation, Larkin Joyner for drawing the figures, Interscience Publishers for his or her cooperation and help, and especially Lipman Bers, who recommended the booklet in its current shape.

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The result is that there are altogether p+i "diasymmetric" and w " orthosymmetric " cases. If we denote as a line of symmetry any line whose points 32 LECTURE IV. remain unchanged by the conformai transformation, the diasymmetric cases contain respectively p, p—i,-"2, I, o lines of symmetry, and the orthosymmetric cases contain p+i, p— I, p — 3>'~ such lines. A surface is called diasymmetric or orthosymmetric according as it does not or does break up into two parts by cuts carried along all the lines of symmetry.

By means of the theory of Riemann. The first problem that here presents itself is to establish the connection between a plane curve and a Riemann surface, as I have done in Vol. 7 of the Math. Annalen (1874)4 Let us consider a cubic curve; its deficiency is p=i. Now it is well known that in Riemann's theory this deficiency is a measure of the connectivity of the corresponding Riemann surface, which, therefore, in the present case, must be that of a tore, or anchor-ring. The question then arises : what has the anchor-ring to do with the cubic curve ?

All this suggests the question whether it would not be possible to create a, let us say, abridged system of mathematics adapted to the needs of the applied sciences, without passing through the whole realm of abstract mathematics. Such a system would have to include, for example, the researches of Gauss on the accuracy of astronomical calculations, or the more recent and highly interesting investigations of Tchebycheff on interpolation. The problem, while perhaps not impossible, seems difficult of solution, mainly on account of the somewhat vague and indefinite character of the questions arising.