HERMANN VON HELMHOLTZ 



four dimensions, because we have no organs by which 

 such a conception can be formed ; but it does not 

 follow that there may not exist space of four dimen- 

 sions, or, indeed, of any number of dimensions. If 

 anything we saw slipped into the fourth dimensional 

 space it would vanish from our eyes, and we 

 would be quite unable either to follow it or to 

 imagine whither it had gone. The point, however, 

 is, that intelligent creatures living in a two-dimen- 

 sional or in a one-dimensional space could develop 

 a geometry of their own, in which the axioms would 

 be like our own Euclidean axioms. Further, such 

 intelligent beings living on a spherical or a pseudo- 

 spherical surface, in which Euclidean axioms do not 

 hold good, might develop a spherical or a pseudo- 

 spherical geometry. Such non-Euclidean geometries 

 have actually been worked out in some detail by 

 Lobatschewsky of Kasan, and Beltrami of Bologna. 

 Helmholtz, from these mathematical speculations, 

 concludes : (i) that Kant's assumption of the a priori 

 nature of the axioms is not proved ; (2) that it is 

 unnecessary ; and (3) that it is useless for the explana- 

 tion of the real world. Still he takes care not to 

 deny that space may be transcendental, even although 

 the axioms of geometry may not be so, but are 

 ultimately founded on our experience of the state of 

 things in the space in which we live. The axioms 

 then are empirical and are derived, like other laws of 

 nature, from observation and induction. Helmholtz, 

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