Academy,  1 (1869–70), 128–30.

[Review of On the Hypotheses Upon Which Geometry is Based, by Georg F B Riemann]

H Helmholtz



Publications reviewed:

Riemann 1867


Mathematics, Metaphysics, Popularization, Ancient Authorities, Hypothesis, Truth, Astronomy

People mentioned:

Euclid , Carl F Gauss , Eugenio Beltrami , Nikolai I Lobachevsky , Thomas Young , James C Maxwell

    Helmholtz claims that the conflicting axioms of geometry 'may be made generally interesting to all who have studied even the elements of mathematics', and attempts 'to give here the general drift and the results of these investigations, as far as it is possible to do so, without entering into mathematical calculations or using formulæ' (128). After examining the non-Euclidean forms of geometry put forward by Georg F B Riemann, Helmholtz concludes that 'the axioms on which our geometrical system is based, are no necessary truths, depending solely on irrefragible laws of our thinking. On the contrary, other systems of geometry may be developed with perfect logical consistency. Our axioms are, indeed, the scientific expression of a most general fact of experience, the fact, namely, that in our space bodies can move freely without altering their form. From this fact of experience it follows, that our space is a space of constant curvature, but the value of this curvature can be found only by actual measurements' (130).

See also:

Richards 1988, 78–85

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