Geometries andSpace 205plJuly lJanuarybgaFig. 10The parallax of a star.pis the width of the Earth’s orbit, and the linesl,l′are perpendicular to the diameter. In January (July)
the line to the star makes an angleα(β)with the line in question. The parallax is the angleα+β, which equalsγin Euclidean
geometry, and so is very small for large distance. In non-Euclidean geometry it can never be smaller thanπ/ 2 −
(p).- Lobachevsky–Bolyai or ‘hyperbolic’ geometry. This was not considered by Riemann, but when
his ideas came to be publicized, particularly by Helmholtz in the 1870s, it had become widely
known and could be seen as another candidate.
Helmholtz wrote a number of articles setting out his view that alternative models for space
should be considered (and tested). In particular, he wrote for the new English journalsNatureand
Mind; and an extract from one of his articles is included as Appendix C. By this time it had been
established that hyperbolic geometry was free from contradiction (the model argument). However,
this did not settle the question of whether it was worth considering, which hinged on whether
space could conceivably have such a geometry. Lobachevsky had already considered the question
of measurements to determine this, and Helmholtz clarified the point:All systems of practical mensuration that have been used for the angles of large rectilinear triangles, especially all
systems of astronomical measurement which make the parallax of the immeasurably distant fixed stars equal to zero
(in pseudospherical space the parallax even of infinitely distant points would be positive), confirm empirically the
axiom of parallels and show the measure of curvature of our space thus far to be indistinguishable from zero. It
remains, however, a question, as Riemann observed, whether the result might not be different if we could use other
than our limited base lines, the greatest of which is the major axis of the earth’s orbit. (Helmholtz 1979, p. 258)The ‘parallax’ of a starS(Fig. 10) is the angleα+βin the diagram, which in Euclidean geometry
equalsγ(and so is vanishingly small when the star’s distance is much bigger thanp, the diameter
of the Earth’s orbit). In non-Euclidean geometry, the smallest possible value ofα+βis (roughly)
π/ 2 −
(p). Interestingly, the fact that this is, for practical purposes, zero was used as an argument
against the Copernican theory; if the Earth moved, it was argued, the stars would have a measurable
parallax. By the nineteenth century it was accepted that the Earth did move, but the parallax was
too small to measure.^7
7. What revolution?
Let us not forget that no serious work toward constructing new axioms for Euclidean geometry had been done until
the discovery of non-Euclidean geometry shocked mathematicians into reexamining the foundations of the former.
We have the paradox of non-Euclidean geometry helping us to better understand Euclidean geometry! (Greenberg
1974, p. 57)- Sirius, the obvious candidate as it is both bright and close, has a parallax of 0.0377 seconds of arc.