On the relativity of magnitudes

Faced with the mathematical possibility of non-Euclidean geometries, 19th Century geometers were tasked
with the problem of determining which among the possible geometries corresponds to that of our space.
In this context, the contribution of the Belgian philosopher-mathematician, Joseph Delboeuf, has been unduly
neglected. The aim of this essay is to situate Delboeuf’s ideas within the context of the philosophies of geometry
of his contemporaries, such as Helmholtz, Russell and Poincaré. We elucidate the central thesis, according to
which Euclidean geometry is given special status on the basis of the relativity of magnitudes, we uncover its
hidden history and show that it is defensible within the context of the philosophies of geometry of the epoch.
Through this discussion, we also develop various ideas that have some relevance to present-day methods in
gravitational physics and cosmology.