Note on the Division of Space into Infinitesimal Cubes

Extract

The proposition that “the only series of surfaces which, together, divide space into cubes are planes and their electric images” presented itself to me twenty years ago, in the course of a quaternion investigation of a class of Orthogonal Isothermal Surfaces (Trans. R.S.E., Jan. 1872). I gave a second version of my investigation in vol. ix., p. 527, of our Proceedings. Prof. Cayley has since referred me to Note vi., appended by Liouville to his edition of Monge's Application de l'Analyse á, la Géométrie (1850), in which the proposition occurs, probably for the first time. The proof which is there given is very circuitous; occupying some eight quarto pages of small type, although the reader is referred to a Memoir by Lamé for the justification of some of the steps. But Liouville concludes by saying :—“l'analyse précédente qui établit ce fait important 'est pas indigne, ce me semble, de l'attention des géométres.” He had previously stated that he had obtained the result “en profitant d'une sorte de hasard.” As Liouville attached so much importance to the theorem, and specially to his proof of it, it may not be uninteresting if I give other modes of investigation.