Quaternions and Elliptical Space: (Quaternions et Espace Elliptique)

This is generally a literal translation, terms such as versor and parataxis in use at the time are not updated to contemporary style, rather defined and briefly compared in an appendix. The decision to translate Lemaître’s 1948 essay arose not merely because of personal interest in Lemaître’s ‘persona’, Physicist - Priest, but from an ever increasing interest in Quaternions and more recent discovery of correspondence with Octonions in additional dimensional (XD) brane topological phase transitions (currently most active research arena in all physics), and to make this particular work available to readers interested in any posited historical value during the time quaternions were still considered a prize of some merit, which as well-known, were marginalized soon thereafter (beginning mid1880’s) by the occluding dominance of the rise of vector algebra; indeed, Lemaître himself states in his introduction: “Since elliptic space plays an increasingly important part …I have thought that an exposition … could present some utility even if the specialists … must bear the judgment that it contains nothing really new”, but also because the author feels quaternions (likely in conjunction with octonions), extended into elliptical and hyperbolic XD spaces, especially in terms of the ease they provide in simplifying the Dirac equation, will be essential facilitators in ushering in post-standard model physics of unified field mechanics. The translator comes to realizes that 3^rd regime Natural Science (Classical → Quantum → Unified Field Mechanics) will be described by a reformulated M-Theoretic topological field theory, details of which will be best described by Quaternion-Octonion correspondence.