Note 15 — Introductory material

It is important to remember that, throughout the period we have just reviewed, the purpose of hyperbolic Geometry is indirect: not the truth of the latter, but the logical independence of the axiom of parallels from the rest, is the guiding motive of the work. If, by denying the axiom of parallels while retaining the rest, we can obtain a system free from logical contradictions, it follows that the axiom of parallels cannot be implicitly contained in the others. If this be so, attempts to dispense with the axiom, like Legendre's, cannot be successful; Euclid must stand or fall with the suspected axiom. Of course, it remained possible that, by further development, latent contradictions might have been revealed in these systems. This possibility, however, was removed by the more direct and constructive work of the second period, to which we must now turn our attention.