"A scruple... has troubled conscientious writers. We take Euclidean space as we know it, we take Cartesian geometry in that space, we set up certain point functions in that space and call them distances, certain transformations and call them motions, and find at last a set of objects which obey the presuppositions of non-Euclidean geometry. But is there not here, perhaps, a vicious circle around which the kitten is chasing its tail? The basis is a Euclidean space, and a Cartesian coordinate system in that space, which is based upon Euclidean measurements, and cross ratios which depend upon distances. How do we know that without all of these it would be possible to erect a consistent non-Euclidean geometry? ... We begin by setting up a system of axioms for a projective geometry in a space of as many dimensions as we please. The undefined elements are point, line as a system of points, and separation of pairs of collinear points. Other choices are possible... The idea of taking separation as fundamental was introduced by Vailati."