=33.= Since these systems are all obtained from a Euclidean plane, by a mere alteration in the definition of distance, Cayley and Klein tend to regard the whole question as one, not of the nature of space, but of the definition of distance. Since this definition, on their view, is perfectly arbitrary, the philosophical problem vanishes--Euclidean _space_ is left in undisputed possession, and the only problem remaining is one of convention and mathematical convenience[42]. This view has been forcibly expressed by Poincaré: "What ought one to think," he says, "of this question: Is the Euclidean Geometry true? The question is nonsense." Geometrical axioms, according to him, are mere conventions: they are "definitions in disguise[43]." Thus Klein blames Beltrami for regarding his auxiliary plane as merely auxiliary, and remarks that, if he had known Cayley's Memoir, he would have seen the relation between the plane and the pseudosphere to be far more intimate than he supposed[44]. A view which removes the problem entirely from the arena of philosophy demands, plainly, a full discussion. To this discussion we will now proceed. =34.= The view in question has arisen, it would seem, from a natural confusion as to the nature of the coordinates employed. Those who hold the view have not adequately realised, I believe, that their coordinates are not _spatial_ quantities, as in metrical Geometry, but mere conventional signs, by which different points can be distinctly designated. There is no reason, therefore, until we already have metrical Geometry, for regarding one function of the coordinates as a better expression of distance than another, so long as the fundamental addition-equation[45] is preserved. Hence, if our coordinates are regarded as adequate for all Geometry, an indeterminateness arises in the expression of distance, which can only be avoided by a convention. But projective coordinates--so our argument will contend--though perfectly adequate for all projective properties, and entirely free from any metrical presupposition, are inadequate to express metrical properties, just because they have no metrical presupposition. Thus where metrical properties are in question, Beltrami remains justified as against Klein; the reduction of metrical to projective properties is only apparent, though the independence of these last, as against metrical Geometry, is perfectly real.