_Die Welt als Vorstellung und Wille; Werke (Frauenstädt) (Leipzig, 1877), Bd. 2, p. 55._ [11] Schopenhauer’s table contains a third column headed “of matter” which has here been omitted. =2004.= The clear possession of the Idea of Space is the first requisite for all geometrical reasoning; and this clearness of idea may be tested by examining whether the axioms offer themselves to the mind as evident.--WHEWELL, WILLIAM. _The Philosophy of the Inductive Sciences, Part 1, Bk. 2, chap. 4, sect. 4 (London, 1858)._ =2005.= Geometrical axioms are neither synthetic _a priori_ conclusions nor experimental facts. They are conventions: our choice, amongst all possible conventions, is guided by experimental facts; but it remains free, and is only limited by the necessity of avoiding all contradiction.... In other words, axioms of geometry are only definitions in disguise. That being so what ought one to think of this question: Is the Euclidean Geometry true? The question is nonsense. One might as well ask whether the metric system is true and the old measures false; whether Cartesian co-ordinates are true and polar co-ordinates false. --POINCARÉ, H. _Non-Euclidean Geometry; Nature, Vol 45 (1891-1892), p. 407._ =2006.= I do in no wise share this view [that the axioms are arbitrary propositions which we assume wholly at will, and that in like manner the fundamental conceptions are in the end only arbitrary symbols with which we operate] but consider it the death of all science: in my judgment the axioms of geometry are not arbitrary, but reasonable propositions which generally have the origin in space intuition and whose separate content and sequence is controlled by reasons of expediency.--KLEIN, F. _Elementarmathematik vom höheren Standpunkte aus (Leipzig, 1909), Bd. 2, p. 384._ =2007.= Euclid’s Postulate 5 [The Parallel Axiom]. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.--EUCLID. _The Thirteen Books of Euclid’s Elements [T. L. Heath] Vol. 1 (Cambridge, 1908), p. 202._ =2008.= It must be admitted that Euclid’s [Parallel] Axiom is unsatisfactory as the basis of a theory of parallel straight lines. It cannot be regarded as either simple or self-evident, and it therefore falls short of the essential characteristics of an axiom....--HALL, H. S. and STEVENS, F. H.