From this difference in the geometrical premises important variations followed. Thus, whereas in Euclidean geometry the sum of the angles of any triangle is always equal to two right angles, in non-Euclidean geometry the value of this sum varies with the size of the triangles. It is always less than two right angles in Lobatchewski’s, and always greater in Riemann’s. Again, in Euclidean geometry, similar figures of various sizes can exist; in non-Euclidean geometry, this is impossible. It appeared, then, that the universal absoluteness of truth formerly credited to Euclidean geometry would have to be shared by these two other geometrical doctrines. But truth, when divested of its absoluteness, loses much of its significance, so this co-presence of conflicting universal truths brought the realisation that a geometry was true only in relation to our more or less arbitrary choice of a system of geometrical postulates. From a purely rational point of [Pg 36] view, there was no means of deciding which of the several consistent sets was true. The character of self-evidence which had been formerly credited to the Euclidean axioms was seen to be illusory. However, there are a number of rather delicate points to be considered, and these we shall now proceed to investigate. Euclid’s parallel postulate and the alternative non-Euclidean postulates reduce to indirect definitions of what we intend to call a straight line in the respective geometries mentioned. If there existed such a universal as absolute straightness, represented, let us say, by a Euclidean straight line, we might claim that Euclidean geometry constituted the true geometry, since its straight line conformed to the ideal of absolute straightness. But this existence of a universal representing absolute straightness is precisely one of the metaphysical cobwebs of which the discovery of non-Euclidean geometry has purged science. To illustrate this point more fully, let us assume that we think we know what is implied by a straight line. Whether we merely imagine a straight line or endeavour to realise one concretely, we are always faced with the same difficulty. For instance, we consider that a rod is straight when it can be turned over and superposed with itself, or else we place our eye at one of its extremities and note that no bumps are apparent. Again, we may realise straightness by stretching a string, viewing a plumb line or the course of a billiard ball. We may also execute measurements with our rigid rods; the straight line between any two points will then be defined by the shortest distance. But whatever method we adopt, it is apparent that our intuitive recognition of straightness in any given case will always be based on physical criteria dealing with the behaviour of light rays and material bodies. We may close our eyes and think of straightness in the abstract as much as we please, but ultimately we should always be imagining physical illustrations.