The whole trouble arises from the continuity of space which precludes us from attributing any absolute meaning to the statement that two lengths situated in different parts of space are equal or unequal. There is no absolute significance in stating that there is as much space between and as between and . We cannot compare lengths by counting the number of points that they contain, since there are just as many points between the extremities of an inch as between the extremities of a mile—an infinite number in either case.[7] In short, we see that in mathematical continua, just as in sensory ones, measurement and comparison of lengths can be considered only as a result of some convention; there is nothing in the continua themselves to suggest any definite metrics or geometry. This is what is meant by saying that space is amorphous and presents us with no means of determining absolute shape and size. [Pg 41] Now, it might appear from the preceding discussions that so far as the mathematician is concerned, since a definition of equal distances is purely conventional, any conceptual rod might be chosen for this purpose. We might take a rod which, when moved about, would squirm like a worm, and elevate it to the position of a standard rod to which all other lengths would have to be compared. But this would be too extreme; for although mathematical space possesses no inherent metrics, yet certain requirements are demanded of rods susceptible of being considered as remaining congruent when displaced. It is these requirements, as postulated by Riemann, which we shall now proceed to discuss. Riemann assumed that the necessary requirements would be as follows: In the first place, he assumed that if we restricted ourselves to infinitesimal volumes of space, congruence would be established by means of Euclidean solids and measuring rods. This postulate is called the postulate of Euclideanism in the infinitesimal. Furthermore, if our rods remained congruent and we made certain constructions with them (pyramids, etc.), we should be able to transfer these rods and make exactly the same constructions, presenting exactly the same numerical relationships, in any other part of space. This postulate is a mere reflection of our belief in the homogeneity and isotropy of measurements in space, the same everywhere and the same in any direction. When presented in a slightly different form, it is often referred to as the postulate of free mobility. [Pg 42] We may also say that if two rods coincide in a certain region of space, they should continue to coincide when separated, then brought together again in any other region.[8] With these restrictions imposed, Riemann discovered that the Euclidean type of measurement was only one among others. Two other types, namely, the Lobatchewskian and the Riemannian, were also possible.