As a result of these investigations it was proved that there existed bodies which, as contrasted with Euclidean solids, would squirm when displaced. Yet, in spite of this fact, these non-Euclidean bodies would present all the mathematical requirements of congruent bodies. These alternative types of bodies may be called Riemannian bodies and Lobatchewskian bodies, respectively.
If we select Euclidean congruence for the purpose of measuring space, non-Euclidean bodies will appear to squirm as they move; if we select non-Euclidean congruence, it will be the Euclidean bodies which appear to vary in form when displaced. It is usual, however, to reserve the term rigid for the Euclidean bodies; but this is only in order to conform to the ordinary understanding of rigidity as derived from experience. If we omit to take into consideration the physical behaviour of material bodies which has forced a certain conception of rigidity upon us, there appears to be no mathematical reason for assigning greater importance to Euclidean measurements than to non-Euclidean ones. It follows that the non-Euclidean bodies are just as much entitled to the appellation “rigid” as are the better-known Euclidean ones. When it comes to deciding which of the types of rigidity represents conservation of absolute shape and size, the problem appears to be entirely meaningless. Accordingly we shall refer to the various types of bodies as Euclideanly or as non-Euclideanly rigid.
Now it is perfectly obvious that when we measure lengths and ratios of lengths with one type of standard rods or another, we shall obtain conflicting numerical ratios and values. Thus, if we fix one extremity of our standard rod and allow it to rotate in all directions, its free extremity will describe points on a spherical surface regardless of the type of rod or geometry with which we are dealing. In a similar way we might obtain a circumference. If, then, we abide by the Euclidean system of measurement, the value of the ratio of the lengths of circumference and diameter would remain ever the same however great the diameter; we should always obtain the same constant number, known as
—equal to 3.141592 ..., first calculated by Archimedes. On the other hand, with Riemannian or Lobatchewskian measurements, we should obtain a variable value for this ratio, always smaller than
in the case of Riemannian geometry, decreasing as the diameter increased; and always greater than
in the event of our having selected Lobatchewskian measurements. Similar discrepancies would attend the measurement of all other geometrical figures and angles.
And here there is a further point to be considered. Even when we have defined a particular type of measurement, be it Euclidean or non-Euclidean, we have by no means fixed the behaviour of our rods and bodies in any absolute sense; and an understanding of absolute rigidity still escapes us. We can only define Euclidean bodies, for example, as those whose laws of disposition yield Euclidean numerical results. The following illustration will show us that these bodies might all vary in absolute shape when displaced, yet still yield the same Euclidean results.