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 [Pg 43] 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. All we have to do is to consider a transparent plane and a man executing geometrical constructions with Euclidean rods on its surface. If we consider a point-source of light casting the shadows of these rods on some other surface, whether plane or curved, these shadows, considered as measuring rods in turn, will still yield exactly the same Euclidean results, their laws of disposition will remain unchanged, and hence they will be rigid Euclidean rods exactly to the same extent as the original ones. Yet as contrasted with these original rods the shadows would squirm when displaced varying in shape and size, and the Euclidean straight lines would appear curved.