For instance, if we received a visual impression corresponding to a circle, and if this impression were followed by one corresponding to a triangle, and if it were impossible for us to re-establish the circular impression, we should have to assume that the body had [Pg 34] changed in shape; whereas, if it were possible to re-establish the circular impression by exerting certain efforts (which would finally be interpreted as a displacement of our point of observation), we should end by assuming that we had witnessed a partial rotation of a rigid cone-shaped object. This discovery of rigid objects in nature is of fundamental importance. Without it, the concept of measurement would probably never have arisen and metrical geometry would have been impossible. But with the discovery of objects which were recognised as rigid, hence as maintaining the same size and shape wherever displaced, it was only natural to appeal to them as standards of spatial measurement. Measurements conducted in this way would soon have proved that between any two points a certain species of line called the straight line would yield the shortest distance; and this in turn would have suggested the use of straight measuring rods. Henceforth, two straight rods would be considered equal or congruent if, when brought together, their extremities coincided. As for a physical definition of straightness, it could have been arrived at in a number of ways, either by stretching a rope between two points or by appealing to the properties of these rigid bodies themselves. For instance, two rods would be recognised as straight if, after coinciding when placed lengthwise, they continued to coincide when one rod was turned over on itself. Finally, parallelograms would be constructed by forming a quadrilateral with four equal rods, and parallelism would thus have been defined. Equipped in this way, the first geometricians (those who built the Pyramids, for instance) were able to execute measurements on the earth’s surface and later to study the geometry of solids, or space-geometry. Thanks to their crude measurements, they were in all probability led to establish in an approximate empirical way a number of propositions whose correctness it was reserved for the Greek geometers to demonstrate with mathematical accuracy. Thus there is not the slightest doubt that geometry in its origin was essentially an empirical and physical science, since it reduced to a study of the possible dispositions of objects (recognised as rigid) with respect to one another and to parts of the earth. In fact, the very word geometry proves this point conclusively.