And now let us return to the problem of measurement. We have mentioned that in certain cases it would be possible, by changing our post of observation, to counteract the apparent modifications in the visual shapes of bodies that had suffered a displacement in our field of vision.
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 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.
Now, an empirical science is necessarily approximate, and geometry as we know it to-day is an exact science. It professes to teach us that the sum of the three angles of a Euclidean triangle is equal to 180°, and not a fraction more or a fraction less. Obviously no empirical determination could ever lay claim to such absolute certitude. Accordingly, geometry had to be subjected to a profound transformation, and this was accomplished by the Greek mathematicians Thales, Democritus, Pythagoras, and finally Euclid.
The difficulty that Euclid had to face was to succeed in defining exactly what he meant by a straight line and by the equality of two distances in space. So long as geometry was in its empirical stage these definitions were easy enough. All that men had to say was, “Two solid rods will be recognised as straight if after turning one of them over they still remain in perfect contact,” or again, “The distance between the two extremities of a material rod remains the same by definition wherever we may transport the rod.”
Euclid, however, could not appeal to such approximate empirical definitions; for perfect rigour was his goal. Accordingly he was compelled to resort to indirect methods. By positing a system of axioms and postulates, he endeavoured to state in an accurate way properties which were presented only in an approximate way by the solids of nature. Euclid’s geometry was thus the geometry of perfectly rigid bodies, which, though idealised copies of the bodies commonly regarded as rigid in the world of experience, were yet defined in such a manner as to be untainted by the inaccuracies attendant on all physical measurements.
But this empirical origin of Euclid’s geometrical axioms and postulates was lost sight of, indeed was never even realised. As a result Euclidean geometry was thought to derive its validity from certain self-evident universal truths; it appeared as the only type of consistent geometry of which the mind could conceive. Gauss had certain misgivings on the matter, but did not have the courage to publish his results owing to his fear of the “outcry of the Bœotians.” At any rate, the honour of discovering non-Euclidean geometry fell to Lobatchewski and Bolyai.
To make a long story short, it was found that by varying one of Euclid’s fundamental assumptions, known as the Parallel Postulate, it was possible to construct two other geometrical doctrines, perfectly consistent in every respect, though differing widely from Euclidean geometry. These are known as the non-Euclidean geometries of Lobatchewski and of Riemann.