In our schools we are taught a magnificent series of geometrical theorems, all solidly interconnected, the principal of which were created by the great Greek genius, Euclid. That is why classical geometry is known as Euclidean geometry. Its theorems are based upon a certain number of axioms and postulates, though these are really only affirmations or definitions. The most important of these definitions is: “A straight line is the shortest distance between two points.” That seems to schoolboys quite simple, because they know that the youth who amuses himself by running in a zigzag on the racing track will be the last to reach the tape; and at the sports ground one is not in a mood or has not time to bother about the validity of the axioms of geometry. What is the precise meaning of this definition of a straight line? There has been a great deal of discussion of that point. Henri Poincaré has written a number of fine and profound pages on it, yet his conclusions are not entirely without an element of uncertainty. In practice we all know what we mean by a straight line: it is the line that we make by means of a good ruler. But how do we know that a ruler is good and correct? By holding it up before the eye, and seeing that both ends of it and all the intermediate points in its edge merge together when we look along it. That is how a carpenter tells if a board is smoothly planed. In a word, in practice we mean by a straight line the line which is taken by the eye of the rifleman looking along his sights. All this amounts to saying that a straight line is the direction in which a ray of light travels. However we look at the matter, we always come back to the same point—to say that the edge of an object is straight means that the delimiting line coincides in its whole length with a ray of light.[10] We may therefore say that practically a straight line is the path followed by light in a homogeneous medium. [10] It goes without saying that in all this we assume that the luminous ray travels in a homogeneous medium. And that gives rise to a question. Is the world in which we live, the universe, in conformity with Euclid’s geometry? Is it Euclidean? It must be understood that Euclid’s geometry is not the only one that has been created. In the nineteenth century there were bold and profound mathematicians—Riemann, Bolyay, Lobatchewski, even Poincaré—who founded new and different and rather strange geometries. They are just as logical and coherent as the classical geometry of Euclid, but they are based upon different axioms and postulates—in a word, different definitions.