However decisive appear to me the reasons already given, I believe I should emphasize this point because here a false idea is profoundly rooted in many minds. 2. If we construct a material circle, measure its radius and circumference, and see if the ratio of these two lengths is equal to [pi], what shall we have done? We shall have made an experiment on the properties of the matter with which we constructed this _round thing_, and of that of which the measure used was made. 3. GEOMETRY AND ASTRONOMY.--The question has also been put in another way. If Lobachevski's geometry is true, the parallax of a very distant star will be finite; if Riemann's is true, it will be negative. These are results which seem within the reach of experiment, and there have been hopes that astronomical observations might enable us to decide between the three geometries. But in astronomy 'straight line' means simply 'path of a ray of light.' If therefore negative parallaxes were found, or if it were demonstrated that all parallaxes are superior to a certain limit, two courses would be open to us; we might either renounce Euclidean geometry, or else modify the laws of optics and suppose that light does not travel rigorously in a straight line. It is needless to add that all the world would regard the latter solution as the more advantageous. The Euclidean geometry has, therefore, nothing to fear from fresh experiments. 4. Is the position tenable, that certain phenomena, possible in Euclidean space, would be impossible in non-Euclidean space, so that experience, in establishing these phenomena, would directly contradict the non-Euclidean hypothesis? For my part I think no such question can be put. To my mind it is precisely equivalent to the following, whose absurdity is patent to all eyes: are there lengths expressible in meters and centimeters, but which can not be measured in fathoms, feet and inches, so that experience, in ascertaining the existence of these lengths, would directly contradict the hypothesis that there are fathoms divided into six feet? Examine the question more closely. I suppose that the straight line possesses in Euclidean space any two properties which I shall call _A_ and _B_; that in non-Euclidean space it still possesses the property _A_, but no longer has the property _B_; finally I suppose that in both Euclidean and non-Euclidean space the straight line is the only line having the property _A_. If this were so, experience would be capable of deciding between the hypothesis of Euclid and that of Lobachevski. It would be ascertained that a definite concrete object, accessible to experiment, for example, a pencil of rays of light, possesses the property _A_; we should conclude that it is rectilinear, and then investigate whether or not it has the property _B_.