MEASUREMENT. We propose in the first place to enter into some detail on the fundamental principles of the theory of measurement, and in doing so it will be necessary to sketch the very remarkable theory established by Riemann and other mathematicians as to the foundations of our geometrical knowledge.
Every system of geometrical measurement, as indeed the whole science of geometry itself, is founded on the possibility of transferring a fixed figure from one part of space to another with unchanged form. We are so familiar with this process that we are apt not to realize its importance until very special attention has been directed to the subject. We therefore propose to make a logical examination of the nature of the assumptions involved in the possibility of moving a figure in space so that it shall undergo no alteration. We shall find that we require to postulate certain suppositions with regard to the nature of space and to the measurement of distances.
It will facilitate the conception of this somewhat difficult subject to consider the case of hypothetical reasoning beings which Sylvester described as being infinitely attenuated bookworms confined to infinitely thin sheets of paper. We suppose such two-dimensional beings to be absolutely limited to a certain surface. They could have no conception of space except as of two dimensions. The movement of a point would for them form a line, the movement of a line would form a surface. They could conduct their measurements and form their geometrical theories. They would be able to draw the shortest lines between two points, these lines being what we would call geodesies. To these two-dimensioned geometers geodesies would possess many of the attributes of straight lines in ordinary space. If the surface to which the beings were confined were actually a plane, then the geometry would be the same as our geometry. They would find that only one straight line could be drawn between two points, that through a point only one parallel to a given line could be drawn, and that the ends of a line would never meet even though the line be prolonged to infinity.
We might also suppose that intelligent beings could exist on the surface of a sphere. Their straightest line between two points would be the arc of the great circle joining those two points. They would also find that a second geodesic could be drawn joining the two points, this being of course the remaining part of the great circle. A curious exception would, however, be presented by two points diametrically opposite. An infinite number of geodesies can be drawn between these points and all those geodesies are of equal length. The axiom that there is one shortest line between two given points would thus not hold without exception. There would be no parallel lines known to the dwellers on the sphere. It would be found by them that every two geodesies must intersect, not only in one, but even in two points. The sum of the three angles of a triangle would for them not be constant. It would always be greater than two right angles, and would increase with the area of the triangle. They would thus have no conception of similarity between two geometrical figures of different sizes. If two triangles be constructed which have their sides proportional, the angles of the larger triangle would be greater than the corresponding angles of the smaller triangle.
It is thus plain that the geometrical axioms of the sphere-dwellers must be very different from those of the plane-dwellers. The different axioms depend upon the different kinds of space which they respectively inhabit, while their logical powers are identical. In one sense, however, the dwellers on the sphere and on the plane have an axiom in common. In each case it will be possible for a figure to be moved about without alteration of its dimensions. A spherical triangle can be moved on the surface of a sphere without distortion just as a plane triangle may be moved in a plane. The sphere-dwellers and the plane-dwellers would be equally able to apply the test of congruence. It is, however, possible to suppose reasoning beings confined to a space in which the translation of a rigid figure is impossible. Take, for instance, the surface of an ellipsoid or even a spheroid such as the surface of the earth itself. A triangle drawn on the earth at the equator could not be transferred to the surface of the earth near the pole and still preserve all its sides and all its angles intact.
If a surface admits of a figure being moved about thereon so as still to retain all its sides and all its angles unaltered, then that surface must possess certain special properties. It can be shown that, if a surface is to possess this property, a certain function known as the “measure of curvature” is to be constant. The measure of curvature is the reciprocal of the product of the greatest and least radii of curvature. We do not now enter into the proof, but it is sufficiently obvious that a sphere of which the radius is the geometric mean between the greatest and least radii of curvature at each point will to a large extent osculate the surface, so that a portion of the surface in the neighbourhood of the point will, generally speaking, have the same curvature as the sphere. If the sphere thus determined be the same at all the different points of the surface, then the curvature of the different parts of the surface will on the whole resemble that of the sphere, and therefore we cannot be surprised that the surface possessing this property will admit the displacement of a rigid figure thereon without derangement of its form.
We are thus conducted to a kind of surface the geometry of which is similar to that of the plane, but in which the axiom of parallels does not hold good. In this surface the radii of curvature at every point have opposite signs, so that the measure of curvature which is zero for the plane and positive for the sphere is negative for the surface now under consideration. This surface has been called the “pseudosphere,” and its nature has been investigated by Beltrami.[1] In the geometry of two dimensions we can thus have either a plane or a sphere or a pseudosphere which are characterized by the property that a surface may be moved about in all directions without any change either in the lengths of its lines or in the magnitudes of its angles. The axiom which assumes that there is only one geodesic connecting two points marks off the plane and the pseudosphere from the sphere. The axiom that only one parallel can be drawn through a given point to a given line marks off the plane from the pseudosphere. The geometry of Euclid is thus specially characterized among all conceivable geometries of two dimensions by the following three axioms—(1) the mobility of rigid figures, (2) the single geodesic between two points, (3) the existence of parallels.
in Clifford's Lectures and Essays, vol. i. p. 317. We shall follow to some extent the method employed by him in order to obtain an idea of the important conception which is called the “curvature of space.” Suppose a geodesic be drawn on a surface of constant curvature. Then a piece of the surface adjoining this geodesic can be slid along the curve so as all the time to fit in close contact therewith. If the piece of surface be turned to the other side of the geodesic it will still fit along this side. A
- ↑ Saggio di Interpretazione della Geometria non-Euclidea, Naples, 1868; “Teoria fondamentale degli sparii di curvatura constante,” Annali di Matematica, ser. ii. tom. ii. pp. 232–55. Both papers have been translated into French by J. Houël (Annales Scientifiques de l'École Normale, tom. v., 1869). An exceedingly interesting account of the whole subject will be found in Helmholtz, Popular Lectures on Scientific Subjects, translated by Atkinson, second series, London, 1881, pp. 27–71.
