Curvature

“ Hence the taking away of any particle of matter from the top of a mountain tends to diminish the curvature of the horizontal surface under the mountain, where it is greatest; and the deposition of the same particle at the bottom of the sea, tends to increase the curvature of this superficies where it is least. The general tendency, therefore, being to increase the curvature where it is least, and to diminish it where it is greatest, must be to bring about an uniform curvature throughout, that is, a spherical figure. ”

“ When the plate is bent in any manner, the curvature produced at each section of the middle surface may be regarded as arising from the superposition of two cylindrical curvatures; and the flexural couples across normal sections through the lines of curvature, estimated per unit of length of those lines, are C (1/R1 + σ/R2) and C (1/R2 + σ/R1) , where R1 and R2 are the principal radii of curvature. ”

“ That is, if we suppose P with its tangent to move along the curve to P', at the point of inflection Q the curvature is zero, the rotation of the tangent is momentarily arrested, and as the direction of rotation changes, the center of curvature moves out indefinitely and the radius of curvature becomes infinite. ”

“ We next find that the three-dimensional space inside the fluid possesses a curvature at every point which is proportional to the density of the fluid at every point. Assuming this density to be the same throughout, we see that the curvature of space is constant, and we obtain a spherical space of three dimensions analogous to a cap on the two-dimensional surface of an ordinary sphere in three-dimensional Euclidean space. This curvature of space inside the sphere has nothing to do with the dimensions of our fluid sphere; the curvature is governed solely by the density of the fluid. ”

“ The point which we have now gained is one of very great importance. In our ordinary conceptions of space the geodesic surfaces are of course our ordinary planes, and the common curvature they possess is zero, but the condition that rigid bodies shall be capable of translation with unaltered features does not require that the curvatures shall be zero, it merely requires that the curvatures shall be constant. ”

“ The expression for the specific curvature, which is in general somewhat elaborate, assumes a very simple form when a system of geodetics and the system of their orthogonal trajectories are chosen for the parameter curves, the parameter u being made the length of the arc of the geodetic, measured from the curve, u=o selected as the standard. ”

“ The curvature inwards would make the pressure inside less, and the curvature outwards would make it more, and as they are equal they just balance, and there is no pressure at all. If we could in the same way examine the bubble with the waist, we should find that this was true not only at the waist but at every part of it. Any curved surface like this which at every point is equally curved opposite ways, is called a surface of no curvature, and so what seemed an absurdity is now explained. ”

“ We approached the problem from the point of view that the actual world with its ten vanishing coefficients of curvature (or its isotropic directed curvature) has a specialisation which requires explanation; we were then comparing it in our minds with a world suggested by the pure mathematician which has entirely arbitrary curvature. But the fact is that a world of arbitrary curvature is a sheer impossibility. If not the directed radius, then some other directed length derivable from the metric, is bound to be homogeneous and isotropic. ”

“ When this is the case, the line generally returns into vigorous curvature at some part of its course, otherwise it is apt to be weak, or slightly rigid; multitudes of other curves, not approaching the right line so nearly, remain less vigorously bent in the rest of their course; so that the quantity88 of curvature is the same in both, though differently distributed.
§ 12. The modes in which Nature produces variable curves on a large scale are very numerous, but may generally be resolved into the gradual increase or diminution of some given force. ”

“ Just in the same way straightness mingled with curvature, that is to say, the close approximation of part of any curve to a straight line, gives to such curve all its spring, power, and nobleness: and even perfect straightness, limiting curves, or opposing them, is often pleasurable: yet, in the abstract, straightness is always ugly, and curvature always beautiful. ”

“ The deflection, or curvature, is continuous and progressive, for it is caused by no external force but only by causes intrinsic in the system. And the whole system is symmetrical: the angles at which the successive shoots are given off being all equal, and the lengths of the shoots diminishing in constant ratio. The result is that the successive shoots, or successive increments of growth, are tangents to a curve, and this curve is a true logarithmic spiral. ”

“ In addition to those here illustrated, examples may be seen of forward or backward, regular or irregular curvature, single or double curvature of one leg, with an outward or inward, regular or irregular, single or double curvature of the other leg. ”

“ The tendency of the primary strata to remain straight in the horizontal direction, and to be bent in the vertical, is a phenomenon which points very directly to the causes from whence it has arisen. A surface of simple curvature, or a surface straight in one direction, is - 231 - what the application of forces to different points of a plane, which is flexible, though with a certain degree of rigidity, will naturally produce. ”

“ As the curvature is towards the stimulus, the phototropic curvature thus remains positive, even under continued stimulation.
The positive curvature is due to the differential action of unilateral stimulus on the proximal and distal sides. But when a strong light is made to act continuously on one side of an organ, the excitation becomes internally diffused, and the differential effect on the two sides is reduced in amount or vanishes altogether. ”

“ The left-hand side gives us the curvatures of space-time from point to point as measured in our mesh-system, and the right-hand side gives us the various aspects of matter, energy and momentum situated at the same points where the curvature is being calculated. ”

“ Here we have a perfect balance between variety and unity. The curvature is varied to an infinite degree, at no point is its curving at the same ratio as at any 139other point; perhaps the maximum amount of variety that can be got in a symmetrical figure, preserving, as it does, its almost perfect continuity, for it approaches the circle in the even flow of its curvature. ”

“ All curvature, however, is not equally agreeable; but 264 the examination of the laws which render one curve more beautiful than another, would, if carried out to any completeness, alone require a volume. ”

“ It is suggested that on the average, and in regions remote from matter, the universe is not quite flat, but has a slight curvature, analogous, in three dimensions, to the curvature of a sphere in two dimensions. ”

“ The special feature of our space, by which it is distin guished from spherical space on the one hand and pseudo- spherical space on the other hand, depends upon what Riemann calls the measure of curvature. If the sum of the three angles of a triangle is to be two right angles, and if the geometrical similarity of large figures and small figures is to be possible, then the measure of curvature must be zero. Now all measurements that can b.e made seem to confirm the axiom of parallels and seem to show the measure of curvature of our space to be indistinguish able from zero. ”

“ Nor, though the instances we have examined are of globular or cylindrical masses chiefly, is it to be thought that breadth can only be secured by such alone: many of the noblest forms are of subdued curvature, sometimes hardly visible; but curvature of some degree there must be, in order to secure any measure of grandeur in a small mass of light. One of the most marked distinctions between one artist and another, in the point of skill, will be found in their relative delicacy of perception of rounded surface ”

“ Nor can they be extended immediately to the case where the inquiry is not for potential surfaces but, say, for surfaces of least area, or surfaces of constant positive gaussian curvature, which are to pass through a prescribed twisted curve or to stretch over a given ring surface. ”

“ And perhaps the most mischievous is the expression, a curvature of space. Now of space as it is generally used, in its accepted significance, there can be no curvature. For space means a system of positions extending uniformly in the number of dimensions we choose to fix upon.
If we take the straight line as our space, we may call it 1 space; then the set of positions follow one on after the other without bending. If the line is bent it becomes a line, not a straight line. It should not be called 1 space, but a thing in 2 space. That is, it is a bent line in a plane. ”

“ The limit tended to is the sphere of closest contact with the curve at (x, y, z) ; its centre and radius are called the centre and radius of spherical curvature. It cuts the osculating plane in a circle, called the circle of absolute curvature; and the centre and radius of this circle are the centre and radius of absolute curvature. The centre of absolute curvature is the limiting position of the point where the principal normal at (x, y, z) is cut by the normal plane at a neighbouring point, as that point moves up to (x, y, z) . ”

“ We might, however, seek an avenue of escape by claiming that the curvature of space would account for the accelerated appearance of these motions, much as the apparent velocity of a body depends on its orientation with respect to our line of vision. To be sure, a curvature of space would entail a visual distortion of this sort, since the light rays emanating from a luminous source would now reach us along curved routes. ”

“ The absolute curvature at any point is measured by a single quantity called the radius of spherical curvature. But space-time is a four-dimensional manifold embedded in—well, as many dimensions as it can find new ways to twist about in. Actually a four-dimensional manifold is amazingly ingenious in discovering new kinds of contortion, and its invention is not exhausted until it has been provided with six extra dimensions, making ten dimensions in all. ”

“ He attributes the gravitation of massive bodies to a curvature of the world in the region surrounding them and so throws a flood of light on the whole problem; but he is not primarily concerned to explain how material bodies produce (or are associated with) this curvature of the world around them, nor how this curvature is made subject to a law. ”

“ It is the tendency of such folded ranges to form arcs of large curvature, which are drawn inwards, where the lateral pressure meets with most resistance, and expand outwards, where it is withstood in a lesser degree. ”

“ Since objects in equilibrium, such as rigid objects, electrons and atoms, all adjust themselves to the radius of curvature of the universe, it is only natural that results of greatest simplicity should be obtained when we select this radius or some definite fraction thereof as a gauge at every point. ”

“ We are not empty space so there is no limit to the kind of curvature entering into our composition; in fact our unusual sort of curvature is what distinguishes us from empty space. We are, so to speak, ridges in the four-dimensional world where it is gathered into a pucker. The pure mathematician in his unflattering language would describe us as “singularities”. These two non-empty ridges are joined by empty space, which must be free from those kinds of curvature described by the ten principal coefficients. ”

“ In the international exhibition of 1851 at Hyde Park was exhibited a small model ellipsoid made in Berlin, on which the lines of curvature were traced after a method invented by Roberts. The lines of curvature and asymptotic lines on the surface, at any point of which the sum of the principal curvatures is zero, were also discussed in Liouville's ‘Journal de Mathématiques,’ 1850. ”