Spherical shape

“ Given two spherical bodies of similar material but of unequal magnitude and originally possessing the same degree of heat, the smaller body will cool more rapidly than the larger, by reason of the greater proportion which the surface of the smaller sphere bears to its volume than that of the larger sphere to its volume—this proportion depending upon the geometrical ratio which the surfaces of spheres bear to their volumes, the contents of spheres being as the cubes and the surfaces as the squares of their diameters. ”

“ Here our spheres must be called "rigid" spheres. Their increase in size as they depart from S is not to be detected by measuring with measuring-rods, any more than in the case of the disc-shadows on E, because the standards of measurement behave in the same way as the spheres. Space is homogeneous, that is to say, the same spherical configurations are possible in the environment of all points. [1] Our space is finite, because, in consequence of the "growth" of the spheres, only a finite number of them can find room in space. ”

“ Let a sphere, of radius a and density w, be made of elastic material whose bulk and rigidity moduli are k and n, and let it be subjected to forces due to a potential wr*S 2 per unit volume, where S 2 is a surface spherical harmonic of the second order. Then it is required to find the strain of the sphere. We refer the reader to the original sources for the methods of solution applicable to spherical shells and to solid spheres. In order to write Thomson's solution we put r, X, I for radius vector, latitude, and longitude, and p, /*, v for the corresponding displacements. ”

“ If we say that the specific gravity of a body is 6, we mean, that its weight is six times as great as that of a portion of water, exactly equal to it in bulk.
Spectrum. That appearance of differently coloured rays, which is produced by the refraction of the solar ray, by means of a prism, is called the prismatic spectrum; it exhibits most distinctly, and beautifully, all the colours seen in the rainbow.
Sphere. A globe, or ball.
Spheroid. Spherical; a body approaching nearly to a sphere in its figure. The earth, is denominated an oblate spheroid ”

“ 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. ”

“ But, unless the two bubbles be of precisely equal size (and therefore of equal curvature) it is obvious that the tangents to the spheres will not meet the plane of their circle of contact at equal angles, and therefore that the partition-wall must be a curved surface: it is only plane when it divides two equal and symmetrical cells. It is also obvious, from the symmetry of the figure, that the centres of the spheres, the centre of the partition, and the centres of the two spherical surfaces are all on one and the same straight line. ”

“ It is well known that the centrifugal force arising from the earth’s rotation is sufficient to change the otherwise spherical form of the earth to an oblate spheroid, with a difference of twenty-six miles between the equatorial and polar diameters. ”

“ And although this spherical form possesses an intensity of power decreasing in proportion as the sphere expands, still it has everywhere the same property of giving existence to its centre, since it has everywhere an intrinsic spherical character essentially connected with a central point as its indispensable term. Whence we see that the substantial form, though virtually extending into an indefinite sphere, is everywhere terminated to its own matter. ”

“ 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. ”

“ If the movement, though over a solid surface, is not constant in direction, but takes place as often on one side as on another, radial symmetry may be again looked for; and if the motions are still more variously directed—if they are not limited to approximately-plane surfaces, but extend to surfaces that are distributed all around with a regular irregularity—an approach of the radial towards the spherical symmetry is to be anticipated. ”

“ The method assumes that gravity is measured on the earth’s surface at a sufficient number of points, and that it is conformably reduced. In order to secure the convergence of the expansions in spherical harmonics, it is necessary to assume all masses outside a surface parallel to the surface of the sea at a depth of 21 km. ”

“ And next, if the including or included fluid have a greater gravity one than another, then will the globular Form be deprest into an Elliptico-spherical: As if, for example, we suppose the Circle A B C D, in the fourth Figure, to represent a drop of water, Quick-silver, or the like, included with the Air or the like, which supposing there were no gravity at all in either of the fluids, or that the contained and containing were of the same weight, would be equally comprest into an exactly spherical body (the ambient fluid forcing equally against every side of it.) ”

“ Thus two hemispheres attract each other with precisely the same force as if they were spheres; and as the force is as the number of attracting points in operation directly, and as the squares of the respective distances inversely, it follows that the attraction between a mere ring and a circular area is no greater than that between two similar rings, and the force between a sphere and an opposed spherical segment of the same curvature is no greater than that of two similar segments, each equal to the given segment. ”

“ Thus the most perfect idea we can possibly acquire of a sphere, is by conceiving an infinite number of straight rays of equal lengths, issuing from the center, as from the eye, spreading every way alike; and circumscribed or wound about at their other extremities with close connected circular threads, or lines, forming a true spherical shell. ”

“ Thickness would be as inconceivable to them as a fourth dimension in space is to us. Transplant these beings to the surface of a sphere, and their planimetry would change to a spherical geometry. Defining a straight line as the shortest distance between two points, all their straight lines would be arcs of great circles, and every straight line, when sufficiently extended, would return to itself. ”

“ For if we stick the paper discs on the globe, so that no disc overlaps another, the surface of the globe will finally become so full that there is no room for another disc. This simply means that the spherical surface of the globe is finite in relation to the paper discs. Further, the spherical surface is a non-Euclidean continuum of two dimensions, that is to say, the laws of disposition for the rigid figures lying in it do not agree with those of the Euclidean plane. ”

“ The fact that two surfaces can be made to fit another in all azimuths proves that they are spherical and of equal curvatures, but one convex and the other concave, the case of perfect flatness not being excluded. ”

“ 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. ”

“ The solid angle subtended at a given point by a closed curve is measured by the area of a spherical surface whose centre is the given point and whose radius is unity, the outline of which is traced by the intersection of the radius vector with the sphere as it traces the closed curve. ”

“ If all the parts of a spherical fluid, homogeneous at equal distances from the centre, lying on a spherical concentric bottom, gravitate towards the centre of the whole, the bottom will sustain the weight of a cylinder, whose base is equal to the superficies of the bottom, and whose altitude is the same with that of the incumbent fluid. ”

“ However, somehow or other the conception of a spherical earth was gradually grasped, and the heavenly bodies were perceived all to revolve round it: some moving regularly, as the stars, all fixed together into one spherical shell or firmament ”

“ The earth is, in fact, flattened at the poles to the amount of about 27 miles in the length of its diameter, and by this amount, as well as by smaller variations due to mountains and valleys, the shape of the earth differs from a perfect sphere. These astronomical measurements of the curvature of the earth's surface furnish by far the most satisfactory proof that it is very approximately a sphere, and furnish as its equatorial diameter 7,926 miles. ”

“ From spherical symmetry, in which we have an infinite number of axes through each of which may pass an infinite number of planes severally dividing the aggregate into equal and similar parts; and from radial symmetry, in which we have a single axis through which may pass an infinite number of planes severally dividing the aggregate into equal and similar parts; we now turn to bilateral symmetry, in which the divisibility into equal and similar parts becomes much restricted. ”

“ This, like the sphere, is called a surface of positive curvature, in reference to the plane, which has no curvature.
Now, just as to the plane corresponds an uncurved or homaloidal space, so to a surface of positive curvature corresponds a space of positive curvature; and if the space in which we live can be proved to have the slightest positive curvature, it instantly follows that the universe is only finite in extent, and that every physical straight line, for example, every ray of light, if sufficiently produced, returns into itself. ”

“ The variation which has been observed in their number, shape, and dimensions, can in no way be better accounted for, than by concluding the planet to be constituted of a number of concentric spheres, of different breadths, revolving on different axes, and with different velocities, so as sometimes to present to our view the verge of one sphere, and sometimes that of another: and the rays of the sun falling on the parts of the verges presented to us, would occasion the diversified appearances which we discover. ”

“ The material crust of the earth rests on this placid sphere of vigor, excepting in a few places, where, as in the crevice we have entered, gaps, or crevices, in matter exist, beginning from near the outer surface and extending diagonally through the medial and inner spheres into the intra-earth space beyond. This medial sphere is a form of pure force, a disturbance of motion, and although without weight it induces, or conserves, gravity. It is invisible to mortal eyes, and is frictionless, but really is the bone of the earth. ”

“ As the true earth’s surface cannot be represented by any simple algebraic formula, it is customary to assume that it is a uniform sphere, neglecting at least partially, if not wholly, the spheroidal shape. We may first assume that there is no friction between the earth and the air, but must afterwards make allowance for its influence. ”

“ The spheres, in many parts of the unfathomable ocean, is believed to be water quite through from the concave or convex surfaces to the great mid-plane-space, and probably the earthy or solid matter of the sphere, may in many places extend quite through from one surface to the [94] other, tending, like ribs or braces, to support the sphere in its proper form. Such a formation of spheres appears to be supported by various facts and phenomena; amongst the most prominent of which are Volcanoes and Earthquakes. ”

“ Even if the spheres were more or less flattened about the poles of their axes, as is the case with all the planets which we have been able carefully to measure, as well as with the sun, provided the axes of rotation were precisely parallel to each other, the mutual attraction of the masses would cause no disturbance of the spheres. The same would be the case if the polar axis of one sphere stood precisely at right angles to that of the other. ”

“ The material portion of the earth is altogether a surface film, an insignificant skin over the sphere of purity, the center of gravitation. Although men naturally imagine that the density and stability of the earth is dependent on the earthy particles, of which his own body is a part, such is not the case. Earth, as man upon the outer surface, can now know it, is an aggregation of material particles, a shell resting on this globular sphere of medial force, which attracts solid matter from both the outer and inner surfaces of earth, forming thereby the middle of the three concentric spheres. ”