Separate spheres

“ And hence it follows that the spheres, whether they are of uniform density, or consist of concentric layers, with densities varying according to any law whatever, will act upon each other in the same manner as if their force resided in their centres alone. ”

“ 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. ”

“ When this system is inverted back again into the form of the two spheres in contact, we have a corresponding series of negative images, the distances of which from the point of contact are of the form , where is positive for the sphere and negative for the sphere . The charge of each image, when the potential of the spheres is unity, is numerically equal to its distance from the point of contact, and is always negative. ”

“ Whether solid globes or hollow spheres, they equally owe their existence to the great first cause, that spoke matter into existence, that arranged it in whatever form and order infinite wisdom dictated ”

“ In investigating the interior of the earth we did not find a central shaft running from pole to pole, such as has been described by some mediums, nor did we find a number of concentric spheres resting upon cushions of steam. ”

“ 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. ”

“ That if the earth be not a solid globe, but a hollow concentric sphere, the quantity of matter being diminished, the attraction of gravitation must be lessened so much that all moveable bodies resting on the earth would be thrown off by centrifugal force, in the line of a tangent from the surface of the sphere. ”

“ The Sphere would willingly have continued his lessons by indoctrinating me in the conformation of all regular Solids, Cylinders, Cones, Pyramids, Pentahedrons, Hexahedrons, Dodecahedrons, and Spheres: but I ventured to interrupt him. Not that I was wearied of knowledge. On the contrary, I thirsted for yet deeper and fuller draughts than he was offering to me. ”

“ In two spheres mutually gravitating each towards the other, if the matter in places on all sides round about and equi-distant from the centres is similar, the weight of either sphere towards the other will be reciprocally as the square of the distance between their centres. ”

“ 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. ”

“ 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. ”

“ If spheres is the progress from the centre to the circumference be however dissimilar and unequable, but similar on every side round about at all given distances from the centre; and the attractive force of every point be as the distance of the attracted body ”

“ If the spheres were entirely removed from each other's influence, the state of stress would be uniform round each sphere, and the pulls on its surface-elements would balance, giving no resultant force on the sphere. ”

“ Later, these remarkable people, in trying to account for the fixed and orderly movement of the stars as they slowly passed from east to west, adduced that they must be fastened in crystal spheres which whirled, one within the other, over and under the motionless disk of the earth. ”

“ For spheres of different sizes, the greatest increase in mass varies as the cube of the radius; hence the ratio of this increase to the whole mass of the sphere is constant for all spheres of the same material ”

“ We imagine a wave-front divided into elementary rings or zones—often named after Huygens, but better after Fresnel—by spheres described round P (the point at which the aggregate effect is to be estimated) , the first sphere, touching the plane at O, with a radius equal to PO, and the succeeding spheres with radii increasing at each step by 1/2λ. There are thus marked out a series of circles, whose radii x are given by x2 + r 2 = (r + 1/2nλ) 2, or x2 = nλr nearly; so that the rings are at first of nearly equal area. Now the effect upon P of each element of the plane is proportional to its area ”

“ The Crystalline Spheres.—We have seen that all the heavenly bodies appear to be situated on the surface of the celestial sphere. The ancients assumed that the stars were really fixed on the surface of a crystalline sphere, and that they were carried around the earth daily by the rotation of this sphere. ”

“ I have proved in an independent manner, and different from that of Père Leray, that two equal, penetrable spheres of ponderable matter, existing in space at a distance apart which is large in proportion to their diameters, will be impelled towards each other by the impact of ethereal matters, according to a law which is approximately that of the inverse square of the distance. When the spheres are brought to within a much shorter distance of each other the law ceases to be approximately true. ”

“ In many parts of the unfathomable ocean there may be communications or passages from the surface of the sphere on the outer side to the surface of the inner, at least all except the great mid-plane space, through which liquid apertures, light and heat may be communicated, perhaps, to the interior surface of the sphere. ”

“ One continent having thus emerged, another would naturally be produced simultaneously on the opposite side of the sphere, as an equipoise to the first, to keep equal the earth's motion; until all the heavy substances should be thrown out and united in a compact sphere. ”

“ Hence also we find the densities of the Planets. For (by proposition 72. book I.) the weights of equal and similar bodies towards similar spheres, are, at the surfaces of those spheres, as the diameters of the spheres. ”

“ A sphere of such a radius would contain as many spheres of ten trillions of miles radius as one could put cubic inches in 850 cubic miles, or 216,000 trillions. This sphere could hold as many suns of one million miles diameter as there might be grass seeds that would cover a million of earth’s for one mile in depth, each earth containing on its surface 200 millions of square miles, and every cubic inch thereof representing 27 millions of grass seeds. When we think that each seed represents two millions of earths can we comprehend the greatness of the universe? ”

“ The laws of conservation may not be violated: the earth will respond.
How are the Spheres constructed?
The constitution of these two retro-acting spheres, and consequently of all the others of [Pg 26] the heavenly host, [3] at this point demands our attention. How are the spheres made up? How speaks the earth? The earth with which we are familiar—our sample—is formed of a slight crust, a core, to a greater or less extent and degree incandescent, and measuring 250,000 millions of cubic miles in dimensions, also an envelope which we call the atmosphere. ”

“ It follows from this that the extremely rapid movement by which all the spheres are revolved in twenty-four hours, belongs to the Primum Mobile, which we call the tenth sphere, and which carries with it all the nine lesser spheres; that the second or planetary movement, that is, from west to east, is communicated to the lesser spheres and belongs to the ninth sphere ”

“ Spherical space is not very easy to imagine. We have to think of the properties of the surface of a sphere—the two-dimensional case—and try to conceive something similar applied to three-dimensional space. Stationing ourselves at a point let us draw a series of spheres of successively greater radii. The surface of a sphere of radius should be proportional to ; but in spherical space the areas of the more distant spheres begin to fall below the proper proportion. There is not so much room out there as we expected to find. ”

“ For this purpose we will first give our attention once more to the geometry of two-dimensional spherical surfaces. In the adjoining figure let K be the spherical surface, touched at S by a plane, E, which, for facility of presentation, is shown in the drawing as a bounded surface. Let L be a disc on the spherical surface. Now let us imagine that at the point N of the spherical surface, diametrically opposite to S, there is a luminous point, throwing a shadow L′ of the disc L upon the plane E. Every point on the sphere has its shadow on the plane. ”

“ 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. ”

“ For in a right sphere the needle is parallel to the plane of the horizon; but when the cœlestial pole is vertically overhead, or when the pole of the earth is itself the place of the region, then the needle is perpendicular to the horizon. ”

“ And the sphere holds the same position among solids. For it alone is embraced by a single surface, while rectilinear solids have several. The sphere is among solids what the circle is among plane figures. Further, those who divide bodies into planes and generate them out of planes seem to bear witness to the truth of this. Alone among solids they leave the sphere undivided, as not possessing more than one surface: for the division into surfaces is not just dividing a whole by cutting it into its parts, but division of another fashion into parts different in form. ”

“ If a corpuscle placed without an homogeneous sphere is attracted by a force reciprocally proportional to the square of its distance from the centre, and the sphere consists of attractive particles; the force of every particle will decrease in a duplicate ratio of the distance from each particle. ”

“ If by any distribution of imaginary sources within the spherical surface we can effect this, the value of the corresponding potential outside the sphere is the true and only one. The potential inside the sphere must really be constant and equal to that at the surface. ”

“ For even a small sphere formed around the centre in that way would be hollow, and would be undone when the different concentric layers approached each other, under proportionate forces of attraction, and formed into one hollow sphere. ”