Let us now try to find out something about the inner half-mass of the earth, and the first thing we have got to bear in mind is, that where it comes in contact with it, its density must be the same as that of the outer half-mass at the same place, and continue to be so for a considerable distance, varying much the same as the other varies in receding from that place, and diminishing at the same rate as it diminishes. This being the case—and we cannot see how it can be otherwise—if we attempt to distribute the inner half-mass over the whole of the inner half-volume, and suppose that its density decreases from its contact with the outer half—where it was found to be 8·8 times that of water—to zero at the centre, in direct proportion to the distance; then, it is clear that at half the distance between that place and the centre, the density must be just 4·4 times that of water. Now, if we divide the outer moiety of the inner half-mass of the earth—that is, the distance between the diameters of 6284·5 miles and 3142·25 miles—into layers of 25 miles thick each, take their volumes from [Table IV]., and multiply each of them by a corresponding density, decreasing from 8·8 to 4·4, we shall obtain a mass far in excess of the whole mass corresponding to the inner half of the earth. This shows that a region of no density would not be at the centre but would begin at a distance very considerably removed from it. It is another notice to us that the earth must be a hollow sphere. But why should there be a zero point or place of no density? And what would a zero of no density be? It would represent something less than the density of the nebulous matter out of which the earth was formed; and all that we have contended for, as yet, is that there is a space at the centre where there is no greater density than that corresponding to the earth nebula; but we must now go farther.
If the earth is a hollow sphere, it must have an internal as well as an external surface. But how are we to find out what is the distance between these two surfaces? Let us, to begin, take a look at the hollow part of the sphere. From the time of Arago it began to be supposed that there is a continual deposit of cosmic matter upon the earth going on, and since then it has been proved that there is a constant and enormous shower of meteors and meteorites falling upon it. But although this is the case on the exterior surface, it may be safely asserted that on the interior surface, where the supply of cosmic matter must have been limited from the beginning, there can be no continual deposit of such matter going on now; nor can there have been from, at least, the time when the earth changed from the form of vapour to a liquid state. We may, therefore, be sure that there is no undeposited cosmic matter of any kind in the hollow of the sphere, and that, as far as it is concerned, there is an absolute vacuum.
[TABLE V].— Calculations of the Volumes and Densities of the Outer Half of the Earth—taken as a Hollow Sphere—at the Diameters specified, and reduced to the Density of Water.
With mean diameter of 7918 miles. Diameter of half-volume at 6284·5 miles, and density there of 8·8 times that of water.
As to how far the internal surface is from the centre, it may be possible to designate a position, or region, from which it cannot be very far distant, although we can never expect to be able to point out exactly where it is. Going back to the time when the whole earth was in a molten liquid state, and just before the outer surface began to become solid, it is certain that the interior surface must have been in the same liquid condition, whatever may have been the condition of the mass of matter between the two surfaces, owing to the pressure of superincumbent matter; nay, we may be sure that whatever may be its state now, it continued liquid long after the other became solid, because it had no outlet by which to get rid of its melting heat by radiation, nor weight of superincumbent matter to consolidate it; and it would always be much hotter than the outer surface. At that time we have every reason to believe that the outer surface was at least as dense as it is now, there being no water upon it to lower its average density, as is the case at the present day; and we have equal reason to consider that the density at the inner surface, whether liquid or solid, is now at least equal to what the outer surface was then. Duly considering, therefore, the absence of water from the interior surface, we shall suppose that the first layer of 25 miles thick upon it will have an average density of 2½ times that of water, terminating at 3 times, which is the density we have taken for the outer surface at 9 miles deep. But there is another contingency, which it will be necessary to take into consideration before going any farther.
It has been understood—as it is certainly the truth—in the calculations made with respect to the outer half of the mass of the earth, that the increase of density in descending was due to the pressure of the superincumbent matter, caused by the attraction for it of the inner half, as well as that of the whole of both the outer and inner halves on the other side of the hollow interior. In the case of the inner half we have now to consider that the attraction of the outer half alone would be the effective agent, and that the superincumbent pressure—that is, of course, the pressure acting from the centre outwards—would be interfered with, or perturbed, by the attraction of the mass on the other side of the hollow interior, so that it would not exert its full power in that direction. But that does not mean that the density would be in any way diminished. The attractions of the planets for each other perturb them in their revolutions around the sun, accelerating or retarding each other, but do not increase or diminish their density or mass; only it will lead us to expect that the same depth of 817 miles will not produce the same amount of pressure outwards at the meeting of the two halves as it does inwards, and that to obtain an equal pressure a greater depth will be required. We believe that an expert mathematician, taking as bases two opposite pyramids in a sphere, similar to those we have used in a former part of our work, could point out, with very approximate accuracy, what ought to be the distance of the inner surface of the shell from the centre—provided a maximum density were determined for the earth—but that goes beyond our powers, and we shall limit ourselves to the use of our own implements; which will cause us to depart from the statement we have made, that the density of the inner half must decrease from the place of meeting of the two halves, at the same rate as the outer half had increased. It must decrease much more rapidly than the other increased. All this premised, and having established a density of 3 for the interior surface, we may proceed to calculate where that surface ought to be, so as to give for the interior half of the earth a mass equal to 735,584,493,738 cubic miles of water.
If we begin our operations with a density of 8·8 times that of water at the meeting of the two halves of the shell, and diminish it for any considerable distance at the same rate as it increased when we were finding the mass of the outer half, that is 0·1812 for each layer, we soon find that before we could make up the whole mass of the inner half of the shell, the density would be decreased to at least that of water, which cannot be, as there can be no liquid or solid matter of any kind of so low density anywhere in the interior half of the shell. Furthermore, if we decrease it at the same rate as the volumes of the different layers of the earth decrease as they approach the centre, it involves a mass of calculation that serves no useful purpose, as such calculations bring no contingent of satisfaction with them; because all the densities with which we are dealing have to be brought to a rational form before we can frame a proper approximate idea of what the interior construction of the earth is, as will be seen hereafter; and because it takes no account of the perturbation—above alluded to—produced by the attraction of the matter on the opposite side of the hollow. But, in order to get such a result as we can with our limited powers, if we begin with a density of 8·8 at the diameter of 6284·5 miles and fix the density of 3—which we have adopted above—at the diameter of 3200 miles, we shall get a mass somewhat less than one-half of the earth; and with a density of 2·91 at 3150 miles diameter we get a mass of 735,713,884,116 cubic miles of water, which is rather greater than one-half of the mass required (see operations of [Table V].). This density of 2·91 reduced to 2·5, as we mentioned, might be done when we were fixing the number 3, would make very little difference on the resulting mass, compared with what we have been in quest of.
Here we may state that we found that, had the calculations been made with documents of density proportioned to the decrease of the volumes of the layers of the earth as they approached the centre, the density would have been reduced to 2·25 at 3150 miles in diameter; which tends to show that should that process be considered to be more accurate, it would not have made any great difference on the result.
With all, we may consider that it has been demonstrated, that the greatest density of the earth is not necessarily greater at any part of its interior than 8·8 times that of water.