The Interior of the Earth and its Density—continued.

When, according to the nebular hypothesis, the ring for the formation of the earth and moon had been thrown off by the nebula, and had broken up and formed itself into one isolated mass—rotating or not on an axis, as the case may have been—it must have been in a gasiform state. What was its density, more or less, may be so far deduced from Table III., where it will be seen that when it had condensed to about one-half of its volume, it must have had a density of only 1/9000th part of our atmosphere, and in which each grain of matter would have for its habitat 16 cubic feet of space, or a cube of 2·52 feet to the side. So that, with an average distance from its neighbours of 2½ feet, a grain of matter could not be looked upon as wedged-in in any way, and would be free to move anywhere. Now, supposing this earth-moon nebula to have been in the form of even an almost shapeless mass, and that it was nearly homogeneous—as it could hardly be otherwise after the tumbling about it had in condensing from a flat ring—its molecules would attract each other in all directions, and as the mass—without having arrived perhaps at the stage of having any well defined centre—would have an exterior as well as an interior, the individual molecules at the exterior would draw those of the interior out towards them, just as much as those at the interior would attract those of the exterior in towards them; but as the number of those at the exterior would—owing to the much greater space there, being able to contain an immensely greater number—be almost infinitely greater than of those nearer to the central part, the latter would be more effectually attracted, or drawn, outwards than the former would be inwards, and there would be none left at the interior after condensation had fairly begun. The mass would speedily become a hollow body, the hollow part gradually increasing in diameter. But let us go deeper into the matter.

Let us suppose that the whole mass had assumed nearly the form of a sphere. We have already shown that, although the general force of attraction would cause all the component particles of the sphere to mutually draw each other in towards the centre, yet the more powerful tendency of the particles at the exterior—due to their greatly superior number—would at first be to draw the particles near the centre outwards towards them, and that there would consequently be a void at the centre, for a time at least. Of course it is to be understood that each part of the exterior surface would draw out to it the particles on its own side of the centre, just in the same manner as the four masses we placed at the centre were shown to be drawn out by those at London, Calcutta, and their antipodes. Now we must try to find out what would be the ultimate result of this action; whether it would be to form a sphere solid to the centre, or whether the void at first established there would be permanent.

In order to show how the heat of the sun is maintained by the condensation and contraction of that luminary, Lord Kelvin—in his lecture delivered at the Royal Institution, on Friday, January 21, 1887—described an ideal churn which he supposed to be placed in a pit excavated in the body of the sun, with the dimension of one metre square at the surface, and tapering inwards to nothing at the centre. In imitation of him, we shall suppose a similar pit of the same dimensions to be dug in the spherical mass, out of which we have supposed the earth to have been formed; only we shall call it a pyramid instead of a pit. This we shall suppose to be filled with cosmic matter, and try to determine what form it would assume were it condensed into solid matter, in conformity with the law of attraction. The apex of our imaginary pyramid would, mathematically speaking, have no dimension at all, but we shall assume that it had space enough to contain one molecule of the cosmic matter of which the sphere was formed. This being so arranged, we have to imagine how many similar molecules would be contained in one layer at the base of the pyramid at the surface of the sphere, and we may be sure that when brought under the influence of attraction, the great multitude of them would have far more power to draw away the solitary molecule from the apex, than the single one there would have to draw the whole of those in the layer at the base in to the centre of the sphere. A molecule of the size of a cubic millimetre would be an enormously large one, nevertheless one of that size placed at the apex of the pyramid would give us one million for the first layer at the base, and shows us what chance there would be of the solitary one maintaining its place at the apex. At the distance of one-twentieth of the radius of the sphere from the centre, the dimension of the base of the pyramid would be one-twentieth of a square metre, and the proportion of preponderance of a layer of molecules there would be as 25 to 1, so that the molecule at the centre would be drawn out almost to touch those of that layer; at one-tenth of the radius from the centre, the preponderance of a layer over the solitary central molecule would be as 10,000 to 1; and so on progressively to 1,000,000 to 1, as we have already said.

Following up this fact, if we divide the pyramid into any number of frusta, the action of attraction will be the same in each of them; the molecules in the larger end of each will have more power to draw outwards those of the small end, than they will have to draw inwards those of the larger end; and then the condensed frusta will act upon each other in the same manner as the molecules did, the greater mass of those at the larger end, or base, drawing down, or out—whichever way it may seem best to express it—a greater number of the frusta at the smaller end of the pyramid, until, in the whole of it, a point would be reached where the number of molecules in the various frusta drawn down from the apex would be equal to those drawn up from the base, leaving a part of the pyramid void at each end, because we are dealing with attraction, not gravitation, and there would be no falling to the base or apex, but concurrence to the point, just hinted at, where the outwards and inwards attractions of the masses would balance each other. This point of meeting of the two equal portions of cosmic matter may be called the plane of attraction in the pyramid. The whole pyramid would thus be reduced to the frustum of a pyramid, whose height would be as much more than double the distance from the plane of attraction to its base, as would be required to make the upper part above the plane of attraction equal in volume, or rather in number of molecules, to the lower part. It would be impossible for us to explain how, in a pyramid such as the one we have before us, the action of attraction could condense, and at the same time cram, the whole of the molecules contained in it into the apex end.

We must not, however, forget that there are two sides to a sphere, as well as to a question, and that we must place on the opposite side to the one we are dealing with, another equal pyramid with apex at the centre and base at the surface, at a place diametrically opposite to the first one, and that the tendency of the whole of this new pyramid would be to draw the whole of the first one in towards the centre of the sphere. But in the second, the law of attraction would have the same action as in the first; the molecules of the matter contained in it near the base would far exceed, in attractive force, those near the apex, and would draw them outwards till the whole were concentrated in a frustum of a pyramid, exactly the same as the one in the first pyramid. And while the whole masses of matter in the two pyramids were attracting each other at an average distance, say, for simplicity's sake, of one-half the diameter of the sphere, the molecules in each of them would be attracting each other from an average distance of one-quarter the diameter of the sphere; their action would consequently be four times more active, and they would concentrate into the frusta as we have shown, before the two pyramids had time to draw each other in to the centre. There would be then two frusta of pyramids attracting each other towards the centre with an empty space between them. Here then we have two elements of a hollow sphere, one on each side of the centre, and if we suppose the whole sphere to have been composed of the requisite number of similar pyramids, set in pairs diametrically opposite to each other, we see that the whole mass of the matter out of which the earth was formed must have—by the mutual attractions of its molecules—formed itself into a hollow sphere.

All that has been said must apply equally well whether we consider the earth to have been in a gasiform state, or when by condensation and consequent increase of temperature it had been brought into a molten liquid condition. For up to that time it must have been a hollow sphere, and we must either consider it to be so still, or conceive that the opposite sides have continued to draw each other inwards till the hollow was closed up; in which case, the greatest density would not be at the centre, but at a distance therefrom corresponding to what has been called the plane of attraction of the pyramid. That the opposite sides have not yet met will be abundantly demonstrated by facts that will meet us, if we try to find out what is the greatest density of the earth at the region of greatest mass or attraction, wherever that may be.

Seeing that the foregoing reasoning forces us to look upon the earth as a hollow sphere, or shell, in which the whole of the matter composing it is divided into two equal parts, attracted outwards and inwards by each other to a common plane, or region of meeting, we shall divide its whole volume into two equal parts radially, that is, one comprising a half from the surface inwards, and the other a half from the centre outwards—that is to say, each one containing one-half of the whole volume of the earth. Referring now to our calculations, Table IV., we find that the actual half volume of the earth is comprised in very nearly 817 miles from the surface, where the diameter is 6284 miles, because the total volume at 7918 miles in diameter is 259,923,849,377 cubic miles. This being the case, we cannot avoid coming to the conclusion, after what has just been demonstrated by the pyramids that if one-half of the whole volume is comprehended in that distance from the surface, so also must be one-half of the mass.

But for further substantiation of this conclusion let us return to the table of calculations. There we find that from the surface to the depth of 817 miles—where the diameter would be 6284 miles—which comprehends one-half of the volume—the mass at the density of water is shown to be only 518,596,945,467 miles instead of 735,584,493,738 cubic miles, which is the half of the whole mass of the earth reduced to the density of water. That is, the outer half of the volume gives only 70·5 per cent. of half the mass, while the inner half of the volume gives not only one-half of the mass but 29·5 per cent. more; or, to put it more clearly, the mass of the inner half-volume is 1·84 times, nearly twice as great, as the mass of the outer half-volume. On the other hand, we have to notice that the line of division of the mass into two halves falls at 1163·25 miles from the surface, where the diameter is 5591·5 miles; so that on the outer half of the earth, measured by mass, 64·74 per cent. of the whole volume of the earth contains only one-half of the mass, whereas on the inner portion, measured in the same way, 35·26 per cent. of the same whole contains the other half. All these results must be looked upon as unsatisfactory, or we must believe that two volumes of cosmic matter which at one time were not far from equal, had been so acted upon by their mutual attractions that the one has come to be not far from double the mass of the other; that the vastly greater amount of cosmic matter at the outer part of a nebula has only one-half of the attractive force of the vastly inferior quantity at the centre. This we cannot believe if the original cosmic, or nebulous, matter was homogeneous; and if it was not homogeneous we have, in order to bring about such result, to conceive that the earth was built up, like any other mound of matter, under the direction of some superintendent who pointed out where the heavier and where the lighter matter was to be placed.

We shall now proceed to find out what would be the internal form, and greatest density of the earth, under the supposition that it is a hollow sphere divided into two equal volumes and masses—exterior and interior—meeting at 817 miles from the surface; but before entering upon this subject we have something to say about the notion of the earth being solid to the centre.

We are forced to believe that, according to the theory of a nucleus being formed at the centre as the first act, the matter collected there must have remained stationary ever since, because we cannot see what force there would be to uniform the nucleus just formed; gravitation, weight falling to a centre, would only tend to increase, condense, and wedge in the nucleus more thoroughly. Attraction, as we have shown, would not allow the matter to get to the centre at all. Convection currents, or currents of any kind, could not be established in matter that was being wedged in constantly. Moreover, when in a gasiform state, it would be colder than when condensed by gravitation to, or nearly to, a liquid or solid state, and heat would be produced in it in proportion to its condensation, that is, gradually increasing from the surface to the centre in the same manner as density, which, when the cooling stage came, would be conducted back to the surface to be radiated into space, but could not be carried—by convection currents—because the matter being heavier there than any placed above it, and being acted upon by gravitation all the time, would have no force tending to move it upwards; and above all, when solidification began at the surface, it is absurd to suppose that the first formed pieces of crust could sink down to the centre through matter more dense than themselves; unless it was that by solidification they were at once converted into matter of the specific gravity of 13·734. Even so the solid matter would not be very long in being made liquid again by meeting with matter not only hotter than itself, but constantly increasing in heat through continual condensation, which would act very effectively in preventing any convection current being formed to any appreciable depth, certainly never to any depth nearly approaching to the centre. If solidification began first at the centre—as some parties have thought might be the case—owing to the enormous pressure it would be subjected to there, before it began at the surface, then, without doubt, the central matter must have remained where it was placed at first, up to the present day. This would suit the sorting-out theory very well, as all the metals would find their way to the centre and there remain; but judged under a human point of view, it would be considered very bad engineering on the part of the Supreme Architect to bury all the most valuable part of His structure where they could never be availed of; or that He was not sufficiently fertile in resources to be able to construct His edifice in a way that did not involve the sacrifice of all the most precious materials in it. Man uses granite for foundations—following the good example He has actually given we believe, and are trying to show—and employs the metals in superstructures; but some people may also think that it was better to keep the root of all evil as far out of man's reach as possible. What a grand prospectus for a Joint Stock Company might be drawn up, on the basis of a sphere of a couple of thousand miles in diameter of the most precious metals, could only some inventive genius discover a way to get at them!

Returning to our pyramids. We know that the centre of gravity of a pyramid is at one-fourth of its height, or distance from the base, and if we lay one of 3959 miles long (the radius of the earth) over a fulcrum, so that 989¾ miles of its length be on one side of it and 2969¼ miles on the other, it will be in a state of equilibrium. This does not mean, however, that there are equal masses of matter on each side of the fulcrum, for we know that the mass of the base part must be considerably greater than that of the apex part, and that it must be counterbalanced by the greater leverage of the apex part, due to its greater distance from the point of support. This being so, in the case of a pyramid consisting of gasiform, liquid, or solid matter, the attractive power of the 989¾ miles of the base part would be greater than that of the 2969¼ miles of the apex part, and the plane of equal attraction of the two parts would be less than 990 miles from the base of the pyramid. This is virtually the same argument we have used before repeated, but it is placed in a simpler and more practical light, and shows that the plane of attraction in a pyramid will not be at its centre of gravity but nearer to its base, and that it must be at or near its centre of volume. Thus the plane of attraction in one of the pyramids we have been considering of 3959 miles in length, and consequently the radial distance of the region of maximum attraction of the earth, would not be at 990 miles from the base or surface, but at some lesser distance.

Now, if we take a pyramid, such as those we have been dealing with, whose base is 1 square and height 3959, its volume would be the square of the base multiplied by one-third of the height, that is 1² × 3959/3 = 1319·66, the half of which is 659·83. Again, if we take the plane of division of the volume of the pyramid into two equal parts to be 0·7937 in length on each side, and consequently (from equal triangles) the distance from the plane to the apex to be 0·7937 the total height of 3959, which is 3142·258; then, as we have divided it into a frustum and a now smaller pyramid, if we multiply the square of the base of this new pyramid by one-third of the height we have 0·7937² × 3142·258/3, or 0·62996 × 1047·419 = 659·83, which is equal to the half-volume of the whole pyramid as shown above. Thus we get 3959 less 3142·258 = 816·74 miles as the distance from the base of the plane of division of the pyramid into two equal parts, which naturally agrees with the division of the earth into the two equal volumes that we have extracted from the table of calculations, where we have supposed the earth to be made up of the requisite number of such pyramids. So that it would seem that we are justified in considering that the greatest density of the earth must be at the meeting of the two half-volumes, outer and inner, into which we have divided it.

Considering, then, that one-half of the volume and mass of the earth is contained within 817 miles in depth from the surface, this half must have an average density of 5·66 times that of water, the same as the whole is estimated to have. Also, as we have seen already, that, taking its mean diameter at 7918 miles, its mass will be equivalent to 1,471,168,987,476 cubic miles, one-half of this quantity, or 735,584,493,738 cubic miles will represent the half-volume of the earth reduced to the density of water. With these data let us find out what must be the greatest density where the two half-volumes meet, supposing the densities at the surface and for 9 miles down to remain the same as in the calculations we have already made, ending with specific gravity of 3 at 7900 miles in diameter.

Following the same system as before when treating of the earth as solid to the centre, and using the same table of calculations for the volumes of the layers: If we adopt a direct proportional increase between densities 3 at 7900 miles and 8·8 at 6284·5 miles in diameter, multiply the volumes by their respective densities, and add about 31 per cent. of the following layer, taken at the same density as the previous or last one of the number, we shall find a mass ( see Table V.) of 735,483,165,215 cubic miles at the density of water, which is as near the half mass 735,584,493,738 cubic miles as is necessary for our purpose. It would thus appear that if the earth is a hollow sphere, its greatest density in any part need not be more than 8·8 times that of water, instead of 13·734 times, if we consider it to be solid to the centre.

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.

TABLE VI.— Calculations of the Volumes and Densities of the Inner Half of the Earth, on the same Data as those for the Outer Half.