New Theories in Astronomy

by William Stirling

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.