To the first theorem no objection can be made: The particle on the outside of the shell will undoubtedly be attracted by every particle in the shell, with the same force as if the attractive power of all the particles composing it were concentrated in the centre. Not so with the second theorem: for it can be objected that it altogether ignores the Law of Attraction laid down by Sir Isaac Newton, where it asserts that the resultant attraction of the shell for the particle will be zero, when it is placed anywhere on the inside. In fact the theorem supposes a case impossible for the Harton Colliery experiments, in order to demonstrate their accuracy; for it makes use of the bob of the pendulum—a particle of matter—as if it were transferable to any part of the interior of the earth instead of being confined within the bounds of its swing. That the attraction of the shell—1260 feet thick all round the earth—on the pendulum bob inside of it continues in all its force, and is only divided into two opposing parts, is made plain by Fig. 1. Supposing O to represent the bob of the pendulum at the bottom of the mine, and the space between the two circles the shell of the earth. Then the line B C will show where the attraction of the shell for the bob is divided into two parts acting in opposite directions. Supposing these two parts to be separated from each other, only far enough to admit the bob—a particle to all intents and purposes—between them; the part B A C will attract the bob as if its whole attractive force were collected at its centre of gravity, and the part B D C as if the whole of its attractive force were collected, not at the centre B of the shell, but at its centre of gravity, a very little distance from B in the direction towards D. This is an incontrovertible fact, because it is in strict accordance with Newton's Law of Attraction, which is: Every particle of matter in the universe attracts every other particle with a force directly as their masses, and inversely as the square of the distance which separates them.
Fig. 1.
If we now suppose the interior of the shell to be filled up solid, that will make no difference, because the mass of the part B D C will only be increased vastly thereby, while the mass of A B C will remain the same; the two parts only increasing their proportion to each other, and thus coming to be for the earth—in the Harton Colliery experiments—what we represented them to be at [page 24]; and we can now proceed to find the attractive force of each of the two masses for the bob of the pendulum which is as the inverse square of their distances from it. These distances may be taken, without any very great stretch of conscience, as one-tenth of a mile and 3999·75 miles; because the centre of gravity of the segment A B C will be about that distance from O, and that of B D C cannot be adequately represented by a greater sum than 3999·75, always supposing the diameter of the earth to be 8000 miles. Thus the squares of these two distances will be 0·01 and 15,898,000 miles respectively, and the relative force of attraction for the pendulum of the two segments A B C and B D C will be as 1 × 0·01 and 772,846,315, and 772,846,315 × 15,898,000; that is as 1 is to 1,228,671,000,000,000,000. Here then we get confirmed the unbelief in the theory we expressed at pages [23 and 24]. Surely no one will be bold enough to assert that by decreasing the total attractive force of the earth by a little less than a 1¼ trillionth part cut off from one side of it, the want of homogeneity in what remains will not only not decrease its attractive force at the centre, but increase it so as to make a pendulum be lessened by 1/38,400th part of its time in beating one second. This fraction of time is quite small enough to inspire doubt of any theory founded upon it; and if there ever is a quantity in mathematics that can be called negligible, the fraction of attractive force found above ought to be included in the same category. We may therefore assert that no human measurements could find a true difference between the beats of a seconds pendulum at the top and bottom of the pit at the Harton Colliery. If all the people who have puzzled themselves with this theory had spent an hour or two in making the above calculations before they began them, there would have been no experiments made, and the theory would have died almost ere it was born. Those who believed in it may have looked upon a particle as a negligible quantity, but as the whole earth is made up of particles a little thought would have put an end to such a notion. What puzzles us is how such a theory could be formed by people who knew nothing whatever of the nature of the interior of the earth at a depth of even one mile, and how they could speculate on its want of homogeneity without knowing anything of how the density of 5·66 is made up in it? To suppose that the earth is made up of strata of different densities, and that each is in some degree elliptical—the ellipticity of one stratum being different from another, as the French mathematician Clairaut did—is all very allowable; but to build up any theory on any such suppositions is to build upon shifting sands without examining the foundations. For anything that is known up to the present time, the density of the earth may go on increasing gradually from the surface to the centre, or it may attain nearly its greatest density at a few miles from the surface, and continue homogeneous or nearly so from there to the centre.
To go further now: it is not true that the attraction of a hollow shell of a sphere for any particle within it, is the same "no matter whereabouts in the interior the particle may be." The only place where the attraction will be the same is when the particle is at the centre. In that position a particle would be in a state of very unstable equilibrium, and a little greater thickness of the shell on one side than the others, would pull it a little, perhaps a great, distance from the centre towards that side; and if we extend our ideas to a plurality of particles within the shell of a sphere, we are led to speculate on how they would be distributed, and to see the possibility of there not being any at all at the centre. This is a point which has never been mooted, as far as we have been able to learn, and we shall have to return to it when the proper time comes.
It is difficult to understand how any man could conceive the notion that a shell of a sphere, such as that shown at [Fig. 1], could have no attraction for each separate one of all the particles which make up the mass of the whole solid sphere within it; for that is the truth of the matter if properly looked into, when it is asserted, as has been done by Messrs. Newcomb and Holden, that "the resultant attraction of the shell will therefore be zero." If such a notion could be carried out in a supposed formation of the earth, an infinity of particles would carry off the whole of the interior, and leave the earth as only a shell of 1260 feet thick, as per the Hartley Colliery experiment; only we are told, or left to understand, that that process could not go on for ever, but would have to come to an end somehow and somewhere; and then we are left to speculate on how the unattracted particles could come back to take part in the composition of the earth. Left to ourselves we can only liken the process to that followed by a man who peels off the outer layer of an onion, eats the interior part, and when he is satisfied throws down the outer layer and thinks no more of it; not even that he might be asked what had become of the interior part.
Curiously enough, there is a way of explaining how, or rather why, the notion was formed—not unlike the one just given—to be found in the third of Sir George B. Airy's lectures on Popular Astronomy, delivered at Ipswich several years before the final experiments were made at the Harton Colliery. In that lecture, while describing how the Greek Astronomers accounted for the motions of the sun and planets round the stationary earth, he says, "It does appear strange that any reasonable man could entertain such a theory as this. It is, however, certain that they did entertain such a notion; and there is one thing which seems to me to give something of a clue to it. In speaking to-day and yesterday of the faults of education, I said that we take things for granted without evidence; mankind in general adopts things instilled into them in early youth as truths, without sufficient examination; and I now add that philosophers are much influenced by the common belief of the common people."
We can agree with Sir George B. Airy in his ideas about education, and now conclude by saying that he has given us a very clear and notable example of a theory being accepted very generally, without being thoroughly examined to the very end, and of how easy it is for such theories to be handed down to future generations for their admiration.