Before astronomers could begin to determine the relative distances from each other, and the relative dimensions and masses of the various members of the solar system, they had to establish scales of measurements appropriate to their undertaking. This entailed upon them, of course, the necessity of determining the form, the different circumferences and diameters, and the weight of the whole earth, as any other scales derived from the only available source, the earth, would have been too small to give even an approximate value of the measures and masses to be sought for.

History tells us that at least one attempt had been made, over two thousand years ago, to find the circumference and necessarily the diameter of the earth, but it says nothing of any to ascertain its weight. There may have been many to determine both diameter and mass, but we know nothing of them; and when we think seriously about this, we cannot help feeling somewhat surprised that no attempt had been made to find out the density and mass till more than a century after Sir Isaac Newton's discovery of the law of Attraction, or Gravitation, as it is more usually called. But perhaps this is an idea that could only occur to one who has been spoilt by witnessing, in great measure, the immense strides in advance that have been made during the nineteenth century in science of all kinds, and does not duly take into account the immense labour, and the incessant meeting with almost insurmountable difficulties, that astronomers have had to encounter and overcome between the birth of modern astronomy and the end of the eighteenth century. Indeed, the difficulties can hardly be looked upon as altogether overcome even yet, as efforts are still being made to find out the exact distance of the sun, and it is not impossible that some small difference may be found, plus or minus, in the density at present adopted for the earth of 5·66 times the weight of water.

The geometer who, more than two thousand years ago, set himself the task of measuring the circumference of the earth, is supposed to have made use of very much the same kind of implements as those employed by modern astronomers. He must have had a very fair instrument for measuring angles, and have known very well how to use it, seeing he was able to determine a value for the obliquity of the ecliptic which agrees so well with that established by modern science, its variations being, for what we know, taken into account; and for length or distance he would doubtless have some implement analogous to the metre, chain, foot-rule, or something called by other name that would, in those days, present facilities for selling a yard of calico. His operations would probably be as plain and simple as those applied to the measuring of a village green—for we are not told that he had any idea of there being any difference between the length of a degree of the meridian at the equator and one nearer either of the poles—and involved no hypotheses or theories, any more than modern operations have done.

When the time came for making efforts to ascertain the density of the earth, science seems to have employed the very simplest means it had at its disposal for attaining its object, and to have gone on refining its implements and operations in conformity with the lessons it went on learning while pursuing its self-imposed task. Every one who, even for recreation, has read a fair amount of the multitude of works and writings that have been published on Popular Astronomy—not to speak of text-books—knows that the first attempts were made by measuring the attraction of steep, or precipitous, mountains for plummets suspended in appropriate positions in their neighbourhood; then—evidently from knowledge acquired during these operations—by the attraction for each other of large and small leaden balls suspended on frames and torsion balances, which go under the name of the Cavendish Experiment; and afterwards by a refinement on this in using the Chemical Balance, where only one large and one small ball of metal are required. All these operations and their results are to be found described in works of various kinds, and are generally reduced to something like the following tubular form, which we reproduce in order to make more intelligible what we have just said, and that we may make a few remarks upon them.

There is no hypothesis, no theory, connected with any of the operations, unless it was the supposition that a plummet—which was naturally believed to point to the centre of the earth—should be pulled to one side by the attraction for it of a mountain in its neighbourhood, and that was found to be a fact.

Methods Employed for Finding the Density of the Earth,
And their Results.

In the case of the plummet deviating from its absolutely straight direction towards the centre of the earth, caused by their attraction, not only the mountains themselves had to be measured and virtually weighed as far as they were measurable, but the weight of the wedge or pyramid between that measurable point, in each case, and the centre of the earth had to be estimated in some way; then the centre of gravity of the whole of this mass had to be ascertained, as well as the respective distances from the centre of the earth of this centre of gravity and that of the plummet, and only after all this and a deep study of the mutual attractions of this mass and the plummet could an estimate be formed of the mass of the earth. It will thus be seen that such measurements and estimates could never be looked upon as very exact and reliable; and nevertheless they have come very near the density of 5·66 finally adopted for the earth.

In the case of the Torsion Balance experiments a very considerable advance was made in consequence, most undoubtedly, of the knowledge acquired from what had been done by Maskelyne. When it was found that the attraction of Schiehallien for the plummets was such a measurable quantity, Cavendish evidently saw that the attraction of manageable leaden balls for each other would be measurable also, and that as no calculations of any kind whatever were necessary to find the masses of the balls, the mutual attraction of large and small balls would furnish a more exact means of measuring the density of the earth, than the roundabout way of having to calculate the weight of a mountain as a beginning; and with the requisite ingenuity, invention, and labour, he found the means of applying the torsion balance, to make the experiments.

After these experiments were revised by Reich and Baily—and the density of 5·66 adopted, we believe—still another set were undertaken by J. H. Pointing, with the Chemical Balance, in which only two metal balls, one large and one small were required, which gave a density of 5·690 as shown opposite, and from its extreme simplicity may perhaps have been the most exact of all.

We have said, we think with truth, that there is no hypothesis or theory involved in any of these experiments, but only the simplest form of—we might almost say—arithmetical calculation. But there is a theory built up on hypothesis which has no foundation whatever, and about which most people, who take the trouble to study it out to the very end, will come to the conclusion that "the less said the better." This, at all events, is our opinion, and we would not have taken any notice whatever of it had it not been that up to the present day, it is published in many works on Popular Astronomy, and even in some text-books, and is looked upon in them, apparently, as an example of the transcendent height to which human science can reach.

We allude, of course, to the theory that the deeper we go down into the earth—at least to an undefined and undefinable depth—the greater is its attraction for the bob of a pendulum at that depth, and the greater the number of vibrations the pendulum is caused to make in a given time. The explanation of the theory is, that were the earth homogeneous throughout its whole volume, the pendulum ought to make the fewer vibrations, the deeper down in the earth it is placed; but as the earth is not homogeneous, it actually makes a greater number of vibrations in a given time, because the attractive force of the earth increases—up to the undefined and undefinable depth—on account of the denser matter beneath the pendulum bob more than overbalancing the loss of attraction from the lighter matter left above it. The author of the theory was the late Astronomer Royal, Sir George B. Airy, who from it endeavoured to calculate the mean density of the earth, and with that view made two experiments which are thus described by Professor C. Piazzi Smythe in his work on the Great Pyramid:—

"Another species of experiment. . . was tried in 1826 by Mr. (now Sir) George B. Airy, Astronomer Royal, Dr. Whewell, and the Rev. Richard Sheepshanks, by means of pendulum observations at the top and bottom of a deep mine in Cornwall; but the proceedings at that time failed. Subsequently, in 1855, the case was taken up again by Sir George B. Airy and his Greenwich assistants, in a mine near Newcastle. They were reinforced by the new invention of sympathetic electric control between clocks at the top and bottom of a mine, and had much better, though still unexpectedly large results—the mean density of the earth coming out, for them, 6·565."

From other sources we have also found that the pit, or mine, was at the Harton Colliery and 1260 feet deep, that the pendulum at the bottom of it gained 2¼ seconds on the similar one at the top, in 24 hours; and that the surrounding country had to be extensively surveyed, the strata had to be studied, and their specific gravities ascertained.

A little unbiassed thought bestowed on this theory will at once show that it begins by violating the law of attraction discovered by Newton, when he showed that the mutually attractive forces of several bodies are the same as if they were resident in the centres of gravity of the bodies. In the case in point this means, that the attraction of the earth for the bob of the pendulum at the top of the mine was the same as if all its force was collected at its (the earth's) centre. In that position the force of the earth's attraction comprehended, most undeniably, the whole of its attractive power, including whatever might be imagined to be derived from the non-homogeneity of the earth, due to its density increasing towards the centre; and we are called upon to believe that when, virtually, the same pendulum was removed to the bottom of the mine, and a segment 1260 feet thick, at the centre as good as cut off from the earth and—as far as the pendulum was concerned—hung up on a peg in a laboratory, the diminished quantity of its matter had a greater attractive force, a very little beyond the centre—non-homogeneity again included—than the whole when the sphere was intact. This we cannot do, because all that we can see in the placing of the pendulum at the bottom of the mine, is that the position of the bob has divided the earth into two sections, one of which has a tendency to pull it up towards the surface, and the other to pull it down towards its centre of gravity; and because the mass of the smaller segment is so insignificant that its entire removal to the laboratory peg, not only could not produce the reverse action, on which the theory is based, but could not be measured by any stretch of human invention or ingenuity; it is far beyond the reach of mathematics and human comprehension of quantity.

The difficulty of belief is increased when we reflect that, were the pendulum taken down towards the centre of the earth, the number of its vibrations in a given time ought gradually to decrease as it approached the centre, and would cease altogether when that point was reached. And we feel confident that no mathematician could calculate where the theoretical acceleration of the vibrations would cease, and the inevitable retardation commence; where the theory would come to an end and the law of attraction begin to assert its rights, simply because he does not know how the non-homogeneity is distributed in the earth. No man can tell, even yet, how the mean density of 5·66 is made up throughout the earth, and without that any theory founded on its non-homogeneity is out of place.

But to follow up our assertion of non-commensurability. Taking the diameter of the earth at 8000 miles, and its mean specific gravity at 5·66, its mass would be represented by 1,517,391,000,000 cubic miles of water. On the other hand, supposing the earth to be a true sphere, the volume of a segment of it cut off from one side, at one quarter of a mile deep—not 1260, but 1320 feet—would be 785·35 cubic miles in volume, and if we suppose its specific gravity to be 2·5—greater most probably than the average of all the strata in the neighbourhood of the Harton Colliery—its mass would be represented by 1963·38 cubic miles of water. Then, if we divide the mass of the section below the pendulum, that is, 1,517,391,000,000 minus the mass of the one above it, 1963·38, viz. 1,517,390,998,036·62 by the mass of 1963·38 just mentioned, we find that the proportion they bear to each other is as 1 to 772,846,315. This being so, we are asked to believe that by removing 1/772,846,315th part of the mass of the earth from one side of it, its force of attraction at the centre will not only not be decreased, but will be so increased that it will cause a pendulum, suspended at the centre of the flat left by the removal of the segment, to vibrate 86,402·25 times in twenty-four hours instead of 86,400 times as it did when suspended at the surface before the segment was removed; that is, that the vibrations will be increased by 1/38,400th part. Again we cannot do so. Had we been asked to believe that the removal of so small a fraction as 1/772,846,315th had decreased the earth's attraction at its centre, so much as to produce a diminution of 1/38,400th part in the number of vibrations of the pendulum, we could not have done so; how much less then can we believe that the central attractive force had increased so much as to produce an augmentation of the vibrations in the same proportions? But more in this strain presently.

We have no doubt whatever that Sir George B. Airy and his assistants satisfied themselves that the pendulum at the bottom of the mine gained 2¼ seconds in twenty-four hours over the one at the top, but they may have been deceived by their over-enthusiastic adoption of what seemed to be a very grandly scientific theory, or by some unperceived changes in the temperature in the pendulums, caused by varying ventilation in the mine or the varying weather outside of it, or by the insidious manifestations of the "sympathetic electric control between clocks at the top and bottom of a mine," called in to assist at the experiments. An error of 1/38,400th part of the time the sympathetic electricity would take to travel from the top to the bottom of the shaft would be sufficient to make the experiments of no value whatever; not to speak of the small errors that may have been made in surveying the surrounding country, calculating the specific gravities of the strata—for we are told that all this had to be done-and applying the elements thus obtained to the solution of the problem they had in hand. We have read of the difficulties met with by Mr. Francis Baily when he began to revise the Cavendish Experiment—some twelve or fifteen years before the final Harton Colliery experiments were made, and suppose it possible that they met with similar difficulties without being aware of it. And 1/38,400th part is such a very small fractional difference in the vibrations in twenty-four hours, of the pendulums of the two separate clocks, that—taking into consideration the circumstances under which it was found—it would hardly be looked upon as reliable at the present day, when the clocks of astronomical observatories are placed in the deepest cellars or even caves available, so as to free them as much as possible from variations of temperature.

Having referred to the difficulties met with by Mr. Baily, we believe it worth while to transcribe Professor C. Piazzi Smythe's account of them, given in his work already referred to at page 22; because it not only has a very direct bearing on what we have been saying of changes of temperature, but is exceedingly interesting, and probably very rarely to be met with in other works. It is as follows:—

"Nearly forty years after Cavendish's great work, his experiment was repeated by Professor Reich of Freyberg, in Saxony, with a result of 5·44; and then came the grander repetition of the late Mr. Francis Baily, representing therein the Royal Astronomical Society, and, in fact, the British Government and the British Nation.

"With exquisite care did that well-versed and methodical observer proceed to his task, and yet his observations did not prosper.

"Week after week, and month after month, unceasing measures were recorded; but only to show that some disturbing element was at work, overpowering the attraction of the larger on the smaller balls.

"What could it be?

"Professor Reich was applied to, and requested to state how he had continued to get the much greater degree of accordance with each other, that his published observations showed.

"'Ah!' he explained, 'he had to reject all his earlier observations until he had guarded against variations of temperature by putting the whole apparatus into a cellar, and only looking at it with a telescope through a small hole in the door.'

"Then it was remembered that a very similar plan had been adopted by Cavendish, who had furthermore left this note behind him for his successor's attention—'that even still or after all the precautions which he did take, minute variations and small changes of temperature between the large and small balls were the chief obstacles to full accuracy.'

"Mr. Baily therefore adopted yet further, and very peculiar, means to prevent sudden changes of temperature in his observing room, and then only did the anomalies vanish and the real observations begin.

"The full history of them, and all the particulars of every numerical entry, and the whole of the steps of calculation, are to be found in the Memoirs of the Royal Astronomical Society, and constitute one of the most interesting volumes (the Fourteenth) of that important series; and its final result for the earth's mean density was announced as 5·675, probable error ± 0·0038."

After reading this story of Baily's experiments with care, one cannot help feeling something stronger than want of confidence in those made at the Harton Colliery, especially after what has been shown of the smallness of the fraction of the earth that was dealt with, and due consideration is given to the insignificant difference of effect that the non-homogeneity of the earth could produce on the remainder after the supposed removal of such a small fraction; and here we might let the theory drop. Perhaps it may be thought that now there is nothing to be gained by spending time and work in showing it to be more truly erroneous than we have yet made it out to be; but if there is error, it cannot be too clearly exposed, and the sooner it is put an end to, the better; more especially as it has been accepted as true by some authors of text-books, and by some competent astronomers who, in trying to explain the anomaly of the increase instead of decrease in the force of attraction at the bottom of a mine compared with the top, have used arguments which are not consistent with the law of gravitation, or rather attraction.

Messrs. Newcomb and Holden in their work, entitled "Astronomy for High Schools and Colleges," sixth edition, 1889, apparently accept the theory, and proceed to explain and support it by showing what would be the action of a hollow spherical shell of any substance on a particle of it, say the bob of a pendulum, placed on the outside and also on the inside of the shell; and give us two theorems which are supposed to comprehend both cases. These are:—

(1) "If the particle be outside of the shell, it will be attracted as if the whole mass of the shell were concentrated at its centre."

(2) "If it be inside the shell, the opposite attractions in every direction will neutralise each other, no matter whereabouts in the interior the particles may be, and the resultant attraction of the shell will therefore be zero."

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.