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Science, "the great measurer," is for ever busy with scales, weights, and measuring-tape. Directly it was settled that the world is round, we find the Alexandrian astronomers attempting to measure its circumference. Hardly had Newton formed his theory of gravitation before his mind was full of schemes for "weighing the earth." From the moment when the modern atomic hypothesis was accepted, and indeed even before, Dalton and his colleagues were as busy as bees trying to weigh invisible, nay, hypothetical, atoms and molecules. And the very discovery of the "electrons" or "corpuscles" in Sir William Crookes's vacuum tubes may almost be said to have consisted in attempts to compare their masses with those of the lightest particles previously known—atoms of hydrogen. Nothing seems too difficult. The weight of the earth, the weight of an atom, the velocity of light—nay, the speed of thought itself, or, at least, the speed with which thought can be translated into action—all these and a thousand other quantities have been brought by science within the compass of her measuring instruments, their values ascertained, stated in familiar terms, and placed, gratis, at the service of man.
Perhaps some of the readers of the "Cornhill" may feel disposed to take a peep into the machinery employed to accomplish the tremendous task of weighing a world? If so, I must ask them, first, to consider this question:
What do we mean by "the weight of the earth"?
When we speak of the weight of such an object as a lump of coal, we mean, of course, the pull of the earth upon that piece of coal; and the quantity of coal we call a pound is that quantity
which is pulled to the earth with a force just equal to the force that pulls a particular piece of platinum, marked "P.S. 1844 1 lb.," and called the "Imperial Avoirdupois Pound," which is kept at the Standards Office in Westminster.
Now it is clear that the earth, as a whole, cannot pull itself to itself. Every particle of it in every direction must pull every other particle, with the result that there is a state of equilibrium and no pull; and thus, in the everyday sense of the term, the earth has no weight at all.
But we all know that though when we weigh bodies we may seem merely to measure the pull of the earth upon them, we not only learn the strength of this pull, but also measure what Newton called "the quantity of matter in them," or, as we say to-day, "their masses." For it has been shown by Newton that at any given point on its surface the earth's pull on an object is proportional to the mass of the object, and quite independent of all such qualities or considerations as its shape or position, whether it is a solid, a liquid, or a gas, and also, as Lavoisier has taught us, independent of its chemical constitution; this being, of course, only a particular case of Newton's law of gravitation, which tells us that every particle of matter in the universe attracts every other particle with a force which depends on their masses and on the distances which separate them; the attraction being proportionately greater between large masses than between small masses, increasing when the masses are brought closer to one another, and decreasing as they recede, in such a manner that if the distances between the centres of two spheres be doubled, then the attraction between
theui is reduced to one-quarter of its original strength.
Returning now to our question, we see that the process familiarly termed "weighing the earth" consists really in measuring the quantity of matter the earth is made of, or, in modern terms, in determining its mass.
Although we cannot even imagine ourselves balancing the earth on a pair of scales against a set of weights, some other way of attacking the problem which is not altogether beyond the range of the imagination may occur to the reader, and help him to grasp its nature and difficulty.
We know, for example, that the diameter of the earth is about 8,000 miles, and we know how to calculate the approximate volume of a sphere when we have measured its diameter. Why, then, should we not calculate the volume of the earth in cubic feet, find the mass of a cubic foot of it in pounds by weighing samples, finally multiply these two quantities, and so determine its mass in pounds? It would not be very difficult to perform these simple operations, but, unfortunately, even if we neglect the irregularity of the earth's surface, there are still some fatal objections. The masses of equal volumes of rock taken from different parts of the earth's crust vary considerably; and, further, even if this were not so, we have no means of getting samples of the material of which the earth is made except by scratching its outer skin, and it would by no means be safe to assume that the average weight of each cubic foot of the rocks which exist below, out of our reach, is the same as the average weight of each cubic foot of the rocks which are familiar to us on its surface. Still, the general idea of the problem presented in the form of this faulty proposal is not unhelpful. It simplifies the matter considerably. We know the volume of the earth more or less
closely, therefore all we have to do is to find its "mean density"—to find, that is, what proportion the mass of the earth bears to the mass of a globe of water of equal size. When this is done, since every cubic foot of water weighs about GiiVfs lbs., we can easily calculate the weight of the earth in the ordinary sense of the term, and state it in pounds or tons, in grams or kilograms, as we may desire.
The process of "weighing the earth," then, may be said to consist in finding its mean density, water, which is said to have the density 1, being taken as the standard substance. Thus stated, the problem seems easy enough, but the solution of this simple problem has occupied the thoughts of many master minds, and taxed to their utmost the powers of many great experimenters from the days of Newton.
It is true that, by taking the earth as their standard, astronomers have been able to draw up a table of densities for the heavenly bodies, from which we learn that the mean density of the sun is about one-fourth as great as that of our globe, that of Venus and Mars about nine-tenths as great, that of Mercury one and a quarter times greater, and so on. But this, though sufficient for many purposes, fails to give us such a clear idea of the matter as we get when we can think of our quantities in familiar terrestrial standards such as the gram or the pound; and so it is necessary to connect the celestial scale of densities, in which the earth is made the standard, with one of the more familiar terrestrial scales. The first attempt to do this was made by Newton. This attempt was a mere estimate—in fact, a guess. I give it in full in his own words, as translated by Motte:
But that our globe of earth is of greater density than it would be if the whole consisted of water only, I thus make out. If the whole consisted of water only, whatever was of less density than water, because of its less specific gravity, would emerge and float above. And upon this account, if a globe of terrestrial matter, covered on all sides with water, was less dense than water, it would emerge somewhere: and the subsiding water falling back would be gathered to the opposite side. And such is the condition of our earth, which, in a great measure, is covered with seas. The earth, if it was not for its greater density, would emerge from the seas, and, according to its degree of levity, would be raised more or less above their surface, the water and the seas flowing backwards to the opposite side. By the same argument, the spots of the sun which float upon the lucid matter thereof are lighter than that matter. And however the planets have been formed while they were yet in fluid masses, all the heavier matter subsided to the centre. Since, therefore, the common matter of our earth on the surface thereof is about twice as heavy as water, and, a little lower, in mines, is found about three or four or even five times more heavy; it is probable that the quantity of the whole matter of the earth maybe five or six times greater than If It consisted all of water, especially since I have before showed that the earth is about four times more dense than Jupiter.
Newton's guess, curiously enough, hits the limits between which the values subsequently fixed by experiments are mostly to be found.
In practice, all the methods of weighing the earth resolve themselves into experiments in which we measure the attraction between two bodies having known masses placed at a known distance from each other on the earth's surface, and then compare this with the attraction of the earth on some known mass of matter, also on its surface. The following illustration, taken from a lecture by Professor J. H. Poynting, will make the idea clearer:
Suppose you hang a weight of 50 lbs. from a spring balance a few feet above
the earth. Then the pull of the earth, whose centre is about 4,000 miles or -'0,000,000 feet away, is 50 lbs. Now suppose you bring a second weight, this time, let us say, a weight of 350 lbs., to a position one foot from the first one. and between the latter and the earth, so that its pull is added to that of the earth. Then, if your balance is sufficiently sensitive, you will find the smaller mass no longer weighs 50 lbs., but a little more—in fact about of a grain more—that is to say, the pull of the 350-lb. weight at the distance of a foot is equal to the -fa of a grain. or TTrJsrff of 1 lD-. or the pull of the earth at a distance of 20,000,000 feet is about ninety million times as great as that of a sphere of 350 lbs. at one foot, for
1,750,000 X 50 = 87,500,000.
If the earth could be placed at an average distance of one foot from the 50-lb. weight, instead of at a distance of 20,000,000 feet, its pull would be proportionately greater—viz. about four hundred billion times greater, so that at equal distances the pull of the earth would be four hundred billion times ninety million times that of a 350-lb. sphere. But, as already explained, at equal distances these pulls are proportional to the masses concerned, and thus, by doing a little more arithmetic, we should find that the earth weighs about 12,500,000,000,000,000,000,000,000 lbs. Finally, if we calculate the mean density of the earth from these figures and from its volume, which can be deduced from its diameter, we find that its mass is about five and a half times as great as that of an equal volume of water, or, to use the technical term, that the "mean density" of the earth is five and a half times as great as that of water. This, however, is only the result of an imaginary experiment. The real thing, though similar in principle, is far more complicated, as will easily
be understood when I mention that a determination of the density of the earth carried out with due precautions to eliminate all sources of error may occupy several years, and that in some cases the necessary operations are of so delicate a character that the mere passage of railway trains in the neighborhood of the apparatus may be a serious source of trouble. Indeed, on one occasion Professor Boys, when working at Oxford, was stopped by an earthquake which occurred thousands of miles away, and was, I believe, only detected in this part of the world through the circumstance that Professor Boys was weighing the earth when the wave reached these regions.
The actual objects whose attractions have been observed in attempts to weigh the earth have varied very widely.
The earliest observers studied the attractions of mountains on objects brought near them; Professor Boys those of small metallic spheres, the largest of which were only four and a half inches, and the smallest one-fourth of an inch in diameter. The methods employed divide themselves into three or four groups.
First come experiments in which the attraction of a mountain or some natural object, such as a zone of known thickness of the upper crust of the earth, is compared with that of the earth as a whole.
Secondly, the famous "Cavendish experiment," in which the attractions between metallic masses quite small in size are investigated by means of what is known as a torsion balance.
And, thirdly, researches in which common but very delicate scales and weights are employed. Some very beautiful experiments falling within this last class were made a few years ago at what was then the Mason College, Birmingham, by Professor Poynting. on whose publications on the sub
ject of the weight of the earth this article is very largely based.
And now, after all these preliminary remarks to clear the way, we come to the real thing, to the actual experiments made for the purpose of weighing the earth, from the time of Newton, who inspired all this work, in which our fellow-countrymen have always played a conspicuous and successful part, till to-day.
We have learnt from the preceding pages that astronomers have succeeded in comparing the densities of various heavenly bodies by means of astronomical observations, and have drawn up tables stating their results in terms of the density of the earth, but that if we wish to get out our results in earthly measures, such as ounces or grams, we must descend from the stars, and compare, for example, the pull of the earth on some object on its surface with the pull of some measurable mass on the same object. All this, of course, was very well understood by Newton, who saw, further, that the power of a mountain to deflect a plumb-line might be employed; unfortunately, he concluded that the effect would be too small to measure, which, indeed, may possibly have been true at that time. Newton also investigated the possibility of measuring the attraction between large spheres, and calculated how long it would take a sphere one foot in diameter, and of equal density with the earth, to draw a second sphere, of the same dimensions and equal density, placed a quarter of an inch away, across this interval of a quarter of an inch. Through a mistake in his arithmetic, he found the required time to be about a month, which is vastly more than the few minutes that would really be needed, and as such a rate of motion was utterly beyond measurement, he confined himself to making the celebrated guess mentioned above. But not very long afterwards both these methods were put to the test of experiment with a considerable degree of success.
Some doubt is said to exist as to whether Newton was the real author of this mistake, but, as Professor Poynting remarked in a lecture at the Royal Institution a few years ago, there is something not altogether unpleasing in the belief that even Newton could make a mistake. His faulty arithmetic showed that there was, at any rate, one quality which he shared with his fallible fellow-men.
When the attractive force of a mountain is to be studied, the experiment, in its simplest form, is somewhat as follows: A weight hanging at the end of a thread—that is, a plumb-line more or less similar to the plumb-line employed by a mason, but far more sensitive and provided with more exact means of measurement—is placed first in some suitable position not too far away from the mountain, but well out of the range of its attraction, and its position noted on a scale of divisions when it hangs freely suspended, and, therefore, perpendicular to the earth's surface. The plumb-line is then brought up as close as may be to one side of the mountain. When this is done the plumb-line is found to be drawn a little to one side of its previous line of suspension—that is to say, a little out of the perpendicular and towards the mountain. The amount of this displacement is measured on the scale of divisions, and the length of the plumb-line is also measured. From these data the astronomer can calculate the ratio of the horizontal pull of the mountain to the pull of the earth.
Finally, the mountain is most carefully surveyed, and the densities of pieces of the rock of which it is composed are measured. Knowing these densities and the volume of the mountain we can estimate the mass of the mountain in pounds or kilograms, according to the system selected; and
when this is done we know the mass of the mountain, the pull of the mountain, the pull of the earth, and their distances, and from these, knowing the law of gravitation, quoted above, we can deduce the other quantity involved, the mass of the earth.
The first investigator to actually determine the mean density of the earth by this method was M. Bouguer, who was a member of one of two scientific commissions sent out by France about 1740 to measure the lengths of degrees of latitude in Peru and Lapland—that is, at points near to and remote from the equator—in order to settle finally the shape of the earth, whether it is flattened at the poles, as Newton supposed, or drawn out, as had then lately been suggested. The members of these commissions, which, by the way, settled the question in favor of Newton's views, did not confine themselves to investigating the shape of the earth; and M. Bouguer, in particular, seized the opportunity of testing the "mountain mass method" of weighing the earth thus afforded him by his visit to the great mountains of the Andes. M. Bouguer made two distinct sets of measurements. In the first he studied the swing of a pendulum at the sealevel, then at a point 10,000 feet higher, on the great plateau on which Quito stands, and, finally, on the top of Pichincha, which is about 6,000 feet above Quito. He knew that if a pendulum were lifted to a great height above a wide plain or over the open sea, say, for example, by means of a balloon, its swing would gradually grow slower as gravity decreased at the higher levels; and he calculated from the swing of his pendulum at Quito that gravity there was greater than the calculated amount for the height at which he worked, owing to the down pull of the great tableland beneath him.
Bouguer's second set of observations