perigee should be greater by 1·5″ than the theoretical motion. E. W. Brown is the first investigator to determine the theoretical motions with this degree of precision; and he finds that there is no such divergence between the actual and the computed motion. There is therefore as yet no ground for regarding any deviation from the law of inverse square as more than a possibility. (S. N.)
Gravitation Constant and Mean Density of the Earth
The law of gravitation states that two masses M1 and M2, distant d from each other, are pulled together each with a force G. M1M2/d2, where G is a constant for all kinds of matter—the gravitation constant. The acceleration of M2 towards M1 or the force exerted on it by M1 per unit of its mass is therefore GM1/d2. Astronomical observations of the accelerations of different planets towards the sun, or of different satellites towards the same primary, give us the most accurate confirmation of the distance part of the law. By comparing accelerations towards different bodies we obtain the ratios of the masses of those different bodies and, in so far as the ratios are consistent, we obtain confirmation of the mass part. But we only obtain the ratios of the masses to the mass of some one member of the system, say the earth. We do not find the mass in terms of grammes or pounds. In fact, astronomy gives us the product GM, but neither G nor M. For example, the acceleration of the earth towards the sun is about 0·6 cm/sec.2 at a distance from it about 15 × 1012 cm. The acceleration of the moon towards the earth is about 0·27 cm/sec.2 at a distance from it about 4 × 1010 cm. If S is the mass of the sun and E the mass of the earth we have 0·6＝GS/(15 × 1012)2 and 0·27＝GE/(4 × 1010)2 giving us GS and GE, and the ratio S/E＝300,000 roughly; but we do not obtain either S or E in grammes, and we do not find G.
The aim of the experiments to be described here may be regarded either as the determination of the mass of the earth in grammes, most conveniently expressed by its mass ÷ its volume, that is by its “mean density” Δ, or the determination of the “gravitation constant” G. Corresponding to these two aspects of the problem there are two modes of attack. Suppose that a body of mass m is suspended at the earth’s surface where it is pulled with a force w vertically downwards by the earth—its weight. At the same time let it be pulled with a force p by a measurable mass M which may be a mountain, or some measurable part of the earth’s surface layers, or an artificially prepared mass brought near m, and let the pull of M be the same as if it were concentrated at a distance d. The earth pull may be regarded as the same as if the earth were all concentrated at its centre, distant R.
Then
| w＝G.43πR3Δm/R2＝G · 43 πRΔm, | (1) |
and
| p＝GMm/d2. | (2) |
By division
If then we can arrange to observe w/p we obtain Δ, the mean density of the earth.
But the same observations give us G also. For, putting m＝w/g in (2), we get
In the second mode of attack the pull p between two artificially prepared measured masses M1, M2 is determined when they are a distance d apart, and since p＝G.M1M2/d2 we get at once G＝pd2/M1M2. But we can also deduce Δ. For putting w＝mg in (1) we get
Experiments of the first class in which the pull of a known mass is compared with the pull of the earth may be termed experiments on the mean density of the earth, while experiments of the second class in which the pull between two known masses is directly measured may be termed experiments on the gravitation constant.
We shall, however, adopt a slightly different classification for the purpose of describing methods of experiment, viz:—
1. Comparison of the earth pull on a body with the pull of a natural mass as in the Schiehallion experiment.
2. Determination of the attraction between two artificial masses as in Cavendish’s experiment.
3. Comparison of the earth pull on a body with the pull of an artificial mass as in experiments with the common balance.
It is interesting to note that the possibility of gravitation experiments of this kind was first considered by Newton, and in both of the forms (1) and (2). In the System of the World (3rd ed., 1737, p. 40) he calculates that the deviation by a hemispherical mountain, of the earth’s density and with radius 3 m., on a plumb-line at its side will be less than 2 minutes. He also calculates (though with an error in his arithmetic) the acceleration towards each other of two spheres each a foot in diameter and of the earth’s density, and comes to the conclusion that in either case the effect is too small for measurement. In the Principia, bk. iii., prop. x., he makes a celebrated estimate that the earth’s mean density is five or six times that of water. Adopting this estimate, the deviation by an actual mountain or the attraction of two terrestrial spheres would be of the orders calculated, and regarded by Newton as immeasurably small.
Whatever method is adopted the force to be measured is very minute. This may be realized if we here anticipate the results of the experiments, which show that in round numbers Δ＝5·5 and G＝1/15,000,000 when the masses are in grammes and the distances in centimetres.
Newton’s mountain, which would probably have density about Δ/2 would deviate the plumb-line not much more than half a minute. Two spheres 30 cm. in diameter (about 1 ft.) and of density 11 (about that of lead) just not touching would pull each other with a force rather less than 2 dynes, and their acceleration would be such that they would move into contact if starting 1 cm. apart in rather over 400 seconds.
From these examples it will be realized that in gravitation experiments extraordinary precautions must be adopted to eliminate disturbing forces which may easily rise to be comparable with the forces to be measured. We shall not attempt to give an account of these precautions, but only seek to set forth the general principles of the different experiments which have been made.
I. Comparison of the Earth Pull with that of a Natural Mass.
Bouguer’s Experiments.—The earliest experiments were made by Pierre Bouguer about 1740, and they are recorded in his Figure de la terre (1749). They were of two kinds. In the first he determined the length of the seconds pendulum, and thence g at different levels. Thus at Quito, which may be regarded as on a table-land 1466 toises (a toise is about 6·4 ft.) above sea-level, the seconds pendulum was less by 1/1331 than on the Isle of Inca at sea-level. But if there were no matter above the sea-level, the inverse square law would make the pendulum less by 1/1118 at the higher level. The value of g then at the higher level was greater than could be accounted for by the attraction of an earth ending at sea-level by the difference 1/1118−1/1331＝1/6983, and this was put down to the attraction of the plateau 1466 toises high; or the attraction of the whole earth was 6983 times the attraction of the plateau. Using the rule, now known as “Young’s rule,” for the attraction of the plateau, Bouguer found that the density of the earth was 4·7 times that of the plateau, a result certainly much too large.
In the second kind of experiment he attempted to measure the horizontal pull of Chimborazo, a mountain about 20,000 ft. high, by the deflection of a plumb-line at a station on its south side. Fig. 1 shows the principle of the method. Suppose that two stations are fixed, one on the side of the mountain due south of the summit, and the other on the same latitude but some distance westward, away from the influence of the mountain. Suppose that at the second station a star is observed to pass the meridian, for simplicity we will say directly overhead, then a
