3M w
[Delta] = --------- . --.
4[pi]Rd^2 p
If then we can arrange to observe w/p we obtain [Delta], the mean
density of the earth.
But the same observations give us G also. For, putting m = w/g in (2),
we get
d^2 p
G = --- . -- . g.
M w
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/d^2 we get at once G = pd^2/M1M2. But we
can also deduce [Delta]. For putting w = mg in (1) we get
g 1
[Delta] = 3/4 -- . -----.
G [pi]R
Experiments of the first class in which the pull of a known mass is
compared with the pull of the earth maybe 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 [Delta] = 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 [Delta]/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.