million
or even billion tons; but that is quite small in astronomy.

But now it may be asked, surely the moon perturbs the earth, swinging it
round their common centre of gravity, and really describing its own
orbit about this point instead of about the earth's centre? Yes, that is
so; and a more precise consideration of Kepler's third law enables us to
make a fair approximation to the position of this common centre of
gravity, and thus practically to "weigh the moon," i.e. to compare its
mass with that of the earth; for their masses will be inversely as their
respective distances from the common centre of gravity or balancing
point--on the simple steel-yard principle.

Hitherto we have not troubled ourselves about the precise point about
which the revolution occurs, but Kepler's third law is not precisely
accurate unless it is attended to. The bigger the revolving body the
greater is the discrepancy: and we see in the table preceding Lecture
III., on page 57, that Jupiter exhibits an error which, though very
slight, is greater than that of any of the other planets, when the sun
is considered the fixed centre.

Let the common centre of gravity of earth and moon be displaced a
distance _x_ from the centre of the earth, then the moon's distance
from the real centre of revolution is not _r_, but _r-x_; and the
equation of centrifugal force to gravitative-attraction is strictly

4[pi]^2 _VE_
--------- (_r-x_) = ------,
T^2 r^2

instead of what is in the text above; and this gives a slightly
modified "third law." From this equation, if we have any distinct
method of determining _VE_ (and the next section gives such a
method), we can calculate _x_ and thus roughly weigh the moon,
since

_r-x_ E
----- = -----,
_r_ E+M

but to get anything like a reasonable result the data must be very
precise.

No. 6. The force constraining the moon in her orbit is the same gravity
as gives terrestrial bodies their weight and regulates the motion of
projectiles.

Here we come to the Newtonian verification already several times
mentioned; but because of its importance I will repeat it in other
words. The hypothesis to be verified is that the force acting on the
moon is the same kind of force as acts on bodies we can handle and
weigh, and which gives them their weight. Now the weight of a mass _m_
is commonly written _mg_, whe