Your book is just being careful, and the authors are perfectly correct in worrying about a small detail like that.
The gravitational force between two point particles, or more
generally, two spherically symmetric objects, has a strength of
G*M1*M2/r^2 where r is the distance between their centers. The farther
the two objects are from each other, the weaker their gravitaitonal
force gets.
If your balance was so big that the end of one arm was noticeably
farther away from the center of the earth than the end of the other
arm, you would expect a difference in the gravitational forces on
identical masses. Most balances are not this big, but a similar
argument explains why there are tides on the earth. The earth is big
enough that the gravitational pull of the moon (and the sun) is
different enough from one side to the other that water sloshes around
in the tides (we should have a more complete explanation elsewhere on
this site).
The book probably said that you can neglect the spatial variation
of the gravitational field strength if the radius of the earth is
*much* longer than the arms of the balance.
Tom
(published on 10/22/2007)
