Experiments were made also with a ball rolling down a curved
gutter, the curve representing the are of a circle. These
experiments led to the study of the curvilinear motions of a
weight suspended by a cord; in other words, of the pendulum.

Regarding the motion of the pendulum, some very curious facts
were soon ascertained. Galileo found, for example, that a
pendulum of a given length performs its oscillations with the
same frequency though the arc described by the pendulum be varied
greatly.[1] He found, also, that the rate of oscillation for
pendulums of different lengths varies according to a simple law.
In order that one pendulum shall oscillate one-half as fast as
another, the length of the pendulums must be as four to one.
Similarly, by lengthening the pendulums nine times, the
oscillation is reduced to one-third, In other words, the rate of
oscillation of pendulums varies inversely as the square of their
length. Here, then, is a simple relation between the motions of
swinging bodies which suggests the relation which Kepler bad
discovered between the relative motions of the planets. Every
such discovery coming in this age of the rejuvenation of
experimental science had a peculiar force in teaching men the
all-important lesson that simple laws lie back of most of the
diverse phenomena of nature, if only these laws can be
discovered.

Galileo further observed that his pendulum might be constructed
of any weight sufficiently heavy readily to overcome the
atmospheric resistance, and that, with this qualification,
neither the weight nor the material had any influence upon the
time of oscillation, this being solely determined by the length