On several manuscript pages Galileo explored the the "Law of the Pendulum" in the context of his theorie of motion and, in particular, its relation to the law of fall. In the first Day of the Discorsi he expresses the "Law of the Pendulum" as follows:

As to the times of vibration of bodies suspended by threads of different lengths, they bear to each other the same proportion as the square roots of the lengths of the thread; or one might say the lengths are to each other as the squares of the times; so that if one wishes to make the vibration-time of one pendulum twice that of another, he must make its suspension four times as long.