The most important of Galileo's discoveries, published in Sidereus Nuncius, was that of Jupiter's satellites that represented “a notable and splendid argument to remove the scruples of those who can tolerate the revolution of the planets round the Sun in the Copernican system, yet are so disturbed by the motion of one Moon about the Earth, (...) that they consider that this theory of the constitution of the universe must be upset as impossible; for now we have (...) four satellites circling about Jupiter, like the Moon about the Earth, while the whole system travels over a mighty orbit about the Sun in the space of twelve years.” [From The Sidereal Messenger of Galileo Galilei, translated by Edward Stafford Carlos, 1880]
Galileo's interest in studying the movements of Jupiter's satellites came from the thought that the knowledge of the satellites’ positions can be read in reverse over time: from the position of the satellites, it can be traced back to the time, i.e. to the time of the observation place, necessary for determining longitudes at sea, a problem still unsolved in that period. While the latitude of a point can be found immediately by measuring the altitude of the North Star from the horizon, for the longitude, it was necessary to know the time on board and the time at that same moment at the port of departure (whose longitude was known). The difference in the two values gave the difference in longitude between the two places: the port of reference and the ship.
As a solution to the problem of longitude at sea was vital, the great naval powers offered generous rewards to anyone who could find a solution. (A solution was found in the XVIII century with the invention of marine chronometers.) Galileo presented his solution to the problem and paved the way for the creation of the jovilabe.
In 1636, Galileo decided to participate in the competition for the 30,000-florin prize announced by the States General of the United Provinces of the Netherlands for the invention of a method capable of determining longitude at sea. To the ephemerides of Jupiter’s satellites—his “celestial clock”—he added a device based on the isochronism of a pendulum’s oscillations that was capable of maintaining the local time between two successive observations of Jupiter's satellites. In 1641, he improved the device by adding a pinwheel escapement. His son Vincenzo made a sketch for a model later elaborated by Viviani.
The development and refinement of these ideas were then thanks to Christiaan Huygens.
The pendulum is a device derived from a plumb line: a weight is connected to a support and made to oscillate. The amplitude of the oscillations varies, continuously decreasing. However, the time it takes to complete a full swing is roughly constant and is known as the “period of a pendulum”. It is proportional to the length of the wire that supports the weight. Galileo was the first to establish the law associating this length to the period and to exploit the pendulum’s features to adjust the escapement of a watch.