157. Another of Huygens’s discoveries revolutionised the art of exact astronomical observation. This was the invention of the pendulum-clock (made 1656, patented in 1657). It has been already mentioned how the same discovery was made by Bürgi, but virtually lost (see chapter V., § 98); and how Galilei again introduced the pendulum as a time-measurer (chapter VI., § 114). Galilei’s pendulum, however, could only be used for measuring very short times, as there was no mechanism to keep it in motion, and the motion soon died away. Huygens attached a pendulum to a clock driven by weights, so that the clock kept the pendulum going and the pendulum regulated the clock.[96] Henceforward it was possible to take reasonably accurate time-observations, and, by noticing the interval between the passage of two stars across the meridian, to deduce, from the known rate of motion of the celestial sphere, their angular distance east and west of one another, thus helping to fix the position of one with respect to the other. It was again Picard (§ 155) who first recognised the astronomical importance of this discovery, and introduced regular time-observations at the new Observatory of Paris. 158. Huygens was not content with this practical use of the pendulum, but worked out in his treatise called _Oscillatorium Horologium_ or _The Pendulum Clock_ (1673) a number of important results in the theory of the pendulum, and in the allied problems connected with the motion of a body in a circle or other curve. The greater part of these investigations lie outside the field of astronomy, but his formula connecting the time of oscillation of a pendulum with its length and the intensity of gravity[97] (or, in other words, the rate of falling of a heavy body) afforded a practical means of measuring gravity, of far greater accuracy than any direct experiments on falling bodies; and his study of circular motion, leading to the result that a body moving in a circle must be acted on by some force towards the _centre_, the magnitude of which depended in a definite way on the speed of the body and the size of the circle,[98] is of fundamental importance in accounting for the planetary motions by gravitation.