The explanation which has been given above of the reasons why a pendulum swings backwards and forwards in a given time independently of the length of the arc through which it swings, that is to say of the amount by which it sways from side to side, is only approximate, because in the proof we assumed that the arc of swing and the line _F B_ were equal, which is not really and exactly true. Galileo never got at the real solution, though he tried hard. It was reserved for another than he to find the true path of an isochronous pendulum and completely to determine its laws. Huygens, a Dutch mathematician, found that the true path in which a pendulum ought to swing if it is to be really isochronous is a curve called a cycloid, that is to say the curve which is traced out by a pencil fixed on the rim of a hoop when the hoop is rolled along a straight ruler. It is the curve which a nail sticking out of the rim of a waggon wheel would scratch upon a wall. I will not go into the mathematical proof of this. Clocks are not made with cycloidal pendulums, because when the arc of a pendulum is small the swing is so very near a cycloid as to make no appreciable difference in time-keeping. I am now glad to be able to say that I have dealt with all the mathematics that is necessary to enable the mechanism of a clock to be understood. It all leads up to this:— (1) A harmonic motion is one in which the accelerating force increases with the distance of the body from some fixed point. (2) Bodies moving harmonically make their swings about this point in equal times. (3) A spring of any sort or shape always has a restitutional force proportional to the displacement. (4) And therefore masses attached to springs vibrate in equal times however large the vibration may be. (5) The bob of a pendulum, oscillating backwards and forwards, acts like a weight under the influence of a spring, and is therefore isochronous. (6) The time of vibration of a pendulum is uninfluenced by changes in the weight of the bob, but is influenced by changes in the length of the pendulum rod. The time of vibration of a mass attached to a spring is influenced by changes in the mass. We have now to deal with the application of these principles to clocks and watches. Clocks had been known before the time of Galileo, and before the invention of the pendulum. They had what is known as balance, or verge escapements. Strictly in order of time I ought to explain them here. But I will not do so. I will go on to describe the pendulum clock, and then I will go back and explain the verge escapement, which, we shall see, is really a sort of huge watch of a very imperfect character.