But in the case of gravity, by altering the mass, you thereby proportionally alter the attraction on it, and therefore the time of swing is unaltered.
Fig. 39.
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.