683. In a good clock an extremely minute force need only be supplied to the pendulum, so that, notwithstanding 86,400 vibrations have to be performed daily, one winding of the clock will supply sufficient energy to sustain the motion for a week.
THE HANDS.
684. We shall explain by the model shown in [Fig. 101], how the hour-hand and the minute-hand are made to revolve with different velocities about the same dial.
Fig. 101.
g is a handle by which I can turn round the shaft which carries the wheel f, and the hand b. There are 20 teeth in f, and it gears into another wheel, e, containing 80 teeth; the shaft which is turned by e carries a third wheel d, containing 25 teeth, and d works with a fourth c, containing 75 teeth, c is capable of turning freely round the shaft, so that the motion of the shaft does not affect it, except through the intervention of the wheels e, f, and d. To c another hand a is attached, which therefore turns round simultaneously with c. Let us compare the motion of the two hands a and b. We suppose that the handle g is turned twelve times; then, of course, the hand b, since it is on the shaft, will turn twelve times. The wheel f also turns twelve times, but e has four times the number of teeth that a has, and therefore, when f has gone round four times, e will only have gone round once: hence, when f has revolved twelve times, e will have gone round three times. d turns with e, and therefore the twelve revolutions of the handle will have turned d round three times, but since c has 75 teeth and d 25 teeth, c will have only made one revolution, while d has made three revolutions; hence the hand a will have made only one revolution, while the hand b has made twelve revolutions.
We have already seen ([Art. 681]) how, by a train of wheels, one wheel can be made to revolve once in an hour. If that wheel be upon the shaft instead of the handle g, the hand b will be the minute-hand of the clock, and the hand a the hour-hand.
685. The adjustment of the numbers of teeth is important, and the choice of wheels which would answer is limited. For since the shafts are parallel, the distance between the centres of f and e must equal that between the centres of c and of d. But it is evident that the distance from the centre of f to the centre of e is equal to the sum of the radii of the wheels f and e. Hence the sum of the radii of the wheels f and e must be equal to the sum of the radii of c and d. But the circumferences of circles are proportional to their radii, and hence the sum of the circumferences of f and e must equal that of c and d; it follows that the sum of the teeth in e and f must be equal to the sum of the teeth in c and d. In the present case each of these sums is one hundred.
686. Other arrangements of wheels might have been devised, which would give the required motion; for example, if f were 20, as before, and e 240, and if c and d were each equal to 130, the sum of the teeth in each pair would be 260. e would only turn once for every twelve revolutions of f, and c and d would turn with the same velocity as e; hence the motion of the hand a would be one-twelfth that of b. This plan requires larger wheels than the train already proposed.