We may now go to the front (or left hand) of the clock, and describe the dial or " motion-work." The minute hand fits on to a squared end of a brass socket, which is fixed to the wheel M, and fits close, but not tight, on the pro longed arbor of the centre wheel. Behind this wheel is a bent spring which is (or ought to be) set on the same arbor with a square hole (not a round one as it sometimes is) in the middle, so that it must turn with the arbor; the wheel is pressed up against this spring, and kept there, by a cap and a small pin through the end of the arbor. The consequence is, that there is friction enough between the spring .and the wheel to carry the hand round, but not enough to resist a moderate push with the finger for the purpose of altering the time indicated. This wheel M, which is sometimes called the minute-wheel, but is better called the hour-wheel as it turns in an hour, drives another wheel N, of the same number of teeth, which has a pinion attached to it; and that pinion drives the twelve-hour tcheel H, which is also attached to a large socket or pipe carrying the hour hand, and riding on the former socket, or rather (in order to relieve the centre arbor of that extra weight) on an intermediate socket fixed to the bridge L, which is screwed to the front plate over the hour-wheel M. The weight W, which drives the train and gives the impulse to the pendu lum through the escapement, is generally hung by a catgut line passing through a pulley attached to the weight, the other end of the cord being tied to some convenient place in the clock frame or seat-board, to which it is fixed by screws through the lower pillars. It has usually been the practice to make the case of house clocks and astronomical clocks not less than 6 feet high ; but that is a very unnecessary waste of space and materials ; for by either diminishing the size of the barrel, or the number of its turns, by increasing the size of the great wheel by one-half, or hanging the weights by a treble instead of a double line; a case just long enough for the pendulum will also be long enough for the fall of the weights in 7? or 8 days. Of courso the weights have to be increased in the same ratio, and indeed ratharmore, to overcome the increased friction; but that is of no consequence.
Pendulum.
The claim to the invention of the pendulum, like the claim to most inventions, is disputed ; and we have no intention of trying to settle it. It was, like many other discoveries and inventions, probably made by various persons independently, and almost simul taneously, when the state of science had become ripe for it. The discovery of that peculiarly valuable property of the pendulum called isochronism, or the disposition to vibrate different arcs in very nearly the same time (provided the arcs are none of them large), is commonly attributed to Galileo, in the well-known story of his being struck with the isochronism of a chandelier hung by a long chain from the roof of the church, at Florence. And Galileo s son appears as a rival of Avicenna, Huyghens, Dr Hooke, and a London clockmaker named Harris, for the honour of having first applied the pendulum to regulate the motion of a clock train, all in the early part of the 17th century. Be this as it may, there seems little doubt that Huyghens was the first who mathematically investi gated, and therefore really knew, the true nature of those properties of the pendulum which may now be found explained in any mathe matical book on mechanics. He discovered that if a simple pen dulum (i.e., a weight or lob consisting of a single point, and hung by a rod or string of no weight) can be made to describe, not a circle, but a cycloid of which the string would be the radius of cur vature at the lowest point, all its vibrations, however large, will be performed in the same time. For a little distance near the bottom, the circle very nearly coincides with the cycloid ; and hence it is that, for small arcs, a pendulum vibrating as usual in a circle is nearly enough isochronous for the purposes of horology ; more espe cially when contrivances are introduced either to compensate for the variations of the arc, or, better still, to destroy them altogether, by making the force on the pendulum so constant that its arc may never sensibly vary.
The difference between the time of any small arc of the circle an any arc of the cycloid varies nearly as the square of the circular arc ; and again, the difference between the times of any two smal and nearly equal circular arcs of the same pendulum, varies nearly as the arc itself. If a, the arc, is increased by a small amount da, the pendulum, will lose IQSOOada seconds a day, which is rather more than 1 second, if a is 2 (from zero) and da is 10 , since the numerical value of 2 is 035. If the increase of arc is considerable, t will not do to reckon thus by differentials, but we must take the difference of time for the day as 5400 (a, 2 a 2 ), which will be j ist seconds if a ? 2 and a t 6. For many years it wag thought of great importance to obtain cycloidal vibrations of clock pendulums, ind it was done by making the suspension string or spring vibrate Between cycloidal checks, as they were called. But it was in time discovered that all this is a delusion, first, because there is and can be no such thing in reality as a simple pendulum, and cycloidal cheeks will only make a simple pendulum vibrate isochronously ; secondly, because a very slight error in the form of the cheeks (as Huyghens himself discovered) would do more harm than the circular error uncorrected, even for an arc of 1 0, which is much larger than the common pendulum arc ; thirdly, because there was always some friction or adhesion between the cheeks and the string ; and fourthly (a reason which applies equally to all the isochronous contrivances since invented), because a common clock escapement itself generally tends to produce an error exactly opposite to the circular error, or to make the pendulum vibrate quicker the farther it swings ; and therefore the circular error is actually useful for the purpose of helping to counteract the error due to the escapement, and the clock goes better than it would with, a simple pendulum, describing the most perfect cycloid. At the same time, the thin spring by which pendulums are always suspended, except in some French clocks where a silk string is used (a very inferior plan), causes the pendulum to deviate a little from circular and to approxi mate to cycloidal motion, because the bend does not take place at one point, but is spread over some length of the spring.
The accurate performance of a clock depends so essentially on the pendulum, that we shall go somewhat into detail respecting it. First then, the time of vibration depends entirely on the length of the pendulum, the effect of the spring being too small for considera tion until we come to differences of a higher order. But the time does not vary as the length, but only as the square root of the length ; i.e., a pendulum to vibrate two seconds must be four- times as long as a seconds pendulum. The relation between the time of vibration and the length of a pendulum is expressed thus : t TT/-, where t is the time in seconds, it the well-known symbol for 3 141 59, the ratio of the circumference of a circle to ita diameter, I the length of the pendulum, and g the force of gravity at the latitude where it is intended to vibrate. This letter g, in the latitude of London, is the symbol for 32 2 feet, that being the velocity (or number of feet per second) at which a body is found by experiment to be moving at the end of the first second of its fall, being necessarily equal to twice the actual number of feet it has fallen in that second. Consequently, the length of a pendulum to beat seconds in London is 39 14 inches. But the same pendulum carried to the equator, where the force of gravity is less, would lose 2J minutes a day.
The seconds we are here speaking of are the seconds of. a common clock indicating mean solar time. But as clocks are also required for sidereal time, it may be as well to mention the proportions between a mean and a sidereal pendulum. A sidereal day is the interval between two successive transits over the meridian of a place by that imagin ary point in the heavens called T, the first point of Aries, at the intersection of the equator and the ecliptic ; and there is one more sidereal day than there are solar days in a year, since the earth has to turn more than once round iii space before the sun can coma a second time to the meridian, on account of the earth s own motion in its orbit during the day. A sidereal day or hour is shorter than a mean solar one in the ratio of 99727, and consequently a sidereal pendulum must be shorter than a mean time pendulum in the squaro of that ratio, or in the latitude of London the sidereal seconds pen dulum is 38 87 inches. As we have mentioned what is or 24 o clock by sidereal time, we may as well add, that the mean day is also reckoned in astronomy by 24 hours, and not from midnight as in civil reckoning, but from the following noon ; thus, what wo call 11 A.M. May 1 in common life is 23 h. April 30 with astronomers.
It must be remembered that the pendulums whose lengths _we have been speaking of are simple pendulums ; and as that is a thing which can only exist in theory, the reader may ask how the length of a real pendulum to vibrate in any required time is ascertained. In every pendulum, that is to say, in every body hung so as to be capable of vibrating freely, there is a certain point, always some where below the centre of gravity, which possesses these remarkable properties that if the pendulum were turned upside down, and set vibrating about this point, it would vibrate in the same time as before, and moreover, that the distance of this point from the point of suspension is exactly the length of that imaginary simple pendulum which would vibrate in the same time. This point is therefore , called the centre of oscillation. The rules for finding it by calcula-
