We have examined the action of a body under the accelerating or speed-quickening force due to gravity, the attractive force of which on any body is always proportional to the mass of that body. Let us now consider another form of acceleration.

Take the case of a strip of indiarubber. If pulled it resists and tends to spring back. The more I pull it out the harder is the pull I have to exert. This is true of all springs. It is true of spiral springs, whether they are pulled out or pushed in, and in each case the amount by which the spring is pulled out or pushed in is proportional to the pressure. This law is called Hooke’s law. It was expressed by him in Latin, “Ut tensio, sic vis”: “As the extension, so the force.” It is true of all elastic bodies, and it is true whether they are pulled out or pushed in or bent aside. The common spring balance is devised on this principle. The body to be weighed is hung on a hook suspended from a spring. The amount by which the spring is pulled out is a measure of the weight of the body. If you take a fishing rod and put the butt end of it on a table and secure it by putting something heavy on the end, then the tip will bend down on account of its own weight. Mark the point to which it goes. Now, if you hang a weight on the tip, the tip will bend down a little further. If you put double the weight the tip will go down double the distance, and so on until the fishing rod is considerably bent, so that its form is altered and a new law of flexure comes into play. Suppose I use a spring as an accelerating force. For example, suppose I suspend a heavy ball by a string and then attach a spiral spring to it and pull the spring aside. The ball will be drawn after the spring. If then I let the ball go, it will begin to move. The force of the spring will act upon it as an accelerating force, and the ball will go on moving quicker and quicker. But the acceleration will not be like that of gravity. There will be two differences. The pull of the spring will in no way depend on the mass of the ball, and the pull of the spring, instead of being constant, like the pull of gravity, will become weaker and weaker as the ball yields to it. Consequently the equations above given which determine the relations between this space passed through, the velocity, and the time which were determined in the case of gravity are no longer true, and a different set of relations has to be determined. This can be easily done by mathematics. But I do not propose to go into it. I prefer to offer a rough and ready explanation, which, though it does not amount to a proof, yet enables us to accept the truth that can be established both by experiment and by calculation.

Let a heavy ball (A) be suspended by a long string, so that the action of gravity sideways on the ball is very small and may be neglected, and to each side attach an indiarubber thread fastened at B and C. Then when the ball is pulled aside a little, say to a position D, it will tend to fly back to A with a force proportioned to the distance A D. What will be the time it will take to do this? If the distance A D is small, the ball has only a small distance to go, but then, on the other hand, it has only small forces acting on it. If the distance A D is bigger, then it has a longer distance to go, but larger forces to urge it. These counteract one another, so that the time in each case will be the same.

The question is this:—Will you go a long distance with a powerful horse, or a small distance with a weak horse? If the distance in each case is proportioned to the power of the horse, then the amount of the distance does not matter. The powerful horse goes the long distance in the same time that the weak horse goes the short distance. And so it is here. However far you pull out the spring, the accelerative pull on the ball is proportioned to the distance. But the time of pulling the ball in depends on the distance. So that each neutralises the other. Whence then we have this most important fact, that springs are all isochronous; that is to say, any body attached to any spring whatever, whether it is big or small, straight or curly, long or short, has a time of vibration quite independent of the bigness of the vibration. The experiment is easy to try with a ball mounted on a long arm that can swing horizontally. It is attached on each side to an elastic thread. If pulled aside, it vibrates, but observe, the vibration is exactly the same whether the bigness of the vibration is great or small. If the pull aside is big, the force of restitution is big; if the pull is small, the force of restitution is small. In one case the ball has a longer distance to go, but then at all points of its path it has a proportionally stronger force to pull it; if the ball has a smaller distance to go, then at all the corresponding points of its path it has a proportionally weaker force to pull it. Thus the time remains the same whether you have the powerful horse for the long journey or the weaker horse for the smaller journey.

Next take a short, stiff spring of steel. One of the kind known as tuning forks may be employed.

The reader is probably aware that sounds are produced by very rapid pulsations of the air. Any series of taps becomes a continuous sound if it is only rapid enough. For example, if I tap a card at the rate of 264 times in a second, I should get a continuous sound such as that given by the middle C note of the piano. That, in fact, is the rate at which the piano string is vibrating when C is struck, and that vibration it is that gives the taps to the air by which the note is produced.

This can be very easily proved. For if you lift up the end of a bicycle and cause the driving wheel to spin pretty rapidly by turning the pedal with the hand, then the wheel will rotate perhaps about three times in a second. If a visiting card be held so as to be flipped by the spokes as they fly by, since there are about thirty-six of them, we should get a series of taps at the rate of about 108 a second. This on trial will be found to nearly correspond to the note A, the lowest space on the bass clef of music. As the speed of rotation is lowered, the tone of the note becomes lower; if the speed is made greater, the pitch of the note becomes higher, and the note more shrill. However far or near the card is held from the centre of the wheel makes no difference, for the number of taps per second remains the same. So, again, if a bit of watch-spring be rapidly drawn over a file, you hear a musical note. The finer the file, and the more rapid the action, the higher the note. The action of a tuning fork and of a vibrating string in producing a note depends simply on the beating of the air. The hum of insects is also similarly produced by the rapid flapping of their wings.

It is an experimental fact that when a piano note is struck, as the vibration gradually ceases the sound dies away, but the pitch of the note remains unchanged. A tune played softly, so that the strings vibrate but little, remains the same tune still, and with the same pitch for the notes.

A “siren” is an ingenious apparatus for producing a series of very rapid puffs of air. It consists of a small wheel with oblique holes in it, mounted so as to revolve in close proximity to a fixed wheel with similar holes in it. If air be forced through the wheels, by reason of the obliquity of the orifices in the movable wheel it is caused to rotate. As it does so, the air is alternately interrupted and allowed to pass, so that a series of very rapid puffs is produced. As the air is forced in, the wheel turns faster and faster. The rapidity of succession of the puffs increases so that the note produced by them gradually increases in pitch till it rises to a sort of scream. For steamers these “sirens” are worked by steam, and make a very loud noise.

It is, however, impossible to make a tuning fork or a stretched piano spring alter the pitch of its note without altering the elastic force of the spring by altering its tension, or without putting weights on the arms of the tuning fork to make it go more slowly. And this is because the tuning fork and the piano spring, being elastic, obey Hooke’s law, “As the deflection, so the force”; and therefore the time of back spring is in each case invariable, and the pitch of the note produced therefore remains invariable, whatever the amplitude of the vibration may be.

Upon this law depends the correct going of both clocks and watches.

Wonderful nature, that causes the uniformity of sounds of a piano, or a violin, to depend on the same laws that govern the uniform going of a watch! Nay, more, all creation is vibrating. The surge of the sea upon the coast that swishes in at regular intervals, the colours of light, which consist of ripples made in an elastic ether, which springs back with a restitutional force proportioned to its displacement, all depend upon the same law. This grand law by which so many phenomena of nature are governed has a very beautiful name, which I hope you will remember. It is called “harmonic motion,” by which is meant that when the atoms of nature vibrate they vibrate, like piano strings, according to the laws of harmony. The ancient Pythagorean philosophers thought that all nature moved to music, and that dying souls could begin to hear the tones to which the stars moved in their orbits. They called it, as you know, the music of the spheres. But could they have seen what science has revealed to man’s patient efforts, they would have seen a vision of harmony in which not a ray of light, not a string of a musical instrument, not a pipe of an organ, not an undulation of all-pervading electricity, not a wing of a fly, but vibrates according to the law of harmony, the simple easy law of which a boy’s catapult is the type, and which, as we have seen, teaches us that when an elastic body is displaced the force of restitution, in other words, the force tending to restore it to its old position, is proportional to the displacement, and the time of vibration is uniform. The last is the important thing for us; we seem to get a gleam of a notion of how the clock and watch problem is going to be solved.

But before we get to that we have yet to go back a little.

About the year 1580 an inattentive youth (it was our friend Galileo again) watched the swing of one of the great chandeliers in the cathedral church at Pisa. The chandeliers have been renewed since his day, it was one of the old lamps that he watched. It had been lit, and allowed to swing through a considerable space. He expected that as it gradually came to rest it would swing in a quicker and quicker time, but it seemed to be uniform. This was curious. He wanted to measure the time of its swing. For this purpose he counted his pulse-beats. So far as he could judge, there were exactly the same number in each pendulum swing.

This greatly interested him, and at home he began to try some experiments. As he got older his attention was repeatedly turned to that subject, and he finally established in a satisfactory way the law that, if a weight is hung to the end of a string and caused to vibrate, it is isochronous, or equal-timed, no matter what the extent of the arc of vibration.

The first use of this that he made was to make a little machine with a string of which you could vary the length, for use by doctors. For the doctors of that day had no gold watch to pull out while with solemn face they watched the ticks. They were delighted with the new invention, and for years doctors used to take out the little string and weight, and put one hand on the patient’s pulse while they adjusted the string till the pendulum beat in unison with the pulse. By observing the length of the string, they were then able to tell how many beats the pulse made in a minute. But Galileo did not stop there. He proceeded to examine the laws which govern the pendulum.

We will follow these investigations, which will largely depend on what we have already learned.

Before, however, it is possible to understand the laws which govern the pendulum, there are one or two simple matters connected with the balance and operation of forces which have to be grasped.

Suppose that we have a flat piece of wood of any shape like Fig.34, and that we put a screw through any spot A in it, no matter where, and screw it to a wall, so that it can turn round the screw as round a pivot.

Next we will knock a tintack into any point B, and tie a string on to B. Then if I pull at the string in any direction B C the board tends to twist round the screw at A. What will the strength of the twisting force be? It will depend on the strength of the pull, and on the “leverage,” or distance of the line C B from A. We might imagine the string, instead of being attached at B, to be attached at D; then, if I put P as the strength of the pull, the twisting power would be represented by P × A D. This is called the “moment” of the force P round the centre A. It would be the same as if I had simply an arm A D, and pulled upon it with the force P. It is an experimental truth, known to the old Greek philosophers, that moments, or twisting powers, are equal when in each case the result of multiplying the arm by the power acting at right angles to it is equal.

Now suppose A B is a pendulum, with a bob B of 10 lbs. weight, and suppose it has been drawn aside out of the vertical so that the bob is in the position B. Then the weight of the bob will act vertically downwards along the line B C. The moment, or twisting power, of the weight will be equal to 10 lbs. multiplied by A D, A D being a line perpendicular to B C.

Now suppose that another string were tied to the bob B, and pulled in a direction at right angles to A B, with a force P just enough to hold the bob back in the position B. The pull along D B × A B would be the moment of that pull round the point A. But, because this moment just holds the pendulum up, it follows that the moment of the weight of the pendulum round A is equal to the moment of the pull of the string B D round A.

Whence P × A B = 10 lbs. × A D.

Whence P = 10 lbs. × (A D)/(A B). But A B is always the same, whatever the side deflection or displacement of the pendulum may be. Whence then we see that when a pendulum is pulled aside a distance E B (which is always equal to A D), then the force tending to bring it back to E is always proportional to E B. But if the pendulum be fairly long, say 39-1/7 inches, and the displacement E B be small,—in other words, if we do not drag it much out of the vertical,—then we may say that the force tending to bring it back to F, its position of rest, is not very different from the force tending to bring it back to E. But F B is the “displacement” of the pendulum, and, therefore, we find that when a pendulum is displaced, or deflected, or pulled aside a little, the amount of the deflection is always very nearly proportional to the force which was used to produce the deflection. This important law can be verified by experiment. If C is a small pulley, and B C a string attached to a pendulum A B whose bob is B. Then if a weight D be tied to the string and passed over a pulley C, the amount F B by which the weight D will deflect the bob B is almost exactly proportional to D, so long as we only make the deflection E B small, that is two or three inches, where say 39-1/7 inches is the length A B of the pendulum.

If F B is made too big, then the line B F can no longer be considered nearly equal to the arc of deflection E B, and the proposition is no longer true. Hence then, both by experiment and on theory, we find that for small distances the displacement of a pendulum bob is approximately equal to the force by which that displacement is produced.

But if so, then from what has gone before, we have an example of harmonic motion. The weight of the bob, tending to pull the bob back to E, acts just as an elastic band would act, that is to say pulls more strongly in proportion as the distance F B is bigger. In fact, if we could remove the force of gravity still leaving the mass B of the pendulum bob, the force of an elastic band acting so as to tend to pull the bob back to rest might be used to replace it. It would be all one whether the bob were brought back to rest by the downward force of its own gravity, or by the horizontal force of a properly arranged elastic band of suitable length.

But the motion of the bob, under the influence of the pull of an elastic band where the strain was always proportional to the displacement, would, as we have seen, be harmonic motion, and performed in equal times whatever the extent of the swing. Whence then we conclude that if the swings of a pendulum are not too big, say not exceeding two and a half inches each way, the motion may be considered harmonic motion, and the swings will be made in equal times whether they are large or small ones. In other words, a clock with a 39-1/7 inch pendulum and side swing on each side if not over two inches will keep time, whatever the arc of swing may be.

This may be verified experimentally. Take a pendulum of wood 39-1/7 inches long, and affix to its end a bob of 10 lbs. weight. The pendulum will swing once in each second. To pull it aside two inches we should want a weight such that its moment about the point of support was equal to the moment of the force of gravity acting on the bob, about the point of support. In other words, the weight required × 39-1/7 inches = 10 lbs. × 2 inches. Whence the weight required = 1/2 lb. (nearly).

Now fix a similar pendulum A B 39-1/7 inches long, horizontally, with a weight B of 10 lbs. on it. Fasten it to a vertical shaft C D, with a tie rod of wire or string A B so as to keep it up, and attach to each side of the rod A B elastic threads E F and E G. Let these threads be tied on at such a point that when B is pulled aside two inches the force tending to bring it back to rest is half a pound. Then if set vibrating the rod will swing backwards and forwards in equal times, no matter how big, the arc of vibration (provided the arc is kept small), and the time of oscillation will be that of a pendulum, namely, one swing in a second. In fact, whether you put A B vertically and let it swing on the pivots C and D by the force of gravity, or put it horizontally, and thus prevent gravity acting on it, but make it swing under the accelerating influence of a pair of elastic bands so arranged as to be equivalent to gravity, in each case it will swing in seconds.

It is this curious property of the circle that makes the vertical force of gravity on a pendulum pull it as though it were a horizontally acting elastic band; that is the reason why a pendulum is equal-time-swinging, or, as it is called, isochronous, from two Greek words that mean “the same” and “time.”

But it must be remembered that this equal swinging is only approximate, and only true when the arc of vibration is small.

Here then we have a proof which shows us that the pendulum of a clock and the balance wheel of a watch depend on exactly the same principles. They are each an example of harmonic motion.

The next question that arises is whether the weight of the pendulum has any influence upon the time of its vibration.

A little reflection will soon convince us that it has none. For we know that the time that bodies take to fall to the ground under the action of gravity is independent of the weight. A falling 2 lb. weight is only equivalent to two pound-weights falling side by side.

In the same way and by the same reasoning we might take two pendulums of equal length, and each with a bob weighing 1 lb. They would, if put side by side close together swing in equal times. But the time would be the same if they were fastened together, and made into one pendulum.

For inasmuch as the fall of a pendulum is due to gravity, and the action of gravity upon a body is proportional to its mass, it follows that in a pendulum the part of the gravitational force that acts upon each part of the mass is occupied in moving that mass, and the whole pendulum may be considered as a bundle of pendulums tied together and vibrating together.

The same would be the case with a pendulum vibrating under the influence of a spring. If you have two bobs and two springs, they will vibrate in the same time as one bob accelerated by one spring. In this case, however, the force of the one spring must be equal to the combined force of the two springs. In other words, the springs must be made proportional in strength to the masses.

Hence, then, you cannot increase the speed of the vibration of a pendulum by adding weight to the bob.

On the other hand, if you have a bob vibrating under the influence of a spring, like the balance wheel of a watch, then if you increase the bob without increasing the spring, since the mass to be moved has increased without a corresponding increase in the accelerating force acting on it, the time of swing will alter accordingly.

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.

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.

As soon as Galileo had discovered that pendulums were isochronous, that is, equi-time-swinging, he set to work to see whether he could not contrive to make a timepiece by means of them. This would be easy if only he could keep a pendulum swinging. When a pendulum is set swinging, it soon comes to rest. What brings it to rest? The resistance of the air and the friction of the pivots. Therefore what is obviously wanted is something to give it a kick now and then, but the thing must kick with discretion. If it kicked at the wrong time, it might actually stop the pendulum instead of keeping it going. You want something that, just as the pendulum is at one end and has begun to move, will give it a further push. Suppose that I have a swing and that I put a boy in it, and I swing him to and fro. I time my pushes. As he comes back against my hand I let him push it back, and then just as the swing turns I give it a further push. But I cannot stand doing that all day. I must make a machine to do it. Now what sort of a machine?

First, the machine must have a reservoir of force. I can’t get a machine to do work unless I wind it up, nor a man to do work unless I feed him, which is his way of being wound up. But then what do I want him to do? I want him, when I give him a push, to push me back harder. I want a reservoir of force such that when a pendulum comes back and touches it, the touch, like the pressure of the trigger of a gun, shall allow some pent-up power to escape and to drive the pendulum forward.

This is the case in a swing. Each time that the swing returns to my hands I give it a push, which serves to sustain the motion that would otherwise be destroyed by friction and the resistance of the air.

Such an arrangement, if it can be contrived mechanically, is called an “escapement.”

An arrangement of this kind was contrived by Galileo. He provided a wheel, as is here shown, with a number of pins round it. The pendulum A B has an arm A H attached to it, and there is a ratchet C D which engages with the pins. The ratchet has a projecting arm E F.

When the pendulum comes back towards the end of its beat, the arm A H strikes the arm E F, and raises the ratchet C D. This releases the wheel, which has a weight wound up upon it, and therefore at once tries to go round. The consequence is, that the pin G strikes upon the arm A H, and thus on its return stroke gives an impetus to the pendulum. As the pin G moves forward it slides on the arm A H till it slips over the point H. The wheel now being free, would fly round were it not that when the pendulum returned, and the arm A H was lowered, the ratchet had got into position again and its point D was ready to meet and stop the next pin that was coming on against it. At each blow of the pins against the pendulum a “tick” is made, at each blow of a pin against the ratchet a “tock” is sounded, so that as it moves the pendulum makes the “tick-tock” sound with which we are all familiar.

Hence then a clock consists of a wheel, or train of wheels, urged by a weight or spring, which strives continually to spin round, but its rotation is controlled by an escapement and pendulum, so contrived as only to allow it to go a step forward at regular equal intervals of time.

But this would make only a poor sort of escapement. For the mode of driving the pendulum adds a complication to the swing of the pendulum. Instead of the pendulum being simply under the accelerative force of gravity, it is also subjected to the acceleration of the pin G. This acceleration is not of the “harmonic” order. Hence so far as it goes it does not tend to assist in giving a harmonic motion to the pendulum, but, on the contrary, disturbs that harmonic motion. Besides this, the impulse of the pin is in practice not always uniform. For if the wheel is at the end of a train of wheels driven by a weight, though the force acting on it is constant, yet, as that force is transmitted through a train of wheels, it is much affected by the friction of the oil. And on colder days the oil becomes more coagulated, and offers greater resistance. Moreover, as will be explained more in detail afterwards, the fact that the impulse is administered by G at the end of the stroke of the pendulum is disadvantageous, as it interferes with the free play of the pendulum. From all these causes the above escapement is imperfect in character, and would not do where precision was required.

It is now time to return to the old-fashioned escapements which were in use before the time of Galileo. These consisted of a wheel called a crown wheel, with triangular teeth. On one side of this wheel a vertical axis was fitted, with projecting “pallets” e f. Across the axis a verge or rod e f was placed, fitted with a ball at each end. When the crown wheel attempted to move on, one of its teeth came in contact with a pallet. This urged the pallet forward, and thereby caused an impulse to be given to the axis, on which was mounted the verge, carrying the balls. These of course began to move under the acceleration of the force thus impressed upon the pallet. Meantime, however, the other pallet was moving in the opposite direction, and by the time the first pallet had been pushed so far that it escaped or slid past the tooth of the crown wheel, which was pressing upon it, the other pallet had come into contact with the tooth on the other side of the crown wheel. This tended to arrest the motion of the verge, to bring the balls to a standstill, and ultimately to impart a motion in a contrary direction to them.

Thus then the arrangement was that of a pendulum not acted on by gravity, for the balls neutralised one another. The pendulum was, however, not subjected to a harmonic acceleration, but alternately to a nearly uniform acceleration from A to B and B to A. As a result, therefore, the time of oscillation was not independent of the arc of swing, but varied according to it, as also according to the driving power of the crown wheel. At each stroke there was a considerable “recoil.” For as each tooth of the wheel came into play it was unable at first to overcome and drive back the pallet against which it was pressing, but, on the contrary, was for a time itself driven back by the pallet.

Of course, so long as the motions of the wheel and verge were exactly uniform, fair time was kept. But the least inequality of manufacture produced differences.

Nevertheless it was on this principle that clocks were made during the thirteenth, fourteenth and fifteenth centuries. They were mostly made for cathedrals and monasteries. One was put up at Westminster, erected out of money paid as a fine upon one of the few English judges who have been convicted of taking bribes.

The time of swing of these clocks depended entirely upon the ratio of the mass of the balls at the end of the verge as compared with the strength of the driving force by which the acceleration on the pallets was produced. They were very commonly driven by a spring instead of a weight. The spring consisted of a long strip of rather poor quality steel coiled up on a drum. As it unwound it became weaker, and thus the acceleration on the verge became weaker, and the clock went slower.

In order, therefore, to keep the time true, it became necessary to devise some arrangement by which the driving force on the crown wheel should be kept more constant.

This gave rise to the invention of the fusee. The spring was put inside a drum or cylindrical box. One end of the spring was fastened to an axis, which was kept fixed while the clock was going; the other was fastened to the inside of the drum. Round the drum a cord was wound, which, as the drum was moved by the spring, tended to be wound up on the surface of the drum. Owing to the unequal pull of the spring, this cord was pulled by the drum strongly at first, and afterwards more feebly. To compensate its action a conical wheel was provided, with a spiral path cut in it in such a way and of such a size and proportion that as the wheel was turned round by the pull of the drum the cord was on different parts of it, so that the leverage or turning power on it varied, becoming greater as the pull of the cord became weaker, and in such a ratio that one just compensated the other, and the turning power of the axle was kept uniform.

In this manner small table clocks were made which kept very tolerable time.

Huygens converted these clocks into pendulum clocks in a very simple manner. He removed one of the balls, lengthened the verge, and slightly increased the weight of the other ball. By this means, while the crown wheel still continued to drive the verge and remaining ball, the acceleration on that ball now no longer depended entirely on the force of the crown wheel. The acceleration and retardation were now almost entirely governed by the force of gravity on the remaining ball, and this acceleration was harmonic. The clock, therefore, was immensely improved as a time-keeper. Still, however, the acceleration remained partly due to the driving power, and this was partly non-harmonic and introduced errors.

Most of the old clocks were converted shortly after the time of Huygens. As there was in general no room for the pendulum inside the clock-case, they usually brought the axle on which the pallets were mounted outside the clock and made it vibrate in front of the face.

Many old clocks exist, of which the engraving in the frontispiece is an example, that have been thus converted. A true old verge escapement clock is now a rarity.

The type of escapement invented by Galileo never came into vogue for clocks, on account of its imperfections, except till after a long interval, when, with certain modifications, it became the basis of a new improvement at the hands of Sir George Airey.

The crown wheel fell into disuse and was replaced by the anchor escapement, which was employed in that popular and excellent timepiece used throughout the eighteenth and the early part of the nineteenth century, and is now known as “The Grandfather’s Clock.” It was after all the crown wheel in another shape. The wheel, however, was flattened out, the teeth being put in the same plane. This made it much easier to construct. The pallets were fixed on an axis, and were a little altered so as to suit the changed arrangement of the teeth. The pendulum was no longer hung on the axis which carried the pallets. A cause of a good deal of friction and loss of power was thus removed. The pendulum was hung from a strip of thin steel spring, which allowed it to oscillate, and which supported it without friction. This excellent manner of suspending pendulums is now universal. It enabled the pendulum to be made very heavy. The bob was usually some eight or nine pounds weight. By this means the acceleration on the pendulum was due almost entirely to gravity acting on the bob, and thus the motion of the pendulum became almost wholly harmonic. Whence it followed that variations in the pendulum swing became of secondary importance, and did not greatly alter the going of the clock.

Therefore when the wheels became worn, and the pivots choked with old oil and dust, the old clock still went on. If it showed a tendency to stop for want of power, a little more was added to the driving weight, and the clock kept as good time as ever.

The swing of the pendulum was by this escapement enabled to be made small, so that the arc of swing of the bob differed but little from a cycloid.

The secret of the time-keeping qualities of these old “Grandfather” clocks is the length of pendulum. This renders it possible to have but a small arc of oscillation, and therefore the motion is kept very nearly harmonic. For practical purposes nothing will even now beat these old clocks, of which one should be in every house. At present the tendency is to abolish them and to substitute American clocks with very short pendulums, which never can keep good time. They are made of stamped metal. When they get out of order no one thinks of having them mended. They are thrown into the ash-pit and a new one bought. In reality this is not economy.

Good “Grandfather” clocks are not now often made. The last place I remember to have seen them being manufactured is at Morez, in the district of the Jura. An excellent clock, enclosed in a dust-tight iron case, with a tall painted case of quaint old design, can be bought for about 55s. The wheels are well cut, and the internal mechanism very good.

I visited the town of Morez in the year 1893. The clock industry was declining. The farmers of France seemed to prefer small clocks of hideous appearance, made in Germany and in America, to the excellent work of their own country. Probably by now the old clockmaking industry is extinct. One I purchased at that time has gone well ever since.