Time and Clocks: A Description of Ancient and Modern Methods of Measuring Time

by Sir Henry H. Cunynghame

Appendix.

To make a sun-dial, procure a circular piece of zinc, about ⅛ inch thick, and say twelve inches in diameter. Have a “style” or “gnomon” cast such that the angle of its edge equals the latitude of the place where the sun-dial is to be set up. This for London will be equal to 51° 30´´. A pattern may be made for this in wood; it should then be cast in gun-metal, which is much better for out-of-door exposure than brass. On a sheet of paper draw a circle A B C with centre O. Make the angle B O D equal to the latitude of the place for London = 51° 30´´. From A draw A E parallel to O B to meet O D in E, and with radius O E describe another circle about O. Divide the inner circle A B C into twenty-four parts, and draw radii through them from O to meet the larger circle. Through any divisions (say that corresponding to two o’clock) draw lines parallel to O B, O C, respectively to meet in a. Then the line O a is the shadow line of the gnomon at two o’clock. The lines thus drawn on paper may be transferred to the dial and engraved on it, or else eaten in with acid in the manner in which etchings are done.

Fig. 18.

The centre O need not be in the centre of the zinc disc, but may be on one side of it, so as to give better room for the hours, etc. A motto may be etched upon the dial, such as “Horas non numero nisi serenas,” or “Qual ’hom senza Dio, son senza sol io,” or any suitable inscription, and the dial is ready for use. It is best put up by turning it till the hour is shown truly as compared with a correctly timed watch. It must be levelled with a spirit level. It must be remembered that the sun does not move quite uniformly in his yearly path among the fixed stars. This is because he moves not in a circle, but in an ellipse of which the earth is in one of the foci. Hence the hours shown on the dial are slightly irregular, the sun being sometimes in advance of the clock, sometimes behind it. The difference is never more than a quarter of an hour. There is no difference at midsummer and midwinter.

Fig. 19.

Civil time is solar time averaged, so as to make the hours and days all equal. The difference between civil time and apparent solar time is called the equation of time, and is the amount by which the sun-dial is in advance of or in retard of the clock. In setting a dial by means of a watch, of course allowance must be made for the equation of time.

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CHAPTER II.

In the last chapter a short description has been given of the ideas of the ancients as to the nature of the earth and heavens. Before we pass to the changes introduced by modern science, it will be well to devote a short space to an examination of ancient scientific ideas.

All science is really based upon a combination of two methods, called respectively inductive and deductive reasoning. The first of these consists in gathering together the results of observation and experiment, and, having put them all together, in the formulation of universal laws. Having, for example, long observed that all heavy things tended to go towards the centre of the earth, we might conclude that, since the stars remain up in the sky, they can have no weight. The conclusion would be wrong in this case, not because the method is wrong, but because it is wrongly applied. It is true that all heavy things tend to go to the centre of the earth, but if they are being whirled round like a stone in a sling the centrifugal force will counteract this tendency. The first part of the reasoning would be inductive, the second deductive. All this reasoning consists, therefore, in forming as complete an idea as possible respecting the nature of a thing, and then concluding from that idea what the thing will do or what its other properties will be. In fact, you form correct ideas, or “concepts,” as they are called, and reason from them.

But the danger arises when you begin to reason before you are sure of the nature of your concepts, and this has been the great source of error, and it was this error that all men of science so commonly fell into all through ancient and modern times up to the seventeenth century.

Of course, if it were possible by mere observation to derive a complete knowledge of any objects, it would be the simplest method. All that would be necessary to do would be to reason correctly from this knowledge. Unfortunately, however, it is not possible to obtain knowledge of this kind in any branch of science.

The ancient method resembled the action of one who should contend that by observing and talking to a man you could acquire such a knowledge of his character as would infallibly enable you to understand and predict all his actions, and to take little trouble to see whether what he did verified your predictions.

The only difference between the old methods and the new is that in modern times men have learned to give far more care to the formation of correct ideas to start with, are much more cautious in arguing from them, and keep testing them again and again on every possible opportunity.

The constant insistence on the formation of clear ideas and the practice of, as Lord Bacon called it, “putting nature to the torture,” is the main cause of the advance of physical science in modern times, and the want of application of these principles explains why so little progress is being made in the so-called “humanitarian” studies, such as philosophy, ethics, and politics.

The works of Aristotle are full of the fallacious method of the old system. In his work on the heavens he repeatedly argues that the heavenly bodies must move in circles, because the circle is the most perfect figure. He affects a perplexity as to how a circle can at the same time be convex and also its opposite, concave, and repeatedly entangles his readers in similar mere word confusion.

Regarded as a man of science, he must be placed, I think, in spite of his great genius, below Archimedes, Hipparchus, and several other ancient astronomers and physicists.

His errors lived after him and dominated the thought of the middle ages, and for a long time delayed the progress of science.

The other great writer on astronomy of ancient times was Ptolemy of Alexandria.

His work was called the “Great Collection,” and was what we should now term a compendium of astronomy. Although based on a fundamental error, it is a thoroughly scientific work. There is none of the false philosophy in it that so much disfigures the work of Aristotle. The reasons for believing that the earth is at rest are interesting. Ptolemy argues that if the earth were moving round on its axis once in twenty-four hours a bird that flew up from it would be left behind. At first sight this argument seems very convincing, for it appears impossible to conceive a body spinning at the rate at which the earth is alleged to move, and yet not leaving behind any bodies that become detached from it.

On the other hand, the system which taught that the sun and planets moved round the earth, and which had been adopted largely on account of its supposed simplicity, proved, on further examination, to be exceedingly complicated. Each planet, instead of moving simply and uniformly round the earth in a circle, had to be supposed to move uniformly in a circle round another point that moved round the earth in a circle. This secondary circle, in which the planet moved, was called an epicycle. And even this more complicated view failed to explain the facts.

A system which, like that of Aristotle and Ptolemy, was based on deductions from concepts, and which consisted rather of drawing conclusions than of examining premises, was very well adapted to mediæval thought, and formed the foundation of astronomy and geography as taught by the schoolmen.

Fig. 20.

The poem of Dante accurately represents the best scientific knowledge of his day. According to his views, the centre of the earth was a fixed point, such that all things of a heavy nature tended towards it. Thus the earth and water collected round it in the form of a ball. He had no idea of the attraction of one particle of matter for another particle. The only conception he had of gravity was of a force drawing all heavy things to a certain point, which thus became the point round which the world was formed. The habitable part of the earth was an island, with Jerusalem in the middle of it J. Round this island was an ocean O. Under the island, in the form of a hollow cone, was hell, with its seven circles of torment, each circle becoming smaller and smaller, till it got down into the centre C. Heaven was at the opposite side H of the earth to Jerusalem, and was beyond the circles of the planets, in the primum mobile. When Lucifer was expelled from heaven after his rebellion against God, having become of a nature to be attracted to the centre of the earth, and no longer drawn heavenwards, he fell from heaven, and impinged upon the earth just at the antipodes of Jerusalem, with such violence that he plunged right through it to the centre, throwing up behind him a hill. On the summit of this hill was the Garden of Eden, where our first parents lived, and down the sides of the hill was a spiral winding way which constituted purgatory. Dante, having descended into hell, and passed the centre, found his head immediately turned round so as to point the other way up, and, having ascended a tortuous path, came out upon the hill of Purgatory. Having seen this, he was conducted to the various spheres of the planets, and in each sphere he became put into spiritual communion with the spirits of the blessed who were of the character represented by that sphere, and he supposes that he was thus allowed to proceed from sphere to sphere until he was permitted to come into the presence of the Almighty, who in the primum mobile presided over the celestial hosts.

The astronomical descriptions given by Dante of the rising and setting of the sun and moon and planets are quite accurate, according to the system of the world as conceived by him, and show not only that he was a competent astronomer, but that he probably possessed an astrolabe and some tables of the motions of the heavenly bodies.

Our own poet Chaucer may also be credited with accurate knowledge of the astronomy of his day. His poems often mention the constellations, and one of them is devoted to a description of the astrolabe, an instrument somewhat like the celestial globe which used to be employed in schools.

But with the revival of learning in Europe and the rise of freedom of thought, the old theories were questioned in more than one quarter.

It occurred to Copernicus, an ecclesiastic who lived in the sixteenth century, to re-examine the theory that had been started in ancient times, and to consider what explanation of the appearance of the heavenly bodies could be given on the hypothesis put forward by Pythagoras, that the earth moved round on its own axis, and also round the sun.

It may appear rather curious that two theories so different, one that the sun goes round the earth and the other that the earth goes round the sun, should each be capable of explaining the observed appearances of those bodies. But it must be remembered that motion is relative. If in a waltz the gentleman goes round the lady, the lady also goes round the gentleman. If you take away the room in which they are turning, and consider them as spinning round like two insects in space, who is to say which of them is at rest and which in motion? For motion is relative. I can consider motion in a train from London to York. As I leave London I get nearer to York, and I move with respect to London and York. But if both London and York were annihilated how should I know that I was in motion at all? Or, again, if, while I was at rest in the train at a station on the way, instead of the train moving the whole earth began to move in a southward direction, and the train in some way were left stationary, then, though the earth was moving, and the train was at rest, yet, so far as I was concerned, the train would appear to have started again on its journey to York, at which place it would appear to arrive in due time. The trees and hedges would fly by at the proper rate, and who was to say whether the train was in motion or the earth?

The theory of Copernicus, however, remained but a theory. It was opposed to the evidence of the senses, which certainly leads us to think that the earth is at rest, and it was opposed also to the ideas of some among the theologians who thought that the Bible taught us that the earth was so fast that it could not be moved. Therefore the theory found but little favour. It was in fact necessary before the question could be properly considered on its merits that more should be known about the laws of motion, and this was the principal work of Galileo.

The merit of Galileo is not only to have placed on a firm basis the study of mechanics, but to have set himself definitely and consciously to reverse the ancient methods of learning.

He discarded authority, basing all knowledge upon reason, and protested against the theory that the study of words could be any substitute for the study of things.

Alluding to the mathematicians of his day, “This sort of men,” says Galileo in a letter to the astronomer Kepler, “fancied that philosophy was to be studied like the ‘Æneid’ or ‘Odyssey,’ and that the true reading of nature was to be detected by the collating of texts.” And most of his life was spent in fighting against preconceived ideas. It was maintained that there could only be seven planets, because God had ordered all things in nature by sevens (“Dianoia Astronomica,” 1610); and even the discoveries of the spots on the sun and the mountains in the moon were discredited on the ground that celestial bodies could have no blemishes. “How great and common an error,” writes Galileo, “appears to me the mistake of those who persist in making their knowledge and apprehension the measure of the knowledge and apprehension of God, as if that alone were perfect which they understand to be so. But ... nature has other scales of perfection, which we, being unable to comprehend, class among imperfections.

“If one of our most celebrated architects had had to distribute the vast multitude of fixed stars over the great vault of heaven, I believe he would have disposed them with beautiful arrangements of squares, hexagons, and octagons; he would have dispersed the larger ones among the middle-sized or lesser, so as to correspond exactly with each other; and then he would think he had contrived admirable proportions; but God, on the contrary, has shaken them out from His hand as if by chance, and we, forsooth, must think that He has scattered them up yonder without any regularity, symmetry, or elegance.”

In one of Galileo’s “Dialogues” Simplicio says, “That the cause that the parts of the earth move downwards is notorious, and everyone knows that it is gravity.” Salviati replies, “You are out, Master Simplicio: you should say that everyone knows that it is called gravity; I do not ask you for the name, but for the nature, of the thing of which nature neither you nor I know anything.”

Too often are we still inclined to put the name for the thing, and to think when we use big words such as art, empire, liberty, and the rights of man, that we explain matters instead of obscuring them. Not one man in a thousand who uses them knows what he means; no two men agree as to their signification.

The relativity of motion mentioned above was very elegantly illustrated by Galileo. He called attention to the fact that if an artist were making a drawing with a pen while in a ship that was in rapid passage through the water, the true line drawn by the pen with regard to the surface of the earth would be a long straight line with some small dents or variations in it. Yet the very same line traced by the pen upon a paper carried along in the ship made up a drawing. Whether you saw a long uneven line or a drawing in the path that the pen had traced depended altogether on the point of view with which you regarded its motion.

Fig. 21.

But the first great step in science which Galileo made when quite a young professor at Pisa was the refutation of Aristotle’s opinion that heavy bodies fell to the earth faster than light ones. In the presence of a number of professors he dropped two balls, a large and a small one, from the parapet of the leaning tower of Pisa. They fell to the ground almost exactly in the same time. This experiment is quite an easy one to try. One of the simplest ways is as follows: Into any beam (the lintel of a door will do), and about four inches apart, drive three smooth pins so as to project each about a quarter of an inch; they must not have any heads. Take two unequal weights, say of 1 lb. and 3 lbs. Anything will do, say a boot for one and pocket-knife for the other; fasten loops of fine string to them, put the loops over the centre peg of the three, and pass the strings one over each of the side pegs. Now of course if you hitch the loops off the centre peg P the objects will be released together. This can be done by making a loop at the end of another piece of string, A, and putting it on to the centre peg behind the other loops. If the string be pulled of course the loop on it pulls the other two loops off the central peg, and allows the boot and the knife to drop. The boot and the knife should be hung so as to be at the same height. They will then fall to the ground together. The same experiment can be tried by dropping two objects from an upper window, holding one in each hand, and taking care to let them go together.

Fig. 22.

This result is very puzzling; one does not understand it. It appears as though two unequal forces produced the same effect. It is as though a strong horse could run no faster than a weaker one.

The professors were so irritated at the result of this experiment, and indeed at the general character of young Professor Galileo’s attacks on the time-honoured ideas of Aristotle, that they never rested till they worried him out of his very poorly paid chair at Pisa. He then took a professorship at Padua.

Let us now examine this result and see why it is that the ideas we should at first naturally form are wrong, and that the heavy body will fall in exactly the same time as the light one.

We may reason the matter in this way. The heavy body has more force pulling on it; that is true, but then, on the other hand there is more matter which has got to be moved. If a crowd of persons are rushing out of a building, the total force of the crowd will be greater than the force of one man, but the speed at which they can get out will not be greater than the speed of one man; in fact, each man in the crowd has only force enough to move his own mass. And so it is with the weights: each part of the body is occupied in moving itself. If you add more to the body you only add another part which has itself to move. A hundred men by taking hands cannot run faster than one man.

But, you will say, cannot a man run faster than a child? Yes, because his impelling power is greater in proportion to his weight than that of a child.

If it were the fact that the attraction of gravity due to the earth acted on some bodies with forces greater in proportion to their masses than the forces that acted on other bodies, then it is true that those different bodies would fall in unequal time. But it is an experimental fact that the attractive force of gravity is always exactly proportional to the mass of a body, and the resistance to motion is also proportional to mass, hence the force with which a body is moved by the earth’s attraction is always proportional to the difficulty of moving the body. This would not be the case with other methods of setting a body in motion. If I kick a small ball with all my might, I shall send it further than a kick of equal strength would send a heavier ball. Why? Because the impulse is the same in each case, but the masses are different. But if those balls are pulled by gravity, then, by the very nature of the earth’s attraction (the reason of which we cannot explain), the small ball receives a little pull, and the big ball receives a big pull, the earth exactly apportioning its pull in each case to the mass of the body on which it has to act. It is to this fact, that the earth pulls bodies with a strength always in each case exactly proportional to their masses, that is due the result that they fall in equal times, each body having a pull given to it proportional to its needs.

The error of the view of Aristotle was not only demonstrated by Galileo by experiment, but was also demonstrated by argument. In this argument Galileo imitated the abstract methods of the Aristotelians, and turned those methods against themselves. For he said, “You” (the Aristotelians) “say that a lighter body will fall more slowly than a heavy one. Well, then, if you bind a light body on to a heavy one by means of a string, and let them fall together, the light body ought to hang behind, and impede the heavy body, and thus the two bodies together ought to fall more slowly than the heavy body alone; this follows from your view: but see the contradiction. For the two bodies tied together constitute a heavier body than the heavy body alone, and thus, on your own theory, ought to fall more quickly than the heavy body alone. Your theory, therefore, contradicts itself.”

The truth is that each body is occupied in moving itself without troubling about moving its neighbour, so that if you put any number of marbles into a bag and let them drop they all go down individually, as it were, and all in the time which a single marble would take to fall. For any other result would be a contradiction. If you cut a piece of bread in two, and put the two halves together, and tie them together with a thread, will the mere fact that they are two pieces make each of them fall more slowly than if they were one? Yet that is what you would be bound to assert on the Aristotelian theory. Hold an egg in your open hand and jump down from a chair. The egg is not left behind; it falls with you. Yet you are the heavier of the two, and on Aristotelian principles you ought to leave the egg behind you. It is true that when you jump down a bank your straw hat will often come off, but that is because the air offers more resistance to it than the air offers to your body. It is the downward rush through the air that causes your hat to be left behind, just as wind will blow your hat off without blowing you away. For since motion is relative, it is all one whether you jump down through the air, or the air rushes past you, as in a wind. If there were no air, the hat would fall as fast as your body.

This is easy to see if we have an airpump and are thus enabled to pump out almost all the air from a glass vessel. In that vessel so exhausted, a feather and a coin will fall in equal times. If we have not an airpump, we can try the experiment in a more simple way. For let us put a feather into a metal egg-cup and drop them together. The cup will keep the air from the feather, and the feather will not come out of the cup. Both will fall to the ground together. But if the lighter body fall more slowly, the feather ought to be left behind. If, however, you tie some strings across a napkin ring so as to make a sort of rough sieve, and put a feather in it, and then drop the ring, then as the ring falls the air can get through the bottom of the ring and act on the feather, which will be left floating as the ring falls.

Let us now go on to examine the second fallacy that was derived from the Aristotelians, and that so long impeded the advance of science, namely, that the earth must be at rest.

The principal reason given for this was that if bodies were thrown up from the earth they ought, if the earth were in motion, to remain behind. Now, if this were so, then it would follow that if a person in a train which was moving rapidly threw a ball vertically, that is perpendicularly, up into the air, the ball, instead of coming back into his hand, ought to hit the side of the carriage behind him. The next time any of my readers travel by train he can easily satisfy himself that this is not so. But there are other ways of proving it. For instance, if a little waggon running on rails has a spring gun fixed in it in a perpendicular position, so arranged that when the waggon comes to a particular point on the rails a catch releases the trigger and shoots a ball perpendicularly upwards, it will be found that the ball, instead of going upwards in a vertical line, is carried along over the waggon, and the ball as it ascends and descends keeps always above the waggon, just as a hawk might hover over a running mouse, and finally falls not behind the waggon, but into it.

So, again, if an article is dropped out of the window of a train, it will not simply be left behind as it falls, but while it falls it will also partake of the motion of the train, and touch the ground, not behind the point from which it was dropped, but just underneath it.

The reason is, that when the ball is dropped or thrown it acquires not only the motion given to it by the throw, or by gravity, but it takes also the motion of the train from which it is thrown. If a ball is thrown from the hand, it derives its motion from the motion of the hand, and if at the time of throwing the person who does so is moving rapidly along in a train, his hand has not only the outward motion of the throw, but also the onward motion of the train, and the ball therefore acquires both motions simultaneously. Hence then it is not correct reasoning to say, because a ball thrown up vertically falls vertically back to the spot from which it was thrown, that therefore the earth must be at rest; the same result will happen whether the earth is at rest or in motion. You can no more tell whether the earth is at rest or in motion from the behaviour of falling bodies than you can tell whether a ship on the ocean is at rest or in motion from the behaviour of bodies on it.

But you will say. Then why do we feel sea-sick on a ship? The answer is, that that is because the motion of the ship is not uniform. If the earth, instead of turning round uniformly, were to rock to and fro, everything on it would be flung about in the wildest fashion. For as soon as the earth had communicated its motion to a body which then moved with the earth, if the earth’s motion were reversed, the body would go on like a passenger in a train on which the break is quickly applied, and he would be shot up against the side of the room. Nay, more, the houses would be shaken off their foundations. Changes of motion are perceptible so long as the change is going on. We are therefore justified in inferring from the behaviour of bodies on the earth, not that the earth is at rest, but that it is either at rest, or else, if it is in motion, that its motion is uniform and not in jerks or variable.

Fig. 23.

For if it were not so, consider what would be happening around us. The earth is about 8,000 miles in diameter, and a parallel of latitude through London is therefore about 19,000 miles long, and this space is travelled in twenty-four hours. So that London is spinning through space at the rate of over 1,000 feet a second, due to the earth’s rotary motion alone, not to speak of the motion due to the earth’s path round the sun. If a boy jumped up two and a half feet into the air, he would take about half a second to go up and come down, but if in jumping he did not partake of the earth’s motion, he would land more than 500 feet to the westward of the point from which he jumped up, and if he did it in a room, he would be dashed against the wall with a force greater than he would experience from a drop down from the top of Mont Blanc. He would be not only killed, but dashed into an indistinguishable mass. If the earth suddenly stood still, everything on it would be shaken to pieces. It is bad enough to have the concussion of a train going thirty miles an hour when dashed against some obstacle. But the concussion due to the earth’s stoppage would be as of a train going about 800 miles an hour, which would smash up everything and everybody.

Thus, then, the first effect of the new ideas formulated by Galileo was to show that the Copernican theory that the earth moved round on its axis, and round the sun, was in agreement with the laws of motion. In fact, he introduced quite new ideas of force, and these ideas I must now endeavour to explain.

Let us consider what is meant by the word “force.” If I press my hand against the table, I exert force. The harder I press, the more force there is. If I put a weight on a stand, the weight presses the stand down with a force. If I squeeze a spring, the spring tries to recover itself and exerts a certain force. In all these cases force is considered as a pressure. And I can measure the force by seeing how much it will press things. If I take a spring, and press it in an inch, it takes perhaps a force of 1 lb. It will take a force of 2 lbs. to press it in another inch. Or again, if I pull it out an inch, it takes a force of 1 lb. If I pull it out another inch, it takes a force of 2 lbs. We thus always get into the habit of conceiving forces as producing pressures and being measured by pressures.

Fig. 24.

This is a perfectly legitimate way of looking at the matter, just as the cook’s method of employing a spring balance to weigh masses of meat is a perfectly legitimate way of estimating the forces acting upon bodies at rest. But when you come to consider the laws of the pendulums of clocks, to which all that I am saying is a preparation, then you have to deal with bodies in motion. And for this purpose a new idea of force altogether is requisite. We shall no longer speak of forces as producing pressures. We shall treat them quite independently of their pressing power. The sun exerts a force of attraction on the earth, but it does not press upon it. It exerts its force at a distance. Hence then we want a new idea of “force.” This idea is to be the following. We will consider that when a force acts upon a body it endeavours to cause it to move; in fact, it tries to impart motion to the body. We may treat this motion as a sort of thing or property. The longer the force acts on the body, the more motion it imparts to it, provided the body is free to receive that motion. So that we may say that the test of the strength of the force is how much motion it can give to a body of a given mass in a given time. It does not matter how the force acts. It may act by means of a string and pull it; it may act by means of a stick and push it; it may act by attraction and draw it; it may act by repulsion and repel it; it may act as a sort of little spirit and fly away with it. In all these cases it acts. The more it acts, the more effect it has. In double the time it produces double the motion. If the mass is big, it takes more force to make the mass move; if the mass of the body is small, it is moved more easily. Therefore when we want to measure a force in this way we do not press it against springs to see how much it will press them in. What we do is to cause it to act on bodies that are free to move and see what motions it will produce in them. Of course we can only do this with things that are free to move. You cannot treat force in this way if you have only a pair of scales; in that case you would have to be content with simply measuring pressures. It is important clearly to grasp this idea. If a body has a certain mass, then the force acting on it is measured by the amount of motion that will in a given time be imparted to that mass, provided that the mass is free to move. This is to be our definition of force.

Therefore, by the action of an attraction or any other force on a body free to move; motion is continually being imparted to the body. Motion is, as it were, poured into it, and therefore the body continually moves faster and faster.

Here is a ball flying through the air. Let us suppose that forces are acting on it. How can we measure them? We cannot feel what pressures are being exerted on it. The only thing we can do is to watch its motions, and see how it flies. If it goes more and more quickly, we say, “There is propelling force acting on it”; if it begins to stop, we say again, “There is retarding force acting on it.” So long as it does not change its speed or direction, we say, “There is no force acting on it.” By this method, therefore, we tell whether a body is being acted on by force, simply by observing its speed or its change of speed. Merely to say a body is moving does not tell us that force is acting on it. All we know is that, if it is moving, force has acted on it. It is only when we see it changing its speed or direction, that is changing its motion, that we say force is acting. Every change of motion, either in direction or speed, must be the result of force, and must be proportional to that force. This is what we mean when we say motion is the test and measure of force.

This most interesting way of looking at the matter lies at the root of the whole theory of mechanics. It is the foundation of the system which the stupendous genius of Newton conceived in order to explain the motion of the sun, moon, and stars.

Forces were treated by him as proportional to the motions, and the motions proportional to the forces, and with this idea he solved a part of the riddle of the universe. Galileo had partly seen the same thing, but he never saw it so clearly as Newton. Great discoveries are only made by seeing things clearly. What required the force of a genius in one age to see in the next may be understood by a child.

Hence then we say a force is that which in a given time produces a given motion in a given mass which is free to move.

You must have time for a force to act in; for however great the force, in no time there can be no motion. You must have mass for a force to act on; no mass, no effect. You must have free space for the mass to move in; no freedom to move, no movement.

But what is this “mass”? We do not know; it is a mystery. We call it “quantity of matter.” In uniform substances it varies with size. Double the volume, double the mass. Cut a cake in half, each half has the same “mass.” But then is mass “weight”? No, it is not. Weight is the action of the earth’s attraction on matter. No earth to attract, and you would have no weight, but you would still have “mass.” What then is matter? Of that we have no idea. The greatest minds are now at work upon it. But mass is quantity of matter. Knock a brick against your head, and you will know what mass is. It is not the weight of the brick that gives you a bump; it is the mass. Try to throw a ball of lead, and you will know what mass is. Try to push a heavy waggon, and you will know what mass is. Weights, that is earth attractions on masses, are proportional to the masses at the same place. This, as we have seen, is known by experiment.

Therefore, when a force acts for a certain time on a mass that is free to move, however small the force and however small the time, that body will move. When a baby in a temper stamps upon the earth it makes the earth move—not much, it is true, but still it moves; nay, more, in theory, not a fly can jump into the air without moving the earth and the whole solar system. Only, as you may imagine they do not show it appreciably. Still, in theory the motion is there.

Hence then there are two different ways of considering and estimating forces, one suitable for observations on bodies at rest, the other suitable for observations of bodies that are free to move. The force of course always tends to produce motion. If, however, motion is impossible, then it develops pressures which we can measure, and calculate, and observe. If the body is free to move, then the force produces motions which we can also measure, calculate, and observe. And we can compare these two sets of effects. We can say, “A force which, acting on a ball of a mass of one pound, would produce such and such motions, would if it acted on a certain spring produce so much compression.”

The attraction of the earth on masses of matter that are not free to move gives rise to forces which are called weights. Thus the attraction of gravitation on a mass of one pound produces a pressure equal to a weight of one pound. Unfortunately the same word “pound” is used to express both the mass and the weight, and has come down to us from days when the nature of mass was not very well appreciated. But great care must be taken not to confuse these two meanings.

But the earth’s attractions and all other forces acting upon matter which is free to move give rise to changes of motion. The word used for a change of motion is “acceleration” or a quickening. “He accelerated his pace,” we say. That is, he quickened it; he added to his motion. So that force, acting on mass during a time, produces acceleration.

From this, then, it follows that if a force continues to act on a body the body keeps moving quicker and quicker. When the force stops acting, the motion already acquired goes on, but the acceleration stops. That is to say, the body goes on moving in a straight line uniformly at the pace it had when the force stopped.

If, then, a body is exposed to the action of a force, and held tight, what will happen? It will, of course, remain fixed. Now let it go—it will then, being a free body, begin to move. As long as the force acts, the force keeps putting more and more motion into the body, like pouring water into a jug, the longer you pour the faster the motion becomes. The body keeps all the motion it had, and keeps adding all the motion it gains. It is like a boy saving up his weekly pocket-money: he has what he had, and he keeps adding to that. So if in one second a motion is imparted of one foot a second, then in another second a motion of one foot a second more will be added, making together a motion of two feet a second; in another second of force action the motion will have been increased or “accelerated” by another foot per second, and so on. The speed will thus be always proportional to the force and the time. If we write the letter V to represent the motion, or speed, or velocity; F to represent the acceleration or gain of motion; and T to represent the time, then V = FT. Here V is the velocity the body will have acquired at the end of the time T, if free to move and submitted to a force capable of producing an acceleration of F feet per second in a unit of time.

V is the final velocity. The average velocity will be 1/2 V, for it began with no velocity and increased uniformly. How far will the body have fallen in the interval? Manifestly we get that by multiplying the time by the average velocity, that is S = 1/2 VT, where V, as I said, is the final velocity, but we found that V = FT. Hence by substitution S = 1/2 FT × T = 1/2 FT².

It is to be carefully borne in mind that these letters V, S, and T do not represent velocities, spaces, and times, but merely represent arithmetical numbers of units of velocities, spaces, and times. Thus V represents V feet per second, S represents S feet, and T represents T seconds. And when we use the equation V = FT we do not mean that by multiplying a force by a time you can produce a velocity. If, for instance, it be true that you can obtain the number of inhabitants (H) in London by multiplying the average number of persons (P) who live in a house by the number of houses (N), this may be expressed by the equation H = PN. But this does not mean that by multiplying people into houses you can produce inhabitants. H, P, and N are numbers of units, and they are numbers only.

Therefore when a body is being acted on by an accelerating force it tends to go faster and faster as it proceeds, and therefore its velocity increases with the time. But the space passed through increases faster still, for as the time runs on not only does the space passed through increase, but the rate of passing also gets bigger. It goes on increasing at an increasing rate. It is like a man who has an increasing income and always goes on saving it. His total mounts up not merely in proportion to the time, but the very rate of increase also increases with the time, so that the total increase is in proportion to the time multiplied into the time, in other words to the square of the time. So then, if I let a body drop from rest under the action of any force capable of producing an acceleration, the space passed through will be as the square of the time.

Now let us see what the speed will be if the force is gravity, that is the attraction of the earth.

Turning back to what was said about Galileo, it will be remembered that he showed that all bodies, big and small, light and heavy, fell to the earth at the same speeds. What is that speed? Let us denominate by G the number of feet per second of increase of motion produced in a body by the earth’s action during one second. Then the velocity at the end of that second will be V = GT. The space fallen through will be S = 1/2 GT².

What I want to know then is this: how far will a body under the action of gravity fall in a second of time?

This, of course, is a matter for measurement. If we can get a machine to measure seconds, we shall be able to do it; but inasmuch as falling bodies begin by falling sixteen feet in the first second and afterwards go on falling quicker and quicker, the measurements are difficult. Galileo wanted to see if he could make it easier to observe. He said to himself, “If I can only water down the force of gravity and make it weaker, so that the body will move very slowly under its action, then the time of falling will be easier to observe.” But how to do it? This is one of those things the discovery of which at once marks the inventor.

Fig. 25.

The idea of Galileo was, instead of letting the body drop vertically, to make it roll slowly down an incline, for a body put upon an incline is not urged down the incline with the same force which tends to make it fall vertically.

Can any law be discovered tending to show what the force is with which gravity tends to drag a mass down an incline?

There is a simple one, and before Galileo’s time it had been discovered by Stevinus, an engineer. Stevinus’ solution was as follows. Suppose that A B C is a wedge-shaped block of wood. Let a loop of heavy chain be hung over it, and suppose that there is a little pulley at C and no friction anywhere. Then the chain will hang at rest. But the lower part, from A to B, is symmetrical; that is to say, it is even in shape on both sides. Hence, so far as any pull it exerts is concerned, the half from A to D will balance the other half from B to D. Therefore, like weights in a scale, you may remove both, and then the force of gravity acting down the plane on the part A C will balance the force of gravity acting vertically on the part C B. Now the weight of any part of the chain, since it is uniform, is proportional to its length. Hence, then, the gravitational force down the plane of a piece whose weight equals C A is equal to the gravitational force vertically of a piece whose weight equals C B. In other words, the force of gravity acting down a plane is diminished in the ratio of C B to C A.

But when a body falls vertically, then, as we have seen, S = 1/2 GT², where S is the space it will fall through, G the number of feet per second of velocity that gravity, acting vertically on a body, will produce in it in a second, and T the number of seconds of time. If then, instead of falling vertically, the body is to fall obliquely down a plane, instead of G we must put as the accelerating force

G × (vertical height of the end of the plane)/(length of the plane).

To try the experiment, he took a beam of wood thirty-six feet long with a groove in it. He inclined it so that one end was one foot higher than the other. Hence the acceleration down the plane was 1/36 G, where G is the vertical acceleration due to gravity which he wanted to discover. Then he measured the time a brass ball took to run down the plane thirty-six feet long, and found it to be nine seconds. Whence from the equation given above 36 feet = 1/2 acceleration of gravity down the plane × (9 seconds)². Whence it follows that the acceleration of gravity down the plane is (36 × 2)/(9)² feet per second.

But the slope of the plane is one thirty-sixth to the vertical. Therefore the vertical acceleration of gravity, i.e., the velocity which gravity would induce in a vertical direction in a second, is equal to thirty-six times that which it exercises down the plane, i.e.,

36 × (36 × 2)/(9)²; and this equals 32 feet per second.

Though this method is ingenious, it possesses two defects. One is the error produced by friction, the other from failure to observe that the force of gravity on the ball is not only exerted in getting it down the plane, but also in rotating it, and for this no allowance has been made. The allowance to be made for rotation is complicated, and involves more knowledge than Galileo possessed. Still the result is approximately true.

Fig. 26.

The next attempt to measure G, that is the velocity that gravity will produce on a body in a second of time, was made by Attwood, a Cambridge professor. His idea was to weaken the force of gravity and thus make the action slow, not by making it act obliquely, but by allowing it to act, not on the whole, but only on a portion of the mass to be moved. For this purpose he hung two equal weights over a very delicately constructed pulley. Gravity, of course, could not act on these, for any effect it produced on one would be negatived by its effect on the other. The weights would therefore remain at rest. If, however, a small weight W, equal say to a hundredth of the combined weight of the weights A and B and W, were suddenly put on A, then it would descend under an accelerating force equal to a hundredth part of ordinary gravity. We should then have

S (the space moved through by the weights) = 1/2 × G/100 × t².

With such a system, he found that in 7½ seconds the weights moved through 9 feet. Whence he got

9 = 1/2 G/100 × (7½)².

From which

G = (2 × 9 × 100)/(7½)² = 32 feet per second nearly.

Thus by letting gravity only act on a hundredth part of the total weight moved, namely A, B, and W, he weakened its action 100 times, and thus made the time of falling and the space fallen through sufficiently large to be capable of measurement. To sum up, when a body free to move is acted upon by the force of gravity, its speed will increase in proportion to the time it has been acted upon, and the space it will pass through from rest is proportional to the square of the time during which the accelerating force has acted on it.

Gravity is, of course, not the only accelerating force with which we are acquainted. If a spring be suddenly allowed to act on a body and pull it, the body begins to move, and its action is gradually accelerated, just as though it were attracted, and the acceleration of its motion will be proportional to the time during which the accelerating force acts. Similarly, if gunpowder be exploded in a gun-barrel, and the force thus produced be allowed to act on a bullet, the motion of the bullet is accelerated so long as it is in the barrel. When the bullet leaves the barrel it goes on with a uniform pace in a straight line, which, however, by the earth’s attraction is at once deflected into a curve, and altered by the resistance of the air.

Fig. 27.

It has been already stated that motions may be considered independently one of another, so that if a body be exposed to two different forces the action of these forces can be considered and calculated each independently of the other. Let us take an example of this law. We have seen if a body is propelled forwards, and then the force acting on it ceases, that it proceeds on with uniform unchanging velocity, and if nothing impeded it, or influenced it, it would go on in a straight line at a uniform speed.

We have also seen that if a body is exposed to the action of an accelerating force such as gravity it constantly keeps being accelerated, it constantly keeps gaining motion, and its speed becomes quicker and quicker.

Fig. 28.

Let us suppose a body exposed to both of these forces at the same time. Shoot it out of a cannon, and let an accelerating force act on it, not in the direction it is going, but in some other direction, say at right angles. What will happen? In the direction in which it is going, its speed will remain uniform. In the direction in which the accelerating force is acting, it will move faster and faster. Thus along A B it will proceed uniformly. If it proceeded uniformly also along A C (as it would do if a simple force acted on it and then ceased to act), then as a result it would go in the oblique line A D, the obliquity being determined by the relative magnitude of the forces acting on it. But how if it went uniformly along A B, but at an accelerated pace along A C? Then while in equal times the distances along A B would be uniform the distances in the same times along A C would be getting bigger and bigger. It would not describe a straight line; it would go in a curve. This is very interesting. Let us take an example of it. Suppose we give a ball a blow horizontally; as soon as it quits the bat it would of course go on horizontally in a straight line at a uniform speed; but now if I at the same instant expose it to the accelerating force of gravity, then, of course, while its horizontal movement will go on uniformly, its downward drop will keep increasing at a speed varying as the time. And while the total distances horizontally will be uniform in equal times, the total downward drop from A B will be as the squares of the times. Here, then, you have a point moving uniformly in a horizontal direction, but as the squares of the times in a vertical direction. It describes a curve. What curve? Why, one whose distances go uniformly one way, but increase as the squares the other way.

Fig. 29.

This interesting curve is called a parabola. With a ball simply hit by a bat, the motion is so very fast that we cannot see it well. Cannot we make it go slowly? Let us remember what Galileo did. He used an inclined plane to water down his force of gravity. Let us do the same. Let us take an inclined plane and throw on it a ball horizontally. It will go in a curve. Its speed is uniform horizontally, but is accelerated downwards. If we desire to trace the curve it is easy to do. We coat the ball with cloth and then dip it in the inkpot. It will then describe a visible parabola. If I tilt up the plane and make the force of gravity big, the parabola is long and thin; if I weaken down the force of gravity by making the plane nearly horizontal, then it is wide and flat.

One can also show this by a stream of peas or shot. The little bullets go each with a uniform velocity horizontally, and an accelerated force downwards.

Instead of peas we can use water. A stream of it rushing horizontally out of an orifice will soon bend down into a parabola.

Thus then I have tried to show what force is and how it is measured. I repeat again, when a body is free to move, then, if no further force acts on it, it will go on in a straight line at a uniform speed, but if a force continues to act on it in any direction, then that force produces in each unit of time a unit of acceleration in the direction in which the force acts, and the result is that the body goes on moving towards the direction of acceleration at a constantly increasing speed, and hence passing over spaces that are greater and greater as the speed increases. This is the notion of a “force.” In all that has been said above it has been assumed that the attraction of gravity on a body does not increase as that body gets nearer to the earth. This is not strictly true; in reality the attractive force of gravity increases as the earth’s centre is approached. But small distances through which the weights in Attwood’s machine fall make no appreciable difference, being as nothing compared to the radius of earth. For practical purposes, therefore, the force may be considered uniform on bodies that are being moved within a few feet of the earth’s surface. It is only when we have to consider the motions of the planets that considerations of the change of attractive force due to distance have to be considered.

I am glad to say that the most tiresome, or rather the most difficult, part of our inquiry is now over. With the help of the notions already acquired, we are now ready to get to the pendulum, and to show how it came about that a boy who once in church amused himself by watching the swinging of the great lamps instead of attending to the service laid the foundation of our modern methods of measuring time.

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CHAPTER III.

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.

Fig. 30.

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.

Fig. 31.

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.

Fig. 32.

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.

Fig. 33.

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.

Fig. 34.

Fig. 35.

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.

Fig. 36.

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.

Fig. 37.

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.

Fig. 38.

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.

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.

(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.

Fig. 40.

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.

Fig. 41.

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.

Fig. 42.

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.

Fig. 43.

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.

Fig. 44.

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.

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CHAPTER IV.

It is now time to give a description of the various parts of an ordinary pendulum clock. We will take the “Grandfather” clock as an example. We shall want an hour hand and a minute hand in the centre of the face, and a seconds hand to show seconds a little above them. There will be a seconds pendulum 39·14 inches long, and the centre of the face of the clock will be about seven feet above the ground, so as to give practically about five feet of fall for the weight.

Fig. 45.

In the first place, we have to consider the axle which carries the minute hand, and which turns round once in each hour. This is usually made of a piece of steel about one-sixth of an inch in diameter. Clockmakers usually call an axle an “arbor,” or “tree,” whence our word axletree.

This “arbor” is turned in the lathe, so as to have pivots on each end, fitted into holes in the clock plates, that is to say, the flat pieces of brass that serve as the body of the clock. The adjoining diagram shows S T the clock faces, and C, the arbor of the minute hand.

Inasmuch as the seconds hand is to turn round sixty times while the minute hand turns round once, it is obvious that the arbor of the minute hand must be connected to the arbor of the seconds hand by a train of cogwheels so arranged as to multiply by sixty. This of course involves us in having large and small cogwheels.

Fig. 46.

The small cogwheels usually have eight teeth, and are for convenience of manufacture, as also to stand prolonged wear, cut out of the solid steel of the arbor. They are nicely polished.

The easiest pair of wheels to use will be two pinions of eight teeth, or “leaves,” as they are called, and two cogwheels, one of sixty-four teeth, the other of sixty teeth.

It is then clear that if the arbor A turns round once in an hour, the arbor B will turn round eight times in an hour, and C will turn round (60 × 64)/(8 × 8) = 60 times in an hour, or once in each minute.

By having 480 teeth on the cogwheel on A, you could, of course, make C go round once in a minute without the use of any intermediate arbor such as B.

Fig. 47.

But this would not be a very convenient plan. For as the wheel on A is usually about two and a quarter inches in diameter, to cut 480 teeth on so small a wheel would involve us in cutting about sixty teeth to the inch. The teeth would thus be microscopically small, and would have to be set so fine that the least dirt would clog them. Moreover, the pinion of eight leaves would have to be microscopic. For these reasons, therefore, it is usual in clocks not to use wheels with teeth more than sixty or sixty-four in number, and to diminish the motion gradually by means, where needful, of intermediate arbors. We have next to consider how the weight is to be arranged so as to turn the arbor A once round in an hour. We know that we have five feet of space for the weights to fall in. If we arrange to have what is called a double fall, as shown in the sketch, then, allowing room for pulley wheels, we shall find that our string may be practically about nine feet in length.

Fig. 48.

The clock will be wanted to go for a week without winding, and as people may forget to wind it at the proper hour of the day, we will give it a day extra, and make an “eight-day” clock of it. Hence then, while nine feet of cord is being pulled out by a weight which falls four and a half feet, the minute hand is to be turned round as many times as there are hours in eight days, viz., 192 times. This could be accomplished, of course, by winding the cord round the arbor of the minute hand. But this would require 192 turns. If our cord is to be ordinary whipcord, or catgut, say one-twelfth of an inch in diameter, in order that the cord could be wound upon it, the arbor would have to be 192/12 inches long = 14⅓ inches long. This would make the clock case unnecessarily deep. We must therefore again have recourse to an intermediate wheel.

Fig. 49.

If we put a pinion of eight leaves on the minute hand arbor c, and engage it with a wheel of sixty-four teeth on another arbor b, then b will obviously turn round once in eight hours, that is to say, twenty-four times in the period of eight days. And, if we fix on b a “drum” or cylinder two inches long, the twenty-four turns of our cord will just fit upon it, since, as has been said, our cord is to be one-twelfth of an inch in diameter. The diameter of the drum must be such that a cord nine feet long can be wound twenty-four times round it. That is to say, each lap must take (9 × 12)/24 = 4½ inches of cord. From this it is easy to calculate that the diameter of the drum must be rather less than one and a half inches. From this then it results that we want for a “Grandfather’s” clock a drum two inches long and one and a half inches diameter, on this a cogwheel of sixty-four teeth working into a minute hand arbor, with a pinion wheel with eight leaves, and a cogwheel of sixty-four teeth, an intermediate or idle wheel with an eight-leaved pinion, and a cogwheel of sixty teeth, engaging with a seconds hand arbor with a pinion of eight leaves. This is called the “train of wheels.” With it a weight such as can be arranged in an ordinary “Grandfather’s” clock case will cause by its fall during eight days the second hand arbor to turn round once in each minute during the whole time, and the minute hand arbor to turn round once in each hour.

Fig. 50.

We must next provide an arrangement for winding the clock up. It is obvious that we cannot do so by twisting the hands back. It is true that this could be done, but it would take about five minutes to do each time and be wearisome. In order to save this trouble, an arrangement called a ratchet wheel and pall must be provided. A ratchet wheel consists of a wheel with a series of notches cut in it, as shown in the figure A. A pall is a piece of metal, mounted on a pin, and kept pressed up against the ratchet wheel by a spring C. It is obvious that if I turn the wheel A round, and thus wind up a weight, fastened to a cord wound round the drum D, that the pall B will go click-click-click as the ratchet wheel goes round, but that the pall will hold it from slipping back again. When, however, I take my hands away, and let the ratchet wheel alone, then the weight E will pull on the drum D, and try and turn the ratchet wheel back the opposite way to that in which I twisted it at first. If the pall B is held fast, it is impossible to move it, but if the pall is fixed to a cogwheel F, which rides loose on the arbor of the drum D, then the pull of the weight E will tend to twist the cogwheel F round, and this, if engaged with a pinion wheel on the minute hand arbor, will therefore drive the clock. As the clock arbors move, of course the weight E gradually runs down, and, at last all the string is unwound from the drum D. The clock is said then to have “run down,” but if I take a clock key, and by means of it wind the string up upon the drum D, then the pall lets the drum and ratchet slip; the clock hands are not affected. When I have given twenty-four turns to the arbor, the nine feet of cord will then be wound upon the drum again, and the clock will be ready to go for eight more days, and will begin to move as soon as I cease to press upon the clock key.

Fig. 51.

I have thus described the winding mechanism. It now remains to describe the escapement.

It is of course obvious that, if the weight and train of wheels were simply let go, the weight would rush down, and the seconds-hand wheel would fly round at a tremendous pace; but we want it to be so restrained as only to be allowed to go one-sixtieth part of its journey round in each second. In fact, we need an “escapement” and a pendulum.

The escapement usually employed in “Grandfather” clocks is the anchor escapement above described. It is not by any means the best sort of escapement, but it is the easiest to make; and hence its popularity in the days sometimes called the “dear, good old days,” when people had to file everything out by hand, and had to take a day to do badly what can now be done well in five minutes.

The escape wheel of an anchor escapement has thirty sharp angular teeth on its rim. The wheel is made as light as possible, so that the shock of stoppage at each tick of the clock may be as slight as possible, for a heavy blow of course wastes power and gradually wears out the clock. The anchor consists of two arms of the shape shown in the illustration ([Fig. 44]). As the escape wheel goes round in the direction of the arrow, the anchor, mounted on its arbor, rocks to and fro. The wheel cannot run away, because the act of pushing one arm or “pallet,” as it is called, outwards, and thus freeing the tooth pulls the other pallet in, and this stops the motion of the tooth opposite to it, but when the anchor rocks back again, so as to disengage the pallet from the tooth that holds it, then the opposite tooth is free to fly forward against the other pallet. This tends to rock the anchor the other way, but at that instant the pallet just engages the next tooth of the wheel, and so the action goes on. The anchor rocks from side to side; the pallets alternately engage the teeth of the wheel, making at each rock of the anchor the tick-tock sound with which we are so familiar. If the anchor were free to rock at any speed it could, the ticking of the clock would be very quick; so, to restrain the vivacity of the anchor, we have a pendulum. The pendulum might be simply hung on to the anchor. But the disadvantage of doing this would be that the heavy bob of the pendulum would cause such a pressure on the arbor of the anchor that there would be great friction, and the arbor would soon be worn out, and the accurate going of the clock disturbed. The pendulum therefore is hung on a piece of steel spring on a separate hook, which lets it go backwards and forwards and carries the weight easily, while a rod projecting from the anchor has a pin, which works in a slot on the pendulum. The pendulum is therefore able to control and regulate the movements of the escapement, and thus the time of the clock.

Of course it is clear that the heavier the driving weight put on the drum of the clock, and the better the cut and finish of the wheels, and the greater the cleanliness and oil, the more will be the pressure tending to drive round the escape wheel, and the harder the pressure on the pallets, and hence the bigger the impulses on the pendulum, and therefore the larger the amplitude of its swing.

If the amplitude of the pendulum’s swing affected the time of its swing, then the time kept by the clock would vary with the weight, and the dirt and friction, and the drying up of the oil. But here precisely is where the value of the beautiful law governing the harmonic motion of the pendulum comes in. The time of the pendulum is (for small arcs) independent of the length of swing, and therefore of the driving force of the clock, and hence within limits the clock, even though roughly made and foul with the dirt of years, continues to keep good time. But the anchor escapement has imperfections. The only way in which a pendulum can be relied on to keep accurate time is by leaving it unimpeded. But the pressure of the teeth on the pallets in an anchor escapement constantly interferes with this.

Fig. 52.

A little consideration will easily show that there are some times during the swing of a pendulum at which interference is far more fatal to its time-keeping than at others. Thus the bob of a pendulum may be regarded as a weight shot outwards from its position of rest against the influence of a retarding force varying as its distance from rest—in fact, shot out against a spring. The time of going out and coming in again will be quite independent of the force exerted to throw it out, quite independent of its original velocity. Therefore a variation in the impulse given to the bob is of no consequence, provided that impulse is given when the bob is near the position of rest. This follows from the nature of the motion. If a ball be attached to a piece of elastic thread, and thrown from the hand, so that it flies out, and then stops and is brought back by the elastic force of the thread, the time of the outward motion and the return is the same whatever be the force of the throw. And so if a pendulum be impelled outwards from a position of rest, the time of the swing out and back is the same, however big (within limits) is the impelling force and the consequent length of the swing. The use of a pendulum as a measure of time is to impel it outwards, and then let it fly freely out and back. But if its motion is not free, if forces other than gravity act upon it while on its path, then its time of swing will be disturbed. It does not matter with what force you originally impel it, but what does matter is, that when it once starts it should be allowed to travel unimpeded and uninfluenced. Now that is what an anchor escapement does not do. The impulse is given the whole way out on one of the pallets, and then when it is at its extreme of swing, and ought to be left tranquil, the other pallet fastens on it. But a perfect escapement ought to give its impulse at the middle point of the swing, when the pendulum is at the lowest, and then cease, and allow the pendulum to adapt its swing to the impulse it has received, and thus therefore to keep its time constant. This is done by an escapement called the dead beat escapement, which, though in an imperfect way, realises these conditions.

The alteration is made in the shape of the pallets of the anchor. The wheel is much the same. Each pallet consists of two faces: a driving face a b and a sliding face b c.

When the tooth b has done its work by pressing on the driving face, and thus driving the anchor over, say, to the left, then the tooth on the opposite side falls on the sliding face of the other pallet. This being an arc of a circle, has no effect in driving the anchor one way or the other; hence the pendulum is free to swing to the left as far as it likes and return when it feels inclined, always with the exception of a little friction of the tooth on the faces of the pallets, but when it returns and begins to move towards the right, the tooth slides back along the face of the pallet till the pendulum is almost at the middle of its swing; then an impulse is given by the pressure of the tooth upon the inclined plane a´ b´. As soon, however, as the tooth leaves b´, another tooth on the other side at once engages the sliding face b c of the other pallet, and so the motion goes on.

This beautiful escapement is at present used for astronomical clocks; the pallets are made of agate or sapphire, and therefore do not grind away the teeth of the wheel perceptibly, and the loss by friction on the sliding surfaces is exceedingly small.

There are several other ways even better than this for securing a free pendulum movement. We have now to return to our clock.

The centre arbor moves round once in an hour, and carries the minute hand. In order to provide an hour hand, which shall turn round once in twelve hours, we fasten a cogwheel and tube N on to the minute hand arbor by means of a small spring, which keeps it rather tight, but allows it to slip if turned round hard (see [Fig. 45]). This spring is a little bent plate slipped in behind the cogwheel on which its ends rest; its centre presses on a shoulder on the minute hand arbor; it is a sort of small carriage spring. The cogwheel n has thirty teeth. This cogwheel engages another cogwheel o with thirty teeth, on a separate arbor, which carries a third cogwheel, p, with six teeth, and this again engages a fourth cogwheel, q, with seventy-two teeth, mounted on a tube which slips over the tube to which the cogwheel a is attached. It is now easy to see that for each turn of the minute hand arbor the arbor p makes one turn, and for each turn of the arbor p the cogwheel d, makes one-twelfth of a turn. From which it follows that for each turn of the minute hand arbor the cogwheel d with its tube, or, as it is sometimes called, its “slieve,” makes one-twelfth of a turn, and thus makes a hand fastened to it show one hour for every complete turn of the minute hand.

The minute hand is attached to the tube or slieve which carries the cogwheel N. The hour hand is attached to the tube or slieve which carries the cogwheel Q, and one goes twelve times as slowly as the other.

But if you want to set the clock it is easy to do so by reason of the fact that the minute hand is not fixed to the arbor, but only to the slieve on the cogwheel that fits on the arbor, and is held somewhat tight to the arbor by means of the spring. The hands can thus be turned, but they are a little stiff. A washer on the minute hand arbor keeps the slieve on the cogwheel pressed tight against the spring, being secured in its turn by a very small lynch-pin driven through a hole in the minute hand arbor.

It remains to explain a few subsidiary arrangements, not always found upon all clocks, but which are useful.

In order to prevent the overwinding of the clock (see [Fig. 43]), which would cause the cord to overrun the drum, an arm is provided, fitted with a spring. As the weight is wound up the free part of the cord travels along the drum or the fusee; and the cord, when it is near the end of the winding, comes up against the arm and pushes it a little aside. This causes the end of the arm to be pushed against a stop on the axis of the fusee, and thus prevents the clock being further wound up. The stop, being ratchet-shaped, does not prevent the weight from pulling the ratchet wheel round the other way, and thus driving the clock; it only prevents the rotation of that wheel when the string is near it, and the winding is finished.

Another arrangement is the “maintaining spring.”

It will be remembered that during the process of winding the clock the hand twisting the key takes the pressure of the ratchet wheel off the pall, so that during that operation no force is at work to drive the clock. In consequence the pendulum receives no impulse, but swings simply by virtue of its former motion. If the process of winding were done slowly enough the clock might even stop. To avoid this, a very ingenious arrangement is made to keep the cogwheel mounted on the winding shaft going during the winding-up process. This is called a maintaining spring.

The arrangement shown in [Fig. 53] will explain it.

Fig. 53.

The cogwheel a and the ratchet wheel are both mounted loosely on the arbor carrying the drum. a is linked to b by a spring c. The ratchet wheel b is engaged by a pall fixed to some convenient place on the body of the clock frame. When the weight pulls on the drum the pull is communicated to the ratchet wheel b, and this acts on the spring c and pulls it out a little. As soon as the spring c is pulled out as far as its elasticity permits, a pull is communicated to the cogwheel a, and the clock is driven round. When the clock is wound the pressure of the weight is removed, and therefore the ratchet wheel e no longer presses on the pall, and thus no pressure is communicated to the ratchet wheel b, or through it to the clock. But here the spring c comes into play. For since the ratchet wheel b is held fast by the pall d, the spring c pulls at the wheel a, and thus for a minute or so will continue to drive the clock. This driving force, it is true, is less than that caused by the weight, but it is just enough to keep the pendulum going for a short time, so that the going of the clock is not interfered with.

If the reader can get possession of a clock, preferably one that does not strike, and, with the aid of a small pair of pincers and one or two screwdrivers, will take it to pieces and put it together again, the mechanism above described will soon become familiar to him. Not every clock is provided with maintaining spring and overwinding preventer.

The cause of stoppage of a clock generally is dirt. Where possible, clocks should always be put under glass cases. “Grandfather” clocks will go much better if brown paper covers are fitted over the works under the cases. In this way a quantity of dust may be avoided. To get a good oil is very important. It will be noticed that pivot-holes in clocks are usually provided with little cup-like depressions. This is to aid in keeping in the oil. The best clock oil is that which does not easily solidify or evaporate. Ordinary machine oil, such as used for sewing machines, is good as a lubricant, but rapidly evaporates. Olive oil corrodes the brass.

It is best to procure a little clock oil, or else the oil used for gun locks, sold by the gunsmiths. The holes should be cleaned out with the end of a wooden lucifer match, cut to a tapering point. The pivots should be well rubbed with a rag dipped in spirits of wine. If the pivots are worn they should be repolished in the lathe. If the cogs of the wheels are worn, there is no remedy but to get new ones. Old clocks sometimes want a little addition to the driving weight to make them go.

The weight necessary to drive the clock depends on its goodness of construction, and on the weight of the pendulum. If the clock is driven for eight days with a cord of nine feet in length with a double fall, then during each beat of the pendulum that weight will descend by an amount =

9/(2 × 24 × 60 × 60 × 8) feet or 1/12800th inch.

Whence, if the clock weight is 10 lbs., the impulse received by the clock at each beat is equivalent to a weight of 10 lbs. falling through 1/12800th of an inch, or to the fall of six grains through an inch.

The power thus expended goes in friction of the wheels and hands, and in maintaining the pendulum in spite of the friction of the air.

The work therefore that is put into the clock by the operation of winding is gradually expended during the week in movement against friction. The work is indestructible. The friction of the parts of the clock develops heat, which is dissipated over the room and gradually absorbed in nature. But this heat is only another form of work. Amounts of work are estimated in pressures acting through distances. Thus, if I draw up a weight of 1 lb. against the accelerative force of gravity through a distance of one foot, I am said to do a foot-pound of work.

One pound of coal consumed in a perfect engine would do eight millions foot-pounds of work. Hence, if the energy in a pound of coal could be utilized, it would keep about 100,000 grandfather’s clocks going for a week. As it is consumed in an ordinary steam engine it will do about half a million foot-pounds of work. One pound of bread contains about three million foot-pounds of energy. A man can eat about three pounds of bread in a day, and, as he is a very good engine, he can turn this into about three-quarters of a million foot-pounds of work. The rest of the work contained in the bread goes off in the form of heat.

Fig. 54.

As has been previously said, the power of the action of gravity in drawing back a pendulum that has been pushed aside from its position of rest becomes less in proportion as the pendulum is longer, and hence as the pendulum is longer the time of vibrations increases. In the [appendix] to this chapter a short proof will be given showing that the length of a pendulum varies as the square of the time of its vibration. A pendulum which is 39·14 inches in length vibrates at London once in each second. Of course at other parts of the earth, where the force of gravity is slightly different, the time of vibration will be different, but, since the earth is nearly a globe in shape, the force of gravity at different parts of it does not vary much, and therefore the time of vibration of the same pendulum in different parts of the earth does not vary very much. The length of a pendulum is measured from its point of suspension down to a point in the bob or weight. At first sight one would be inclined to think that the centre of gravity of the pendulum would be the point to which you must measure in order to get its length. So that if B were a circular bob, and the rod of the pendulum were very light, the distance A B to the centre of the bob would be the length of the pendulum. But if we were to fly to this conclusion, we should be making the same error that Galileo made when he allowed a ball to roll down an inclined plane. He forgot that the motion was not a simple one of a body down a plane, but was also a rolling motion. The pendulum does not vibrate so as always to keep the bob immovable with the top side C always uppermost. On the contrary, at each beat the bob rotates on its centre and makes, as it were, some swings of its own. Therefore in the total motions of the pendulum this rotation of the bob has to be taken into account. Of course, if the pendulum were so arranged that the bob did not rotate, and the point C were always uppermost, as, for instance, if the pendulum consisted of two parallel rods, A B and C D, suspended from A and C, then we might consider the bob as that of a pendulum suspended from E, and the pendulum would swing once in a second if A B = C D = E F were equal to 39·14 inches, for by this arrangement there would be no rotation of the bob. But as pendulums are generally made with the bob rigidly fixed to the rod E F, the rotation must be taken into account.

Fig. 55.

It wants some rather advanced mathematical knowledge to do this. But in practice clockmakers take no account of it. The correction is not a large one, so they make the rod as nearly true as they can, arrange a screw on the bob to allow of adjustment, and then screw the bob up and down until in practice the time of oscillation is found to be correct.

Fig. 56.

The mode of suspension of a pendulum of the best class is that shown in [Fig. 56], which allows the pendulum to fall into its true position without strain. A is a tempered steel spring, which bends to and fro at each oscillation. It is wonderful how long these springs can be bent to and fro without breaking. Inasmuch as lengthening the pendulum increases the time, so that the time of vibration t varies as the square of the length of the pendulum, a very small lengthening of the pendulum causes a difference in the time. In practice, for each thousandth of an inch that we lengthen the pendulum we make a difference of about one second a day in the going of the clock. If we cut a screw with eighteen threads to the inch on the bottom of the pendulum rod, and put a circular nut on it, with the rim divided into sixty parts, then each turn through one division will raise or lower the bob by 1/1080th of an inch, and this first causes an alteration of time of the clock by one second in the day. This is a convenient arrangement in practice, for it affords an easy means of adjusting the pendulum. We need only observe how many seconds the clock loses or gains in the day, and then turn the nut through a corresponding number of divisions in order to rectify the pendulum.

Fig. 57.

Another needful correction of the pendulum is that due to changes in temperature. If the rod of the pendulum be made of thoroughly dried mahogany, soaked in a weak solution of shellac in spirits of wine, and then dried, there will not be much variation either from heat or moisture. But for clocks required to have great precision the pendulum rod is usually made of metal. A rod of iron expands about 1/160000th of its length for each degree Fahrenheit; and therefore for each degree Fahrenheit a pendulum rod of 39·14 inches will expand about 1/4000 thousandths of an inch, and thus make a difference in the going of the clock of about one-fourth of a second per day. The expansion will, of course, make the clock go slower. It would be possible to correct this expansion if some arrangement could be made, whenever it occurred, to lift up the bob of the pendulum by an amount corresponding to it, as, for instance, to make the bob of some material which expanded very much more by heat than the material of which the pendulum rod was made.

Fig. 58.

Thus if we hang on to the end of a pendulum of iron a bottle of iron about seven inches long, and almost fill it with mercury, then, as soon as the heat increases, the iron of the rod and of the bottle expands, and the centre of oscillation of the pendulum is lowered. But as the linear expansion of mercury contained in a bottle is about five times that of iron, the mercury rises in the bottle, and thus the expansion downwards of the pendulum rod is compensated by the expansion upwards of the mercury in the bottle. The rod may be fastened to the mouth of the bottle by a screw, so that as the bottle is turned round it may be raised or lowered on the rod, and thus the length of the pendulum may be adjusted. The bottle is made of steel tube, screwed into a thin turned iron top and bottom. Of course no solder must be used to unite the iron, for mercury dissolves solder. A little oil and white-lead will make the screwed joints tight. This is an excellent form of pendulum. Another plan is to use zinc as the metal which is to counteract the expansion of the iron. The expansion of zinc is about three times that of iron.

Fig. 59.

Hence a zinc tube, about twenty inches long (shown shaded in [Fig. 59]), is made to rest upon a disc fastened to the lower part of the iron pendulum rod. On the top of the zinc rests a flat ring A, from which is suspended an iron tube A, which carries the bob B. The expansion of the zinc tube is large enough to compensate the expansion both of the rod and the tube, and the bob consequently remains at the same depth below the point of suspension, whatever be the temperature.

There is, however, a new method which is far superior to all these, and this is due to the discovery by M. Guilliaume, of Paris, of a compound of nickel and steel which expands so little that it can be compensated by a bob of lead instead of by a bob of mercury. This material is sold in England under the name of “invar.” An invar rod with a properly proportioned lead bob makes an almost perfect pendulum, the expansion of the invar and the lead going on together. The exact expansion of the invar is given by the makers, who also supply information as to the size and suspension of the bob proper to use with it.

It has been already shown that the uniformity of time of swing of a pendulum is only true when the arc through which it swings is very small. If the total swing from one side to another is not more than about two inches very little difference in time-keeping is made by putting a little more driving weight on the clock, and thus increasing its arc of swing; but when the arc of swing becomes say three inches, or one and a half inches on each side of the pendulum, then the time of vibration is affected. At this distance each tenth of an inch increase of swing makes the pendulum go slower by about a second a day.

The resistance of the air, of course, has a great influence on a pendulum, and is one of the chief causes that bring it ultimately to rest. Even the variations of pressure of the atmosphere which the barometer shows as the weather varies have an effect on the going of a clock. Attempts have been made by fixing barometers on to pendulums with an ingenious system of counter balancing to counteract this, but these refinements are not in common use, and are too complicated to be susceptible of effective regulation.