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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.