CHAPTER X.

MOTION.

180. =Universality of Motion.=--The world is full of motion. The rising and setting of the sun, the changes of the seasons, the falling of the rain, the running of rivers into the ocean, the ascent of water into the air by evaporation, the wind moving in silence or rushing on in its might, are familiar examples of motion constant and every where present. But with all this motion, sometimes in conflict and often variable, order and regularity reign. The causes of motion, though various in their operation, are kept by the Creator from producing confusion and disorganization by a few simple laws, which regulate the movements both of atoms and of worlds. The principal of these causes I will now briefly notice.

181. =Causes of Motion.=--Attraction is the most universal of the causes of motion in the universe. While it binds atom to atom, it also binds system to system throughout the immensity of space; and while it makes the stone fall to the ground, it moves the countless orbs forever onward in their courses. It is this which causes the tides to flow and the rivers to run down their slopes to the ocean, and thus by keeping up the never-ending motion of water all over the earth in seas, lakes, rivers, and the millions of little streamlets, diffuses life and beauty over the vegetable world, and gives to man the vast resources which we see developed in the numberless applications of water-power and navigation.

Heat pervades all matter, and is every where uniting its influence with the other causes of motion. It is heat that produces all the motions of the air, termed winds. It is heat that causes the rise of the water all over the earth in evaporation, so that it may be collected in clouds, again to descend to moisten the earth and keep the ever-flowing rivers full. Heat applied to water gives to man one of his best means of producing motion in machinery.

The agencies which Chemistry reveals to us are ever at work causing motion among the particles of matter; and though they generally work in silence, they sometimes show themselves in tremendous explosions, and in convulsions of nature.

Busy life is every where producing motion, more especially in the animal world. It gives to the myriads of animals, great and small, that swarm the earth not only the power of moving themselves, but also the power, to some extent, of moving the material world around them.

182. =Action and Reaction Equal.=--When any of the causes of motion act, the action is met by an opposite and equal reaction. If, for example, a blow be given, an equal blow is received in return. For this reason, if one in running hits his head against the head of another both are equally hurt. When a child knocks his head against a table, there is sound philosophy in the common saying that he has given the table as good a blow as he has received, though it may afford him no comfort. Many very interesting illustrations of this law of motion suggest themselves, of which I will give a few.

A swimmer, pressing the water downward and backward with his hands and feet, is carried along forward and upward by the reaction of the water. And in this case, as in every other, the greater the action the greater is the reaction; in other words, the more strongly he presses with his hands and feet, the more rapidly is he borne along by the reaction of the water against the pressure. A boat advances in proportion to the force with which the oars press against the water. So the rapidity of a steamboat depends on the force with which the paddles drive the water astern. Birds rise in the air by the reaction of the air against their wings as they are pressed downward. A sky-rocket pursues its rapid flight because a large quantity of gaseous matter issues from its lower end, and, being resisted by the reaction of the air, by its pressure throws the rocket upward. So if a ship fire guns from the stern its advance will be accelerated, but if from the bow it will be retarded. When a broadside is fired the ship inclines to the other side. In Fig. 126 (p. 136) is represented the plan of Barker's mill. It consists of a cylinder, _c_, arranged in a frame in such a way that it can revolve on the point upon which it rests. Water runs into it by a tube, _p_, and escapes by the branches, _a_ and _d_. These are so arranged that the reaction upon the issuing water makes the cylinder revolve rapidly, causing the ends of the branches to whirl around as indicated by the dotted lines and arrows.

If a spring be compressed between two equal bodies, it will throw them off with equal velocities. If they are unequal, the velocity of the smaller body will be greater than that of the larger, and in proportion to its smallness. For this reason, when a ball issues from a cannon, though the cannon and the ball are equally acted upon by the elastic or expansive force of the gases set free by lighting the powder, the gun is moved but very little because the force is diffused through so large a mass, while the ball being so much smaller moves with great velocity. When a volcano throws stones from its crater the earth may be compared to the cannon, the stones to the ball, and the explosive materials throwing the stones to the exploding powder projecting the ball. As the cannon is moved as much as the ball, so is the earth moved as much as the stones, the only reason that it does not move as far and as rapidly as the stones being that the force is diffused through so large a bulk. These examples illustrate very well the relation of action and reaction, for whenever there is an action of one body upon another it is as if a spring were between the two bodies, acting equally upon both. When a man jumps from the ground it is as if a spring were compressed between him and the earth, and this expanding moves the earth exactly as much as it does the man. He really kicks the earth away from him. The motion of the earth is not obvious because it is diffused through so large a mass. The case is parallel to that of the ball and cannon. The same force is exerted upon the man and the earth, but the man, like the ball, moves the most, and in proportion to his comparative smallness. So when a bird hops from the ground, the earth moves as really as the bird. If the bird hop from a twig, you perceive that the twig is moved by the pressing down of the bird as it rises. When it starts from the ground it exerts the same downward pressure, and moves the earth as really as in the other case it did the twig.

183. =Inertia Shown in the Communication of Motion.=--What is meant by the inertia of matter you have already learned in § 48. This property is exemplified in the communication of motion to any body, or, in other words, in setting it in motion. Of this I will give some illustrations. When the sails of a vessel are first spread to the wind the vessel does not move swiftly at once, for some time is required for the force applied to overcome the inertia of so large a mass, and to put it in rapid motion. Horses make a greater effort to start a load than they do to keep it in motion after it is started. If one be standing up in a carriage, and the horses start off suddenly, he falls backward, because his body, from its inertia, does not readily and at once partake of the motion of the carriage. If a person start forward quickly with a waiter filled with glasses in his hands, the glasses will slide backward. So if a person start quickly from his chair with a cup of tea in his hand, the tea will be thrown backward upon him.

You see from the foregoing illustrations that it requires some time to communicate motion to any body. I will give some illustrations of this fact of a more striking character. If a ball be thrown against an open door it will move the whole door, and perhaps shut it; but the same ball if fired will pass through the door without moving it perceptibly. In the latter case its velocity is so great that there is not time enough to communicate motion to the whole door, and it moves only that part of it with which it comes in contact. A bullet thrown with but little force against a window will crack a whole pane of glass; but if shot from a pistol it merely makes a round hole. So, also, a cannon-ball having a great velocity may pass through the side of a ship, doing perhaps comparatively little damage, while one moving with much less velocity may do vastly more damage by splintering the wood to a considerable extent. For the same reason a rapid ball hitting a person may occasion less suffering and do less harm than a slow ball; for a rapid ball kills merely the parts which it touches, leaving the flesh around in a sound state, while the slow ball bruises over a large space. If a large pitcher filled with some heavy liquid be quickly taken up the handle will break, leaving the pitcher behind. Large dishes are often broken in this way when heavily loaded.

184. =Inertia Shown in the Disposition of Motion to Continue.=--Of this I will cite some illustrations. As in the case of the ship, in the first illustration in § 183, it takes time to communicate motion to the whole ship, or, in other words, to overcome its inertia, so when the ship is once in rapid motion it does not stop suddenly when the sails are taken down, but its inertia tending to keep it moving is gradually overcome by the resistance of the water. If one be standing up in a carriage in motion, and the horses suddenly stop, he will be thrown forward, for his body has a motion in common with the carriage, and from inertia is disposed to go on when the carriage stops. When you strike your foot against any thing to get the snow off, you give the foot and the snow a common motion together, then arresting the motion of the foot, the snow from inertia passes on. The same thing is illustrated in striking a book against any thing to get the dust off. If a ship strike upon a rock every thing on board which is loose is dashed forward. The earth as it revolves on its axis has a velocity at the equator of about 1000 miles an hour. If this revolution should be suddenly arrested every thing loose on its surface, having acquired the motion of the earth, would be at once thrown eastward, just as the furniture, etc., on board ship are dashed forward when it is stopped by running against a rock. All the houses, and monuments, and structures of every kind would fall prostrate eastward. All the cities on our Atlantic coast would be plunged into the ocean; and while the waters would leave the western shores of the Atlantic, they would overflow its eastern shores, and deluge the continent of Europe, as water in a vessel on board a ship that had struck an obstacle would be thrown forward over its side.

185. =An Equestrian Feat.=--In the feat represented in Fig. 127 the only exertion made by the rider is to raise himself sufficiently to pass over the cord, and he comes down again upon the horse's back, simply because of the motion which he has in common with the horse, his feet going in the path represented by the dotted line. If he should attempt to throw himself forward, as in leaping from the ground, he would go too far, and perhaps strike upon the horse's neck instead of his back. Skill in jumping from a moving carriage consists in making proper allowance for the forward motion which is had in common with the carriage. Most persons are apt to overdo the matter, and so come to the ground prostrate, and with more violence than is necessary.

186. =A Case in Court.=--A dashing young man driving a light phaeton ran against a heavy carriage. His father was induced by his son's representations to prosecute the driver of the carriage for driving too fast. A knowledge of motal inertia very readily decided the case. The son and his servant both declared that the shock of the carriage was so great against the phaeton that they were thrown over the horses' heads. They thus proved themselves guilty of the fast driving, for it was their own rapid motion that threw them out when the phaeton was stopped by running against the carriage. The following case is a parallel one. If two boats, the one of large size sailing slowly up stream, the other a small one sailing rapidly down, run against each other, a man standing in the bow of the one going down will be thrown much farther forward than one standing in the bow of the other.

187. =Course of Bodies Thrown into the Air.=--It results from the principle that I have illustrated that when any body, as a stone, is thrown, as we say, straight upward, it does not, in reality, go up or come down perpendicularly. If it did it would come down at a great distance from us. Suppose it takes two seconds for it to go up and to reach the ground. If we are at the equator, in that two seconds we move from the point where we threw up the stone nearly 3000 feet eastward, and therefore if the stone rose and fell perpendicularly it would fall 3000 feet westward of us. Why, instead of this, does it fall at our feet? Because that when thrown into the air it has not only the upward motion given by the hand, but also the forward motion of the earth. It is a case similar to that of the rider in Fig. 127, the horse representing the surface of the earth and the rider the stone. For the same reason a man on board of a steamboat, though it move fifteen miles an hour, tosses up his ball or orange and catches it as well as if he were on land. This he could not do if both he and his orange did not have the same forward motion that the boat does. So, also, if a man fall from a mast-head he reaches the deck at the foot of the mast when the vessel is sailing rapidly, just as he would if it were lying still at the wharf. If he did not by inertia retain the forward motion which he had in common with the vessel he would fall at some distance behind the mast.

188. =The Earth and the Atmosphere.=--The air being held to the earth by attraction, § 151, it has a motion in common with the earth. It revolves with the earth just as the tire of a wheel revolves with the wheel. This being so, our winds are nothing but slight variations of this constant rapid whirl of the aerial coating of the earth. If the atmosphere were suddenly to stop whirling round with the earth we should move through it with a velocity of 1500 feet a second; and the destructive effect upon us would be the same as it would were the earth standing still while the air moved over its surface with this fearful velocity. A wise man, not reflecting that the atmosphere moved with the earth, proposed rising in a balloon, and waiting till the country to which he wished to go should be passing under him.

189. =Motion and Rest.=--Though we use the term rest in opposition to motion, it is obvious from some of the illustrations given that rest is merely a relative term, for not a particle of matter in the universe is at rest. Though when we are sitting still we call ourselves at rest, we are moving every hour, by the revolution of the earth on its axis, 1000 miles eastward, and 68,000 miles in our annual journey round the sun. Why, then, are we so insensible to these rapid motions? It is partly because the motions are so uniform, but chiefly because all things around us, our houses, trees, and even the atmosphere, are moving along with us. If we were moving along alone, even at a slow rate, while all these objects were standing still, we should be conscious of our motion, as we are when, as we ride along in a carriage, we see the objects at the road-side not moving along with us.

190. =A Comparison.=--The above can be made more clear and impressive by a familiar comparison. A man on board of a steamboat, by confining his attention to things within the boat, may, after a while, be almost unconscious of the boat's moving, if the water be smooth, though the boat may be going at the rate of fifteen miles an hour. If he be reading in the cabin he will think as little of his motion as he would were he reading in his parlor at home. If he should be blindfolded, and turned around a few times, it would be impossible for him to tell the direction in which the boat is going. Now it is with a man on the earth as it is with the man in the boat. He is unconscious of the motion of the earth for the same reason that the man in the boat is unconscious of the boat's motion. All objects around him are moving along with him, as the objects around the man in the cabin of the boat are moving along with him. We can carry the parallel farther. While the man sits in the cabin he knows not how fast the boat moves, nor even whether it moves at all. He must look out to decide this, and even then he may not be able to tell whether the boat moves, or whether he merely sees the water running by it. We often are actually deceived in this respect. A steamboat struggling against wind and wave may appear to those on board to be advancing when it is really stationary, or even when it is losing ground. So when we look at the sun we know not whether it is the sun or the earth that is moving. Mere vision, without reasoning on the subject, leads one to think that it is the sun that moves. For the same reason, if a child should be placed in a carriage for the first time without seeing the horses, but with its eyes fixed on objects at the road-side, he would probably think that all the fences and trees and rocks and houses are in motion.

191. =Absolute and Relative Motion.=--The motion of a body is said to be _absolute_ when it is considered without relation to the position of any other body. Its motion is said to be _relative_ when it is moving with respect to some other body. Absolute rest is unknown, for no body in the universe is known to be without motion. But a body may be relatively at rest, that is, in a fixed relative position to other bodies. Every body is in a state of absolute motion, and yet it may be in a state of relative rest. All objects that appear to us to be at rest have a very rapid absolute motion. They appear to be at rest merely because they have the same rapidity and direction of absolute motion that we have ourselves. And all the motions which are apparent to the eye are only slight differences in the common absolute motions, of which, though they are so exceedingly rapid, we are entirely unconscious. Thus, if I stand still, and another at my side walks at the rate of three miles an hour eastward, we both of us have a common absolute motion of 1000 miles in every hour, and he merely adds three miles to his thousand--I move 1000 miles, and he 1003. So if I sit still in my parlor, and my friend travels eastward at the rate of 20 miles an hour, I move every hour 1000 miles, and he 1020. And if he travel westward at this rate he really travels slower than I do--he has an absolute motion eastward of 980 miles, and I of 1000. At the same time we are both whirling on in our annual journey around the sun at the rate of 68,000 miles an hour.

192. =Obstacles to Motion.=--As motion is naturally disposed to continue (§ 49 and § 184), whenever it is stopped it does not spend itself, but is stopped by obstacles. The principal of these obstacles are: gravitation; the resistance of opposing substances--solids, liquids, and gases; and friction. When a stone is thrown into the air its upward motion is gradually destroyed by the attraction of the earth and the resistance of the air. Observe, now, why it descends. It is from the action of one of the causes which arrested its upward flight--the attraction of the earth. In its descent it is retarded by the resistance of the air, as it was in its ascent. This retardation is very obvious in the case of substances which present a large surface to the air, as a feather. A small piece of lead will outweigh many feathers, and therefore, as its quantity of matter is so much greater in proportion to its surface than that of a feather, it will fall to the ground much more quickly. That this is owing wholly to the resistance of the air can be proved with the air-pump.

Suppose that you have a tall receiver, Fig. 128, on the air-pump, and a piece of lead and a feather are placed at its upper part in such a way that they can be made to fall at the same instant. Exhaust the air, and then let them fall. They will go down side by side, as represented by the figure, and reach the bottom of the receiver at the same time, because there is no air there to resist the progress of the feather. The toy called the water-hammer illustrates the same thing. When water falls through the air the resistance of the air tends to separate its particles, as we see in the falling of water thrown up by a fountain. In the water-hammer, which is a closed tube containing a little water and no air, when the water is made to fall from one end to the other, as there is no air to divide it, it falls as one mass, and gives a sharp sound like the blow of a hammer. An instrument essentially like this can be made with a thin glass flask. Put a little water in it, and, after heating it to boiling over a spirit-lamp, cork the flask tightly, and then leave the water to cool. As all the space above the water was filled with steam when the flask was corked, it is a vacuum now that the steam is condensed.

193. =Relation of Bulk to the Resistance of Liquids and Gases.=--You have already seen, in § 192, that the more surface a body has in proportion to its weight the greater is the resistance of the air to its motion. This truth, which applies to liquids as well as to airs or gaseous substances, explains the fact that small bodies meet with proportionately more resistance than large ones. The body B, Fig. 129, you see is made up of eight cubes of the size of the cube _a_, that is, it has eight times the quantity of matter that _a_ has. Now if B were moving through air or water, any of its sides pushing the water before it would meet with only four times the resistance that a side of _a_ would, for its surface is only four times as large, and yet the body is eight times as large as _a_. And the greater the difference of size the greater is the difference of resistance. If B were a cube twenty-seven times as large as _a_ it would meet with only nine times as much resistance. You see here the reason that shells and cannon-balls can be thrown much farther than bullets and small shot. The sportsman does not throw away his shot by foolishly aiming at birds at great distances, and yet shells and large cannon-balls can be thrown the distance of several miles. The difference is not in the degree of velocity which the powder produces, but in the resistance of the air. It is for the same reason that rain falls with greater rapidity than drizzling mist.

As liquids and aeriform substances resist solids in motion in proportion to the amount of surface which the solids present to them, so also when they strike against solids they cause motion in them in proportion to the amount of surface acted upon. Thus a violent wind could not move a lump of tin, but could blow along a sheet of it, or tear up a roofing of it if it got beneath. So clouds of sand are raised into the air in the deserts of Africa, although the particles are of the same material as stones, and therefore have the same specific gravity. For the same reason dust, feathers, the down and pollen of flowers, etc., are blown about, although they are heavier than the air. A pebble is moved more easily by a current of water than a stone, because it has a larger surface, in proportion to its weight, to be acted upon by the water. For the same reason sand is moved more easily than pebbles, and fine mud than sand, though stones, pebbles, sand, and mud may all be of the same material. This explains why you will find mud where the current is slow, sand where it is faster, pebbles and stones where it is still faster, and where the current is exceedingly rapid you find nothing but large rocks--sand, pebbles, and stones not being able to resist its force. For the same reason, in the process of winnowing, the chaff is carried away by the wind; while the grain, presenting less surface in proportion to its weight to be acted upon by the air, falls to the floor.

In all the above cases the moving water or air may be considered as acting in opposition to the attraction of the earth, the latter pulling the substance down to the earth, and the former pushing it away from the earth. Of course, the more surface the water or air has to push upon the greater is the effect; and it is to be remembered that the attraction of gravity is as the quantity of matter, without any regard to amount of surface in the body attracted.

194. =Relation of Force to Velocity.=--It would seem at first thought that the motion produced in any body must be in exact proportion to the force producing it; that is, that twice the force which produces a given velocity would double that velocity, and three times would treble it, etc. This is true where there are no obstacles to motion, as in the case of the heavenly bodies moving in their orbits. But in all motions here upon the earth there are obstacles; and as reaction is always equal to action, the greater the velocity the greater is the reaction of the obstacle. If, therefore, you increase the velocity of any body, you not only have to communicate more motion to it, but you must overcome also the increased reaction. The rate of increase of force for increased velocities has been very accurately ascertained. This I will explain. A boat moving from B to A, Fig. 130, we will suppose, displaces a quantity of water represented by the space between the two lines extending from B to A. Now if it move from B to C, it displaces twice the bulk of water B C; and as it is displaced in the same time that B A was, each particle is displaced with twice the velocity. Double the force is required to displace a double portion of water, and to do this with double the velocity the force must be doubled again. So if the boat is made to move three times as far in the same time, that is from B to D, three times the quantity of water is displaced, and each of these three portions, B A, A C, and C D, is displaced with three times the velocity. The force required, then, to do this is nine times that required to carry the boat from B to A in the same time. It is plain, therefore, that with velocities represented by the numbers 1, 2, 3, 4, etc., the forces requisite to produce these velocities must be as the squares of these numbers; viz., 1, 4, 9, 16, etc. This law is a very important one in a practical point of view. For example, it shows us how much larger a quantity of coal is required to produce in steamboats a high velocity than a moderate one. Its application too to the science of gunnery is important.

195. =Relation of Shape to Velocity.=--The resistance of air or water to a flat surface is greater than to a convex one, because the latter readily turns the particles to the one side and the other. So, also, a concave surface is resisted much more than a flat one, because the particles of the air or water can not so easily escape sideways. Fishes are of a spindle-like and slender shape, that they may have as little resistance as possible from the water. It is for this reason that a fish has no neck, for if it had one the upper portion of its body would, from the resistance of the water striking against it, prove a serious impediment to rapidity of motion. Mankind have in some measure imitated the shape of fishes in their boats and ships. Boats which are intended to bear light burdens and go swiftly are made very long and narrow. The webbed feet of water-fowls, when they are moved forward, are folded up so as to meet with as little resistance as possible; but when they are moved backward they are spread out so as to press against the water a broad concave surface. For the same reason the wings of a bird are made convex upward and concave downward; and when it moves its wing upward it makes it cut the air somewhat edgewise, but in moving it downward it presses directly with the whole concave surface.

196. =Friction.=--Friction is generally an obstacle to motion. When we roll a ball, the more rough is the surface on which we roll it the greater is the friction and the sooner is the ball stopped. Friction lessens the rapidity of motion in machinery, and to prevent this as far as possible oiling and other expedients are employed. But sometimes friction is a cause of motion, as, for example, the friction of the driving-wheels of a locomotive upon the rails. In this case the wheel pushes backward on the rail at each successive point of contact. To make this clear, suppose a common wheel is deprived of its rim and is made to revolve on the ends of its spokes. The end of each spoke gives a backward push as it strikes the ground. Now the rim of a wheel makes the same pushes, but they are more numerous--they are continuous, being made by all the successive points in the rim. Sometimes the rails of a railroad are too smooth from frost or some other cause, and then sand is thrown upon them to give the locomotive a start. The sand serves to prevent the wheels from sliding by enabling them to get some hold upon the rails in their backward pushes.

197. =Friction of Liquids in Tubes.=--So easily does water flow along that we should not at first view suppose that it would be delayed much from friction as it passes through pipes or along channels. But the retarding influence is considerable. An inch tube 200 feet long, lying horizontally connected with a reservoir, will discharge water not one quarter as fast as an inch orifice in the side of the reservoir. Sudden turns in a pipe should be avoided, because they occasion so much friction against the sides of the pipe and among the particles of water by disturbing the regularity of the current. In the entrance of the arteries into the brain, in order to prevent the blood from flowing too rapidly into this organ, there are sudden turns in the arteries to retard the blood; and in grazing animals, as there is special danger that the blood will flow too freely to the brain as the head is held down in eating, there is a special provision to prevent this in a net-work of arteries. If the arteries of the brain in such animals were straight tubes they would continually be dying of congestion of the brain or of apoplexy.

Friction in a small pipe is greater in proportion to its size than in a large pipe. In a pipe an inch in diameter water will not move more than one-fifth as fast as in a tube two inches in diameter. This may be made clear by Fig. 131, in which is represented the area of a small tube inside of the area of a tube of twice its diameter. Suppose the effect of the friction in the large tube to extend in to _a_. In the small one it will extend in as far, that is, to _b_. But _e a_ is about five times as long as _e b_, so that there is full five times as much water clear of friction in the large tube as there is in the smaller one.

198. =Friction in Streams.=--The retarding effect of friction is very obvious in brooks and rivers. The water in the middle of a stream runs much more rapidly than it does near its banks. When a river is very shallow at its sides the water there scarcely moves, though in the middle the water may be running at a rapid rate. A tide, therefore, flowing up a river, moves more freely near its banks than it does in the middle of the stream, because it meets with less resistance there from the downward current. Water moves less rapidly at the bottom of a river than it does at the surface. For this reason, if a stick be so loaded at one end as to stand upright in water, in the current of a river its upper end will be carried along faster than its lower end, and therefore it will incline forward, as in Fig. 132. As the sea rolls in over a beach, each wave at length pours over its crest and breaks, because the lower part of the wave is retarded by friction on the beach. Were it not for the constant retardation of friction at the sides and bottom of rivers, and at their bends, those rivers which have their rise at a considerable height above the level of the sea would acquire an immense velocity. Thus the Rhone, drawing its waters from 1000 feet above the level of the ocean, would pour them forth with the velocity of water which had fallen perpendicularly the same height, that is, at the rate of 170 miles an hour, did not friction continually diminish the velocity.

199. =Waves.=--Waves are generally formed by the friction of air upon water. Observe how they are formed. As soon as any portion of water is raised above the general surface it tends by gravity to fall to a level with the water around it, and in doing so the portion next to it is forced upward, forming another wave; and so one wave produces another, each one being smaller than the preceding, till at length the motion is wholly lost. This is always the process when the cause of the motion is a single impulse, as when a stone is dropped into the water. But when the waves are produced by a succession of impulses, as when wind makes them, they are mostly of the same size. It is quite a common notion that the water moves as rapidly as the waves appear to do; but the water really remains nearly stationary, rising and falling, while merely the form of the wave advances. The same wave is made up continually of a succession of different portions of water, or rather it is a succession of different waves. This is very well illustrated by the waving of a rope or carpet. In an open sea a wave slopes regularly on either side; but when it comes near the shore, for the reason given in § 198, it grows more and more nearly perpendicular on the side toward the shore, till at length it falls over, and if it be very large the roar thus caused by its breaking is heard to a great distance.

200. =Height of Waves.=--"So awful," says Dr. Arnot, "is the spectacle of a storm at sea that it is generally viewed through a medium which biases the judgment; and lofty as waves really are, imagination pictures them loftier still. Now no wave rises more than ten feet above the ordinary sea-level, which, with the ten feet that its surface afterward descends below this, gives twenty feet for the whole height from the bottom of any water-valley to an adjoining summit. This proposition is easily verified by a person who tries at what height on a ship's mast the horizon remains always in sight over the top of the waves--allowance being made for accidental inclinations of the vessel, and for her sinking in the water to much below her water-line, at the time when she reaches the bottom of the hollow between two waves. The spray of the sea, driven along by the violence of the wind, is of course much higher than the summit of the liquid wave; and a wave, coming against an obstacle, may dash to a great elevation above it. At the Eddystone Light-house, when a surge breaks which has been growing under a storm all the way across the Atlantic, it dashes even over the lantern at the summit."

201. =Momentum.=--The momentum of a body is its force when in motion. In estimating the momentum of any body two things must be considered--its velocity, and its quantity of matter or weight. A bullet fired from a gun has a vastly greater force, or power of overcoming obstacles, than one thrown by the hand, from its greater velocity. Now suppose the weight or quantity of matter to be increased ten times, and that it moves with the same velocity as before, it will have ten times as much force as before, and will overcome ten times as great an obstacle. For this reason a small stone dropping upon a man's head may do but little harm, while one ten times as large, falling from the same height, may stun and perhaps kill him. But if the large stone could fall with only one-tenth of the velocity of the small one, the effect of both would be the same. Let this example illustrate the rule for calculating the momentum of moving bodies, viz., multiply the quantity of matter into the velocity: Let the weight of the small stone be 1 ounce, and that of the large one 10 ounces. If they fall from a height of 16 feet the force with which the large one will strike will be expressed by 160 (16×10), that of the small one by 16 (1×16). Suppose, now, that by some force in addition to gravity the small one could be made to move ten times as fast as the large one, the force with which it would strike would be equal to that of the large one, and would be expressed by the number 160.

I will illustrate this in another way. Let _a_ and _b_, Fig. 133, be two balls of clay of equal size hanging over a graduated arc. Now if _b_ be let fall from the top of the arc, 6, on striking against _a_ it gives half of its motion to _a_, and they both move on together. But how far will they go? To 3, on the other side of the arc. Why? Let the quantity of matter in each ball be called 1, and the motion of _b_ 6. The momentum will therefore be 6. Now the momentum of the two together will be the same after the blow as that of _b_ was before it. But the quantity of matter is twice as great, and must be called 2. Therefore the motion must be represented as 3, to make the momentum 6 (2×3). But suppose that _b_ is twice as large as _a_. Falling from 6, its momentum would be represented by 12 (2×6). After it has struck _a_, the momentum of the two together would be the same as that of _b_ before the stroke; but the quantity being 3, the motion would be represented by 4. They would therefore move to 4 on the arc.

202. =Examples.=--A few examples illustrating momentum as being compounded of quantity of matter and velocity will suffice. If a musket-ball of an ounce weight were so much spent as to move with only a velocity of a foot in a second, its force would be so small that if it hit any one it would do no harm. But a cannon-ball weighing a thousand ounces moving at this slow rate would have a very great force--equal, in fact, to the momentum of an ounce ball moving 1000 feet in a second.--If a plank push a man's foot against a wharf he will scarcely feel it; but if the plank, instead of being alone, is one of a thousand planks fastened together in a raft, and the whole move with the same velocity, the force will be increased a thousand-fold, and the plank will crush the foot. So, also, if the one plank when alone should move a thousand times as fast as the whole raft, the same result would follow.--So soft a substance as a candle can be fired through a board from the momentum given to it by an immense velocity.--Perhaps there is no better example of the great force given to a substance by an enormous velocity than we have in the wind. So light a thing is air that people think of it as almost nothing. But let it be set in rapid motion, and the velocity gives to it a force, a momentum, which will drive ships upon the shore, throw over buildings, and tear up trees by the roots. In this last example we see beautifully illustrated the meaning of the expression quantity of motion. In the moving air each particle does its share of the work in the destructive effects mentioned. Each particle, therefore, may be considered as a _reservoir_ of motion, and the quantity of motion in any case depends upon the quantity which each particle has and the number of the particles.

203. =Production of Great Velocities.=--When there are no obstacles to motion great velocities may be produced by a single impulse. Thus at the beginning the Creator gave a single impulse to each of the heavenly bodies, producing enormous velocities, which continue unaltered year after year and age after age, because these bodies fly in their orbits through space where there is no resistance of any thing like air to retard the motion. But in all the motions that we see around us there are obstacles continually retarding them; and therefore no very rapid motion is produced by any single impulse, but a succession of impulses is required to accumulate sufficient momentum so to overcome the obstacles as to secure a great velocity. I will give a few examples in illustration. One of the best examples we have in the fall of bodies to the earth. You know that the greater the elevation from which a body falls the greater is its velocity, and therefore the greater the force with which it strikes. Why is this? If it fell because of a single impulse making it go toward the earth, this would not be the case, and if there were no air in the way the velocity would be uniform; but the resistance of the air would retard the velocity, so that if a number of bodies should receive the same impulse at different elevations, the one the farthest off would be the most retarded, and therefore come down slower than all the rest. In this case, the higher the elevation from which a man should fall the less would be the injury. But a body does not come to the ground by a single impulse, but by a succession of impulses, or rather a continued impulse. Every moment that the body is coming down it is drawn upon by the attraction of the earth, and this continued action of the cause of the motion makes it continually increase in rapidity. It is on the same principle of continued action that a man lifts his hammer high when he wishes to inflict a heavy blow. In this case both gravitation and the muscular power of the arm exert their force on the hammer through the whole space. A horse in kicking does the same thing, and by the great length of the leg the velocity given to the foot by this continued action of the muscles is very great. An arrow is not shot by a single momentary impulse of the bow-string, but the string, by following it through a considerable space, gives it a continued impulse. The action of gunpowder upon a bullet issuing from a gun is apparently an instantaneous and single impulse, but it is not really so. The great velocity given to the bullet is given to it by the continued impulse of the expansive force of gases produced from the powder, and it therefore depends much on the length of the barrel. If this be short, the force of the powder is not confined long enough to the bullet to give it a great velocity.

204. =Arrest of Great Velocities.=--As a continued force is required to produce great velocities, so a continued resistance is necessary to arrest them. It is by the gradual or continued resistance of the air that the motion of a cannon-ball is destroyed. Now if instead of this gradual resistance any hard substance, as a block of granite, were opposed to the progress of the ball, it would be at once broken asunder. We see then the reason that a hard substance of moderate thickness does not offer so effectual a resistance to a body moving very rapidly as some substance of a more yielding kind and of greater bulk. For example, a bale of cotton will arrest a ball which would pass through a plank, for the cotton yielding easily permits the force of the ball to be felt and resisted by a larger bulk, while the wood, not yielding, opposes but a small portion of its whole bulk to the force of the ball, and therefore does not arrest it; in other words, the momentum of the ball is communicated to a much larger quantity of matter in the cotton than in the wood. These principles afford a ready explanation of a feat which is sometimes performed. A man lies upon his back, and, having an anvil carefully placed upon his chest, allows some one to strike a heavy blow with a hammer upon the anvil, and no injury is received. Why? Because the momentum or force of the hammer is diffused throughout the bulk of the anvil, and then again through the bulk of the yielding chest. The man takes good care to have his lungs well filled with air at the moment of the blow, for this increases the bulk and elasticity of the chest, and thus promotes the diffusion of the momentum. If the blow of the hammer were received directly upon the chest great injury would be done, for the force would now be spent upon one small spot alone.--The principles above elucidated are applied by men instinctively in their common labors and efforts. You see a man catching bricks that are tossed to him. As he receives the bricks into his hands he lets his hands and the bricks move together a little way, so that he may gradually arrest the motion of the bricks. To do it suddenly would give him a painful lesson on momentum. So when a man jumps from a height he does not come to the ground in a straight position. This would cause a sudden and therefore a painful arrest of the motion of the whole body. To avoid this he comes to his feet with all the great joints of his body bent, so that the different portions approach the ground successively, his head having its motion arrested last.

205. =Communication of Motion in Elastic Bodies.=--Momentum is transferred from one body to another very differently in elastic from what it is in non-elastic bodies. As you saw in § 201, when one non-elastic body strikes upon another the momentum is divided between them, and both move on together. Now if _a_ and _b_, Fig. 133, were elastic bodies, as ivory balls, and _b_ should be let fall against _a_, it would give all its momentum to _a_. Therefore _b_ would stop, and _a_ would move on to the same height from which _b_ came. The reason is, that the velocity lost by _b_ and received by _a_ is just double what it would be if the balls were non-elastic. For the same reason, if _a_ and _b_, being elastic, meet each other from equal heights on the arc, they will both rebound, and return to the same heights from which they came. But if non-elastic they simply destroy each other's momentum and stop. The effect produced in the former case is just twice as great as in the latter, as you may see by reckoning on the arc. For the same reason, too, if you have a row of elastic balls, as in Fig. 134, and let _a_ fall from the point _i_ upon _b_, it will stop there; and communicating all its momentum to _b_, this momentum will pass from _b_ to _c_, and so on through all the row of balls to _e_, the last one, which will fly off to the point _h_, at the same height with _i_, the point from which _a_ fell. If _b_ be held still, and _a_ be let fall upon it, _a_ will rebound to the height from which it fell, for then the compressed elastic spring (§ 39) of each ball, as _b_ is immovable, communicates all the motion to _a_. It is for this reason that an elastic ball, on being thrown against any thing fixed, rebounds. If what it is thrown against be perfectly elastic it rebounds with a force equal to that with which it is thrown.

206. =Reflection of Motion.=--If an elastic body be thrown perpendicularly upon a surface it rebounds in the same path in which it is thrown. But if it hit the surface obliquely it is thrown off or reflected in a different direction. Thus a ball thrown from _b_ upon _c_, Fig. 135 (p. 158), will return in the line drawn to _b_. But if it be thrown from _d_ it will be reflected in the line _c a_. Now the angle _d c b_, called the angle of incidence, is always equal to _b c a_, the angle of reflection. The same, you will find in other parts of this book, is true of sound and light and heat.

207. =Uniformity of Motion.=--Since motion, when once begun, is disposed to continue unless arrested by obstacles, it is naturally uniform both in its velocity and its direction. I will speak now only of velocity. Suppose a body to be set in motion, and to meet with no opposition from friction, or the resistance of air, or attraction, it would move on forever, and with the same velocity with which it began. Now precisely these circumstances we have in the motion of the heavenly bodies in their orbits. They are, it is true, under the influence of attraction, but in such a way, as you will soon see, as not to interfere with the uniformity of their motion. Were it not for this uniformity we should have no regularity of times and seasons. It is only by the uniform motion of the earth round the sun, and round its own axis, that we can calculate for to-morrow, or next week, or next year. If these motions were irregular it would throw confusion into all our calculations for the future and all our recollections of the past. We can measure time by nothing else but regular motion, and were there no regular motion we should have merely the very inaccurate measure furnished by our sensations. To measure time with accuracy we take some great and extensive uniform motion as our standard. Thus, the revolution of the earth around the sun we take as one division of time, and call it a year. We observe that during this time it whirls around on its own axis 365 times, and the time occupied by each of these revolutions we call a day.

208. =The Pendulum.=--Various modes of measuring time have been adopted by mankind. At first time was inaccurately divided by merely observing the sun. But after a while man resorted to various contrivances to measure short periods of time with accuracy. All of these depend upon the uniformity of motion alone. The sun-dial measures time by the uniform movement of the shadow on its face, caused by the uniform movement of the earth in relation to the sun. The hour-glass measures time by the uniform fall of sand produced by the attraction of gravity. The best measurement of time is by the comparatively modern invention of clocks and watches, in which time is divided into very minute periods by the uniform motion of the pendulum or the balance-wheel. The pendulum furnishes an interesting example of motion kept up by the influence of gravity. It was not till the time of Galileo, less than three centuries ago, that its operation was understood and appropriated to the measurement of time. He observed that chandeliers hanging from lofty ceilings vibrated very long and uniformly after they were accidentally agitated, and the thought of the philosopher evolved from this phenomenon the most important results. Though it had been before men's eyes in some shape or other since the creation, it was reserved for Galileo to observe its significance, and the result is that the pendulum has become man's time-keeper over the whole earth.

209. =Explanation of its Operation.=--A pendulum consists commonly of a ball or weight at the end of a rod suspended so as to vibrate with little friction at the point of the suspension. Let _a b_, Fig. 136, represent such a pendulum. When it is at rest it makes a plumb-line hanging toward the centre of the earth. If it be raised to _c_ and be left to fall, the force of gravity will not only carry it to _b_, but, by the accelerated velocity or accumulated momentum which it gives it in its descent, it will carry it to _d_. The same would be true of its return from _d_. And it would vibrate forever in this way if it could be entirely freed from the resistance of the air and friction. But, as it is, the pendulum left to itself gradually loses its motion from these obstacles. In the common clock the office of the weight is to counteract the influence of these obstacles, and keep the pendulum vibrating. In the watch the mainspring performs the same office to the balance-wheel.

The times of the vibrations of a pendulum are nearly equal whether the arc it describes be great or small. For when the vibration is a large one the velocity which the pendulum acquires in falling is greater than when the vibration is of small extent. The reason is that the higher it rises the more steep is the beginning of its descent. Thus _a c_, Fig. 137, is steeper than _c b_.

210. =Gridiron Pendulum.=--The longer a pendulum is the longer time does its vibration occupy. It requires a pendulum of the length of a little over thirty-nine inches to vibrate seconds. Cold weather, by contracting the pendulum, makes it vibrate quicker than in summer, and so makes the clock go faster. Various contrivances have been resorted to in order to counteract the variation of length in pendulums by heat and cold, but what is called the gridiron pendulum is the best. In this pendulum an ingenious use is made of the fact that heat expands brass nearly twice as much as it does steel. A simple form of this pendulum is given in Fig. 138. The middle rod is made of brass, and the side rods, _b_ and _c_, of steel. Suppose that the brass rod expands or increases in length half an inch. The rod _c_ would be drawn upward by it, and the rod _b_ downward, each one quarter of an inch; but this effect is counteracted by the expansion of each steel rod, which is half that of the brass, that is, one quarter of an inch. The ball _d_, therefore, always retains the same distance from the point of suspension, _e_. In Fig. 139 you have a gridiron pendulum of a more compound character, a part of the bars being steel, and a part brass.

211. =Motion Disposed to be Straight.=--When a body is set in motion, if it be left to itself--that is, if nothing interfere with its motion--it will move in a perfectly straight line. It requires some interference from some force to bend the motion. You will readily see from the views which I have given you that there never is any motion that is, strictly speaking, straight, because every motion is in some measure compound; that is, each cause of motion is modified in its action by other causes of motion. But we can approximate very nearly to straight motion by making one cause preponderate very much over other causes. This I will illustrate. If we fire a bullet horizontally from a gun it is acted upon by three forces: the propulsive force of the powder, the resistance of the air, and the attraction of the earth. The action of the second of these is in direct opposition to the first, and therefore only retards the motion, and does not tend at all to turn it from its straight course. This is seen in the fact that the ball is turned neither to the right hand nor to the left. But the third force tends to make the ball bend its course toward the ground. It does this from the instant that the ball leaves the gun throughout its flight, but so slightly that practically we can consider the ball as going straight for short distances. When we take a long range we must make allowance for this bending down of the motion. Accordingly, for the sake of precision, a double sight is provided in modern guns, as seen at A and B, Fig. 140. This you see secures the pointing of the gun a little above the level of the object aimed at, that level being indicated by the dotted line.

The greater is the propulsive force the more nearly to a straight line is the path of the propelled body. This may be seen very clearly in Fig. 141, representing the issuing of water at different points from a vessel. As pressure in a liquid is as depth, § 121, the force with which the water is thrust out is greater at C than at B, and at D than at C. The issuing stream, therefore, is most nearly straight at the lowest point, D.

The motion of projectiles, thus alluded to, will be more particularly noticed farther on.

212. =Compound Straight Motion.=--We call that motion compound which is produced by two or more forces acting upon the body. This may be straight or curved. I will first speak of the straight. If a man attempt to row a boat straight across a river, the point which he will reach will not be directly opposite to that from which he started, but below. Two forces act upon the boat: the current tending to carry it straight down the stream, and his rowing tending to carry it straight across. The boat will go in neither of these directions, but in a line between them. Let A B, Fig. 142, represent the bank of the river, from which he starts at A, with the bow of the boat pointing to C, on the opposite bank. Suppose now that in the time that it takes him to row across the current would carry him down to B if he did not row at all. He will in this time, by the two forces together, reach the point D, opposite to B, his course being the line A D. So if the wind blow upon a vessel in such a way as to carry it eastward, and a current is pushing it southward, the vessel will run in a middle line, viz., southeast. For the same reason if a boy kick a foot-ball already in motion, it will not be carried in the direction in which he kicks it, but in a line between that direction and the direction in which its former motion was carrying it. In swimming, flying, rowing, etc., we have examples of compound motion, the middle line between the directions of the forces always being taken by the body moved.

If we take Fig. 142, illustrating the movement of the boat, and draw two lines, one from A to C and the other from B to D, we shall have the parallelogram A C D B, Fig. 143, in which the line A C represents the force of the rowing, A B the force of the current, and A D the path of the boat. You see, then, that if we wish to find in what direction and how far in a given time a body acted upon by two forces will move, we are to draw two lines in the direction of these forces, and of a length in proportion to the distances to which they would move it in that time; then by drawing two lines parallel to these we shall have a parallelogram, and the diagonal of this will represent the distance and the course of the moving body. If a body be acted upon by two equal forces and at right angles to each other, the figure described will be a square, as you see in Fig. 144. If they vary from being at right angles to each other the figure will vary in the same proportion from the square figure, as seen in Figs. 145 and 146. In the three figures A B and A D represent the two forces, and A C the resulting motion. You observe by these diagrams that the nearer the two forces come to being in the same direction the farther will they move the body.

You see this in the different lengths of the diagonals in Fig. 144 and Fig. 146. The more nearly, therefore, the wind coincides with the current the more rapidly will a vessel be carried along before the wind. When, on the other hand, the angle at which two forces act upon a body is much greater than a right angle, they will propel it but a small distance. Thus if two forces act on a body in the directions D A and D C, Fig. 147, they will move it only the distance represented by the diagonal D B. This diagram represents the motion of a vessel sailing almost directly against a current by a wind the force of which is equal to that of the current, while Fig. 146 represents the motion of a vessel where wind and current being of equal force very nearly coincide. In the above diagrams I have supposed the forces to be equal; but the same truth can be shown in regard to unequal forces as seen in Fig. 143.

213. =Curved Motion.=--No single impulse can produce a curved motion. Neither can two or more impulses communicated at one time. In both of these cases the motion would be in a straight line. Curved motion may be produced by two forces, one of which gives it a single impulse, and the other acts upon it continuously. A familiar example you have in a ball whirled around at the end of a string. You can give it an impulse, and then, holding it in your hand, let it whirl. Here the impulse you give the ball is one force, and the tension of the string is the other, the latter acting continuously. Your hand holding the end of the string is the centre about which the motion revolves; the impulse which you have given the ball tends to make it fly away from the centre in a straight line, and hence is called the _centrifugal_ force; the tension of the string keeps it from thus flying off, and so is called the _centripetal_ force. When the earth, at the creation, was put in motion it would have moved in a perfectly straight line, were it not constantly drawn toward the sun by attraction, the continuous action of this latter force being the same as the tension of the string in the case of the whirling ball. The force of attraction, then, is the centripetal force of the earth, and the impulse which was given to it by the Creator in the beginning is its centrifugal force; and, balanced between these two forces, the earth and all the heavenly bodies move uniformly onward in their orbits. The centrifugal force you see in these illustrations is simply the tendency of motion to a straight line from the inertia of matter; and this is constantly counteracted by the centripetal force.

214. =Illustrations of Centrifugal Force.=--When a wet mop is whirled the water flies off in every direction by its centrifugal force. On the same principle a dog, coming out of the water, shakes off the water by a semi-rotary motion.--When a suspended bucket of water is turned swiftly around the water rises high on its sides, and leaves a hollow in the middle. It is the tendency to fly away from the centre of motion that causes this.--Large wheels, revolving with great velocity, have been broken by the centrifugal force of its particles, and hence the necessity of having such wheels made very strong. The immense grindstones used in gun-factories have sometimes been broken through in the middle, or have flown into pieces from the same cause.--A man riding horseback on turning a sharp corner inclines his body toward the corner, to avoid being thrown off by the centrifugal force. So, in the feats of the circus, a man standing on a horse running at full speed around the ring inclines his body strongly inward, as you see in Fig. 148 (p. 167). The horse also instinctively inclines in the same direction for the same reason. If the rider finds himself in danger of falling, by making the horse go a little faster, thus adding to the centrifugal force, the difficulty is relieved.--The centrifugal force is made use of in milling. The grain is admitted between two circular stones by a hole in the centre of the upper one, and as the stone revolves it constantly moves toward the circumference, and there escapes as flour.

215. =Bends in Rivers.=--We see the operation of the centrifugal force in the bends of rivers. When a bend has once commenced in a river it is apt to increase, for as the water sweeps along the outer bank of the bend it presses strongly against it, just as the water in the whirled bucket, § 214, presses against its sides, by its centrifugal tendency, or, in other words, its tendency to assume a straight motion. Of course the result is a wearing away of this outer bank, and in proportion to the looseness of the material of which it is composed and the velocity of the river's current. And when one bend is formed another is apt to form below, but in an opposite direction. The water, by sweeping along the bend _a_, Fig. 149, is directed by it toward the opposite bank at _b_, and makes a bend there also.

It is in this way that a river, running through a loose soil, the Mississippi, for example, acquires a very serpentine course. As the water in the whirled bucket rises around the sides, so in the river the water will be higher against the bank _a_ than on the opposite side. Eddies and whirlpools are produced on the same principles, when water is obliged to turn quickly around some projecting point. If a current were moving swiftly along the shore _a_ toward the point _b_, Fig. 150, it would be directed outward by the resistance of this projection, and so a depression would be left at _c_, just behind it, and this depression would be surrounded by a revolving edge of water.

216. =Application of the Centrifugal Force in the Arts.=--Much use is made of the centrifugal force in the arts, but I will give but two examples. In the art of pottery the clay is made to revolve on a whirling table, the workman at the same time giving the clay such shape as he chooses with his hands and various instruments. In doing this he constantly has reference to the centrifugal force, giving the table a velocity proportioned to the amount of this force which is needed in each stage of the operation. The most beautiful application of this force that I have ever witnessed is in the manufacture of common window-glass. The glass-blower gathers up on the end of his iron tube a quantity of the melted glass, and blows it out into a large globe. When it is of sufficient size and thinness he places it on a rest, as you see in Fig. 151 (p. 169). A second man now comes with a rod having some melted glass on the end, and attaches this to the globe at a point opposite to that where the tube of the first man is joined to it. There now comes a boy, and, giving this tube a quick blow, severs its connection with the globe, leaving a hole in the globe where the glass breaks out. The second man, having the globe attached to his rod, carries it to a blazing furnace, and resting the rod on a bar at its mouth, puts the globe directly into the flame. The glass is soon softened, and he whirls the globe continually around. The hole in the globe enlarges by the centrifugal force, and at length by this force the globe is changed into a flat, circular disk. Panes of glass which are called bull's-eyes are cut from the centres of these disks.

217. =Steam-Governor.=--The operation of the centrifugal force is beautifully exemplified in this regulator of the steam-engine. It consists of two heavy balls, Fig. 152, suspended by bars from a vertical axis, the bars being connected to the axis by hinges. The bars have also a hinged connection at their lower ends with two smaller bars, and these latter have a similar connection with a collar that slides up and down on the axis. Now the faster the axis turns the farther the balls fly out from it, from the centrifugal force, and the higher the collar slides up on the axis. From the collar extends, as you see, a lever. This is connected with a valve in the steam-pipe, and so regulates the amount of steam that enters the working part of the engine. The object of this ingenious contrivance is to make the engine regulate its own velocity. When it is not working too fast the valve in the steam-pipe is wide open. But the moment that it works too rapidly the balls extend out far from the axis, so that the collar rises, and by the lever partly closes the valve. Less steam, therefore, can come to the engine, and the engine working in consequence less rapidly, the balls fall again, opening the valve. You see, then, that the regulation of this valve by the governor effectually prevents the action of the engine from becoming too rapid.

218. =Shape of the Earth Influenced by the Centrifugal Force.=--If the potter should make a ball of soft clay revolve rapidly around on a stick run through it, the ball would bulge out at the middle, where the centrifugal force is greatest, and would be flattened at the ends where the stick runs through it. This is precisely what has happened to the earth. At the equator, where the centrifugal force is greatest, it has bulged out about thirteen miles, while it is flattened at the poles. This shape was of course assumed before the earth became solid. In Fig. 153 we have the shape of the earth represented, N S being the polar diameter, and E E' the equatorial diameter. The tendency to take this shape from the centrifugal force may be illustrated by the instrument represented in Fig. 154. It consists of a set of circular hoops of brass, with an axis, _b a_. The hoops are fastened to the axis at _a_, but are left free at _b_. By a little machinery at the top they can be made to revolve rapidly, and bulging out at the sides by the centrifugal force, they slide down on the axis at _b_.

219. =Projectiles.=--I have already spoken of projectiles in § 211. You saw there that any body, as a cannon-ball, which is projected horizontally, falls to the earth in a curved line. Two forces act on the ball; viz., the projectile force given by the powder and the force of gravitation. The force of gravity being always the same, the shape of the curve which the projected body describes must depend on the force with which it is projected. This is very strikingly exemplified in the curves described by the different streams of water in Fig. 141. But whether the projectile force be great or small, the moving body thrown horizontally will in every case reach the ground in the same time. Thus if two cannons stand side by side on a height, one of which will send a ball a mile and the other half a mile, the two balls, if fired together, will reach the ground at the same instant, though at first thought it would seem that the ball which travels twice as far as the other would take a longer time to do it in. This is because the _horizontal_ force of the ball does not oppose in the least the _downward_ force of gravity. If it were thrown upward instead of horizontally, the projectile force would be opposed to gravity, and in proportion as the direction came near to being vertical. As horizontal force does not interfere with the action of the force of gravity, it follows that a ball dropped at the instant at which another is fired will reach the ground at the same instant that the fired ball does. This can be made clear by Fig. 155. Suppose it takes three seconds for a ball to fall from the top of a tower to its foot. In the first second it falls to _a_. The ball projected horizontally from the cannon, being operated upon by the same force of gravity, will fall just as far, and will be on a level with it at _b_. Both balls fall farther and farther each second, both being accelerated in the same degree because it is done by the same force. The projected ball will reach _d_ when the falling ball is at _c_, and the plain at _f_ when the falling ball is at _e_, the foot of the tower. The same holds true in all cases. A bullet dropped from a level with the barrel of a gun, paradoxical as it may seem, will fall to the ground no sooner than one which is shot from the gun.

220. =All Falling Bodies really Projected.=--When a body falls from any height, it does not, as you have already seen in § 187, fall in a straight line, as it appears to do. It falls in a curved line, for, like all projectiles, it is acted upon by a horizontal force as well as the force of gravity. But what is this horizontal force? It is the motion which the body has in common with the earth in its rotation on its axis. In this rotation the height from which the body falls goes to the eastward 1500 feet in a second. If, therefore, the body did not partake of the motion of the earth, and went to the ground in a _straight_ line in a second, it would be when it reached the ground 1500 feet westward from the foot of the height from which it fell. But it does partake of the earth's motion, and goes eastward as fast as the height does, and so describes the curved line of a projectile. Suppose a ball falls from a height A, Fig. 156, and in a second of time that height passes to C. The forward or projectile force would tend to carry the ball to C, and the force of gravity would tend to carry it to B. But both forces acting together, it pursues a middle path, and this path is a curved line, because one of the forces is a continued force, § 213. For the same reason, if a ball be dropped from a railway car in motion, and it takes a second for it to fall, it will be at the end of that second just under that part of the car from which it fell. Although the car may have moved a considerable distance, the dropped ball, partaking of its motion, goes along with it in its fall. So a ball dropped from a mast-head when a ship is in motion goes along with the ship in its fall. The ball in each of these cases describes in its fall a curved line.

221. =Motion in Orbits.=--Why is it, let us ask, that a cannon-ball shot horizontally from some great height will not revolve around the earth like the moon. It has the same two forces acting upon it as the moon has--viz., a projectile force, and the attraction of the earth--and both ball and moon describe a curve in their motion. But the curve of the ball bends to the earth, while that of the moon ever sweeps around the earth. Why is this? First, there is the resistance of the air continually retarding the velocity of the ball. But, secondly, even if the ball could be projected from an elevation sufficiently high to be outside of the atmosphere, the force of the projection would not be great enough. We know, from the rate of progress of the heavenly bodies in their orbits, that it would require an immense velocity to keep the ball from being brought to the earth by its attraction. The Creator of these worlds, when he launched them into their orbits, gave them precisely that impulse which is needed to balance the centripetal force of attraction, and so they pursue a middle course between the two directions in which these two forces tend to carry them. And as their velocities have never been retarded by the resistance of air or any other substance, they have been ever the same from the beginning.