Some of these velocities have been given by Peschel; others are cased on calculations derived from reliable observations.

"Uniform motion or velocity."

105. A body has a uniform motion or velocity when it passes over the same distance in each second or minute of its duration; and we get the same idea of the rate, whether we consider the space traversed in a long or in a short time.

Our standard of uniform motion, with which we compare and measure all other motions, is that of the earth round its own axis. Here we have a huge spinning-top, which, not for hours or days, but for unknown ages, has kept up its original speed practically undiminished. All our notions of time are based on the regularity with which the earth turns round. We can tell, roughly, by looking at the position of the sun, and more accurately by noting the position of any star, when our earth has made a complete revolution. Any phenomenon lasting while the earth makes one or ten of its revolutions, is said to have a duration of one or ten days; and if it last while the earth makes one twenty-fourth part of a turn, it is said to be of an hour's duration.

It would be rather difficult to tell from the sun or stars when the carth has made a twenty-fourth or forty-eighth part of a turn; but various inventions have been made which enable us to say with very great nicety what fraction of a day or revolution of the earth has passed. Watches and chronometers are but contrivances for producing a practically uniform motion which can be readily compared with that of the earth, but whose subdivisions can be more easily referred to and noted than the latter. And to such perfection

has this artificial uniformity attained, that modern astronomical clocks and chronometers are constructed which scarcely vary a single second in the course of a whole year.

In point of fact, the motion of the minute hand of a watch is not uniform, but consists of a succession of little hops, which may be seen when we look at the seconds hand, but which are no more considered than the steps of a horse when we speak of riding at a uniform rate.

In the more antiquated contrivances for measuring time, such as the sand-glass, and the water-dropping clepsydra, or in the flow of mercury through a small opening in the bottom of a funnel, which is used even at the present day, we have a motion that is really continuous and free from interruptions, though it may not be so generally convenient for measuring as that of a clock.

"Variable motion."

106. When the motion of a body is not uniform, as in the case of a body falling to the earth, it is obvious that we shall have a truer idea of the rate at any instant the shorter the interval of time through which we can measure the space passed over.

The captain of a steamboat may form a very incorrect idea of the actual speed of his vessel if he merely know the distance passed within the last two hours; he comes much nearer the actual speed if he know the distance traversed within the last minute; still nearer if he can tell how far he has gone during the last second, or tenth or hundreth of a second.

In other words, the exact velocity of the steamboat at any instant may be quite a different thing from its average velocity for the past hour or two hours.

If the velocity be a rapidly changing one, our power of noting time must be correspondingly nice if we wish to measure the velocity at any moment. Thus, the explosion of gunpowder within a cannon gives motion to the bullet gradually, yet so rapidly, that the whole time occupied in its passage through the cannon is inappreciable by ordinary means of time measurement. But, in recent years, an electric chronoscope (or time-detector) has been invented that indicates the very small fraction of a second taken by the ball to pass from point to point within the cannon; and so the law of the rate according to which the motion is imparted can be ascertained—a point of great importance in gunnery.

Of the cases of variable motion the most important are those

Absolute and Relative Motion.

47

where the change is uniform or regular, because only then is the motion calculable.

Uniformly accelerated or retarded motion.”

107. If, by accident, a carriage gets detached from a train which is standing on an incline, it will move down the incline with a ccnstantly increasing speed. At the end of the first second the velocity would not be great, and there would be little difficulty in stopping the runaway carriage. In three seconds its speed will be triple what it was at the end of the first second; at the end of a minute it will be sixty times as great, and may defy the power of the brake to arrest its impetuosity.

A stone allowed to drop from a height to the earth will have a velocity of about thirty-two feet at the end of the first second, of sixty-four feet at the end of the next second, of one hundred and sixty feet at the end of the fifth second of its fall, and so on.

These are examples of uniformly increasing (or accelerated) motion, because the velocity is constantly increasing by the same

amount.

108. Uniformly retarded motion is seen when we shoot an arrow vertically up. Its speed gets less and less by the same amount during each second of its flight, till at last it is brought to rest, and the operation begins to be reversed.

The application of the brake to a train may reduce its speed from thirty miles an hour to twenty-eight, twenty-six, or twenty-four miles an hour in one, two, or three seconds, in which case the effect of the brake is a uniform retardation of the motion.

109. Nearly all motions that we meet with in nature are accelerated or retarded. The production of motion, as will be more fully considered afterwards, is never absolutely instantaneous, but always more or less gradual; the stoppage of motion takes place also by degrees, though the time occupied may not always be appreciable.

The acceleration or retardation of motion is not, however, in every case, or even in many cases, uniform. For example, the velocity of a rifle ball at the end of the first half-second after the explosion will be much more than double of what it was at the end of the first quarter of a second.

"Absolute and relative motion."

110. A man sitting on the deck of a sailing ship has common motion with the ship, though relatively at rest as regards it; if

48

Co-existence of Motions.

walking on deck, he has one motion relatively to the vessel, and another relatively to the land. If he walk towards the stern just as fast as the ship sails, he is at rest relatively to the sea-bottom or shore.

A boat rowed against the stream as fast as it flows is at rest as regards the river and the earth just as much as if it were moored. Absolute motion would be change of position compared with some absolutely fixed point of reference. In reality we cannot know such motion, because there is no spot in the universe absolutely at rest, so far as we can tell. All nature is in ceaseless movement, and thus every motion is only relative to some other moving body. When the term absolute is used, therefore, it must be understood as taken in a limited sense, the motion of the point or place of reference not being considered as affecting the conditions of motion.

Thus, in comparing the motions of the hour and minute hands of a watch, we may say that their real or absolute motions are five minutes and sixty minutes respectively per hour, but that the relative motion of the minute hand compared with the hour hand is fifty-five minutes per hour.

"Co-existence of motions."

111. A body may partake of two or more motions at the same time.

Our earth keeps turning once a day round itself, and at the same time wheeling round the sun at the rate of once a year; in all probability it is also moving with the sun and its sister planets round some other greater central sun.

A top set spinning on a plate of glass will rotate and at the same time travel over the glass.

Smoke ascending from a chimney, or a balloon rising from the ground, is at the same time driven away in a cross direction by the wind.

112. When a body possesses two simultaneous motions, either in the same or in different directions, it is often of much consequence to know the conjoint effect of the motions, or the actual movement relatively to some object independent of both.

"Simultaneous motions in the same direction."

113. If a person walk towards the prow or the stern of a steamer in motion, he has at the same instant two velocities; and the actual

Co-existing Motions.

49

rate of his motion, referred to the land or an object independent of both motions, is the sum or difference of his velocity and that of the steamer, according as he walks towards the prow or the stern.

A clown leaping forward when his horse is at full speed, will light on the neck of the horse just as he would if it were standing still, because, by leaping, he simply adds motion to the motion which he already has.

"Co-existing motions in different directions."

11. A ferry-boat rowed straight across a river may at the same time be borne by the current as fast down the stream; and the resulting motion will be neither right across nor down the stream, but in an intermediate direction.

If the velocity down the stream be three times as great as that across the stream, the direction of the real or resultant motion will be more inclined to the bank of the stream than to the line right across, and may be found in the way shown by fig. 2 :—

B

Take a line, A B, to represent the direction of the flow of the stream three times the length of a line, A D, in the direction of the breadth of the river, and complete the parallelogram by drawing lines D C and B C parallel to A B and A D.

e

f

Fig. 2.

borne a distance, A e, during any instant, it is rowed one-third as far across in the direction of A D, so that at the end of that instant it will be at f; during the next similar instant it will be carried a distance, fg, in the one direction, and one-third of that, g h, across that direction, being at h at the end of the second instant. Thus, at the end of every such instant the boat will be in the line A C; and this is true for the smallest conceivable instant. Hence, we see that the boat will not be out of the line A C for the smallest conceivable instant. In other words, the true course of the boat will be seen to be along a C.

115. Nor is it necessary that the one motion be right across the other.

A ship sailing in the direction A B (fig. 3), may at the same time be drifted by the tide in the direction A D. At the end of successive seconds or instants it will have gone for the distances A e, ƒ g, &c.,