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of what it was at first; and so on. This principle or law is easily understood for all cases; it is technically expressed by saying that the attraction varies inversely as the square of the distance: thus, if the distance is made ten times as great, since the square of 10 is 100, the attraction will be only 100 of what it was before.

78. The law stated in the preceding Article is true of the spheres whatever be the size and the substance of each. It is not strictly true of other bodies; but there are cases in which it may be extended to bodies not spherical. Thus, if the bodies are excessively small it will be true without regard to the shape of them; and therefore it is frequently given as the law of attraction of particles. If the bodies are not particles, still, if the distance between them is very great compared with the size of the bodies, the law will be practically true.

79. As to the kind of attraction called cohesion we do not know accurately in what way the change of distance is connected with the change in the amount of the attraction. The attraction appears to be very intense between particles that are very close together, and to become feeble as soon as the distance between the particles is large enough to be practically sensible. It is possible that science may hereafter shew that cohesion and gravitation are really attractions of the same kind; the law being that established by Newton when the particles are at a sensible distance, and taking some other form when the particles are extremely close.

80. Many substances which occur in nature present themselves under three forms, namely, the solid, the liquid, and the aeriform. Thus it is one and the same substance with which we are familiar under the names of ice, water, and steam. Mercury also, which is usually a liquid, can be frozen, and can be turned into a vapour. There are grounds for believing that every substance can take these three forms, and that the change from the solid state to the liquid state is produced by the application of heat, and the change from the liquid state to the aeriform state by the further application of heat.

81. In solids the cohesion is strong, aud keeps the particles in contact; in liquids the cohesion is very weak, and indeed scarcely sensible, so that the particles may be separated by the slightest effort; in aeriform bodies there is no cohesion whatever, but on the contrary the particles repel each other, and some external force is required in order to keep them near each other. The distinction between the three forms of matter is sometimes expressed with technical precision as follows. Solid bodies have an independent volume and an independent shape. Their parts do not move easily among themselves; it requires always more or less effort to disturb them or to separate them. When once separated they do not unite by being merely placed again in contact. Liquids have an independent volume, but not an independent shape. They take the shape of the vessel in which they are placed. The least effort can move or separate the parts; but after separation the parts unite again when placed in contact. Aeriform bodies have neither an independent volume, nor an independent shape; they spread themselves through any space open to them, until restrained by some external obstacle.

82. It is, as we have said, by the application of heat that the cohesion of solid bodies is destroyed and the liquid state assumed; and by a further application of heat the cohesion is changed into repulsion and the aeriform state assumed. Hence some writers have been inclined to consider heat mainly as a repulsive power opposite in character to that attractive power of which we have already spoken.

After this general notice of our subject, of its connection with other parts of science, and of the necessary preliminary mathematical knowledge, we proceed in this volume to treat in detail of the various mechanical properties of solid and fluid bodies; that is of properties which belong to all such bodies, connected with the operation of force.

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IV. MOTION. FALLING BODIES.

83. Objects in motion present themselves readily to our notice as we look around us; and moreover we soon learn that motion is a thing which may be measured. Thus we are told that a man whom we know walks four miles an hour, that a certain horse trots nine miles an hour, and that a railway train in which we made a journey moved through thirty miles in an hour. In these cases we understand that the motion is uniform, that is the motion is kept up steadily, not becoming sometimes faster and sometimes slower. It will then be an easy question in arithmetic to find the distance moved through by any of these bodies in whatever interval of time may be mentioned; as, for instance, to find the distance moved through by the railway train in one second. In 30 miles there are 30 times 5280 feet, that is 158,400 feet; and in an hour there are 60 times 60 seconds, that is 3600 seconds. Divide 158,400 by 3600; the quotient is 44: so that the railway train moved through 44 feet in one second.

84. There are two words much used when we speak and write about motion, of which the meaning is perhaps familiar, but for clearness should be mentioned; these words are space and describe. The word space is used as equivalent to length or distance; thus we talk about a space of 44 feet, meaning a length or distance of 44 feet. The word describe is used in the sense of moving through; thus we say a body describes a space of 44 feet, meaning that it moves through a space or distance of 44 feet. Sometimes we omit the word space and say that a body describes 44 feet.

85. When a body describes 44 feet in a second we say that its rate of motion is 44 feet a second; or we may say that its velocity is 44 feet a second: it is very customary to employ a Latin preposition and say 44 feet per second. We see that velocity is a thing which admits of exact measurement. Thus, if a child can walk at the rate of one mile in an hour, and his father at the rate of four miles in an hour, the velocity of the father is four times that of the child. If a railway train goes 45 miles in an hour, and a

steamer 15 miles in an hour, the velocity of the railway train is three times that of the steamer.

86. Uniform motion is the simplest kind of motion, and that with which we first become familiar; but we soon find that it is not the only kind of motion. Thus, if an arrow be shot straight upwards the eye can easily see that as the arrow gets nearer to the highest point which it reaches it moves more slowly than when it first left the bowstring; and if a cricket-ball be driven a long way over the ground by a stroke from a bat it moves more slowly at last than at first. Such motion is called variable motion.

87. One of the simplest cases of variable motion, and at the same time one of the most important, is that of falling bodies. The fact, that bodies if not supported will fall to the ground, must have been known from the earliest ages, but what we call the laws of falling bodies were not discovered by any person before Galileo, a famous Italian philosopher who lived from 1564 to 1642. We proceed to state these laws.

88. The motion of a falling body is not uniform; the longer a body falls the more quickly it moves at the end of the time. In the following Table the first column gives the number of seconds since the beginning of the motion, and the second column gives the space through which the body has fallen since the beginning of the motion:

In one second.....

in two seconds

in three seconds

in four seconds

.....

and so on.

..16 feet;
.2 × 2 × 16 feet;

..3 × 3 × 16 feet;
.4 × 4 × 16 feet;

Thus we have an easy Rule for finding the number of feet fallen through since the beginning of the motion: take the number of seconds, multiply that number by itself, and the product by 16. If we remember the meaning of the word square in Arithmetic, as stated in Art. 7, we may put the Kule more briefly thus: multiply the square of the number of seconds by 16. For example, to find the number of feet fallen through in 10 seconds: the square of 10 is 100, and 100 × 16=1600.

89. A reader of a cautious turn of mind may perhaps think that no person can ever have dropped a stone down 1600 feet and observed the time of motion to be 10 seconds; and thus he may suppose that we are here saying more than we strictly know to be true. And indeed it must be confessed that this precise experiment never has been made and probably never will be made: but still we may feel confident that if it could be made the result would be just what has been stated: the grounds of this confidence will become to some extent known as we proceed. We may say briefly here that by observation and experiment we gain the conviction that there are laws of nature, and that these laws are permanent and universal; then when by long investigation we have discovered such a law, we believe that it will hold even in circumstances which do not admit of obvious trial by experiment.

90. The Rule given in Art. 88 will also apply if we wish to find the space fallen through in a time which is not a whole number of seconds. For example, required the space fallen through in two seconds and a half. We have 5 5 5 25

5

=

2 2 2 4

25

4

2}= 2; the square of X-= ; and x 16=100; thus the space fallen through is 100 feet.

91. The Rule will also enable us to determine the space fallen through in any time which may be specified, even when not beginning with the beginning of the motion. For example, suppose we want to know the space fallen through during the fourth second of the motion. In four seconds the space fallen through is 4 × 4 × 16 feet, that is 256 feet; in three seconds the space fallen through is 3 x 3 x 16 feet, that is 144 feet. Hence, to find the space fallen through in the fourth second we must subtract 144 feet from 256 feet; the result is 112 feet. Again, suppose it required to find the space fallen through in one-tenth of a second, occurring at the end of three seconds after the beginning of the motion. We have 3= the square

31 31 31 961
10 100 and

of

10

= X
10

961

100

31

103

x16 15318. Thus in three

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