When a slow rider increases his speed we recognize at once that he has applied additional power to the wheel, and when this speed is slackened it equally shows that force has been applied against the motion. It is force alone which can produce a change in either velocity or direction of motion; but simple as this law now appears it required the genius of Galileo to discover it and of Newton to give it the form in which it is stated above. 35. THE SECOND LAW OF MOTION, which is also due to Galileo and Newton, is: "Change of motion is proportional to force applied and takes place in the direction of the straight line in which the force acts." Suppose a man to fall from a balloon at some great elevation in the air; his own weight is the force which pulls him down, and that force operating at every instant is sufficient to give him at the end of the first second of his fall a downward velocity of 32 feet per second--i. e., it has changed his state from rest, to motion at this rate, and the motion is toward the earth because the force acts in that direction. During the next second the ceaseless operation of this force will have the same effect as in the first second and will add another 32 feet to his velocity, so that two seconds from the time he commenced to fall he will be moving at the rate of 64 feet per second, etc. The column of figures marked _v_ in the table below shows what his velocity will be at the end of subsequent seconds. The changing velocity here shown is the change of motion to which the law refers, and the velocity is proportional to the time shown in the first column of the table, because the amount of force exerted in this case is proportional to the time during which it operated. The distance through which the man will fall in each second is shown in the column marked _d_, and is found by taking the average of his velocity at the beginning and end of this second, and the total distance through which he has fallen at the end of each second, marked _s_ in the table, is found by taking the sum of all the preceding values of _d_. The velocity, 32 feet per second, which measures the change of motion in each second, also measures the _accelerating force_ which produces this motion, and it is usually represented in formulæ by the letter _g_. Let the student show from the numbers in the table that the accelerating force, the time, _t_, during which it operates, and the space, _s_, fallen through, satisfy the relation s = 1/2 gt^{2}, which is usually called the law of falling bodies. How does the table show that _g_ is equal to 32? TABLE _t_ _v_ _d_ _s_ 0 0 0 0 1 32 16 16 2 64 48 64 3 96 80 144 4 128 112 256 5 160 144 400 etc. etc. etc. etc. If the balloon were half a mile high how long would it take to fall to the ground? What would be the velocity just before reaching the ground?