(123.) If velocity be estimated by the space through which the body would move uniformly in one second, then the final velocity of a body falling for one second will be 2; for with that final velocity the body would in one second move through twice the height through which it has fallen. (124.) Since the final velocity increases in the same proportion as the time, it follows that after two seconds it is twice its amount after one, and after three seconds thrice that, and so on. Thus, the following table exhibits the final velocities corresponding to the times of descent: +---------------+---+---+---+---+----+----+----+----+----+----+ |Time | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | +---------------+---+---+---+---+----+----+----+----+----+----+ |Final velocity | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | +---------------+---+---+---+---+----+----+----+----+----+----+ The numbers in the second row express the spaces through which a body with the final velocity would move in one second, the unit being, as usual, the space through which a body falls freely in one second. (125.) Having thus developed theoretically the laws which characterise the descent of bodies, falling freely by the force of gravity, or by any other uniform force of the same kind, it is necessary that we should show how these laws can be exhibited by actual experiment. There are some circumstances attending the fall of heavy bodies which would render it difficult, if not impossible, to illustrate, by the direct observation of this phenomenon, the properties which have been explained in this chapter. A body falling freely by the force of gravity, as we shall hereafter prove, descends in one second of time through a height of about 16 feet[1]; in two seconds, it would, therefore, fall through four times that space, or 64 feet; in three seconds, through 9 times the height, or 144 feet; and in four seconds, through 256 feet. In order, therefore, to be enabled to observe the phenomena for only four seconds, we should command an height of at least 256 feet. But further; the velocity at the end of the first second would be at the rate of 32 feet per second; at the end of the second second, it would be 64 feet per second; and towards the end of the fall it would be about 120 feet per second. It is evident that this great degree of rapidity would be a serious impediment to accurate observation, even though we should be able to command the requisite height. It appears therefore that the number expressed by _g_ in the preceding formulæ is 16·083. [1] More exactly through 16-1/12 feet, or 193 inches.