By these formularies, if the height through which a body falls freely in one second be known, the height through which it will fall in any proposed time may be computed. For since the height is proportional to the square of the time, the height through which it will fall in _two_ seconds will be _four_ times that which it falls through in _one_ second. In _three_ seconds it will fall through _nine_ times that space; in _four_ seconds, _sixteen_ times; in _five_ seconds, _twenty-five_ times, and so on. The following, therefore, is a general rule to find the height through which a body will fall in any given time: “Reduce the given time to seconds, take the square of the number of seconds in it, and multiply the height through which a body falls in one second by that number; the result will be the height sought.” The following table exhibits the heights and corresponding times as far as 10 seconds: +-------+---+---+---+----+----+----+----+----+----+-----+ |Time | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | +-------+---+---+---+----+----+----+----+----+----+-----+ |Height | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | +-------+---+---+---+----+----+----+----+----+----+-----+ Each unit in the numbers of the first row expresses a second of time, and each unit in those of the second row expresses the height through which a body falls freely in a second. (122.) If a body fall continually for several successive seconds, the spaces which it falls through in each succeeding second have a remarkable relation among each other, which may be easily deduced from the preceding table. Taking the space moved through in the first second still as our unit, four times that space will be moved through in the first two seconds. Subtract from this 1, the space moved through in the first second, and the remainder 3 is the space through which the body falls in the _second_ second. In like manner if 4, the height fallen through in the first two seconds, be subtracted from 9, the height fallen through in the first three seconds, the remainder 5 will be the space fallen through in the third second. To find the space fallen through in the fourth second, subtract 9, the space fallen through in the first three seconds, from 16, the space fallen through in the first four seconds, and the result is 7, and so on. It thus appears that if the space fallen through in the first second be called 1, the spaces described in the second, third, fourth, fifth, &c. seconds, will be expressed by the odd numbers respectively, 3, 5, 7, 9, &c. This places in a striking point of view the accelerated motion of a falling body, the spaces moved through in each succeeding second being continually increased.