But since the velocity which the body receives in any time, counted from the beginning of its descent, is in the proportion of that time, it follows that the velocity of the body after half the whole time of descent is half the final velocity. From whence it appears, that the height from which a body falls in any proposed time is equal to the space through which a body would move in the same time with half the final velocity, and it is therefore equal to half the space which would be moved through in the same time with the final velocity. (120.) It follows from this reasoning, that between the three quantities, the height, the time, and the final velocity, which enter into the investigation of the phenomena of falling bodies, there are two fixed relations: _First_, the time, counted from the beginning of the fall and the final velocity, are proportional the one to the other; so that as one increases, the other increases in the same proportion. _Secondly_, the height being equal to half the space which would be moved through in the _time_ of the fall, with the _final velocity_, must have a fixed proportion to these two quantities, viz. the _time_ and the _final velocity_, or must be proportional to the product of the two numbers which express them. But since the time is always proportional to the final velocity, they may be expressed by equal numbers, and the product of equal numbers is the square of either of them. Hence, the product of the numbers expressing the time and final velocity is equivalent to the square of the number expressing the time, or to the square of the number expressing the final velocity. Hence we infer, that the height is always proportional to the square of the time of the fall, or to the square of the final velocity. (121.) The use of a few mathematical characters will render these results more distinct, even to students not conversant with mathematical science. Let S = the height from which the body falls, expressed in feet. V = the velocity at the end of the fall in feet per second. T = the number of seconds in the time of the fall. _g_ = the number of feet through which a body would fall in one second. It will therefore follow that the velocity acquired in one second will be 2_g_, and the velocity acquired in T seconds will therefore be 2_g_ × T; so that V = 2_g_ × T [1] Since the space which a body falls through in T seconds is found by multiplying the space it falls through in one second by T^2, we shall have S = _g_ × T^2 [2] from which, combined with [1] we deduce S = V^2/(4_g_) [3] S = (1/2)V × T [4]