These, whether they be feathers, paper, gold-leaf, pieces of money, &c. all descend with the same speed, and strike the bottom at the same moment. (118.) Every one who has seen a heavy body fall from a height, has witnessed the fact, that its velocity increases as it approaches the ground. But if this were not observable by the eye, it would be betrayed by the effects. It is well known, that the force with which a body strikes the ground increases with the height from whence it has fallen. This force, however, is proportional to the velocity which it has at the moment it meets the ground, and therefore this velocity increases with the height. When the observations on attraction in the last chapter are well understood, it will be evident that the velocity which a body has acquired in falling from any height, is the accumulated effects of the attraction of terrestrial gravity during the whole time of the fall. Each instant of the fall a new impulse is given to the body, from which it receives additional velocity; and its final velocity is composed of the aggregation of all the small increments of velocity which are thus communicated. As we are at present to suppose the intensity of the attraction invariable, it will follow that the velocity communicated to the body in each instant of time will be the same, and therefore that the whole quantity of velocity produced or accumulated at the end of any time is proportional to the length of that time. Thus, if a certain velocity be produced in a body having fallen for one second, twice that velocity will be produced when it has fallen for two seconds, thrice that velocity in three seconds, and so on. Such is the fundamental principle or characteristic of _uniformly accelerated motion_. (119.) In examining the circumstances of the descent of a body, the time of the fall and the velocity at each instant of that time are not the only things to be attended to. The spaces through which it falls in given intervals of time, counted either from the commencement of its fall, or from any proposed epoch of the descent, are equally important objects of enquiry. To estimate the space in reference to the time and the final velocity, we must consider that this space has been moved through with varying speed. From a state of rest at the beginning of the fall, the speed gradually increases with the time, and the final velocity is greater still than that which the body had at any preceding instant during its descent. We cannot, therefore, _directly_ appreciate the space moved through in this case by the time and final velocity. But as the velocity increases uniformly with the time, we shall obtain the average speed, by finding that which the body had in the middle of the interval which elapsed between the beginning and end of the fall, and thus the space through which the body has actually fallen is that through which it would move in the same time with this average velocity uniformly continued.