As the force down an inclined plane is less than that which urges a body falling freely in the vertical direction, the space through which the body must fall to attain a certain final velocity must be just so much greater as the accelerating force is less. On this principle we shall be able to determine the final velocity in descending through any space on a plane, compared with the final velocity attained in falling freely in the vertical direction. Suppose the body P, _fig. 27._, placed at the top of the plane, and from H draw the perpendicular H C. If B H represent the force of gravity, B C will represent the force down the plane (131). In order that the body moving down the plane shall have a final velocity equal to that of one which has fallen freely from B to H, it will be necessary that it should move from B down the plane, through a space which bears the same proportion to B H as B H does to B C. But since the triangle A B H is in all respects similar to H B C, only made upon a larger scale, the line A B bears the same proportion to B H as B H bears to B C. Hence, in falling on the inclined plane from B to A, the final velocity is the same as in falling freely from B to H. It is evident that the same will be true at whatever level an horizontal line be drawn. Thus, if I K be horizontal, the final velocity in falling on the plane from B to I will be the same as the final velocity in falling freely from B to K. (135.) The motion of a heavy body down a curve differs in an important respect from the motion down an inclined plane. Every part of the plane being equally inclined to the vertical direction, the effect of gravity in the direction of the plane is uniform; and, consequently, the phenomena obey all the established laws of uniformly-accelerated motion. If, however, we suppose the line B A, on which the body P descends, to be curved as in _fig. 28._, the obliquity of its direction at different parts, to the direction P F of gravity, will evidently vary. In the present instance, this obliquity is greater towards B and less towards A, and hence the part of the force of gravity which gives motion to the body is greater towards B than towards A (130). The force, therefore, which urges the body, instead of being uniform as in the inclined plane, is here gradually diminished. The rate of this diminution depends entirely on the nature of the curve, and can be deduced from the properties of the curve by mathematical reasoning. The details of such an investigation are not, however, of a sufficiently elementary character to allow of being introduced with advantage into this treatise. We must therefore limit ourselves to explain such of the results as may be necessary for the development of the other parts of the science.