But the operations may be contracted by resolving the ordinates into converging series. As if to a base A the length B be ordinately applied in any given angle, and that length be as any power of the base ; and there be sought the force with which a body, either attracted towards the base or driven from it in the direction of that ordinate, may be caused to move in the curve line which that ordinate always describes with its superior extremity; I suppose the base to be increased by a very small part O, and I resolve the ordinate into an infinite series &c., and I suppose the force proportional to the term of this series in which is of two dimensions, that is, to the term . Therefore the [Pg 243]force sought is as , or, which is the same thing, as . As if the ordinate describe a parabola, m being = 2, and n = 1, the force will be as the given quantity , and therefore is given. Therefore with a given force the body will move in a parabola, as Galileo has demonstrated. If the ordinate describe an hyperbola, m being = 0 - 1, and n = 1, the force will be as or ; and therefore a force which is as the cube of the ordinate will cause the body to move in an hyperbola. But leaving this kind of propositions, I shall go on to some others relating to motion which I have not yet touched upon. SECTION XIV. Of the motion of very small bodies when agitated by centripetal forces tending to the several parts of any very great body. PROPOSITION XCIV. THEOREM XLVIII. If two similar mediums be separated from each other by a space terminated on both sides by parallel planes, and a body in its passage through that space be attracted or impelled perpendicularly towards either of those mediums, and not agitated or hindered by any other force; and the attraction be every where the same at equal distances from either plane, taken towards the same hand of the plane; I say, that the sine of incidence upon either plane will be to the sine of emergence from the other plane in a given ratio.