J04 THE MATHEMATICAL PRINCIPLES [BOOK 11 COR. 5. And universally, the quantity of matter in the pendulous body is as the weight and the square of the time directly, and the length of the pendulum inversely. COR. 6. But in a non-resisting medium, the quantity of matter in the pendulous body is as the comparative weight and the square of the time directly, and the length of the pendulum inversely. For the comparative weight is the motive force of the body in any heavy medium, as was shewn above ; and therefore does the same thing in such a non-resisting medium as the absolute weight does in a vacuum. COR. 7. And hence appears a method both of comparing bodies one among another, as to the quantity of matter in each ; and of comparing the weights of the same body in different places, to know the variation of its gravity. And by experiments made with the greatest accuracy, I have always found the quantity of matter in bodies to be proportional to their weight. PROPOSITION XXV. THEOREM XX. Funependulous bodies that are, in, any 'medium, resisted in the ratio oj the moments of time, and funepetidulons bodies that move in a nonresisting medium of the same specific gravity, perform their oscilla tions in. a cycloid in the same time, and describe proportional parts oj arcs together. Let AB be an arc of a cycloid, which a body D, by vibrating in a non-re sisting medium, shall describe in any time. Bisect that arc in C, so that C may be the lowest point thereof ; and the accelerative force with which the body is urged in any place D, or d or E, will be as the length of the arc CD, pressed by that same arc ; and since the resistance is as the moment of the time, and therefore given, let it ba expressed by the given part CO of the cycloidal arc, and take the arc Od in the same ratio to the arc CD that the arc OB has to the arc CB : and the force with which the body in d is urged in a resisting medium, being the excess of the force Cd above the resistance CO, will be expressed by the arc Od, and will therefore be to the force with which the body D is urged in a non-resisting medium in the place D, as the arc Od to the arc CD ; and therefore also in the place B, as the arc OB to the arc CB. Therefore if two bodies D, d go from the place B, and are urged by these forces ; since the forces at the beginning are as the arc CB and OB, the first velocities and arcs first described will be in the same ratio. Let those arcs be BD and Ed, and the remaining arcf