Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/110



Every body, that, by a radius drawn to the centre of another body howover moved, decribed areas about that centre proportional to the times, is urged by a force compounded out of the centripetal force tending to that other body, and of all the accelerative force by which that other body is impelled.

Let L repreent the one, and T the other body; and (by Cor. 6 of the laws) if both bodies are urged in the direction of parallel lines, by a new force equal and contrary to that by which the econd body T is urged, the firt body L will go on to decribe about the other body T, the ame areas as before: but the force, by which that other body T was urged, will be now detroyed by an equal and contrary force; and therefore (by Law 1.) that other body T, now left to it elf, will either ret, or move uniformly forward in a right line: and the firt body L impell'd by the difference of the forces, that is, by the force remaining, will go on to decribe about the other body T; areas proportional to the times. And therefore (by Theor. 2.) the difference of the forces is directed to the other body T as its centre. Q. E. D.

Hence if the one body L, by a radius drawn to the other body T, decribes areas proportional to the times; and from the whole force, by which the firt body L is urged (whether that force is