In 1679 Newton knew with approximate correctness the value of the radius of the earth. He repeated his calculations, and found the results to be in accordance with his former hypothesis. He then proceeded to the general theory of the motion of a particle under a force directed to a fixed point, and showed that the vector to the particle would sweep over equal areas in equal times. He also proved that, if a particle describes an ellipse under a force directed to a focus, the law must be that of the inverse square of the distance from the focus, and conversely, that the orbit of a particle projected in free space under the influence of such a force must be a conic. The application to the solar system was obvious, since Kepler had shown that the planets describe ellipses with the sun in one focus, and that the vectors from the sun to them sweep over equal areas in equal times. This investigation was made for his own satisfaction and was not published at the time. In it he treated the solar bodies as if they were particles, and he must have realized that the results could be taken as being only approximately correct. In 1684 the subject of the planetary orbits was discussed in London by Halley, Hooke, and Wren. They were aware that, if Kepler's conclusions were correct, the attraction of the sun or earth on a distant external particle must vary inversely as the square of the distance, but they could not determine the orbit of a particle subjected to the action of a central force of this kind. It was suggested that Newton might be able to assist them. Accordingly in August, Halley went to Cambridge for a talk on the subject, and then found that Newton had solved the problem some five years previously, and that the path was necessarily a conic. At Halley's request Newton wrote out the substance of his argument, and sent it to London. Halley at once realized the importance of the communication, and later in the autumn returned to Cambridge to urge Newton to prosecute the theory further. He found that Newton had already done something in the matter, the results being contained in a manuscript which he saw. Probably this reference is to the holograph manuscript, still preserved in the University Library at Cambridge, of Newton's lectures in the Michaelmas Term, which served as the basis of his memoir sent to the Royal Society a few months later. The great value of these investigations was recognized, and Newton was persuaded to attack the more general problem. His results are given in the _Principia_.