As Newton attempted to escape from problems of mathematics and mechanics, others kept calling him back. Hooke, who had continued to work on the problem of orbital motion, wrote to Newton in 1679 to encourage him to send something to the Royal Society. In his reply, Newton discussed an experiment to detect the rotation of the earth by dropping an object from a tower. Since the top of the tower is moving faster than the surface of the earth (because it is moving in a larger circle), the falling object should land to the east, in the direction of rotation.
This much is correct. But Newton went on to sketch the path the object would follow if it could penetrate the earth, showing it as a spiral toward the center of the earth. Hooke caught the mistake, and suggested instead that it would follow an elliptical path.
Newton, annoyed that he had been caught, admitted the error, but then claimed (correctly) that, if the gravitational force were taken as constant, the path would not be an ellipse but rather a cloverleaf shape, in which the points of minimum and maximum distance from the center are about 120 degrees apart. Hooke agreed, but said he had not been thinking of a constant gravity, but of an inverse-square law. This correspondence would be the basis for Hooke's later charge of plagiary against Newton.
In November of 1680, a comet appeared, heading toward the sun. In mid-December, another comet appeared, moving away from the sun. The Royal Astronomer, John Flamsteed, suggested that the two were the same comet. Here Newton made his biggest blunder of all. He wrote to Flamsteed and argued that Flamsteed was wrong about the two comets. Amazingly, although Newton had solved the problem of orbital mechanics a year before, he made no attempt to apply his equations of planetary motion to the comet's path.
Flamsteed had suggested that some sort of magnetic force deflected the comet as it passed the sun, and Newton made a similar assumption in his reply. Both men assumed that comets obeyed different laws than planets: planets were permanent members of the solar system, while comets were strange visitors with a dynamics all their own.
This shows beyond a doubt that Newton was not yet thinking in terms of universal gravitation in 1680. But Hooke's letters and the discussions surrounding the comet(s) had re-invigorated his interest in problems of mechanics. He plunged back into the study of problems of motion, and, for once, managed to complete the project he had begun. That project produced the most amazing scientific treatise that the world had ever seen, the Principia.