Not long after his return from this tour, Halley was led to that undertaking for which we owe him the greatest debt of gratitude, and which must be regarded as his greatest achievement. Some fifty years before, the great Kepler had brought out the third of his well-known laws of planetary motion. These laws stated that the planets move round the sun in ellipses, of which the sun occupies one of the foci; that the straight line joining any planet with the sun moves over equal areas of space in equal periods of time; and, lastly, that the squares of the times in which the several planets complete a revolution round the sun are proportional to the cubes of their mean distances from it. These three laws were deduced from actual examination of the movements of the planets. Kepler did not work out any underlying cause of which these three laws were the consequence. But the desire to find such an underlying cause was keen amongst astronomers, and had given rise to many researches. Amongst those at work on the subject was Halley himself. He had seen, and been able to prove, that if the planets moved in circles round the sun, with the sun in the centre, then the law of the relation between the times of revolution and the distances of the planets would show that the attractive force of the sun varied inversely as the square of the distance. The actual case, however, of motion in an ellipse was too hard for him, and he could not deal with it. Halley therefore went up to Cambridge to consult Newton, and, to his wonder and delight, found that the latter had already completely solved the problem, and had proved that Kepler's three laws of planetary motion were summed up in one, namely, that the sun attracted the planets to it with a force inversely proportional to the square of the distance. Halley was most enthusiastic over this great discovery, and he at once strongly urged Newton to publish it. Newton's unwillingness to do so was great, but at length Halley overcame his reluctance; and the Royal Society not being able at the time to afford the expense, Halley took the charges upon himself, although his own resources had been recently seriously damaged by the death of his father.