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great delight, Newton replied that he had proved the proposition completely; though oddly enough he could not find the paper on which he had written it out. Newton was not careful and tidy, as we all are, I hope. But he was very, very able: and soon wrote out a new proof and sent it to Halley, who had already hastened to London to tell the Royal Society the good news that the great problem was solved. This was, however, rather premature exultation, though no one suspected it but Newton. He knew that there was still the formidable difficulty we have noticed: he had not yet found out where to measure the distance from, whether from the nearest point of the Earth (or other attracting body) , or its centre, or some other point. But so much had he been stimulated by Halley's visit and genial influence, that he forthwith attacked this difficulty again, this time with success: he found the astonishing result that you must measure always from the Earth's centre.

We may therefore summarize the history of this great discovery in three steps—

(i) In 1665 Newton saw the apple fall, and was led to think of the Law of Gravity: "the attraction is inversely as the square of the distance." But he could not see how to measure the distance accurately.

(ii) In 1679 he made the suggestion of proving the Earth's rotation; and Hooke's reply irritated him into solving "the problem of the ellipse"; the solution, however, he kept to himself.

(iii) In 1685 Halley's visit elicited the solution and stimulated Newton into finding that the distance must be measured from the centre of the attracting object.