mately, and for great distances, the intensity of the gravitating force must depend upon the inverse square. The second episode was simultaneous, as we have just seen, with the correspondence with Hooke at the end of 1679 or early in 1680, when he discovered how to calculate the orbit of a body moving under a central force, and showed that if the force varied as the inverse square, the orbit would be an ellipse with the centre of force in one focus. But for five years no one was told of this splendid achievement, and it was not till August 1684 that Halley learnt the secret in Cambridge.
Halley's account of the matter is given in a letter to Newton (29 June 1686, ib. App. p. 35). ‘And this know to be true, that in January 1684, I, having from the consideration of the sesquialterate proportion of Kepler concluded that the centripetal force decreased in the proportion of the squares of the distances reciprocally, came on Wednesday to town, where I met with Sir Christopher Wren and Mr. Hooke, and, falling in discourse about it, Mr. Hooke affirmed that upon that principle all the laws of the celestial motions were to be demonstrated, and that he himself had done it. I declared the ill-success of my own attempts, and Sir Christopher, to encourage the inquiry, said he would give Mr. Hooke or me two months' time to bring him a convincing demonstration thereof, and, besides the honour, he of us that did it should have from him a present of a book of 40 shillings. Mr. Hooke then said that he had it, but he would conceal it for some time, that others, trying and failing, might know how to value it when he should make it public. However, I remember that Sir Christopher was little satisfied that he could do it; and though Mr. Hooke then promised to show it him, I do not find that in that particular he has been as good as his word. The August following, when I did myself the honour to visit you, I then learned the good news that you had brought this demonstration to perfection; and you were pleased to promise me a copy thereof, which the November following I received with a great deal of satisfaction from Mr. Paget,’ mathematical master at Christ's Hospital (Brewster, Life of Newton, i. 255; Ball, Essay on the Principia, p. 162).
In the later letter to Halley of 14 July 1686, part of which has been already quoted, Newton says that it was Halley's request which induced him to search for the paper in which he had solved the problem five years earlier, but which he had then laid aside. The original paper could not be found, but, ‘not finding it,’ Newton ‘did it again, and reduced it into the propositions’ shown to Halley by Paget. As soon as Halley had read them he paid another visit to Newton at Cambridge, and induced him to forward an account of his discoveries to the Royal Society. On 10 Dec. 1684 Halley informed the Royal Society ‘that he had lately seen Mr. Newton at Cambridge, who had showed him a curious treatise, “De Motu,” which upon Mr. Halley's desire was promised to be sent to the Society to be entered on their register.’ A tract by Newton entitled ‘Propositiones de Motu’ was registered in the Royal Society archives in February 1685, with the date 10 Dec. 1684 affixed to the margin (see Edleston, Cotes Corr. n. 74–5, p. lv.).
This set of propositions (four theorems and seven problems) has been printed by Rigaud (Historical Essay on Newton's Principia, App. i.) and by Ball (Essay on the Principia, p. 35) from the Register of the Royal Society, vi. 218. Three other papers entitled ‘Propositiones de Motu,’ differing in many ways from that in the Royal Society Register, are among the Portsmouth MSS (viii. 5, 6, 7).
Meanwhile the subject of Newton's Lucasian lectures in the October term 1684 was also entitled ‘De Motu Corporum;’ these lectures are preserved in Newton's autograph in the Cambridge University Library (Dd. ix. 46). They must be carefully distinguished from the ‘Propositiones’ sent to the Royal Society, although some of the chief propositions are the same in both. The lectures ‘De Motu’ differ very little from the first ten sections of the published ‘Principia,’ of which they formed the first draft. Cotes refers to them in writing to Jones on 30 Sept. 1711 (Newton and Cotes Correspondence, ed. Edleston, p. 209): ‘We have nothing of Sir Isaac's that I know of in Manuscript at Cambridge, besides the first draught of his “Principia” as he read it in his lectures.’
Newton was away from Cambridge from February to April 1685. During that year, however, he made the third great discovery which rendered the writing of the ‘Principia’ possible. The discovery is referred to in the letter to Halley of 20 June 1686 (ib. p. 27). ‘I never extended the duplicate proportion lower than to the superficies of the Earth, and before a certain demonstration I found last year have suspected that it did not reach accurately enough down so low.’
This demonstration forms the twelth section of book i. of the ‘Principia,’ ‘De Corporum Sphæricorum Viribus Attractivis.’ According to Newton's views, every particle of matter in the universe attracts every other particle with a force which is inversely proportional to the square of the distance between them. ‘Gravitatio in singulas corporis