Later in the year (1676) in which Newton's important optical papers were communicated to the Royal Society he began a correspondence on his methods of analysis with Leibnitz, through his friends Collins and Oldenburg, to which, at a later date, very great importance attaches in the celebrated controversy respecting the invention of fluxions. The correspondence with Leibnitz was continued to the summer of 1677, when the death of Oldenburg put a stop to it.
For the next two years (1678–9) we know little of Newton's life. He took part in various university functions. On 8 Nov. 1679 Charles Montagu, afterwards Lord Halifax, Newton's firm friend and patron, entered as a fellow commoner at Trinity College. In December 1679 he received a letter from Hooke, asking his opinion about an hypothesis on the motion of the planets proposed by M. Mallement de Messanges. His reply has only recently been discovered, though many pages were previously written as to its contents; it was bought by Dr. Glaisher for Trinity College at a sale at Messrs. Sotheby's in 1888, and is now in the library. In this letter Newton, after alluding briefly to M. Mallement de Messanges's theory, proceeds, in response to a request from Hooke for some philosophical communication, to suggest an experiment by which the diurnal motion of the earth could be verified, namely, ‘by the falling of a body from a considerable height, which he alleged must fall to the eastward of the perpendicular of the earth moved’ (Birch, Hist. of Roy. Soc. iii. 512). Newton's words are: ‘And therefore it will not descend in the perpendicular AC, but, outrunning the parts of the earth, will shoot forward to the east side of the perpendicular, describing in its fall a spiral line ADEC.’ A figure shows the path of the falling body relative to the earth from a point above the earth's surface down to the centre of the earth. The portion of the path above the earth does not differ much from a straight line slightly inclined to the vertical, but near the centre the path is drawn as a spiral, with one convolution closing into the centre. Writing to Halley at a later date (27 May 1686), Newton admitted that he had ‘carelessly described the descent of the falling body in a spiral to the centre of the earth, which is true in a resisting medium such as our air is.’ But Hooke, as will be seen in the sequel, seized upon this spiral curve as proof that Newton was ignorant of the true law of gravitation, and wrote explaining (ib. iii. 516) that the path ‘would not be a spiral line, as Mr. Newton seemed to suppose, but an excentrical elliptoid [sic], supposing no resistance in the medium; but supposing a resistance, it would be an excentric ellipti-spiral.’ He also called attention to the fact that the deviation would be south-east, which is right, and more to the south than to the east, which is wrong. After a short interval Hooke wrote again (6 Jan. 1680, manuscripts in Trinity College Library, in Hooke's hand): ‘In the celestial motions the sun, earth, or central body are the cause of the attraction, and though they cannot be supposed mathematical points, yet they may be supposed physical, and the attraction at a considerable distance computed according to the former proportion from the centre;’ while in a further letter (17 Jan. 1680, same manuscripts) he says: ‘It now remains to know the properties of a curve line, not circular or concentrical, made by a central attracting power, which makes the velocity of descent from the tangent or equal straight motion at all distances in a duplicate proportion to the distance reciprocally taken. I doubt not that by your excellent method you will easily find out what that curve must be and its properties, and suggest a physical reason of the proportion. If you have had any time to consider of this matter a word or two of your thoughts will be very grateful to the Society, where it has been debated, and more particular to, sir, your very humble servant.’ All these letters are printed in Ball's ‘Essay on Newton's Principia,’ 1893, p. 139.
Newton does not appear to have replied till 3 Dec. 1680, when, writing about another matter, he thanked Hooke for the trial he had made of the experiment (Edleston, Cotes Corr. p. 264). The correspondence ceased, but Hooke's letters and his statement that the motion would be elliptical had started Newton in a train of thought which resulted in the first book of the ‘Principia.’ ‘This is true,’ he says, writing to Halley on 14 July 1686 (App. to Rigaud's Essay on the First Publication of the Principia, p. 40), ‘that his letters occasioned my finding the method of determining figures which when I had tried in the ellipsis, I threw the calculations by, being upon other studies, and so it rested for about five years, till upon your request I sought for that paper.’ On 27 July (ib. p. 44) he wrote again, Hooke's ‘correcting my spiral occasioned my finding the theorem by which I afterwards examined the ellipsis.’
Two episodes, says Dr. Glaisher in his bicentenary address, preceded the composition of the ‘Principia.’ One of these happened in 1665, when the idea of universal gravitation first presented itself to his mind. At that time too he knew that, at any rate approxi-
