442 the Royal Society on January 11, 1672, Oldenburg the secretary read a letter from Paris describing the method followed by Picard in measuring a degree, and specifically stating the precise length that he calculated it to be. It is probable that Newton had become acquainted with this measurement of Picard s, and that he was therefore led to make use of it when his thoughts were redirected to the subject. This estimate of the earth s magnitude, giving 69 ! miles to one degree, made the two results, the discrepancy between which Newton had regarded as a disproof of his conjecture, to agree so exactly that he now regarded his conjecture as fully established. In January 1684 Sir Christopher Wren, Halley, and Hooke were led to discuss the law of gravity, and, although probably they all agreed in the truth of the law of the inverse square, yet this truth was not looked upon as established. It appears that Hooke professed to have a solution of the problem of the path of a body moving round a centre of force attracting as the inverse square of the distance ; but Halley, finding, after a delay of some months, that Hooke " had not been so good as his word " in showing his solution to Wren, started in the month of August 1684 for Cambridge to consult Newton on the subject. Without mentioning the speculations which had been made, he went straight to the point and asked Newton what would be the curve described by a planet round the sun on the assumption that the sun s force diminished as the square of the distance. Newton replied promptly, " an ellipse," and on being questioned by Halley as to the reason for his answer he replied, " Why, I have calculated it." He could not, however, put his hand upon his calculation, but he promised to send it to Halley. After the latter had left Cambridge, Newton set to work to reproduce the calculation. After making a mistake and producing a different result he corrected his work and obtained his former result. In the following November Newton redeemed his promise to Halley by sending him, by the hand of Mr Paget, one of the fellows of his own college, and at that time mathe matical master of Christ s Hospital, a copy of his demon stration ; and very soon afterwards Halley paid another visit to Cambridge to confer with Newton about the problem; and on his return to London on December 10, 1684, he informed the Royal Society "that he had lately seen Mr Newton at Cambridge, who had showed him a curious treatise De Motu," which at Halley s desire he promised to send to the Society to be entered upon their register. " Mr Halley was desired to put Mr Newton in mind of his promise for the securing this invention to himself, till such time as he could be at leisure to publish it," and Paget was desired to join with Halley in urging Newton to do so. Newton was not slow in responding to the wish of the Society. By the middle of February he had sent his paper to Aston, one of the secretaries of the Society, and in a letter to Aston dated February 23, 1685, we find Newton thanking him for " having entered on the register his notions about motion." Newton adds, " I designed them for you before now, but the examining several things has taken a greater part of my time than I expected, and a great deal of it to no purpose. And now I am to go to Lincolnshire for a month or six weeks. Afterwards I intend to finish it as soon as I can con veniently." This treatise De Motu was the germ of the Principia, and was obviously meant to be a short account of what that work was intended to embrace. It occupies twenty-four octavo pages, and consists of four theorems and seven problems, some of which are identical with some of the most important propositions of the second and third sections of the first book of the Principia. The years 1685 and 1686 will ever be memorable in the history of science. It was in them that Newton composed almost the whole of his great work. During this period Newton had a very extensive correspondence with Flamsteed, who was then the astronomer-royal. Many of the letters are lost, but it is clear from one of Newton s, dated September 19, 1685, that he had received many useful communications from Flamsteed, and especially regarding Saturn, "whose orbit, as defined by Kepler," Newton "found too little for the sesquialterate propor tions." In the other letters written in 1685 and 1686 he applies to Flamsteed for information respecting the orbits of the satellites of Jupiter and Saturn, respecting the rise and fall of the spring and neap tides at the solstices and the equinoxes, respecting the flattening of Jupiter at the poles (which, if certain, he says, would conduce much to the stating the reasons of the precession of the equinoxes), and respecting the difference between the observed places of Saturn and those computed from Kepler s tables about the time of his conjunction with Jupiter. On this last point the information supplied by Flamsteed was peculiarly gratifying to Newton ; and it is obvious from the language of this part of his letter that he had still doubts of the universal application of the sesquialteral proportion. "Your information," he says, "about the errors of Kepler s tables for Jupiter and Saturn has eased me of several scruples. I was apt to suspect there might be some cause or other unknown to me which might disturb the sesqui alteral proportions, for the influences of the planets one upon another seemed not great enough, though I imagined Jupiter s influence greater than your numbers determine it. It would add to my satisfaction if you would be pleased to let me know the long diameters of the orbits of Jupiter and Saturn, assigned by yourself and Mr Halley in your new tables, that I may see how the sesquialteral proportion fills the heavens, together with another small proportion which must be allowed for." Upon Newton s return from Lincolnshire in the begin ning of April 1685, he seems to have devoted himself to the preparation of his work. In the spring he had determined the attractions of masses, and thus completed the law of universal gravitation. In the summer he had finished the second book of the Principia, the first book being the treatise De Motu, which he had enlarged and completed. Excepting in the correspondence with Flamsteed, to which we have already referred, we hear nothing more of the preparation of the Principia until April 21, 1686, when Halley read to the Royal Society his Discourse concerning Gravity and its Properties, in which he states " that his worthy countryman Mr Isaac Newton has an incomparable treatise of motion almost ready for the press," and that the law of the inverse square " is the principle on which Mr Newton has made out all the phenomena of the celestial motions so easily and naturally, that its truth is past dispute." The intelligence thus given by Halley was speedily confirmed. At the very next meeting of the Society, on April 28, " Dr Vincent presented to the Society a manuscript treatise entitled Philosophise Naturalis Principia Mathematica, and dedicated to the Society by Mr Isaac Newton." Although this manuscript contained only the first book, yet such was the confidence the Society placed in the author that an order was given " that a letter of thanks be written to Mr Newton ; and that the printing of his book be referred to the consideration of the council ; and that in the meantime the book be put into the hands of Mr Halley, to make a report thereof to the council." Although there could be no doubt as to the intention of this report, yet no step was taken towards the publication of the work. At the next meeting of the Society, on May 1 9, some dissatisfaction seems to have been expressed at the delay, as it was ordered " that Mr Newton s work should be printed forthwith in quarto, and
