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===Work begins===
[[Image:NewtonsPrincipia.jpg|thumb|250px|right|Newton's own copy of his ''Principia'', with hand written corrections for the second edition.]]
Between 1685 and 1686, Newton had a very extensive correspondence with [[John Flamsteed]], who was then the astronomer-royal.
Between 1685 and 1686, Newton had a very extensive correspondence with [[John Flamsteed]], who was then the astronomer-royal. Many of the letters are lost but it is clear from one of Newton's, dated 19 September 1685, that he had received many useful communications from Flamsteed, especially regarding [[Saturn]], "whose orbit, as defined by [[Johannes Kepler|Kepler]]," Newton "found too little for the sesquialterate proportions." Newton refers to [[Kepler's laws of planetary motion|Kepler's third law]], that the orbital period is proportional to the distance from the sun to the power of 3/2 ("sesquialteral" comes from the Latin word for the ratio 3/2).
In the other letters written in 1685 and 1686, he asks Flamsteed for information about the orbits of the moons of [[Jupiter]] and Saturn, the rise and fall of the spring and neap tides at the solstices and the equinoxes, about 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 about the universal application of Kepler's third law. "Your information 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 {{sic|hide=y|sesqui|altera}} proportion. 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 [[Edmond Halley|Halley]] in your new tables, that I may see how the {{sic|hide=y|sesqui|plicate}} proportion fills the heavens, together with another small proportion which must be allowed for." <ref>(Letter of mid-January (before 14th) 1684|1685 (Old Style), published as #537 in Vol.2 of ''The Correspondence of John Flamsteed'', ed. E.G. Forbes et al., 1997. (This reference was supplied after original compilation of the present article, and gives original spellings; but most spellings and punctuations in the text above have been modernised. The words 'sesquialtera' and 'sesquiplicate', now archaic, refer to the relation between a given number and the same multiplied by its own square root: or to the square root of its cube, which comes to the same thing: the 'one-and-a-half-th' power, as it were.)</ref>
Upon Newton's return from [[Lincolnshire]] in the beginning 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 corporum in gyrum]]'', which he had enlarged and completed. Except for correspondence with Flamsteed we hear nothing more of the preparation of the ''Principia'' until 21 April 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."
At the next meeting of the Society, on 28 April 1686, "Dr Vincent presented to the Society a manuscript treatise entitled ''Philosophiae 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, no step was taken towards the publication of the work. At the next meeting of the Society, on 19 May 1686, 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 that a letter should be written to him to signify the Society's resolutions, and to desire his opinion as to the print, volume, cuts and so forth." Three days afterwards Halley communicated the resolution to Newton, and stated to him that the printing was to be at the charge of the Society. At the next meeting of the council, on 2 June 1686, it was again ordered "that Mr Newton's book be printed," but, instead of sanctioning the resolution of the general meeting to print it at their charge, they added "that Mr Halley undertake the business of looking after it, and printing it at his own charge, which he engaged to do."
In order to explain to Newton the cause of the delay, Halley in his letter of 22 May 1686 alleges that it arose from "the president's attendance on the king, and the absence of the vice-president's, whom the good weather had drawn out of town"; but there is reason to believe that this was not the true cause, and that the unwillingness of the council to undertake the publication arose from the state of the finances of the Society. Halley certainly deserves the gratitude of posterity for undertaking the publication of the work at a very considerable financial risk to himself.
In the same letter Halley found it necessary to inform Newton of [[Robert Hooke|Hooke]]'s conduct when the manuscript of the Principia was presented to the Society. [[Sir John Hoskyns, 2nd Baronet|Sir John Hoskyns]] was in the chair when Dr Vincent presented the manuscript, and praised the novelty and dignity of the subject. Hooke was offended because Sir John did not mention what he had told him of his own discovery. Halley only communicated to Newton the fact "that Hooke had some pretensions to the invention of the rule for the decrease of gravity being reciprocally as the squares of the distances from the centre," acknowledging at the same time that, though Newton had the notion from him, "yet the demonstration of the curves generated thereby belonged wholly to Newton." "How much of this," Halley adds, "is so, you know best, so likewise what you have to do in this matter; only Mr Hooke seems to expect you should make some mention of him in the preface, which 'tis possible you may see reason to prefix. I must beg your pardon that 'tis I that send you this ungrateful account; but I thought it my duty to let you know it, so that you might act accordingly, being in myself fully satisfied that nothing but the greatest candour imaginable is to be expected from a person who has of all men the least need to borrow reputation."
Many of the letters are lost, but it is clear that Flamsteed was helpful, especially regarding Kepler's definition of Saturn. The publication of Newton's discoveries led to controversies involving English
natural philosopher [[Robert Hooke]], Anthony Lucas, mathematical professor at Liege; [[Francis Line]],
a physician in Liége, and many others. English astronomer and mathematician [[Edmond Halley]] attempted to mediate and get Newton to agree that Hooke deserved some credit for the invention of "the rule for the decrease of gravity being reciprocally as the squares of the distances from the centre". Halley acknowledged that although Hooke should be credited for the theory, "the demonstration of the curves generated thereby belonged wholly to Newton". In a letter, Newton responded to Halley as follows:
[[Image:Principia Page 1726.jpg|thumb|right|A page from the ''Principia'']]
In thus appealing to Newton's honesty, Halley obviously wished that Newton should acknowledge Hooke in some way. Indeed, he knew that before Newton had announced the inverse law, Hooke and [[Christopher Wren|Wren]] and himself had spoken of it and discussed it, and therefore justice demanded that Hooke especially should receive credit for having maintained it as a truth of which he was seeking the demonstration, even though none of them had given a demonstration of the law. On 20 June 1686 Newton wrote to Halley the following letter:
<blockquote>
</blockquote>
On 20 June 1686, Halley wroterespinded to Newton:
<blockquote>
"I am heartily sorry that in this matter, wherein all mankind ought to acknowledge their obligations to you, you should meet with anything that should give you unquiet"; and then,
''(after an account of Hooke's claim to the discovery as made at a meeting of the Royal Society, he concludes: )''
"But I found that they were all of opinion that nothing thereof appearing in print, nor on the books of the Society, you ought to be considered as the inventor. And if in truth he knew it before you, he ought not to blame any but himself for having taken no more care to secure a discovery, which he puts so much value on. What application he has made in private, I know not; but I am sure that the Society have a very great satisfaction, in the honour you do them, by the dedication of so worthy a treatise. </blockquote>
<blockquote>
Sir, I must now again beg you, not to let your resentments run so high, as to deprive us of your third book, wherein the application of your mathematical doctrine to the theory of comets and several curious experiments, which, as I guess by what you write, ought to compose it, will undoubtedly render it acceptable to those, who will call themselves Philosophers without Mathematics, which are much the greater number. Now you approve of the character and paper, I will push on the edition vigorously. I have sometimes had thoughts of having the cuts neatly done in wood, so as to stand in the page with the demonstrations. It will be more convenient, and not much more charge. If it please you to have it so, I will try how well it can be done; otherwise I will have them in somewhat a larger size than those you have sent up. </blockquote>
On 30 June 1686 the council resolved to license Newton's book, entitled ''Philosophiae Naturalis Principia Mathematica''.
On 14 July 1686, Newton wrote to Halley approving of his proposal to introduce woodcuts among the letterpress, stating clearly the differences which he had from Hooke, and adding, "And now having sincerely told you the case between Mr Hooke and me, I hope I shall be free for the future from the prejudice of his letters. I have considered how best to compose the present dispute, and I think it may be done by the inclosed scholium to the fourth proposition." This scholium was "The inverse law of gravity holds in all the celestial motions, as was discovered also independently by my countrymen Wren, Hooke and Halley." After this letter of Newton's the printing of the ''Principia'' was begun. The second book, though ready for the press in the autumn of 1686, was not sent to the printers until March 1687. The third book was presented to the Society on 6 April and the whole work published about midsummer in that year, 5 July 1687.<ref>[[Richard S. Westfall]], ''Never at Rest'', {{isbn|0-521-27435-4}} (paperback) Cambridge 1980..1998.</ref> It was dedicated to the Royal Society, and to it was prefixed a set of Latin hexameters addressed by Halley to the author. The work, as might have been expected, caused a great deal of excitement throughout Europe, and the whole of the impression was very soon sold. In 1691 a copy of the Principia was hard to obtain.
===Illness in 1693===
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