# The writing of Principia Mathematica

This article is part
of a series on
Isaac Newton.
Early life
Writing Principia
Later life

The years 1685 and 1686 will ever be memorable in the history of science. It was during these years that Newton composed almost all his great work, the Principia Mathematica.

 Contents

## Authoring Principia

### Newton's major work—Principia Mathematica

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 September 19, 1685, that he had received many useful communications from Flamsteed, especially regarding Saturn, "whose orbit, as defined by Kepler," Newton "found too little for the sesquialterate proportions." Newton refers to 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 satellites 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 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 Kepler's third law. "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 sesquialteral 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 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, which he had enlarged and completed. Except for correspondence with Flamsteed 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."

At the next meeting of the Society, on April 28, "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 May 19, 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 June 2, 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 May 22 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 Hooke's conduct when the manuscript of the Principia was presented to the Society. 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."

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 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 June 20, 1686 Newton wrote to Halley the following letter: "Sir, In order to let you know the case between Mr Hooke and me, I give you an account of what passed between us in our letters, so far as I could remember; for 'tis long since they were writ, and I do not know that I have seen them since. I am almost confident by circumstances, that Sir Chr. Wren knew the duplicate proportion when I gave him a visit; and then Mr Hooke (by his book Cometa written afterwards) will prove the last of us three that knew it. I intended in this letter to let you understand the case fully; but it being a frivolous business, I shall content myself to give you, the heads of it in short, viz, that I never extended the duplicate proportion lower than to the superficies of the earth, and before a certain demonstration I found the last year, have suspected it did not reach accurately enough down so low; and therefore in the doctrine of projectiles never used it nor considered the motions of the heavens; and consequently Mr Hooke could not from my letters, which were about projectiles and the regions descending hence to the centre, conclude me ignorant of the theory of the heavens. That what he told me of the duplicate proportion was erroneous, namely, that it reached down from hence to the centre of the earth.

"That it is not candid to require me now to confess myself, in print, then ignorant of the duplicate proportion in the heavens; for no other reason but because he had told it me in the case of projectiles, and so upon mistaken grounds, accused me of that ignorance. That in my answer to his first letter I refused his correspondence, told him I had laid philosophy aside, sent him, only the experiment of projectiles (rather shortly hinted than carefully described), in compliment to sweeten my answer, expected to hear no further from him; could scarce persuade myself to answer his second letter; did not answer his third, was upon other things; thought no further of philosophical matters than, his letters put me upon it, and therefore may be allowed not to have had my thoughts of that kind about me so well at that time. That by the same reason he concludes me then ignorant of the rest of the duplicate proportion, he may as well conclude me ignorant of the rest of that theory I had read before in his books. That in one of my papers writ (I cannot say in what year, but I am sure some time before I had any correspondence with Mr Oldenburg, and that's above fifteen years ago), the proportion of the forces of the planets from the sun, reciprocally duplicate of their distances from him, is expressed, and the proportion of our gravit to the moon's conatus recedendi a centro terrae is calculated, though not accurately enough. That when Hugenius put out his Horol. Oscill., a copy being presented to me, in my letter of thanks to him I gave those rules in the end thereof a particular commendation for their usefulness in Philosophy, and added out of my aforesaid paper an instance of their usefulness, in comparing the forces of the moon from the earth, and earth from the sun; in determining a problem about the moon's phase, and putting a limit to the sun's parallax, which shows that I had then my eye upon comparing the forces of the planets arising from their circular motion, and understood it; so that a while after, when Mr Hooke propounded the problem solemnly, in the end of his attempt to prove the motion of the earth, if I had not known the duplicate proportion before, I could not but have found it now. Between ten and eleven years ago there was an hypothesis of mine registered in your books, wherein I hinted a cause of gravity towards the earth, sun and planets, with the dependence of the celestial motions thereon; in which the proportion of the decrease of gravity from the superficies of the planet (though for brevity's sake not there expressed) can be no other than reciprocally duplicate of the distance from the centre. And I hope I shall not be urged to declare, in print, that I understood not the obvious mathematical condition of my own hypothesis. But, grant I received it afterwards from Mr Hooke, yet have I as great a right to it as to the ellipse. For as Kepler knew the orb to be not circular but oval, and guessed it to be elliptical, so Mr Hooke, without knowing what I have found out since his letters to me, can know no more, but that the proportion was duplicate quam proximè at great distances from the centre, and only guessed it to be so accurately, and guessed amiss in extending that proportion down to the very centre, whereas Kepler guessed right at the ellipse. And so, Mr Hooke found less of the proportion than Kepler of the ellipse.

"There is so strong an objection against the accurateness of this proportion, that without my demonstrations, to which Mr Hooke is yet a stranger, it cannot be believed by a judicious philosopher to be any where accurate. And so, in stating this business, I do pretend to have done as much for the proportion as for the ellipsis, and to have as much right to the one from Mr Hooke and all men, as to the other from Kepler; and therefore on this account also he must at least moderate his pretences.

"The proof you sent me I like very well. I designed the whole to consist of three books; the second was finished last summer being short, and only wants transcribing, and drawing the cuts fairly. Some new propositions I have since thought on, which I can as well let alone. The third wants the theory of comets. In autumn last I spent two months in calculations to no purpose for want of a good method, which made me afterwards return to the first book, and enlarge it with diverse propositions some relating to comets others to other things, found out last winter. The third I now design to suppress. Philosophy is such an impertinently litigious lady, that a man has as good be engaged in lawsuits, as have to do, with her. I found it so formerly, and now I am no sooner come near her again, but she gives me warning. The two first books, without the third, will not so well bear the title of Philosophiae Naturalis Principia Mathematica; and therefore I had altered it to this, De Motu Corporum libri duo

"But, upon second thoughts, I retain the former title. It will help the sale of the book, which I ought not to diminish now it's yours. The articles are with the largest to be called by that name. "If you please you may change the word to sections, though it be not material. In the first page, I have struck out the words uti posthac docebitur as referring to the third book; which is all at present, from your affectionate friend, and humble servant,

"Is. NEWTON."

On June 20, 1686 Halley wrote to Newton: "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. 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. I am, Sir, your most affectionate humble servant, E. HALLEY."

On June 30, 1686 the council resolved to license Newton's book, entitled Philosophiae Naturalis Principia Mathematica.

On July 14, 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 April 6th, and the whole work published about midsummer in that year, July 5th, 16872. 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.

### Conflict between the University and James II

While Newton was writing the second and third books of the Principia, a very important event occurred at Cambridge which had the effect of bringing him before the public in a new light. James II had already, in 1686, in open violation of the law, conferred the deanery of Christ Church at Oxford on John Massey, a person whose sole qualification was that he was a member of the Church of Rome; and the king had boasted to the pope's legate that "what he had done at Oxford would very soon be done at Cambridge." In accordance with this boast, in February 1687 he issued a mandate directing that Father Alban Francis, a Benedictine monk, should be admitted a master of arts of the university of Cambridge, without taking the oaths of allegiance and supremacy.

Upon receiving the mandamus Dr Pechell, the master of Magdalene College, who was vice-chancellor, sent a messenger to the duke of Albemarle, the chancellor, to request him to get the mandamus recalled; and the registrary and the bedell waited upon Francis to offer him instant admission to the degree if only he would take the necessary oaths. Both the king and the monk were inexorable. The court and the university were thus placed on a collision course. A menacing letter was despatched by Sunderland to shake the firmness of the university—but, though humble and respectful explanations were returned, the university showed no sign of compliance, nor even of a desire to suggest a compromise. In consequence the vice-chancellor and deputies from the senate were summoned to appear before the High Commission Court at Westminster. Newton was one of the eight deputies appointed by the senate for this purpose.

The deputies, before starting for London, held a meeting to prepare their case for the court. A compromise which was put forward by one of them was stoutly and successfully resisted by Newton. On April 21 the deputation, with their case carefully prepared, appeared before the court. Lord Jeffreys presided at the board. The deputation appeared as a matter of course before the commissioners, and was dismissed. On April 27 they gave their plea. On May 7 it was discussed, and feebly defended by the vice-chancellor. The deputies maintained that in the late reign several royal mandates had been withdrawn, and that no degree had ever been conferred without the oaths having been previously taken. Jeffreys spoke with his accustomed insolence to the vice-chancellor, silenced the other deputies when they offered to speak, and ordered them out of court. When recalled the deputies were reprimanded, and Pechell was deprived of his office as vice-chancellor, and of his salary as master of Magdalene.

Newton returned to Trinity College to complete the Principia. While thus occupied he had an extensive correspondence with Halley, a very great part of which is extant. The following letter from Halley, dated London, July 5th, 1687, announcing the completion of the Principia, is of particular interest:

"I have at length brought your book to an end, and hope it will please you. The last errata came just in time to be inserted. I will present from you the book you desire to the Royal Society, Mr Boyle, Mr Paget, Mr Flamsteed, and if there be any else in town that you design to gratify that way; and I have sent you to bestow on your friends in the University 20 copies, which I entreat you to accept. In the same parcel you will receive 40 more, which having no acquaintance in Cambridge, I must entreat you to put into the hands of one or more of your ablest booksellers to dispose of them. I intend the price of them, bound in calves' leather, and lettered, to be [OCR error] shillings here. Those I send you I value in quires at 6 shillings, to take my money as they are sold, or at 5 sh. for ready, or else at some short time; for I am satisfied there is no dealing in books without interesting the booksellers; and I am contented to let them go halves with me, rather than have your excellent work smothered by their combinations. I hope you will not repent you of the pains you have taken in so laudable a piece, so much to your own and the nation's credit, but rather, after you shall have a little diverted yourself with other studies, that you will resume those contemplations wherein you had so great success, and attempt the perfection of the lunar theory, which will be of prodigious use in navigation, as well as of profound and public speculation... You will receive a box from me on Thursday next by the wagon, that starts from town tomorrow."

### Illness in 1693

In 1692 and 1693 Newton seems to have had a serious illness, the nature of which has given rise to very considerable dispute. In a letter dated the September 13, 1693, addressed to Samuel Pepys, he writes: "Some time after Mr Millington had delivered your message, he pressed me to see you the next time I went to London. I was averse, but upon his pressing consented, before I considered what I did, for I am extremely troubled at the embroilment I am in, and have neither ate nor slept well this twelvemonth, nor have my former consistency of mind. I never designed to get any thing by your interest, nor by icing James's favour, but am now sensible that I must withdraw from your acquaintance, and see neither you nor the rest of my friends any more, if I may but have them quietly. I beg your pardon for saying I would see you again, and rest your most humble and obedient servant." And in a letter written to John Locke in reply to one of his about the second edition of his book, and dated the 15th of October 1693, Newton wrote: "The last, winter, by sleeping too often by my fire, I got an ill habit of sleeping; and a distemper, which this summer has been epidemical, put me farther out of order, so that when I wrote to you, I had not slept an hour a night for a fortnight together, and for five days together not a wink. I remember I wrote to you, but what I said of your book I remember not. If you please to send me a transcript of that passage, I will give you an account of it if I can."

The loss of sleep to a person of Newton's temperament, whose mind was never at rest, and at times so wholly engrossed in his scientific pursuits that he even neglected to take food, must necessarily have led to a very great deal of nervous excitability. It is not astonishing that rumours got abroad that there was a danger of his mind giving way, or, according to a report which was believed at the time, that it had actually done so. Pepys must have heard such rumours, as in a letter to his friend Millington, the tutor of Magdalene College at Cambridge, dated September 26, 1693, he wrote: "I must acknowledge myself not at the ease I would be glad to be at in reference to excellent Mr Newton; concerning whom (methinks) your answer labours under the same kind of restraint which (to tell you the truth) my asking did. For I was 10th at first dash to tell you that I had lately received a letter from him so surprising to me for the inconsistency of every part of it, as to be put into great disorder by it, from the concern I have for him, lest it should arise from that which of all mankind I should least dread from him and most lament for I mean a discomposure in head, or mind, or both. Let me, therefore, beg you, Sir, having now told you the true ground of the trouble I lately gave you, to let me know the very truth of the matter, as far at least as comes within your knowledge."

On September 20, 1693, Millington wrote to Pepys that he had been to look for Newton some time before, but that "he was out of town, and since," he says, "I have not seen him, till upon the 28th I met him at Huntingdon, where, upon his own accord, and before I had time to ask him any question, he told me that he had written to you a very odd letter, at which he was much concerned; added, that it was in a distemper that much seized his head, and that kept him awake for above five nights together, which upon occasion he desired I would represent to you, and beg your pardon, he being very much ashamed he should be so rude to a person for whom he hath so great an honour. He is now very well, and though I fear he is under some small degree of melancholy, yet I think there is no reason to suspect it hath at all touched his understanding, and I hope never will; and so I am sure all ought to wish that love learning or the honour of our nation, which it is a sign how much it is looked after, when such a person as Mr Newton lies so neglected by those in power."

The illness of Newton was very much exaggerated by foreign contemporary writers. Christiaan Huygens, in a letter dated June 8, 1694, wrote to Leibniz, "I do not know if you are acquainted with the accident which has happened to the good Mr Newton, namely, that he has had an attack of phrenitis, which lasted eighteen months, and of which they say his friends have cured him by means of remedies, and keeping him shut up." To which Leibniz, in a letter dated the 22nd of June, replied, "I am very glad that I received information of the cure of Mr Newton at the same time that I first heard of his illness, which doubtless must have been very alarming."

### Initial election to Parliament

The active part which Newton had taken in defending the legal privileges of the university against the encroachments of the crown had probably at least equal weight with his scientific reputation when his friends chose him as a candidate for a seat in parliament as one of the representatives of the university. The other candidates were Sir Robert Sawyer and Mr Finch. Sir Robert headed the poll with 125 votes, Newton next with 122 and Mr Finch was last with 117 votes. Newton retained his seat only about a year, from January 1689 till the dissolution of the Coventry Parliament in February 1690. During this time Newton does not appear to have taken part in any of the debates in the House, but he was not neglectful of his duties as a member. On April 30, 1689 he moved for leave to bring in a bill to settle the charters and privileges of the University of Cambridge, just as Sir Thomas Clarges did for Oxford at the same time, and he wrote a series of letters to Dr Lovel, the vice-chancellor of the university, on points which affected the interests of the university and its members.

Some of the members of the university who had sworn allegiance to James had some difficulty in swearing allegiance to his successor. On February 12, 1689, the day of the coronation of William and Mary, Newton intimated to the vice-chancellor that he would soon receive an order to proclaim them at Cambridge. He enclosed a form of the proclamation, and expressed a hearty "wish that the university would so compose themselves as to perform the solemnity with a reasonable decorum."

### Notes

Note 1: For information on Newton's later life and post-Principia work, see the Isaac Newton's later life.

Note 2: Richard S. Westfall, Never at Rest, ISBN 0-521-27435-4 (paperback) Cambridge 1980..1998.

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