# Papers relating to the origin of the dispute

*Origin of Commercium Epistolicum*^{[1]}

In the year 1704 M^{r} Iohn Bernoulli 1671 M^{r} Newton was upon a designe of publishing his Theory of colours & Method of fluxions, but upon \wrangling/ disputes arising about \the/ theory of colours, \he chose the sake of a quiet life {he} to/ laid|y| his designe aside before the Tract about the Method of fluxions was finished. There wanted that part w^{ch} in w^{ch} the method \was to be taught/ of resolving Problems was cannot be reduced to Quadratures w^{ch} related to the solution of Problemes not reducible to colours Quadratures. In this state things rested till the year 1676, & then he d out of what he had written five years before & other older Papers he composed the Book of Quadratures extracting it out of his old Papers, & in the year 1676 1704 published this Tract & his Theory of colours, And considering that D^{r} Wallis had said in the intro preface to the first Volume of his works \without then being then contradicted/ that M^{r} Newton had in his Letters of 13 Iune & 24 Octob. 1676 explained to M^{r} Leibnitz the Method of fluxions found by him ten years before that or above & in the Introduction of the Tract of Quadratures said that he found the method of fluxions nine years gradually in the years 1665 & 1666, this being not so much as D^{r} Wallis had said nine years before \(in the Preface to the first Volume of his works)/ without being then contradicted. But And the next \year/ in the Acta Eruditorum for Ianuary, M^{r} a Paper was published without the name of the Author w^{ch} is giving an account of the said Tract accuses it of plagiary in these words. *Ingeniosissimus deinde Author [Newtonus] antequam ad Quadraturas Curvarum [ (vel potius figurarum curvilinearum) veniat præmittit brevem Isagogem. Quæ ut MELIVS INTELLIGATVR, sciendum est, cum magnitudo aliqua continuò crescit, incrementa illa momentanea appellari differentias, nempe inter magnitudinem quæ antea erat et quæ pen mutationem momentaneam est producta; at hinc natum este Calculum integralem ei reciprocum summatorium, cujus elementa ab INVENTORE D. Godefrido Guilielmo Leibnitio in his Actis sunt tradita, varij usus tum ab ipso tum a fratibus Bernoulijs tum a D. Marchione Hospitalio — sunt tradita, varij usus tu \ostensi/ Pro diffentijs igitur \IGITVR/ Leibnitianis D. Newtonus adhibet semper [pro ijsdem adhibuit Fluxiones — ijs tum in suis Principijs Mathe Naturæ Mathematicis tum in alijs postea editis* [pro differentijs Leibnitianis] *eleganter est usus, QVEMADMODVM et Honoratus Fabrius in sua synopsi Geometrica, motuum progressus Cavallerianæ methodo substituit* SVBSTITVIT. D^{r} Menkenius who printed this \defatory {sic}/ Paper ought to discover the name of the author that he may either prove the accusation or be looked upon as guilty of calumny.

This accusation gave a beginning to the late controversy about the author of the Method. For when the accusation was retorted \contradicted/ by D^{r} Keill, M^{r} Leibnitz wrote twice to the R. S. against the D^{r} & in his second Letter dated 29 Decem 1711 justified the accusation saying that the Acta Lipsiensia had given every man his due & claimin|ed|g a right to the Invention & affir representin|ed|g that n{o} body had gone before him in it; & only palliated the accusation by saying that he & his friends allowed that M^{r} Newton also attained the method by him self. And all this as much as to say that M^{r} Newton was only second \not first/ inventor, & therefore committed a Act of Plagiary in pretending to have found the Method gradually in the years 1665 & 1666 & t|w|hereby giving |he gave| himself a right to the method. For second Inventors have no right. It lay upon M^{r} Leibnitz therefore to prove that he was the first inventor.

M^{r} Leibnitz pressing the R. Society to condemn & silence D^{r} Keill they appointed a Committee to {illeg} search out old Letters & Papers relating to this matter & report what they found in them, & then ordered the Papers & Report to be publist. And M^{r} Leibnitz avoided answering this Commercium Epistolicum to the day of his death: For the Book is matter of fact & uncapable of an answer, & establishes M^{r} Newton the first Inventor.

To avoid answering it he pretended the first year that he had not seen it nor had leasure to examin it, but had desired M^{r} Newton an emin Mathematician & unpartial Iudg \appealed to the judgment of the first rank & well skilled in these things & impartial desiring him/ to examin it & give his judgment upon it. And his judgment upon it dated 13 Iune 16 1713 was inserted into a defamatory Letter dated Iuly 29 following, & published in a flying paper without the names of the Authors or printers or city where it was printed & dispersed \it/ over all the western parts of Europe: a back biting infamous way of proceeding w^{ch} in England is punishable by the civil Magistrate. This Paper has been since translated into French & inserted into another abusive Letter & answered by D^{r} Keill in Iuly 1714 & no Answer has \yet/ been given to the Doctor. In his paper the Auth M^{r} Newton is accused of plagiary by both the authors in a more open manner & in a higher degree then before, & therefore the Au by the laws of all nationes the Authors ought to have proved their acusations upon pain of being deemed guilty of calumny. The Iudge pretends that when the Letters & Paper published in the Commercium were written M^{r} Newton did not so much as dream of his calculus of fluxions because there are no prickt letters in them; no no {sic} when he wrote his book of Principles, these letters not \first/ appearing til in the third Volume of the book of D^{r} Wallis many years after the differential calculus had obteined in {al} every where, & M^{r} Newton not understanding {sic} \knowing/ how to find the differences of differences long after it was familiar to others. But this \impartial/ Iudge gave judgment contrary to the evidence w^{ch} lay before him. For the Method of fluxions is taught in the Itrod without prickt letters in the Introduction to the very Book de Quadratura Curvarum, & prickt letters w^{th} the Rule {illeg} for finding finding all degrees of fluxions was|er|e {p} published A.C. 1693 three year in the second Volume of the works of D^{r} Wallis A.C. 1693, three years before any Rule for finding all degrees of differences came abroad. This candid Iudge cited M^{r} Iohn Bernoulli by the name of an eminent Mathematician as if he were M^{r} Bernoulli was not the author of the Iudgment & yet M^{r} Bernoulli was no Leibnitz in \his/ Letters to M^{r} l'Abbé Conti, Ba the Comtess of Kilmansegge & Baron Bothmar that M^{r} Bernoulli himself was the author.

About November or December 16 1715

M^{r} Leibnitz in a Letter to M^{r} Chamberlain dated from Vienna 28 Apr. 1714, wrote that he had not yet seen the Commercium Epistolicum but & so could not make such a|n| defence \Appology/ as the thing required: but & a little after he wrote again to the to M^{r} Chamberlain that when he came to Hanover might \print/ another Commercium & for that end desired that the Original Letters might be sent to him to be printed entire{ly} & impartially. But upon reading this Letter to the Society, it was represented that this was a reflexion upon their Committee, the the Originalls ought to be kept for justifying what had been published by the Order, & that M^{r} Newton was so far from publishing the Commercum {sic} himself that he did not so much as produce some ancient Letters in his own custody, & it as improper that M^{r} Leibnitz himself should be trusted with printing a Commercium, at least not untill the ancient Letters in his Custody should be examined & approved by them who knew the hands, & at the same time M^{r} Newton produced two ancient Letters w^{ch} he had in custody without producing them to p|b|e published in the Commercium, the one written to him by M^{r} Leibnitz himself from Hanover $\frac{7}{17}$ Martij 1693, the other writter to him by D^{r} Wallis from Oxford Apr. 10^{th} 1695. The first shews that M^{r} Leibnitz himself in \the beginning of/ the year 1693 gave M^{r} Newton the preference at w^{ch} time {illeg} till the beginning of the year 1693 at which time he knew nothing more of M^{r} Newtons Method then what he had learnt from his Letters & Papers writ in or before the year 1676 & from his book of Principles & the second (compared with the Preface to the Doctor first Volume of the Doctors works) shews what opinion the English Mathematicians had of this matter before when they first heard that the Differential Method began to be celebrated in Holland as invented by M^{r} Leibnitz. These two Letters being examined & approved befor the Society were ordered to be laid up in their Archives & an Anwer was given to D|M|^{r} Chamberlain that if M^{r} Leibintz had any ancient Letters {illeg} relating to this matter & would send them to any friend in London to be examined before the R. Society by them who knew the hands, after they were approved he might either print them himself or have them printed in the Philosophical Transactions if he pleased: but nothing has been sent.

About November or December 1715 M^{r} Leibnitz in a Letter to M^{r} Abbe Conti wrote a large Postcript relating to these matters, railling at the Commercium Epistolicum as attaquing his candor by false interpretations & omittion|ng| of what made for him or against me M^{r} Newton, & \saying/ that his adversaries should not have the pleasure to see him return an answer to their slender reasonings, & endeavouring to run the dispute into a squabble about universal gravity, & Gods being |& occult qualities & miracles & Gods being not the soul of the world but being| intellegentia supramundana & not the soul of the world nor having need of a sensorium, & about atoms & the nature of time space & time, & about solving mathematical Problems. All whi{s}|c|h are prevarications serving \digressions prevarications & evasions serve/ to no other purpose then to avoid answering the Com̄ercium Epistolicum|.| [by running the dispute into a squabble about other matters.] |And when he was pressed to let these digressions alone & return an Answer to the Commercium he replied in a Letter from Hannover Apr {illeg}|9|^{th} 16 1716 that:| M^{r} Newton being pressed to write an Answer to this Postscript that both might be shewed to the King, wrote his Letter of Feb 26 17^{15}_{16}, & therein told M^{r} Leibnitz of his an endeavouring to avoid answering the Commercium Epistolicum Epistolicum first by pretending the first year that he had not seen the Book & afterwards by running into all these digressions & saying that th|hi|s adversaries should not have the pleasure to see him answer it. He pressed M^{r} Leibnitz therefore to answer the book & told him that he was the aggressor & had accused M^{r} Newton \brought an accusation/ of plagiary & by the laws of all nations was bound to prove his accusation upon pain of being deemed guilty of calumny.

M^{r} Leibnitz in his answer dated \from Hanover/ 9 Apr. 1716 st. nov. pretended that he was not the aggressor nor had accused M^{r} Newton of plagiary, & refused to answer the Commercium Epistolicum saying that to answer it from point to point would require another book as bigg at the least as that /&\ that he must for that end enter into a great examination of many minutes | particulars passed 30 or 40 years agoe, of w^{ch} he remembred but little; \&/ that he must search his old Letters many of w^{ch} {illeg}|w|ere lost, besides that for the most part he had kept no minutes of his own Letters, & the others were buried in a great heap of Papers which he could not unravel but with time & patience. But he had little or no leasure for that, being enga{illeg}|g|ed at present w^{th} \in/ business of a very different nature. [And in the end of his Letter he s|a|greed w^{th} M^{r} Newton that the Accusers ought to prove their accusations upon pain of being deemed guilty of calumny. but said that he himself was accused]

In the Elogium of M^{r} Leibnitz published in the Acta Eruditorum for I Iune|ly| 1717, M^{r} \pag. 335/ there are these words \{pa}/. Quo perspicerent intelligentes quid de tota illa controversia sentiendum sit Commercio Epistolico Anglorum, \[D. Leibnitius]/ aliud quoddam suum idem amplius opponere decreverat & paucis ante obitum diebus Cl. Wolfio significavit se Anglos famam ipsius lacessentes reipsa refutaturum: quamprimum enim a laboribus historicis vacaturus sit, daturum se aliquid in Analysi prorsus inexpectatum & cum inventis quæ hactenus in publicum prostant sive Newtoni sive aliorum nil quicquam affine habens. Here M^{r} Leibnitz a few days before his death {illeg} said wrote that he would refute the English with by out by a new & wonderful analytical invention of a different kind from any thing yet extant, & the Author of the Elogium thinks it was not by a new Commercium Epistolicum. But what ever it was, we were to stay for it till his historical labours were \should be/ at an end. And then parturient m{on}tes [And the world was to suspend their judgments \about this matter/ till M^{r} Leibnitz could \be at leasure to/ be heard. [And yet the whole series of \the/ Letters between M^{r} Leibnitz one the one ha{n} & |&| \& the English by means of/ M^{r} Oldenburgh is already published so far as it relates to this matter, unless & it does not appear that he had any correspondence with the English in those days but by means of M^{r} Oldenberg And the Commercium already published is plane p matter of fact & uncapable of being confuted by a contrary Commercium.

It has been said that in this Commercium several things w^{ch} made against M^{r} Newton have been omitted|.| but \But/ M^{r} Leibnitz endeav^{r}ing to prove this produced two instances: {illeg}|b|ut failed in them both.

It has been said that M^{r} Lei the Letters are interpreted falsly & maliciously: But no instances of this kind have been produced when M^{r} {p} would \Leibnitz/ named an instance of this kind it proved a mistake of his own. For {h} in his Letter of 9^{th} of April 16 1716 he said that where the Author of the Remarks upon the Commercium Epistolicum said (pag. 108) Sensus verborum est, *quod Newtonus fluxiones differentijs Leibnitianis substituit:* this interpretation was a malicious one: but M^{r} Newton \in his Observations upon this Letter/ has shewed that M^{r} L. has|im|self has misinterpreted the place. {illeg}

It has been said that the Author of th M^{r} Leibnitz found the Commercium Epistolicum differential Method by himself. And so he might not withstanding any thing {to} in the Commercium Epistolicum to the contrary. The Committee of the R. S. say they take the proper question to be, not who invented this or that Method, but who was the first inventor of the method. D^{r} Wallis in the Pr Wh D^{r} Wallis in the Preface to the first Volume of his works said that M^{r} Newton in his Letters of Iune 13 & Octob 24 1676 that M^{r}|h|ad explained to M^{r} Leibnitz the Method found by him ten years before that time or above: but he meant nothing more then that M^{r} Newton had given him so much light into it as made it easy \for him/ to find it out. And M^{r} Newton seems to have been sensible |there|of that wher|n|e he said that he wrote his Letter of 24 Octob. 1676. For{illeg} there \he/ said that the \foundation of the/ Method being obvious he would conceale it in an Ænigma. However

Seing therefore that M^{r} Leibnitz & his friends have deserted the Question: for putting an end to this dispute, we will only add a few observations in justification for wiping off the aspersion u

— In this state things rested till the year 1676, & then he composed the Book de Quadratura Curvarum extracting it out of the afores^{d} Tract & other older papers & in his Letter of Iu|Oc|tob 24 1676 cited the first Proposition of the Book verbatim {illeg} in an Ænigma w^{ch} he th as the foundation of the method, & said that this foundation gave him Theorems for squaring Curvilinear figures, the invention of w^{ch} is explained in the first six Propositions of the Book, & in the same letter copied the Ordinates of \the/ Curves w^{ch} in the end of the tenth Propositions are compared w^{th} the squared by the Conic Sections, & wrote a Letter to M^{r} Collins dated 8 Novem. 1676 relating to the 7^{th} 8^{th} 9^{th} & 10^{th} Propositions of the book: all w^{ch} make it sufficiently appear that the Book was then in MS.

In autumn 1690 D^{r} Halley & M^{r} Raphson took coming to Cambridge took the Book w^{th} them to London & in the end of the year 1692 D the first Proposition thereof w^{th} the solution & illustra{ted} by examples in first & second fluxions was printed almost verbatim in the second Volume of |t|his|e| works \of D^{r} Wallis/ & came abroad the next year. And at length

Whereas a Paper was published in favour of the Acta Eruditorum af{o}r {sic} Iuly 1761 in favour of M^{r} Iohn Bernoulli against D^{r} Keill, & therein M^{r} Bernoulli is called *excelsum ingenium* as|&| if he were no *vir ad abstrusa et abdita detegenda natus*, as if he were not the author of that Paper, & yet the author thereof ascribes \it to M^{r} Bernoulli by/ calls|i|ng M^{r} Bernulli's formula of an Equation formulam *meam formulam* \pag 314/ & thereby ascribes that Paper to M^{r} Bernoulli. And whereas in a Letter dated 13 Iune 1713 M^{r} Bernoulli is & inserted into a defamatory Paper \Letter {sic}/ \another Letter/ dated 29 Iuly 1713 & dispersed al M^{r} Bernoulli is cited by the name of an fam eminent Mathematician as if he were not the auther of that Letter & yet M^{r} Leibnitz in several Letters has affirmed that he was the author thereof [& the designe of these Letters is to lay aside the ancient Records published in the Commercium Epistolicum] & the designe of this shuffling is to propagate an opinion that the Book of Quadratures published by S^{r} Isaac Newton in the year 1704 is a piece of plagiary & that M^{r} Leibnitz was the first inventor of the direct method of Fluxions & M^{r} Bernoully the first inventor of the inverse method thereof; & for g{illeg}ing compassing this end M^{r} Ber designe all endeavours have been made use of to lay aside the ancient Letters & Papers published in the Commercium Epistolicum \without answering them/ & to bring the Question to a squabble about \universal gravity &/ occult qualities, \& miracles/ & the Vacuum, \& the sensorium of God & perfection of the world/ & the nature of time & space & the solving of Probles|m|s &c all w^{ch} are nothing to the purpose{illeg}: & at length we are told that M^{r} Leibnitz these are \therefore/ to give notice that since the|i|s \M^{r} Leibnitz & his/ friends of M^{r} Leibnitz are of s\have for above five years/ declined \& still decline/ answering the Comercium Epistolicum, the Mathematicians at London \friends of D^{r} Keill will henceforward/ decline medling \any furt/ with their squabbles. {illeg}

The first Proposition of the Book of Quadrature *de Quadratura Curvarum* with it's solution & examples in first & second fluxions was printed \almost varbatim/ in the seco second Volume of D^{r} Wallises works in the end of the year \autumn/ 1692 & came abroad {illeg} the \next/ year, & therefore the book was then in Manuscript D^{r}

The first Proposition of the Book de Quadratura Curvarum with its Solution & examples in first & second fluxions was published almost verbatim by the D^{r} Wallis in the second volume of his Works A.C. 1693, (pag. 391, 392, 393, &c) being sent to y^{e} D^{r} & printed off the year before; & therefore this Book was then in Manuscript. M^{r} Ralpson has testified publickly that he & D^{r} Halley had it in their hands at Cambridge about the year 1691 in order to bring it up to London & D^{r} Halley remembers that this was in A.C. 1690, & thence it may be understood that this Book was MS. before the differential method was spread abroad began to be spread abroad by M^{r} Iohn Bernoulli & his brother. In M^{r} Newton's Letters of Iune 13, Octob. 24 & Novem. 8^{th} 1676, there are many things relating to this Book, & \particularly the first Proposition is there cited verbatim &/ therefore it was in MS before M^{r} Leibnitz knew any thing of the Method differential Method.

In this Book are many things w^{ch} had they been proposed as Problemes to be solved by others, might have puzzeled all the Mathematicians in Europe. As for instance, to t|r|educe the integration of the following equations to the quadrature of the conic sections. $\frac{\mathrm{d}\stackrel{.}{\mathrm{z}}{\mathrm{z}}^{2\mathrm{n}-1}}{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{n}}+\mathrm{g}{\mathrm{z}}^{2\mathrm{n}}}=\stackrel{.}{\mathrm{y}}$. $\mathrm{d}\stackrel{.}{\mathrm{z}}{\mathrm{z}}^{\frac{1}{2}\mathrm{n}-1}=\mathrm{e}\stackrel{.}{\mathrm{y}}+\mathrm{f}\stackrel{.}{\mathrm{y}}{\mathrm{z}}^{\mathrm{n}}+\mathrm{g}\stackrel{.}{\mathrm{y}}{\mathrm{z}}^{2\mathrm{n}}$. $\mathrm{d}\stackrel{.}{\mathrm{z}}{\mathrm{z}}^{\frac{3}{2}\mathrm{n}-1}=\mathrm{e}\stackrel{.}{\mathrm{y}}+$$+\mathrm{f}\stackrel{.}{\mathrm{y}}{\mathrm{z}}^{\mathrm{n}}+\mathrm{g}\stackrel{.}{\mathrm{y}}{\mathrm{z}}^{2\mathrm{n}}.\mathrm{d}\stackrel{.}{\mathrm{z}}\sqrt{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{n}}+\mathrm{g}{\mathrm{z}}^{2\mathrm{n}}}=\mathrm{z}\stackrel{.}{\mathrm{z}}.\mathrm{dz}\sqrt{\frac{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{n}}}{\mathrm{g}+\mathrm{h}{\mathrm{z}}^{\mathrm{n}}}}=\mathrm{z}\stackrel{.}{\mathrm{y}}$. Or to reduce the integration of the following equations to the simplest cases of quadratures. $\mathrm{a}{\mathrm{z}}^{\mathrm{p}+\mathrm{q}}{\mathrm{z}}^{\mathrm{m}}+\mathrm{b}{\stackrel{.}{\mathrm{z}}}^{\mathrm{q}}{\mathrm{z}}^{\mathrm{n}}{\stackrel{.}{\mathrm{y}}}^{\mathrm{p}}=\mathrm{c}{\stackrel{.}{\mathrm{z}}}^{\mathrm{p}}{\stackrel{.}{\mathrm{y}}}^{\mathrm{q}}$. $\mathrm{a}{\stackrel{.}{\mathrm{z}}}^{\mathrm{q}}{\mathrm{z}}^{\mathrm{m}}+\mathrm{b}{\stackrel{.}{\mathrm{z}}}^{\mathrm{q}-\mathrm{p}}{\mathrm{z}}^{\mathrm{n}}{\stackrel{.}{\mathrm{y}}}^{\mathrm{p}}=\mathrm{c}{\stackrel{.}{\mathrm{y}}}^{\mathrm{q}}$.

But the pedantry of proposing Problems to be solved by others is not in fashion in England.

And after all this light received from England (besides what he saw in the hands of M^{r} Collins when he was last in London) the {sic} was reason to print {illeg}

In the end of the year 1679 I found the demonstration of Keplers Proposition

{illeg}

At the Request of D^{r} Halley I sent to him in October or November 1684 the following Propositions demonstrated, & soon after gave him leave to to {sic} communicate them to the R. Society
1
2
3
4
5
6
7
8
9
10
11
And by consulting the Minute Books \Iournal Books/ of the R. Society of \And/ find that |t|he|y| {illeg} communicated them to the R. Society |were| \read before the R. Society entered in the Register Book of the Society Decem 10 1684/ Decem. |10| following. And there upon a|A|t {sic} \And at/ the request of the R. Socity {sic} that they might be printed I soon after set upon {wri} writing the Book of Principles {illeg} & sent it to the Society \Apr / & they \ordered it to be printed/ May 19 1686 as I find by another {illeg} a minute in their Iornal book. And their President oredered it to be printed \wrote an Imprimatur/ Iuly 5 following. And the book came abroad the next year in as I find by the Account given of it in the Philosophical Transactions

In the end of the year 1679 I com in answer to a Letter from D^{r} Hook I wrote that then Secretary of the R.S. I wrote that whereas it had been objected against the diurnal motion of the earth that it would cause bodies to fall in the west, the contrary was true. For bodies in falling would keep the {illeg} motion from west to east which they had from west to east before they began to fall, & this motion being added to their motion of falling would carry them to the east D^{r} Hook replied that soon after that |t|it|he||y| would be|do| so under the line & that \Equator/ but in our Latitude they would fall not exactly to the east by|u|t decline from the east a little to the south. And that he had made some experiments thereof & found that they did fall in that manner. And he added that they they would not fall down to the center of the earth but rise up again & describe an Oval as the Planets do in their orbs. Whereupon I computed what would be the Orb described by the Planets. For I had found before that the by the sesquialterate proportion of the tempora periodica of the Planets {illeg} \with respect/ to their distances from the Sun, that the forces w^{ch} kept them in their Orbs were about the Sun were as the squares of their mean distances from the Sun reciprocally: & I found now that whatsoever was the law of the forces w^{ch} kept the Planets in their Orbs, the areas described by a Radius drawn from them to the Sun would be proportional to the times {illeg} in w^{ch} they were described. And upon \by the help of/ these two Propositions I found that their Orbs would be such Ellipses as Kepler had described.

In the end of the year 1679 I communicated to D^{r} Hook then Secretary of the R. Society, that whereas it had been objected \represented/ {illeg} that if the Earth had a diurnal motion from west to east, it would leave falling bodies behind & cause them to fall to the west; the contrary was rather true. \In falling/ They would \keep/ the motion w^{ch} they had from west to east before they began to fall, & this motion compounded with the motion of descent arising from gravity would carry them to the east.

In Spring 1684 D^{r} Halley coming to Cambridge & asking me if I knew what figure the Planets described in their Orbs about the Sun was very desirous to have my Demonstration & in autumn following I sent to him the following Propositions demonstrated & soon after gave him leave to {illeg}|c|ommunicate them to y^{e} R.S. And they were entred in the Register Book of the R. Society Decem 10 1684. D^{r} Hook complained that I had the hint from him, but he producing no demonstration the R. Society desired that the Paper conteining to|h|os Propositions might be printed. And thereupon I soon after set upon writting the Book of Principles, & sent it to the R. Society & they made an order May 19 1686 that it should be printed, as I find by a minute in their journal book; & their President wrote an Imprimatur Iuly 5 following.

By reason of the short time in w^{ch} I wrote it, & of its being copied by an Emanuensis who understood not Mathematicks there were some faults besides those of the Press. By measuring the quātity of water w^{ch} ran out of a vessel through a round hole in the borrom of a given magintude I found that the velocity of the water in the hole was s|t|hat w^{ch} a body would acquire in falling half the height of the vessel water in the vessel. And by other Experiments I found afterwards that the water accelerated after it was out of the vessel untill it arrived at a distance from the vessel equal to the diameter of the hole, & by accelerating acquired a velocity equal to the v equal to that w^{ch} a body would acquire in falling \almost/ through the whole height of the water in the vessel. or thereabouts. In {illeg} The Demonstration of the first Corollary of the 11^{th} 12^{th} & 13^{th} Propositions I {omm} being very obvious I omitted it in the first edition & instead thereof inserted \contented myself with/ adding the 17^{th} Proposition whereby it is proved that a body in all cases \in/ going from any place with any velocity will in all cases describe a conic Section: w^{ch} is that \very/ Corollary. But at the desire of M^{r} Cotes I In the 10^{th} Proposition of the second Book the tangent of the Arch GH was drawn from the wrong end of the arch, but is now put right w^{ch} made some think that there was a error in second differences fluxions.

The Ancients had two Methods in Mathematics w^{ch} they called Synthesis & Analysis {illeg} or Composition & Resolution. By the method of Analysis they found their inventions & by the method of Synthesis they published them composed them for the publick. The Mathematicians of the last age have very much improved Analysis & Analysis & laid aside the method of Synthesis but stop there in so much as & think they \have/ solved a Problem when they have only resolved it it, & by this means the method of Synthesis is almost laid aside. The Propositions in the following book were invented by Analysis. But considering that they were the Ancients (so far as I can find) admitted nothing into Geometry but wha before it was demonstrated by c|C|omposition I composed what I invented by Analysis to make it \Geometrically authentic &/ more fit for the publick And this is the reason why this Book was written \in words at length/ after the manner of the Ancients without Analytical calculations. But if any man who undestands Analysis will resolve de reduce the Demonstrations of the Propositions book from their composition back into Analysis (w^{ch} is \very/ easy to be done,) he will esily see by \what/ method of Analysis they were invented. \And/ By this means the Marquess de l'Hospital was able to affirm that this Book was [pres tout de ce Calcule] almost wholy of the infinitesimal calculus Analysis. [And M^{r} Leibnitz in a letter to me dated $\frac{7}{17}$ 17 Mar. 16 1693 st. vet. Mirifice ampliaveras Geometriam tuis seriebus sed edito Principiorum opere ostendisti patere tibi etiam quæ Analysi receptæ non subsunt. Conatus sum ego quo. Notis commodis adhibitis, quæ Differentias & Summas exhibent, Geometriam illam quam Transcendentem appello, Analysi quodammodo subjcere {sic}. And in the Acta Eruditorum for May |1700| 1699 he allowed that I was the first who by a specimen made publick (meaning in the Scholium upon the 35^{th} Proposition of y^{e} second Book) had proved that I had the method of maxima & minima in infinitesimals. And when he himself had composed the Pro first & second sections of the second Book of my Principles without Analysis in another form of words without calculations he concluded: {Om nia} ‡]

In the second Lemma of the second Book I set down the elements of this \Analytic/ method & demonstrated the{m} \Lemma/ by composition. And in order to make use of it in some fo the demonstration of some following Proposition. And I added a Scholium not to give away the Lemm because M^{r} Leibnitz had published those Elements a year \about eighteen |a year & some|/ months before without making any mention of the Correspondence w^{ch} I had with him before By means of M^{r} Oldenburg ten years before \that time/, I added a Scholium not to give away the Lemma but to put him in mind of that correspondence. For in my Letter of 24 Octob \1676/ I told him th{a} mentioned an Analysis w^{ch} proceeded without stopping at surds & gave me the method of tangents of Slusius & \{illeg}{ciate}d Quadratures of Curves & gave me/ the Quadrature of Curves by Series w^{ch} brake converging series w^{ch} brake off & became infinite equations when the Curve can be squared by such \finite/ equations & {to} I said that I had wrote a Treatise upon this method & the method of Series together. five years before, that is in the year 1671, w^{ch} was two years before M^{r} Leibnitz began to study the higher Geometry. And that the Problem upon which this method was grounded I set down enigmatically. And in a former Letter dated 13 Iune 1676 I said that the method of converging series in conjunction with some other methods (meaning the methods of fluxions \that of extracting fluents & {illeg}t of/ |of| arbitrary series) extended to almost all Questions except perhaps some numeral ones. of like those of Diophantus. And in hs|i|s answer \Letter Answer/ dated Iune Aug 27 1676 he sa replied that he did not beleive that my methods were so general there being many Problems w^{ch} could not be reduced to equations or Quadratures. And in mine dated 24 Octob. 1676, I represented that my Analysis \in m{illeg}/ proceeded without stopping at surds, readily gave me the faciliated the quadrature of Curves & \readily/ gave the method of tangents of Slusius, faciliated Quadratures & gave me converging series for squaring of Curves w^{ch} become finite when ever the Curve could be squared by a finite equation. \& extended also to Problems w^{ch} could not be reduced to Quadratures./ And I said also that I had written a Treatise of on this subject five years before tha five years before, that is, in the year 1671, & w^{ch} was two years before M^{r} Leibnitz began to study the higher Geometry. And in the Problem upon w^{ch} \foundation of this/ this Method was grounded I \I said was obvious/ |&| set \it/ down enigmatically in this sentence: *Data æquatione fluentes quotcun quantitates involvente invenire fluxiones: & vice versa*. And in both my Letters I said that I had then absteined from this subject five years, being tired w^{th} it before. And to put M^{r} Leibnitz in mind of all this & of what he had further received from M^{r} Oldenburg & M^{r} Collins in these days \in relation to thes matters/, was the designe of this Scholium.

|For| At the same time that M^{r} Oldenburg sent my Letter of 13 Iune 1676 to M^{r} Leibnitz, he sent also a (w^{ch} was Iune 26 following) he sent also (at the request of M^{r} Leibnitz) a collection of extracts of the papers & Letters of M^{r} Iames Gregory then deceased. And among those extracts was a Letter of M^{r} Gregory dated to M^{r} Collins dated 5 Sept 1670 in w^{ch} M^{r} Gregory represented that he had improved the method of Tangents beyond what D^{r} Barrow so as had done, so as to draw Tangents to \all/ Curves without calculation. There was also a Copy of a Letter written by me to M^{r} Collins 10 Decem 1672 in w^{ch} I wrote that the Methods of Gregory & Slusius were only *Corollarium Methodi generals|i|s quæ extendit se citra molestum ullum calculum non modo ad ducendum Tangentes ad quasvis Curvas sive Geometricas sive Mechanicas vel quomodocum rectas lineas vel curvas alias|ve| Curvas respicien{illeg}tes; verum etiam ad resolvendum alia abstrusiora Problematum genera de Curvitabus. Areis Longitudinibus, Centris gravitatum Curvarum* &c. Ne (quamademodum Huddenij Methodus de Maximis et Minimis) ad solas restringitur æquationes illas quæ quantitatibus surdis sunt immunes. {illeg} &c. |These letters he received in Iuly or August 1676 & in October following coming from Paris to London he there \met/ with D^{r} Barrows Lectures as he had informed us ☉| & in October following he saw my Letter o procured D^{r} Barrows Lectures as above & saw my Letter of Iuly|Octob| 24, [& therein had notice of my Compendium of Series w^{ch} was the Analysis per series numero terminorum infinitas, [& wanted the demonstration] & consulted M^{r} Collins to see his correspondence with Gregory & me] And all this ☉ And all this was enough to put him upon considering how to improv{e} the method of Tangents of D^{r} Barrow as Gregory had done before, so as to make it proceed without draw Tangents without calculation. And \then/ how to improve the method of Tangents of Gregory & Slusius so as to make he |it| proceed without stopping at fractions & surds, & to extend \it/ not only to Tangents & Maxima & Minima but |a|to|ls||o| \to/ all other Quadratures & all other sorts of Problems so as to become such a general as I described in my Letters of 10 Decem 1672, 13 Iune 1676 & 24 Octob. 1676.

In October following M^{r} Leibnitz came to London &

In this same year in a Letter dated May 12 M^{r} Leibnitz desired M^{r} Oldenburg to procure from M^{r} Colling|s| the Demonstration of two of my series for finding the Arc of a circle whose sine was \is/ given & the sine whose Arc was \is/ given; & that is, the method {illeg}|o|f finding them. And in October follow this Letter occasioned {illeg} my Letter in October following he de coming to London he consulted M^{r} Collins to see the commerce Letters the Mathematical Letters & Papers w^{ch} M^{r} Collins had received from M^{r} Iames Gregory & me. And no doubt he would desire to see the demonstration of the two series w^{ch} he wanted, that is, the *Analysis per series numero terminorum infinitas*, w^{ch} D^{r} Barrow had sent \from me/ to M^{r} Collins in Iuly 1669, w^{ch} Analysis consisted in reducing quantities to converging series & applying those series to the solution of Problems by the Method of Moments & Fluxions. For the direct method of fluxions \described in the three first four first Propositions of the Book of Quadratures is in this/ is there described under this Title: *Inventio Curvarum quæ quadrari possunt* And there are examples of the inverse method in finding the Areas &c & Length of the Quadratic Trochoides & Quadratix & the lengths of some Curves & demostrating the first Proposition of the Book. And the universality of the Method is described in these words. *Quicquid vulgaris Analysis per æquationes ex finito terminorum numero constantes (quando id sit possibile) perfici\a/t hæc per æquationes infinitas semper perficit: ut nil dubitaverim nomen Analyseos etiam huic tribuere {illeg}* — *Deni ad Analysin \merito/ pertinere* merito *censeatur cujus beneficio Curvarum areæ & longitudines &c (id modo fiat) exacte et Geometrice determinentur. Sed ista narrandi non est locus.* This last is explain comprehended in the fift & sixt Propositions of the Book of Quadratures. And therefore the Method comprehended in the first six Propositions of the Book of Quadratures in the year 1669. And all this may suffice to justify me in publishing the second Lemma of the second Book of Principles as my own.

By measuring the quantity of water w^{ch} ran out of a vessel in a given time through a given round hole in the bottom of the vessel: I found that the velocity of the water in the hole was that w^{ch} a body would acquire in falling half the height of the water stagnating in the vessel. And by other experiments I found afterwards that the water accelerated after it was out of the vessel untill it arrived at a distance from the vessel equal to the diameter of the hole; & by accelerating acquired a velocity equal to that w^{ch} a body would acquire in falling through a l the hei almost the whole height of the stagn water stagnating in the vessel, or thereabouts.

The Demonstration of the first Corollary of the 13^{th} Proposition of the first Book being very obvious I omitted \it/ in the first Edition of this Book & contented my self with adding \determining in/ the 17^{th} Proposition what will be the Conic Section described in all cases by a body going from any place with the square of the distance.

In the X^{th} Proposition of the second Book there was a mistake by drawing the Tangent of the Arch GH from the wrong end of the Arch. But the mistake was rectified in the last \second/ Edition. And there may have I been some other mistakes occasioned by the shortness of the time in w^{ch} the book was written & partly by its being copied by an Emanuensis who understood not what he copied; besides the Press faults. For after I had found eight or tenn \tenn or twelve/ of the Propositions relating to the heavens, & they were communicated to the R.S. in in December \10^{th}/ 1684, & at their request that the Book \Propositions/ might be printed I set upon composing this Book & sent it to the \R.S./ in May 1686 [& they ordered it to be printed May 19,] as I find entred in their Iournal Books Decem 10 1684 & May 19^{th} 1686. And I was enabled to make the greater dispatch by means of the Book of Quadratures composed some years before|.| & annexed to this Edition

*Multa ex his deduci possent praxi accommodata, sed nobis nunc fundamenta Geometrica jecisse sufficeatrit, in quibus maxima consistebat difficultas. Et fortasse|i|s attente consideranti vias quasdam novas vel autem satis certe satis antea impeditas aperuisse videbimur. Omnia autem respondent nostræ Analysi infinitorum, hoc est, calculo (cujus elementa quædam in his Actis dedimus) communibus quoad licuit veb|r|bis hic expresso*.

M^{r} Leibnitz has claimed a right to {illeg}

And whereas M^{r} Leibnitz has introduced \encouraged/ a practise of claiming a right to \other mens/ inventions as second inventor & offered M^{r} Newton such a right to the differential method & put in such a claim \has claimed such a right himself/ to y^{e} differential method of Mouton, & offered M^{r} Newton such a right to the Differential \infinitesimal/ M|m|ethod of fluxions \or method of fluxions/ And yet second Inventors have no right. The sole right is in the first inventor untill another man finds out the same thing. and then to take away the right of the first inventor & divide it between him & that other would be an Act of injustice. It lies upon M^{r} Leibnitz \therefore/ {his} to \renounce/ all right to the differential method of Mouton & to all other inventions whatsoever \as second inventor/ & to forbeare this practise as tend to encourage this practise \in others/ as tending to plagiary.

M^{r} Leibnitz has claimed a right to a property of a series of |num|\bers/ natural, triangular, pyramidal, triangulo-triangular numbers &c set down be published before by M^{r} Paschal, & to make it his own has represented that he wondered that M^{r} Paschal should omit it. He is therefore to renounce all right to the invention of this property.

M^{r} Leibnitz in his Letter of 21 Iune 1677, when is further desired to explain what he meant by these words in his Letter of 21 Iune 1677 : *Clarissimi Slusij Methodum Tangentium ge nondum esse absolutam Newtono assentior: et jam a multo tempore rem Tangentium generalius tractavi scilit|c|et per differentias Ordinataru:*: whether he \Hee might/ meant|e| only that he had long ago understood the method of tangents of D^{r} Barrow or \further/ that he had long ago changed the letters a & e used by D^{r} Barrow into the symbols dy & dx, or \further/ that he had \long ago/ made it still more general so as to proceed without taking away fractions or surds. or whe that he had \long ago/ made it still more general so as to be able to reduce the inverse Problems of extend it into Quadratures & other Problems besides those of drawing Tangents & curvatures \of Curves/ & inverse Problems of tangents & others more & into the reduction of inverse Problemes of tangents \& others/ to converging series & to differential aquations & others Quadratures [for whether he intended only by those ambiguous words to l these are the words by which he began first to lay hands upon the method w^{ch} \intr{illeg}re with pretend to the method w^{ch}/ M^{r} Newton had been describing to him the year before] & \And he is desired also to tell us whether/ whether by these words he intended to make himself the first or second inventor of the Differential method & if he will not explain them \himself/ he authorises us to take them in the most obvious sence o that sence which seems most pe obvious to us, viz^{t} that he had found th long be a multo tempore, long before those days found \out/ the Differentiall method so far \that he had found it/ above nine years before he published it. And then he is to beg M M^{r} Newton's pardon for pretending to have found the differential method long before he had found it, in order to make himself the first inventor. For its most certain that he had it not when he wrote his Letter of 27 Aug. 1676.

M^{r} Leibnitz in his Answer to M^{r} Fatio published in the Acta Eruditorum for May 1700, has said: *Certe cum elemente calculi mea edidi anno 1684, ne constabat quidem mihi aliud de inventis ejus \Newtoni/ in hoc genere quam quod ipse olim significaverat in literis, posse se tangentes invenire non sublatis irrationalibus, quod Hugenius quo se posse mihi significavit postea, etsi cæterorum istius calculi adhuc expers: sed majora multu|o|m consecutum esse Newtonum, viso demum libro Principiorum ejus, satis intellexi*. He did know also And yet it's certain that he by his Letter of 27 Aug. 1676 that he did then know that M^{r} Newton's method extended not only to the Quadratures of drawing of Tangents but also faciliated the Quadrature of curves. His words are Arbitror quæ celare voluit Newtonus de Tangentibus ducendis ab his non abludere Quod addit, ex hoc edem fundamento, quadraturas quo reddi faciliores me in hac sententia hac confirmat, nimirum semper figuræ illæ sunt quadrabiles quæ sunt ad Æquationem Differentialem. Æquationem Differentialem voco talem qua valor ipsius dx exprimitur quæque ex alia deriva{ll}atur|a| qua est qua valor ispius x exprimebatur. M^{r} Leibnitz knew \was told/ also \in the same Letter/ that by the same method M^{r} Newton \determined maxima & minima without sticking \taking away irrationals/ & solved other problems also &/ had found general Theorems for squaring of Curves which gave the Quadrature by converging series w^{ch} brake off & became finite æquations in certain cases. And the first of the Theoremes M^{r} Newton sent him \in that Letter/ with several examples thereof. And therefore M^{r} Leibnitz ought to beg M^{r} Newton's for {illeg} tellin saying \pardon for pretending/ that when he in y^{e} year 1684 when he published his method he knew nothing more of M^{r} Newtons inventions in this kind, then that he could draw tangents without taking away irrationalls. He ought to When he published his method he ought \in point of candor/ to have told the world what light he \had/ received into it from England, & to have let them know \at that time/ that by the words AVT SIMILI he meant such a method of M^{r} Newton like invented by M^{r} Newton & signified him by \described by him in/ his Letter of 10 Decem 1672, 13 Iune 1676 & 24 Octob 11|6|76 \copies of/ w^{ch} had been sent to him by M^{r} Oldenburg. And he ought also to have acknowledged at the same time what light he had received also from D^{r} Barrows method of tangents. But to conceale all this, \to conceal his whole correspondence w^{th} M^{r} Oldenb: to conceal every thing that he had received from England/ & afterwards to pretend that he \then/ knew nothing more of M^{r} Newton's method \at that time/ then that it extended to the drawing of Tangents , is without taking away surds, is a sort of behavior that wants an apology pardon.

And since he has told us that his friends kno in his Letter to D^{r} Sloan dated Decem 29 1711 he has told us that his friends know how he came by the Differential Method, & his way of coming by it in question & yet he \but is silent about this matter \it/ pretending that/ {bon hach} but M^{r} Keil being haud satis exercitatus {illeg} artis Inveniendi arbite{r} he \[M^{r} Leibnitz]/ is not bound to teach him his methods & yet his way of coming by it is in question: it lies upon him \in point of candor/ openly & plainly \& without any further hesitation/ to tell the world how he came by this method

And since in the same Letter he has told us that he had this method above nine years before he published it, & it follows from thence that he had it in the year 1675 or before: it lies upon him to prove that he had it before he wrote his Letters to M^{r} Oldenburg dated Aug 27, 1676 wherein he affirmed that he had Problemes of the Inverse Method of tangents & many others could not be reduced to infinites series nor to Equations or Quadratures. Or rather it lies upon \him/ in point of candor to tell us what he means by pretending to have found make us understand that by p make us understand \tell us/ what he means by pretending to have found the method before he had found it. For its most certain that he had it not before |by that affirmation that when| he wrote that Letter, he had it not.

And he is also to tell us what he meant formerly by pretending to the {illeg} saying in his Letter of 21 Iune 16

And whereas in his Letter of Iune 29 1676|7| he wrote: *Clarissimi Slusij methodum Tangentium nondum esse absolutam Celeberrimo Newtono assentior. Et jam a multo tempore rem Tangentium longe generalius tractavi; scilicet per differentias Ordinatarum:* Which is as much as to say that he had long ago improved the method of Tangent beyond what Slusius had done, And made it non general a so as to include that method as a par \particular/ branch thereof, & not to stick at fractions surds & yet in his Letter to M^{r} Oldenburg dated $\frac{18}{28}$ Novemb. 1676 \w^{ch} was but half a year before,/ he was contriving to improve the method of Slusius & make it general \make it more general/ \& extend it to all sorts of Problems, not by the Differential method but by/ by means of a Table of Tangents. he is des it lies upon him \in point of candor/ to explain to us what he meant in those days, by pretending to have fo \Iune following/ when he was but newly fallen into y^{e} Differential method, to pretend that he had thereby improved the method \of tangents beyond that/ of Slusius {C}, jam multo tempore long before that time \those days/, & to satisfy that he did it b either by mad{oe}{illeg} accident \or inadvertancy/ or with some other designe then to rival M^{r} Newton & to make us beleive that he had it before M^{r} Newton explained it to him in his Letter of 13 Iune & 24 Octob 1676, & before he received a copy of M^{r} Newton's Letter of 1672 10 Decem 1672 whereby it was further explained.

And whereas M^{r} Leibnitz

M^{r} Leibnitz in his Answer to M^{r} Fatio — sent him in that Letter with several examples thereof. And therefore M^{r} Leibnitz is to tell us what he meant by concealing all this when he published his Differentiall method, & \excusing himself afterwards by/ telling the world afterwards that he knew the when he published it he knew nothing more of M^{r} Newton's methods \inventions in this kind/ then that he could draw Tangents without taking away surds. All that he then published of the Differential method was the manner of drawing tangents & determining maxima & minima without taking away fractions & surds. He knew that M^{r} Newton's method would do all this & therefore ought \in candor/ to have acknowledged what he then knew. He added \in general/ that his method extended to other Probl difficul Problemes w^{ch} were not to be resolved without his method AVT SIMILI. If he knew any thing of a methodus SIMILIS such another method it lies upon him candor to let the to tell us why he did not then discover what he knew how he came by his knowledge & whose was the methodus similis METHODVS SIMLIS |do M^{r} Newton justice by saying concerning it & acknowledging whose was the method & what he had method. For if learnt from England concerning it.| [It was impossible in those days to understand the meaning of the words AVT SIMILI without an interpreter & therefore they were not inserted in favour of M^{r} Newton] If the words AVT SIMILI were not to be understood untill M^{r} Newton should put in his cla put in his claim, they were intended not to do him justice but only save the reputation of M^{r} Leibnitz in case of a claim the method should be claimed from him.

But when M^{r} Newton did not put in a claim, M^{r} Leibnitz began to omit the methodus similis, as may be seen in his Paper De Geometria Recondita published in the Acta Eruditorum for Iune 1686 [where the Inventions of several Mathematicians are recited & nothing more then the method of converging series is allowed to M^{r} Newton]

When M^{r} Leibnitz first published his Differential Method he ought in candor to have acknowledged what he knew of M^{r} Newtons method for doing the same things. All that he then explained concerning \of/ his \own/ method was how to draw tangents & determin maxima & minima without taking aways fractions & surds. He knew that M^{r} Newtons method would do all this & therefore ought in candor to have acknowledged it. He After he had thus far explained his own method he added that this method extended to the most difficult & notable problems w^{ch} were scarce to be resolved without the Differential calculus AVT SIMILI, W or another like it. What he meant by the words AVT SIMILI it was impossible for the Germans to understand without an interpreter. He ought to have done M^{r} Newton justice in plain intelligible language & tho told them that his\the proposed/ method \w^{ch} he there published/ extended to such difficult Problemes as were not to be resolved without his calculus differentialis or another calculus of M^{r} Newton of w^{ch} he had received some notice by his correspondence w^{th} M^{r} Oldenburg & w^{ch} he took to be like his own. But on the contrary in his Answer to M^{r} Fatio he has said \he ha/ published in the Acta Eruditorum for May 1700 he has affirmed t said: Certe cum elementa . . . . . satis intellexi. [It lies upon him therefore in point of candor to tell us why in the year 1684 when he first published his method, he concealed his knowledge of what had been communi cated to him from England of the same kind before the method was know to him; & why in the year 1700 in his answer to M^{r} Fatio he denyed his knowledge of almost all that he had received from England: & why he now {sic} \now denys what he then acknowledged &/ contradicts himself, & telling us that the it doth not appear by the book of Principles that M^{r} Newton knew any thing of this method.] In his boo When by his correspondence w^{th} Newton \M^{r}/ Oldenburg he received the first notices of M^{r} Newtons method he could acknowledge|d| \in part/ saying: Arbitror quæ celare voluit Newtonus de Tangentibus ducendis ab his non abludere. Quod addit, ex hoc eodem fundamento quadraturas quo reddi faciliores, me in sententia hoc confirmat, nimirum semper figuræ illa sunt quadrabiles quæ sunt ad æquationem differētialem. When he published his method he conceled all this tho he knew it. f as is manifest by the words AVT SIMILI. When M^{r} Fatio taxed For he understood his own words tho they were not inteded to be understood by \intelligible to/ others. A When M^{r} Fatio taxed him he denyed that he knew any thing more of M^{r} |N's| method before the publishing of his Principles, & now he denys that he whereby he understood that it was of much greater extent. And now he denys that the Principles contein any thing \make any discovery/ of the method. It lies upon him therefore in point of candor to give an account of all himself in all this m his candor in all this management, & at length to acknoledge publickly, that before he wrote his Letter of 21 Iune 1677 he did know that M^{r} Newton had a method not only for drawing of tangents & determining maxima & minima without sticking taking away surds but also faciliating the quadrature of curves \& determining their lengths & curvatures/, for squaring them /Curves\ by series w^{ch} break off & become finite in certain cases, for solving \inverse/ Problemes of the inverse m Tangents & other more difficult, for For comparing the areas of Curves w^{th} those of the Conic Sections, & w^{ch} M^{r} Newton joyned w^{th} his method of series & r by w^{ch} his method of series became so general as to extend to almost all Problemes except perhaps some numeral ones like those of Diophantus

$\begin{array}{rr}47,35.& 6856,59\\ 335\phantom{35.}& 413\\ 1645\phantom{35.}& 3304\end{array}$

## Observations upon the Proposal

Lin 1. *very scarce.* Obs. They are very scarce only in London, & sixty Tonns of new farthings would be enough for that city.

Lin 3. *One hundred Tons per an for ten years necessary.* Obs. It never was though{t}necessare to coyn to much. In coyning the last copper money in never was thought one hundred Tons per an in five years made such a clamour as gave occasion to the Parliament to stop that coynage for a year &c Six hundred Tons are was sufficient for were then found sufficient for all England, & now will

l. 6 *for private advantage only.* Obs. The Officers of the Mint have always opposed the proposals made for private advantage, & in their Reports upon them represented that the next copper money should be coyned \as neare as could be/ to y^{e} intrinsic value ({illeg}{ludin incl} & charge that there might be no more temptation to counterfeith them was necessary, that they be well coyned to make it difficult to counterfeit them, & that there should be no more coyned then was necessary for the uses of the nation for fear of clamours & that the coynage should be upon account so that if any thing were to be got by *it it might go to* the king. And that it be set upon a \put into a/ standing method

We have shewed that M^{r} Leibnitz in his Letter return \the ende of the year 1676 in returning/ home from France was med through En by England & Holland was meditating how to improve the method of Slusius & extend it to all sorts of Problems & for this end proposed a general Table of Tangents, \& therefore had not yet found out the true improvement: but/ & about half a year after, viz^{t} Iune 21 16677 when he was newly fallen upon the true improvement, wrote back: *Clarissimi Slusij methodum Tangentium nondum esse absolutam celeberrimo Newtono assentior. Et jam a multo tempore rem Tangentium generalius tractavi scilicet per differentias Ordinatarum.* Which is as much as to say that he had long ago improved made this improvement \long ago/. It lies upon him in point of candor to make us understand that he pretended this antiquity of his invention with some other designe then to rival \& supplant/ M^{r} Newton & make us \him & his friends/ /us\ beleive that he had the Differential method before M^{r} Newton explained it to him in his Letters of 13 Iune & 24 Octob 1676 & before M^{r} Oldenburg sent him a copy of M^{r} Newtons Letter of 10 Decem 1672 concerning it

In Mathematical Sciences the Ancients had two Methods which they called Synthesis & Analysis or Composition & Resolution. By the Method of Analysis they found their inventions & by the Method of Synthesis they composed them for the publick. The Mathematicians of the last age have very much improved Analysis, but stop there & think they have solved a Problem when they have only resolved it, & by this means the method of Synthesis is almost laid aside. The Propositions in the following Book were invented by Analysis: but considering that the Ancients (so far as I can find) admitted nothing into Geometry before it was (for the greater certainty) demonstrated by composition, I composed what I invented by Analysis, to make it Geometrically authentic & fit for the publick. And this is the reason why this Book was written in words at length after the manner of the Ancients without Analytical Symbols & Calculations. But if any man who understands Analysis, will reduce the Demonstrations of the Propositions from their composition back into Analysis (w^{ch} is easy to be done) he will see by what method of Analysis the Propositions were invented. And by this means the Marquess de l'Hospital was able to affirm that this Book was [presque tout de ce calcule] almost whole of the Infinitesimal Calculus Analysis.

In the second Lemma of the second Book of these Principles, I set down the Elements of the Analytic Method & demonstrated the Lemma by Composition in order to make use of it in the Demonstration of of some following Propositions. And because M^{r} Leibnitz had published those elements a year & some months before without making any mention of the Correspondence w^{ch} I had with him by means of M^{r} Oldenburg ten years before that time, I added a Scholium not to give away the Lemma, but to put him in mind of that Correspondence. \in order to his making a publick acknowledgm^{t} thereof before he proceeded to claim that Lemma from me./ For in my Letter dated Iune 13^{th} 1676 I said that the Method of converging series in conjunction with some other methods (meaning the Methods of Fluxions & Arbitrary Series) extended to almost all Questions except perhaps some numeral ones like those of Diophantus. And in his Answer dated Aug. 27 1676, he replied that he did not beleive that my method were so general, there being ma\n/y Problemes w^{ch} could not be reduced to Qu Equations or Quadratures. And in mine dated 24 Octob. 1676 I represented that my Analysis proceeded without stopping at surds, & readily gave the method of Tangents of Slusius & faciliated Quadratures \& extended also to Problems which could not be reduced to Quadratures/ & gave me converging series for squaring of Curves which become finite whenever the Curve can be squared by a finite equation. & which extended also to Problems which could not be reduce to Quadratures. And I said also that I had written a Treatise on this subject five years before that time, that is in the year 168|7|1: w^{ch} was two years before M^{r} Leibnitz began to study the higher Geometry. And the foundation of this method I said was obvious, & wrote it down enigmatically in this sentence: *Data æquatione fluente quotcun æquationes quantitates involvente invenire fluxiones, & vice versa.* And in both my Letters I said that I had joyned this method & the method of Series together & that I had then absteined from this subject five years, being tyred with it before. When this Letter \of 24 Octob/ arrived at London M^{r} Leibnitz was there the second time & saw it, & procured D^{r} Barrows Lectures, \wherein was his method of Tangents invented above 12 years before, & M^{r} Leibnitz/ & in his way to Hanover was considering how to make the Method of Tangents of Slusius general by a Table of Tangents \as I find by a Letter of his to M^{r} Coll M^{r} Oldenburg 18 N dated at Amsterdam Novem 18 \1676/. st vet/. And when M^{r} Leibnitz \Oldenburg/ heard that M^{r} Leibnitz was arrived at Hanover, w^{ch} was in March following, he t sent to him a copy of my last Letter. And M^{r} Leibnitz in his Answer dated 21 Iune 1677 sent back D^{r} Barrows Method of Tangents under the differential notation & how this method might be improved so as to give the method of Tangents of Slusius; & then how it might be further improved so as to g proceed without taking away fractions & surds: & then added Arbitror quæ celare voluit Newtonus, ab his non abludere. Quod addit, ex hoc eodem fundamento, nimirum semper figuræ illæ sunt quadrabiles, quæ sunt ad æquationem differentialem. And to put M^{r} Leibnitz in mind of all this was the meaning of that|e| Scholium above mentioned.

At the same time that M^{r} Oldenburg sent my Letters of 13 Iune 1676 to M^{r} Leibnitz (w^{ch} was the Iune 26^{th} day of the same Iune) he sent also (at the request of M^{r} Leibnitz) a collection of extracts of the Letters & papers of M^{r} Gregory to M^{r} Collins dated 5 Sept. 1670 in w^{ch} M^{r} Gregory represented that he had improved the Method of Tangents beyond what D^{r} Barrow had done, so as to draw Tangents to all Curves without Calculation. There was also a Letter Copy of a Letter written by me to M^{r} Collins 10 Decem 1672 in which I wrote that the methods \of Tangents/ of Gregory & Slusius were only Corollarium Methodi generalis quæ extendit se citra molestum ullum calculum non modo ad ducendum Tangentes ad quasvis Curvas sive Geometricas sive Mechanicas, vel quomodocun rectas lineas aliasve curvas respicientes; verum etiam ad resolvendum alia abstrusiora Problematum genera de Curvitatibus, Areis, Longitudinibus, Centris gravitatum Curvarū &c. Ne (quemadmodum Huddenij Methodus de Maximis et Minimis) ad solas restringitur æquationes illas quæ quantitatibus surdis sunt immunes Hanc methodum intertextui alteri isti, qua Æquationum Exegesi{t}|n| instituo, reducendo eas as Series infinitas. These Letters M^{r} Leibnitz received in Iuly 168|7|6. And in October following he procured D^{r} Barrows Lectures at London as above & saw my Letter of Octob 24. And all this was enough to put him upon considering how to improve the Method of Tangents of D^{r} Barrow as Gregory had done before & so as to draw Tangents without calculation; And then \and/ how to improve the Method of Tangents of Gregory & Slusius so as to make it proceed without stopping at fractions & surds, & to extend it not only to Tangents & Maxima & Minima, but also to Quadratures & all other sorts of Problemes, so as to become such a general Method as I described in my Letters of 10 Decem. 1672, 13^{th} Iune 1676 & 24 Octob 1676. And after all this light received from England (besides what he saw in the hands of M^{r} Collins when he was last in London) there was reason to put him in mind, of his of his correspondence with by the Scholium above mentioned, \I had great reason by the Scholium above mentioned to put him in mind/ of his correspondence w^{th} M^{r} Oldenburg & M^{r} Collins.

By measuring the quantity of water w^{ch} runns out of a vessel in a given time through a given round hole in the bottom of the vessel I found that the velocity of the water in the hole was that which a body would acquire in falling half the height of the water stagnating in the vessel. And by other experiments I found afterwards that the water accelerated after it was out of the vessel & untill it arrived at a distance from the vessel equal to the diameter of the hole, & by accelerating acquired a velocity equal to that of which a body would acquire in falling the whole height of the water stagnating in the vessel or thereabouts. That the streame might not accelerate by its weight, it ran out horizontally, & that its diameter at several distances from the hole might be measured with more exactness it was above half an inch thick.

In the tenth Proposition of the second Book there was a mistake in the first edition by drawing the Tangent of the Arch GH from the wrong end of the Arch \which caused an error in the conclusion/: but in the second Edition I rectified the mistake And there may have been some other mistakes occasioned by the shortness of the time in which the book was written & by its being copied by an Emanuensis who understood not what he copied; besides the press faults. For I wrote it in 17 or 18 months, beginning in the end of December 1684 & sending it to y^{e} R. Society in May 1686: excepting that about ten or twelve of the Propositions were composed before, viz^{t} the 1^{st} & 1{0}|1|^{th} in December 167 1679, the 6^{th} 7^{th} 8^{th} 9^{th} 10^{th} 12^{th}, 13^{th} 17^{th} Lib. I & Prop \the/ 1, 2, 3 & 4 Lib. II, composed in the summer time of the year 1684 \in 1684 Iune & Iuly 1684./

## {In} this is the account how long Sr I. was writing his principia

<107r>Quæ in \hoc/ libro {illeg}

Veteres duplici methodo tractabant res Geometricas, Analysi \scilicet/ et Synthesi \seu Resolutione\et compositione/ ex Pappo liquet/. Per Analysin investibant Propositiones suas et per Synthesin demonstrabant \inventas/ ut in Geometriam admitterentur. Laus enim Geometriæ in ejus certitudine consistit, & synthetica Demonstrationes ideo nihil in ipsam \prius/ admitte|i|batur /debet\ quam \debet qu{ā} redda{illeg}tur certissimum. Hæc certitudo oriebatur/ demonstrabitur, & /ex\ demonstrationi|bu|e|s|sset & et Veterum Demonstrationes omnes erant syntheticæ. Algebra nihil aliud est quam Arithmetica ad res Ge\o/metricas applicata, et ejus operationes complexæ sunt & erroribus nimis obnoxiæ & a solis Algebræ peritis legi possunt. Propositiones autem in Geometria {illeg} sic proponi debent ut a plurimis legantur, et mentem claritate sic maxime afficiant, ideo synthetice demonstrandæ sunt. Vtilis est Analysis ad veritates inveniendas, sed certitudo \inventi/ duplicatur \examinari debet/ per compositionem Demonstrationis & quam fieri potest perspicua reddi \clara/ & omnibus manifesta reddi: Et \præsertim cum/ Proportio quæ non demonstratur synthetice, ex mente Veterum {illeg} non demonstratur, ideo in Geometriam admitti non debet. Problemato quo quorum constructiones no innotescunt tantum per Analysin, nondum soluta sunt sed tantum resol{u}ta, et \nec prius soluta dici debent quam/ ea|o|rum Constructiones prius demonstram|en|/tur\ debent synthetice|.| quam solut

His de causis Librum Principior Pr{illeg}|o|positiones quas in hoc Libro \in Libris Principiorū/ quas inveni synth per Analysin demonstravi per Synthesin, si forte Prop. XLV Libri primis excipiatur & Prop. X Lib II excipiantur. Et Mathematicis autem hujus sæculi, qui fere toti in Algebra versantur in Algebra, conquesti sunt quod hæc non scripsissem Algebraice, quasi hoc genus \hocce syntheticum/ scribendi minus placet, quasi seu quod nimis prolixum sit videatur & methodo veterum nimis affine, seu quod ratio|nē| inveniendi minus appareat{illeg}i patefacui maximis incumbunt, min \minus patefaciat/ Et certe minori cum labore potuissem scribere sp Analyticè quàm p{ro}{illeg} ea componere quæ Analytice invenere|a|m: sed propositum not|n| erat Analysin docere|.| sed Philosophiam mathemati Scribebam ad Philosophos Elementis Geometriæ imbutos & Philosophiæ naturalis fundamenta Geometrice demonstrata{illeg} {sic} ponebam. Ideo Geometrica \Et/ inventa \Geometrica {illeg}/ quæ ad Astronomiam et Philosophiam non spectabant, vel \penitus/ prætere|i|bam, vel leviter tantum attingebant|m|{ur}. Sed c|C|um autem de Analysi disputetur qua usus sum, visum est hanc paucis exponere.

1 Geometræ Veteres quæsita investigabant Analysin, inventa demonstrabant per Synthesin, demonstrata edebant \ut/ in Geometriam recipe\r/endu|tur|. Resoluta non statim recipiebantur in Geometriam: opus erat solutione per compositionem demonstrationum. Nam Geometriæ vis et laus omnis in certitudine rerum, certitudo in demonstrationibus luculenten compositis constabat. In hac scientia non tam brevitati quam scribendi quam certitudim rerum consulendum est. Ideo Pro in sequenti Tractatu Propositiones per Analysin inventas demonstravi synthetice.

2. Geometria Veterum versabatur quidem circa magnitudines: sed Propositiones de magnitudinibus nonunquam demonstrabantur per \mediante/ motu locali: ut cum triangulorum æqualitas in Propositione quarta libri primi elementorum Euclidis demonstraretur transferendo trangulum alterutrum in locum alterius. Sed et genesis magnitudinum per motum continuum recepta fuit in Geometria: ut cum linea recta duceretur in lineam rectam ad generandam aream, & area duceretur in lineam rectam ad generandum solidum. Si recta quæ in aliam ducitur datæ sit longitudinis generabitur area parallelogramma. Si longitudo ejus lege aliqua certa continuo mutetur generabitur area curvilinea: \Si magnitudo areæ in rectam ductæ continuo mutetur generabitur solidum superficie curva terminatum./ Si tempora, vires, motus et velocitates motuum exponantur per longitudines linear|s|um vel \per/ magnitudines arear|s|um \angularum,/ aut solidarum \vel anguli|o|s/, tractari etiam possunt hæ quantitates in Geometria.

Quantitates continuo fluxu crescentes vocamus fluentes & velocitates crescendi vocamus fluxiones, & incrementa momentanea vocamus momenta, et methodum qua tractamus ejusmodi quantitates vocamus methodum fluxionum et momentorum: est hæc methodus vel synthetica vel analytica.

Methodus synthetica fluxionum et momentorum in Tractatu sequente passim occurrit, et ejus elementa posui {illeg}|i|n Lemmatibus undecim primis Libri primi & Lemmate secundo Libri secundi.

Methodus analyticæ \specimina/ occurrit|un|t in Prop XLV & Schol Prop. XCII Lib. I & Prop. X{illeg} & XIV Lib. II. \et præterea describitur in Scholio ad Lem. II & Lib. II./ Sed et ex Demonstrationibus compositis Analysis qua Propositiones inventæ fuerunt, addisci potest regrediendo. [Et præterea \describitur in Scholio ad Lem. II Lib: II./ [Tractatum de hac Analysi ex chartis antea editis desumptam, Libro Principiorum subjunxi.]

Scopus Libri Principiorum non fuit ut methodos mathematicas edocerem, non ut difficilia omnia ad magnitudines figuras motus & vires spectantia tractarem eruerem; sed ut ea tantum tractarem quæ ad Philosophiam naturalem et apprime ad motus cœlorum spectarent: ideoque quæ ad hunc finem parum conducerent, vel penitus omisi, vel leviter tantum attigi, omissis demonstrationibus.

In Libris duobus primis vires generaliter tractavi, eas si in centrum aliquod seu immotum seu mobile tendunt, centripetas vocavi, (nomine generali) vocavi, non inquirendo in causas vel species virium, sed earum quantitates determinationes & effectus tantum considerando. In Libro tertio quamprimum didici Lunam in vires — quibus Planetæ in orbibus suis retinentur, recede|a|ndo a Planetis in quorum centra vires illæ tendunt, decrescere in duplicata ratione g{illeg} distantiarum a centris, & vim qua Luna retinetur in Orbe suo circum Terram, descendendo ad superficiem Terræ æqualem evadere vi gravitatis nostræ, cœpi gravitatem tractare ut vim quæ corpora cœlestia adeo vel gravitatem esse vel vim gravitatis duplicare: cœpi gravitatem tractare ut vim qua corpora cœlestia in orbibus suis retine\a/ntur. Et in eo versatur Liber iste tertius \tertius/, ut Gravitatis proprietates, vires, directiones & effectus edoceat.

Planetas in orbibus fere concentricis & Cometas in orbibus valde excentricis circum Solem revolvi, Chaldæi olim crediderunt, Et hanc philosophiam Phythagorei in Græciam introduxerunt invexerunt. Sed et Lunam gravem esse in Terram, \& stellas graves esse in se mutuo,/ et corpora omnia in vacuo æquali cum velocitate in Terram descend cadere, adeo gravia esse pro quantitate materiæ in singulis notum fuit Veteribus. Defectu demonstrationum hæc philosophia intermissa fuit et|a|ndem non inveni sed vi demonstrationum in lucem tantum revocare conatus sum. Sed et Præcessionem Æquinoxiorum, & fluxum & refluxum maris et motus inæquales Lunæ {illeg} & {illeg} et orbes Cometarum & perturbationem orbis Saturni per gravitatem ejus in Iovem ad ijsdem Principijs consequi, et quæ ab his Principijs consequuntur Cum Phænomenis probe congruere, hic ostensum est. Causam gravitatis ex phænomenis nondum didici.

Qui leges et effectus Virium electricarum pari successu et certitudine eruerit, philosophiam multum promovebit, etsi \forte/ causam harum Virium ignoraverit. Nam Phænomena primo consideranda \spectanda |observanda| sunt/ sunt, {sic} dein horum causæ proximæ, & postea causæ causarum \eruendæ eruendæ; ac/ {illeg} tandem a causis \supremis causarum/ per phænomena stabilitis, ad caus phænomena & \causus p eo|a|rum effectus,/ \eorum causas proximas,/ argumentando a priori, descendere licebit. Et inter Phænomena numerandæ sunt actiones mentis quæ nobis innotescunt quarum conscij sumus Philosophia naturalis non in opinionibus Metaphysicis, sed in Principijs proprijs fundanda est; & hæc

Consessus Arbitrorum a Regia Sociètate constitutus Commercij subsequentis Epistolici exemplasia tantum pauca Anno 1712 imprimi curavit et ad mathematicas mitti qui soli de his rebus judicare possent. Cum vero D. Leibnitius huic Libro minime responderet sed Quæstionem des\er/eret & ad Quæstiones Methaphysicas ilias ad hanc rem nihil spectantes {descæ}\sophistice/ confugoret rixando, et ejus amici quidam adhuc rixentur, visum est hunc librum una cum ejus Recensione quæ in Tansactionibus

Et cum

In scribendis Philosophiæ Principijs Mathematicis Newtonus hoc Libro \hocce de quadraturis/ pluronum est usus est, ideo eundem Librio Principiorum subjungi voluit. Scripsit Investigavit uti Propositiones in Libro Principiorum per Analysin, investigatas demonstravit per per Synthesin pro lege Veterum qui nihil \Propositiones suas non/ prius in Geometriam admittebant quam demonstratæ essent synthetics. Hæ Analysis hodierna nihil aliud est quam Arithetica in specibus. Hæc Arithmetica ad res Geometricas applicare potest, et Propositiones sic inventæ sunt Arithmetice inventæ. Demonstrari debent Synthetice more veterum et tum demun pro Geometricis haberi.

Vox D. Fatio de Duillier incidit in \hanc/ Methodum differentialem anno 1687 ed visis postea Newtoni MSS antiquis ille \{is}/ anno 1699 Testimonium in Tractatu de solido minimæ resistentiæ testimonium pro Newtono exhibuit his verbis: Newtonum tamen primum ac pluribus annis vetustissimum hujus calculi inventorum ipse verum evidentia coactus agnosco. A quo utrum quicquam mutuatus sit Leibnitius ejus secundus \ejus/ inventor, malo eorum quam meum sit judicium quibus visæ fuerent Newtoni Litteræ alij ejusdem manuscripti Codices.

At quamvis Liber Principiorum sunt su synthetice scriptus sit, tamen Analysis per Synthesin elucet, et plerun erui potest resolvendo Propositiones synthetice demonstratas regrediendo a Synthesi ad Analysin \ad Analysin/ & quærendo Analysin aqua Synthesis derivate fuit. Hac retione Marchi *Hospitalius* didiu intellexit librum Principiorum *fere toc|t|um \esse/ de hocce* Analysi [momentorum] *calculo.* L \Et/ Leibnitius in Epistola ad Newtonum 7 Mar. 1693 data scipsit:*Mirifice ampliaveras Geometriam tuis seriebus sed edito Principiorum opere ostendisti patere Tibi quæ Analysi recepte non subsunt. Conatus sum Ego quo Notis commodis adhibitis quæ differentias & summas exhibent Geometriam illam quam Transcendentem appello, Analysi quodammodo subjicere nec res male processit.*

## Ad Lectorem.

Quam primum Wallisius noster Methodum differentialem in Hollandia celebriri audivit, is in Præfatione ad Volum{e}na secundum \priman|m|dum prima/ operum suorum \anno 1695 editum/ inseruit hanc admonitionem inseruit Quæ in secundo Volumine habentur — dictum esse. Hæc Wallisius. Cum vero \Et quamvis/ Leibnitius et Menkenius hæc cognoserent, et nemo \tamen/ per ea tempora contradi{illeg}|xit|{illeg}, credidi {m} tuto ho \At cum/ in Itroductione {sic} ad Librum sequentem anno 16{illeg} 1704 (postquam annis 28 in MS latuisset) edebam idem tacto dicere posse me incidisse paulatim in methodum Fluxionum qua ill in hoc Libro \illic/ usus sum in Quadratura Curvarum edebam, di{xi} dixissem \dicerem/ me incidisse paulatim in methodum Fluxionum annis 1665 et 1666 in Methodum Fluxionū qua illic usus essem in Quadratura Curvarum:

## Annotationes. Vel Schol. in Prop. V.

✝ NB Pag. 41 Hoc, Wallisius \vir/ prius affirmavit et in Præfatione ad Volumen primum Operum suorum. Et sic idem sic ostendit|u|p|r|otest Wallisius noster Newtonus literis ad Wallisium datis Aug 27 & Sep. 17 1692 mittebat Propositionem|s| prima duas hujus Libri, primam et quintam una cum \Propositione/ Extractere|hendi| fluxionum ex Æquationibus involventibus radicis|em| ex Æquatione fluxionem radicis involvente; et Wallisius hæc edidit anno 1693 in secundo Volumine operum suorum, Et scripsit \ibi dixit notavit/ has \tres/ Propositiones, in Epistola Newtoni ad Oldenbergium 24 Octob. 17|6| 1676 data, describi. In eadem Epistola Propositio Prima \hujus Libri/ dicitur esse fundamentum Methodi ge\ne/ralis de qua \Newtonus anno 1671/ tractatum scripseram|t| anno 1671. Proposito {sic} secunda extat in Analysi per Series Æqua Series sub finem et pendet a Propositione prima ideo Propositiones duæ Primæ \hujus Libri/ Newtono innotuere anno 167|6|9 ubi \quo uti/ Barrovius hanc Analysin ad Collinium misit. Sed et Propositio quinta Newton eodem tempore \anno/ Newtono innotuit. Nam in Analysi illa dicitur, quod |i|ej|ll|u|iu|s beneficio curvarum areæ et longitudines &c (id modo fiat) exacte et Geometrice fieri determinantur. Et hoc fit per Propositionem illam quintam. Propositio autem secunda et tertia \et quarta/ sunt tantum exempla Propositionis secundæ \ut in hoc libro dicatur/. Ideo Analysis per series Methodus Fluxionum quatenus habetur in Propositionibus quin primis Libri de Quadraturis Newtono innotuit anno 1669. Porro \Deni/ Collinius in Litteris Epistola ad Tho Strode 26 Iulij 1672 data, scripstit {sic} *quod ex Analysi per* |quod *mense Septembri anni 1668 Mercator Logarithmotechniam suam edidit* {illeg} quod ex Analysi per| *series et chartis alijs quæ olim a Newtono cum Barrovio communicatæ fuerant pateret illam methodum a dicto Newtono aliquot annis antea excogitatam \[et antequam Logarithmotechnia/ & modo universali applicatam fuisse: ita ut ejus ope in quavis figura Curvilinea proposita quæ una vel pluribus proprietatibus definitur, Quadratura vel Area dictæ Figuræ accurata si possibile sit, sin minus infinite vero propinqua — obtineri queat.* Hoc fit per Propositionem illam quintam. Ideo methodus fluxionum quatenus exhbitur habetur in Proposionibus quin primis Libri de Quadraturis Newtono innotuit anno (testibus Barrovio et Collinio) annis aliquot antequam prodiret Mercatoris Logarithmetechnia \testibus Barrovio et Collinio,/, {sic} id est anno 1666 aut antea, ut etiam Wallisius affirmavit.

Porro Corollarium secundum Propositionis decimæ habetur in Epistola Newtoni ad Collinium \Novem. 8 1676. data et/ a Ionesio in Analysi per Quantitatum Series, Fluxiones ac Di{f}ferentias edita. Et Tabularum in Scholio ad Prop. X, pars difficilior Et Tabula \ultima/ Ordinatarum |ad| Curvar|s|um in Scholio ad Prop. X, \posita/ recitatur in Epistola Newtoni \ad/ Oldenburgum Octob 24 1676. Et inde, colligitur Propositionem illam deciman Newtono innotuisse anno 1676 aut antea. Et Ordinatæ Curvarum quæ in Tabula ultima in Scholio ad Prop. X pen habentur, recitantur \eodem ordine & ijsdem literis/ in Epistola Newtoni ad Oldenburgium Octob. 24 1676. \☉/ ☉ et ibi dicitur figuras illas Curvilineas quarum Ordinatæ ibi ponuntur recitantur Cum sectionibus comparari posse per Theoremata quæ in Tabulam illam \dicitur/ tunc OLIM conditam fuisse (id est anno 1671 aut antea) conditam fuisse. ☉ et ibi Tabula illa Catalogus Theorematum dicitur pro comparatione Curvarum cum Conicis Sectionibus dudum conditus id est diu ante annum 1676, et propterea anno 1671 aut antea. Inde vero colligitur Propositionem illam deciman Newtono innotuisse anno 1671. Extractus uti fuit hic liber circa annum 1676 ex libro antiquiore quem Newtonus scripsit anno 1671. & Inde \vero/ colligitur Propositionem illam decimam Newtoni|o| innotuisse anno 1678|1| Extractus uti fuit hic Liber \circa/ anno|u|m 1676 ex Libro antiquiore quem Newton{illeg}|u|s scripsit anno 1671.

Beneficio hujus methodi contuli \didici/ anno 1664 vim centr vires quibus Planetæ primarij retinentur in orbibus circa Solem esse in ratione duplicata distantiarum \mediocrium/ a sole inverse et vim qua Luna retinetur in Orbe cirum Terram esse in eas|d|em \fere/ ratione ad gravitatē in superficie Terræ. Deinde anno 1679 ad finem vergente inveni demonstrationem Hypotheseos Kepleri quod Planetæ primarij revolvuntur in Ellipsibus Solem in foco inferiore habentibus, & radijs ad Solem ductis areas describunt temporibus proportionales Tandem anno 1684|5| et parte anni 1686 scripsi beneficio hujus methodi \& subsidio libri de Quadraturis/ scripsi libros duos primos me Principl|i|orum mathematicorum Philosophiæ. Et propterea Librum de Quadraturis subjunxi Libro Principiorum.

Anno 1666 incidi in Theoriam colorum, et anno 1671 parabam Tractatum de hac re, alium de se methodo serierum & fluxionum ut in lucem ederentur. Sed subortæ mox disputationes aliquæ me a consilio deterruerunt usad annum 1704.

|Interea| D. Leibnitius in methodum momentorum incidit anno 1677.

Fatum et necessitas \et vim necessitas/ non |e|st|n||t| causa sufficiens nis{illeg}|i| per productionem Agentis Entis \ubi omnipræsentis/ intelligentes, et libere agentis per voluntatem \volentis/ & actiones suas eligentis. Hujusmodi ens est Natura illa sepientissima \quæ nihil facit frustra &/ quam|| omnes prœdicant ex phænomenis prædicant, [nemo nisi per phænomena cognoscit]\&/ quæ rectius fons naturæ quam natura \ipsa/ dici deberet.

cum ipse prius in Actis Eruditorum pro mense anni 16712 pro Leibnitio contra Newtono scripsisset

Et ab eo tempore Newtonum aggressus est propositis novis disputationibus

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