Newton's statement of the case in dispute between Leibniz and himself

M^r Newton published his Treatise of Quadratures in the year 1704. This Treatise was written long before, many things being cited out of it in his Letter of 24 Octob. 1676. T|I|t relates to the method of fluxions & that it might not be taken for a new piece M^r Newton repeated what D^r Wallis had published nine years before without being then contradicted, namely that this method was invented by degrees in the years 1665 & 1666. Hereupon the Editors of the Acta Lipsiensiā in Ianuary 1705, in the style of M^r Leibnitz, repr{e} in giving an Account of this book represented that M^r Leibnitz was the first inventor of the method & that M^r Newton had substituted fluxions for differences. And M^r Keil in an Epistole published in the Philosophical Transactions for May & Iune 1708 retorted the accusation, saying: Fluxionum Arithmeticam sine omni dubio primus invenit D. Newtonus, ut cuilibet ejus Epistolas a Wallisio editas legenti facile constabit. Eadem tamen Arithmetia {sic} postea mutatis nomine & notationis modo a Domino Leibnitsio in Actis Eruditorum edita est. And this was the beginning of the present controversy.

M^r Leibnitz understanding this in a stronger sense then M^r Leibnitz Keill intended it, gave his reasons against it in a Letter to D^r Sloan dated 4 May|r|ch st. n. 1711, & d moved that the R. Society would cause M^r Keill to make a publick recantation. M^r Keill chose rather to explain & defend what he had written. And M^r Leibnitz in a second Letter to D^r Sloan dated 29 Decem. 1711 instead of making \good/ his accusation as he was bound to do that it might not be deemed a calumny, insisted only upon his own candor as if it would be injustice to question it, & said that the Acta Lipsiensia had given every man his due, & called M^r Keill a Novice unacquainted with things past & one that acted without authority from M^r Newton & a clamorous man, & desired that M^r Newton himself would give his opinion in the matter. He knew that M^r Newton had already given his opinion in the Introduction to the book of Quadratures but M^r Newton must retract that opinion & allow that M^r Leibnitz was the first he had substituted fluxions for differences or not be quiet.

The Royal Society therefore having as much authoritiy over M^r Leibnitz as over M^r Keil & being now twice pressed by M^r Leibnitz to interpose & seing no reason to condemn or censure M^r Keil without inquiring into the matter; & that neither M^r Newton nor M^r Leibnitz (the only persons alive who remembred knew & remembred any thing of what had passed in these matters 40 years ago) could be witnesses for or against M^r Keill, appointed a \numerous/ Committee to search old Letters & Papers & report their opinion upon what they found, & ordered the Letters & Papers with the opinion \Report/ of their Committee to be published. And by those Letters & Papers it appeared to the Committee that M^r Newton had the Method \& that the methods of/ in or before the year 1669, & it did not appear to them that M^r Leibnitz had it \the method of differences/ before the year 1677 |& that the methods of|

[M^r Leibnitz began his second Letter to D^r Sloan with these words. Quæ D. Iohannes Keillius nuper ad te scripsit candorem meum apertius quam ante oppugnant: quem ut ego hac ætate post tot documento vitæ, Apologia defendam, & cum homine docto sed novo, & parum perito rerum anteactarum cognitare, nec mandatum habente ab eo cujus interest, tanquam <255v> pro tribunali litigem, nemo prudens æquus probabit. Thus he declined to make good his accusation against M^r Keill which he should have pursued for avoyding the censure of calumny{illeg} He & told the Society that they would be unjust if they should question suffer his candor to be questioned; & \{illeg}/ refused to contend with any body but M^r Newton or those imployed by him, justifying in the same Letter the Acta Lipsiensia against him & pressing him to declare his mind, that is, to submit to the judgment of the Editors of the Acta|.| Lipsiensia & retract what he had published in the Introduction to his Principles. But those Editors \are not competent Iudges. They/ have sometimes imployed the pen of M^r Leibnitz himself in mathematical affairs & no man \how candid soever/ can be either Iudge or witness in his own cause. M^r Leibnitz is a party & it lies upon him to prove his assertions. before the R. Society can censure M^r Keill It lies upon him to prove that he had the differential method a multo tempore before the year 1677|.| {tho} he was of opinion the year before that inverse problemes were not reducible to equations. It lies upon him to prove that he had the series of Gregory before he received it from Greg M^r Oldenburg A.C. 1675, at w^ch time he did not know it to be his own, & even before Gregory sent it to M^r Collins, A.C. 1671. He affirmed (in his Letters of 28 Decem 1675 & 27 Aug 1676) that this Series was communicated by him to his friends at Paris A.C. 1673, & was the series whereof he had written to M^r Oldenburg sometimes before December 1675, viz in Iuly & October 1674: it lies upon him to prove that the series which he then wrote of to M^r Oldenburg was this series & not the series for finding the Arc by the sine. In his said Letter of October 1674 he affirmed that he had a method of finding the Arc by the sine in a series of rational numbers, whether the proportion of the Arc to the whole circumference was known or not, & that the same method gave him the series for the whole circumference; & yet in his Letter of 12 May 1676 he desired M^r Oldenburg to procure from M^r Collins the method of finding the series: it lies upon him to prove that he had the method in or before the year 1674. At his request M^r Newton in his Letter of Iune 13, 1676 sent him his methods of finding that & such like series direct & inverse & illustrated that methods with examples of Series; & upon the receipt of that Letter M^r Leibnitz wrote back that he had found some of those series w^ch brought him before the receipt of that letter w^ch brought him the desired methods of finding them. He understood M^r Newtons inverse methods of Series with difficulty & so soon as he understood them he wrote back {illeg} that he had found one of them before & forgot it as he perceived by his old papers: it lies upon him to prove that he had found it & forgot it before, he wrote for it that is, before 12 May 1676. \$\overline{)+}$/ He pretends to be the first inventor of that method of series w^ch consists in assuming the terms of a series & determining them gradually by the conditions of the Probleme: it lyes upon him to prove that he had this method before M^r Newton sent it to him in cyphers in his Letter of Octob. 24 1676. A year or tow after M^r Newton had published his Principia Philosophiæ, M^r Leibnitz published these papers relating <256r> to the principal Propositions in that book, pretending that he had found them before that book was published: it lies upon him to prove that he had them before. He pretended that the Propositions concerning the motions of bodies in resisting Mediums were for the most part found out by him twelve years before while he was yet at Paris, that is, before he had the differential method: it lies upon him to prove it. It lies upon him also to prove In the 19^th Article of his Tentamen de motuum cœlestium causis it lies upon he has pretended to the invention of M^r Newtons Proposition that a Body w^ch revolves in an Ellipsis & with a Radius drawn to the lower f{oc} focus describes equal areas in equal times, is attracted towards that focus by a force which is reciprocally as the square of that Radius. The demon His demonstration is erroneous, And it lies upon him to make us understand how he could find out that Proposition by an erroneous demonstration, & then to prove that he did find it out in such a manner before the wr M^r Newtons Principia came abroad. By a Proposition for transmuting figures into one another he had|s| laid claim to the method of converging series as a coinventor. It lies upon him to prove that the method by this Proposition is either general or usefull or that any thing can be done by it w^ch cannot be better done without it or that it was invented before the year 1675, or that the Proposition is any part of the method of Series or any thing more then a Lemma in some particular cases, or that any one series of moment has been invented by it w^ch was not invented before without it. His differential method of Tangents published in the Acta Lipsiensia of November 167|8|4 is nothing else then D^r Barrows method of Tangents published improved so as not to stick at fractions & surds & disguised with a new Notation & a new name without acknowledging any part of the method to be D^r Barrows or that the Author had received any light into it from M^r Newton's Letters. It lies upon him prove that what he then published was intirely his own, without being obliged to either of them for any thing. All the pretenses above mentioned tend to diminish the right of invention due to M^r Barrow M^r Newton & M^r Gregory & to let in M^r Leibnitz for a share. M^r Leibnitz has hitherto claimed this share without any proof. His candor cannot make him a witness for himself. By the laws of all nations he must either prove his claim or quit it; & this he ought to do without delay in point of candor & justice.]

For making himself the first inventor of the method of flu{e} \differential method/ & deriving M^r Newtons method from his own he has further represented that M^r Newton at first used the letter o in the vulgar manner for the given increment of x, w^ch destroys the advantages of the differential method; but after the writing of his Principles, changed o into $\stackrel{.}{\mathrm{x}}$, substituting $\stackrel{.}{\mathrm{x}}$ for $d\mathrm{x}$. It lies upon him to prove that M^r Newton ever changed o into $\stackrel{.}{\mathrm{x}}$ or used $\stackrel{.}{\mathrm{x}}$ for $d\mathrm{x}$, or left off the use of the letter o. M^r Newton used the letter o in his Analysis written in or before y^e year 1669, & \also/ in his book of Quadratures, & in his Principia Philosophiæ, & still uses it in the very same sense as at first. In his book of Quadratures he used it in conjunction with the symbol $\stackrel{.}{\mathrm{x}}$, & therefore did not use that symbol in its room. These symbols o & $\stackrel{.}{\mathrm{x}}$ are put for things of a different kind. The one is a moment, the other a fluxion or velocity as has been explained above. When the letter x is put for a quantity w^ch flows uniformly the symbol $\stackrel{.}{\mathrm{x}}$ is an <256v> unit & the letter o a moment. Prickt letters never signify moments unless when they are multiplied by the moment o either exprest or understood to make them infinitely little, & then y^e rectangles are put for moments. {sic} |& $\stackrel{.}{\mathrm{x}}\mathrm{o}$ & $d\mathrm{x}$ signify the same moment.|

M^r Newton doth not place his method in forms of symbols nor confine himself to any particuli|a|r sort of symbols for fluents or fluxions. When he puts the Areas of Curves for fluents he \usually/ puts the Ordinates for fluxions & denotes the fluxions by the symbools of the Ordinates, as in the Analysis. Where he puts lines for fluents he puts any symbols for the velocities of the points w^ch describe the lines, that is, for the first fluxions, & any other symbols for the increase of those velocities, that is, for the second fluxions, as is frequently done in his Principia Philosophiæ. And where he puts the letters x, y, z for fluxion|ent|s, he denotes their fluxions either by other letters as p, q, r or AB, CD, EF, or by the same letters in other forms as X, Y, Z, or $\stackrel{.}{\mathrm{x}}$, $\stackrel{.}{\mathrm{y}}$, $\stackrel{.}{\mathrm{z}}$. And this |is| evident by his book of Quadratures where he uses for \represents/ fluxions \by/ prickt letters in the first Proposition, \by/ the Ordinates of Curves in the last Proposition & \by/ other symbols in explaining the method & illustrating it with examples in the Introduction. M^r Leibnitz has no symbols of fluxions in his method & therefore M^r Newtons symbols of fluxions are the oldest in the kind. M^r Leibnitz began to use the symbols of moments or differences $d\mathrm{x}$, $d\mathrm{y}$, $d\mathrm{z}$ in the year 1677: M^r Newton has represented moments by the rectangles under the fluxions & the moment o ever since the writing of his Analysis which was at least 45 years ago. M^r Leibnitz has used the symbols $\int \mathrm{x}$, $\int \mathrm{y}$, $\int \mathrm{z}$ for the summ of Ordinates ever since the year 1686: M^r Newton represented the same thing \in his Analysis 45 years ago/ by inscribing the Ordinate in a square or rectangle. in his Analysis \many years before/. All M^r Newtons symbols are the oldest in their several kinds by many years.

And whereas M^r Leibnitz has represented that the use of the letter o is vulgar & destroys the advantages of the Differential Method: on the contrary the method of fluxions as used by M^r Newton has the advantage of the differential in all respects. It is more elegant because in his calculus there is but one infinity little quantity represented by \a symbol/ the letter \symbol/ o. It is more natural & geometrical because founded upon the primæ quantitatum nascentium rationis w^ch have a being in Geometry whilst indivisibles upon which the Differential method is founded have no being either in Geometry or in nature. There are rationes primæ quantitatum nascentium; but not quantitatis primæ nascentes. Nature generales \quantities/ by continual flux or increase; & the ancient Geometers admitted such a generation of areas & solids when they drew one line into another by local motion to generate an area & the area into a line \by local motion/ to generate a solid. But the summing up of indivisibles to generate \compose/ an area or solid was never yet admitted into Geometry. M^r Newtons method is also of a greater extent \use & certainty/ being adapted either to the ready finding out a Proposition or to the Demonstrating it: M^r Leibnitz's is only for finding it out. When the work succeeds not in finite equations M^r Newton has recourse to converging series {illeg} & thereby his method becomes incomparably more universal then that of M^r Leibnitz w^ch is confined to finite equations \For he has a share in the method of infinite series/. And when the law of the fluxions is not known but the fluxions are had only in a few particular cases, M^r Newton finds that law quamproximè by drawing a curve line through and number of given points & thence deduces the solution of the Probleme. And to this degree of perfection M^r Newton had brought the|hi|s method sometime before the year 1676 as appears by his Analysis & his Letters of 10 Decem. 1672, 13 Iune 1676 & 24 Octob 1676.] By the help of this Analysis fou{illeg}nd out \M^r Newton found out most of/ the Propositions in his Principia Philosophiæ. But because the Ancients admitted nothing into Geometry before it was demonstrated synthetically, M^r Newton |he| demonstrated the Propositions synthetically that the systeme of the heavens might be founded upon good Geometry. And this makes it \now/ difficult for unskillful men to see the Analysis by w^ch those Propositions were found out.