Historical Annotations on the Elogium of Leibniz

Part of Hist Annotations in Elog of Leibnitz a{illeg} 1717?

Pag 44. l. 1. What M^r Newton calls fluxions \& ma{illeg}|r|ks sometimes with prickt letters & sometimes w^th other \marks// M^r Leibnitz doth not call fluxions \differences/, nor hath any name or mark for them. What M^r Newton calls moments & marks sometimes with prickt letters sometimes with {sic} \& particles/ other marks, M^r Leibnitz calls different|c|es & marks by prefixing the letter d M^r Leibnitz calls differences; And|But| \&/ fluxions are {motions} of increase quantities of another kind. They are motions \motions or velocities/ of increasi|{e}|ng They are no & not small of quantities & not parts of the increasing quantities.

Ib. l. 3, 4, 5. When the Marquess de l'Hospital represented that the Characters of M^r Leibnitz were \more/ convenient then those of M^r Newton, he had seen {illeg} only the marks used by M^r Newton in the second Lemma of the second Book of his Principles \& therefore meant those Characters & not prickt letters/. Of Of his marks by prickt letters &c he knew nothi The marks by prickt letters & the letter o are the most useful & the best adapted to true reasoning. But They that have been used to the characters of M^r Leibnitz will be apt to think them the best.

Ib. lin. penult. Before this, D^r Wallis in the Preface to the two first Volumes of his Works published A.C. 1695 wrote that M^r Newton in his two Letters of 1676 explained to M^r Leibnitz the Method found found by him ten years before or above, that is in the year 1665|6| or 1665 \before/. And an anonymous {illeg} the Editors of the Acta Eruditorum in giving an account of those two Books replied \falsly/ that M^r Newton had both publickly & privately allowed the invention of the Method to M^r Leibnitz. And even before this when M^r Leibnitz first published the elements of the differential method & made no mention of his corresp any thing he had re correspondence he had w^th M^r Newton six years before by means of M^r Oldenbeg: M^r Newton added the wrote the Scholium to \described the Elements of the Method in/ y^e second Lemma of the second book of his Principles not to give away the e & wrote the Scholium to that Lemma, not to give away the Lemma as M^r Leibnits has pretended, but to assert it to himself.

P. 48. {illeg} lin. 15. M^r Newton was first accused of Plagiary in the Acta Eruditorum for Ianuary 16 1705 pag 34 & 35 & the accusation was justified by M^r Leibnitz himself in his Letter of 29 Decem. 1711 to D^r Sloan |And when y^e Committee of the R S| |And for obviating this accusation the Committe insinuated that there was more colour to suspect M^r Leibnit{z} nec lex est {illeg}|j|ustior ulla.|

P. 49. l. 20. M^r Leibnitz has affirmed that he found the differentiall method in the year 1676, & therefore could not undertake to prove that he by a new Commercium that he found it before. And if he did not find it before, he was not the first inventor. And second Inventors have no right. The first Inventor has the sole right till another Inventor arises: & then to take away the right of w^ch he was once justly possessed & share it with others would be injustice. All

p. 47. lin. 6|5|, 6 A copy of This Letter was writ to M^r Collins in 1672 & a copy of it sent to M^r Leibnitz in 1676 Iune 1676 pag 34 & 35

p. 51. lin. 21. M^r Leibnitz went from London to Paris in \{illeg}/ March 1673 & continued to write Letters to M^r Oldenburg till Iune 8^th fo about Arithmetical Matters till Iune 8^th following. Then he intermitted his correspondence for a year & in the mean time applied himself to {illeg} learn the higher Geometry by means of M^r Huygens whose Horologium oscillatorium came abroad that year in April. And by \reading/ this first this & the{illeg}|n| some \the/ other books I |here| mentioned, he got light into the method w^ch gave him the Theoremes here mentioned like those of Barrow & Gregory. And now he had his Method by the Differentiæ generatrices in Arithmetic, & that by the Triangulum characteristicum in Geometry. But he had not yet the Differential method in Algebra, for he tells us in one <372v> of his Letters that he found that in the year 1676.

Pag. 53 l. 6. He did not see M^r Newtons papers before the year 1676. Vpon the death of M^r {illeg} Collins Gregory he desired \that/ a collection of Gregory correspondence might be made & sent to Paris. It was sent Iune 26 1676 & therin was, 1 M^r Gregories Letter to M^r Collins dated 15 Feb. 1672|1| conteining Gregories Series for squaring the circle, published afterwards by M^r Leibnitz without ackcnowledge|i|ng what he had received from London. 2 A letter of M^r Gregories Letter of 5 Sept 1670, in w^ch Gregory mentions that from by the help of D^r Barrows method of Tangents he had found a general method of Tangents without calculus, viz^t a method like that of Slusius. 3 M^r Newtons Letter of 10 Decem 1672 in w^ch M^r Newton described his general method of Analysis & said that the method of Grego Tangents of Gregory & Slusius was but a Corollary or branch thereof. And at the same time M^r Newton|s| Letter of 13 Iune 1676 was also sent in the end of w^ch he writes that Analysis by the method of series & anot{illeg} some other method|s| becomes very {illeg} (meaning the methods of fluxions & arbitrary series, becomes very universal. And in October \following M^r Leibnitz coming to London saw/ {illeg} M^r Newtons Letter of 24 October {illeg} wherein {illeg} M^r Newton further describes the Method of fluxions, & mentions also his \m {illeg} his/ Compendium of the Method of series communicated by D^r Barrow to M^r Collins presently in the ye upon the publishing of Mercators Logarithmotechnia, A.C. 1669, that is the Analysis per series numero terminorum infinitas. And having {illeg} the year before written to M^r Oldenberg to procure M^r Newtons method of series & from M^r Collins & now meeting with this further notice of it: he could scarce miss of seeing it. For he {illeg} app went to ap consulted M^r Collins {illeg}|a|t that very time to see what he could meet with of the correspondence of M^r Gregory & M^r Newton \with Collins/ concercerning {sic} these series. And if he there saw this Analysis he had a sufficient \plane/ description of the method of fluxions at the end of it.

P. 53. l. 22. M^r Newton inserted the \inserted having occasion to use the/ Elements of the method \of mo \of Moments// into y^e second Lemma of this|e| second Book of his Principles, & added a Scholium to the Lemma justify his doing so. And last And this he did without under \in a/ manner without g{illeg} or offend And when a Letter of M^r Leibnitz written neare then end of t in Autumn 16|7|15 had been handed about some time among persons of note in London & M^r Newton was pressed to answer it, he returned an answer dated Feb 26 1616 & M^r Leibnitz sent these Letters with his Answer to

P. 48. l. 6 A Letter writ by M^r Leibnitz in Autumn 16 1715 being handed about some time among persons of note in London, M^r Newton was at length pressed \prevailed with/ to answer it, which w^ch he did by an Letter dated Feb 26 16|7|16. M^r Leibnitz wrote answered it \that Letter/ by another dated 9 Apr. 1716. But [considering that he \because he/ began this Letter \Answer/ with a pretence that |M|I|^r| |N| had challenged him & that he would |now| enter the lists & give him satis] \&/ instead of sending this Letter to|d|irectly to London he sent it with the two former \Letters/ to Paris, \And {illeg}/ |✝| < insertion from the bottom of the page > ✝ The third came from thence to London from whence \&/ it came \thence/ to to {sic} London: M^r Newton returned no answer, but only shewed privately to some of his friends < text from f 372v resumes > M^r Newton \returned {illeg}|n|o/ answered him no further \to this/, but only drew up shewed some of his friends in private what {illeg} answer might have been answered: & afte upon the death of M^r Leibnitz, becaus the three first Letters had been dispersed, published them, with what the answer what was drwn {sic} up, but not sent.

P. 53 l 22 M^r Newton published the elements of the method in the 2^d Schol of the 2^d bo

1 What M^r Leibnitz calls Differences M^r Newton calls not Flu \not fluxions but/ Moments & \small/ particles. Fluxions are quantities of another kind & M^r Leibnitz hath no marks nor name for them. They are motions or velocities of increase & not parti|s|cles of the increasing quantity.

2 to think them best: but that M^r Newtons are \more universal &/ better adapted to just reasoning may appear by this instance. Let x, y, z be fluents $\stackrel{.}{\mathrm{x}}$, $\stackrel{.}{\mathrm{y}}$, $\stackrel{.}{\mathrm{z}}$ their fluxions & $\stackrel{.}{\mathrm{x}}\times \mathrm{o}$, $\stackrel{.}{\mathrm{y}}\times \mathrm{o}$ $\stackrel{.}{\mathrm{z}}\times \mathrm{o}$ their moments: & {let} any equation \expressing the relation of the fluents,/ as $\mathrm{b}\mathrm{b}\mathrm{x}-\mathrm{c}\mathrm{x}\mathrm{y}+\mathrm{z}\mathrm{z}=0$ being given let it be required to find the relation of their fluxions. Put {illeg} find in this equal Let x flow uniformly & it fluxion |$\stackrel{.}{\mathrm{x}}$| be 1 & its moment will be o & after one moment of x the fluents will become $\mathrm{x}+\mathrm{o}$, $\mathrm{y}+\stackrel{.}{\mathrm{y}}\mathrm{o}$, $\mathrm{z}+\stackrel{.}{\mathrm{z}}\mathrm{o}$, \w^ch being substituted for the former flu{illeg}/ & the equation will become $\mathrm{b}\mathrm{b}\mathrm{x}+\mathrm{b}\mathrm{b}\mathrm{o}-\mathrm{c}\mathrm{x}\mathrm{y}-\mathrm{c}\mathrm{o}\mathrm{y}-\mathrm{c}\mathrm{x}\stackrel{.}{\mathrm{y}}\mathrm{o}-\mathrm{c}\stackrel{.}{\mathrm{y}}\mathrm{o}\mathrm{o}+\mathrm{z}\mathrm{z}+2\mathrm{z}\stackrel{.}{\mathrm{z}}\mathrm{o}+\stackrel{.}{\mathrm{z}}\stackrel{.}{\mathrm{z}}\mathrm{o}\mathrm{o}=0$. And striking out $\mathrm{b}\mathrm{b}\mathrm{x}-\mathrm{c}\mathrm{x}\mathrm{y}+\mathrm{z}\mathrm{z}=0$, there \will/ remains $\mathrm{b}\mathrm{b}\mathrm{o}-\mathrm{c}\mathrm{o}\mathrm{y}-\mathrm{c}\mathrm{x}\stackrel{.}{\mathrm{y}}\mathrm{o}-\mathrm{c}\stackrel{.}{\mathrm{y}}\mathrm{o}\mathrm{o}+2\mathrm{z}\stackrel{.}{\mathrm{z}}\mathrm{o}+\stackrel{.}{\mathrm{z}}\stackrel{.}{\mathrm{z}}\mathrm{o}\mathrm{o}=0$. d|D|ivide the whole by o & the Quotient will be $\mathrm{b}\mathrm{b}-\mathrm{c}\mathrm{y}-\mathrm{c}\mathrm{x}\stackrel{.}{\mathrm{y}}-2\mathrm{c}\stackrel{.}{\mathrm{y}}\mathrm{o}+2\mathrm{z}\stackrel{.}{\mathrm{z}}+\stackrel{.}{\mathrm{z}}\stackrel{.}{\mathrm{z}}\mathrm{o}=0$. Let the moment o become infinitely little & the æqua terms $\mathrm{-2}\mathrm{c}\stackrel{.}{\mathrm{y}}\mathrm{o}+\stackrel{.}{\mathrm{z}}\stackrel{.}{\mathrm{z}}\mathrm{o}$ will become infi vanish, & the remaining equation will give the {illeg} $\mathrm{b}\mathrm{b}-\mathrm{c}\mathrm{y}-\mathrm{c}\mathrm{x}\stackrel{.}{\mathrm{y}}+2\mathrm{z}\stackrel{.}{\mathrm{z}}=0$ will give the relation of the fluxions \to w^ch the moments are proportional/. Here the whole operation is exact & as evident as any thing in mathematicks & all the quantities are finite \(as they ought to be in Geometry)/ to|Un|til the operation is {illeg} finished & \then/ o is made to decrease in infinitum & vanish And if o be at any time {illeg}|d|uring the operation supposed indefinitely or infinitely little yet it is the only \infinietly {sic} small/ quantity in the whole operation w^ch is infin supposed infinitely little & it is always supposed to be a constant {illeg} given quantity: whereas in the notation of M^r Leibnitz & the Artist is at liberty to take o for a quantity finite or infinitely small as he shall find it best for dispatch \his purpose whether he has a mind to \be/ exact or to make dispatch by approximation./ But in the differential notation \of M^r Leibnitz/, there are more infinitely small quantititis|e|s then one. always \two or more/ infinitely small quantities & more of them then one & all of them \& one or more of them are always/ inconstant & but one, & the calculations are \generally/ grounded upon approximations. The errors indeed are infinitely small, but Geometry admits of no errors. In these calculations a right line may be considered as touching a Curve in more points then one, but Ge & an {illeg} chord infinitely short may be considered as coincident with its Arch: but Geometry admits of no such Propositions.

M^r Leibnitz might become acquainted w^th M^r L Huguens in 1672 but when he was in London the first time w^ch was in Ian. & Feb. 1673 he knew nothing of the higher Geometry. The Horologium oscillatorium of M^r Huguens was published in April 1673

The Letter writ to Collins in 1672 M^r Leibnitz did not see till 1676.

Pag. 53. lin. 6. A copy of M^r Newtons Letter of 10 Decem. 1672 was not sent to M^r Leibnitz in Iune the end o Iune 26 1676. And M^r Leibnitz was not accused by the Committee of the R. S. of understanding all that he had seen, but only that \in/ what he saw \had seen/, the Method of Fluxions was sufficiently described to any intelligent person. And M^r Tschurnhause & M^r Iames Bernoulli thought it almost described in D^r Barrows works alone. The words of Bernoulli in the Acta Erud. for Ian. 1691 pag. 14 are these Qui calculum Barrovianum — intellexerit alterum a Dn Leibnitio inventum ignorare vix poterit; utpote qui in priori illo fundatus est & nisi forte in differentialium notatione & operationis aliquo compendio ab eo non differt. The Marquess de l'Hospital in the Preface to his works Book said that D^r Barrows calculus stuck at fractions & surds & that where D^r Barrow left off M^r Leibnitz went on. He had not yet seen M^r Newtons Letters of 10 Decem. 1672 & 24 Octob 1672 in w^ch M^r Newton wrote that his general method \did not/ stu|i|ck not at surds. But M^r Leibnitz had seen them before he found the differential method in the year 1676. before he found the Differential Method, & thereby knew that M^r Newton was before him in this particular. He might learn from Gregories Letter of Sept 5 1670 that Gregory had improved Barrow's Method of Tangents so as to draw Tangents without calculation, & from Newton's of 10 Decem 1672 that this Method was the same with that of Slusius & that both Methods was|er|e a branch or Corollary or Branch of Newtons general Method & by consequents|ce| that Newton's general Met Barrows Method was a step to that of Gregory & Slusius & both of them were steps to Newtons general method, & this might make him write in the beginning of his Letter of 21 Iune 169|7|7 Clarissimi Slusij Methodum Tangentium nondum esse absolutam Celeberrimo Newtono assentior; & then set down a|n| specimen example of D^r Barrows method of Tangents with the notation altered, & conclude the calculation with these words Quod coincidit cum Regula Slusiana, ostendit eam statim occurrere hanc Methodum intelligenti. And a little after: Methodo autem Slusiana omnes ordine irrationales tollendas esse nemo not|n| videt. <373r> Arbitror quæ celare voluit Newtonus de Tangentibus ducendis, ab his non abludere Quod addit, ex hoc eodem fundamento Quadraturas quo reddi faciliores me in hac sententia \hac/ confirmat; nimirum semper figuræ illæ sunt quadrabiles quæ sunt ad æquationem differentialem. Now M^r Newton |L.| having seen all this in M^r Newton's Letters & Papers abovementioned before he understood the method & understanding all this out of them so soon as he understood the Method, the Committee of the R. S. may be excused for saying that in what M^r Leibnitz had seen the Method was sufficiently described to any intelligent person.

Pag 44 T lin 10, 11. The Editors of the Acta Erudit. would print nothing in other characters.

Pag 45. l 8, 9. The {illeg}ing \assuming/ of right by second Inventors upon a supposition \after it appears that he was not y^e first/ is plagiary notwithstanding th{e}|a|t the second Inventor may have invented apart without knowing what the first had done before.

Pag 49 lin 9, 10. This writing without {a}ff{ow} \denyed in effect/ that M^r Newton understood the first Proposition in his own Book of Quadratures, or that the Elements of the Differential Method are conteined in the second Lemma of the second Book of Principles, or that the sentence Data æquatone {sic} fluentes quotcun quantitates involvente invenire fluxiones et vice \versa/, denotes the method of fluxions, or that the series for squaring of Curves w^ch breaks off & becomes finite when the Curve may be squared by \a/ finite equation, & wh wh|a|s found by tha|e|t Method of fluxions altho M^r Newton said in his Letter of 24 Octob. 1676 that it was found by that method & in his book of Quadratures shewed how it was found by that Method, & there is no other method yet known by w^ch it can be found. He denied these things not in express words but by denying that there was any

p. 54. l. 7, 8. The Editors of the Acta Leipsica never would print any thing with other characters then those of M^r Leibnitz, w^ch against M^r Leibnitz or w^th other characters then the Differential. When M^r Fatio sent them a Paper against M^r Leibnitz they would not print it. Wh{e} When In giving an Account of D^r Wa the two first Volumes of D^r Wallis's works they would not let the reader know what objection the {illeg} that the D^r ascribed the ascribed the D^r ascribed the Differential method to M^r Newton as being found by him in the year 1666 or before, \the oldest Inventor/ but shuffled of the Objection by {illeg}s{uls} pretending \feigning/ that Newton had always b{illeg}|o|th publickly & privately acknowledged M^r Leibnitz the inventor. N In giving an Account of M^r Newtons book of Quadratures instead of giving an account of M^r Newto the characters used by M^r Newton used in that Book they gave an account of the characters used by M^r Leibnitz. When D^r Keil sent them a paper with writ in M^r Newtons characters they would not print it in those characters.

Pag. 53. lin 22. When \In the year 1684/ M^r Leibnitz published the elements of his calculus \Method/ without making any mention of his the correspondence between him & M^r Newton eight years before as in candour he {illeg} ought to have done. For he knew by that correspondence that M^r Newton had then & five ye such another Method, as is plain by his Letter of 12 Iuly 1677 & that \he/ had it \& wrote of it/ in the year 1671 or before & therefore was the oldest Inventor, as is plain by M^r Newton's Letters of 13 Iune & 24 October 1676. M^r Newton therefore inser in the year 1686 inserted \set down/ the Elements of this method \as his own/ in the second Scholium \Lemma/ of the second book of his Principles & [in a Scholium added to that Lemma \he/ told the Reader that he did not give away that Lemma to M^r Leibnitz \as M^r Leibnitz has pretended/ but asserted his right to it as] added a Scholium to that Lemma not to g{illeg}|i|ve away that Lemma \to M^r Leibnitz/ as has been pretended but to assert his \own/ right to it as first inventor. And yet M^r Leibnitz co And yet M^r Leibnitz continued to claim it \upon a pretence that he found it apart/, tho he knew that he was not the first inventor.

Historical Annotations on the Elogium of M^r Leibnitz.

|1| Pag. 44. lin. 1. Fluxions are not differences. They are quantities of another kind, & M^r Leibnitz hath no marks nor name for them. They are motions or velocities of increase & not parts of the increasing quantity. What M^r Leibnitz calls Differences M^r Newton calls Moments & particles|& designes them by fluxions multiplied by the moment o.|

|2| Ib. lin. 3, 4, 5. When the Marquess de l'Hospital represented that the characters of M^r Leibnitz were more convenient then those of M^r Newton, he had seen only the characters used by M^r Newton in the second Lemma of the second book of his Principles & therefore meant no other.

|3| Ib. lin. 22. M^r Fatio was not the first. D^r Wallis, so soon as he heard that the method of M^r Newton was spreading in Holland by the name of the Differential Method of M^r Leibnitz, wrote in the Preface to the first volume of his works, published in April 1695, that M^r Newton in his two Letters of Iune 13^th & Octob. 24, 1676, had explained to M^r Leibnitz the method of fluxions or differences found by him ten years before or above; that is, in the year 1666 or before.

|8| Pag |5|45. lin. 6|3|. A letter writ by M^r Leibnitz in autumn 1715 being handed about among persons of note in London, M^r Newton was pressed to return an answer, & at length did so by a Letter dated Feb. 26, 1716, & M^r Leibnitz replied Apr 9^th 1716, but sent his Reply to Paris (with Copies of the two former Letters) to be communicated to his friends there before it came to London. For which reason M^r Newton wrote to him no more, but only shewed privately to a few of his friends what might have been writ in answer.

|4| Ib. \Pag. 48./ lin. 15. M^r Newton was accused of Plagiary in the Acta Eruditorum for Ianuary 1705 pag. 34 & 35, & the accusation was justified by M^r Leibnitz himself in his Letter to D^r Sloane dated 29 Ian. 1711. And for obviating this accusation the Commercium Epistolicum insinuated that the contrary was more to be suspected, M^r Leibnitz having seen severall of M^r Newton's Papers \Letters/ in the year 1676, it not appearing that he had the Method before he saw them.

|5| Pag 49. lin. 20. M^r Leibnitz pretended that the Committee of the R. Society had in the Commercium Epistolicum omitted many things w^ch ought to have been published, & upon that account pretence proposed to print a new Commercium impartially, & in autumn 1714 desired that the original Letters in the custody of the R. Society might be sent{illeg} to him for that purpose. M^r Newton replied before the R. Society that the Commercium was printed out of Originals which had been constantly in their Archives & those of M^r Collins, & had been also examined by those who knew the hands, & that they ought still to be kept safe for justifying what had been printed out of them. He added further that he did not print the Commercium himself, nor so much as produce any Letters or Papers which he had then in his own custody, because he would not make himself a witness in his own cause. And to prove the truth of what he said he produced two Letters written to himself the one by M^r Leibnitz March 7^th 1693, the other by D^r Wallis Apr. 10^th 1695. These Letters were examined before the R. Society by them who knew the hands & then they were read & laid up in the Archives of the R. Society as authentic. At the same time it was allowed that if M^r Leibnitz had any Letters in his custody which he thought material to be published, & pleased to send them to any friend in London whom he could trust with them that they might be publickly examined by those tha|who| knew the hands, & attested copies be taken of them: they might then be published either in the Phil. Transactions or by M^r Leibnitz himself. But nothing has been sent. On the contrary M^r Leibnitz having <374v> been since pressed to return a direct Answer to the Commercium, replied in his Letter of April 9^th 1716. Pour repondre done de point en point a l'Ouvrage publié contre moi, il falloit une autre Ouvrage aussi grand pour l{a} moins que celuy là, il falloit entrer dans une grand detail de quantité de Minuties passés|e|s il y a 30 ou 40 ans dont je ne me souvenois gueres; il me falloit chercher mes vel|i|lles Lettes|re|s dont plusieurs \se/ sont perdues, outre que le plus souvent; je n'ay pas g{illeg}|a|rdé les Minutes des miennes; et les autres sont ensevelies dans une grand tas de Papiers, qui je ne pouvois de|d|ebrouiller qu|'|{illeg} avec du Temps & de la patience. Mais je n'en avois gueres le loisir, etant chargé presentmant d'Occupations d'une toute autre nature. And the author of the Elogium of M^r Leibnitz published in the Acta Eruditorum for Iuly 1717 pag. 335, writes: Quo tamen perspicerent intelligentes quid de tota illa controversia sentiendum sit, Commercio Epistolico Anglorum aliud quoddam 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 wrote only that he would refute the English (he doth not say with a new Commercium, as the Author of the Elogium seems to understand, but) with a new surprizing invention in Analysis of another kind. And this he wrote a few days before his death, & we were to stay for the performance till his Historical labours should be at an end.

Pag 51. lin. 22. M^r Leibnitz might become acquainted with M^r Hugens in the year 1672; but when he was in London the first time \w^ch/ was in Ianuary & February 1673, & while he kept a correspondence w^th M^r Oldenburg about Arithmetical matters which was till Iune following, he knew nothing of the higher Geometry The Horologium oscillatorium of M^r Huguens was published in April 1673, & in studying the higher Geometry he read first this book & then the works of Paschal & Gregory of S^t Vincent & then fell upon his Triangulum characteristicum & his many Theorems, & went on to his Method founded on his Analytical Tables of Tangents & his Combinatory Art & this was the top of his skill when he wrote his Letter of 27 August 17 1676. For in that Letter he said of one par{illeg}|t| of this Letter \Method/ Nihil est quod norim in tota Analysi momenti majoris. And of another part: Cujus vim ac potestatem nescio an quisquam hactenus sit consecutus. Ea verò nihil diffe{r}t ab Analysi illa suprema ad cujus intima Cartesius non pervenit. Est enim ad eam constituendam opus Alphabeto cogitationum humanarum. In the same Letter he used the common Algebra where the Differential Calculus would have been proper had he then known it, & said that i|I|nverse Problemes of Tangents could not be reduced to Equations. In October following he was again in England where he met with Barrows works & Newton's Letter of 24 Octob. 1676, & saw the correspondence of Gregory & Newton in the hands of Collins, having seen Newton's Letters of 10 Decem 1672 & 13 Iune 1676 three mon & Gregories of 5 Sept. 1670 & 15 Feb 1671 three months before. And it doth not appear that he found the Differential method before he saw all this. See his Letter of 9 April 16|7|16.

Pag. 53. lin. 6. The Letter was writ to M^r Collins in 1672 & seen by M^r Leibnitz in 1676.

NB. M^r Leibnitz in his Letter of Iune 21. 1677 acknowledged a similitude between the Methods &

NB M^r Leibnitz observed \& acknowledged/ this similitude of the Methods long before the publishing of the two books here mentioned, namely in his Letter of 13 Iune 167|9|7. Here he acknowledges it again; but contends He acknowleg|d|ged it also in his {sic} letter to me date And he And|lso| in th{is}|{e}| Paper in which he published the {illeg} elents of the Differential method A.C. 1684 he allowed a methodus similis. Also in his Letter to me dated {illeg} $\frac{7}{17}$ March 1693 & printed above he allowed the same thing & e{illeg} that by the Principia Philosophiæ mathematia|c|a I had shewed that I had such a method, & \here he/ exprest himself in such a manner as with some deference to me, as if he also had endeavoured to do the like. And now after the publication of the edition of the two first Volumes of the Works of D^r Wallis he repeats the acknowledgment allowing that the Methods are so far the same that he calls them both by the common name of the infinitesimal method And the|But| as the Analysis of Vieta & Cartes were called by the common name of Analysis speciosa & yet differed by some improvements w^ch Cartes had made to the method of Vieta, so the the Diferential method h my infinitesimal Analysis & his might differ in some improvements which he had made to {illeg}|m|ine & which were not yet come to the knowledge of D^r Wallis. And three of those improvements he names, 1 the reduction of transcendent quantities to Equations. 2 The reduction of such Curves to Equations as Cartes called Mechanical. 3 The invention of Exponential Equations. [And thus the general \infinitesimal/ method common to us bo|b|oth, & here allowed to be mine & the improvements {illeg}|m|ade to it by M^r Leibnitz are h{illeg}|i|s.] While M^r Leibnits contended from {illeg}d{illeg} that my method & his differed in somethings its plain that he di{illeg}|d| not yet bega|i|n to claim the whole [Nor did D^r Wallis claim for |me| the improvements w^ch M^r Leibnitz had made to the Method. [The D^r told him \published/ that I had invented the Method in the year 1666 or before & \M^r Leibn./ in all the Letters which afterwards past between them, the did not deny it. [The D^r told him published that in the year 1676 I had by Letters explained \this method/ to M^r Leibnitz th & he answers that he had the first light into the method from the consideration of the summs & differences in Numbers: the Summs \differences/ answering to tangents & the summs to Quadratures] I nor pretends that he had it before the year 1676. And second inventors have nor right]. M^r Leibnitz claimed the improvements & D^r Wallis claimed the rest for me as the oldest inventor by many years, & was not complained \of/ for this by M^r Leibnitz. De te autem queri, saith M^r Leibnitz nunquam mihi in mentem venit quem facile apparet nostro \[our improvements]/ in Actis Lipsientibus pra|o|dita, \[our improvem^ts]/ non satis vidisse. {illeg} — for saying that I invented the method in the year 1666 or before & explained it in my Letters ten years after in my Letters.

The M^r Leibnitz did not deny that {I} complain of the D^r for saying that invented y^e method in the year 1666 or before & explained it ten years after in my letters, but

And so the question w^ch \then/ remained \between them/ was about these improvements & particularly whether the symbols $\int \frac{\mathrm{a}\mathrm{a}}{\mathrm{b}+\mathrm{x}}$, $\int \frac{d\mathrm{x}}{\sqrt{2\mathrm{x}-\mathrm{x}\mathrm{x}}}$ & such like by w^ch he designed transcendent quantit{illeg} & introduced \them/ into equations or mine {illeg} for the same purpose $\boxed{\frac{\mathrm{a}\mathrm{a}}{\mathrm{b}+\mathrm{x}}}$. $\boxed{\frac{d\mathrm{x}}{\sqrt{2\mathrm{x}-\mathrm{x}\mathrm{x}}}}$ & the like, or \&c/ were the oldest, or whether the summatory method or the \inverse/ method of fl fluxions was the oldest. For my part I said in my Letter of 24 Octob 1676 that in the year 1671 I wrote a Tract in the year 16 five years before, that {illeg} concerning the method of series & another method together, & that {o}the{r} {illeg}e \said there that the/ other method proceeded without sticking at surds & was founded in this Propositions: Data Æquatione fluentes quotcun quantitates involvente invenire fluxiones & vice versa, that is in the direct & inverse methods of fluxions \improperly/ called by M^r Leibnitz the methods differential & summatory methods. And \I/ said further that this method extended to Problems about maxima &minima, \& to/ tangens|t|s \directly & inversely, & to/ Quadratures, & some others & gave \me/ the series there set down for squaring of Curves & some others of like nature. And in my Letter of 10 Decem. 1672 I ({illeg}|a| copy of w^ch was sent to M^r Leibnitz & came to his hands) I mentioned some others \sorts of Problemes/ & particulary {sic} the working in mechanical curves & the finding the curvatures of Curves & the tangents of the Mechanical Curv{s} and in my letter of 13 Iune 1676 I said that my Analys {sic} (composed of the method of Series & another methods, (described afterward in my Letter ( \viz^t/ the <375v> methods of fluxions & arbitrary series] extended to almost all sorts of Problems exp except perhaps some numeral ones like those of Diophantus. And M^r Leibnitz in his Letter of 27 Aug. 1676, replied that he did not beleive that my method could be so general, nor extend to inverse Problems of Tangents & many others tho he had reduced one inverse probleme of Tangents (that proposed by Beaune) to a quadrature. And if all this & the method of extracting fluents out of In my Letter of 24 Octo. 1676 equations involving the{illeg} fluxions & that of arbitrary series set down in my Letter of 24 Octob {illeg}|1|676 as known to me some years before, be not sufficient to secure these thing security: it no many must hereafter communicate any thing in writing till he has before he has secured \it/ to himself by printing; This & so there's an end of correspondence in writing about sciences. And the next step is to make void all Indentures Bonds & other writings in law which are not in print: or at least to give the preference to those which are in print.

– Copies of the two Letters of M^r Leibnitz here desired & of some others

Pag. 44. l. 1. Fluxions are not differences. They are quantities of another kind, & M^r Leibnitz hath no marks nor name for them. They are motions or velocities of increase & not parts of the increasing quantity.

Ib. l. 3, 4, 5. When the Marquess de l'Hospital represented that the characters of M^r Leibnitz were more convenient then those of M^r Newton, he had seen only the characters used by M^r Newton in the second Lemma of the second book of his Principles & therefore meant no other.

Ib. l. 22. M^r Fatio was not the first. D^r Wallis in the Preface to y^e first volume of his works dated published in April 1695 wrote that M^r Newton in his two Letters of Iune 13 & Octob 26|4| 1676 sent to M^r Oldenburg to be communicated to M^r Leibnitz, had explained to M^r Leibnitz the method of fluxions ca{illeg} found by him ten years before or above, that is, in the year 1666 or before. And M^r Newton in the year 1686 inserted the Elements of this Method into the second Lemma of the second Book of his P

Pag. 48. lin. 15. M^r Newton was accused of Plagiary in the Acta Eruditorum for Ianuary 1705 pag 34 & 35 & the accusation was ratified \justified/ by M^r Leibnitz himself in his Letter f to D^r Sloan dated 29 Decem 1711. And for obviating this accusation the Committee \Commercium Epistolicum/ insinuated that there was in the year 1676 \the contrary was more to be suspected/ several of M^r M^r Newton's papers \relating to this matter/ were \having been/ seen by M^r Leibnitz|.| w{illeg} relating to this matter in the year 1676.

Pag 49. l. 20. The whole correspondence of M^r Leibnitz & M^r Oldenburg from the begining of the year 1673 to the death of M^r Oldenburg is in the hands of the R. S. & the{y}

Pag 48. l. 6. A l|L|etter writ by M^r Leibnitz in Autumn 1715 being handed about among persons of note in London, M^r Newton was at length prevailed with to answer it This Answer was dated \His Letter was dated/ Feb. 26. 1716, & M^r Leibnitz wrote an Answer|ed| to M^r Newton \the letter/ |it| 9 Apr. 1716, but sent it \his Answer/ to Paris (w^th copies of the two former Letters) to be communicated to his friends at Paris \there/ before it came to London, & began it with a pretence that M^r Newton's Letter was a challenge & {illeg} \as if M^r Newton had been the Aggressor./ For w^ch reasons M^r Newton wrote to him no more, but yet drew up in writing what might \hav/ be said in answer & shewed it privately to a few of \only/ shewed privately to a few of his friends what might have been said in Answer \some Observations upon the last letter |what might have been writ in answer|/ And because the three Letters had been communicated in France, M^r Newton upon \both France & England published them/ |upon| the d\e/ath of M^r Leibnitz published them with together w^th the said Observations.

Pag 49. lin. 20. All the Letters of M^r Leibnitz to M^r Oldenburg are from the beginning of the year 1673 to the death of M^r Oldenburg \are in the hands of the R. S/ & so are copies of all M^r Oldenburgs & {M} Letters to M^r Leibnitz from Decem 8 1674 to y^e end of the year 1696. M^r Leibnitz complained that And a Commercium of a later date can signify little. M^r Leibnitz complained that the Committee of the R. S. had extracted out of them|se| old Letters every thing that made against him & omitted every thing that made for him but foreigners who have since viewed the Letters \have/ fo\u/nd it otherwise. And when \lately wⁿ/ M^r Le{illeg}|i|bnitz \pretended to/ ga|i|ve two instances of things omitted in them hath be he was found mistaken in them both. And in Autumn 16 1714 M^r Leibnitz desiring that the original Letters {illeg} in their custody \of y^e R. S./ might be sent |to| him in order to \his/ print|ing| a new Commercium, the was answered, that if he had any thing in his custody w^ch he thought material to be printed, & pleased to send the original \MS/ to any of his friends in London; they would {illeg}|o|rder it to be printed in the P. Transactions after the hand had been examined. But MSS not examined by \unconcerned/ persons who knew the hands {illeg} they would not regard.

P. 51. lin. 22. M The Horologium oscillatorium of M^r Huguens was published in April 1673. Af M^r Leibnitz began presently after to study the higher Geometry Then he fell upon the triangulum characteristicum w^ch might help him to many new Theorems like those of Gregory & Barrow. Then he fell upon his Analysis \& his many new Theorems, & his a{illeg}/ \& went on to his method {illeg} Method/ by analytical Tables of Tangents & |the| Combinatory Art, & this was the top of his skill when he wrote his Letter of 27 Aug. 1676. For in that Letter he saith \{illeg}/ of these things one part of this method: Nihil est quod norim in tota Analysi momenti majoris. And of ano <377v> ther part: Cujus vim ac potestatem nescio an quisquam hactenus sit consecutus. Ea vero nihil differt ab Analysi illa suprema ad Cujus intima Cartesius non pervenit. Est enim ad eam constituendam opus Alphabeto cogitationum humanarum. ✝ < insertion from the bottom of the page > ✝ And there are other indications in the Letter by w^ch it may be gathered that the differential method was not yet known to him. He seems to have learnt it in the Autumn or winter following. ✝ \And soon/ After this, he fell upon the differential method Analysis|.| w^ch was of a nature above t < text from f 377v resumes > A The method \therefore/ by differential equations he was not yet acquainted with, for he adds sunt Quod dicere b{illeg} as may be gathered also from his words w^ch follow in the same Letter: Quod dicere videmini pleras difficultates ad Series infinitas reduci, id mihi non videtur. Sunt enim multa us adeo mira et implexa ut ne ab æquationibus non pendeant. Qualia sunt (ex multis alijs) Problemata methodi tangentium inversæ. M^r Leibnitz in his Letter of 9 Apr 16 1716 reprehen{illeg} said that he judged that by the words certa Analysi solvi in his Letter of 27 Aug. 1676 that he \must have/ had them some light into the Differential calculus. And at that time. But w|W|hether the certa Analysis were the Differential method or the Analysis illa suprema ad cujus intima Cartesius non pervenit may be doubted: but how ever nothing more can be affirmed then that M^r Leibnitz according to the best of his knowledge & memory bega got some light into the Me Differential method in the year 1676.

Pag. 53. l. 6. A copy of M^r Newtons Letter dated 10 Decem 1672 did not come to |t|his|e| hands \of M^r Leibnitz/ before Iuly 1676 \was not sent to him/ before Iune 26 1676.

Pag. 49. lin. 20. All the Letters of M^r Leibnitz to M^r L{illeg}ds Oldenburg from the beginning of the year 1683 are in the east{illeg} to the dearth {sic} of M^r Oldenburg are in the custody of the R. Society In autumn 1714 M^r Leibnitz wrote a Letter \to London signifying that he intended to print a {illeg} new Commercium &/ desir{ii}ng desiring that the Original Letters in the custody of the R. Society might be sent to him in|fo||r that| order to his printing an impartial Commercium \purpose/. The Letter being read before the |R.| Society M^r Newton replied that when the|ir| Committee of the R. S printed the Commercium \out of their archives & those of M^r Collins/ he did not e{n}t{illeg} so much as produce the Letters in his own custody because he would not make himself a witness in his own cause. And to satisfy them of the truth of this he produced two Letters \written to himself/ the one writ to him by M^r Leibnitz himself the oth{illeg} March 7^th 1693 the other by D^r Wallis Apr. 10^th 169{illeg}|5|. The Letters were examined \before the R. S/ by those that knew the hands & then read & of M^r Leibnitz pretended that the Committee of the R. Soc. had omitted many things w^ch ought to have bee{illeg}|n| published, & upon that pretence proposed to publish a new Commercium. And in his Letters of Novem. 16 1716|5| & Apr. 9 1716 he to pr to make good his complaint he instanced in two things that they had omitted but was mistaken in them both. The Original Letters have been seen by forreigners & nothing material has been found omitted.

Pag 49. lin. 20. M^r Leibnitz pretended that the Committee of the R. S. had in their Commercium \Epist/ omitted many things w^ch ought to have been published & upon that pretence proposed to print a new Commercium impartially: And in autumn 1714 wrote to London t desired that the original Letters in the custody of the R. Society might be sent to him for that purpose. His Letter being read before the R. Society; M^r Newton replied that {illeg}the{ir} Commit Commercium was printed out of Originals w^ch had been kept \remained from the beginning/ in the Archives of the R. S. & those of M^r Collins, from the beginning & that he did not thi \the hands being examined by those that knew them & were still to be kept for justifying {illeg} \{w} also had been printed out of them// & that he himself did not \intermeddle with those Originals nor/ so much as produce the letters & {illeg}|P|apers in his own custody because he would not make himself a witness in his own cause. And to prove the truth of this he produced two \shewed them two old/ Letters written to himself the one by M^r Leibnitz March 7^th 1693 the other by D^r Wallis Apr. 10^th 1695. These Letters were examined – – – – as authentic unattested papers. And now if M^r Leibnitz has left any orig For making good his accusation of the Committee as if they had pu ommitted what made against M^r Newton, he lately charged them with two such omissions but was mistaken in them both. The Original letters have been seen by forreigners & nothing material has been found omitted.

– and that for further before printing the|y| had been also \been/ examined by those who knew the hands; & that they ought still to be kept by those {illeg} knew the hand \by the R. Society/ for justifying what had been printed out of them. He added further that he did not intermeddle with those Originals nor so much as produce what \papers/ he had in his own custody because he would not

Historical Annotations on the Elogium of M^r Leibnitz.

|1| Pag. 44. lin. 1. Fluxions are not differences. They are quantities of another kind, & M^r Leibnitz hath no marks nor name for them. They are motions or velocities of increase & not parts of the increasing quantity. /What M^r Leibnitz calls Differences M^r Newton calls particles & Moments & parts.\

|2| Ib. l. 3, 4, 5. When the Marquess de l'Hospital represented that the characters of M^r Leibnitz were more convenient then those of M^r Newton, he had seen only the characters used by M^r Newton in the second Lemma of the second book of his Principles & therefore meant no other.

|3| Ib. l. 22. M^r Fatio was not the first. D^r Wallis, so soon as he heard that the method of M^r Newton was spreading in Holland by the name of the differential method of M^r Leibnitz, wrote in the Preface to the first volume of his works published in April 1695, that M^r Newton in his two Letters of Iune 13 & Octob. 24 1676, sent to had explained to M^r Leibnitz the method of fluxions \or differences/ found by him ten years before or above; that is, in the year 1666 or before.

|5| Pag. 48. lin. 15. M^r Newton was accused of Plagiary in the Acta Eruditorum for Ianuary 1705 pag. 34 & 35. & the accusation was justified by M^r Leibnitz himself in his Letter to D^r Sloan dated 29 Ian 1711. And for obviating this accusation the Commercium Epistolicum insinuated that the contrary was more to be suspected, M^r Leibnitz having seen several of M^r M^r Newtons Papers in the year 1676.

|4| Pag. 48. lin 6. A Letter writ by M^r Leibnitz in autumn 1715 being handed about among persons of note in London, M^r Newton was pressed to return an answer, & did so by a Letter dated Feb. 26. 1716 & M^r Leibnitz answered him \replied/ Apr. 9^th 1716, but sent his Answer \Reply/ to Paris (with Copies of the two former Letters) to be communicated to his friends there before it came to London. For w^ch reason M^r Newton wrote to him no more, but only shewed privately to a few of his friends what might have been writ in answer.

|6| Pag. 49. lin. 20. M^r Leibnitz pretended that the Committee of the R. Society had in the Commercium Epistolicum omitted many things w^ch ought to have been published, & upon that pretence proposed to print a new Commercium impartially, & in autumn 1714 desired that the original Letters in the custody of the R. Society might be sent to him for that purpose. M^r Newton replied \before the R. Society/ that the Commercium was printed out of Originals which had been constantly in their Archives of the R. Society & those of M^r Collins & had been also examined by those who knew the hands, & that they ought still to be kept by the R. Society for justifying what had been printed out of them. He added further that he did not print the Commercium himself nor so much as produce the papers w^ch he had \then/ in his own custody, because he would not make himself a witness in his own cause. And to prove the truth of what he said he shewed them \produced/ two Letters written to himself the one by M^r Leibnitz March 7^th 1693, the other by D^r Wallis Apr. 10 1695. These Letters were examined before the R. Society by them who knew the hands & then they were read & laid up in the Archives of the R. Society as authentic. At the same time it was allowed that if {illeg}|M|^r Leibnitz had any Letters in his custody w^ch he thought material to be published & pleased to sent|d| them to any friend in London whom he could trust with them that they might be publickly examined by those who knew the hands & attested copies be taken of them: they might then be published either in the Ph. Transactions or by M^r Leibnits himself. But nothing has been sent. On the contrary M^r Leibnitz having been since pressed to ‡ < insertion from f 379v > ‡ to return a direct Answer to y^e Commercium, replied in his Letter of April 9^th 1716 Pour repondre done de point en point a l'Ouvrage publié contre moi, il falloit une autre Ouvrage aussi grand pour le moins que celuy là, il falloit entrer dans une grand detail de qualite de Minuties passees il y a 30 ou 40 ans dont Ie ne me souvenois gueres, il me falloit chercher mes vieilles Lettres, dont plusieurs sont perdues, outre que le plus souvent, je nay pas g{illeg}ardé les Minutes des Miens miennes, et les autres sont ensevelies dans une grand tas de Papiers, qui je ne pouvois de brouiller qui avec du Temps & de la patience. Mais je n'en avois gueres le loisir, etant chargé presentmant d'Occupations d'une toute autre nature. And in \the author of/ this|e| Elogium \of M^r Leibnitz/ published in the Acta Eruditorum for Iuly 1717 pag. 335 the Author of the Elogium write{illeg}|s|: Quo tamen perspicerent intelligentes quid de tota illa controversia sentiendum sit, Commercio Epistolico Anglorum aliud quoddam suum idem amplius opponere decreverat, & paucs|i|s ante obitum diebus Cl. Wolfio significavit se Anglos famam ipsius lacessentes reipsa refutaturum: quam primum 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. He doth no We were to stay till his Historical labours were at an end, & then he would confute the English; he doth not say with a new Commercium, but with a new surprizing invention in Analysis \of another kind then the differential method/; & this he wrote a few days before his death. And perhaps he meant his Alph{illeg}|a|betum cogitationum humanarum or something like it. M^r L The author of the Elogium tells us of an aliud quoddam \[Commercium]/ idem amplius, but M^r Leibnitz himself wrote only that \so soon as historical labours were at an end/ he would refute the English (he doth not say with a new Commercium \as the Author of the Elogium seems to understand/ but) with a new surprizing invention in Analysis of another kind; & this he wrote a few days before his death, & we were to stay for this till his historical labours were over \should be/ at an end. < text from f 379r resumes >

Pag 51. lin. 22. |He returned from London to Paris in Febr. 1673 & began to study the higher Geometry a few months after.| The Horologium oscillatorium of M^r Huygens was published <379v> in April 1673. M^r Leibnitz \soon after read this Book & then some others/ [began presently after to study the higher Geometry] & then fell upon his Triangulum characteristicum & his many Theorems, & went on to his Method by Analytical Tables of Tangents, & \by/ his Combinatory Art. And this was the top of his skill when he wrote his Letter of 27 Aug. 1676. For in that Letter he said of one part of this method: Nihil est quod norim in tota Analysi momenti majoris. And of another part: Cujus vim ac potestatem nescio an quisquam hactenus sit consecutus. Ea vero nihil differt ab Analysi illa suprema ad cujus intima Cartesius non pervenit. Est enim ad eam constituendam opus Alphabeto cogitationum humanarum. And soon after this, he fell upon the Differential Analysis.

Pag. 53. l. 6. A copy of M^r Newton's Letter of dated 10 Decem. 1672 was not sent to him before Iune 26, 1676.

M^r Leibnits [Might be acquainted w^th M^r Hugens in 1672 but he] came to London in the beginning of the next year \1673/ & returned to Paris in March \following/ & kept a correspondence with M^r Oldenburg about Arithmetical questions till Iune 8 following, & hitherto he was unacquainted with the higher Geometry. The Horologium ascillatorium w{ere} of M^r Huygens was published in April 1673 & when he began to study the higher Geometry he read first this book & then the works of Paschal & Gregory of S^t Vincent & then fell upon his Triangulum characteristicum & his many Theorems, \&/ went on to his method by Analytical Tables of Tables of Tangents & the combinatory Art. And this was the top of his skill when he wrote his Letter of 27 Aug. 1676. – – – humanarum. And In the same Letter he used the common Algebra where the Differential calculus would have been proper had he then known it, & said \{illeg}/ that inverse problems of Tangents & many others could not be reduced to equations. But soon /soon\ \not be |It was therefor|/ after the writing of this Letter \that/ he fell upon the Differential calculus. In October following he was in|a|gain in England where he might mee|t|t w^th the \Barrows/ works of Gregory & \D^r/ Barrow &|c| learn D^r Barrows method of Tangents \&/ \& M^r Newton's Letter of 24 Octob 1676/. [And Iames Bernoulli in the Acta Eruditorum for Ian 1691 pag 14 wrote, Vt verum fatear, q|Q|ui calculum Barrovium — intellexerit, alterum a Dn. L. inventum ignorare vix poterit; utpote qui in prior illa|o| fundatus est & nisi forte in differentialium notatione & operationis aliquo compendio ab eo non differt. And the Marques de l'Hospital \in his Preface/ that D^r Barrow's calculus taught not how to take away fractions & surds; A|a|nd \that/ where he & others left off M^r Leibnitz began. [And M^r Newton in his Letters of 10 Decem. 1672 & 24. Octob. 1676 that his method proceeded wit did not stop at fractions & surds. proceeded in equations involving surds.] But the Marquess had not seen M^r Newtons Letters of 10 Dec. 1672 & 24 Octob. 1676] & \saw/ the correspondence of Gregory & Newton in the hands of Collins.

Criticism of Fontenell {Cl}{illeg}

to Varignon {endof} 1718 or beginning of 1719

S^r

I am much obliged to you for your large Letter in order principally to bring M^r Iohn Bernoulli & me to a better understanding. M^r Leibnitz has attributed to him the Letter of Iune 7^th 1713 & in that Letter inserted into a flying Paper dated 29 Iuly following & is|t| that Letter he |h|is cited by the title of \he cites sti cites himself by the title of/ Eminens quidam Mathematicus {illeg}en effect denies |& took upon him to {illeg}|give| \as/ {a} judge {al} so twice between M^r Leibnitz & the Committee of the R. Society & in that sentence denied (in effect that| that {sic} when I wrote the book of Quadratures, (w^ch I can assure you was above 40 years ago, (except the Introduction & Conclusion)|)| I did not understand the first Proposition of the Book

✝ And I return you my hearty thanks for your kind present of the Elogia of the Academiciens. In that of M^r Leibnitz M^r Fontenelle has been very candit. There are some mistakes in matter of fact but not by designe. I reccon that M^r Fontenell was not sufficiently informed. He \seems to/ follows the Marquess de l'Hospital in saying that the differential characters are the more convenient, but the Marquess whom I reccon a very candid person when he published his book of infinitely littles had seen no other characters of mine then those in the Book of Principles. In pag. 44 lin 22 he should have mentioned D^r Wallis before M^r Fatio if he had known what the D^r wrote in the Preface to the two first Volumes of his works. Pag. 48 lin 15 he should have mentioned that I was first accused of Plagiary in the Acta Eruditorum for Ianuary 1705 pag. 34 & 35 & by M^r Leibnitz himself in his Letter to D^r Sloan dated 29 Ian. 1711, had he been advised of it. & For I {illeg} this was in opposition to my saying in the Introduction to the book of Quadratures that I found the method of fluxions gradually in the years 1665 & 1666. Pag. 49, l. 20 they should have told have given him a wrong information from Germany. For in the Elogium of M^r Leibnitz printed in the Acta Eruditorum for Iuly 1717 page 335 were are told that a few days before his death he wrote to M^r Wolfius that he would refute the English after his historical labours were over. Pag. 51 lin. 22 M^r Leibnitz became acquainted w^th M^r Hugens in 1672 came to London \& returned to Paris/ in the beginning of 1673, kept a correspondence with M^r Oldenburg till Iune follow about Arithmetical questions till Iune following & hitherto knew nothing of the higher Geometry. The Horologium oscillatorium of M^r Hugens was published in April 1673 & he begun with this book. {to}he But doth A{illeg} \And it doth not appear that he knew any thing of the Differential Method before/ his second coming to London \at w^ch time/ he met with my Letter of 24 Octob 1676 & D^r Barrows works & my Letter of 24 Octob 1676 & saw in the hands of Collins & the Letters & Papers sent to him by me & Gregory{illeg} having received Pag. 53. lin. 6, the Letter here referred unto was writ by Colling|s| in 1672 & see but not seen by M^r Leibnitz till 1676. [Pag. 53 l. 22 M^r Leibnitz is blamed here \in point of candor/ for concealing what he knew of other mens inventions. He received Gregories series twice from England & by a copy of Gregories Letter dated 15 Feb 1671 knew that Gregory had invented it before that time \the date of that Letter/ & yet printed it as his own \in 1682/ without making any mention that he ha G he had received it from London & that it was first invented by Gregory. He might have said that he invented it himself before he knew that Gregory had invented it, bef but he should not have {illeg}|c|oncealed his having receiving it from London, & much less should he have concealed his knowledge that Gregory had invented it before him. And so when he printed the elements of the Differential Method he should not have concealed his knowledge of what he seen in my Leitters concerning such a method. For its very plane by his Letters of 21 Iune 1677 that he then understood that my method was of the same kind with his own. And much less should he have concealed his knowledg after I had in the second Lemma of the second Book of Principles set down the Elemt|e|nts of my method & in the Scholium upon that Lemma explained the sentent|c|e in in|w|^ch I had couched that Lemm Mem|th|od of w^ch I said in my Letter of 24 Octob. 1676 \where I said/ that I had explained it \in/ a Tract written in the five years before. He knew therefore that I was the firs before him & should not have concealed his knowledge. H In his Letter of 7 March 1693 he acknowledged of his own that I had shewed by my Book of Principles that I had such a method \when I wrote it/ & to deny this afterwards proceeded from the want of candor.] All th D^r Barrows works & Gregories Letters of 5 Sept 1670, & 15 Sept 1671 & mine of 10 Decem 1672 <381v> 13 Iune 1676 & 24 Octob 1676 were all of them seen by him when he was the second time in England or within three months less the {sic} four months before & it doth not ap appear that he understood the Differential Method before he saw all these Papers. And the Committee of the R. Society have affirmed nothing more that|en| that he saw them & that I was before it appears that he had s{illeg} had the Method, & that in them the Method of fluxions it sufficiently described to any intelligent person. But if the friends of M^r Leibnitz beleive that he did not understand those Papers the Gentlemen of

M^r Leibnitz has told us that M^r Iohn Benoulli was the author of the Letter dated 7 Iune 1713 & \&/ inserted into the flying Paper of 29 Iuly following. In \was written by M^r Iohn Bernoulli. And the author of/ that Letter M^r Benoulli {illeg} to conceale himself, \For he was/ cites|d| himself \has told the world that he is not it was not written by I. B. M^r Bernoulli/ by the name of an Eminent Mathematician, & \In that Letter he has/ to te|o|ll|d|s the world that I d (in effect) that I did not understand the first Proposition of the Book of Quadratures & th{illeg}|a|t in the Introduction to that Book I d{illeg}|i|d not write of the Method of Fluxions because there are no prickt letters in it & therefore that in the years 1665 & 1666 (in w^ch I said I|t|hat in that Introduction that I found tha|e|t Method \of fluxions/ gradually) I did not so much as dream of that Method it. And for p{illeg} M^r Leibnitz {illeg} \hath/ subscribed to these opinions \telling us that/ as \these are/ the judgment of a primary candid \& a{illeg}i{illeg} {illeg}/ \& impartial/ Mathematician, & for a \further/ Demonstration thereof sent a \had|s|/ challenged|in||g| to the English Mathematicians to solve a Problem of M^r Iohn Bernoulli{illeg}.

T|W|hen the Commer. Epist. came abroad M^r {illeg} Leibnitz appealed from the judgment of the Committee of the R. Society to that of a primary & impartial Mathematician & the judgement of that Mathematician was inserted into the dated 7 Iune|ly| 167|71|3 was inserted into a flying paper dated 29 Iuly 1713 & \which was dated 24 Iuly 1713 & was/ written by somebody who knew {illeg} what passed \at Paris/ above 36 years before between M^r {illeg}|L|eibnitz & M^r Hugens 37 years before,|.| & dated 29 Iuly 1713. And the Mathematician \distinguished himself/ denyed that he was M^r Iohn Bernoulli by citing cited M^r Iohn Bernoulli by the name of an Eminent Mathematician [& [thereby denyed that M^r Iohn Bernoulli was the author of that judgm Iudge to whom M^r Leibnitz appealed,] but M^r Leibnitz has since told us again & again that M^r Bernoulli was that Iudge \the Mathematician/. I This \impartial/ Iudg in that Paper of 7 Iune 1613 has told the world (in effect) \has given a Iudment {sic} which amounts to this our people look/ \upon as if he had said/ that I did not understand the first Proposition of my book of Quadratures & that in the Introduction to that book \& in the 2^d Lemma of the 2^d Book of Principles & the Scholium upon it/ I did not write of {illeg} not Lette \I said nothing of/ the Method of Fluxions because there are no prickt letters in them, & that in the years 1665 & 1666, {illeg} in which I said in that Introduction that I found the Method of fluxions gradually, I did not so much as dream of it. |nor knew any thing of the method till before the third Volume of D^r Barrow's works ca understood the Method till it was grown commonly known by others.| And M^r Leibnitz hath subscribed to |this judgment| this judgment & dem{illeg} these opinions & \endeavoured to/ demonstrated \propagate support it/ them, by telling the world they are \that it is/ the judgment of an Eminent a primary & impartial Mathematician \& that is the judgm^t of M^r Iohn/ M^r Iohn {sic} Bernoulli \himself/, & by sending a challenging the English Mathematicians to solve a Problem of that great man. Th{illeg} And now M^r Bernoulli writes \And that \he/ beleived that M^r {illeg} me in my saying/ that I found hte Method by my self till M^r Bernulli gave a contrary judgment. And thus M^r Leibnitz has made M^r Iohn Bernulli the Principal in this controversy. There have been several defamato detracting papers printed in the Acta Eruditorum without the names of the authors, & such Papers are in England accounted {illeg} \unfair/, uncandid & of ill repute & condemned by the name of Libells.