Historia methodi infinitesimalis

It is represented that in the third fourth & fift Centuries some in Afric took the water spirit & blood to represent the father {illeg}|W|or{t}|d| & holy spirit, & that Tertullian probably introdued {sic} this interpretation, & might have it from y^e School of the Montanists, & that

The{illeg} testimony of the three in heaven (1 Iohn 5. 7) is wanting in \all/ the Greek Manuscripts now extant \hitherto coll hitherto examined/ & in [the Syriac & Ethiopic, versions \&/ & Coptic, Arabic \&/ Armenian & in the & Coptic versions {illeg}s] all the old Versions now extant in MSS except the vulgar Latin made by Ierome. It was originally wanting in this Version \also/ for it is wanting in most of the oldest MSS of this Version but is got into almost all those w^ch have been maded since the twelf Century.

pag 191 lin 19. – quence not before the yeare 1677 or not above a month or two before.

Ib. lin 23. – year 1670, & M^r Ia. Gregory in a letter to M^r Collins dated 5 Sept. 1670 was|ro|te thus. Barrovij Lectiones summa cum voluptate legi & attentione legi pers|l|egi; at omnes qui unquam hisce de rebus scripserunt infinito intervallo superasse comperio. Ex ejusdem [Barrovij] methodis Tangentes ducendi cum quibusdam e proprij collatis inveni Methodum generalem & Geometricam ducendi Tangentes ad omnes Curvas sine calculo. [A copy of this Letter was sent to M^r Tschurnhause in May 1675 & \another copy thereof was sent/ to M^r Tsch Leibnitz in Iune 1676 amongst in the collection of M^r Gregorys papers Letters above mentioned]. M^r Newton sent in my letter of dated Decem 10 1672 \having n|N|otice of the method & of that of Slusius, being sent to M^r Newton he/ communicated his method of Tangents \in a Letter/ to M^r Collins \dated Decem 10 1672/ saying that he took it to be the same with that of Slusius & Gregory, & adde after he had explained his method & illustrated his m it by an example he added: Hoc est unum particulare — — —

Pag 192. lin 7. A Copy|ie|s of M^r Gregories Letter of Sept 5. 1670 & of M^r Newtons of 10 Decem 1672 were {illeg}|s|ent to M^r Oldenb Leibnitz by M^r Oldenburg — — — — of Iune 13 1676; & M^r Leibnitz in October following next \be{illeg} coming to London met there/ with the math Lectures of D^r Barrow & with M^r Newtō Letter of 24 Octob 1676, & |M^r Newton having in his Letter & in that of 10 Decem. 1672 described that he had a very general Analysis — — — M^r Leibnitz in his returning| in his returning home from London through Holland was meditating upon the improvement of the method of Slusius. For in a Letter to M^r Oldenburg — —

Pag. 193 l. 7. M^r Newton in his Letter dated Octob. 24. 1676.

Pag. 194. l 6. M^r Leibnitz saw this Letter before he lefte London but not having {illeg}|tim|e to {illeg} get it copied a copy thereof was sent to him so soon as M^r Oldenburg had notice that he was arrived at Hannover: & M^r Leibnitz soon after viz^t in a Letter dated Iune 21, 1677

Pag. 196 lin. ult. — {ry} of his general method, & M^r Gregory that the method of Barrow might be improved t|s|o as to draw {sh}|Ta|ngents without calculation, that is, so as to give the method of Tangents \Rule/ of Slusius.

Pag 191 {illeg}|li|n. 22|1| & 22. D^rBarrow published his Method of Tangents in the year 1670. & M^r Iames Gregory in a Letter to M^r Collins dated 5 Sept 1670 wrote that he had improved the method of Barrow so as to be able to draw Tangents to all Curves without calculation. M^r Collins in autum 1672 gave notice of this method to M^r Newton & that M^r Slusius had offered to <173v> offere writ to M^r Oldenburg about such another Method. M^r Newton in a letter dated 10 Decem 1672 wrote back \to M^r Collins/ that he beleived took the methods of Gregory & Slusius to be the same with his own & after he had described his own Method & illustrated it with an example he added: Hoc est unum particulare vel Corollarium potius methodi generalis —

pag. 198. lin. 10. Occasio fuit hæc. Cum D. Leibnitius \Viennæ agens/ ad amicum scrib{illeg}|psit|t probabile esse se cum quod cum probabile esset {Eq} literas aliquas {illeg} archivis nondum publicatas inter eas Oldenburgi et Collinij latere, ipse optaret ut hæ sibi a Regia Societate mitterentur. Se enim cum Hanoveram rediret posse Cis|o|mmercium novum Epistolicum \etiam/ in lucem emittere & velle publicare non minus litteras quæ contra ipsum allegari possent quam illas quæ ipsi faverent. Newtonus autem respondit, se non edidisse Commercium Epistolicum |D| Leibnitium non admitte debere testem in propria an se non edidisse Commercium Epistolicum, ne \ne dum/ literas {illeg} quasdam quas ipse habuit in propria custodia \archivis suis habuit/ in lucem prod|t|u{illeg}|l|isse ut in Commercio edito pulicarentur, ne se{u} testem faceret \n|s|e s{illeg}p{illeg} enim seipsum/ in propria causa \testem {illeg} faceret, nolluisse &/ sunt D. Leibnitium {illeg} in propria causa testem admitti \non/ debere. [Siquid in Archivis R. Societatis adhuc lateret quod luce dignum, hoc imprimi posse in Transactionibus Philosophicis] Et subinde in hujus rei testimonium is Epistolas duas protulit, utram ad seipsum scriptam, unam a D. Wallisio Apr. 10 1695, alteram a D. Leibnitio Martij $\frac{7}{17}$ 167|9|3, quæ postquam \ab ijs/ {illeg}ot{æ} sunt {illeg} examinatæ sunt ab ijs qui scrip \& pro genuinis agnitæ ab ijs sunt ab ijs qui/ utrius autographa ex scribendi formulis noverant, et pro genuinis agnitæ, lectæ sunt coram Societate Regia et subinde in {illeg} Societatis Archivis repositæ. Addidit re{illeg} insuper Newtonus quod siquid in Archivis R. Societatis adhuc inveniri posset luce dignum, hoc in Transactionibus Philosophis|c|is imprimi posset, & siqu{illeg} Epistolæ quas Et siquid amplius D. Leibnius {sic} imprimi vellet vel ex suis archivis haberet quod imprimi vellet, deberent autographa prius examinari. Et examinari quidem possent coram Regia Societate {illeg} et subinde vel in Phil. Transactionibus vel alias imprimi si Leibnitio placeret Sed nihil missum est ut examinat|r|etur, nihil amplius in archivis R. Societatis \nondum editum/ inventum \est/ ad hanc controversiam spectans, quod luce qu{illeg} \quod vel pro Leibnitio vel contra Newtonum faceret./

Pag 207 lin penult. After 393 & 396 add. And in the year 391 D^r Halley & M^r Ralphson borrowed this Book in MS \of Quadraturs at Cambridge/ & carried it with them to London as D^r Halley hath declared before the R. Society & M^r Ralphson has affirmed in his Treatise of              And therefore th{illeg}|is| Book was in MS in those days & M^r Newton had not then forgotten {illeg} the method of second fluxions.

And M^r Leibnitz himself Ego vero libentur,, saith he, ingentia Newtoni merita oblatis . . . . . . . satis intellexi. mihi mature transmisit. And a little after he added Quam [meth{o}dum] ante Dominum Newtonum et me - - - - - Compounding Hitherto therefore Hitherto there Here M^r Leibnitz allowed that \before he published his meth{ors}/ he understood by his Lette M^r Newtons Letter \of 1676/ that he \M^r Newton/ could draw tangents without taking away surds & by his Principia \publi published 1687/ that he had gone much further in this Ans method. And M^r Newton had told him in those Letters that his methods \Analysis/ extended th to y^e quadrature of Curves & \to/ inverse Problemes of tangents & others more difficult: And M^r Newton \& {illeg} that he had the method five years before or above &/ had demonstrated the elements of his calculus in the Principia, D{illeg} & described & that he had those this co He had to And before M^r Newto he began to lay any claim to the Method M^r Newton had told him that had wrote a tract concerning that meth it five years before the writing of those Letters. {illeg} And a copy of a Letter dated 10 Decem 1672 concerning the method was sent him by M^r Oldenburg as was said above. Hitherto therefore M^r Leibnitz did not pretend to be the first inventor, but only excused himself for not having made a fuller acknowledgm^t of what M^r Newton had done. He did not begin to put in a claim of first inventor till after the death of D^r Wallis. the last

The occasion of inserting this intimation D^r Wallis signified also to M^r Leibnitz in his a Letter dated Decem 1. 1696 \& published by D^r {illeg} in the third Volum of the Doctors Works./. {sic} Cum Præfationis (præfigendæ) postremum folium erat sub prælo ejus typos jam posuerant Typothetæ; me monuit amicus quidam (harum rerum grarus) qui peregræ fuerant, tum talem methodum in Belgio prædicari tum illam cum Newtoni methodo fluxionum quasi coincidere. Quod fecit ut (translatis typis jam positis) id monitum in \ter/ seruerim

p. 199 And in a Letter to M^r Leibnitz dated 1 Decem. 1696, he wrote thus gave this Account of it Cum præfationis – – – – intersererem. And in a Letter to M^r Newton dated Apr 10. 1695, he wrote thus about it. I wish

p. 201 And in his Letter of Aug. 27 1676 he denyed \that such problemes/ what M^r Newton had affirmed concerning the reduction of Problemes to such æquations, & therefore was then an absolute stranger to th{illeg} method equations this method

p. 198. l. 14. And again in his answer to M^r Fatio printed in the Acta Lipsi Eruditorum of May 169 1700 pag 203 lin 21 he allowed \acknowledged/ the same thing.

p. 198 l 31 Method, & illustrated it with examples of drawing tangents & squaring curves, that it could not be difficult.

p. 200 l 27 — so early; did not so much as affirm that he had it so early|ie||r| tho D^r Wallis had {illeg}d him {illeg} publis affirmed that M^r Newton had it ten years before that time or above; allowed

p. 201. l. 13. [And in making this {h}|D|efence he pretended that \in the year 1684/ when he published his method of tangents he knew {illeg} nothing more of M^r] & published it without knowing what M^r Newton had done before him. {illeg} And in making this Defense

p. 203 \l. 5./ – Novices, M^r Leibnitz had prepossessed his country men that he was the first inventor

p. 202 l. 2|3|. represented not only that M^r Leibnitz was the first in fluxions for differences & by consequence taken his method from that of M^r Leibnitz. And this Hitherto he had upon several occasions acknowledged that M^r Newton had found the Method apart. N{illeg}

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$\begin{array}{r}\mathrm{16∟783}.\mathrm{1∟1455}\colon\colon \mathrm{43,817}\phantom{0(\mathrm{0∟000∟000}}\\ 43817\phantom{(\mathrm{0∟000∟000}}\\ 17527\phantom{(\mathrm{0∟0}}\mathrm{43∟817}\\ 2192\phantom{(\mathrm{0∟00}}2\phantom{∟}99\phantom{0}\\ 219\phantom{(\mathrm{0∟0}}40\phantom{∟}827\\ \mathrm{1∟6783}\left)\mathrm{50∟1935}\right(\mathrm{2∟99}\phantom{∟0000}\\ 33566\phantom{0(\mathrm{0∟000∟000}}\\ 166275\phantom{(\mathrm{0∟000∟000}}\\ 151047\phantom{(\mathrm{0∟000∟000}}\\ 185228\phantom{(\mathrm{0∟000∟000}}\\ 151\phantom{(\mathrm{0∟000∟000}}\end{array}$

M^r Leibnitz had hitherto allowed \acknowledged/ upon several occasions that M^r Newton had invented the Method apart

\p. 202 l. 17/ complaining that in these words that what M^r Keil had written amounted to a calumny & moved that y^e R. S. &c    l 21 leave to do so, being now sensible that

allowed that M^r Newton had found the method apart & did not deny – – – then that he also had found the method apart

& when he published it knew little or nothing of what M^r Newton had done. And in making this d|D|efen s{,}|ce| he added —

In calling it injustice to put {illeg} question his candour he

He was bound in justic candour & justice to make good his accusation against M^r Keill that it might not go for a calumny.

He knew that no man can be a witness in his own cause & yet he {a}lls calls it injustice to question his candor. He knew that the R. Society could not in justice condemn M^r Keil without some proof against him & yet he insists an excuses himself fr insists only upon his own candour. He tells & calls it injustice to expect that he should plead defend it. He tells the R. Society that his friends know how he came by the method, but he should have told the R. Society how he came by it. He saith saith that the Editors of the Acta Lipsiensia had given every man his due & in the next words that he & his friends wer had some times shewn that they were willing to beleive that M^r Newton found the method apart. & yet that M^r Leibnits had but without depriving M^r Leibnitz of his right to the invention. He cites the opinion of Hugenius in his behalf {illeg} tho Hugenius knew never had an opportunity to examin the matter He saith that

He had seen three of M^r Newtons

M^r Newton had given his opinion in this matter in his Analys Letters of d|D|ecem. 10 1672 Iune 13 1676 & Octob 24 1676 copies of all w^ch had been sent to M^r Leibnitz. {illeg} And thereupon therefore M^r Leibnitz gave him seven years time to print his method. & refused to tell how he came by the method

$44⁤\frac{3}{4}.$

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$\begin{array}{rr}\text{♍ }23.1\phantom{0\text{ mens.}}& \phantom{0}\\ \text{♎ }\phantom{0}8.15\phantom{\text{ mens.}}& 15.14\\ \phantom{\text{♍ }}13.8\phantom{0\text{ mens.}}& 20.\phantom{0}7\\ \phantom{\text{♍ }}18.44\phantom{\text{ mens.}}& 25.43\\ \phantom{\text{♍ }}23.2\phantom{0\text{ mens.}}& 30.\phantom{0}1\\ \phantom{\text{♍ }}27.54\phantom{\text{ mens.}}& 35.\phantom{0}0\\ \text{♏ }\phantom{0}3.00\phantom{\text{ mens.}}& 40\phantom{.00}\\ \phantom{\text{♍ }}13.\phantom{00\text{ mens.}}& 50\phantom{.00}\\ \underline{1722⁤\frac{1}{2}}\phantom{00\text{ mens.}}& \phantom{0}\\ 574.2\phantom{0}\text{ mens.}& \phantom{0}\end{array}$

$\begin{array}{rr}3906\phantom{.6}\mathrm{m}\phantom{⁤\frac{1}{2}}\text{Iul. }& 539\phantom{⁤\frac{1}{2}}\text{comp}\phantom{\text{l.}}\\ 5579\phantom{.6\mathrm{m}⁤\frac{1}{2}}\text{Dec}\phantom{\text{. }}& 1680\phantom{⁤\frac{1}{2}}\text{compl}\phantom{\text{.}}\\ {1723.6}^{\mathrm{m}}\phantom{⁤\frac{1}{2}\text{Iul. }}& 1141\phantom{⁤\frac{1}{2}\text{compl.}}\\ {574.6}^{\mathrm{m}}⁤\frac{1}{2}\phantom{\text{Iul. }}& 570⁤\frac{1}{2}\phantom{\text{compl.}}\\ \phantom{0}& 1680\phantom{⁤\frac{1}{2}}\text{comp}\phantom{\text{l.}}\\ \phantom{0}& 574⁤\frac{1}{2}\phantom{\text{compl.}}\\ \phantom{0}& \overline{1105⁤\frac{1}{2}}\phantom{\text{compl.}}\\ \phantom{0}& \underline{1149\phantom{⁤\frac{1}{2}}}\phantom{\text{compl.}}\\ \phantom{0}& 531\phantom{⁤\frac{1}{2}}\text{compl.}\end{array}$

For S^r Isaac Newton

Motum Cometarum Leibnitius non attigit, Planetæ et Cometæ ijsdem legibus revolvuntur apud Newtonum. Vortices ali\cu/bi harmonice alibi non harmonice moveri Leibnitius pro lubitu finxit: harmonicæ circulationi vortices satellitum Saturni Iovis at|c| Terræ plane obstant: Motibus Cometarum Vortices Leibnitianæ non favent. Motum harmonicum Planetarum Leibnitius non probavit sed in Artic Tentaminis Artic 6 assumpsit: Newtonus motum harmonicum corporum omnium in centrum immotum attractorum demonstravit. Motum circulationis et motum paracentricum Leibnitius nunc a diversis causis nunc ab eadem deducit: Newtonus utrum ab eadem causa semper deducit. Vim centrifugam sinui verso anguli circulationis proportionalem esse Leibnitius {illeg}|i|n T{a}|e|ntaminis Artic 11 assumpsit contra veritatem; et in motum paracentricum Planetarum a differentia virium centrifugarum & centripetarum (quæ nulla est) oriri finxit inde deduxit vim centrifugam mobilis harmonice circulantis esse in ratione radiorum reciproca triplicata (Artic 12) newtonus demonstravit hanc vim mobilis in Ellipsi circa focum harmonice circulantis esse in ratione radiorum reciproca duplicata. \Leibnitius/ Motum paracentricum Planetarum a differentia virium centrifugarum et centripetarum oriri finxit (Artic 15, 21, 25) Newtono differentia illa nulla est. Ex errantibus Articulis 12 et 15 Leibnitius deduxit Artic 19: et Proposiones {sic} Mathematicæ sic inveniri non solent.

Dicunt aliqui D. Leibnitium \in Tentamine suo/ Propopsitiones 19 et 20 Tentamini a falsis Propositionibus falsis \non veris/ /falsis\ (nempe 14, 12 & 15) \per calculū s{illeg}/ deduxisse \su{illeg}|u|m/ {illeg} [C|A|t per calculum errantem Propositiones Mathematicæ inve{nr}i non solent] Ac talis calculus ad Propositiones \quidem/ prius invente|a|s aptam potuit non autem invenotrem constituere.

Dicunt aliqui quasdam \falsas est|s|e/ Tentaminis Propositiones quasdam, nempe 11, 13 & 15, & D. Leibnitium ab his per calculus|m| suum deduxisse Propositiones 19 et 20 \ejusdem Tentaminis/. Talis autem calculus ad Propositiones prius inventas quidem inventas aptari quidem potuit, non autem inventorem constituere.

Gravitas in corpore cadente et sp{a} tempore $\sqrt{\mathrm{F}}\mathrm{G}$ spatium FG generante cadendo describente generat velocitatem qua duplum illud spatium eodem tempore describi posset id est velocitatem $\frac{2\mathrm{F}\mathrm{G}}{\sqrt{\mathrm{F}}}$ seu $2\sqrt{\mathrm{F}}\mathrm{G}$; at in corpore &c $\frac{\mathrm{H}\mathrm{N}}{\mathrm{C}\mathrm{N}+\mathrm{F}\mathrm{G}}-\frac{\mathrm{C}\mathrm{F}}{2\mathrm{F}\mathrm{G}}${illeg}= $\frac{\mathrm{H}\mathrm{n}}{\mathrm{C}\mathrm{N}+\mathrm{F}}-\frac{\mathrm{C}\mathrm{F}}{2\mathrm{F}\mathrm{G}}=\frac{\mathrm{a}\mathrm{e}\mathrm{e}}{2\mathrm{n}\mathrm{e}\mathrm{e}+\mathrm{a}{\mathrm{a}}^3\mathrm{o}}=\frac{\mathrm{a}}{2\mathrm{n}}$. $\frac{\mathrm{F}\mathrm{I}}{\mathrm{C}\mathrm{F}}=\frac{\mathrm{a}}{\mathrm{n}}$ Resistentia ad Gravitatem ut $\frac{\mathrm{a}}{2\mathrm{n}}+\frac{\mathrm{a}}{\mathrm{n}}$ ad 1, seu 3a ad 2n. Densitas Medij ut $\frac{3\mathrm{a}}{2\mathrm{n}}$ in $\frac{\mathrm{F}\mathrm{G}}{\mathrm{C}\mathrm{F}\mathrm{q}}$ id est {illeg} ut a. Et velocitas ut L {illeg} {illeg} seu $\frac{\mathrm{C}\mathrm{F}}{\sqrt{\mathrm{F}\mathrm{G}}}$ id est ut $\sqrt{\mathrm{e}}$.

And by those Letters & Papers it appeared to the Committee that M^r Newton had the Method in \or before/ the year 1669 & it did not appear to them that M^r Leibnitz had it before the year 1677.

M^r Leibnitz {illeg} \began/ his Second Letter to D^r Sloan dated 29 Decem. 1711, nor|w^th| these words. Quæ D. Iohannes Keillius nuper ad te scripsit, candorem meum apertius quam ante oppugnant: quem ut ego hac ætate cum h post tot documenta vitæ, Apologia defendam & cum homine docto, sed novo, & parum perito rerum anteactarum cognitore, nec mandatum habente ab eo cujus interest, tanquam pro Tribunali litigem, nemo prudens æquus probabit. That is That is, he tells the Society that they are unjust unless they a ⨳ < insertion from f 178v > ⨳ Thus he declined \to/ maki|e|ng good his accusation against M^r Keil as he was bound to do for avoyding the censure of calumny & told the R. Society that they would be unjust if they should question his candor, that is, if they should not allow him to be a witness in his own cause contrary to the laws all nations, & refused to contend with any body but M^r Newton or those imployed by him, justifying the Acta Lipsiensia against him in the same letter, & pressing him to declare his mind, that is, to retract what he had published in the {A} P{illeg} Introduction to his Principles & \to/ submit to the jus|d|gment of the Editors of those Acta. But those editors have sometimes imployed the pen of M^r Leibnitz \himself/, & the Motto of th sometimes the pens of inferior men & the Motto of the R. Society is NVLLIVS IN VERBA, \of other men of less note/ & by the law of all nations no man can be either Iudge or Witness in his own cause, & by the law of all n the Motto of the R. Society is NVLLIVS IN VERBA|.|, & th{illeg}i{illeg} It lies upon M^r Leibnitz \as well as upon M^r Keill/ to prove{illeg} his assertions. It lies upon him &c < text from f 177r resumes > Thus he de{t}|c|lined making good his charge accusation against M^r Keil |as he was bound to do for avoyding y^e censure of calumny &| refused to contend with any body but M^r Newton or those imployed by him |pressing M^r Newton in the same Letter to give his {illeg} declare his opinion mind & justifying the Acta Lipsiensia against him, &|, & tells the R. Society that they are unjust if they question his candor, that is, if they do not allow him to be a witness in his own cause contrary to the la{w o}f all nations. But the Motto of the R. Society is NVLLIVS IN VERBA. It lies upon him \M^r Leibnitz/ to prove that he had the Differential Method \jam a multo tempore/ before the year 1677. It lies upon him to prove that he had the series of Gregory before he received it from M^r Oldenburg A.C. 1675 \at w^ch time h{illeg}|e| did not {illeg} know it to be his own/ & even before Gregory sent it to Collins A.C. 1671.  < insertion from f 178v >  He has affirmed that (in his Letters of 28 Decem 1675 & 27 Aug. 1676) that the|i|s series for finding the Arc by the Tangent was communicated by him to his friends at Paris in 1673 & was the series whereof he had sometimes writ to M^r Oldenburg, {illeg} It lies viz^t in {y} y^e year 1674 \before December 1675 viz^t in Iuly & October A.C 1674/ It lies upon him to prove that the series w^ch he then wrote of to M^r Oldenburg was this series & not the series for finding the Arc by the sine. In his Letter of October 1674 he has affirmed that he had a method of finding the Arc by the sine & that the same me \in a series of rational numbers/ whether the proportion of the Arc to y^e whole circumference In his Letter of 21 Iune 1677 he wrote was known or not, & that y^e In his Letter of same method gave him \a series for/ the whole circumference in a series. H{illeg} \&/ in his Letter of 12 May 1676 he desired M^r Oldenburg to procure from M^r Collins the method of finding this Series: it lies upon him to prove that he had such a \the/ method in \or before/ y^e year 1674. At his request M^r Newton \in his Letter of 13 {Iu}/ sent him his method of finding that & such like series direct & inverse |& illustrated his method w^th examples of series| & upon the receipt of that Letter M^r Newton Leibnitz pretended that he had found some of those series before it < text from f 177r resumes > It lies upon him to prove that he had a method of finding the series for the Arc by the Sine in the year 1674. It lies upon him to prove that before the receipt of M^r Newtons Letter of 13 Iune 166|7|6 he had some of the series sent him therein with the method of finding them. It lies upon him to prove that he had found out one of M^r Newtons inverse methods of series so early as to have forgot it before y^e year 1676. {illeg} He pretends to \be the inventor of/ the method of series by assuming the terms thereof gradually & determining them gradually by the conditions of the Problem: it lies 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 two after M^r {illeg} Newton had published his Principia Philosophiæ M^r Leibnitz published three papers relating to to {sic} the principal Propositions in that Book pretending that he had found them before the publishing of that book: it lies upon him to prove that he had found them before. He pretended that the Propositions concerning the motion of Bodies in resisting Mediums were for y^e 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 that it. It lies upon him also to prove that by a{illeg}|n| fals \erroneous/ Demonstration he could find out the Proposition that a Body revolving in an Ellipsis & with with a radius \drawn to the lower focus/ describing equal areas in equal times, is attracted towards that focus by a force w^ch is reciprocally as the square of that radius. ♀ < insertion from f 178v >  For \{illeg}/ making himself a coinventor of the method of series He has laid claim to it by a Proposition for transmuting \any one/ figures into \{illeg}|a|n/others equal to it{illeg}. It lies upon him to prove that this method \of Series/ is either general or useful or that any thing can be done by it which cannot be better done without it or that he had it before y^e year 1675 or that he has done any thing of moment by it w^ch was not done before by other æquations where the indices of dignities were fract or surd: M^r Leibnitz in his reply proposed æquations where the indices of dignities were indeterminate & calling these æquations exponential, & \{illeg}/ recons these|n| equations the most perfect \of all others/. It lies upon him to shew that \any/ use that {in q} hath been \made or can/ or may be made of them. < text from f 177r resumes > He has represented that M^r Newton \For making M^r Newtons method of fluxions his own/ He has represented that M^r Newton at first used the letter o for the \given/ increment of x in the vulgar manner for the unifo {ever} given \constant given/ increment of x, w^ch destroys the advantages of the differential method, but at length \after the writing of his Principles/ changed o into $\stackrel{.}{\mathrm{x}}$ substituting $\stackrel{.}{\mathrm{x}}$ for dx: & thereby found the differential of it lies upon him to prove that he ever M^r Newton ever changed o into $\stackrel{.}{\mathrm{x}}$ or doth not use the letter o to this day. |or used $\stackrel{.}{\mathrm{x}}$ for dx or left off the use of the letter o.|

M^r Newton used the letters o in his book of Quadratures {illeg} Analysis \& in his Principia Philosophiæ/ & in his book of Quadratures & still uses it in the very same sense \as at first/. He In his book of Quadratures he used it in conjunction with the {illeg} symbol $\stackrel{.}{\mathrm{x}}$ & therefore did not use \substitute/ the one for the other \use that symbol in ints room./ These symbols \o & $\stackrel{.}{\mathrm{x}}$/ are put for things of a different kind. [The letter o is p signifies a moment of time or a {illeg} small part of any quantity by w^ch time is represented. The marks $\stackrel{.}{\mathrm{x}}$, $\stackrel{.}{\mathrm{y}}$, $\stackrel{.}{\mathrm{z}}$ are not parts generated but velo fluxions or velocities of increase] as has been explained] The symbo \The one is a moment the other a fluxion or velocity. The one i as has been explained above./ When the letter x is put for a quantity w^ch flows uniformly, the let symbol $\stackrel{.}{\mathrm{x}}$ is an unit & the letter o a moment {illeg}|&| \the symbols/ $\stackrel{.}{\mathrm{x}}\mathrm{o}$ & dx agre \signify/ the same moment; but $\stackrel{.}{\mathrm{x}}$ without the coefficient o either exprest or understoot|d| to make it infinitely little is|d|oth not signify a moment. Prickt letters {illeg}|n|ever signify moments, unless when they are multiplied by the letter o moment o either exprest or understood \to make them infinitely little/, & then the rectangles are put for moments.

^[1]M^r Newton doth not place his method in forms of symbols nor confine himself to any particular sort of symbols for \fluents or/ fluxions. Where he puts the Areas of Curves for fluents he puts the ordinates for fluxions & denotes the fluxions by the symbols of the Ordinates. When 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 increases of those velocities, that is, for the second fluxions: instances of w^ch are frequently to be met with in his {ba}{illeg} Principia Philosophiæ: And where he puts the letters x, y, z for fluents, 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 $\stackrel{.}{\mathrm{x}}$, $\stackrel{.}{\mathrm{y}}$, $\stackrel{.}{\mathrm{z}}$, \a{illeg} x{illeg} {illeg}/ And this he doth to this day, as is And this is eviden{t b}y his book of Quadratures where he uses \for fluxions/ prickt letters in the first {Pr}oposition, the Ordintes of Curves in the second & last Proposition & other symbols in solving several Problems in the Introduction. M^r Leibnitz has no \symbols of/ fluxions in his method, \& therefore/ all M^r Newton's symbols \of fluxions/ are the oldest in the kind. M^r Leibnitz began to use the symbols of moments dx & dy in the year 1677. M^r Newton used the rectangle moment o & the rectangles under this Moment & the fluxions ever \has represented|s| moments by y^e fluxions & the moment o & has done rectangles under y^e fluxions & the moment{illeg} o/ \& has done so ever/ since the writing of his Analysis, w^ch was eight years before or above \at least 45 years ago/. M^r Leibnitz has used the symbols $\int \mathrm{x}$, $\int \mathrm{y}$, $\int \mathrm{z}$ for the summs of the ordinates ever since the year 1686: M^r Newton represented the same thing {illeg} by inscribing the Ordinate in a square or rectangle in his Analysis written 17 years before or above. 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 he uses \in his calculus there is/ but one infinitely small quantity signified by the Letter o. It is more natural & gemetrical because founded upon the primæ quantitatum nascentium rationes which have a being in Geometry whilst indivisibles upon w^ch the Differential method is founded have no being either in Geometry or in Nature. There are rationes primæ quantitatum nascentium, but not quantitates primæ nascentes. Nature generates 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 to generate a solid. But the summing up of indivisibles to generate & {sic} area or solid was never yet admitted into Gem|o|metry. M^r Newtons method is also fo a greater extent being adapted either to the ready finding of out of 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 recouse {sic} to converging series & thereby his method becomes incomparably more universal then that of M^r Leibnitz w^ch is confined to finite equations. & when 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 quamproxime by drawing a Curve line through any number of given points, & thence deduces the solution of the Probleme. And to this degree of perfection M^r Newton had brought his method |sometime| before the year 1676, as appears by his Analysis & his Letters of 10 Decem 1672, & 13 Iune \1676/ & 24 Octob 1676. And when he wrote his Principia Philosophiæ, had re he had recourse upon all occasions to this method & most frequently to that part of it w^ch is conteined in his book of Quadratures And because by the law of the Ancients Propositions invented by Analysis were not admitted into Geometry till they were composed & demonstrated synthetically <178r> he w the Propositions which he found out by his Analysis, he composed which \that the systeme of the heavens might be founded upon good Geometry. And this/ makes it now difficult for unskilful men to see the Analysis by w^ch they|os|e \Propositions/ were found out. But this Book \compounded with the Tentamen de motuum cœlestium causis/ will be \prove/ a lasting {d}emonstration to all posterity that M^r Newton when he wrote it was master of an Analysis w^ch carried him through many such difficulties as M^r Leibnitz would have stuck at.

Phil. Trans. p. 206.

resist$=\mathrm{A}{\mathrm{B}}^{\mathrm{q}}+\mathrm{C}{\mathrm{B}}^{\mathrm{q}}$. {illeg} Grav \ut/ AC.

In the 19^th Article of his Tentamen de Motuum cœlestium causis he has pretended to the invention of M^r Newtons Proposition that if a body moving \revolving/ in an Ellipsis describes & with a radius drawn to the

His differential method of Tangents published \at Leip{iu}/ in the year 1684 is nothing es{illeg}|ls|e then D^r Barrows differential method improved so as not to stick at factions {sic} & surds & disguised with new notation & a new name without \wer/ acknowledging any part of y^e method to be D^rBarrow's, He ought [It lies upon him in point of justice to tell the word how much of the Differential method is D^r Barrows.] or that the \Author/ had received any light into it from M^r Newtons Letters. It lies upon him to prove that what he then published was intirely his own without being obliged to either of them for any thing

All the All the avertions \pretensio{illeg}|n|s/ above mentioned tend to if they be not true tend to diminish the right of invention due to D^r Barrow \M^r Greg. &/ M^r Newton & \to/ let in M^r Leibnitz for a share. And by the M^r Leibnitz has hitherto claimed this share without any proof. His candor cannot make him a witness in his own cause {illeg}he must cla for himself. By the laws of all nations he must claim prove his claim or quit it, & this he ought to do {illeg}a{illeg} w^thout delay in point of candor & justice.

It lies upon him to prove that

M^r Newton at his request sent him two inverse methods of Series, \A.C. 1676/ & he understood then w^th difficulty & then pretended \wrote back in Iune/ that he had found one of them so long ago \before/ as to have 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. He pretends to be the first inventor of that method of series w^ch consists in assuming the terms of a series & determining them &c

In his letter of 27 Aug. 1676 he denyed that M^r Newtons method of series extended to the invers me Problems of tangents & when M^r Newton replied that it did extend to them \extend to them/ he answered that M^r Newton spake of \a solution by/ his method of series but {illeg} whereas he d had desired a Geometric solution of such P those Problems: \this implies that he knew th{a} before that it might be solved by the method of series/ it lies upon him to prove that he had desired a Geometric solution & M^r Newton had been speaking of any other solution then by the method of series. \knew it./

It seems therefore that as he learnt the method of fl{u} differential method by means of M^r Newtons aforesaid three Letters {illeg} \compared with/ D^r Barrows method of Tangents: so ten years after \when the Principia Philosophiæ came abroad/ he improved his knowledge in these matters by trying to extend this method \calculus/ to the prima Proposit principal Propositions in that Book. And publish{illeg} \&/ by this means composed the said three Tracts. And yet in the end of the first Tract he published the It lies upon him in candor to acknowledg \And th {sic}/ For the Propositions conteined in them (errors \& trifles/ exceps|t|ed) are M^r Newtons, being published by him in other forms of words before. And yet M^r Leibnitz has published \them/ as invented by himself before M^r Newtons published them \long before/|.| \Pricipia Philosophiæ came abroad./ For in the end of the first Tract he re \For/ In the end of the first Tract he represents that he invented them all before M^r Newton's Principia \Philosophiæ/ came abroad & some of them before he left Paris, that is before the end of the year 1676. And the second Tract

Here he acknowledges \after represents allows/ that the fundamenta Geometrica in quibus maxima consistebat difficultas were layd by \first |now| laid by him in/ what he \{now}/ published in this second Tract, but he should have acknowledged that they were laid \{f} more fully laid before/ by what Newton published before in his Principles. Here he acknowledges \allows/ that he had by this Tract opened vias quasdam novas satis antea impeditas, but he should have acknowledged that M^r Newton had opened them \more fully/ before by his Principles. Here he allows that {illeg}|a|ll that he had now published answered to the calculus summarum & differentiarum, tho it {illeg}{e}{illeg} was here expressed communibus quoad licuit verbis, {illeg}|i|n vulgar words with|out| calculations & \but in candor/ he should \also/ have acknowledged that what M^r Newton had published in his Principles answered to the calculus fluxionum tho writ communibus \quod licuit/ verbis [in the language of the vulgar as far as could be \done/ conveniently. It remains therefore th in point of candor & justice that he now acknowledge at length do justice to M^r Newton by acknowledging these things that he was the first who laid the fundamenta Geometriæ in quibus maxima consistebat difficultas & opened vias quasdam novas satis antea impeditas, & that the composition of his Principles tho written {as} communibus quoad licuit verbis expressa answers throughout to this calculus infinitesimalis \communibus quoad licuit verbis, answered in all things to his the calculus infinitesimalis./

Here he represents pretends that the fundamenta Geometrica in quibus maxima consistebat difficultas were now \first/ laid by him & in this {illeg} \self in this Tract/ & that he had \himself had in this \other/ {illeg}|Tr|act/ opened vias quasdam novas satis antea impeditas. [& that what he ha now published answered to the calculus infinitesimalis tho it writ written communibus quoad licuit verbis]: & yet [he knew & ought to have acknowledged that M^r Newton had done all this before in his Principia Philosophiæ. He cannot pretend \acknowledge/ that he did those thing laid those foundations & opened those new ways before because he by this Tract Tract &S] since] & \yet/ since \And yet/ M^r Newton's Principia \Philosophiæ/ came abroad almost two years before & gave occasion to the writing of this Tract, & conteing conteins all these Principles & \all these/ new ways. And therefore M^r Leibnitz when he published that Tract knew & ought to have acknoledged that M^r Newton was the first who laid the fundamenta satis antea impe{illeg} Geometrica in quibus maxima consistebat difficultas & opened \the/ vias novas satis antea impeditas. [And \For in his Answer to M^r Fatio he allows/ if he objects that this|e| Principles were written \not Analytically but/ Communibus quoad licuit verbis, so was this Tract; & \yet/ all things in it answered to the new Analysis of infinites he tells us the same thing of this \his own/ Tract & yet affirms that in all things it answers to the new Analysis] \He makes no objection against the Book being communibus quo written not in the way of Analysis but communibus quod {sic} licut {sic} verbis. For his own Tract was written in the same manner & yet in all things answered to the new Analysis./ In his Answer to M^r Fatio he tells us that Nemo Ante allows y^t No man before M^r Newton specimine publice dato [hanc Methodum] se habere probavit, [proved that he had this method by publishing a specimen thereof \made publick/. And if he knew that the Principia Philosophiæ were a specimen \specimen/ of this method & that M^r Newton was the first who had proved by the published \such/ a specimen of this method & thereby proved that had the method, he ought in candor & justice where h to have acknowledged \this/ when he published those Tracts & ought still to have acknowledged it.] For silencing D^r Wallis & M^r Fatio he could was forced to allow that the Infinitesimal method was M^r Newtons {illeg}ted was the first \M^r Newton was the first \who had/ Inventor|ed| of the/ the infinitesimal method & {illeg} had \the first who had/ proved by a publick specimen \(meaning the Book of Princip)/ that it was his \he had this method/: {illeg} at other times he calls the infinitesimal \method/ his own & at length begins to pretend that M^r Newton had the method from him. What he acknowledged in writin his Answers to D^r Wallis & M^r Fatio, he ought {illeg} & in his Letter to M^r Newton dated March $\frac{7}{17}$ 1693 he ought {illeg} in candor & justice \still/ to acknowledge upon all occasions.

when he wrote those Tracts therefore he was but a learner, & this he improved his knowledge by M^r Newtons Principles & this he ought \in candor/ to acknowledge.

He pretended indeed that to D^r Wallis that he had added some things to the infinitesimal Analysis of M^r Newton, & particularly the infintesi Differential Æquations: but by their Letters written in y^e year 1676 its abundantly manifest that M^r Newto Leibnitz had th knew not \did not then know/ how to reduce Problems to differential Equations & M^r Newton did then know how to do it, & tho he did not use the diffential {sic} characters w^ch M^r Leibnits invented afterwards.

M^r Newton in his Letter of Octob 24 1676, wrote – – – – wrote his said Letter.

D^r Wallis interpoled the indices of Dignities 0, 1, 2, 3, 4, 5 &c by this series $0.\frac{1}{2}.1.1⁤\frac{1}{2}.2.2⁤\frac{1}{2}.3.3⁤\frac{1}{2}\mathrm{\&c}$. And M^r Newton introduced into his Analytical computations fract the fract, surd, indefinite & negative indices of d|D|ignities; & in – – – – – its usefulness to the world

It lies upon M^r Leibnitz \also/ to make a publick acknowlegment of his Receipt of M^r Oldenburghs letter of Apr. 15. 1675 – – – – – – series as his own

It lies upon him also to make – – – – Method of fluxions.

It lies upon him also to tell the World – – – – from M^r Collins

It lies upon him also to tell the world what was the Method – – – from M^r newton.

And whereas M^r Newton sent him at his own request – – – name of candor & justice – – – – Letter of 10 Decem. 1672 concerning it.

The Editors of the Acta Eruditorum for Iune 1696 – – – – – beg his pardon.

When M^r Leibnitz first published – – – – of candor & justice.

It lies upon him also to satisfy the world why in his Answers to D^r Wallis & M^r Fatio – – – Guilt of that crime

It has been said that the Royal Society – – – – by Persons of Note. And in the mean time – – – – – expulsion to defame them.

The Philosophy of M^r Newton – – – – part of Europe against him.

It is true – – – – rail without arguing.

|21| M^r Leibnitz in his Letter to D^r Sloan dated 29 Decem 1711 retu returned this answer to D^r Keill. Quæ D. Ioannes Keillius nuper ad te scripsit, candorem meum apertius quam ante oppugnant: quem ut Ego hac ætate, post \tot/ documenta vitæ, Apollogia defendam, & cum homine docto, sed novo, et parum perito rerum ante octarum cognitore, nec mandatum habente ab eo cujus interest, tanquam pro Tribunali litigem, nemo prudens æquus probabit. &c. Thus M^r Leibnitz endeavoured to enga{illeg} strip M^r Newton {o} {sic} of \all/ his voluntary friends, & engage him in person w^th [either \to/ he wrote himself or employed \to desired & authore{illeg}{illeg}/ a friend to do it for him] & then to set his own friends upon him, as appeared by the consequence. D^r Wallis & M^r Fatio{illeg} had wro|i|te|n| |in| for him |in| in his behalf \of M^r Newton/, but they did it voluntarily without being desired by M^r Newton to do it, \to write/ & therefore M^r Newton \Leibnits/ could not fall out with \him/ for w|t|hat they did {illeg} D^r Keill did the like by two Letters & made him \in such a manner as made M^r Leibnitz/ cry out both times: But the \Doctor/ had no \wanted/ authority from M^r Newton, & therefore M^r Leibnits would not enter the lists \engage/ with him.|,| M^r but challenged M^r Newton himself, & then fell upon him with h \when the Commercium Epistolicum came abroad/ withdrew, pretending that he had not seen that Commercium Epistolicum \Book/, & employin|ed|g (or pretendin|ed|g to employ) a very great \& candid/ Mathematician to examin it {illeg}, & another nameless person to ruffle \Here to assault/ M^r Newton |with hard words.| And by these little Arts & the tot documenta vitæ above described & tells in {illeg} having prepared the way to this engagement by telling And by these little \c{illeg}ious/ Arts & the tot documenta vitæ above mentioned \described/ you \may/ judge whether it be more prudent & just to expect that this|e| \famous man/ should defend \maintain/ /defend\ his candor \or to place this|e| candor above a{ll} humane of so great a man abo/ or to allow him \so great a man so famous a man/ /him\ to be both witness & Iudge in his own \cause/ contrary to the prudence justice & known laws of all nations \& just to expect that he should defend his/ candor, or to place the candor of so great a man \any prudent & just man will place him above/ above {sic} all humane judicature & allow him to be both witness & judge in his owne cause contrary to the laws of all nations prudence justice & laws of all nations, Nosce teips or not rather expect that he should defend his candor.

Pag. 208. lin. 8. after the words [will vanish] add this Paragraph. |this sentence. But the like Objection made against M^r Leibnitz \by D^r Keil/, namely that before the publishing of the Principia Philosophiæ M^r Leibnitz was unskilled in the method of second & third differences, & has| On the other hand, when M^r Leibnitz Newtons Principia came abroad & gave occasion to M^r Newton not \yet/ sufficiently learnt it \wants an Answer/ By the Letters of M^r Newton & the method of Tangents Before the |of| D^r Barrow he fell into the Calculus differentialis, & by the Principia Philosophiæ received further light into it, but is not yet a Master therein.

In the Commercium – – – – without the assistance of M^r Newton. In his second Paper \[In his second Paper/ The Sixt Article of his second Paper, the Proposition \is erroneous viz^t/ that the motion of a projected body is compounded of the motion which the body would have by gravity alone without projection & the motion which it would have by projection alone without gravity. And the error arises from his not knowing how to argue well about 2^d & 3^d Differences. And \in/ his third Paper he|i|s 10 11^th & 12^th Propositions In his third Paper he founds his infinitesimal Analysis upon his {th} the 10 11^th & 12^t Propositions of that Paper measures the conatus centrifugus by the angle of contact \or versed sine of the arc descri/ as if the body revolved about the center of the Orbs crookedness. perpendicular dix{illeg} upon it \drawn/ to y^e tangent from y^e end of the arch des{illeg}cribed in a given momen moment of time] \3/ |2| By his errors in this Proposition D^r Keil has shewed that when he wrote thise three Tracts he did not well understand the ways of working in second Differences. And this is further manifest by his 10^th 11^th & 12^th A Propositions of this Paper or \third/ Tract. For these he lays down as the foundation of his infinitesimal Analysis about centrifugal & centripetal forces.|,| And \&/ proposes \the first of/ them with relation to the center of curvity but uses them \it in the two next/ with relation to the center of center of circulation. And by confounding these two centers with one another in the fundamental Propositions of this calculus, it is manifest that he was hitherto unacquainted with the manner of arguing about 2^d Differe centrifu{illeg}|g|al & centripetal forces by the infinitesimal method, & was now attempting to learn it. \by \trying to/ adapting {sic} a calculus to M^r Newtons Propositions/ And the|i||s| thing is further confirmed by his atte{illeg}p the sixt Article of this|e| second of these his three Papers o Tracts. For that Article is an erroneu|o|us Proposition & the error arises from his not knowing how to argue well about second \& third/ differences. By M^r Newton's Letters he learnt to improve D^r Barrows method of Tangents & there in the year 1677, [& \then/ gave the name \of the Differential method/ to this improvement,] & ten years after upon the publishing of the Principia Philosophiæ tried to extend this method to the P{illeg} Propositions conteined in that Book: & in the end of the first T of the first published these three Tracts upon the chief of those Propositions not in his usual \as invented by himself/ representing in the end of the first Tract that he found them all before M^r Newton's Book came abro{illeg}|ad|e, A & some of thē even before he left Paris, w^ch {illeg} & concludes|in||g| the second Tract with these words: Multa ex his deduci possent praxi accommodata, sed nobis nunc fundamenta Geometrica jecisse suffer|c|ere{illeg}|it|, in quibus maxima consistebat difficultas. Et fortassis attente consideranti vias quasdam novas satis antea impeditas apperuisse videbimur. Omnia autem respondent nostræ Analysi Infinitorum, hoc est calculo summarum & differentiarum (cujus elementa quædam in his Actis dedimus) communibus quoad licuit verbis hic expresso. \But what he here ascribes to himself he should in justice have ascribed to M^r Newton/ [M^r Newton had writ his Principles communibus verbis: M^r Leibnitz immitated him in these Tracts, tho he met in these Tracts \in writing this Tract communibus verbis/ tho he had laid aside this way of writing long before. And tho he wrote communibus verbis yet he & wrote nothing more (of truth) then what M^r Newton & \had/ written in other words before, yet he calls his performance fundamenta Geometrica \a se jacta/ in quibus maxima consistebat difficultas & said that he had opened vias quasdam novas & satis antea impeditas, & that his performance answered in all this|n||gs| to his infinitesimal Analysis.] Thus he made the Germans in those days beleive that he had laid was the first inventor of all this, [& now with the same sort of candor he tells them that the fundamenta Geometrica in quibus maxima consis the Principia Philosophiæ doth \did/ not contein \lay/ {illeg}|th|e fundamenta Geometrica in quibus maxima consistebat difficus|l|tas, nor \opened/ the vias novas & satis antea impeditas nor answers to the infinitesimal Analysis, but is written in the \vulgar/ way of the Ancients without the help of the|i||s| infinitesimal Analysis.] \Whereas/ He ought in those days to have acknowledged \in those tracts/ that his book M^r Newton by |in| \his Letters & book/ his Princip|i|le|a| \Philosophiæ/ had laid the fundamenta Geometrica in quibus maxima consistebat difficultas, & \had/ opened the vias novas & satis antea impeditas & in \that the composition of this Book in/ all things answered to the infinitesimal method Analysis described by him ten \many/ years before or above in his three Letters above mentioned And since he has yet acknowledged all this, he {ou}ght \lies upon him/ now to acknowledge it in candor & justice to acknowledg it \make an {illeg} express & full/ <180r> full acknowledgment therefor] & further described in the second Lemma of the second book of these Principles & in the Scholium upon that Lemma, & illustrated with a|n| plain example \in demonstrating the 14^th Proposition of that book & with another example/ in solving the 31^th Probleme of the first b|B|ook & with another \good/ Example of solving in the Scholium upon the 103^d Proposition of the same Book|.| & manifested And since he has not yet acknowledged these things but \his on the contrary has/ of late began to deny thi{s}|em|: it lyes upon him {now} in point of candor & justice to make an express & full acknowledgment thereof.

4

|15| The Editors of the Acta Eruditorum, \in Iune 1696/ in giving account of M^r the two first Volumes of the opera mathematical \works/ of D^r Wallis, use thes{illeg} wrote thus in the style of M^r Leibnitz. Cæterum ipse Newtonus non minus candore quam præclaris in rem mathematicam insignis meritis in{illeg}|s|ignis, publice et privatim agnovit, Leibnitium tum cum (interveniente celeberrimo Viro Henrico Oldenburgo|i|o Bremensi, Societatis Regiæ Anglicanæ tunc Secretario) inter ipsos (ejusdem jam tum Societatis Socios) Commercium intercederet), id est jam fere ante annos viginti et amplius Calculum suum differentialem, series infinitas, et pro ijs quo Methodos generales habuisse; quod Wallisius, in Præfatione operis|u|m, factæ inter eos communicationis mentionem faciens, præterijt, quoniam de eo fortasse non satis ipsi constabat. Cæterum Differentiarum consideratio Leibnitiana cujus mentionem facit Leibnitius Wallisius (ne quis scilicet, ut ipse ait, causaretur de Calculo Differentiali nihil ab ipso dictum fuisse) meditationes aperuit, quæ aliunde non æque nascebantur. &c. ‡ < insertion from the bottom of the page > ‡ \M^r Leibnitz here pretends that M^r Newto had therefore seen the Preface of D^r Wallis/ where M^r Newton is said to have explained to \[/M^r Leibnitz) {illeg} in the year 1676 the method of fluxions found by him ten years before or abou|v|te & in answer thereunto, pretended only that M^r Newton \had/ acknowledged only \& D^r Wallis should have acknowledget had he known it/ that he (M^r Leibnitz) had the . . . . . Ordinatarum] differential method in the year 1676 or before: whereas M^r Newton acknowledged nothing more th{illeg}|e|n what he still acknowledges, viz^t that M^r Leibnitz had it when he wrote his Letter of dated 21 Iune 1677. And this confirms me in a|t|he opinion \a suspicion/ that M^r Leibnitz in that Letter used the words Et jam above mentioned Et jam a multo tempore &c, with a designe to make his \the Germans/ beleive that he had found the differential method long before the writing of that Letter, & by consequence before the writing of the Letters w^ch passed between him & M^r Newton in the year 1676, & that he has been evers since carrying on a designe to make this be beleived And yet b its \very/ certain by what we shewed above that he himself knew that he did not invent it before the {illeg} his return from France through England & Holland into Germany in the year 16 end of the year 1676. In his Answer to M^r Fatio printed in Acta Eruditorum anni 167 1700 pag 203 he writes: Newt Ipse [Newtons|u|s] scit unus omnium optime, satis indicavit publice Cum sua Mathematica Naturæ Principia publicaret Anno 1687, nova quædam inventa Geometrica quæ ipsi communia mecum fuere, NEVTRVM LVCI AB ALTERO ACCEPTÆ, sed meditationibus quem suis debere, et a me jam decennio ante exposita fuisse [i.e. anno 1677] exposita fuisse. I And yet M^r Newton \in his Letter of 24 Octob. 1676 did represent that the ground of his method was sufficiently obviou was suff/ < insertion from f 179v > iciently obvious by what he had written, & < text from the bottom of the page resumes > never did allow that M^r Leibnitz found out that Method without receiving light into it from the \three/ Letters sent t above mentioned sent to him by M^r Oldenburg, in the years 1676 & 1677. M^r Newton \indeed/ out of an aversness from disputes forbore to contend with abo enter into a contention with him about it. \complain/ & M^r Leibnitz has abused his candor by pretending that he allowed that M^r Leibnitz found out therby pretendin|ed| not only that he found the method apart before the writing of the Let without receiving any light into it from M^r Newton & even before the {illeg}t{illeg} he received any of M^r Newton's Letters; {illeg}|b|ut also that M^r Newton himself has allowed all this. And for abusing M^r Newtons candor in this manner, he ought in justice to beg his pardon. < text from f 180r resumes > Here M^r Leibnitz pretends that M^r Newton \had therefore seen the Preface of D^r Wallis (in the Scholium to the second Lemma of his Principles)/ acknowledged that he (M^r Leibnitz) had the differential method in the year 1676 or before: whereas Newton acknowledged nothing more then what he still acknowledges, {illeg}t viz^t that he had it in the year 1677 when he wrote his Letter dated 21 Iune 16 1677. And this confirms me in m|th|e opinion that it was with a designe to make himself the oldest inventor that he wrote in the beginning of that Letter: \Clarissimi Slusij methodum Tangentium nondum esse absolutam Newtono assentior;/ Et jam a multo tempore s{it} rem tangentium generalis tractavi scilicet per differentias Ordinatarum....

$\begin{array}{c}\phantom{-}4.10\phantom{\mathrm{.}}0\\ -\phantom{4.}11.3.\\ +\phantom{4.}1.1⁤\frac{1}{2}\end{array}$ and the result, not related in any clear visual way to the sum: $3.19.10⁤\frac{1}{2}$

\Philos. Tra{nsacti}/ p. 210 after lin. 20 add

|9 p. 215| And whereas M^r Newton sent him at his own request a method of Regression, w^ch upon the first reading he did not known to be his own, nor understood it; but so soon as he understood it, he claimed it as his own by pretending that he had found it long before & had forgot it, as he perceived by his old papers: it lies upon \him/ in point of candor & justice, either to prove that he {illeg} was the first inventor of this method, or to renounce his claim to it for preventing future disputes

|10 p. 215| M^r Leibnitz in his Letter to M^r Oldenburg dated 3 Feb. 167$\frac{2}{3}$, claimed a right to a \certain/ property of a Series of numbers natural pyramid{a} triangular pryamidal, triangulo-triangular &c: & to make it \his/ own represented that he wondred that M^r Paschal should omit it. M^r Paschal in his book entituled Triangulum Arithmeticum should omit it. That book was published in the year 1665 & conteins this property of the series & M^r Leibnitz has not yet done him the justice to renounce his claim to this property & acknowledge M^r Paschal the first inventor.

|11 p    | He is also to renounce all right to the Differential Method of Mouton as second Inventor. For second Inventors have no right. The sole right is in the first Inventor untill another finds out the same thing apart. In w^ch case to take away the right of the first Inventor & divide it between him & that second other would be an Act of injustice.

|12 p. 215| In his Letter to D^r Sloan dated 29 Decemb. 1711, he has told us that his friends know how he came by the Differential Method. It lies upon him in point of candor openly & plainly to tell the world how he came by it & without further hesitation to tell \satisfy/ the world how he came by it.

|13 p. 216| In the same Letter he has told us that he had this method above above nine years before he published it, & it follows from thence that he had it in the year 1675 or before. And yet its certain that he had it not when he wrote his Letter to M^r Oldenburgh dated 27 Aug. 1676 wherein he affirmed that Problems of the Inverse Method of Tangents & many others could not be reduced to infinite series nor to Equations or Quadratures. It lies upon him therefore in point of candor to tell us what he means by pretending to have found the Method before he had found it.

In his Letter of 21 Iune 1677 he

|14 p. 216|We have shewed that M^r Leibnitz in the end of the year 1676 in returning home from France through England & Holland was meditating how to improve the Method of Slusius \for Tangents/ & extend it to all sorts of problems, & for this end proposed the making of a general Table of Tangents, & therefore had not yet found out the true improvement: but about half a year after, 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 this improvement long before those days. It lies upon him in point of candor to make us understand that he pretended to this antiquity of his invention with some other designe then to rival t{illeg} & & supplant M^r Newton & to make us beleive that he had the Differential Method before M^r Newton explained it to him by his <181v> Letters of 13 Iune & 24 Octob 1676, & before M^r Oldenburgh sent him a copy of M^r Newton's Letter of 10 Decem. 1672 concerning it.

M^r Leibnitz in his Answer to M^r Fatio published in the Acta Eruditorum

|16.| |p. 219 220| 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 con of his own method, was how to draw Tangents & determin maxima & minima without taking away surds fractions or surds. He certainly knew that M^r Newton's method would do all this & therefore ought in candor to have acknowledged it. After he had thus far explained his own method, he added, that what he had laid down were the Principles of a much sublimer Geometry reaching to the most difficult & noble \valuable/ Problems which were scarce to be resolved without the differential calculus AVT SIMILI or another like it. What he meant by the words AVT SIMILI was impossible for the Germans to understand without an interpreter. He ought to have done M^r Newton justice in plain intelligible language & told his Readers that the Method which he there published extended to such difficult Problemes as were not to be resolved without {illeg}his Differential Calculus {illeg} Differentialis or another calculus invented by M^r Newton some notice of which he had received by his correspondence w^th M^r Newton Oldenburg & which \by reason of performances/ he took to be like his own \& upon which was invented some years before the year 1676/. But on the contrary in his Answer to M^r Fatio published in the Acta Eruditorum for May 1700, he denyes \almost/ all this. His words are: Certe cum elementa Calculi mea edidi anno 1684, ne constabat quidem mihi aliud de inventis New ejus [sc. 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 ejus calculi adhuc expers. Sed majora multo consecutum Newtonum, viso demum libro Principiorum ejus, satis in intellexi. And he is since gone back \even/ from what he acknowledged here, & tells us now that the book of Principles is written in the manner of the Ancients, & hath nothing of the new Analysis in him |it| nor makes it appear that M^r Newton knew any thing of \understood/ these methods when he wrote it. And yet that Book is \very/ full of such Problemes as L|M|^r Leibnitz himself, when he first published his method, called difficillima et pulcherrima etiam mistæ Matheseos Problemata quæ sine Calculo Differentiali AU AVT SIMILI non temere quisquam pari facilitate tractabit. And M^r Leibnitz himself in his Letter of 21 Iune 1677 after he had explained his method & shewed how it readily gave the method of Tangents of Slusius & proceeded without sticking at surds taking away surds, declared himself of opinion that since M^r Netons method did the same things it was of the same kind; especially since both{illeg} the methods also facili|t|ated Quadratures. th{es} Arbitror, saith he, quæ celare voluit Newtonus de Tangentibus ducendis ab his non abludere. Quod addit, ex hoc eodem fundamento Quadraturas quo reddi faciliores, me in sententia hac confirmat, nimirum semper figuræ illæ sunt quadrabiles quæ sunt ad Æquationem differentialem. And M^r Newton had told him further in his three Letters above mentioned that his Analysis \(w^ch proceeded in equations both finite & infinite)/ determined extended to \problems about/ the lengths & curvatures of curves & to \inverse Problemes of Tangents & to/ almost all sorts of Problemes & represented it so general that M^r Leibnitz himself \(in his letter of Iune 21, 1676)/ exprest his disbeleif of it. It lies upon him therefore in candor & justice to acknowledge this & to give an account why he was silent about all \this/ when he first published <182r> the Differential method. For it was not enough to mention a methodus SIMILIS without saying whose it was & of what extent & antiquity (according to the notices he had \received/ from England,) & acknowledging that his own method was not so ancient. For nothing less then this deserves the name of Candor & Iustice. For Nothing elss than this could fully deserve the name of Candor & Iustice.

Notes & emendations upon the Account of y^e Commercium Epistolicum published in the Transactions for Ian. & Feb. 171$\frac{4}{5}$.

Pag. 175. lin.       Add. The Logarithmotechnia came out in September 16{illeg}|6|2 & thereExercfore by the testimony of D^r Barrow the Analysis per series was invented & generally applied before Septem. 167|6|6. The Exercitationes Geometricæ came out towards the end of the yeare 167|6|8 & D^r Barrow sent the said Compendium or Analysis to M^r Collins in the Iuly following, {in}|as| appears &c.

Pag. 176. \lin 20/ After the words haud integrum ducit \conferatur add/, add {sic}: Testimonijs igitur this Paragraph.

In the first of these two citations the words Quadratura vel Area dictæ figuræ, accurata si possibile sit, sin minus infiniè vero propinqua relate to the words of the Analysis: Hujus cujus [Analyseos] beneficio Curvarum areæ & longitudines &c (id modo fiat) exacte et Geometrice determinantur. Sed ista narrandi non est locus. How this {is} done is explained in the first six Propositions of the Book of Quadratures. And without the method there explained it is not to be done. And therefore by the testimonies of D^r Barrow & M^r Collins Newton in the ye{so} had improved this Method of Analysis some years before Iuly 1669 (that is (as M^r {illeg} as they del two years before the Logarithmotechnia came abroad, or in the year 1666, had impro as was expest {sic} above) had improved this method of Analysis so far \at least/ as it is described in the first or six Propositions of the said book of Quadratures.

Pag. 194. lin. 5. Add this Paragraph.

This was the furst Rule for made publick finding {illeg} second third & fourth differences & is the best.

Rixarum scopus fuit ut

Ad Lectorem

When M^r Newton wrote his Book de Philosophiæ naturalis Principia Mathematica, he found the

The ancients \had their \method of/ Analysis but/ admitted nothing into Geometry before it was demonstrated by Composition. The moderns \neglect Composition &/ are intent only upon Analysis, & so soon admit analytical inventions into Gem\o/etry {sic} before they are demonstrated by Composition. M^r New M^r Newton that his Synthetical Demonstrations \are easier to be read/ render Propositions more certain, & convey them better to posterity. {illeg} For the character For the symbols used in Analysis are apt to be changed from time to time. every author of note M^r Newton for these reasons & that th{illeg}|e| Propositions \in his Principia Philosophiæ/ might be \fit to be/ received into Geometry, after he had invented them by Analysis, composed demonstrated them by composition. But the Analysis is so conspicuous through the composition that the Marquess de l'Hospital said that this Book was almost wholy of the Infinitesimal calculus, & M^r Leibnitz in a Letter to M^r Newton dated 7 Mart. 1693 wrote thus of {illeg} it. Mirifice ampliaveras Geometriam tuis seriebus, sed edito Principiorum opere ostendisti patere tibi etiam quæ Analysi receptæ non subst|u|nt. Conatus sum Ego quo notis commod|is| adhibitis quæ summas differentias & summas exhibent, Geometriam illam quam transcendentem appello, Analysi quodam modo subjicere, nec res male successit.

The Elements of this method are conteined in the first Proposition of the Book of Quadratures & \also/ in the second Lemma of the second book of Principles, with the Scholium thereupon|.| & The Proposition \is/ in the Scholium & \the/ solution therof in the Scholium Lemma. The \same/ Elements are also conteined in M^r Newton's Letter to M^r Collins dated 10 Decem 1672: a copy of w^ch was sent to him M^r Leibnitz by M^r Oldenburg among the extracts of Gregorius Letters 26 Iune 1676. For there M^r Newton described a method of Tangents w^ch he conjectured to be the same – – – so is the Ordinate $\mathrm{y}=\mathrm{B}\mathrm{C}$ to the subtangent $-\mathrm{B}\mathrm{D}$, & so is the increase or fluxion of the Ordinate $\stackrel{.}{\mathrm{y}}$ to the fluxion of of {sic} the Abscissa $-\stackrel{.}{\mathrm{x}}$. And by multiplying the extreames & means you have the Equation $3\mathrm{x}\stackrel{.}{\mathrm{x}}-4\mathrm{x}\mathrm{y}\stackrel{.}{\mathrm{x}}+2\mathrm{b}\mathrm{x}\stackrel{.}{\mathrm{x}}-\mathrm{b}\mathrm{b}\stackrel{.}{\mathrm{x}}=\mathrm{+2}\mathrm{x}\mathrm{x}\stackrel{.}{\mathrm{y}}-2\mathrm{b}\mathrm{y}\stackrel{.}{\mathrm{y}}++3\mathrm{y}\mathrm{y}\stackrel{.}{\mathrm{y}}$, or $3\mathrm{x}\mathrm{x}\stackrel{.}{\mathrm{x}}-4\mathrm{x}\mathrm{y}\stackrel{.}{\mathrm{x}}+2\mathrm{b}\mathrm{x}\stackrel{.}{\mathrm{x}}-\mathrm{b}\mathrm{b}\stackrel{.}{\mathrm{x}}-2\mathrm{x}\mathrm{x}\stackrel{.}{\mathrm{y}}+2\mathrm{b}\mathrm{y}\stackrel{.}{\mathrm{y}}-3\mathrm{y}\mathrm{y}\stackrel{.}{\mathrm{y}}=0$. And this equation is the same with that w^ch is produced by the first Proposition of the Book of Quadratures w^ch|he|n there are but two unknown Quantities proposed. If there be more then two the same operation must be applied to them all gives the solution of that Proposition.

In M^r Newtons Letters of 13 Iune & 24 October 1676 there was a further des{illeg}|c|ription of this method in w^ch gave occasion to D^r Wallis {illeg} in the Preface to the two first volumes of the {illeg} his works, to write that the Quam [Methodum] ego descripsi ex binis Newtoni Literis (Algebræ cap. 91 )|&|c præsertim cap 95) ex binis Newtoni Literis (aut earum alteris) Iunij 13 &c Octob. 24 1676 ad Oldenburgium datis, cum Leibnitio tum communicandis (ijsdem fere verbis, saltem leviter mutatis quæ in {illeg} illis literis habentur;) ubi methodum hanc Leibnitio exponit, tum ante decem annos nedum plures ab ipso excogitatam.

By these M^r Newtons Letters M^r Leibnits was put upon thinking how to improve the Method of Tangents of Slusius & in his way from England into Germany wrote to M^r Collins \Oldenburg/ 18 Novem 1676 st. v. that it might be done by a Table of Tangents, &|A|nd that he had discoursed w^th Hudden at Amsterdam & that the method of Tangents of Slusius was known to him long ago, & was {illeg} more ample then that w^ch Slusius had published. And we understand since that Hudens had improved his method did not stop at surds.

was seen by M^r Leibnitz in London & a copy thereof came to his hands at Hanover in the beginning of Spring following. Pag 194. lin. 6.                So {illeg} in his Analysis he uses the symbol $\boxed{\frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}}$ in the same sense in w^ch M^r Leibnitz uses the symbols $\int \frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$. pag. 25. lin 16.

M^r Newtons Rule for finding second third & fourth Differences published by D^r Wallis in y^e second Volume of his works w^ch came abroad in Spring 1693, was the first Rule made public for doing this. And it doth not appear that M^r Leibnitz had any Rule for second \& third/ differences before. p. 208. lin. 8.

Philosoph. Transact. for Ian & Feb. 16{illeg} 171$\frac{4}{5}$

Pag 174. lin. 21 for ten write fifteen.

P. 176. lin 21 for being write from a copy

Ib. l. 24. After of M^r Newton add: The impression was finished in December 1710.

P. 184. lin 3. dele and M^r Collins

Pag 191 lin 22 After year 1670 add From this Method & his own M^r Gregory Deduced a method of Tangents without computation, & notified it to M^r Collins in a Letter dated 5 Novem 1670. M^r Ne

Ib. between lin 23 & 24 insert, conjecturing that it was the same with that of Gregory & Slusius.

Pag 194 lin 6 for [came to the hands of M^r Leibnitz \in the end of winter or] beginning of the/ write was seen by M^r Leibnitz in London in November, & a copy thereof came to his hands in|at| Hanover in the beginning of Spring following.

Pag 195 lin 23. after 1696 add none of them being printed before the year 1699.

Pag 198 lin penult.

Pag 199 lin 14 after [he gave the] add following, & put an asterisk after the word Letter.

Pag. 201 lin 6 for his write the

Ib. lin 28 after he added insert concerning a branch of this method by w^ch M^r M^r {sic} Newton A.C. 1686 found the solidum minimæ resistentiæ: Quam

Pag. 203 lin. 4 after Acta Leipsica insert had not in the least detracted from anybody but

Ib. lin. 9 for Novice write new man

Pag. 205 lin 16. after Rectangle add For there he uses the symbol $\boxed{\frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}}$ in the same sense in w^ch M^r Leibnitz uses the symbol $\int \frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$.

Pag. 208|7| lin {illeg}|3|3. This after & 396, add. This was the first Rule made publick for finding second third & fourth differences & is the best.

Pag. 209 lin. 21. after compared insert with Gregories Letter of 5 Novem. 1670 &

Pag. 212 lin 199. for many years after write in the Acta Eruditorum for April \Aug./ 1693.

Ib. lin 21. after thereof add, & calling it Methodus universalissima pro seriebus.

Pag. 224. lin 34. after Arguments for insert the being of

Of the Notes.

Annotationes quæ in Epist{illeg}|o|las scriptæ sunt nullius sunt authoritatis nisi quam ab ipsis epistolis derivant. De his jactæ sunt \factæ|a| est/ quærela sed quasi malignæ essent sed malignitas est in ip Epis nondum ostensum est quod aliquid an{illeg} continent \contra D. Leibnitz in ipsis sit/ quod non sit in Epistolis.

Commercium \hocce/ Epistolicum lucem vidit sub finem anni 1712, et subinde per biennium D. Leibnitius, ne {e}os eidem responsum daret, \cum eidem respondere e|n|on posset,/ prætendit \per biennium/ se librum non vidisse, sed a magno cum ipse per occupationes diversas rem tunc discutere non satis posset \et/ ad judicium primarij Mathematici provocasse, cum ipse per occupationes diversas rem tunc discutere non satis posset. Et sententia hujus Mathematici \per Epistolam/ 7 Inuy {sic} 1713 data fuit, & \dari fingebatur & Epistola/ /datam\ in alia charta scurrili \die/ 29 Iulij ejusd sequentis descripta \fuit/ \data descripsit/, et per Europam sparg{i}|{a}| \curavit/, sine nomine \vel/ Mathematici vel Impressoris vel oppidi \urbis/ in qua impressa fuit. Et anno proximo in|per| Epista|o|la s|m| ad D. Chamberlai|y|ne Viennæ 25 Aug. 1714 \1714/ datam, postulavit ut Epistolæ nondum editæ Societas Regia Epistolas nondum editas ad ipsum mitterent. Nam, inquit, cum Hanoveram \inquit/ rediero possum etiam in lucem emittere Commercium aliud Epistolicum quod Historiæ l|L|itterariæ inservire possit. Et Et litteras quæ contra me allegari possunt non minus publici juris faciam qu quam quæ pro me faciun{illeg}|t|{illeg}. et judicium \Hæc Leibnitius./ Sed ficta erat insinuatio ut ex jam dictis dicendis patebit. Non cum D. |accusatio cum totum inter ipsum et oldenburgum Commercium continua serie impressum fuit, ut jam dictum dictum fuit est. præter Epistolas duas quæ non extant. S{illeg} Sub finem anni proximi cum D.| Leibnitius probare vellet quod Concessus a R. Societate constitutus omnia omissent quæ contra Newtonum facerent, scripsit \is/ in Epistola sua prima ad Abbatem de Comitibus, quod in secundo ejus in Angliam itinere, Collinius ostendit ipsi – – – ad ipsum Leibnitium 8 Decem 1674. [Porro Libri Epistolici qui in Archivis R. Societatis asservantur, exam{ina} anno 1675 \de novo/ examminati fuere \anno 1675/ et null in Epistolis \omnibus/ inter Leibnitiū et Oldenburgum \scriptis/ nihil inventum est quod ad hanc rem spectans quod Concessus a R. Societate constitutus omisit.]

Sub finem anni 1715

In eadem ad Abbatem de Comitibus Epistola, scripsit D. Leibnitius quod qui $\stackrel{..}{\mathrm{y}}$ Concessus qui contra ipsum scripsissent (is|d| est Concessus {illeg} a Societate Regia constitutus) candorem ejus aggressi essent per interpretationes duras & male fundatas, \et quod {illeg}|h|i/ illi voluptatem non habebunt \habe {sic}/ respondendi videndi ne respondentem ad responsa e{illeg}|jus| ad parvas rationes eorum qui {illeg}e \ipsum/ tam male utuntur \tractassent./ Si hoc ita esset; ostendere debuisset in casibus aquot aliquot quod eorum interpretationes essent duræ et male fundatæ. Notæ in Epistolas nullius sunt authoritatis nisi quam ab ipsis Epistolis derivant, et facile confutantur si male fundatæ sunt. Sed ne Epistolis responsum fuit ne Notis in easdam: ideo utre|o| jam denuo impressæ sunt ut quibus responderi non potest.

Newtonus in Epistola sua ad Abbatem de Comitibus 26 Feb 171$\frac{5}{6}$ data, quo D D. Leibnitij Literis \ita/ respondebat. D. Leibnitius hactenus respondere recusavit, bene intelligens impossibile esse factis respondere|.| impossibile esse. Silentium suum hac in re excusat, alle{illeg} impræsentia allegando \impræsentia/ quod librum nondum vidi\sse/t, et quod otium illi non esset ad examinandum, sed quod orasset Mathematicum celebrem ut hoc negotit|u|m in se susciperet &c Et D. Leibnitius in proxima sua ad Abbatem de Comitibus Epistola 9 Apr. 1716 data, pergebat se excusare ne responderet, dicendo: Vt respondeam perfecte respondeam operi contra me edito perfecte respondeam, opus erit alio opere non minore quam hoc est. opus erit percurrere corpus magnum minutorum quorum ante annos 30 vel 40 præteritorum, quorum perparvum reminiscor, opus erit examinare veteres epistolas quarum plurimæ sunt perditæ, quorū præterquam quod maxima ex parte non conservavi minuta mearum, & reliquæ sunt sepultæ s{in}t|un|t in maximo chartarum acervo quem non possum examinare ni{illeg} sine tempore et patientia sigillatim examinare. Sed otium minime mihi suppetit alijs negotijs occupato alterius prorsus generis occupato. Hactenus Leibnitius. Attamen post mortem ejus, quæ accidit proximo mense Septembri, amici \ejus/ famam sparserunt \scriptum est ejus in ejus Elogio/ quod Commercio Epistolico Anglorum aliud quoddam suum, idem ol amplius opponere decreverat. Vide Elogium Sic enim scripserunt in Elogio ejus quod in Actis Eruditorum pro mense Iulio 1717 impressum est.] {sic}

Mathematicus Iohannem Bernoullium tanquam a se diversum citaverat. Set post annos s|d|uos cum semisse Leibnitius citationem omisit & Bernoullium esse Mathematicum illum prætendit|er|e cœpit.

Sed accusationem Leibnitius probare non potuit ut jam vidimus, et \Et/ Totum inter ipsum et Oldenburgum Commercium continua serie \jam ante/ impressum fuit, {illeg} præ ut jam vidimus præter Epistolas duas quæ non extant.

Methodus momentorum

Elementa methodi momentorum & fluxionum & momentorum Wallisius in secundo operum Volumine edidit ineunte anno 169{illeg}|3| & {illeg} Regu in prima Propositione libri de Quadratur{um}|is| h] Anno 1691 Halleus noster & Ralphsonus Librum \MS/ Newtoni de Quadraturis |figurarum manibus suis inverunt ut| Regula inveniendi fluxiones primas secundas tertias cæteras in infinitum in Propositione prima Libri de Quadraturis habetur. Hanc Regulam Wallisius \Propositionem exemplis in/ \fluxionibus primis et secundis illustratam Wallisius/ in secundo operum Volumine in lucem edidit ineunte anno 1693 & exemplis in fluxionibus et hæc fuit Regulæ omnium prima pro fluxionibus secundis tertijs alijs in infinitum inveniendisquæ\quæ/ lucem vidit \ed{illeg} |vidit.|/. {sic} [Anno 1695 Wal ineunte Wallisius audivit quod Methodus fluxionum Newtoni celebrari inciperet in Hollandia sub nomine methodi differentialis Leibnitij, & me{illeg} in Præfatione ad \operum suorum/ volumina duo prima {illeg} admonitionem inseruit qua|o|d hæ duæ methodi una et eas|d|em esset Methodus sub diversis tantum formulis, quam Newtonus Literis 13 Iunij & 24 Octob. exponit 1676 ad \Oldenburgum datis/ Leibnitio exponit tum ante decem annos nedum plures ab ipso excogitatam. Anno 1695 Editores Actorum Leipsiensium in Epitome operum Wallisij nihil nihil in contrarioni disputarunt. Sed ne Leibnit{us}|ius| in Le|i|teris quæ inter ipsum et Wallisium mox intercedebant de hac re \Wallisio/ questus est. Anno 1697 Literæ] Propositionem quintam libri Principiorum de quadraturis edidit etiam Wallisius in eodem Volumine pag.      eadem \Et hæc Propositio/ continet methodum quadrandi Curveis|ilin||eas| ubi fieri potest. Hanc methodum anno 1669 Newtono innotuisse patet ex Ejus Analysi per Series quam Barrovius eo anno ad Collinium misit, in qua uti dicitur quod hujus Analyseos beneficio Curvarum areæ et longitudines &c (id modo fiat) exacte et geometrie determinantur. Sed et aliquot annis antequam Mercatoris Logarithmotechnia prodiret innotuit hæc method{illeg}|u|s Newtono, {illeg} innotuit Barrovio et Collinio testibus. Sic enim Collinius ad D. Thomam Strode 26 Iulij 1672 scripsit. Mense septembri 1668, Mercator Logarithmotechniam edidit s{illeg}|u|am, quæ specimen – – – obtineri queant. Quinta autem Propositio \Libri de Quadraturis/ pendet a quatuor prioribus. Et propterea methodus fluxionum quatenus continetur in Propositionibus quin primis libri de quadraturis Newtono innotuit anno 1666 aut antea. Hoc Wallisius in Præfatione ad operum suorū volumina duo prima asseruit nemine per ea tempora contradicente. Newtonus in Introductione ad Epi librum de Quadraturis nihil amplius asseruit. Et tamen inde nata est hæc controversia.

In Analysi per series Newtonus utitur symbol $\boxed{\frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}}$ in eodem sensu quo Leibnitius symbolo $\int \frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$, et symbolis o, ov, oy in eodem quo Leibnitius symbolis dx dy dz.

In methodum inversam fluxionum ingressum \tantum/ dedit sed &

Aliud quoddam Commercium Epistolicum non habuit. Inventum novum in Analysi ad rem nil spectat.

{Ed} Mathematicus \autem/ Bernoullium citabat tanquam a se diversum, Leibnitius vero sub finem anni 1715 citationem emisit \delevit/ et Mathematicum esse Bernoullium ipsum scripsit; eum tamen Bernoullius judex constitui non posset nisi ipse juri omni in methodum infinitesimalem \prius/ renunci{illeg}t|as|set. \Huic judice opponendus est/ Wallisius vir antiquus qui litras Newtoni anno 1676 ab Oldenburgo acceperat judicium pro Newtono, \& rem intellexerat ab tulit initio &/ in Præfatione ad operum suorum volumina duo prima anno 1695 judicium \pro Newtono tulit/ Dominis Leibnitius & Menkenio per ea tempora non mussitantibus.

Porro at D. Leibnitius, ad anno 1714 per Epistolam 25 Aug ad D. Chamberlayn 25 Aug. datam postulavit ut Societas Regia – – – quæ non extant. Et ne respondeat {illeg}atur eti{a}m me aggreditur insuper d{illeg} proponendo disputationes \novas/ Philosophicas & Problemata solvenda quæ duo ad rem nil spectant.

Scribendis literis \quam plurimis/, disputationibus philosophicis \instituendis/, & Problematibus tractandis vacab{illeg}|a|t, Commercio Epistolico confutando me|i|\nime/ vacabat non vacabat.

— Dixit uti Newtonum anno 1676 \1676/ Methodum infinitessimalem Leibnitio per Literas \anno 1676/ exposuisse tum ante decem annos nedum plures inventam ab ipso excogitatam. Sed et in secundo ejus Volumine quod lucem vidit anno 1693, Propositionem primam \ut/ et quintam libri de Quadraturis in lucem adidit, Vnde manifestum est quod Liber ille tunc \{illeg}/ in MS extabat. Nam et Colli Halleius et Ralphsonus nostrates anno 1691 \librum illum/ in MS manibus suis triverunt ut alter eorum testatum reliquit, alter adhuc testatur.

In Propositione illa prima habetur Regula inveniendi fluxiones primas secundas tertias alias omnes in infinitum. Et hæc fuit Regula omnium prima \quæ lucem vidit/ pro fluxionibus secundis {illeg} ac tertijs \tertijs et superioribus/ inveniendis quæ lucem vidit est omnium optima.

In Propositione 8 quinta habetur methodus quadravi cu{m} figuras accurate et Geometrice {ub}i {illeg} \illa/ cujus beneficio Curvarum areæ ({illeg}|i|d modo fiat) exacte et Geometrica|e| determina|e|ntur, {d}e qua \ac de Et hanc methodum uti/ New Newtonus in Analysi \{illeg}/ sua per series locutus est anno 1669 a Barrovio ad Collinium missa locutus est \se habere dixit./ Hæc Propositio pendet a quautor prioribus: ideo method{illeg}|u|s f fluxionum quatetenus {sic} in Propositionibus quin primis Libri de Quadraturis habe cotinetur {sic} \anno 166{illeg} 1669/ Newtono innotuit: anno 1669{illeg} {at} \imo/ et annis aliquot antea {t}este Barrovi{illeg} Epistola Collinij ad \quam Mercatori Logarithmotechnia prodiret/ [mense septenbris {sic} anni 1668] teste|i||bus| Barrovio \Et Collinio/ in Epistola Collinij ad Thomam Strode [26 Iulij 1672 data], id \inferius impressa, id/ est anno 1666 aut antea, ut Wallisius affirmavit.

Dixit Iudex ille literas punctis notatas lucem non vidisse antequam tertium Wallisij Volumen prodiret quod fuit anno 1699. Lucem autem viderunt in secunde ejus Volumine quod prodijt anno 1693. In hoc secundo Volumine \pag. 391 392, 393/ habetur Propositio prima Libri \Newtoni/ de Quadraturis Curvarum, constructa et exemplis in fluxionibus primis & secundis illustrata. Et hoc fuit Regula omnium prim{illeg}|a| quæ lucem vidit \& prio in lucem edita/ pro d{illeg} fluxionibus \& momentis/ secundis tertijs quartis cæteris in infinitum inveniendis, est \omnium/ optima. In eodem secundo Volumine edita \pag. 391/ etiam fuit Propositio quinta libri de Quadraturis. \Et/ In hac Propositione habetur methodus illa cujus beneficio

– pro dz Newtonus utitur symbolo ov & pro $\int \frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$ symbolo $\boxed{\frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}}$. In Introductione ad

After 1693 write. Et in ejus anno 1669 \Newtonus/ in ejus Analysi per series Newtonus utitur pro symbolo dz utitur symbolo ov et pro symbolo $\int \frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$ utitur \Newtonus/ \usus est/ symbolis ov & $\boxed{\frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}}$ in eodem sensu quo Leibnitius postea utitur \usus est/ symbolis dz & $\int \frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$.

sed ubi Epistola ne unquam ora comparavit; sed ubi \priusquam/ Epistolā 12 Maij 1676 datam \ad Oldenburgium scripsit/ oblitus est se series acc illas accepisse

|—| Leibnitius vero sub finem anni 1715, in versione Gallica in Hollandia edita citationem illam m{ini}t (nescio qua fide) omisit et Mathematicum esse Bernoullium ipsum scripsit, & problemata Bernoullij Analystis Anglis solvenda proposuit, & chartam illam volantem denuo dispersit, et ad Bernoul |ad| sententiam Bernoullij allegando \appellando/, amicos suos rerum . . . . conatus est

Post tres annos Marchio Hospitalius Regulam similem edidit

Eodem spectat quod Newtonus eandem Regulam demonstraverit synthetice in l|L|emmate secundo — id est anno 1671. Sunto – – – fluxiones invenire.

puta ${\mathrm{x}}^3-4\mathrm{x}\mathrm{y}\mathrm{z}+\mathrm{a}\mathrm{y}\mathrm{y}-\mathrm{z}$

Et ejusdem Propositionis Solutionem Newtonus \po/ demonstravit synthetice in Lem 2 Lib 2 Principiorum anno 1686|.| cum Propositionem prius sine Sunto quantit verbi gratia quantitates datæ a, b, c, fluentes x, y, z, fluxiones p, q, r, \&/ momenta op, oq, or. et proponatur æquatio quævis fluentes involvens, \puta/ ${\mathrm{x}}^4-\mathrm{a}\mathrm{x}\mathrm{y}\mathrm{y}+\mathrm{b}{\mathrm{y}}^3-{\mathrm{z}}^4+\mathrm{b}\mathrm{b}\mathrm{c}\mathrm{c}=0$. Fluat Et per Lemma prædictum si sola fluat x, erit momentum totius erit $4{\mathrm{x}}^3\mathrm{o}\mathrm{p}-\mathrm{a}\mathrm{y}\mathrm{y}\mathrm{o}\mathrm{p}$: si sola fluat y, momentum totius erit $\mathrm{-2}\mathrm{a}\mathrm{x}\mathrm{y}\mathrm{o}\mathrm{q}+3\mathrm{b}\mathrm{y}\mathrm{y}\mathrm{o}\mathrm{q}$. si sola fluat z, momentum totius erit $\mathrm{-4}{\mathrm{z}}^3\mathrm{o}\mathrm{r}$: si fluant omnes, momentum totius erit $4{\mathrm{x}}^3\mathrm{o}\mathrm{p}-\mathrm{a}\mathrm{y}\mathrm{y}\mathrm{o}\mathrm{p}-2\mathrm{a}\mathrm{x}\mathrm{y}\mathrm{o}\mathrm{q}+3\mathrm{b}\mathrm{b}\mathrm{y}\mathrm{y}\mathrm{o}\mathrm{q}-4{\mathrm{z}}^3\mathrm{o}\mathrm{r}$. Et quoniam totum semper est æquale nihilo, momentum totius est semper erit æquale nihilo. Hæc est æquatio invelvens fluentium momenta. Et si \Si/ e|E|adem dividatur per o \&/|,| habebitur æquatio involvens fluxiones. Sic Propositio, Data æquatione fluentes quotcun quantitates involvente \invenire fluxiones/ Solvitur per hoc Lemma, \et in hujus solutione est et fundatur Methodus {Flu}x/ uti di{illeg}|{a}|t{illeg}|{illeg}|{illeg} /est\ in Scholio quod eidem |Prop.| subjungitur. Et Eadem Propositio ponitur \extat/ in Epistola Newtoni ad Oldenburgum 24 Octob. 1676, et ibi dicitur esse fundamentum methodi generalis de qua Newtonus \Tractatum/ scripserat tum ante annos quin, id est anno 1671|.| {illeg} In h|H|ujus autem \autem/ Propositionis solutione habetur \exhibet/ Algorithmus|m| seu calculus|m| Arithmeticus|m| Methodi fluxionum ejusdem, ideo Algorithmus ille Newtono innotuit anno 1671.

Anglice edita in Actis Regiæ Soc. A.D. 1715,          et A.D. 1712 publicatu|a|s est et ex Anglico in Latinum versa.

Introductio ad Recensionem Libri de Commercio Epistolico Collinij et aliorum.

Historia Methodi quem Newtonus \quam Newtonus Methodum/ fluxionum quam Leibnitius \methodum/ differentialem vocavit ex {illeg} Literis antiquis in Commercio Epistolico Collinij et aliorum jussu Regiæ Societatis \collecto/ anno 16 1712 editis, et eruta \& Recensione Commercij/ in Actis Philosophicis ejus|dē| Societatis anno 1715, (anno et aliquot mensibus ante obitum Leibnitij) recensitio edita

Historia Methodis fluxionum \Analyseos/ quam Newtonus Methodum fluxionum, Leibnitius methodum differentialem vocavit, in Commercio Epistolico Collinij et aliorum & Recensione Commercij contenta; quorum prius anno 16|7|12 \ex antiquis Literis/ jussu Regiæ Societatis editum fuit \collectum & editum/ anno 1712, altera anno 1715 (anno et aliquot mensibus ante {illeg} obitum Leibnitij) collectum fuit in Actis Philosophis L ejusdem Societatis edita fuit anno 1715 (anno et aliquot mensibus ante obitum Leibnitij) lucem vidit.

Historia Methodi Analyseos
Quam Newtonus Methodum Fluxionum \& Momentorum/ Leibnitius Methodum differentialem vocavit,
in Commercio Epistolico Collinij et aliorum & Recensione Commercij
contenta:
Quorum prius ex antiquis Litt|e|ris jussus Societatis Regiæ collectum fuit
|collectum fuit| et editum anno 16 1712
Altera in Actis Philosophis|c||is| ejusdem Societatis anno 1715
(anno et aliquot mensibus ante obitum Leibnitij)
lucem vidit.

Historia Methodi Analyseos
per fluxiones et momenta
a D. Leibnitio \in Commercio Epistolico/ Collinij et alio
Methodus Differentialis vocatæ

Historia Methodi Analyseos
per Fluxiones et Momenta \Augmenta, auctus/ /Differentias\
ex scriptis antiquis
eruta.

Historia Methodi Differentialis
ex antiquis Le|i|tteris
ex|in| Commercio Epistolico Collinij et aliorum ex antiquis Literis \anno 171{3}/ jussu R. Societ < insertion from f 188r > as tempore postquam alijs fuisset familiaris. < text from f 187v resumes >
anno 1712 collecto editis anno 1712
et in|est| Recensione Commercij e|a|nno 1715 (anno et aliquot mensibus ante
ante {sic} obitum Leibnitij) in Actis Philosophicis ejusdem societatis

Historia Methodi Analyseos
per fluxiones & momenta a D. Newtono inventa|æ||,|m
et a D. Leibnitio Differentialem|is| nominata|æ|m
ex literis antiquis deducta.

Historia Methodi Fluxionum
extracta
extracta ex Litteris antiquis Anno 1712
In Commercio Epistolico Collinij et aliorum, jussu Regiæ Societatis editis
editis anno 1712 \jussu R. Societatis editis/
et {illeg} Anno 1715 (anno et aliquot mensibus ante obitum Leibnitij)
et in Recensione Commercij in Actis Philosophicis ejusdem \ejus{illeg}/ Societatis
explicatis.

Historia Methodi fluxionum
composita
ex Commercio Epistolico Collinij et aliorum Anno 1712 jussú Regiæ Societatis {illeg}|co|llecto
et ex Recensione Commercij Anno 1715
(anno et aliquot mensibus ante obitum Leibnitij)
in Actis Philosophicis ejusdem Societatis
edita.

Historia methodi fluxionum et momentorum
ex|in| Commercio Epistolico Collinij et aliorum
et Recensione Commercij
tradita:
contenta:
Quorum prius jussu Regiæ Societatis
ex antiquis Litteris collectum fuit
Anno 1712.
Altera in Actis Philosophicis ejusdem Societatis
Anno 1715
(anno et aliquot mensibus ante obitum Leibnitij)
lucem vidit.

Historia methodi Fluxionem | Differentialis
In Commercio Epistolico et Recensione Commercij
contenta:
Quorum prius jussu R. Societatis editum fuit
Anno 1712
Altera in Actis Philosophicis ejusdem Societatis
Anno 1715
(Anno et aliquot mensibus ante obitum Leibnitij)
lucem vide|i|t.

Epistola|æ|rum Leibnitij ad Oldenburgium \datæ sunt/ 3 Feb. 20 Feb. 30 Mar. 26 Apr. 24 Maij, \&/ 8 Iunij 1673; 15 Iulij & 26 Octob. 1674; 30 Mar. 26 Apr. 2{illeg}|0| Maij, 11|2| Iul. & 28 Decem 1675; 12 Maij, 27 Aug & 18 Novem 1676; 21 Iun. & 12 Iulij 1677 datarum \& harum omnium/ Autographa adhuc asservantur si duas tantum excipias 27 Aug. & 18 Novem 1676 scriptas et in tertio operum Wallisij volumine impressas ubi etiam eæ 15 Iul. & 26 Octob. 1674; 12 Iul. & 28 Decem 1675 & 21 Iun. & 12 Iul. 1677 leguntur. Harum etiam septendecim Epistolarū Apographa (si tertiam et ultimas quin excipias) extant in libris antiquis Epistolicis Regiæ Societatis Num 6, pag 35, 34, * 101, 115, 137 & N^o 7, pag 93, 110, 213, 235, 149, 189. {illeg} Et