Printed Copy of a Letter from John Bernoulli to Sir Isaac Newton, with Newton's Observations on it

29. Iulii 1713.

L. . . . us nunc Viennæ Austriæ agens ob distantiam locorum nondum vidit libellum in Anglia nuper editum, quo N . . . . o primam inventionem Calculi differentialis vindicare quidam conantur. Ne tamen commentum mora invalescat, quam primum retundi debere visum est. Equidem negare non potuerunt novam hanc Analyticam Artem primum a L . . . . o fuisse editam (cum diu satis pressisset) & publice cum, amicis excultam; & post complures demum annos a N . . . . o aliis notis & nominibus, quendam quem vocat Calculum Fluxionum, Differentiali similem, fuisse productum; qui tamen tunc nihil contra L . . . . um movere ausus est. Nec apparet quibus argumentis nunc velint L . . . . um hæc a N . . . . o didicisse, qui nihil tale unquam cuiquam quod constet communicavit, antequam ederet. L . . . . us tamen ex suo candore alios æstimans, libenter fidem habuit Viro talia ex proprio ingenio sibi fluxisse dictanti; atque ideo scripsit N . . . . um aliquid calculo differentiali simile habuisse videri. Sed cum postremo intelligeret, facilitatem suam contra se verti, & quosdam in <34:i(v)> Anglia præpostero gentis studio eousque progressos, ut non N . . . um in communionem inventi vocare, sed se excludere non sine vituperii nota vellent, & N . . . . um ipsum (quod vix credibile erat) illaudabili laudis amore contra conscientiæ dictamen tandem figmento favere; re attentius considerata, quam alias præoccupato in N . . . . ni favorem animo examinaturus non fuerat, ex hoc ipso processu a candore alieno suspicari coepit, Calculum Fluxionum ad imitationem Calculi Differentialis formatum fuisse. Sed cum ipse per occupationes diversas rem nunc discutere non satis posset, ad judicium primarii Mathematici, & harum rerum peritissimi, & a partium studio alieni recurrendum sibi putavit. Is vero omnibus excussis ita pronuntiavit literis 7. Iunii 1713. datis:

Videtur N . . . . us occasionem nactus serierum opus multum promovisse per Extractiones Radicum, quas primus in usum adhibuit, & quidem in iis excolendis ut verisimile est ab initio omne suum studium posuit, nec credo tunc temporis vel somniavit adhuc de Calculo suo fluxionum & fluentium, vel de reductione ejus ad generales operationes Analyticas ad instar Algorithmi vel Regularum Arithmeticarum aut Algebraicarum. Ejusque meæ conjecturæ [primum] validissimum indicium est, quod de literis x vel y punctatis, uno, duobus, tribus, &c. punctis superpositis, quas pro dx, ddx, d³x; dy, ddy, &c. nunc adhibet, in omnibus istis Epistolis [Commercii Epistolici Collinsiani, unde argumenta ducere volunt] nec volam, nec vestigium invenias. Imo ne quidem in principiis Naturæ Mathematicis N . . . . i, ubi calculo suo fluxionum utendi tam frequentem habuisset occasionem, eius vel verbulo fit mentio, aut notam hujusmodi unicam cernere licet, sed omnia fere per lineas figurarum sine certa Analysi ibi peraguntur more non ipsi tantum, sed & Hugenio, imo jam antea [in nonnullis] dudum Torricellio, Robervallio, Cavallerio, aliis, usitato. Prima vice hæ literæ punctatæ comparuerunt in tertio Volumine Operum Wallisii, multis annis postquam Calculus differentialis iam ubique locorum invaluisset. Alterum indicium, quo coniicere licet Calculum fluxionum non fuisse natum ante Calculum differentialem, hoc est, quod veram rationem fluxiones fluxionum capiendi, hoc est differentiandi differentialia, N . . . . us nondum cognitam habuerit, quod patet ex ipsis Princi <34:iv(r)> piis Phil. Math. ubi non tantum incrementum constans ipsius x, quod nunc notaret per x punctatum uno puncto, designat per o [more vulgari, qui calculi differentialis commoda destruit] sed etiam regulam circa gradus ulteriores falsam dedit [quemadmodum ab eminente quodam Mathematico dudum notatum est] . . . . . . . . . . . . Saltem apparet, N . . . . o rectam Methodum differentiandi differentialia non innotuisse longo tempore, postquam aliis fuisset familiaris &c. Haec ille.

Ex his intelligitur N . . . . um, cum non contentus laude promotæ synthetice vel linealiter per infinite parva, vel (ut olim minus recte vocabant,) indivisibilia Geometriæ; etiam inventi Analytici seu calculi differentialis a L . . . . o in Numeris primum reperti, & (excogitata Analysi infinitesimalium) ad Geometriam translati, decus alteri debitum affectavit, adulatoribus rerum anteriorum imperitis nimis obsecutum fuisse, & pro gloria, cujus partem immeritam aliena humanitate obtinuerat, dum totam appetit, notam animi parum æqui sincerique meruisse: de quo etiam Hookium circa Hypothesin planetariam, & Flamstedium circa usum observationum, questos ajunt.

Certe aut miram ejus oblivionem esse oportet, aut magnam contra conscientiæ testimonium iniquitatem, si accusationem (ut ex indulgentia colligas) probat, qua quidam ejus asseclæ etiam seriem, quæ arcus circularis magnitudinem ex tangente exhibet, a Gregorio hausisse L . . . . um volunt. Tale quiddam Gregorium habuisse, ipsi Angli & Scoti, Wallisius, Hookius, Newtonus & junior Gregorius, prioris credo ex fratre nepos, ultra triginta sex annos ignorarunt, & L . . . . i esse inventum agnoverunt. Modum quo L . . . . us ad seriei Nicolai Mercatoris (primi talium inventoris) imitationem invenit seriem suam, ipse statim Hugenio B. Lutetiæ agenti communicavit, qui & per Epistolam laudavit. Eundem sibi communicatum laudavit ipse mox N . . . . us, fassusque est in litteris hanc novam esse Methodum pro Seriebus, ab aliis quod sciret nondum usurpatam, Methodum deinde generalem series inveniendi, pro curvarum etiam transcendentium ordinatis in Actis Lipsiensibus editam, non per Extractiones dedit, quibus N . . . . us usus est, sed ex ipso <34:iv(v)> fundamento profundiore Calculi differentialis L . . . . us deduxit. Per hunc enim calculum etiam res serierum ad majorem perfectionem deducta est. Ut taceam Calculi exponentialis, qui transcendentis perfectissimus est gradus, quem L . . . . us primus exercuit, Johannes vero Bernoullius proprio marte etiam assecutus est, nullam N . . . . o aut ejus discipulis notitiam fuisse: & horum aliquos, cum etiam ad Calculum differentialem accedere vellent, lapsus subinde admisisse, quibus eum parum sibi intellectum fuisse prodiderunt, quemadmodum ex junioris Gregorii circa Catenariam paralogismo patet. Cæterum dubium non est, multos in Anglia præclaros viros hanc N . . . . ianorum Asseclarum vanitatem & iniquitatem improbaturos esse; nec vitium paucorum genti imputari debet.

29. Iulii 1713.

L. . . . us nunc Viennæ Austriæ agens ob distantiam locorum nondum vidit libellum in Anglia nuper editum, quo N . . . . o primam inventionem Calculi differentialis vindicare quidam conantur. Ne tamen commentum mora invalescat, quam primum retundi debere visum est. Equidem negare non potuerunt novam hanc Analyticam Artem primum a L . . . . o fuisse editam (cum diu satis pressisset) & publice cum, amicis excultam; & post complures demum annos a N . . . . o aliis notis & nominibus, quendam quem vocat Calculum Fluxionum, Differentiali similem, fuisse productum; qui tamen tunc nihil contra L . . . . um movere ausus est. Nec apparet quibus argumentis nunc velint L . . . . um hæc a N . . . . o didicisse, qui nihil tale unquam cuiquam quod constet communicavit, antequam ederet. L . . . . us tamen ex suo candore alios æstimans, libenter fidem habuit Viro talia ex proprio ingenio sibi fluxisse dictanti; atque ideo scripsit N . . . . um aliquid calculo differentiali simile habuisse videri. Sed cum postremo intelligeret, facilitatem suam contra se verti, & quosdam in <34:ii(v)> Anglia præpostero gentis studio eousque progressos, ut non N . . . um in communionem inventi vocare, sed se excludere non sine vituperii nota vellent, & N . . . . um ipsum (quod vix credibile erat) illaudabili laudis amore contra conscientiæ dictamen tandem figmento favere; re attentius considerata, quam alias præoccupato in N . . . . ni favorem animo examinaturus non fuerat, ex hoc ipso processu a candore alieno suspicari coepit, Calculum Fluxionum ad imitationem Calculi Differentialis formatum fuisse. Sed cum ipse per occupationes diversas rem nunc discutere non satis posset, ad judicium primarii Mathematici, & harum rerum peritissimi, & a partium studio alieni recurrendum sibi putavit. Is vero omnibus excussis ita pronuntiavit literis 7. Iunii 1713. datis:

Videtur N . . . . us occasionem nactus serierum opus multum promovisse per Extractiones Radicum, quas primus in usum adhibuit, & quidem in iis excolendis ut verisimile est ab initio omne suum studium posuit, nec credo tunc temporis vel somniavit adhuc de Calculo suo fluxionum & fluentium, vel de reductione ejus ad generales operationes Analyticas ad instar Algorithmi vel Regularum Arithmeticarum aut Algebraicarum. Ejusque meæ conjecturæ [primum] validissimum indicium est, quod de literis x vel y punctatis, uno, duobus, tribus, &c. punctis superpositis, quas pro dx, ddx, d³x; dy, ddy, &c. nunc adhibet, in omnibus istis Epistolis [Commercii Epistolici Collinsiani, unde argumenta ducere volunt] nec volam, nec vestigium invenias. Imo ne quidem in principiis Naturæ Mathematicis N . . . . i, ubi calculo suo fluxionum utendi tam frequentem habuisset occasionem, eius vel verbulo fit mentio, aut notam hujusmodi unicam cernere licet, sed omnia fere per lineas figurarum sine certa Analysi ibi peraguntur more non ipsi tantum, sed & Hugenio, imo jam antea [in nonnullis] dudum Torricellio, Robervallio, Cavallerio, aliis, usitato. Prima vice hæ literæ punctatæ comparuerunt in tertio Volumine Operum Wallisii, multis annis postquam Calculus differentialis iam ubique locorum invaluisset. Alterum indicium, quo coniicere licet Calculum fluxionum non fuisse natum ante Calculum differentialem, hoc est, quod veram rationem fluxiones fluxionum capiendi, hoc est differentiandi differentialia, N . . . . us nondum cognitam habuerit, quod patet ex ipsis Princi <34:iii(r)> piis Phil. Math. ubi non tantum incrementum constans ipsius x, quod nunc notaret per x punctatum uno puncto, designat per o [more vulgari, qui calculi differentialis commoda destruit] sed etiam regulam circa gradus ulteriores falsam dedit [quemadmodum ab eminente quodam Mathematico dudum notatum est] . . . . . . . . . . . . Saltem apparet, N . . . . o rectam Methodum differentiandi differentialia non innotuisse longo tempore, postquam aliis fuisset familiaris &c. Haec ille.

Ex his intelligitur N . . . . um, cum non contentus laude promotæ synthetice vel linealiter per infinite parva, vel (ut olim minus recte vocabant,) indivisibilia Geometriæ; etiam inventi Analytici seu calculi differentialis a L . . . . o in Numeris primum reperti, & (excogitata Analysi infinitesimalium) ad Geometriam translati, decus alteri debitum affectavit, adulatoribus rerum anteriorum imperitis nimis obsecutum fuisse, & pro gloria, cujus partem immeritam aliena humanitate obtinuerat, dum totam appetit, notam animi parum æqui sincerique meruisse: de quo etiam Hookium circa Hypothesin planetariam, & Flamstedium circa usum observationum, questos ajunt.

Certe aut miram ejus oblivionem esse oportet, aut magnam contra conscientiæ testimonium iniquitatem, si accusationem (ut ex indulgentia colligas) probat, qua quidam ejus asseclæ etiam seriem, quæ arcus circularis magnitudinem ex tangente exhibet, a Gregorio hausisse L . . . . um volunt. Tale quiddam Gregorium habuisse, ipsi Angli & Scoti, Wallisius, Hookius, Newtonus & junior Gregorius, prioris credo ex fratre nepos, ultra triginta sex annos ignorarunt, & L . . . . i esse inventum agnoverunt. Modum quo L . . . . us ad seriei Nicolai Mercatoris (primi talium inventoris) imitationem invenit seriem suam, ipse statim Hugenio B. Lutetiæ agenti communicavit, qui & per Epistolam laudavit. Eundem sibi communicatum laudavit ipse mox N . . . . us, fassusque est in litteris hanc novam esse Methodum pro Seriebus, ab aliis quod sciret nondum usurpatam, Methodum deinde generalem series inveniendi, pro curvarum etiam transcendentium ordinatis in Actis Lipsiensibus editam, non per Extractiones dedit, quibus N . . . . us usus est, sed ex ipso <34:iii(v)> fundamento profundiore Calculi differentialis L . . . . us deduxit. Per hunc enim calculum etiam res serierum ad majorem perfectionem deducta est. Ut taceam Calculi exponentialis, qui transcendentis perfectissimus est gradus, quem L . . . . us primus exercuit, Johannes vero Bernoullius proprio marte etiam assecutus est, nullam N . . . . o aut ejus discipulis notitiam fuisse: & horum aliquos, cum etiam ad Calculum differentialem accedere vellent, lapsus subinde admisisse, quibus eum parum sibi intellectum fuisse prodiderunt, quemadmodum ex junioris Gregorii circa Catenariam paralogismo patet. Cæterum dubium non est, multos in Anglia præclaros viros hanc N . . . . ianorum Asseclarum vanitatem & iniquitatem improbaturos esse; nec vitium paucorum genti imputari debet.

Whereas a scandalous Paper was published in G{erm}{any} {illeg} \in A{ug}{illeg}{ber} {illeg}/ without the name of the Author or Place where it was published.

Whereas a Letter pretended to be writ by M^r Iohn Bernoulli \dated 7 Iune 1713 &/ published in Germany in Iuly or August 1673 without the name of the Author or Printer or Place in Germany where it was published \printed/ was then \{illeg}/ dispersed by the correspondents of M^r Leibnitz & his correspondents as is & is now reprinted \{at the}/ in the Novells Literairs at Hage & under the title of a Letter written by M^r Iohn Bernoulli of Basil, & to

Whereas a Letter \Libel/ dated 7 Iune 1713 & {illeg} was & \was/ pretended to be written {illeg} to M^r Leibnitz; \&/ was inserted into another Litter \Libel/ dated 29 Iuly {17}13 & both of them by the procurement of M^r Leibnitz \were/ published {wi}thout the names of the authors or writer or place in Germany \in Germay/ where they were printed \in Germany/ & \were/ secretly dispersed by M^r Le{illeg} the correspondents of M^r Leibnitz \& his correspondents & {wolu}{illeg}{an}s/ & the said Libell of 7 Iun. 1713 is newly reprinted {illeg} in the Novelles Le|i|tterairs at the Hage under the title of a Letter written by M^r Iohn Bernoulli of Basil: these are to give notice that the said Letter of 7 Iune 1713 was at first pretended to have been written by some other Mathematician then M^r Iohn Bernoulli, because \For/ the authour thereof] \the |the true| Author of libel the said Letter/ refers to I \speaks of/ M^r I. Bernoulli as distinct from himself in these words [quemadmodum ab eminente quodam Mathematico dudum notatum est], w^ch words {we}re omitted by the B{illeg} Authors of \fraudulently omitted in/ the Novelles Litterairs The aforesaid two Libells are both of them written in the style of M^r Leibnitz & it lies upon if he did not write if he pretends that he did not write them himself he knows the authors & i{t} lies upon him {to} to discover them his confederates in writing this {illeg} Libel {illeg} It lies upon M^r Bernoulli also to cleare himself from being the author {of the} Libel \Letter/ here laid to his charge or else to justify it ag^t \{the a}nswer made to it by D^r Iohn Keill/ And it lies also upon the Editors of the Novelles Litteraires to let the world know by what \whose/ authority they have ascribed the s{a}id Libell of 7 Iune 16|7|13 to M^r Iⁿ Bernoulli & omitted the clausee in the|is| Letter|ibell| w^ch distinguishes the author & M^r Bernoulli {from} one another, [& why they did not let the world k{no}w that this Letter has be was printed before in the Iournall litteraire & effectually answered by D^r Keill, & M^r Bernoulli \{illeg}s demonstratively/ {illeg}ted of being a false accuser.]

Whereas a Letter dated it has been pretended that a great M^r Leibnitz wrot desired a great Mathematician of the first rank to examin the Commercium Epistolicum published by order of the R. S. & that the Mathematician wrote back his opinion in a Letter dated 7 Iuly 1713. And whereas the said Mathematician distinguishes himself by the from \in that Letter quotes the authority of/ M^r Iohn Bernoulli {illeg} {illeg} sand{illeg} by referring to his authority in these words [quemadmodum ab eminente quodam Mathematico dudum notatum est] & yet the those are to give notice that the said Letter translated into French is newly prin reprinted in French in the Novelles Literairs|e|s pag. 414 & there said to be written by M^r Iohn Bernoulli of Basil & the {illeg} \the/ words [quemadmodum ab eminente quodam Mathematico dudum notatum est] are omitted that the fraud may not be perceived.

Cum Charta quædam rixatoria \rixis & f{a}l{illeg} accusationibus plena/ impressa fuerit per D. Menkenium in Actis Eruditorum pro mense Iulio anni 1716 pro D. Ioanne Bernoullio contra D. Keill ab sine nomine Authoris; et in eadem D. Bernoullius vocatus fuit|er|it excelsum ingenium & vir ad abstrusa & abdita detegenda natus, quasi is non fuisset ejusdem a|A|uthor, & Author tamen formulam quandam æquationis a Bernoullianæ formulam meam vocando, Chartam illam ipsi D. Bernoullio attribu\er/it, & D. Bernoullius eandem a se {illeg}d{illeg} scrips|t|am fuisse nondum publice negavit|er|it: D. Bernoullius & D. Menkenius postulantur ut quis fuerit chartæ illius verus author notum faciant.

Et cùm \in/ Charta alia rixatoria rixis et accusationibus plena 29 Iulij 1713 data & sine nomine auctoris impressa, Epistola

Et cum D. Bernoullius in Epistola quadam ad D. Libnitium 7 Iu{illeg} \Iunij/ 1713 data citetur nomine eminentis cujusdam Mathematici quasi ipse non esset author istius epistolæ, et hoc non obstante D. Leibnitius in Epistolis pluribus affirmavit|er|it D. Bernullium ejusdem authorem f{ui}s fuisse: postulatur D. Bernoullius ut ipse declcaret quis sit ejusdem verus author.

Et cùm in charta alia a D. Menkenio in Actis Eruditorum pro Ianuario 1705

In Synopsi Libri de Quadratura Curvarum a D. Menkenio in Actis Eruditorum pro \mense/ Ianuario anni 1705, liber ille plagiarij accusator his verbis: Ingeniosissimus deinde Author [Newtonus] antequam ad Quadraturas Curvarum (vel potius Figurarum Curvilinearum) veniat, præmittit brevem Isaga|o|gem: Quæ, ut|VT| MELIVS INTELLIGATVR, sciendum est, cum magnitudo aliqua continuo crescit, incrementa illa momentanea appellari differentias, nempe inter magnitudinem quæ antea erat & quæ per mutationem momentaneam est producta; atque hinc natum esse Calculum Differentialem ei reciprocum summata|o|orium cujus elementa ab INVENTORE D. Godefrido Guilielmo Leibnitio in his Actis sunt tradita varij usus tum ab ipso tum a D. Marchione Hospitalio — sunt ostensi. Pro diffentijs IGITVR Leibnitianis D. Newtonus adhibet semper [pro ijsdem] adhibuit fluxiones — ijs tum in suis Principijs \Naturæ/ Mathematicis tum in alijs postea editis [pro differentijs Leibnitianis] eleganter est usus, QVEMADMODVM et Honoratus Fabrius in sua Synopsi Geometrica, motuum progressus Cavallerianæ methodo substituit SVBSTITVIT. Hæc calumnia \accusatio/ initium dedit controversiæ huic de auctore hujus methodi, proinde D. Menkenius qui eandem edidit rogatur ut auctorem accusationis hujus prodat.

In epistola prædicta septimo Iunij 1713 data & a \quam/ D. Leibniti{illeg}|u|s D. Bernoullio asscript attribuita, accusatio eadem plurius repetitur, & affirmatur quod Newtonus ubi scripsit antiquas Epistolas in Commercio epistolico impressas ne quidem somniavit de calculo su{a}|o| fluxionū & fluentium, propterea quod in ijsdem nullæ sunt literæ punctis notatæ, {illeg}d. Imo ne quidem in Principijs naturæ Mathematicis: sed prima vice hæ literæ punctatæ Comparuerunt in tertio Volumine Operum Wallisij, multis annis postquam Calculus differentialis jam ubi locorum <474v> invaluisset. Affirmatur etiam Newtono rectam Methodum differe{t}|n|tiandi differentia non innotuisse, longo tempore postquam alijs fuisset familiaris. Rogatur igitur D. Bernoullius ut consulat Volumen secundum operum Wallisij pag. 391, 392, 393 &c et tunc agnocat publice, literas punctatas {illeg} lucem vidisse anno 1693, annis nimirum sex antequam Volumen tertium prodijt. Agnoscato etiam Propositionem primam libri de Quadratura Curvarum & solutionem ejus cum exemplis in fluxionibus primis & secundis propemodum verbatim ibi impressam fuisse, id propemodum verbatim \at/ adeo Librum illum tunc in M.S. latuisse. Agnoscat etiam Regulam , {A} ibi traditam pro inveniendis fluxionibus tertijs secundis tertijs quartis alijs in infinitum veram esse et lucem vidisse tra annis tribus antequam Regula aliqua pro inveniendis differentijs secundis tertijs quartis alijs lucem videre

Whereas a squabbling \defamatory/ paper was \Paper was printed \by D^r Menkenius/ in favour of M^r \I./ Bernoulli/ against D^r Keill was printed in the Acta Eruditorum for Iune 167 Iuly        16|7|16 without the name of the Author, \& therein M^r Bernoulli is called excelsum ingenium & vir &|ad| abstrusa & abditas detegenda natus, as if he were not the Author of that Paper/ & yet the Author of the Paper by citing a naming M^r Bernoulli's formula of an Equation meam formulam (pag. 314) he ascribed that Paper to M^r Iohn Bernulli \himself/|,| a& thereby & M^r Bernoulli is desired by not \has not yet/ disowning\ed/ it publickly: has given \{illeg}/ occasion to the world he is \& D^r Menkenius are/ desired to tell \dis{illeg}te/ \tell/ the world who was the author thereof, & /D^r Menkenius who published the said Paper is also desired to declare the name {illeg}{ve}\ And whereas in a{illeg} Letter dated 13 Iune 1713 another squab defamatory paper dated 29 Iuly 1713 \& printed without the name of the author/ a Letter \to M^r Leibnitz/ is inserted dated 7 Iuly 1713 in w^ch M M^r Bernoulli is cited by the name of an eminent Mathematician & ye as if he were not the author of that Paper \Letter/ & yet M^r Leibnitz has \in several Letters/ affirmed that M^r Bernoulli was the author thereof that Letter: M^r Bernoulli is desired to tell the world who was the author of that Letter. And whereas in another paper published \by D^r Menkenius/ in the Acta Eruditorum for Ian. 16|7|05, {illeg} \in giving an Account of/ the book De quadratura Curvarum \the Book/ is accused of as a piece of Plagiary, in these words; Ingeniosissimus deinde author \[Newtonus]/ antequam ad Quadraturas Curvarum vel potius Figurarum Curvilinearum) veniat, præmittit brevem Isagogem. Quæ ut melius \MELIVS/ intelligatur, sciendum est, cum magnitudo aliqua continuo crescit, verbati Linea exempli gratia incrementa illa momentanea appellari differentias, nempe inter magnitudinem quæ antea erat & quæ per mutationem momentaneam est producta; at hinc natum est c|C|alculum Differentialem, ei reciprocum summatorium; cujus elementa ab INVENTORE D. Godefrido Guilielmo Leibnitio in his Actis sunt tradita, varij usus tum ab ipso tum a \D. D/ fratribus Berno\u/llijs tum a D. Marchione Hospitalio — sunt ostensi. Pro diffentijs IGITVR Leibnitianis D. Newtonus adhibet semper adhibuit Fluxiones, [pro Differentijs illis] adhibuit Fluxiones — ijs tum in suis Principijs Metho Naturæ Mathematicis tum in alijs postea editis [pro differentijs Leibnitianis] eleganter est usus, QVEMADMODVM et Honoratus Fabrius in sua synopsi Geometrica, motuum progressus Cavallerianæ methodo SVBSTITVIT. D^r Menkenius who published this \defamatory/ Paper is desired to discover the name of \tell the world who was/ the Author.

And

And because \whereas/ in the aforesaid Letter of 7 Iune 1713 ascribed by M^r Leibnitz to M^r Iohn Bernoulli, this accusation of plagiary is pursued, & the world is told that M^r Newton did not so much as dream of the Calculus of fluxions when he wrote the ancient Letters \& Papers/ published in the Commercium Epistolicum, \②/ no nor in \when he wrote/ his Principia Naturæ Mathematica \nor long after/, |①| because there are no prickt letters in them \③/ & that these \because {illeg} these/ Letters First appeared in the third Volume of Wallises works many years after the Differential Calculus was every where known,] & because \that/] M^r Newton when he wrote his Principles did not \then/ understand \how to find/ second differences as he pretends{illeg} till it was familiar to others: M^r Bernoulli is desired to look into the second Volume of the Works of D^r Wallis pag. 391 392, 393 &c & then tell the word|l|d whether prickt Letters did not come abroad in the year 1693 six years before the third Volume was published & whether the first Proposition of the book of Quadratures with the solution {illeg} & examples in first & second differences fluxions be not there published almost verbatim & whether the Rule there given for finding second third & fourth fluxions &c \be not genuine &/ did not come abroad some years before any Rule for finding second third & fourth differences <476v> & whether the method of fluxions be not taught in y^e Introduction to the |very| book of Quadratures without the use of prickt letters: For as he has publickly accused M^r Newton of Plagiary by feigned pretences so he ought to justif make a publick Recantation if his accusation be not true.

And whereas M^r Bernoull it is pretended that an Italian named Ricatti has made many experiments with Prisms & L Lenses & Prisms of crystal of y^e Rock, w^ch destroy the Theory of Hypothesis of M^r Newton about \apparent/ colours: & /yet\ \yet M^r Newton assumes no expe hypothesis & Prisms of crystall are unfit for {exp} such {illeg}/ yet instead of sending an experiment of that kind \to prove his assertion/ has sent a chalenge to solve a Mathematical Probleme & thereby discovered that he is in a confederat|c|y w^th M^r Bernoulli, & \yet/ M^r Newton assumes no hypothesis \nor meddles with apparent colours/ & Prisms of Crystall are unfit for s by reason of a double refraction are unfit for such purposes \experim^ts/ & M^r Ricatti instead of sending an experiment to prove his assertion, has sent a ma challenge to solve a mathematical Probleme & thereby discovered that he is in a confederacy with M^r Bernoulli, & upon these considerations \he/ is looked upon here by some as \a Pretender &/ a bullying Pretender: he is desired to: for clearing himself from this imputation he is desired to publish communicate to the world {illeg} \any one/ of his experiments w^ch contradicts M^r Newton's Theory.

M^r Raphson a little before his death published that the Book of Quadratures published \w^ch came abroad/ in the year 16{illeg} 1714 1704, was about the year 1676 extracted written from a former Tract, & that about the year 1691 M|D|^r Halley & he had it in their hands \at Cambridge/ about the year 1691 in order to bring it up to London; & D^r Halley remembers that this was presently after the election of M^r Raphson into the R. Society, w^ch was in 1690. And therefore the Book was written before M^r Bernoulli knew any thing of the Differential Method, & for he|i|m {illeg} to accuse it of Plagiary is to tell the wo{l}|r|ld that he has a mind to something that is in it. It is easy to add to inventions, & if he has improved it \the method/ |t|he is desired p{illeg} improvements are his own, & \he is desired/ to publish them & such he is at liberty to publish them when ever he pleases. But with And But the h{onn}our \practise/ of challenging vanity introduced by him & M^r Leibnitz \of challenging every body/ to solve their Problems, is not in England has not yet obteined in England, & we look up it with contempt.

Advertisement

Whereas a wrangling Paper was published in the Acta Eruditor{ū} for Iuly 1716 — who is the author thereof. And whereas in the same paper. And whereas in the same paper M^r Iohn Bernoulli is said to have composed invented Rules for integrating differential quantities, & given {illeg} copies of his MS to the Marquess de l'Hospital, & to M^r Herman & other {illeg} &: if he has improved the \inverse/ method of fluxions, the improvements will be allowed to be his when ever he pleases to publish them. But it will not follow from them that the book of Quadratures written \published/ by M^r Newton \A.C. 1704/ is a piece of plagiary as in his Letter to M^r Leibnitz dated 13 Iune 1713 he as affirmed \represented/, & as the author of a Paper another \scandalous/ Paper printed in the Acta Eruditorum A.C. 1705 pag.     (suspected by some {to} be also M^r Iohn Bernulli or one of his disciples) has also feigned. M^r Raphson has testified publickly in p testified publ publickly that this Book (w^ch about the year 1676 M^r Newton wrote from a former Tract) D^r Halley & he had in their hands at D^r

The first Proposition of this Book was in with it solution & examples in first & second fluxions was published \almost verbatim by D^r Wallis/ in the second Volume of this D^rs works A.C. 1693, being sent to the Doctor the year before, & therefore th & printed off the year before; & therefore the Book was then in manuscript. M^r Raphson has published that he D^r Halley & he had it in their hands at Cambridge about the year 1691 in order to bring it up to London & D^r. Halley rembers {sic} that it was in        A.C. 1690 & thence it may be understood that it was in MS before the Rules invented \& composed/ by M^r Bernoulli. There In M^r Newtons Letters of Iune 13 Octob. 24^th & Novem 8^th 1676 there are many things relating to this Book & therefore it was in Manuscript before M^r Leibnitz knew any thing of the Differential Method.

In this Book are many things which had they been proposed as Problems to be solved, might have puzzelled all the Mathematicians in Europe, as for instance to integrate the quantities, ${\mathrm{x}}^{\mathrm{m}}+{\mathrm{x}}^{\mathrm{n}}{\mathrm{y}}^{\mathrm{p}}$+ c y {illeg} the areas of the Curves exprest by the following equations whose areas are {illeg} [integrate |reduce the integration of $\stackrel{.}{\mathrm{y}}$ $\stackrel{.}{\mathrm{y}}$ following to the quadrature of find the Conic Sections in the following equations| $\stackrel{.}{\mathrm{y}}$ in the following equations $\mathrm{a}{\mathrm{z}}^{\mathrm{m}}+\mathrm{b}{\mathrm{z}}^{\mathrm{n}}{\stackrel{.}{\mathrm{y}}}^{\mathrm{p}}={\stackrel{.}{\mathrm{y}}}^{\mathrm{q}}$ $\frac{\mathrm{d}\stackrel{.}{\mathrm{z}}{\mathrm{z}}^{2\mathrm{n}-1}}{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{n}}+\mathrm{g}{\mathrm{z}}^{2\mathrm{n}}}=\stackrel{.}{\mathrm{y}}.\frac{\mathrm{d}\stackrel{.}{\mathrm{z}}{\mathrm{z}}^{\frac{1}{2}\mathrm{n}-1}}{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{n}}+\mathrm{g}{\mathrm{z}}^{2\mathrm{n}}}=\stackrel{.}{\mathrm{y}}.\frac{\mathrm{d}\stackrel{.}{\mathrm{z}}{\mathrm{z}}^{\frac{3}{2}\mathrm{n}-1}}{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{n}}+\mathrm{g}{\mathrm{z}}^{2\mathrm{n}}}=\stackrel{.}{\mathrm{y}}.\frac{\mathrm{d}\stackrel{.}{\mathrm{z}}}{\mathrm{z}}\sqrt{2+\mathrm{f}{\mathrm{z}}^{\mathrm{n}}+\mathrm{g}{\mathrm{z}}^{2\mathrm{n}}}=\stackrel{.}{\mathrm{y}}$. Or to reduce the integration of $\stackrel{.}{\mathrm{y}}$ to the simplest cases of quadratures in the following equations $\mathrm{a}{\stackrel{.}{\mathrm{z}}}^{\mathrm{p}+\mathrm{q}}{\mathrm{z}}^{\mathrm{m}}+\mathrm{b}{\stackrel{.}{\mathrm{z}}}^{\mathrm{q}}\stackrel{.}{\mathrm{z}}{\stackrel{.}{\mathrm{y}}}^{\mathrm{p}}=\mathrm{c}{\stackrel{.}{\mathrm{z}}}^{\mathrm{p}}{\stackrel{.}{\mathrm{y}}}^{\mathrm{q}}$.

S^r

I received your Letter w^th the Pro\po/sals inclosed & desire you to acquaint Mons^r Bernsdorf that

The Longitude is {illeg} already found at {illeg} Land by Astronomy & good clockwork \together/ & the method is p\r/actised every day \with good success/ for rectifying Geometr|graph|y & there is no other way of for I have \been/ m{any}th{o}d{ea}rs \I am of opinion/ of {sic} opinion that there is no other way to find it by \at/ sea \but |then| by improving this/. This \has been my/ opinion I have reported \had above these th{e}rty years & I reported it lately to/ to {sic} t{illeg} to|h|e Committee of the House of Commons \to whom the buisines of y^e {l vas}/ & {L}{illeg} |now repeat the report being| am now too old to retract it. & {illeg} \change my opinions, or to rec{illeg}{am} in them./ If any \{illeg}/ man {sic} /serious man\ hath any other method to propose, it is improper \he would endeavour that it should not/ be referred i{n} it|{t}|o me because I have made a report already agai\n/st his project;|.| besides that I am not a proper the proper Officer to who His besides that I am not an Officer for sea affairs & His proper {illeg}t way \& am \now/ too old to change my opinions./ Proposals relating to sea affairs are usually referred to the Trinity House [& the proper \way/ to obtein such a Reference is to apply to the board of the Admiralty] But th if Projectors will not keep in the kings high way but go out of the road to get this|e| {sic} And if any Projector declines such a Reference his business is something else then the Longitude [& for that reason I desire to have nothing to do with him].

S^r

I received your Letter with the Paper w^ch Mon|^r|{illeg} |de| Bernsdorf ordered you to send to me, & desire you to acquaint him that \the/ Longitude is already found at Land by Astronomy & \good/ Clockwork together, & the method is practised everd|y|-day with success {illeg}|f|or rectifying Geography: & I am \have been many years/ of opinion that there is no other method of way of finding it by sea then by pursuing & improving this method. [This has been my opinion above these thirty years & I reported it to the Committee of the House of Common to whom the buisiness of the Longitude who summoned me to attend them about this matter the Longitude this matter & I can make no other report \then what I have made alreay upon this occasion/. at present {illeg} for I am too old to chang my opinions or to take these matters into fresh considerations.] And therefore I am not \having reported this opinion I am uncapable of/ meddling with the Paper which you sent me /it being contrary to my report.\. {sic} But if other Gentlemen have a mind that it should be examined, I will not oppose them.

S^r

I received the Proposals w^ch Mon^r de Bernstorff put into your \desired you to/ send to me, & beg the favour of you to acqui|a|int \him/ that I do not approve of them & desire that the person who makes them may not be sent to me. I am

29 Iulij 1713.

L . . . . us nunc Viennæ agens Austriæ agens ob distantiam locorum nondum vidit libellum in Anglia nuper editum, quo N . . . . o primam inventionem Calculi differentialis vindicare quidam conantur. Ne tamen commentum mora invalescat, quam primum retundi debere visum est. Equidem negare non poterunt novam hanc Analyticam Artem primum a L . . . . o fuisse editam (cum diu satis pressisset) & publice cum amicis excultam; et post complures demum annos a N . . . . o alijs notis & nominibus, quendam quem vocat calculum fluxionum, Differentiali similem, fuisse productum; qui tamen tunc nihil contra Leibnitium L . . . . um movere ausus est. Nec apparet quibus argumentis nunc velint L . . . . um hæc a N . . . . o didicisse qui nihil tale unquam cuiquam quod constet communicavit, antequam ederet. L . . . . us tamen ex suo candore alios æstimans, libenter fidem habuit Viro talia ex proprio ingenio sibi fluxisse dictanti; at ideo scripsit N . . . . um aliquid calculo differentiali simile habuisse videri. Sed cum postremo intelligeret, facilitatem suam contra se verti, & quosdam in Anglia præpostero gentis studio eous progressos, ut non Newtonum N . . . um in communionem inventi vocare, sed se excludere non sine vituperij nota vellent, et N . . . . um ipsum (quod vix credibile erat) illaudabili laudis amore contra conscientiæ dictamen tandem figmento favere; re attentius considerata, quam alias præoccupato in Newt N . . . . ni{um} favorem animo examinaturus non fuerat, ex hoc ipso processu a candore alieno suspicari cœpit, Calculum Fluxionum ad imitationem Calculi Differentialis formatum fuisse. Sed cum ipse per occupationes diversas rem nunc discutere non satis posset, ad judicium primarij Mathematici, et harum rerum peritissimi et a partium studio alieni recurrendum sibi putavit. Is v{illeg}|er|o omnibus excussis ita pronuntiavit literis 7 Iunij 1713 datis:

Videtur N . . . . us occasionum nactus serierum opus multum promovisse per extractiones r|R|adicum, quas primus in usum adhibuit, & quidem in ijs excolendis ut verisimile est omne suum studium posuit, nec credo tunc temporis vel somniavit adhuc de calculo suo fluxionum {illeg}et fluentium, vel de reductione ejus ad generales operationes analyticas ad instar Alorithmi vel regularum Arithmeticarum aut Algebraicarum. Ejus meæ conjecturæ [primum] validissimum indicium est, quod de literis x vel y punctatis, uno, duobus, tribus, &c punctis superpositis quas pro dx, ddx, d³x; dy, ddy, &c nunc adhibet, in omnibus istis Epistolis <478v> [Commercij Epistolici Collinsiane|i|, unde argumenta ducere volunt] nec volam nec vestigium invenias. Imo ne quidem in Principijs Naturæ Mathematicis N . . . . i, ubi calculo suo fluxionum utendi tam frequentem habuisset occasionem, ejus vel verbulo fit mentio, aut notam hujusmodi unicam cernere licet sed omnia fere per lineas figurarum sine certa Analysi ibi peraguntur more non ipsi tantum, sed et Hugenio imo jam antea [in nonnullis] dudum Torricellio, Robervallio, Cavallerio, alijs, usitato. Prima vice hæ literæ punctatæ comparverunt in tertio Volumine Operum Wallisij, multis annis postquam Calculus differentialis jam ubi locorum invaluisset. Alterum indicium quo conjicere licet Calculum fluxionum non fuisse natum ante Calculum Differentialem, hoc est, quod veram rationem fluxiones fluxionum capiendi hoc est differentiandi differentialia, N . . . . us nondum cognitam habuerit, quod patet ex ipsis Principijs Phil. Math. ubi non tantum incrementum constans ipsius x quod nunc notaret per x punctatum uno puncto, designat per o [more vulgari qui calculi differentialis commoda destruit] sed etiam regulam circa gradus ulteriores falsam dedit [quemadmodum ab eminente quodam Mathematico dudum notatum est] . . . . . . . . . . . . Saltem apparet, Ne . . . . o rectam methodum differentiandi differentialia non innotuisse longo tempore, postquam alijs fuisset familiaris. &c. Haec ille.

Ex his intelligitur N . . . . um, cum non contentus laude promotæ synthetice vel linealiter per infinite parva, vel (ut olim minus recte vocabant) indivisibilia Geometriæ; etiam inventi Analytici seu calculi differentialis a L . . . . o in numeris primum reperti, & (excogitata Analysi infinitesimalium) ad Gemetriam translati, decus alteri debitum affectavit, adulatoribus rerum anteriorum imperitis nimis obsecutum fuisse, et pro gloria, cujus partem immeritam aliena humanitate obtinuerat, dum totam appetit, notam animi parum æqui sinceri meruisse: de quo etiam Hookium circa Hypothesin Planetariam, et Flamstedium circa usum observationum, quæstos aiunt.

Certe aut miram ejus oblivionem esse oportet, aut magnam contra conscientiæ testimonium iniquitatem, si accusationem (ut ex indulgentia colligas) probat, qua quidam ejus asseclæ etiam seriem, quæ arcus circularis magnitudinem ex tangente exhibet, a Gregorio inventam volunte L . . . . um hausisse L . . . . um volunt. Tale quiddam Gregorium habuisse ipsi Angli & Scoti, Wallisius Hookius Newtonus & junior Gregorius, prioris credo ex fratre nepos, ultra triginta sex annos ignorarunt, & L . . . . i esse inventum agnoverunt. Modum quo L . . . . us ad seriei Nicolai Mercatoris (primi talium inventoris) imitationem invenit seriem suam, ipse statim Hugenio B. Lutetiæ \agenti/ communicavit, qui et per Epistolam laudavit. Eundem sibi communicatum laudavit ipse mox N . . . . us fassus est in Literis hanc novam esse Methodum pro Seriebus, ab alijs quod sciret nondum usurpatam. Methodum deinde generalem series inveniendi, pro curvarum etiam transcendentium ordinatis <479r> in Actis Lipsiensibus editam, non per extractiones dedit, quibus N . . . . us usus est, sed ex ipso fundamento profundiore Calculi differentialis L . . . . us deduxit. Per hunc enim Calculum etiam res serierum ad majorem perfectionem deducta est. Vt taceam calculi exponentialis, qui transcendentis perfectissimus est gradus, quem L . . . . us primus exercuit, Iohannes vero Bernoullius proprio marte etiam assecutus est, nullam Newtono N . . . . o aut ejus discipulis notitiam fuisse: & horum aliquos, cum etiam ad calculum differentialem accedere vellent, lapsus subinde admisisse, quibus eum parum sibi {b} intellectum {illeg}|fu|isse prodiderunt, quemadmodum ex junioris Gregorij circa Catenariam paralogismo patet. Cæterum dubium non est, multos in Anglia præclaros viros hanc N . . . . ianorum Asseclarum vanitatem & iniquitatem improbaturos esse; nec vitium paucorum genti imputari debet.

The following Letter was to M^r Leibnitz was written originally in Latin, & we have met with the ensuing Observations upon it

Videtur N . . . . us occasionum nactus serierum opus multum promovisse per Extractiones Radicum, quas primus in usum adhibuit, et quidem in ijs excolendis ut verisimile est ab initio omne suum studium posuit, nec credo tunc temporis¹ vel somniavit adhuc de calculo suo fluxionum et fluentium, vel de reductione ejus ad generales operationes Analyticas ad instar Algorithmi vel regularum Arithmeticarum aut Algebraicarum. Ejus meæ conjecturæ [primum] val{e}|i|dissimum indicium est,² quod de literis x vel y punctatis, uno, duobus, tribus, &c punctis superpositis, quas pro dx, ddx, d³x; dy, ddy, &c nunc adhibet, in omnibus istis Epistolis [Commercij Collinsiani, unde argumenta ducere volunt] nec volam, nec vestigium invenias. Imo ne quidem in Principijs Naturæ Mathematicis N . . . . i,³ ubi calculo suo fluxionum utendi tam frequentem habuisset occasionem, ejus vel verbulo fit mentio, aut notam hujusmodi unicam cernere licet, sed omnia fere per lineas figurarum sine certa Analysi ibi peraguntur more non ipsi tantum, sed et Hugenio imo jam antea [in nonnullis] dudum Torricellio, Robervallio, Cavallerio, alijs, usitato. Prima⁴ vice hæ literæ punctatæ comparuerunt in tertio Volumine Operum Wallisij, multis annis postquam Calculus differentialis jam ubi locorum invaluisset. Alterum indicium, est quo conjicere licet Calculum fluxionum non fuisse natum ante Calculum differentialem, hoc est,⁴ quod veram rationem \fluxiones fluxionum capiendi, hoc est/ differentiandi differentialia, N . . . . us nondum cognitam habuerit, quod patet ex ipsis Principijs Phil. Math. ubi⁶ non tantum incrementum constans ipsius x, quod nunc notaret per x punctatum uno puncto, designat per o [more vulgari, qui calculi differentialis commoda destruit] sed etiam regulam circa gradus ulterio{illeg}|r|es falsam dedit [quemadmodum⁷ ab eminente quodam Mathematico dudum notatum est] . . . . . . . . . . . . Saltem apparet Newtonum N . . . . o rectam methodum differentiandi differentialia non innotuisse longo tempore, postquam alijs fuisset familiaris &c.

Observations upon the foregoing Letter.

|Artic.| 1. Videtur N . . . . us occasionem nactus, serierum multum promovisse per Extractiones Radicum, quas primus in usum adhibuit, et quidem in ijs excolendis ut verisimile est ab initio omne suum studium posuit{es}, nec credo tunc temporis vel somniavit adhuc de calculo suo fluxionum et fluentium vel de reductione ejus ad generales operationes Analyticas ad instar Algorithmi vel Regularum Arithmeticarum aut Algebraicarum. Ejus meæ conjecturæ [primum] validissimum indicium est &c

Ob

Obs. M^r Newton is here accused of Plagiary by a mere conjecture jecture contrary to the authority \testimony/ of an abler & older Mathematician D^r Wallis in the Preface to the first volume of his works cited above pag. 105, 106.

Artic 2. {a} \Ejus meæ conjecturæ [primum] validissimum argumentum \indicium/ est/ quod de literis punctatis x vel y punctatis, uno, duobus, tribus &c punctis superpositis, quas pro dx, ddx, d³x; dy, ddy &c nunc adhibet, in omnibus {illeg}æ|isti|s Epistolis Commercij Collinsiani unde argumenta ducere volunt] nec volam nec vestigium invenias. Imo ne quidem in Principijs

Obs. In the second Lemma of the second Book of Principles the Elements of the Book Method of fluxions are taught, & in the Introduction to the Book of Quadratures the method it self is expresly taught & illustrated by Examples & all this is done w^thout the use of prickt letters. And in M^r Newton's Letter 10 Decem. 1672 the method is sufficiently described.

Artic. 3.

Observatons upon the foregoing Letter.

D^r Wallis an older & abler Mathematician has attested in the Preface to the first Volume of his works that this Method was M^r Newton explained this Method \to M^r L . . . . z/ in his Letters of 13 Iune & 24 Octob. 1676 & invented it ten years before or above In the second Lemma of the second Book of Princi Mathematical Principles the Elements of this Method of fluxions are demonstrated taught & demonstrated; & in the Introduction to the book of Quadratures the Method it self is taught & il expresly taught & illustrated by examples, & in M^r Newton|s| Letter of 10 Decem 1672 the Method is plainly described & illustrated with an example {illeg} of Drawing Tangents thereby: & all this is done without the use of prickt letters. The Book of Principles was writ by Composition & therefore therefore there was no occasion of using the calculus of fluxions in it. Prickt letters appeared in the second Volume of the works of D^r Wallis, & this Volume was almost all printed in the year 1692 & came abroad {illeg} the next spring before the Differential method began to make a noise. The Manuscript of the Book of Quadratures came abroad was in the hands of D^r Halley & M^r Ralps|h|son in the year 1691 as both of them have attested. And this Book is sufficiently described in M^r Newtons Letters of Octob. 24 & Novem 8 1676, & the Quadratures there cited out of it are not to be attained without the Method in dispute. The constant fluxion of x M^r Newton denotes by $\stackrel{.}{\mathrm{x}}$ w^th a point above it; but the constant increment of or moment of x M^r Newton denotes not by $\stackrel{.}{\mathrm{x}}$ but by o & still uses this Notation as convenient: M^r Leibnitz hath no Notation for fluxions, & therein his Method is defective. The Only Rule for finding {illeg} w^ch M^r Newton has given for finding first second third fourth & other differences \fluxions/ is conteined in the first Propoosition of the Book of Quadratures & was pub is a very true one & was published with examples in the second Volume of the works of D^r Wallis before any other Rule for finding second third & fourth Differences came abroad. And the very words of the Proposition were set down in the Scholium upon the second Lemma of the second Book of Principles, & in M^r Newtons Letter of 24 Octob. 1676, as the foundation of the method of fluxions. And the inverse of this Rule is the first Rule in the Analysis per series numero terminorum infinitas \demonstrated by the method {illeg}/ communicated by D^r Barrow to M^r Collins in the Iuly 169|6|9 \& that Rule is d demonstrated by the method of fluxions in the end of that Analysis/. And without this Rule the Series for Quadratures w^ch break off & become finite when the Curve can be squared by a finite equation, & w^ch are mentioned in the said Analysis, are not to be attained |4.| In translating this Letter \of 7. Iune 1713/ into French & printing it in the Novelles Literairs Decem. 28 1715, we are told that the Author of this Letter was M^rIohn Bernoulli, & to make this credible, the citation of the Eminent Mathematician is omitted in the body of the Letter. But For if M^r I. Bernoulli be the Eminent Mathematician \there/ {illeg} cited by the author of the Letter he cannot be the Author himself. And M^r I. Bernoulli in a Letter to M^r Newton hath positively declared that he was not the author thereof. And if he had been the author thereof he would not have said that the Eminent Mathematician charged M^r Newton with a false Rule. That Mathematiciam {sic} ch noted \only that there was/ an error in a {illeg}l{illeg}ion the Scholium upon Solution of Prob. III Lib. II Princip. Philos. & {illeg}|s|uspected that it lay in second differences. M^r Newton Nicolas Bernoulli told M^r Newton that there was what his Vnkle had observed. M^r Newton upon examining the Solution found that the error lay in drawing the Tangent of the \Arch GH/ from the wrong end of the arch, \GH/ & corrected the error \himself/ & told him that the corrections solution should be set right in y^e new edition of the Principles. The Tangents of the Arcs GH & HI are the first moments of the arcs \areas/ /arcs\ FG & FH & should \have/ be|en| drawn the same way with y^e motion of the body describing {the}se the Curve FGHIK, whereas \through in advertency/ the tangent of the Arc FG had been drawn backward through inadvertency \from the point of contact the contrary way from the point of contact./. There is an error of greater consequence committed \in second dif{f}erences/ by M^r Leibniz in his Tentamen de motuum cœlestium causis, (sect 15.) which tho often \o{illeg}/ complained of is not yet set right, And his friends will not see clearly to pull motes out of other mens eyes till they have pulled the beam out of their own nor so much as acknowledged.

M^r Leibniz appealed from the Commercium Epistolicum to the judgment of the anonymous Author of this Letter: but D^r Wallis an older & abler Mathema-

|5.| M^r Nicholas Bernoulli told M^r Newton from \in/ his Vnkles name \in autumn 16 1712/ that there was an error in the \conclusion of the/ solution of Prob. III Lib. II Princip. Philsoph. Math. M^r w^ch his he suspected to lye in second differences M^r Newton corrected the error himself, shewed him the correction & told him that the Solution Proposition should be reprinted in the E new Edition w^ch was then coming abroad. The Tangents of the Arcs GH & HI are \first/ moments of the Arcs FG & FH & should have been drawn the same way with the motion describing those arcs, whereas \through inadvertency/ one of them had been drawn the contrary way, & this was \occasioned/ the error in conclusion.

There is an error of much greater consequence committed in second Differences by M^r Leibnitz L . . . . z \A.C. 1689/ in his Tentamen de motuum cœlestium causis sect 15: which tho off often complained, of is not yet corrected nor so much as acknowledged. Nor doth it appear by any instance that M^r L . . . . z knew how to work in second differences before M^r Newtons Book of Principles came abroad: whereas by it \plainly/ appears by Prop. XIV. Lib II Princip. that M^r Newton wⁿ \he wrote that Book/ knew how to work in second differences w^ch \in demonstrating that Proposition |he there|/ he there /he {he there}\ calls Differentia Momentorum & \id est/ momentum differentiæ. And M^r Leibnitz himself in the Acta Eruditorum for Iu May 169 1700 has acknowledged that M^r N. was the first who shewed by as specimen made publick, that I {illeg} he had the method of maxima & minima in infinitesimals, & there called it a method of the highest moment & largest \greatest/ extent. And in the Analysis per series numero terminorū infinitas communicated by D^r Barrow to M^r Collins in the year 1669 M^r Newton mentions that by that method of Analysis Curvarum areæ & longitudines &c id modo fiat) exacto et Geometrice determinantur. And how this done \by the Method of F{illeg}/ is explained in the first six Propositions of the Book of Quadratures. And without that Method it cannot be done.

And M^r Collins in a Letter to M^r Tho. Strode dated 26 Iuly 1672, & \still/ extant in \&/ printed fro{illeg}{t}|m| the Original in the Commercium Epistolicum; mentioning the Papers w^ch w^ch he had received from D^r Barrow in the Iuly 1669, subjoyns: Ex quibus [chartis] et alijs quæ olim ab autore cum Barrovio communicata fuerant, patet illam methodum a dicto Newtono aliquot annis antea excogitatam et modo universali applicatam fuisse: ita ut ejus ope in quavis Figura Curvilinea proposta {sic} quæ una vel pluribus proprietatibus definitur, Quadratura vel Area dictæ figuræ, acc ACCVRATA si possibilis SI POSSIBILIS SIT, sin minus infinite vero propinqua; Evolutio vel longitudo lineæ curvæ, Centrum gravitatis Figuræ, Solida ejus rotatione genita, & eorum superficies: sine ulla radicum extractione [lege exterminatione] obtineri queant. Thus by the testimony of D^r Barrow & M^r Collins, as well as by that of D^r Wallis, M^r Newton had the method described in the first six Propositions of the Book of Quadratures, some years before Iuly 1669. And these three ancient & able Mathematicians knew what they wrote: but the author of the Letter of 13 Iune 1713 knew {illeg} above mentioned, {illeg} wrote only by conjecture. His words are Ejus conjecturæ meæ primum validissimum indicium est &c. He accused M^r Newton of plagiary without any better proof then conjecture & therefore is guilty of calumny, even by the concession of M^r Leibnitz. See above pag in the end of his Letter of 9 April 1716 \to M. Conti/ printed above pag

M^r Newton in the Introduction to the {illeg} Book of Quadratures {illeg}|s|aid that he invented the Method of fluxions gradually in the years 1665 & 1666. This Book was published in the year 1704 which was nothing more then what \not so much as/ D^r Wallis had published in the Introduct Preface to the first Volume of his works 1695 & notified to M^r Leibnitz \in a Letter to him dated 1 Decem. 1696 without being then contradicted/ without being then contradicted {sic} [as I understand by the Acta Eruditorum for        169      . This Book was published by M^r Newton in the year 1704 & the next year in giving an Account of it M^r Leibnitz wr in the Acta Eruditorum for the month of        pag      M^r Leibnitz is called the Inventor of the Method & the improvement there of is ascribed to M^r Bernoulle|j| & the Marque{illeg}|s|s de l'Hospital, & thence is inferred that M^r Newton always used fluxions instead of the Leibnitian Differences \just/ as Honoratus Faber substituted the progress of motions for the indivisibles in the Acta Philosophical Transactions f A.C. 170     represented that M^r Leibnitz had the Method from M^r Newton. M^r Leibnitz] In the Acta Eruditorum for Iune 1696 an Account is given of the two first Volumes of the works of D^r Wallis, & notice is taken of what w|is|as {sic} published in the Preface to them|ose| \two Volumes/ relating to this meethod, without denying what was there said or complaining of the D^r for saying it. The Author of that Account {s}{illeg} in relation to what is said in the Preface has|t|h only \indeed/ these words. Cæterum ipse Newtonus non minus candore quam præclaris in rem Mathematicam meritis insignis, publice et privatim agnovit Leibnitium tum cum (inter veniente celeberrimo Viro Henrico Oldenburgo 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 & pro ijs quo Methodos generales habuisse; quod Wallisius in præfatione Operum factæ inter eos communicationis mentionem faciens, præterijt, quoniam deo eo fortasse non satis ipsi constabat. And upon these words there is this Observation in the Commercium Epistolicum. Methodum Differentialem Moutoni – – – – N . . . . s nondum agnovit publice. I add that I never did acknowledge it either publickly or privately. M^r L . . . z printed the published the Elements of his Method A.C. 1684 \without making any mention of the correspondence he had formerly had with me/. M^r N . . . . n demonstrated the Elements of this d Method of moments \the next year/ in the second Lemma {illeg}|o|f y^e second book of Principles & added a Scholium not to give away this {illeg} Lemma to M^r L . . . z but to assert it to himsel in a civil manner, M by putting him in mind of that correspondence & of making making that ackledgment {sic} publickly w^ch he made privately in his Letter of 13 Iune 21 1677 wherin he first began to communicate this Differential method, & acknowledg that M^r Newton's Method did the same things mentioned in his Letters of 13 Iune & 24 Octob 1676 as a part of a Treatis wrote by him in 1671, did the same things.

The Acta Eruditorum for Iune 167|9|6 came to the hands of Doctor|^r| Wallis in the end of November 171 following, & the D^r {illeg} in a letter wro|i|te|n| th to M^r Leibnitz the next day Decem. 1^st, has these words. Neque Calculi Differentialis vel Nomen audivisse me memini – – – – – id monitum inseruerim. Thus M^r Leibnitz had fresh notice of the Paragraph inserted into the said Preface of the two first Volumes of the D^rs Works, & yet neither denied \in the Letters w^ch afterwards passed between them denied not/ what the D^r \had/ there p had there published nor exprest himself in the least affended at it, [but on the contrary in his Letter to D^r Wallis dated $\frac{19}{29}$ Mar. 1697 said De Te autem quæri mihi {illeg}|n|unquam mihi in Mentem venit; quem facile apparet nostra in Actis Lipsiensibus prodita, non satis vidisse. He only took notice \represented that/ that the D^r had said but little about the|i|s performance|s| of M^r Leibnitz published in the Acta Eruditorum but|&| acquiesed in the D^rs excuse who seldome read those Acta; but blamed not the D^r for saing that M^r Newton in his Letters of Iune 13 Iune & 24 Octob. 1676 had explained to M^r Leibnitz the Method in dispute found by him ten years before or above.]

In autumn 16|7|13 I received from M^r Chamberlain (who then kept a correspondence w^th M^r Old Leibnitz) a flying Paper \in Latin/ dated 29 Iuly 1713 in w^ch it was pretended that M^r Leibnitz being then at Vienna had not then the Commercium Epistolicum nor had time to examin it himself but had referred it to the judgment of a very e|f|amous Mathematician who was impartial & very able to judge of it, & had received his judgment in a \the ther following/ |a| Letter of \dated/ Iune 7^th 1713,|.| as f{illeg} Then \And/ this Letter was inserted into the flying Paper, & in the end of the Letter M^r Iohn Bernoulli was cited by the author of the Letter as a Person different from himself, in these words [quem admodum ab enminente quodam Mathematico dudum notatum est. For these words referred to a Paper publisheed by M^r Iohn Bernoulli in the Acta Eruditorum of {illeg} Feb. et Mar. 1713.

This flying Paper was print translated into French \& inserted into nother Letter of the same stile of y^e former/ & printed in|at| the Hague in Holland in the \M^r Iohnsons/ Iournal Literaire of M^r Iohnson {A}{illeg} of Novem & Decem 1713 pag 448, 449, 450 & 451. And two years after it was printed again in Holland in the Novelles Litteraires & in this Imp{r} \for Decem. 28 1715/ pag 414 & in|th|e sentence [quemadmodum ab eminente quodam Mathematico dudum notatū est] was \now/ left out & the world was told that M^r Iohn Bernoulli was the the author of the \aforesaid/ Letter of Iune 7 1713. [And the next year M^r Leibnitz in a Letter {s}ent b{y} written by himself \Apr. 18. 176|1|6/ & sent by Madam Pelniz to Madam Kilmanseg, he int|s|ertd the same translation of the Letter said Letter of Iune 7 1713 omitting the words [quemadmodum ab eminente quodam Mathematico dudm notatum est,] & affirming that M^r Bernoulli was the author of the Letter. And I have been told] |And about two years after M^r Leibnitz began to father the said Letter upon M^r Bernoulli & for tht end to omitt the said citation in the copies of that Letter w^ch he then sent to his friends. For|

In November or December 1714|5| M^r Leibnitz \he/ wrote a Letter to Sen^r Conti with a Postscript {illeg} relating to me. And in the Postscript was this sentence. Il ne paroit point que M. Newton ait eu avant moi la Characteristique & l'Algorithm infinitesimas|l|, \in which were these words/ suivant ce que M. Bernoulli a tres-bien jugé. [And thereby I knew that M^r Leibnitz was ascribed the L to M^r Iohn Bernoulli the aforesaid Letter of Iune 13 7 1713. But I doubted of it & in my Answer dated 26 Feb. 171$\frac{5}{6}$ & {illeg} I called that Letter the Answer of the Mathematican|ia|n \or pretended mathematician/ meaning dated Iune 7 1713: meaning the Answer of the Mathematician {illeg}|w|ho either was & by the p|w|ords pretended Mathematician I meant the Mathematician pretended to be the author of the said Letter. For because [the words [quemadmodum ab eminente quodam Mathematico dudum notatum est] were omitted in order to fit] the author of the Letter cited M^r Bernoulli as a man different from himself I doubted whether M^r Bernoulli was the true author or only the prended {sic} author of the Letter.]

|And| About the same time [that M^r {illeg} Leibnitz sent the said Postscript to Sen^r Conti] he sent \a letter with/ the {t}{illeg}{n} aforesaid flying paper to the author of the Novelles Litterairs in Holland who printed it \them both/ in that Collection Decem 28 1715 pag {illeg}|4|14: & the sentence [quemadmodum ab eminente quodam Mathematico dudum notatum est] was now left out, & the wo\r/ld was told that M^r Iohn Bernoulli was the author of the afors|e|said Letter of 1{7} 7 1713. And the next Spring \soon after in/ in {sic} a Letter written by himself & Apr 18 1716 & sent by Madam Pelniz to Madam Kilmanseg he ascribed the same in{illeg}|s|erted a copy of the same Letter of Iune 17 1713 & ascribed it to M^r Be{illeg}|r|noulli omitting the aforesaid sentence by w^ch the Author of the Letter had distinguished himself from \cited/ M^r Bernoulli as a man different from himself. And \again/ in the Postscript of a Letter written about the same time to Count Bathmarke he affirmed that M^r Bernoulli was the author. [{illeg}|Bu|t I yet I doubted of this (for the reason above mentioned untill I received M^r Bernoulli's <482v> of Iuly 5^th, 172|1|9 & by that Letter I became fully satisfied that M^r Bernoulli was not the Author, he therein asserting per omnia humanitatis sacra that he wrote no such Letter.
The Author of the Letter inciting M^r Bernoulli as a person different from himself is one witness that M^r Bernoulli did not write that Letter. M^r Leibnits who sent that Letter to the Presse with that citation in it is another witness. And M^r Bernoulli is a third. [And he that falsified the Letter by striking out the citation in order to father it \the Letter/ /it\ upon M^r Leibnitz Bernoulli is a witness that it cannot be fathered upon \him/ without a falsification] It cannot be fathered upon M^r Bernoulli without falsifying the Letter by striking out that citation.] And \But/ these two \last/ Letters were not printed \not published till/ about two years after, & I had no hand in publishing them. The three or four \first four or five/ f{illeg} sheets of the second part of M. Desmaizeans collection {illeg} (in w^ch these letters are conteined, were printed off in Holland before I knew any thing of the designe to publish them.

About the time that those Letters went into the Press, M^r Des Maizeaus received from Se^r Conti then at Paris several manuscript pieces of M^r L|eibnit| as he mentions in his Letter to Sen^r Abbe Conti dated 21 Aug. 1718, \& printed in the said second part pag. 362, 363./ saying that \In that Letter he said that/ he would add them|se| \pieces/ to the other writings of M^r Leibnitz w^ch he was going to print in Holland. And you have them in the said second Part. M^r Des-Maizeaus had a correspondence with M^r Leibnitz & was his friend & on that account published this collection of his Remains.

About |e|t|l|e{n}|v||en| months after the writing of this Letter, when the Collection was almost printed off except the {illeg}|p|reface, I received \from you/ M^r Bernoullis Letter dated Iulij 5. 1719 in w^ch he assured me that he wrote no such Letter \to M^r Leibnitz/ as that dated 7 Iune 1713, \notwithstand/ & in my Answer I acquiesced in that Declaration notin & have ever since told my f{illeg}s|rien|ds that I am satisfied that M^r Iohn Bernoulli was not the author of that Letter, [notwithstanding what M^r Leibnitz had written in the affirmative.] The author of the Letter in citing M^r |I.| Bernoulli as a person different from himself is one \a/ witness that M^r I. Bernoulli was not the author thereof. M^r Leibnitz himself in sending that Letter to the Press both in Germany & in {illeg} Holland with the in the year 1713 with that citation in it, is another witness that he then knew that M^r Iohn Bernoulli was not the author M^r Iohn Bernoulli in affirming per omnia humanitatis sacra that he wrote no such {L} anonymous Letter is a third wit{h}nes, & the leaving \out/ that citation in order to father the Letter upon M^r I. Bernoulli is a falisification of the Letter. And for these reasons I \have been &/ am still of opinion that M^r I. Bernoulli was not the author.^[1]

S^r

The author of the Letter of Iune 7^th 16|7|13, as it was at the first sent to the Press by M^r Leibnitz, cited M^r Iohn Bernoulli as a person different from himself in these words [quemadmodum ab eminente quodam mathematico dudum notatum est.]

In the winter between the years 1715 & 1716 M^r Leibnitz began to father the said Letter upon M^r Leibnitz & for that end to leave out the citations abovementioned in the copies of it w^ch he then sent to his friends: as I find by his Letter to Sen^r Conti writ in November or December 1715 where he hus {sic} this expression suivant ce que M. Bernoulli a tres bien jud|g|e. & by his Letter sent at the same time to the author of the Novelles Litterairs in Holland & his Letter of Apr 18 16|7|16 [in both w^ch he inserted copies of the said Letter of 16 7 Iune 16] sent to Madam Kilmanseg & his Letter sent about the same time to Count Bothmar.

But by consid

Observations upon the foregoing Letter.

a Here M^r Newton is directly accused of plagiary & the whole Letter tends to prove it.

1. In the second Lemma of y^e second Book of Principles the Elements of the Method of fluxions are taught, without \with/ prict letters & in the Introduction to the Book of Quadratures the method it self is expresly taught & illustrated by examples \& \all/ this is done/ without the use of prickt Letters.

2 It is falsly alledged \that in the Book of Principles/ there was frequent occasion of using the calculus of fluxions: for the book was written by the method of Composition.

3 \The third Volume of D^r Wallis came abroad in Spring 1699 &/ Prickt letters appeared in the second Volume of the works of D^r Wallis w^ch was almost all printed in the year 1692 & came abroad the next Spring before the Differential method began to make a noise. |And the MS of D was in the hands of D^r Halley & M^r Raphson in the year 1691, & is sufficiently described in M^r Ns Letters of 24 Oct 1676 & Novem 8. 1676 & the Quadratures there cited out of it are not to compassed without the method of fluxions.|

4 The first Proposition of the Book of Quadratures conteins the true method \rule/ of finding the first second third & fourth fluxions &c and was published \with examples/ in the second Volume of the works of D^r Wallis with{illeg} examples i first & {illeg} second fluxion & by consequence some years before any other rule for finding second & third \& fourth/ differences came abroad.

5 The constant moment of x w^ch I \M^r Newton/ sometimes notes w^c|t|h x pointed I|he| still note|s| sometimes w^th o & finds it advantageous

6 By the an Eminent Mathematician \here cited/ the author of the Letter meant M^r Bernoulli somebody \a Mathemation/ different from himself, and {to} nor could \&/ M^r Leibnit {sic} \{co}uld not/ take them for \one &/ the same \person/ when he first published this Letter. And therefore if M^r Iohn Bernoulli was the eminent Mathematician he was not the Author of the Letter, as he himself assures us that he was not This Letter was \translated into French &/ reprinted in \in French Decem. 1713 in/ the Iournal Literair Lib. II. Part II. pag 450 Anno Chr. 171 in Decem. 1713. And again in the Novelles Litterairs Decem 28 1715 pag 414. And in this last edition we{re} are told that M^r Iohn Bernoulli was the Author of the Letter, & to make the Reader beleive this, the citation of the Eminent Mathemation is omitted in the body of the Letter as it is also in the same Letter inserted by M^r Leibnitz into his Letter to Madam Kilmansegg dated 18 Apr. 1716 & {illeg}.^[2] [M^r Newton in his Letter of 26 Febr. \176|1|6/ printed above pag 17 called the Author of this Letter a Mathematician or pretended Mathematician & M^r Leibnitz in his Answer calls him dated 9 Apr. 1716 blames \him/ for calling a Mathematician of the first rank a pretended Mathematician And M^r Newton in his Remarks dated 29 May S|1|716 St. n. replied that it was \above/ two years before we told that after the writing of that Letter, before we were told that |t|he was M^r Iohn B the Mathematician was M^r Iohn Bernoulli, & that he did not call|ed| him a Mathematician or pretended Mathematician, t|n|ot to disparage his skill, but only to express his do{illeg}|u|bt whether M^r Iohn Bernoulli was really the author of that Letter or only pretended to be the Author thereof. And therefore what he said in the Papers printed above in relation to M^r I. Bernoulli's|h||is| being the author of that Letter was only upon a supposition that he was the author. But it appears now \by his Letters to M^r Newton/ that he was not the author: & therefore all that was said upon that supposition falls to the grownd & ought not in the least to reflect upon M^r Bernoulli For him] But on the contrary M^r Iohn Bernoully has told us solemnly that he was not the Author] M^r Leibnitz has done very ill therefore to omit this citation in or{illeg}|d|er to father the the Letter upon M^r Iohn Bernoulli. For as it appl{i}{illeg}|ea|rs by this citation that M^r |I.| Bernouly was not the Author so M^r |I.| Bernoulli has in a Letter to M^r Newton solemnly declared that he was not the Author. And since, no other author then M^r Leibnitz appears either of this Letter or of the Letter of the Letter of 29 Iuly 167|1|3 into w^ch this was inserted: M^r Leibnitz ought to pass for the Author of them both. For this last mentioned \named/ Letter mentions what passed between M^r Leibnitz & M^r Hugens at Paris in the year 1675 w^ch no body about 40 years before which no body in Germany knew besides M^r Leibnitz & therefore no body else could write the Letter; A & the word illaudabilis used in this Letter is peculiar to M^r Leibnitz. Had not the two Letters been feigned by M^r Leibnitz, but really written by other Authors of note, it would have been the interest of M^r Leibnitz to have named his Authors when he first published the Letters. And therefore since he did not <484r> name his authors but endeavoured above two years after to father one of them upon M^r Bernoulli & {ce} & certainly wrote the other himself: its plain that he wrote them both himself & by consequence had the Commercium Epistolicum by him when he wrote them, but t{illeg} found it unanswerable & therefore betook himself to the writing of Libels against i. pretended that he had not seen it nor had leasure to answer it, & betook himself to the writing of \defamatory/ Libels against it.

6 By the eminent Mathematian here cited the Author of the Letter meant a Mathematician different from himself. And M^r Leibnitz could not take them for one & the same person when he first received this Letter & sent it to the Press. And therefore if M^r Iohn Bernoulli was this eminent Mathematician he was not the auther of the Letter. How ever in translating \printing/ this Letter translated into French in the Novelles Literairs Decem 28, 1715, we{illeg} are told that this the Author of this Letter was M^r Iohn Bernoulli, & {illeg} to make this credible the citation of the Eminent Mathematician is omitted in the body of the Letter. But M^r Iohn Bernoulli in a Letter to M^r Newton hath positively declared that he was not the author thereof.

A Letter dated 7 Iune 16|7|13 being mentioned above with some as written by M^r Iohn Bernoulli; to set that matter right we will here set down that Letter with the following Observations upon it.

Videtur Ne N . . . . us occasionem nactus, serierum opus multum promovisse per Extractiones Radicum, quas primus in usum adhibuit, & quidem in ijs excolendis ut verisimile est ab initio omne suum studium posuit, nec credo tunc temporis^{2 1} vel somniavit adhuc de Calculo suo fluxionum & fluentium vel de reductione ejus ad generales operationes Analyticas ad instar Algorithmi vel Regularum Arithmeticarum aut Algebraicarum. Ejus meæ conjecturæ [primum] validissimum, indicium est² quod de literis x vel y punctatis uno duobus tribus &c punctis superpositis, quas pro dx, ddx, d³x; dy, ddy, &c. nunc adhibet, in omnibus istis Epistolis [Commercij Epistolici Collinsiani unde argumenta ducere volunt] nec volam nec vestigium invenias. Imo ne quidem^{2} in Principijs Naturæ Mathematicis N . . . . . i, 3 ubi calculo suo fluxionum utendi tam frequentem habuisset occasionem, ejus vel verbulo fit mentio, aut notam hujusmodi unicam cernere licet, sed omnia fere per lineas figurarum sine certa Analysi peraguntur more non ipsi tantum, sed & Hugenio, imo jam antea [in nonnullis] dudum Torricellio, Robervallio, Cavallerio, alijs, usitato. Prima⁴ vice hæ literæ punctatæ comparuerunt in tertio Volumine operum Wallisij, multis annis postquam Calculus differentialis jam ubi locorum invaluisset. Alterum indicium, quo conjicere licet Calculum fluxionum non fuisse natum ante Calculum differentialem, hoc est,^4|5| quod veram rationem fluxiones fluxionum capiendi, hoc est differentiandi differentialia, N . . . . . . us nondum cognitam habuerit, quod patet ex ipsis Principijs Phil. Math.⁵ ubi⁵ non tantum incrementum constans ipsius x quod nunc notaret per x punctatum uno puncto, designat per o [more vulgari qui calculi differentialis commoda destruit] sed etiam⁵ regulam circa gradus ulteriores falsam dedit [quemadmodum^6|7| ab eminente quodam Mathematico dudum notatum est:] . . . . . Saltem apparet, N . . . . . . o^4|5| rectam Methodum differentiandi differentialia non innotuisse longo tempore postquam alijs fuisset familiaris. &c.

Observations

1 See the Preface to D^r Wallis his the first Volume of the

1. M^r Newton is here accused of plagiary \by a mere conjecture/ & the whole Letter tends to prove it \make it probable/ contrary to the testimoney of \an abler & older Mathematian {sic}/ D^r Wallis in the Introduct Preface to the first Volume of his works.

2 In the second Lemma of the second Book of Principles the Elements of the secon Method of fluxions are taught, & in the Introduction to the book of Quadratures the Method it self is expresly taught & illustrated by examples, & all this is done without the use of prickt Letters.

3 In the Book of Principles there was not occasion of using the calculus of fluxions. For the book was writ by the Method of Composition.

4 {illeg} The third Volume of the works of D^r Wallis came abroad in Spring 1699. And prickt letters appeared in the second Volume of his works w^ch was almost all printed in the year 1692, & came abroad the next Spring before the Differential method began to make a noise. The Manuscript of the book of Quadratures was in the hands of D^r Halley & M^r Raphson in the year 1691 as both of them have attested And this book is sufficiently described in M^r Newtons Letters of Octob 24 & Novem 8. 1676, & the Quadrature|s| there cited out of it are not to be compassed \attained/ without the Method in dispute.

5. The first Proposition of the Book of Quadratures conteins the true Rule of finding first second third fourth & {illeg} \& other/ fluxions &c and was published with examples in the second Volume of the Works of D^r Wallis some years before any other Rule for finding second third & fourth differences came abroad. And the \very/ words of the Rule were inserted {illeg} set down in the s|S|cholium upon the second Lemma of the second Book of Principles, & in M^r Newtons Letter of 24 Octob 1676, as the foundation of the Method of fluxions.

6 M^r Newton denotes fluxions by pricks|t| letters & moments by fluents by any letters, their fluxions by the same letters with pricks set over them, & their moments by their fluxions multiplieed by the letter o & its powers ${\mathrm{o}}^2$ ${\mathrm{o}}^3$ ${\mathrm{o}}^4$. And where he puts the letter o for the constant moment of x he puts an unite for \$\stackrel{.}{\mathrm{x}}$ that is for/ the constant fluxion of x, & still uses this notation as very advantageous. M^r Leibnitz hath no notation for fluxions, w^ch is a defect in his to denote the the {sic} moments of fluents (w^ch he calls their differences) præfixes d, dd, ${\mathrm{d}}^3$, ${\mathrm{d}}^4$ &c \to the fluent/ but has no notation for their fluents, w fluxions; w^ch is a defect in his method.

7 By the eminent Mathematician here cited, the Author of the Letter meant a Mathematician different from himself. And M^r Leibnitz could not take them for one & the same person when he first received this Letter & sent it to the Press. And therefore if M^r Iohn Bernoulli was the Author of this Letter he could not be eminent Mathematician, he was not the author of the Letter. However, in printing this Letter translated into French, in the Novelles Literaires Decem 28 1715, we are told that the Author of this Letter was M^r Iohn Bernoulli, & to make this credible, the citation of the eminent Mathematician is omitted in the body of the Letter. But M^r Iohn Bernoulli is a Letter to M^r Newton hath positively declared that he was not the author thereof.

$\begin{array}{cccc}16\prime .57⁤\frac{1}{3}\times 60\colon\colon 1.14⁤\frac{1}{3}\times 15=143⁤\frac{1}{3}=215.& 1935\phantom{⁤\frac{1}{2}}& 2114\times \overline{)\frac{10000}{790}}& 79\left)2114000\right(26759⁤\frac{1}{2}\\ \phantom{16\prime .57⁤\frac{1}{3}\times 60\colon\colon 1.14⁤\frac{1}{3}\times 15=}\phantom{1}71⁤\frac{2}{3}\phantom{=215.}& 107⁤\frac{1}{2}& & \phantom{79)}158\phantom{4000(26759⁤\frac{1}{2}}\\ & \overline{2042⁤\frac{1}{2}}& & \phantom{79)}\phantom{1}534\phantom{000(26759⁤\frac{1}{2}}\\ & 81⁤\frac{1}{2}& & \phantom{79)}\phantom{1}474\phantom{000(26759⁤\frac{1}{2}}\\ & \overline{2114.}& & \phantom{79)}\phantom{10}600\phantom{00(26759⁤\frac{1}{2}}\\ & & & \phantom{79)}\phantom{10}553\phantom{00(26759⁤\frac{1}{2}}\\ & & & \phantom{79)}\phantom{100}470\phantom{0(26759⁤\frac{1}{2}}\\ & & & \phantom{79)}\phantom{100}395\phantom{0(26759⁤\frac{1}{2}}\\ & & & \phantom{79)}\phantom{1000}750\phantom{(26759⁤\frac{1}{2}}\\ & & & \phantom{79)}\phantom{1000}711\phantom{(26759⁤\frac{1}{2}}\\ & & & \phantom{79)}\phantom{10000}39\phantom{(26759⁤\frac{1}{2}}\end{array}$

$\begin{array}{c}32.60,60,4=\phantom{00}7200,64\\ \phantom{32.60,60,4=}\phantom{0}57600\phantom{,64}\\ \phantom{32.60,60,4=}460800\phantom{,64}\end{array}$

Diam ☉ = ${17}^{\mathrm{IV}}$.

Diam Orbis M. = $30‴$

$\begin{array}{cc}\text{As the semidameter □ of Saturns Orb }& \hfill 953800\square \\ \text{To the semidiameter □ of the ☉ }& \hfill 942\square \end{array}$

$\begin{array}{cccc}\text{So is the density of the Suns light at his surface }& \hfill 10000000000& \text{or}& \hfill 1000\square \\ \text{To the density of the Sun's light at Saturns Orb. }& \hfill 10000& & \hfill 1\square \end{array}$

On the Charta Volans of 1713

Since the printing of the foregoing Letters I hav the folloing Observations upon the a flying paper dated                have

Observations upon the foregoing Letter

1. The author of the Letter asc cited|s| M^r Iohn Bernoulli as a person different from himself, & doth it with an Elogium by the title of a certain Eminent Mathematician, & thereby denyed that M^r Iohn Bernoulli was the Author.

2 Two years & an half after the publishing of this Letter it was translated into F a translation of it \into French/ was published in Holland the Novelles Literaires in Holland & the there \(as also in a Letter |o|t|f|o M^r Leibtz to Madam Kilmansegg)/ this Letter was ascribed to M^r Iohn Bernoulli & to \make this credible the/ aforesaid citation was omitted. Whereas if M^r \And M/ And the same three And M^r Leibn^z in 3 M^r Iohn Bernoulli {illeg}d not but know that that the first a Letter to Madam Kilmang|s|eg has \Which act of omitting it is not justifiable but savours of falsification & this/

3 M^r Iohn Bernoulli \could not but/ know that \in/ the book of Mathematical Principles the Elements of the Method of fluxions were set down without the use of prickt Letters in the second Lemma of the second book of \math./ Principles \of Philos./, & that there was no occasion to use prickt Letters in that book because it was written by the method of Composition, & that the true first Proposition of the Book of Quadratures was pri|u|blished \with prickt letters/ in the second Volume of the secon works of D^r Wallis w^ch came abroad in Spring 1693 before \at w^ch time at w^ch time/ the Differential Method \was only/ began|ining| to be talked of in England \make a noise/ {As} that that the|i||s| Proposition conteins the true Rule of finding first second third fourth &c differences \other fluxions/ & was there illustrated with examples in first & second d|f|luxions & \& {sic} that th{is} was/ the first Rule made publick for this purpose. And for{illeg} these reasons he could not \finding second & third/ \differences. And all this makes it highly improbable that M^r Bernoulli could should/ be the author |of| the said Letter.

4 M^r Iohn Bernoulli has in a letter written M^r Iohn Newton 5 Iuly N. S. 1719 hath decle|a|red that he was not the Author of the said Letter. His words are.

About half a year after the publication of the Commercium Epistolicum M^r Leibnitz \it was pretended/ /{illeg}\ In a flying paper dated {illeg} 29 Iuly 1713 \it was/ pretended that he \the Leibnitz/ had not seen the book but hearing th had written to an eminent |a| Mathematian {sic} of the first rank, who \very skilfull in these matters who/ upon examining all things had given his opinion of the matter in the follo in a Letter dated 7 Iune 1693 as follows.

Videtur N . . . . s — postquam alijs fuisset familiaris.

Vpon this

This Letter therefore conteins the judgment of an eminent Math or pretended judgment of a Mathematician or pre to whom M^r Leibnitz appealed from the judgment \report/ of the Committee of the R. S. It was printed in Germany without the name of the author or printer \or Mathematician/ or city where it was printed & \it was/ dispersed over Europe two years & an half before we were told who was the author that M^r Iohn Bernoulli was the author of it. {illeg} This were told {in} This \That he was the Author/ we were told in the Novelles Literairs for the month of             1716 \& afterwards is|n| s{illeg}|o||me| Letters of M^r L. to m{illeg}{d}{illeg}/ & to make it probable, the citation of M^r Iohn Bernoulli in the body of the by w^th an Elogium in |y^e| body of the Letter was omitted. But

But in that citatio we have the testimony of the author of the Letter that M^r Iohn Bernoulli was a person different from himself. \not the author./                  And The omission of the citation cannot he justified. It amounts to a falsification of the Letter for suppressing the|is| evidence that testimony of the author of the Letter that M^r Iohn Bernoulle|i| was not the author thereof.

M^r Iohn Bernoulli could not but know — — — be y^e author of the said Letter.

M^r Iohn Bernoulli in a letter — — — his words are — And so we have two witnesses the author the Letter & M^r Iohn Bernoulli that M^r Iohn Bernoulli was not the author.

The Letter therefore is of no credit the Author of it being still unknown.