Fragment of "An account of the Differential Method from the year 1677 inclusively"

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And Account of the Me Differential Method from the y{e}ar 1677 inclusively^[1]

|②| M^r Iames Bernoulli in the ge Acta Eruditorum mensis Ianuarij 1691 pag. 14, gave the\an/ account of the Differential method in \these words/ Quanquam, ut verum fatear, qui calculum Barrovianum (quem decennio ante [i.e. ante annum 1784] in Lectionum inibi contentarum farrago) intellexerit alterum a Dn. Leibnitio inventum ignorare vix poterit; utpote qui in priori illo fundatus est, & nisi forte in differentialum numero notatione & operationis aliquo compendio ab eo non differt.. ① And that very candid Gentleman the Marquis de L'Hospital in the Introduction to his Analysis, tells us that M^r Fe\r/mat found a method of Tangents w^ch Des Cartes allowed to be often better then his own. That D^r Barrow made it more simple & adopted a proper calculation to it, but wanted this was \still/ wanting, \viz/ to exclude fractions & radicals in {disit} in the application in d in using this method. In default of w^ch the calculus of M^r Leibnitz succeeded who began where D^r Barrow left off.

These two Gentlemen knew nothing of what M^r Leibnitz might\had/ received from England by the means of D^r Olden M^r Oldenburg. M^r Leibnitz never acknowledged to them any thing of that knid. In the Acta Leipsica \Eruditorum/ he never made any acknowledgm^t of any thing \advantage/w^ch he had received either from D^r Barrow or from M^r Newton \or from M^r Gregory/ or from M^r Collins or M^r Oldenburg or any body \else/ in England, unless where he could not avoid it. He never acknowledged any thing more of that knid then what was published by D^r Wallis, & therefore it is but just that the world should know it from other hands what he has further received from England M^r Oldenburgh

M^r Newton in his Letter of 10 Decem. 1672 & \(a copy of/ w^ch was sent to M^r Leibnitz by M^r Oldenburge amongst the Papers of M^r Gregory in Iune 1676,) & in his Letters of \13 Iune &/ 24 Octob. 1676 w^ch M^r Leibnitz received the next year in spring gave notice to M^r Leibnitz that he had a method whereof the method of Tangents of Slusius was but a branch or Corollary, & that this Method stuck not at extended to the abstruser sorts of Problemes about the curvatures, areas lengths, areas, solid contents, \&/ centers of gravity &c of lines & figures \& to inverse problemess of Tangents & others more difficult & succeeded in/ & extended to mechanical curves as well as o\in/ others & proceeded without sticking at surds, & made the method of Series so universal as to reach to almost all Problemes except perhaps some numeral ones like those of Diophantus. And \he gave examples of this Method in drawing of Tangents & squaring of Curves &/ the foundation of this method he compehended in this sentence exprest a enigmatically, Data æquatione Æ quotcun fluentes quantitates involvente fluxiones invenire & vice versa. And a part of the inverse method he exprest \enigmatically/ in this sentence Ex æquatione fluentes quantitates involvente fluxiones invenire et vice versa Ex æquationes fluxiones involvente flux{illeg}|en|tes|m| extrahere \Extrahere Fluentem quantitatem ex æquatione simulo involvente fluxionem ejus/. In both w^ch sentences the word fluxiones relates to the second third & following fluxions as well as to y^e first. And if all this be added to D^r Barrows Lectures published N.C 1670, there will be nothing more left for M^r Leibnitz but a new \notation/ Leibni

When M^r Leibnitz received this information he could not at first believe M^r Newtons Method was so general: for he wrote back in his Letter dated 27 Aug 1676 th Quod dicere videmino pleras difficultates (exceptis Problematibus Diophantæis) ad Series infinitas reduci, id mihi non videtur. Sunt enim multa us adeo mira et implexa ut ne ab æquationibus pendeat ne ex Quadraturis: qualia sunt (ex multis alijs) Problemata methodi tangentium inversæ; quæ etiam Cartesius in potestate non esse fossus est. And M^r Newton \in his Letter of 24 Octob. 1676/ made answer: Vbi dixi omnia pene Problemata solubilia existere; volui de ijs præsertim intelligi circa quæ Mathematici se hactemus occuparunt vel saltem in quibus ratiocinia Mathematica locum aliquem obtinere possunt. Nam alia sane adeo perplexi conditionibus implicata excogitare liceat, ut non satis comprehendere valeamus; et multo minus tantarum computationum anus sustinere quod ista requirerent. Attamen ne nimium dixisse videar, inversa de Tangentibus Problemata sunt in potestate alio illis difficiliora. Ad quæ solvenda usus sum duplici methodo; una concinniori altera generaliori. Vtram visum est impræsentia literis transpositis consignare <145v> {ne} propter alios idem obtinentes, institutum in aliquibus mutare cogar [Una method{us} consistit in extractione fluentis quantitatis ex æquatione simul involvente fluxionem ejus: altera tantum in assumptione seriei pro quantitate qualibet {illeg} cognita ex qua cætera commode derivari possunt, et in collatione terminorum homologorum æquationis resultantis ad eruendos terminos assumptæ seriei] And {of} all these things being added to D^r Barrows Lectures & method of Tangē{ts} published A.C. 1670, 12 there will be \was/ very little more remaining besides a ne left for M^r Leibnitz \to find out/ besides a new notation & a new name of the method. [Certainly if the Marquess de L'Hospital had seen these thre Letters of M^r Leibnitz left off \began/ where D^r Barrow left off, the other & tang \& improved the method by teaching/ to avoyfractions & radicals & the other would not have ascibed to M^r Leibnitz that little w^ch was wanting to make D^r Barrows method compendious.] By the words of M^r Leibnitz Sunt enim multa us adeo mira et implexe ut ne ab æquationibus pendeant ne ex quadraturis: qualia sunt (ex multis alijs) problemata methodi tangentium inversœ, its manifest that he did not understand when he wrote his Letter of 27 Aug. 1676 he did not understand any thing more the differential method.

Ioachim professed that the father son & {h G.} were una essentia una substantia una natura, but made said that this unity was collective as many men are collectively said to be one people. The Council said that there was not a quaternity because & that Peter Lombard made a quaternity \three persons & one essence/. The \IV/ Lateran Council said that there was not a quaternity because each of y^e persons were that divine essence substance essence or divine nature ut sint distinctiones in personis et unitas in natura.

Peter Lombard \who flourished in \about/ the middle of the 12^th century wrote/ in his sentences had written wrote that \the divine essence was/quædam summa res \quæ/ est Pater est filius est spiritus sanctus & illa non est generani ne genita ne{} procedens. He called that summa res essentia divina speaking of not as a specific nature in w^ch & non genuit Divina essentia. \And gives this reason for his op. Divina essentia/ Non genuit essentiam ne seipsan generareta genuerit. Peter Lombo Ioachim the Abbot. By these last words he seems to have taken the Divine essence not for an \individual/ substance but for a species For \because/ a substance generates a substance but a species does not generate a species \& yet he spaks of this substance essence as a subsistence or being./. /[Thereupon\ Ioachim the Abbot published a book against \him/ calling him a heretick & a mad man & saying because he made as \if/ he had made a quaternity in the Deity instead of a Trinity. And sometime after the Lateran Council w^ch met A.C. 1215, decreed that thre was one summa res w^ch was truly the father son & holy ghost three persons joyntly & severally & therefore there was in God only a Trinity & not a quaternity, each of three persons being that summa res namely \& that summa res being/ the substance essence or divine nature : & that summa res b \w^ch/ neither generates nor is generated nor proceeds ut distinctiones sint in personis, in natura unitas. Which is as much as to say that one & the same nature exists in all the three persons not as a fourth being but as the very nature \& essence/ of the persons, & this \unity of/ nature the Council L calls identitatis in natura unitas.

Ioachim the Abbot who placed the unity of the persons in the unity of will & mind, \& consent/ as man like that (as when many are peop men become \are called/ one people & many faithful one Church,) of ob wrote reprehended him as if he made a quaternity in the Deity instead of an unity a Trinity, three persons & one summa res And sometime after the Lateran Council \composed of 412 Bishops/ w^ch met at Rome A.C. {illeg} 1215. \condemned the opinion of Ioachim &/ defined that there was a summa res & that the sum w^ch neither I begot nor was begotten nor proceeded, but \& that it/ was each of the three persons joyntly & severall & not any fouth thing, t &that it\which/ was the substance essence or divine nature of the persons, not a fourth thing, but the very persons them selves joyntly & severally, ut distinctiones sint in personis et unitas, in natura w^ch unity they call unitas in identitatis in natura unitas|t|i \an unity consisting in the identity of nature/, & say that the father in generating the son gave him his nature substance, not part thereof but the whole, by substance understanding his \so that the father & son have the same substance, meaning the same/ nature or essence. Th|An|d Theodoret: tells us Some of the Montanists deny the hypostases of the divinity as Sabellius, saying \like noetus/ that the father the son & the H.G. are y^e same. They meane the Montanists K{ātæ}

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