Additional notes on the Leibniz Controversy

to the world that he was the first inventor thereof. But for the future {illeg} {illeg} when ever he represents himself the first inventor of exponential equations he ought at the same time to acknowledge that M^r Newton gave him light into the invention by teaching him the use of dignities whose indices were fract surd {illeg}|or| indeterminate. [And whereas he pretends to have enlarged Geometry by Exponential Equations he ought to consider that the indices of all dignities are numbers & where the numbers are fluents, the æquations are Arithmetical & have no place in Geometry. Geometrical quantities may be represented by numbers but are not numbers & Arithmetic may be applyed to the resolution of Geometrical Problems: but such resolutions are only Arithmetical, & have no place in Geometry untill they are demonstrated by Geometrical Proposition & thereby become Geometrical Solutions. Nothing is legally admitted into Geometry \Geometry is composed of nothing else then/ except \besides/ Definitions Axioms Lemmas & Propositions \Lemmas & Corollaries/ demonstrated synthetically, And Analysis is noth whether ancient or modern is only \only/ a mean of resolving Problemes: {illeg} \and/ the Resolutions ought to be turned into Geometrical Solutions by Compos \Geometrical/ Synthesis or Composition before the Problem Propositions be admitted into Geometry.]

M^r Ne S^r M|I|. Newton in his Letter of 24 Octob. 1676 wrote that he had two methods of resolving Invers Problemes of tangents & such like difficult ones, one of w^ch \methods/ consisted in assuming a Series for any unknown quantity from w^ch the other unknown quantities might easily be conveniently be deduced & in the collection of the homologous terms of the resulting equation, for determining the terms of the assumed series. M^r Leibnitz has \s{illeg}|o|me years after/ published this method as his own claiming to himself the first invention thereof. If remains that he either renounce this claim publickly \& acknowledg that M|S|^r I. Newton was the first Innovator/ or prove that he invented it before 24 Octob. 1676 S^r M|I|. Newton wrote the said Letter.

S^r I. Newton in his said Letter of 24 Octob. 1676 used \introduced into Analysis the use of/ fract surd & indefinite indices of dignities & in his letter of 24 Octob 1676 {illeg} proposed \represented/ to M^r Leibnitz th{illeg}|a|t {sic} {use} Problem of resolving \his methods extended to the resolution of/ affected æquations involving dignites|ie|s whose indices were fract or surd. M^r Leibnitz in his answer dated 21 Jun{e} 1677 de mutually desired M^r newton to tell him what he thought of the resolution of æquations {illeg}|in|volving æquation dignities whose indices were surd indetermined quantities as M^r Newton calls fluents, such as are \as in the manner of the following examples such as were/ \the æquations/ ${\mathrm{x}}^{\mathrm{y}}+{\mathrm{y}}^{\mathrm{x}}=\mathrm{x}\mathrm{y}$ & ${\mathrm{x}}^{\mathrm{x}}+{\mathrm{y}}^{\mathrm{y}}=\mathrm{x}+\mathrm{y}$. And these æquations he calls exponentia{illeg}|l|, & has represented to

|pag. 183. \l. 11/| In {illeg}|th|e year 1672 going to Paris he fell acquainted w^th M^r Hugens & in the beginning of the next year came to London, & in February meeting D^r Pell at M^r Boyles.

Pag. 184. l. 1. In the end of February or beginning of March 1672/3 M^r Leibnitz went from London to Paris \{illeg} carrying with him the Logarithmotechnia of Mercator of w^ch D^r Pell had given him \notice// & kept a correspondence with M^r Oldenburg about Arithmetical matters being hith till June{illeg} \following/ being hitherto unacquainted w^th the higher Geometry. But the Horologium Oscillatorum of M^r Hugens being published in April, by the reading first of that book first & then of {illeg} Paschal's Letters & Gregory of s^t Vincent's Book de quadratura circuli &c {illeg} & by the i{illeg} he became acquainted with this Geometry, {illeg} & after some intermission of his correspondence w^th M^r Oldenburg, wrote to him in July 1684 that he had a wonderful Theoreme, w^ch gave &c

Pag. 191. l. 1{illeg}|2|. He found out therefore this new Analysis after Aug. 27 1676. or rather after his being in England \w^ch was/ in October following.

Pag. 189. lin. 25. After the words — he wrote for M^r Newton's — add this section. After he began to study the higher geometry he fell upon h{illeg}|is| Triangulum characteristicum, And {illeg} & invented many particular Theorems like those of Gregory & Barrow, & went on to his Method founded on his Analytical Tables of Tangents & his Combinatory Art. And this was the top of his skill when he wrote his Letter of 27 Aug. 1676. For in that Letter he said of one part of this Method: Nihil est quod norim in tota Analysi momenti majoris. And of another part: Cujus vim ac potestatem nescio an quisquā hactenus sic consecutus. Ea vero nihil differt ab Analysi illa suprema ad {illeg} cujus intima Cartesius non pervenit. Est enim ad eam costituendam opus Alphabeto cogitationum humanarum.

Cæterum per vocem {illeg}|C|orporis hic accipio {illeg} \non pro solido Mathematico sed/ in sensu non mathe vulgi \vulgi id est seu \id est// non Mathematico {illeg} vulgari pro pro substantia tangibili seu quæ tangendo sentiri potest \vel resistentiam creare/; cujus generis sunt [Elementis quatuor vulgaris vulgo sic {dicto} id est] metalla, lapi{illeg}|d|es, arena, lutum, {illeg} salia, \lignum, caro, Argentum vivum,/ aqua, oleum, lac ventus, fumus, \exhalatio/ flamma, &c. |Et quic{illeg}|q|uid su|a|b Elementis quatuor comprehendi potest vel quicquid ab his exhalando emanat & in hæc per condensationem rediret potest| Solida mathematica non sentiuntur tangendo, adeoqꝫ {illeg}|n|eqꝫ apud vulgus corpora nominantur. Siquis contende{illeg}|r|it {illeg} corpora dari quæ tangendo sentiri nequeunt, {illeg} \disputa{illeg}|n|{illeg}tur tantum/ de significatione vocis. Et {illeg} malim loqui cum vulgo, cujus \utiqꝫ/ e{illeg}|s|t lex et norma loquend{illeg}|i|. Quæ tangendo sentiuntur & quæ {illeg} non sentiuntur ita ab invicem distingui poss{illeg}|un|t ut hæc vocentur corpora, illa quovis alio nomine {illeg} \designentur/ & horum genus commune tertio nomine designentur gaudent. Sic /enim\ evitetur \ambiguitas verborum ex qua contentiones oriri solent/ confusio. \& ambiguitas {sic}/ In {illeg} Hoc \Hoc igitur/ sensu {illeg}|v|ocem corporis semper intelligo accipio ubi dico corpora omnia \omnia mobilia esse et tangibilia \& impenetrabilia/ & gravia & vi{illeg}|r|e inertiæ habere/ gravia esse pro quantitate materiæ in singulis & vim inertiæ corporis cujusqꝫ proportionalem esse quantitati materiæ in eodem, & spatia dari corporibus vacua.

Cæterum vocem Corporis hic accipio in sensu vulg{illeg}|i|, nempe pro substan\tia/ tangibili, seu quæ tangendo \resistentiam creat & satis resistendo/ sentiri potest vel resistentiam \motu suo/ crear|t|e: cujus generis sunt metalla, lapides, arena, \argilla/ lutum, salia, lignum, \ossa,/ caro, argentum vivum, aqua, oleum, lac, \sanguis,/ ventus, fumus, exhalatio, flamma, & quicquid sub elementis quatuor comprehendi potest vel ab his exhalando emanare & in hæc per condensationem redire. Solida mathematica non sentiuntur tangendo neqꝫ {illeg} resistentiam creant, ideoqꝫ nec apud vulg{illeg}|us| corpora nominantur. Siquis contenderit corpora dari quæ tangendo nec sentiri nec resistentiam creare qu{illeg}|e|ant, hic jam disputat de vocis signif{illeg}|ica|tione Grammatica {illeg} corpora nominando quæ vulgus corpora non vocat; & malim cum Vulgo loqui, cujus utiqꝫ est l{illeg}|e|x et norma loquendi. Quæ ita tangi possunt et sentiantur vel resistendo sentiantur & quæ non possunt ita tangi, tangendo <97v> premunt et {illeg}unt & \premendo agunt in alia premunt/ /{illeg}\ quæ non premunt \& premendo non agunt in alia/, ita ab invicem distingui possunt, ut {illeg} hæc /priora\ vocentur {illeg} corpora, {al{illeg}|tera|} quovis alio nomine designentur, & horum genus commune, tertio nomine gaudeat. Sic enim evitabi{illeg}|t| verborum \confusio ideorum &/ ambiguitas verborum ex qua \confusio idearum/ contentiones oriri soleant. {illeg} h|H|oc \igitur/ sensu{illeg} vocem corporis semper intelligo \accipio in his Principijs præsertim/ ubi dico corpora omnia mobilia esse, et tangibilia, et impenetrabilia, et gravia, et vires inertiæ habere, & spatia dari corporibus vacua. De natura corporea \substantiarum rerum/ in spatijs resistentia omni destitutis, disputent, per me licet, metaphysici. Hoc ad \Principia {illeg}/ mea nil spectat.

DEFINITIONES.

Def. 1

Corpus voco rem omnem tangibilem [quæ resistentiam tangentibus {sic} creat &] satis resistendo sentiri potest |qua tangentibus resistitur , et \cujus resistentiam \actio/ si satis magna sit /sentiri potest.\/| \Hoc \enim/ vulgus vocem corporis semper accipit. Et/ Hujus generis sunt \astra Planetæ, Cometæ/ metalla, lapides, arena, argilla, lutum, salia, lignum, ossa, caro, aqua, oleus|m|, lac, sanguis, \aer/ ventus, fumus, exhalatio, flamma, et quicquid sub elementis quatuor comprehendi potest, vel ab his exhalando manare & in hæc per condensationem redire. \Planetæ et Cometæ agunt in lucem, & a partibus suis incumbentibus premuntur/ Solida mathematica non sentiuntur tangendo nec resistentiam creant neqꝫ corpora dici solent. # < insertion from f 98r > # Vapores et Exhalationes amittendo & propter \ob/ raritatem suam amitt{illeg}|endo| resistentiam prope omnem sensibilem, & apud vulgus \sæpe/ amittunt \etiam/ nomen corporum & spiritus vocantur. neque eo Quatenus vero neqꝫ {illeg}|C|orpora hic nominantur {illeg}si autem vocari possunt quatenus resistentia{m} habeant densitati proportionalem. Quod si effluvia corporum ita formis mutarentur ut vim resistendi amitterent, hæc non amplius corpora vocarem. < text from f 97v resumes > Pla

Def. 2

Vacuum & locus omnes \voco locum omnem/ sine \absque/ resistentia movetur. Hæ Sic enim vulgus loqui solet.

Siquis contender|at|et corpora dari quæ tangendo nec sentiri nec resistentiam creare queant, hic jam disputa{illeg}|t|{illeg} de vocis significatione Grammatica, corpora nominando quæ vulgus corpora non vocat: et malim cum vulgo loqui, cujus utiqꝫ est vis et norma loquendi. Quæ tangendo premunt & premendo agunt in alia, quæqꝫ non premunt et premendo non agunt in alia, ita ab invicem distingui possunt, ut priora vocentur corpora, altera \vo|di|centur \res/ intangibile{illeg}|s| |vel|/ quovis alio nomine designentur, & horum genus commune tertio {ter.} nomine gaudeat \quale est {illeg} Substantia vel Eus {illeg} vel res mobilis, vel Agens/. Sic enim evitabitur ambiguitas \illa/ verborum ex qua \utiqꝫ/ confusio idearum & contentiones oriri solent. Et cum de corporibus in hoc sensu tanquam \{illeg}/ Phænomenis hic disputetur, spatium omne {quod} {hi{illeg}|c|} destituitur ut Vacuum considero. De natura rerum in spatijs resistentia omni destitutis, disputent, per me licet, Metaphysici. Hoc ad Principia mea nil spectat, in quibus utiqꝫ Phænomena tantum tracto.

Hæc lucem {illeg}|emittent| et reflectunt instar aliorum corporum & in motibus suis observant leges corporum, & a partibus suis incumbentibus premuntur idqꝫ æqualiter undiqꝫ & per æqualitatem pressionis formantur in globos et inter plures corporea naturæ phænomena numerari solent.

Deut 16. 18 Judges & Officers shalt thou make in all thy Gates &c
Deut 25. 7 Go up to the Gate unto the Elders. & Ruth 4. 7
Lam. 5. 14. Elders have ceased from the Gate.
Deut 22. 1{illeg}|5|. unto the Elders of the city in the gate. — \chastise &/ amerce him. — stone her with stones.
Deut 21. 19 & bring him unto the Elders of his city unto the Gate of his place — — — stone him.
Amos. 5. 10,12.