Catalogue Entry: NATP00354

Collations for the History of the Infinitesimal Analysis

Author: Isaac Newton

Source: MS Add. 3968, ff. 113r-144v, Cambridge University Library, Cambridge, UK

[Normalized Text] [Diplomatic Text]

[1] NB < insertion from above the line of f 126v > Id est seribus in finitas æquationes (eo <127r> modo fiat) migrantibus <126v> Exempla habes taliu{m} serierum in <127r> Epistola Newtoni <126v> data 24 Octob. 1676, ubi dicit se <127r> ejusmodi series <126v> per methodum fluxionum inveni <127r> sse. < text from f 126r resumes >

[2] NB b Calculus ita se habet. Sint o et ov momenta ipsorum x et z ut supra et erit aa+xx=zz et aa+xx+x2zo+oo=zz+ov+oovv et deletis æqualibus reliquisque per o divisus fit 2xo+o=2zv+ovv & deletis in finite parvis prodit x=zv et v=xz=xaa+xx. Et similibus computis ex data relatione inter abscissam & aream curvæ cujuscunque ad 1 applicatam, invenietur ordinata, id est ex data quavis æquatione fluentes quantitates involvente invenientur fluxiones.

[3] NB. Hæcce transmutationum methodus per methodum differentialem jam multum abbreviari et ellegantior reddi potest. Namque habilis x=2r3r2+z2 y=2zr2rr+zz, ex æquali{illeg}nt {illeg}ore per methodum illam fit dxdz=4r3z[2]rr+zz. Et ob æquales areas Q1B1DC & C1N1P3P est NP×dz=B1D1×dx. seke NP=yxdxdz=8r5z2[3]rr+zz. =8r5z2r6+r4z2+r2z4+z6. Mirum est quod Leibnitius methodum transmutationum per ambages hactenus tractavit si forte methodum differentialem jam invereat.

[4] NB Mr Leibnitz therefore had not as yet begun to apply differential Equations to inverse Problems of tangents.. But Mr Newton in his answer represents that he had two methods of solving these Problems by the fluxional Equations.

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