Catalogue Entry: NATP00354

^[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 $\mathrm{o}\mathrm{v}$ momenta ipsorum x et z ut supra et erit $\mathrm{a}\mathrm{a}+\mathrm{x}\mathrm{x}=\mathrm{z}\mathrm{z}$ et $\mathrm{a}\mathrm{a}+\mathrm{x}\mathrm{x}+{}^2\mathrm{x}\mathrm{z}\mathrm{o}+\mathrm{o}\mathrm{o}=\mathrm{z}\mathrm{z}+\mathrm{o}\mathrm{v}+\mathrm{o}\mathrm{o}\mathrm{v}\mathrm{v}$ et deletis æqualibus reliquisque per o divisus fit $2\mathrm{x}\mathrm{o}+\mathrm{o}=2\mathrm{z}\mathrm{v}+\mathrm{o}\mathrm{v}\mathrm{v}$ & deletis in finite parvis prodit $\mathrm{x}=\mathrm{z}\mathrm{v}$ et $\mathrm{v}=\frac{\mathrm{x}}{\mathrm{z}}=\frac{\mathrm{x}}{\sqrt{\mathrm{a}\mathrm{a}+\mathrm{x}\mathrm{x}}}$. 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 $\mathrm{x}=\frac{2\mathrm{r}3}{{\mathrm{r}}^2+{\mathrm{z}}^2}$ $\mathrm{y}=\frac{2\mathrm{z}\mathrm{r}2}{\mathrm{r}\mathrm{r}+\mathrm{z}\mathrm{z}}$, ex æquali{illeg}nt {illeg}ore per methodum illam fit $\frac{\mathrm{dx}}{\mathrm{dz}}=\frac{4\mathrm{r}3\mathrm{z}}{\left[2\right]\mathrm{r}\mathrm{r}+\mathrm{z}\mathrm{z}}$. Et ob æquales areas Q₁B₁DC & C₁N₁P₃P est $\mathrm{NP}\times \mathrm{dz}={}_1\mathrm{B}{}_1\mathrm{D}\times \mathrm{dx}$. seke $\mathrm{NP}=\frac{\mathrm{y}\mathrm{x}\mathrm{dx}}{\mathrm{dz}}=\frac{8{\mathrm{r}}^5{\mathrm{z}}^2}{\left[3\right]\mathrm{r}\mathrm{r}+\mathrm{z}\mathrm{z}}$. $=\frac{8{\mathrm{r}}^5{\mathrm{z}}^2}{{\mathrm{r}}^6+{\mathrm{r}}^4{\mathrm{z}}^2+{\mathrm{r}}^2{\mathrm{z}}^4+{\mathrm{z}}^6}$. Mirum est quod Leibnitius methodum transmutationum per ambages hactenus tractavit si forte methodum differentialem jam invereat.

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

^[5] 1

^[6] 2

^[7] 5

^[8] 4

^[9] 5