Copy of an extract of a letter to John Collins, dated 10 December 1672

<45r>

Extractum
Ex Dni Newtoni Epistola, datâ Cantabrigia Dec. 10. 1672. ad Dn. Collinium.

Perquam gaudeo, Dni Barrovij Lectiones Mathematicas adeo acceptus esse {illeg} Mathematicis exteris; nec {parvùm} me afficiebat, quod intelligerem, i{illeg}|ll|os in eandem mecum nicidisse methodum Tangentes ducendi. Quam ipsorum methodum {conjùram}, hoc Exemplo videbis;

Suppone CB, applicatam {illeg}d \ad/ AB ad \in/ qu{illeg}|o|libet angul{illeg}|o| datu|o|m, terminari ad quamvis curvam AC, {illeg} et AB appellato x, et BC y, relatio inter x et y exprimatur æquatione qualibet, puta $${\mathrm{x}}^3-2\mathrm{x}\mathrm{x}\mathrm{y}+\mathrm{b}\mathrm{x}\mathrm{x}+\mathrm{b}\mathrm{y}\mathrm{y}-{\mathrm{y}}^3=0$$ {sic} quâ ipsâ determinatꝫ curva. Ad ducendam Tangentem Regula hæc est: Multiplica terminos æquationis per quamvis progressionem Arithmetica juxt{illeg}|a| dimensiones y, puta hoc modo; $$\begin{array}{ccccccccccc}0& & 1& & 0& & 0& & 2& & 3\\ {\mathrm{x}}^3& -& 2\mathrm{x}\mathrm{x}\mathrm{y}& +& \mathrm{b}\mathrm{x}\mathrm{x}& -& \mathrm{b}\mathrm{b}\mathrm{x}& +& \mathrm{b}\mathrm{y}\mathrm{y}& -& {\mathrm{y}}^3\\ 3& & 2& & 2& & 1& & 0& & 0\end{array}$$ Productum primum erit Numerator, et postremum pe divisum per x, \erit/ Denominator fractionis, quæ exprimit longitudinem BD, ad cujus finem ducanda{illeg} est Tangens;
{&} Longitudo BD est $\frac{-2\mathrm{x}\mathrm{x}\mathrm{y}+2\mathrm{b}\mathrm{y}\mathrm{y}-3{\mathrm{y}}^3}{3\mathrm{x}\mathrm{x}-4\mathrm{x}\mathrm{y}+2\mathrm{b}\mathrm{x}-\mathrm{b}\mathrm{b}}$.

Hoc p{illeg} est unum {illeg}|pa|rticulare, sive potius Corollarium generali methodi, quæ seipsam extendit, abs modesto ullo calculo \{non} solum/ {ad} ducendu Tangentes ad omnes Curvas, sive Geometricas, sive Mechanicas, vel quodmodocun relatas ad lineas Rectas, vel <45v> ad alias Curvas, sed etiam ad resolvenda alia problemata abstrusioris familiæ de curvitate, areis, longitudinibus, centris gravitatis Curvarum etc. Ne (ut Huddenij methodus de maximis et minimis, adeo nec ut Slusij nova methodus de Tangentibus, ut arbitror) limitatur ad æquationes quæ immunes sunt á quantitatis surdis. Methodum hanc intertexui illi alteri, {illeg} operandi {scil.} in Æquationibus eas ad \series/ infinitas reducendo. Memini, me aliquando, occasion{illeg}|e| datâ, dix \dixisse/ Dno Barrovio, dic {cum} in procinctu erat lectiones suas edendi, me instructum me esse istiusmodi methodo Tangentes ducendi; sed aliis rebus impediebar, quò minus tum temporis {cum} ipsi describerem.

Circa resolvendas, Cardani regularum beneficio, Æquationes ejusmodi quæ habent 3 radices possibiles; Exempla strui possunt ad libitum; verùm nisi Brasserùs {directu{m}}{direct{u}} ostendat methodu id præstandi, quod Fergusonus non facit non admitta|e|tur \esse/ scientifica. Quâ ratione id præstandu sit directé, ex occasione forsan ostendam.

<45ar>

In hac æquatione
${\mathrm{z}}^3-21\mathrm{z}\mathrm{z}+120\mathrm{z}$

$\begin{array}{cccc}& & & \text{Resolvenda.}\\ \text{Radices}& 1& —& 120\hfill \\ & 2& —& 164\hfill \\ & 3& —& 198\hfill \\ & 4& —& 225\hfill \\ & 5& —& 200\hfill \\ & 6& —& 180\hfill \\ & 7& —& 154\hfill \\ & 8& —& 128\hfill \\ & 9& —& 108\hfill \\ & 10& —& 100\hfill \\ & 11& —& 110\hfill \end{array}$

Resolvenda supponuntur punctata deorsùm ab O versus R et radices excitatæ tanquam ad ad {{illeg}|o|rdinatæ} ad ea, per <45av> quarum summitates transit Locus æquationis FOBE.

Radices Limitum sunt $\mathrm{GB}=4$. Resolvendo sive Limite ad eas exi BC, existente 225: At radix alterius Limitis est $\mathrm{OD}=10$; cujus Resolvendum $\mathrm{DE}=100$.

Jam verò, quandoquidem Curva hæc habet flexuras, Dni Tschurnhaus rogandus est, ut suam Tangentium Doctrinam ita adaptet, ut portionem quam ducit, \ducatur/ à pede \d/ ordinatæ C vel \ad/ D, ubi demum cum i{illeg}|d| accidit, non verò ab O ad H, vel alioqui sursu vel deorsim a linea ROS.

Si $\left.\begin{array}{c}\mathrm{OL}\\ \mathrm{OS}\end{array}\right\}$ sit Resovendu, radix est $\left\}\begin{array}{c}\mathrm{LW}\\ \mathrm{ST}\end{array}\right\}$. Has inventas supponimus oper per operat{illeg}|i|onem tentativam. At supposito, Resovendu datu esse OK, ad quod requiritꝫ radix. Supposito, Chordam TW ductam, pl liquet, KN nimis parvam esse ad id ut sit radix, et ducto tangente TZ, KO nimis magna est; quâ ratione propinqua est \sit/ approximatio. At inter B et E est aliud punctu flexûs contrarij.