Letter from John Collins to Newton, dated 31 August 1676

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S^r

Writing to you last Weeke I accquainted you that I had received somewhat more from Mr Baker in two Letters in the former he accquaints me with the Paines he hath been taking viz, what he hath written namely
A Treatise of Trigonometry done 26 yeares since, not only simple and vulgar, but such as Bartholinus pretends to have found out that is compound or a new way of resolving two things at once, A Demonstration of Vietas Way and the Prostaphereticall way in one sheete Geometrically done
A Treatise of Angular Sections
A Treatise of Conick Sections in an Analyticall method
A Treatise of Cubick æquations &c, with a Miscellany of Problems of Deschartes, Monsieur de Montfort and diverse other besides, with an infinite Company of his owne Invention
And having lately facilitated the Solution of such æquations even of many Dimensions whose rootes are in Arithmeticall or Geometricall Progression he thereupon undertooke the Solution of Dr Davenants Probleme, about finding those 4 continuall Proportionalls, the Summes of whose Squares and Cubes are given which you have over the leafe after the same manner done, as those Papers you have already seene, there is a very notable large Letter from Leibnitz lately arrived, which is not yet in my hands, I hope next Weeke to transcribe, and send a Duplicate, of it to you, I remaine

Your most humble thankfull Servitor

John Collins

Farthing Office
fanch{urch} streete
the 3{1} of August 1676

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To M^r Isaac Newton fellow
of Trinity Colledge

In Cambridge

these

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$\begin{array}{c}\begin{array}{c}\text{Æquatio Prob: 1° et 2° inserviens}\\ \phantom{0}\\ \begin{array}{ccc}\begin{array}{c}\text{Probl: 1}\\ \phantom{.}\\ \begin{array}{cc}\text{Dat}& \{\begin{array}{l}\mathrm{b}=\mathrm{a}\mathrm{a}+\mathrm{m}\mathrm{m}+\mathrm{n}\mathrm{n}+\mathrm{e}\mathrm{e}\\ \mathrm{d}={\mathrm{a}}^3+{\mathrm{m}}^3+{\mathrm{n}}^3+{\mathrm{e}}^3\end{array}\\ {\text{Q}}^{\text{r}}& \phantom{\{}\begin{array}{l}\mathrm{a},\mathrm{m},\mathrm{n},\mathrm{e}∺\phantom{{\text{Q}}^{\text{r}}}\end{array}\hfill \end{array}\}\end{array}& \begin{array}{cccccccccccccc}& & & \mathrm{p}& & \mathrm{q}& & \mathrm{r}& & \mathrm{s}& & \mathrm{t}& & \mathrm{x}\\ {\mathrm{x}}^7& \ast & -& 3\mathrm{b}{\mathrm{x}}^5& +& 16\mathrm{d}{\mathrm{x}}^4& -& 21\mathrm{b}\mathrm{b}{\mathrm{x}}^3& +& 12\mathrm{b}\mathrm{d}\mathrm{x}\mathrm{x}& -& \underset{‾}{9{\mathrm{b}}^3+8\mathrm{d}\mathrm{d}\mathrm{x}}& +& 12\mathrm{b}\mathrm{b}\mathrm{d}& =& 0& \end{array}& \begin{array}{c}\text{Prob 2}\\ \phantom{.}\\ \left\{\begin{array}{cc}\text{Dat}& \{\begin{array}{l}\mathrm{b}=\mathrm{a}\mathrm{a}+\mathrm{m}\mathrm{m}+\mathrm{n}\mathrm{n}+\mathrm{e}\mathrm{e}\\ \mathrm{d}={\mathrm{a}}^3-{\mathrm{m}}^3+{\mathrm{n}}^3-{\mathrm{e}}^3\end{array}\\ {\text{Q}}^{\text{r}}& \phantom{\{}\begin{array}{l}\mathrm{a},\mathrm{m},\mathrm{n},\mathrm{e}\phantom{{\text{Q}}^{\text{r}}}\end{array}\hfill \end{array}\right\}\end{array}\end{array}\\ \phantom{0}\end{array}\\ \begin{array}{c}\phantom{0}\\ \underset{‾}{\text{Æquatio Prob: 3° et 4° inserviens}}\\ \phantom{0}\\ \begin{array}{ccc}\begin{array}{c}\text{Probl: 3}\\ \phantom{.}\\ \begin{array}{cc}\text{Dat}& \{\begin{array}{l}\mathrm{b}=\mathrm{a}\mathrm{a}+\mathrm{m}\mathrm{m}+\mathrm{n}\mathrm{n}+\mathrm{e}\mathrm{e}\\ \mathrm{d}={\mathrm{a}}^3-{\mathrm{m}}^3-{\mathrm{n}}^3+{\mathrm{e}}^3\end{array}\\ {\text{Q}}^{\text{r}}& \phantom{\{}\begin{array}{l}\mathrm{a},\mathrm{m},\mathrm{n},\mathrm{e}∺\phantom{{\text{Q}}^{\text{r}}}\end{array}\hfill \end{array}\}\end{array}& \begin{array}{c}\begin{array}{cccccccccccccccccc}& & & \mathrm{p}& & \mathrm{q}& & \mathrm{r}& & \mathrm{s}& & \mathrm{t}& & \mathrm{x}& & \mathrm{y}& & \mathrm{z}\\ {\mathrm{x}}^9& \ast & & 3\mathrm{b}{\mathrm{x}}^7& +& 8\mathrm{d}{\mathrm{x}}^6& -& 9\mathrm{b}\mathrm{b}{\mathrm{x}}^5& -& 12\mathrm{b}\mathrm{d}{\mathrm{x}}^4& +& \underset{‾}{23{\mathrm{b}}^3+8\mathrm{d}\mathrm{d}{\mathrm{x}}^3}& -& 12\mathrm{b}\mathrm{b}\mathrm{d}\mathrm{x}\mathrm{x}& -& 12{\mathrm{b}}^4\mathrm{x}& +& 8{\mathrm{b}}^3\mathrm{d}& =& 0\end{array}\\ \phantom{.}\\ \text{Innotescat vero}\mathrm{x}=\mathrm{X}\text{; Dico}\end{array}& \begin{array}{c}\text{Probl 4}\\ \phantom{.}\\ \{\begin{array}{cc}\text{Dat}& \}\begin{array}{l}\mathrm{b}=\mathrm{a}\mathrm{a}+\mathrm{m}\mathrm{m}+\mathrm{n}\mathrm{n}+\mathrm{e}\mathrm{e}\\ \mathrm{d}={\mathrm{a}}^3+{\mathrm{m}}^3-{\mathrm{n}}^3-{\mathrm{e}}^3\end{array}\\ {\text{Q}}^{\text{r}}\text{;}& \phantom{\}}\begin{array}{l}\mathrm{a},\mathrm{m},\mathrm{n},\mathrm{e}?\phantom{{\text{Q}}^{\text{r}}}\end{array}\hfill \end{array}\phantom{\}}\\ \phantom{0}\end{array}\end{array}\end{array}\\ \begin{array}{r}\phantom{.}\\ \begin{array}{c}\phantom{0}\text{Solutio Prob}{\text{1}}^{\text{i}}\text{et}{\text{3}}^{\text{ij}}\\ \begin{array}{c}\frac{\mathrm{x}}{2}+\frac{\mathrm{b}}{4\mathrm{x}}-\frac{1}{2}\sqrt{\frac{\mathrm{b}\mathrm{b}}{4\mathrm{x}\mathrm{x}}+\frac{\mathrm{x}\mathrm{x}}{2}-\frac{\mathrm{b}}{2}},\pm \sqrt{:\frac{\mathrm{b}}{4\mathrm{x}}\sqrt{\frac{\mathrm{b}\mathrm{b}}{4\mathrm{x}\mathrm{x}}+\frac{\mathrm{x}\mathrm{x}}{2}-\frac{\mathrm{b}}{2}},-\frac{\mathrm{b}\mathrm{b}}{8\mathrm{x}\mathrm{x}}+\frac{3\mathrm{b}}{8}-\frac{\mathrm{x}\mathrm{x}}{8}}:=\{\begin{array}{c}\mathrm{a}\\ \phantom{\_\_}\\ \mathrm{e}\end{array}\\ \frac{1}{2}\sqrt{\frac{\mathrm{b}\mathrm{b}}{4\mathrm{x}\mathrm{x}}+\frac{\mathrm{x}\mathrm{x}}{2}-\frac{\mathrm{b}}{2}},-\frac{\mathrm{b}}{4\mathrm{x}}\pm \sqrt{:\overline{\frac{\mathrm{x}}{2}+\frac{\mathrm{b}}{4\mathrm{x}}}\sqrt{\frac{\mathrm{b}\mathrm{b}}{4\mathrm{x}\mathrm{x}}+\frac{\mathrm{x}\mathrm{x}}{2}-\frac{\mathrm{b}}{2}},-\frac{\mathrm{b}\mathrm{b}}{8\mathrm{x}\mathrm{x}}+\frac{\mathrm{b}}{8}-\frac{3\mathrm{x}\mathrm{x}}{8}}:=\{\begin{array}{c}\mathrm{m}\\ \phantom{\_\_}\\ \mathrm{n}\end{array}\end{array}\}\end{array}& \begin{array}{r}\text{Solutio Prob}{\text{2}}^{\text{i}}\text{et}{\text{4}}^{\text{i}}\phantom{0}\\ \{\begin{array}{c}\sqrt{:\frac{\mathrm{b}}{4\mathrm{x}}\sqrt{\frac{\mathrm{b}\mathrm{b}}{4\mathrm{x}\mathrm{x}}+\frac{\mathrm{x}\mathrm{x}}{2}-\frac{\mathrm{b}}{2}},-\frac{\mathrm{b}\mathrm{b}}{8\mathrm{x}\mathrm{x}}+\frac{3\mathrm{b}}{8}-\frac{\mathrm{x}\mathrm{x}}{8}}:\pm \frac{\mathrm{x}}{2}+\frac{\mathrm{b}}{4\mathrm{x}}-\frac{1}{2}\sqrt{\frac{\mathrm{b}\mathrm{b}}{4\mathrm{x}\mathrm{x}}+\frac{\mathrm{x}\mathrm{x}}{2}-\frac{\mathrm{b}}{2}}=\{\begin{array}{c}\mathrm{a}\\ \phantom{\_\_}\\ \mathrm{e}\end{array}\\ \sqrt{:\overline{\frac{\mathrm{x}}{2}+\frac{\mathrm{b}}{4\mathrm{x}}}\sqrt{\frac{\mathrm{b}\mathrm{b}}{4\mathrm{x}\mathrm{x}}+\frac{\mathrm{x}\mathrm{x}}{2}-\frac{\mathrm{b}}{2}},-\frac{\mathrm{b}\mathrm{b}}{8\mathrm{x}\mathrm{x}}+\frac{3\mathrm{b}}{8}-\frac{\mathrm{x}\mathrm{x}}{8}}:\pm \frac{\mathrm{b}}{4\mathrm{x}}-\frac{1}{2}\sqrt{\frac{\mathrm{b}\mathrm{b}}{4\mathrm{x}\mathrm{x}}+\frac{\mathrm{x}\mathrm{x}}{2}-\frac{\mathrm{b}}{2}}=\{\begin{array}{c}\mathrm{m}\\ \phantom{\_\_}\\ \mathrm{n}\end{array}\end{array}\phantom{\}}\\ \phantom{0}\end{array}\end{array}\\ \begin{array}{c}\phantom{0}\\ \underset{‾}{\text{Æquatio Prob: 5° et 6° inserviens}}\\ \phantom{0}\\ \begin{array}{ccccc}\begin{array}{c}\text{Probl 5}\\ \phantom{.}\\ \begin{array}{cc}\text{Dat}& \{\begin{array}{l}\mathrm{b}=\mathrm{a}\mathrm{a}-\mathrm{m}\mathrm{m}-\mathrm{n}\mathrm{n}+\mathrm{e}\mathrm{e}\\ \mathrm{d}={\mathrm{a}}^3+{\mathrm{m}}^3+{\mathrm{n}}^3+{\mathrm{e}}^3\end{array}\\ {\text{Q}}^{\text{r}}& \phantom{\{}\begin{array}{l}\mathrm{a},\mathrm{m},\mathrm{n},\mathrm{e}?\phantom{{\text{Q}}^{\text{r}}}\end{array}\hfill \end{array}\}\end{array}& \phantom{0000000000}& \begin{array}{c}\begin{array}{ccccccccccccc}& & & \mathrm{p}& & \mathrm{q}& & \mathrm{r}& & \mathrm{s}& & \mathrm{t}& \\ {\mathrm{x}}^6& \ast & +& 7\mathrm{b}{\mathrm{x}}^4& -& 16\mathrm{d}{\mathrm{x}}^3& +& 3\mathrm{b}\mathrm{b}\mathrm{x}\mathrm{x}& +& 8\mathrm{b}\mathrm{d}\mathrm{x}& -& 3{\mathrm{b}}^3& =& 0\end{array}\\ \phantom{.}\\ \text{Innotescat verò}\mathrm{x}=\mathrm{X}\text{; Dico}\end{array}& \phantom{0000000000}& \begin{array}{c}\text{Probl 6}\\ \phantom{.}\\ \begin{array}{cc}\text{Dat}& \{\begin{array}{l}\mathrm{b}=\mathrm{a}\mathrm{a}-\mathrm{m}\mathrm{m}-\mathrm{n}\mathrm{n}+\mathrm{e}\mathrm{e}\\ \mathrm{d}={\mathrm{a}}^3-{\mathrm{m}}^3+{\mathrm{n}}^3-{\mathrm{e}}^3\end{array}\\ {\text{Q}}^{\text{r}}\text{,}& \phantom{\{}\begin{array}{l}\mathrm{a},\mathrm{m},\mathrm{n},\mathrm{e}\phantom{{\text{Q}}^{\text{r}}}\end{array}\hfill \end{array}\phantom{\}}\end{array}\end{array}\\ \phantom{0}\end{array}\\ \begin{array}{c}\begin{array}{c}\phantom{0}\\ \underset{‾}{\text{Æquatio Prob: 7° et 8° inserviens}}\\ \phantom{0}\\ \begin{array}{ccccc}\begin{array}{c}\begin{array}{c}\text{Probl: 7}\\ \phantom{.}\\ \begin{array}{cc}\text{Dat}& \{\begin{array}{l}\mathrm{b}=\mathrm{a}\mathrm{a}-\mathrm{m}\mathrm{m}-\mathrm{n}\mathrm{n}+\mathrm{e}\mathrm{e}\\ \mathrm{d}={\mathrm{a}}^3-{\mathrm{m}}^3-{\mathrm{n}}^3+{\mathrm{e}}^3\end{array}\\ {\text{Q}}^{\text{r}}& \phantom{\{}\begin{array}{l}\mathrm{a},\mathrm{m},\mathrm{n},\mathrm{e}\phantom{{\text{Q}}^{\text{r}}}\end{array}\hfill \end{array}\}\\ \phantom{0}\end{array}\\ \begin{array}{c}\phantom{0}\\ \text{Solutio Prob}{\text{5}}^{\text{i}}\text{et}{\text{7}}^{\text{i}}\\ \frac{\mathrm{x}}{4}+\frac{\mathrm{b}}{4\mathrm{x}}\pm \overline{\frac{3\mathrm{x}}{4}-\frac{\mathrm{b}}{4\mathrm{x}}}\sqrt{\frac{\mathrm{b}}{2\mathrm{x}\mathrm{x}-\mathrm{b}}}=\{\begin{array}{c}\mathrm{a}\\ \phantom{\_\_}\\ \mathrm{e}\end{array}\\ \frac{\mathrm{x}}{4}-\frac{\mathrm{b}}{4\mathrm{x}}\pm \overline{\frac{\mathrm{x}}{4}-\frac{\mathrm{b}}{4\mathrm{x}}}\sqrt{\frac{\mathrm{b}}{2\mathrm{x}\mathrm{x}-\mathrm{b}}}=\{\begin{array}{c}\mathrm{m}\\ \phantom{\_\_}\\ \mathrm{n}\end{array}\end{array}\end{array}& \phantom{000}& \begin{array}{c}\begin{array}{c}\phantom{0}\\ & & \mathrm{p}& & \mathrm{q}& & \mathrm{r}& & \mathrm{s}& & & \mathrm{x}& \\ {\mathrm{x}}^6& -& \frac{16\mathrm{d}}{9\mathrm{b}}{\mathrm{x}}^5& +& \frac{\mathrm{b}}{3}{\mathrm{x}}^4& +& \frac{8\mathrm{d}}{9}{\mathrm{x}}^3& -& \frac{5\mathrm{b}\mathrm{b}}{9}\mathrm{x}\mathrm{x}& \ast & +& \frac{{\mathrm{b}}^3}{9}& =& 0\end{array}\\ \phantom{.}\\ \text{Innotescat vero}\mathrm{x}=\mathrm{X}\text{Dico}\end{array}& \phantom{000}& \begin{array}{c}\begin{array}{c}\text{Probl 8}\\ \phantom{.}\\ \begin{array}{cc}\phantom{\text{[Dat}}& \phantom{\{\begin{array}{l}\mathrm{b}=\mathrm{a}\mathrm{a}-\mathrm{m}\mathrm{m}-\mathrm{n}\mathrm{n}+\mathrm{e}\mathrm{e}\\ \mathrm{d}={\mathrm{a}}^3+{\mathrm{m}}^3-{\mathrm{n}}^3-{\mathrm{e}}^3\end{array}}\\ \phantom{{\text{Q}}^{\text{r}}}& \phantom{\{}\begin{array}{l}\phantom{\mathrm{a},\mathrm{m},\mathrm{n},\mathrm{e}?\text{]}{\text{Q}}^{\text{r}}}\end{array}\hfill \end{array}\phantom{\}}\\ \phantom{0}\end{array}\\ \begin{array}{c}\phantom{0}\\ \text{Solutio Probl:}{\text{6}}^{\text{i}}\text{et}{\text{8}}^{\text{i}}\\ \overline{\frac{\mathrm{b}}{4\mathrm{x}}-\frac{3\mathrm{x}}{4}}\sqrt{\frac{\mathrm{b}}{2\mathrm{x}\mathrm{x}-\mathrm{b}}},\pm \overline{\frac{\mathrm{b}}{4\mathrm{x}}+\frac{\mathrm{x}}{4}}=\{\begin{array}{c}\mathrm{a}\\ \phantom{\_\_}\\ \mathrm{e}\end{array}\\ \overline{\frac{\mathrm{b}}{4\mathrm{x}}-\frac{\mathrm{x}}{4}}\sqrt{\frac{\mathrm{b}}{2\mathrm{x}\mathrm{x}-\mathrm{b}}},\pm \overline{\frac{\mathrm{b}}{4\mathrm{x}}-\frac{\mathrm{x}}{4}}=\{\begin{array}{c}\mathrm{m}\\ \phantom{\_\_}\\ \mathrm{n}\end{array}\end{array}\end{array}\end{array}\\ \phantom{0}\end{array}\\ \end{array}\end{array}$

Heveningham
Henningham