Letter from Newton to John Collins, dated 20 June 1674

<37r>

S^r

I thank you for yo^r kind present. M^r Andersons book is very ingenious, & may prove as usefull if his princi{illeg}|p|les be true. But I suspect one of them, namely y^t y^e bullet moves in a Parabola. This would be so indeed were y^e horizontal celerity of y^e bullet uniform, but I should think its motion decays considerably in y^e flight. Suppose for instance a bullet shot horizontally moves from A moves in y^e line AE, & AI being perpendicular to y^e horizon in it take AF
AG, AH, AI, &c in proportion as y^e square numbers 1, 4, 9, 16 &c: & its certain y^t if in one moment of time y^e bullet descend as low as F, in y^e next moment it shall descend as low as G, in y^e 3^d as low as H &c. And therefore dra{illeg}|w|ing y^e horizontall lines FB, GC, HD, IE; y^e bullet at y^e end of y^e first moment will be somewhere in y^e line FB suppose at B, & at y^e end {illeg}|of| y^e 2^d moment it will be somewhere in y^e line GC suppose at C &c. But that FB, GC, HD & IE are in Arithmeticall progression (w^ch is y^e condition of y^e Parabola) se{illeg}|e|ms not probable; for if it were so, y^e celerity of y^e bullet would increas becaus y^e spaces AB, BC, CD, DE described in equall times are y^e latter bigger yⁿ y^e former: whereas I should rather think y^t y^e celerity decreases very considerably. And perhaps this rule for its d{illeg}|ec|reasing may pretty nearly approach y^e truth, viz: Letting fall y^e perpendiculars BK, CL, DM &c to make IK, KL, LM &c, a decreasing Geometricall progression. If you should have occasion to speak of this to y^e Author, I desire you would not mention me becaus I have no mind to concern my self further about it.

As for y^e method of extracting y^e roots of literal equations, the root of this ${\mathrm{y}}^3+\mathrm{a}\mathrm{a}\mathrm{y}-{\mathrm{b}}^3=0$ (w^ch may be a form for all cubic equations) may be thus extracted. Suppose l pretty nearly equal to y^e desired root & put l in y^e quotient for y^e first term. Then proceed as follows.

$\begin{array}{l}\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}\phantom{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}& \phantom{\_\_\_\_\_\_\_\_\_\_}\end{array}& \left(\mathrm{l}-\frac{\mathrm{m}}{\mathrm{n}}-\frac{3\mathrm{l}\mathrm{m}\mathrm{m}}{{\mathrm{n}}^3}+\frac{{\mathrm{m}}^3}{{\mathrm{n}}^4}\right.\\ \phantom{.}\end{array}& \end{array}\\ \begin{array}{cc}\begin{array}{c}\phantom{.}\\ \mathrm{l}+\mathrm{p}=\mathrm{y}& {\mathrm{y}}^3& {\mathrm{l}}^3+3\mathrm{l}\mathrm{l}\mathrm{p}+3\mathrm{l}\mathrm{p}\mathrm{p}+{\mathrm{p}}^3\text{.}\hfill \\ & +\mathrm{a}\mathrm{a}\mathrm{y}& +{\mathrm{a}}^2\mathrm{l}+\mathrm{a}\mathrm{a}\mathrm{p}\text{.}\hfill \\ & -{\mathrm{b}}^3& -{\mathrm{b}}^3\text{.}\hfill \\ \phantom{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}& \phantom{\_\_\_\_\_\_\_\_\_\_}\end{array}& \begin{array}{l}\text{For brevities sake I have put}{\mathrm{l}}^3+\mathrm{a}\mathrm{a}\mathrm{l}-{\mathrm{b}}^3=\mathrm{m}\\ \text{\&}3\mathrm{l}\mathrm{l}+\mathrm{a}\mathrm{a}=\mathrm{n}\text{.}\end{array}\end{array}\\ \begin{array}{cc}\begin{array}{c}\phantom{.}\\ -\frac{\mathrm{m}}{\mathrm{n}}+\mathrm{q}=\mathrm{p}& {\mathrm{p}}^3& -\frac{{\mathrm{m}}^3}{{\mathrm{n}}^3}+\frac{3\mathrm{m}\mathrm{m}\mathrm{q}}{\mathrm{n}\mathrm{n}}-\frac{3\mathrm{m}\mathrm{q}\mathrm{q}}{\mathrm{n}}+{\mathrm{q}}^3\text{.}\hfill \\ & +3\mathrm{l}\mathrm{p}\mathrm{p}& +\frac{3\mathrm{l}\mathrm{m}\mathrm{m}}{\mathrm{n}\mathrm{n}}+\frac{6\mathrm{l}\mathrm{m}}{\mathrm{n}}\mathrm{q}+3\mathrm{l}\mathrm{q}\mathrm{q}\text{.}\hfill \\ & +\mathrm{n}\mathrm{p}& -\mathrm{m}+\mathrm{n}\mathrm{q}\text{.}\hfill \\ & +\mathrm{m}& +\mathrm{m}\text{.}\hfill \\ \phantom{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}& \phantom{\_\_\_\_\_\_\_\_\_\_}\end{array}& \end{array}\\ \begin{array}{cc}\begin{array}{c}\phantom{.}\\ -\frac{3\mathrm{l}\mathrm{m}\mathrm{m}}{{\mathrm{n}}^3}+\mathrm{r}=\mathrm{q}& {\mathrm{q}}^3& -\frac{27{\mathrm{l}}^3{\mathrm{m}}^6}{{\mathrm{n}}^9}+\frac{27\mathrm{l}\mathrm{l}{\mathrm{m}}^4}{{\mathrm{n}}^6}\mathrm{r}-\frac{9\mathrm{l}\mathrm{m}\mathrm{m}}{{\mathrm{n}}^3}\mathrm{r}\mathrm{r}+{\mathrm{r}}^3\text{.}\hfill \\ & -\frac{3\mathrm{m}}{\mathrm{n}}\mathrm{q}\mathrm{q}& -\frac{27\mathrm{l}\mathrm{l}{\mathrm{m}}^5}{{\mathrm{n}}^7}+\frac{18\mathrm{l}{\mathrm{m}}^3}{{\mathrm{n}}^4}\mathrm{r}-\frac{3\mathrm{m}}{\mathrm{n}}\mathrm{r}\mathrm{r}\hfill \\ & +3\mathrm{l}\mathrm{q}\mathrm{q}& +\frac{27{\mathrm{l}}^3{\mathrm{m}}^4}{{\mathrm{n}}^6}-\frac{18\mathrm{l}\mathrm{l}\mathrm{m}\mathrm{m}}{{\mathrm{n}}^3}\mathrm{r}+3\mathrm{l}\mathrm{r}\mathrm{r}\hfill \\ & +\frac{3\mathrm{m}\mathrm{m}}{\mathrm{n}\mathrm{n}}\mathrm{q}& -\frac{9\mathrm{l}{\mathrm{m}}^4}{{\mathrm{n}}^5}+\frac{3\mathrm{m}\mathrm{m}}{\mathrm{n}\mathrm{n}}\mathrm{r}\hfill \\ & -\frac{6\mathrm{l}\mathrm{m}}{\mathrm{n}}\mathrm{q}& +\frac{18\mathrm{l}\mathrm{l}{\mathrm{m}}^3}{{\mathrm{n}}^4}-\frac{6\mathrm{l}\mathrm{m}}{\mathrm{n}}\mathrm{r}\hfill \\ & +\mathrm{n}\mathrm{q}& -\frac{3\mathrm{l}\mathrm{m}\mathrm{m}}{\mathrm{n}\mathrm{n}}+\mathrm{n}\mathrm{r}\hfill \\ & -\frac{{\mathrm{m}}^3}{{\mathrm{n}}^3}& -\frac{{\mathrm{m}}^3}{{\mathrm{n}}^3}\text{.}\hfill \\ & +\frac{3\mathrm{l}\mathrm{m}\mathrm{m}}{\mathrm{n}\mathrm{n}}& +\frac{3\mathrm{l}\mathrm{m}\mathrm{m}}{\mathrm{n}\mathrm{n}}\text{.}\hfill \\ \phantom{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}& \phantom{\_\_\_\_\_\_\_\_\_\_}\end{array}& \end{array}\end{array}$

<37v>

There may be other ways of extracting y^e root of this equation: as for instance if b be greater then a, the root may be thus exprest $\mathrm{y}={\mathrm{b}}^3-\frac{\mathrm{a}\mathrm{a}}{3\mathrm{b}}+\frac{{\mathrm{a}}^6}{81{\mathrm{b}}^5}$$+\frac{{\mathrm{a}}^8}{243{\mathrm{b}}^7}\text{\&c}$. Or thus $\mathrm{y}=\frac{{\mathrm{b}}^3}{\mathrm{a}\mathrm{a}}-\frac{{\mathrm{b}}^9}{{\mathrm{a}}^8}+\frac{3{\mathrm{b}}^{15}}{{\mathrm{a}}^{14}}-\frac{12{\mathrm{b}}^{21}}{{\mathrm{a}}^{20}}\text{\&c}$ if a be considerably greater then b.

The extraction of y^e simple cube root may be done after y^e same manner omitting only y^e 2^d terme $\mathrm{a}\mathrm{a}\mathrm{y}$. Or els it may be done as in y^e following example |numerall Arithmetic|. Suppose ${\mathrm{y}}^3={\mathrm{a}}^3+{\mathrm{b}}^3$, & that a is bigger then b, & let y^e divisor for finding y^e terms of y^e Quotient be always tripp|l|e y^e square of y^e first term, that is $3\mathrm{a}\mathrm{a}$. And y^e form of y^e work will be this.

$\begin{array}{l}\begin{array}{lllll}{\mathrm{a}}^3& +& {\mathrm{b}}^3& & \left(\mathrm{a}+\frac{{\mathrm{b}}^3}{3\mathrm{a}\mathrm{a}}-\frac{{\mathrm{b}}^6}{9{\mathrm{a}}^5}+\frac{5{\mathrm{b}}^9}{81{\mathrm{a}}^8}\text{\&c}\right.\end{array}\\ \begin{array}{lll}{\mathrm{a}}^3& & \\ 0& +& {\mathrm{b}}^3\end{array}\\ \begin{array}{lll}\phantom{0^0}& \phantom{+}& \begin{array}{lllll}{\mathrm{b}}^3& +& \frac{{\mathrm{b}}^6}{3{\mathrm{a}}^3}& +& \frac{{\mathrm{b}}^9}{27{\mathrm{a}}^6}\\ 0& -& \frac{{\mathrm{b}}^6}{3{\mathrm{a}}^3}& -& \frac{{\mathrm{b}}^9}{27{\mathrm{a}}^6}\end{array}\end{array}\\ \begin{array}{llll}\phantom{0^0}& \phantom{+}& \phantom{0^0}& \begin{array}{llllllllll}-& \frac{{\mathrm{b}}^6}{3{\mathrm{a}}^3}& -& \frac{2{\mathrm{b}}^9}{9{\mathrm{a}}^6}& & \ast & +& \frac{{\mathrm{b}}^{15}}{81{\mathrm{a}}^{12}}& +& \frac{{\mathrm{b}}^{18}}{729{\mathrm{a}}^{15}}\\ & \hfill 0\hfill & -& \frac{5{\mathrm{b}}^9}{27{\mathrm{a}}^6}& & \ast & -& \frac{{\mathrm{b}}^{15}}{81{\mathrm{a}}^{12}}& -& \frac{{\mathrm{b}}^{18}}{729{\mathrm{a}}^{15}}\end{array}\end{array}\end{array}$

So if ${\mathrm{y}}^3=\mathrm{c}$, suppose any letter as a to be pretty nearly equall to y^e cube root of c, & putting $\mathrm{c}-{\mathrm{a}}^3=\mathrm{b}$ or ${\mathrm{a}}^3+\mathrm{b}=\mathrm{c}$ extract y^e cube root out of ${\mathrm{a}}^3+\mathrm{b}$ as in y^e former example; w^ch root $\left(\text{viz:}\mathrm{a}+\frac{{\mathrm{b}}^3}{3\mathrm{a}\mathrm{a}}-\frac{{\mathrm{b}}^6}{9{\mathrm{a}}^5}\right.$$\left.+\frac{5{\mathrm{b}}^9}{81{\mathrm{a}}^8}\text{\&c}\right)$ being once extracted may be kept as a rule {illeg}|f|or extracting numeral cube roots. But yet (as you suggest) these infinite series are only usefull where y^e roots of equations cannot be attained accurately. For when they can be accurately attaind recours must be had to other methods, this only performing it by approximation.

S^r I am

Yo^r obliged humble Servant

Is. Newton

Cambridge. June 20.
1674.

<37av>

These

For M^r John Collins at the
Farthing Office in Fenchurch
Street in

London.

|2|

|M^r Newton about Andersons
Parabola and extracting of
rootes in Species|