Draft response from Newton to John Collins's letter of 5 July 1671

<2r>

S^r

I purposed to have given you a visit at y^e late solemni{illeg}|t|y of o^r Chancellors creation; but that when I was prevented {illeg} in y^t Journey by y^e suddain surp{illeg}|ri|sall of a fit of sicknesse, w^ch (God bee thanked) I have now recoved|re|d. But since I am prevented {illeg}|fr|ō making a verball acknowledgment of yo^r favors undeservd favors, I must bee yet conte{illeg}|nt|ed to do it in writing. In w^ch respect I find by yo^r last letter, y^t y{illeg} I must still become {illeg}|m|ore your debtor both for the care you take about my concernes & for Borrellius de motionibus. The last winter I reveiwed the introduction & made some few additions to it. & partly upon D^r Barro{ws} instigation, I began to new methodis{illeg}|e| y^{illeg}|e| discourse{illeg} of infinite series, designing to illustrate it w^th such problems as may (some of y^m perhaps) be more acceptable then the the invention of working by such series. But being suddeinly diverted by some buisiness out of Cambridg I have not yet had leisure to return to thos{e} thoughts & I feare I shall not renew them {illeg}|bef|ore winter. I am glad therefore \But since/ you are {illeg} \you informe me there needs no/ hast I hope I may get into y^e humour of completing them before y^e impression of y^e introduction because I if I must helpe to fill up {illeg} /its\ title page I had rather annex somthing that \also w^ch I may call my owne & w^ch/ may bee acceptable to Artists{illeg}, as well as y^e other to Tyro's.

There haveng {sic} som {sic} things past betweene us concerning musicall progressions, & as I remember you desiring me to communit|c|ate somthing w^ch I had hind|t|ed to you about it, w^ch I then had not (nor have yet) adjusted to practise: I shall in its stead of|fer| you somthing else w^ch I think more to y^e purpose.

Any musicall progression $\frac{\mathrm{A}}{\mathrm{B}}.\frac{\mathrm{A}}{\mathrm{B}+\mathrm{C}}.\frac{\mathrm{A}}{\mathrm{B}+2\mathrm{C}}.\frac{\mathrm{A}}{\mathrm{B}+3}$ $\frac{\mathrm{a}}{\mathrm{b}}.\frac{\mathrm{a}}{\mathrm{b}+\mathrm{c}}.\frac{\mathrm{a}}{\mathrm{b}+2\mathrm{c}}.\frac{\mathrm{a}}{\mathrm{b}+3\mathrm{c}}.\frac{\mathrm{a}}{\mathrm{b}+4\mathrm{c}}\text{\&c}$ being propounded whose last terme is {illeg} $\frac{\mathrm{a}}{\mathrm{d}}$. Suppose e {illeg} \=/ meane proportion twixt b & d, or any integrall \or broken/ number betweene them interceding these limits {illeg} /$\mathrm{a}+\mathrm{d}$\. {illeg} e is a meane proportion twixt $\mathrm{b}-\frac{1}{2}\mathrm{c}$ & $\mathrm{d}-\frac{1}{2}\mathrm{c}$ or any integrall or or {sic} broken number greater then it but not greater then $\sqrt{}${illeg} |that it is convenient by guesse {illeg} w^ch differs not considerably from it.| |suppose it intercede the \limits $\sqrt{}\mathrm{b}\mathrm{d}$ & $\sqrt{}\overline{\mathrm{b}-\mathrm{c}}\times \overline{\mathrm{d}-\mathrm{c}}$/ very near it.| (supose not more differing then \the difference/ bee not greater then {$+\frac{\mathrm{b}}{\mathrm{c}\mathrm{d}}$}) & this proportion will give you y^e summe of all the termes very nearely.

As y^e Logarithm $\frac{\mathrm{e}+\frac{1}{2}\mathrm{c}}{\mathrm{e}-\frac{1}{2}\mathrm{c}}$ to y^e logarithm of $\frac{\mathrm{d}+\frac{1}{2}\mathrm{c}}{\mathrm{b}-\frac{1}{2}\mathrm{c}}$, so {illeg} is $\frac{\mathrm{a}}{\mathrm{e}}$ to the desired summe.

Examp suppose {illeg}|y|^e progression bee $\frac{100}{5}.\frac{100}{6}.\frac{100}{7}.\frac{100}{8}.\frac{100}{9}.\frac{100}{10}$. Then is $\mathrm{a}=100$ $\mathrm{b}=5$. $\mathrm{c}=1$. $\mathrm{d}=10$. & {illeg} $6\frac{1}{2}$ intercedes {illeg} /{$\frac{2\mathrm{b}-\mathrm{c}\times \mathrm{d}+\frac{1}{2}\mathrm{c}}{\mathrm{b}+\mathrm{d}}$}\ & $\sqrt{\mathrm{b}-\frac{\mathrm{c}}{2}\times \mathrm{d}+\frac{\mathrm{c}}{2}}$ (y^t is {illeg} {$\begin{array}{c}6\text{\&}\sqrt{}50\\ 6\frac{3}{10}\text{\&}\sqrt{}47\frac{1}{4}\end{array}$}) w^ch I therefore put for e. And work as follows.
$\begin{array}{cc}\begin{array}{cc}\begin{array}{ccccc}\underset{‾}{\mathrm{e}+\frac{1}{2}\mathrm{c}}& =& \underset{‾}{\phantom{\frac{1}{1}}7\phantom{\frac{1}{1}}}& \text{its Log}& \mathrm{0,77815}\text{.}\\ \mathrm{e}-\frac{1}{2}\mathrm{c}& =& 6& \text{its Log}& \mathrm{0,84510}\text{.}\\ & & & & ‾‾‾‾‾‾‾‾‾‾‾‾\\ & & & \text{Diff.}& \mathrm{0,06705}\text{.}\\ \phantom{0}\\ & & & \hfill \text{Log of}{\text{y}}^{\text{e}}\text{Diff.}& \mathrm{1,17360}\text{.}\\ \phantom{0}\\ & & & & \mathrm{3,82640}\text{.}\end{array}& \begin{array}{ccccc}\underset{‾}{\mathrm{d}+\frac{1}{2}\mathrm{c}}& =& \underset{‾}{10\frac{1}{2}}\text{,}& \text{its Log}& \mathrm{1,02119}\\ \mathrm{b}-\frac{1}{2}\mathrm{c}& =& 4\frac{1}{2}\text{,}& \text{its Log}& \mathrm{0,65321}\\ & & & & ‾‾‾‾‾‾‾‾‾‾‾‾\\ & & & \text{Diff}& \mathrm{0,36798}\\ \phantom{0}\\ & & & \hfill \text{L of}{\text{y}}^{\text{e}}\text{Diff.}& \mathrm{4,56584}\\ \phantom{0}\\ & & & & \mathrm{3,82640}\end{array}\end{array}& \begin{array}{c}\text{Log of}\frac{\mathrm{a}}{\mathrm{e}}\\ \text{or}\frac{100}{6\frac{1}{2}}\\ \mathrm{1,18709}\end{array}\end{array}$

|$\begin{array}{ccc}\begin{array}{c}\begin{array}{r}\phantom{0}\\ 45\phantom{\mathrm{,00}}\\ 2\phantom{\mathrm{,}}25\\ \mathrm{94,5}\phantom{0}\end{array}\\ \begin{array}{r}15\phantom{\mathrm{,00}}\end{array}\end{array}& \begin{array}{c}\phantom{0}\\ (\mathrm{6,3}\end{array}& \begin{array}{cc}\begin{array}{r}\begin{array}{r}\phantom{0}\\ \mathrm{47,25}\end{array}\\ \mathrm{1,75}\\ 1\phantom{\mathrm{,}}41\\ 34& \text{.}\end{array}& (\begin{array}{c}14\\ 7\\ -& 12\end{array}\end{array}\end{array}$|

<2v>

$\begin{array}{cc}\begin{array}{cccc}\begin{array}{c}\frac{\mathrm{e}+\frac{1}{2}\mathrm{c}}{\mathrm{e}-\frac{1}{2}\mathrm{c}}=\frac{7}{6}\end{array}& \text{its log. is}& \mathrm{0,066917}\text{\& The Log of that Logarithm,}& \mathrm{4,825731}\\ \frac{\mathrm{d}+\frac{1}{2}\mathrm{c}}{\mathrm{b}-\frac{1}{2}\mathrm{c}}=\frac{10\frac{1}{2}}{4\frac{1}{2}}& \text{its log. is}& \mathrm{0,367977}\text{\& The Log: of that Logarithm}& \mathrm{5,565820}& \\ \frac{\mathrm{a}}{\mathrm{e}}=\frac{100}{6\frac{1}{2}}& \text{its logarithm is}& \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_& \mathrm{1,187087}\\ & & & ‾‾‾‾‾‾‾‾‾‾‾‾‾\\ & & & \mathrm{1,927176}\\ \phantom{.}\\ & & \text{the Logarith of the desired aggregate}& 84\underset{‾}{|562}\end{array}& \begin{array}{l},\text{substract}\begin{array}{c}\phantom{\frac{\mathrm{e}+\frac{1}{2}\mathrm{c}}{\mathrm{e}-\frac{1}{2}\mathrm{c}}}\end{array}\\ \}\text{add.}\begin{array}{c}\phantom{\frac{\mathrm{d}+\frac{1}{2}\mathrm{c}}{\mathrm{b}-\frac{1}{2}\mathrm{c}}}\\ \phantom{\frac{100}{6\frac{1}{2}}}\end{array}\\ \phantom{‾}\\ \phantom{0}\text{the result, which is}\\ \phantom{.}\\ \phantom{0\underset{‾}{|0}}\end{array}\end{array}$
/Whence\ the \y^e desired/ aggregate desired is found to be $84\underset{‾}{|5621}$
And of {illeg} \The same/ by adding y^e \severall/ seve termes together will bee found \more justly/ to bee $84\underset{‾}{|5636}$

But note y^t were there more termes inserted into y^e progression, (as suppose it was $\frac{100}{5}.\frac{100}{5\frac{1}{2}}.\frac{100}{6}.\frac{100}{6\frac{1}{2}}.\frac{100}{7}.\text{\&c}$) the rule would still more approac{illeg}|h| to truth.

2^dly that the difference of the termes \denominators/ ought to be two or

| $\begin{array}{cccc}\begin{array}{l}\frac{1}{{\mathrm{x}}^3}.\frac{1}{2\mathrm{x}\mathrm{x}}-\frac{1}{2\mathrm{y}\mathrm{y}}.\mathrm{y}-\mathrm{x}::\frac{1}{2\mathrm{z}\mathrm{z}}-\frac{1}{2\mathrm{z}\mathrm{z}+4\mathrm{o}\mathrm{z}}.\mathrm{o}.\\ \frac{\mathrm{o}\mathrm{y}\mathrm{y}-\mathrm{o}\mathrm{x}\mathrm{x}}{2\mathrm{x}\mathrm{x}\mathrm{y}\mathrm{y}}=\frac{\overline{\mathrm{y}-\mathrm{x}}\times \mathrm{o}\mathrm{z}}{{\mathrm{z}}^4+2\mathrm{o}{\mathrm{z}}^3}\\ \mathrm{y}\mathrm{y}-\mathrm{x}\mathrm{x}\times {\mathrm{z}}^3=\mathrm{y}-\mathrm{x}\times 2\mathrm{x}\mathrm{x}\mathrm{y}\mathrm{y}\\ {\mathrm{z}}^3=\frac{2\mathrm{x}\mathrm{x}\mathrm{y}\mathrm{y}}{\mathrm{y}+\mathrm{x}}\end{array}& 61.71& \begin{array}{c}\mathrm{4,5}.\mathrm{10,5}\\ \mathrm{47,25}\end{array}& \begin{array}{r}\begin{array}{c}\phantom{0.}3010300\\ 1.3064250\\ 2.0423786\end{array}\\ \begin{array}{c}\mathrm{3,6498436}\\ \\ 2\phantom{\mathrm{,}}4637523\\ \\ \phantom{0.}82125\phantom{00}\end{array}\end{array}\end{array}$|

|$\begin{array}{ccc}\begin{array}{c}\begin{array}{c}\begin{array}{ccc}\phantom{\mathrm{0,0000000}}& & \phantom{\mathrm{0,000}}0731\phantom{0}\\ \mathrm{0,0659285}& .& \mathrm{4,81907965}\end{array}\\ \begin{array}{cc}\mathrm{0,3679767}.& \begin{array}{c}\mathrm{5,56582007}\\ \mathrm{1,1804561}\phantom{0}\end{array}\end{array}& \}\begin{array}{c}\phantom{1}\\ \phantom{1}\end{array}\end{array}\\ \begin{array}{cc}\begin{array}{ccc}& & \underset{‾}{\overset{‾}{\mathrm{6,74627617}}}\\ 84566\phantom{0000}& & \mathrm{1,92719652}\\ 845674\phantom{000}& & \mathrm{1,92720302}\\ \phantom{00}566\phantom{0000}& & \phantom{0}\end{array}& \phantom{\}}\begin{array}{c}\phantom{1}\\ \phantom{1}\end{array}\end{array}\end{array}& \begin{array}{c}\mathrm{}\\ 18512583\\ 17853298\\ \phantom{00}659285\end{array}& \begin{array}{c}6746275\\ \mathrm{}\\ \phantom{000}7198\end{array}\end{array}$ |

To M^r Isaac
Newton fellow of
Trinity Colledge
In
Cambridge