# Letter from Newton to John Collins, dated 20 July 1671

July 20^{th} 1671.

S^{r}

I purposed to have given you a visit at y^{e} late solemnity of o^{r} Chancellors creation; but I was prevented in y^{t} Journey by y^{e} suddain surprisall of a fit of sicknesse, w^{ch} not long after (God be thanked) I again recovered of. And since I am prevented from making a verball acknowledgment of yo^{r} undeserved favours, I must bee yet contented to do it in writing. In w^{ch} respect I find by yo^{r} last letter, y^{t} I am still become more yo^{r} debtor both for the care you take about my concernes, & for Borellius de motionibus. But for Borrellius I beg that I may bee accomptable to you a{illeg}|t| o^{r} next meeting, & that you would not for y^{e} future put yo^{r} selfe to y^{e} like trouble in sending any more books. I shall take it for a great favour if in yo^{r} letters you will onely inform mee of y^{e} names of y^{e} best of those bookes w^{ch} newly come forth.

The last winter I reveiwed the Introduction & made some few additions to it: And partly upon D^{r} Barrows instigation, I began to new methodiz y^{e} discourse of infinite series, designing to illustrate it w^{th} such problems as may (some of them perhaps) be more acceptable then y^{e} invention it selfe of working by such series. But being suddainly diverted by some buisinesse in the Country, I have not yet had leisure to return to those thoughts, & I feare I shall not before winter. But since you informe me there needs no hast, I hope I may get into y^{e} humour of completing them before y^{e} impression of the introduction, because if I must helpe to fill up its title page, I had rather annex somthing w^{ch} I may call my owne, & w^{ch} may bee acceptable to Artists as well as y^{e} other to Tyros.

There having some things past between us concerning musicall progressions, & as I remember you desiring mee to communicate somthing w^{ch} I had hinted to you about it, w^{ch} I then had not (nor have yet) adjusted to practise: I shall in its stead offer 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\mathrm{c}}.\frac{\mathrm{a}}{\mathrm{b}+4\mathrm{c}}$ &c being propounded whose last terme is $\frac{\mathrm{a}}{\mathrm{d}}$: for y^{e} following operation choose any convenient number \e/ (whither whole broken or surd) which intercedes these limits {illeg} $\frac{2\mathrm{m}\mathrm{n}}{\mathrm{b}+\mathrm{d}}$ & $\sqrt{}\mathrm{m}\mathrm{n}$; supposing $\mathrm{b}-\frac{1}{2}\mathrm{c}$ to bee m, & $\mathrm{d}+\frac{1}{2}\mathrm{c}$ to bee n. And this proportion will give you the aggregate of the termes very neareby the truth.

As y^{e} Logarithm $\frac{\mathrm{e}+\frac{1}{2}\mathrm{c}}{\mathrm{e}-\frac{1}{2}\mathrm{c}}$
to y^{e} Logarithm of $\frac{\mathrm{n}}{\mathrm{m}}$, so is $\frac{\mathrm{a}}{\mathrm{e}}$ to y^{e} {illeg}|d|esired summe.

Example. Suppose y^{e} progression bee $\frac{100}{5}.\frac{100}{6}.\frac{100}{7}.\frac{100}{8}.\frac{100}{9}.\frac{100}{10}$. That is $\mathrm{a}=100$. $\mathrm{b}=5$. $\mathrm{c}=1$. $\mathrm{d}=10$. $\mathrm{m}=\mathrm{4,5}$. $\mathrm{n}=\mathrm{10,5}$. $\frac{2\mathrm{m}\mathrm{n}}{\mathrm{b}+\mathrm{d}}=\mathrm{6,3}$. $\sqrt{}\mathrm{m}\mathrm{n}=\mathrm{6,9}$, & $\mathrm{e}=\mathrm{6,6}$ y^{e} number equally interceding those limits 6,3 & 6,9. And the operation will bee as follows.

$\begin{array}{ll}\frac{\mathrm{e}+\frac{1}{2}\mathrm{c}}{\mathrm{e}-\frac{1}{2}\mathrm{c}}=\frac{\mathrm{7,1}}{\mathrm{6,1}}\text{; its Log: is}\phantom{\rule{0.5em}{0ex}}\mathrm{0,065929}\phantom{\rule{0.5em}{0ex}}\text{. \& the Log: of that Logarithm is}& \hfill \mathrm{4,819076}\\ \phantom{0}\frac{\mathrm{n}}{\mathrm{m}}=\frac{\mathrm{10,5}}{\mathrm{4,5}}\text{; its Log: is}\phantom{\rule{0.5em}{0ex}}\mathrm{0,367976}\phantom{\rule{0.5em}{0ex}}\text{. \&}\phantom{\rule{0.5em}{0ex}}{\text{y}}^{\text{e}}\phantom{\rule{0.5em}{0ex}}\text{Log: of}\phantom{\rule{0.5em}{0ex}}{\text{y}}^{\text{t}}\phantom{\rule{0.5em}{0ex}}\text{Logarithm is}& \hfill \mathrm{5,565819}\\ \phantom{0}\frac{\mathrm{a}}{\mathrm{e}}=\frac{100}{\mathrm{6,6}}\text{; its Logarithm is}\phantom{\rule{1em}{0ex}}\text{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}& \hfill \mathrm{1,180456}\\ & \hfill \text{\_\_\_\_\_\_\_\_\_\_\_}\\ \hfill \phantom{\rule{1em}{0ex}}\text{And hence the fourth proportionall its Logarithm is}\hfill & \hfill \mathrm{1,927199}& \phantom{\rule{1em}{0ex}}\text{which}\end{array}$

indicates 84,566
to bee y^{e} desired aggregate. The same by adding y^{e} severall termes together will bee found more justly to bee 84,5636
. But note that if there were 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}$ &c) the rule would still more approach to truth. And so it will in y^{e} examples of usury $\frac{100}{106}.\frac{100}{112}.\frac{100}{118}.\frac{100}{124}$ &c or $\frac{100}{108}.\frac{100}{116}.\frac{100}{124}.\frac{100}{132}$ &c. Or in any other where the difference of the {illeg}|d|enominators beares a lesse proportion to the {illeg} denominator of the first terme. The ground of this rule I beleive you will easily apprehend by contemplating y^{e} Hyperbola, what relation its area beares to such a musicall progressions. Farewell

Yo^{r} much obliged Servito^{r}

I. Newton.

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