11.
July 20th 1671.

SrSir

I purposed to have given you a visit at yethe late solemnity of orour Chancellors creation; but I was prevented in ytthat Journey by yethe suddain surprisall of a fit of sicknesse, wchwhich not long after (God be thanked) I again recovered of. And since I am prevented from making a verball acknowledgment of yoryour undeserved favours, I must bee yet contented to do it in writing. In wchwhich respect I find by yoryour last letter, ytthat I am still become more yoryour 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 at orour next meeting, & that you would not for yethe future put yoryour selfe to yethe like trouble in sending any more books. I shall take it for a great favour if in yoryour letters you will onely inform mee of yethe names of yethe best of those bookes wchwhich newly come forth.

The last winter I reveiwed the Introduction & made some few additions to it: And partly upon Dr Barrows instigation, I began to new methodiz yethe discourse of infinite series, designing to illustrate it wthwith such problems as may (some of them perhaps) be more acceptable then yethe 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 yethe humour of completing them before yethe impression of the introduction, because if I must helpe to fill up its title page, I had rather annex somthing wchwhich I may call my owne, & wchwhich may bee acceptable to Artists as well as yethe other to Tyros.

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

As yethe Logarithm $\frac{\mathrm{e}+\frac{1}{2}\mathrm{c}}{\mathrm{e}-\frac{1}{2}\mathrm{c}}$
to yethe Logarithm of $\frac{\mathrm{n}}{\mathrm{m}}$, so is $\frac{\mathrm{a}}{\mathrm{e}}$ to yethe desired summe.
Example. Suppose yethe progression bee $\frac{100}{5}.\frac{100}{6}.\frac{100}{7}.\frac{100}{8}.\frac{100}{9}.\frac{100}{10}$. That is $\mathrm{a}=100$. $\mathrm{b}=5$. $\mathrm{c}=1$. $\mathrm{d}=10$. $\mathrm{m}=\mathrm{4,5}$. $\mathrm{n}=\mathrm{10,5}$. $\frac{2\mathrm{m}\mathrm{n}}{\mathrm{b}+\mathrm{d}}=\mathrm{6,3}$. $\sqrt{}\mathrm{m}\mathrm{n}=\mathrm{6,9}$, & $\mathrm{e}=\mathrm{6,6}$ yethe number equally interceding those limits $\mathrm{6,3}$ & $\mathrm{6,9}$. 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 $\mathrm{84,566}$
to bee yethe desired aggregate. The same by adding yethe severall termes together will bee found more justly to bee $\mathrm{84,5636}$
. But note that if there were more termes inserted into yethe 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 yethe 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 denominators beares a lesse proportion to the denominator of the first terme. The ground of this rule I beleive you will easily apprehend by contemplating yethe Hyperbola, what relation its area beares to such a musicall progressions. Farewell
YorYour much obliged Servitorur
I. Newton.