# Letter from Newton to John Collins, dated 6 February 1669/70

Trin Coll Feb 6 1669.

Cambridge.

S^{r}

Mr Barrow shewed mee some of yo^{r} papers in w^{ch} I was much pleased at Mons^{r} Cassinis inventio{illeg}|n| for finding y^{e} Apogœa & excentricitys of y^{e} Planets

For your Annuity Problem I have sent you the following solution to
consider of w^{t} use it b may bee. To know at what rate (N per cent) an Annuity of B is purchased for 31 yeares at y^{e} price A. The rule is $\frac{6\phantom{\rule{0.5em}{0ex}}Log\phantom{\rule{0.2em}{0ex}}\text{: of}\phantom{\rule{0.5em}{0ex}}\left(\frac{31\mathrm{B}}{\mathrm{A}}\right)}{100-50\phantom{\rule{0.5em}{0ex}}Log\phantom{\rule{0.5em}{0ex}}\text{of}\phantom{\rule{0.5em}{0ex}}\left(\frac{31\mathrm{B}}{\mathrm{A}}\right)}=Log\phantom{\rule{0.2em}{0ex}}\text{: of}\phantom{\rule{0.5em}{0ex}}\left(\frac{100+\mathrm{N}}{100}\right)$.

As for example if ${1200}^{\stackrel{\u203e}{\mathrm{li}}}$ bee payd at present for ${100}^{\stackrel{\u203e}{\mathrm{li}}}$ yearely for 31 yeares then is y^{e} Logarithm of $\left(\frac{31\mathrm{B}}{\mathrm{A}}\right)=\mathrm{0,41218}$. And consequently by the rule $\frac{\mathrm{2,47308}}{\mathrm{79,39100}}$ or 0,03111 is the Logarithme of $\frac{100+\mathrm{N}}{100}$; whence $\frac{100+\mathrm{N}}{100}=\mathrm{1,0743}$. And $\mathrm{N}=\mathrm{7,43}$ or ${7}^{\stackrel{\u203e}{\mathrm{li}}}$, ${8}^{\mathrm{s}}$, ${7}^{\mathrm{d}}$. Soe that the Annuity was bought at ${7}^{\stackrel{\u203e}{\mathrm{lib}}}$ y^{e} rate of ${7}^{\stackrel{\u203e}{\mathrm{lib}}}$. ${8}^{\mathrm{s}}$, ${7}^{\mathrm{d}}$ per cent.

This rule is not exact but yet soe exact as never to faile above ${2}^{\mathrm{d}}$ or ${3}^{\mathrm{d}}$ at the most w{illeg}|h|en the rate is not above ${16}^{\mathrm{lib}}$ per cent. And if the rate bee above 16 or ${18}^{\stackrel{\u203e}{\mathrm{lib}}}$ per cent, or w^{ch} is all one if {illeg} $\mathrm{A}\phantom{\rule{0.5em}{0ex}}\stackrel{\u203e}{\phantom{\u203e}\underset{\u203e}{\phantom{\mathrm{\u203e}}|}}\phantom{\rule{0.5em}{0ex}}6\mathrm{B}$, then this rule $\frac{\mathrm{A}+\mathrm{B}}{\mathrm{A}}=\frac{100+\mathrm{N}}{100}$ will not \err/ above ${2}^{\mathrm{s}}$. You may try the truth of these rules by the equation ${\mathrm{x}}^{32}=\frac{\mathrm{A}+\mathrm{B}}{\mathrm{A}}{\mathrm{x}}^{31}-\frac{\mathrm{B}}{\mathrm{A}}$; putting $\frac{100+\mathrm{N}}{100}=\mathrm{x}$, & working in logarithms.

You seeme to apprehend as if I was about writing elaborate Notes upon Kinck-huyson: I understood from M^{r} Barrow y^{t} yo^{r} desire was only to have y^{e} booke reveived: that if any thing were defective or amisse it might be amended, & to that purpose about two Months since I reveived it & made some \such/ observations upon it to that purpose. But though the booke bee a good introduction I think it not worth the paines of a formall comment, There being nothing new or notable in it w^{ch} is not to bee found in other Authors of better esteel|m|e.

You make mention of another book of the same Author translated badly into latten by a German Gunner; w^{ch} you would have mee correct. I understand not Dutch & would not willingly doe the Author soe much wrong as to undertake to correct a translation where I understand not the originall: I suppose there want not Mathematitians in London y^{t} understand Dutch.

In finding y^{e} Aggregate of the termes of a Musicall Progression there is one way by Logarithms very obvious (viz by by|su|bducting y^{e} log^{s} of each denominator from that of the Numerator &c) w^{ch} I supposed to bee the ordinary way in Practise & therefore mentioned it not in my Letter. If you meane another way I would bee glad to have it communicated. Thus Sr I am

Yo^{r} troublesome Freind & Servant

Isaac Newton.