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Trin Coll Feb 6 1669.
Cambridge.

Sr

Mr Barrow shewed mee some of yor papers in wch I was much pleased at Monsr Cassinis inventio{illeg}|n| for finding ye Apogœa & excentricitys of ye Planets

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

As for example if ${1200}^{\stackrel{‾}{li}}$ bee payd at present for ${100}^{\stackrel{‾}{li}}$ yearely for 31 yeares then is ye Logarithm of $\left(\frac{31B}{A}\right)=0,41218$. And consequently by the rule $\frac{2,47308}{79,39100}$ or 0,03111 is the Logarithme of $\frac{100+N}{100}$; whence $\frac{100+N}{100}=1,0743$. And $N=7,43$ or ${7}^{\stackrel{‾}{li}}$, ${8}^{\mathrm{s}}$, ${7}^{\mathrm{d}}$. Soe that the Annuity was bought at ${7}^{\stackrel{‾}{lib}}$ ye rate of ${7}^{\stackrel{‾}{lib}}$. ${8}^{\mathrm{s}}$, ${7}^{\mathrm{d}}$ 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 ${16}^{lib}$ per cent. And if the rate bee above 16 or ${18}^{\stackrel{‾}{lib}}$ per cent, or wch is all one if {illeg} $A\phantom{\rule{0.5em}{0ex}}\stackrel{‾}{\phantom{‾}\underset{‾}{\phantom{‾}|}}\phantom{\rule{0.5em}{0ex}}6B$, then this rule $\frac{A+B}{A}=\frac{100+N}{100}$ will not \err/ above ${2}^{\mathrm{s}}$. You may try the truth of these rules by the equation ${x}^{32}=\frac{A+B}{A}{x}^{31}-\frac{B}{A}$; putting $\frac{100+N}{100}=x$, & working in logarithms.

You seeme to apprehend as if I was about writing elaborate Notes upon Kinck-huyson: I understood from Mr Barrow yt yor desire was only to have ye 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 wch 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; wch 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 yt understand Dutch.

In finding ye Aggregate of the termes of a Musicall Progression there is one way by Logarithms very obvious (viz by by|su|bducting ye logs of each denominator from that of the Numerator &c) wch 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

Yor troublesome Freind & Servant

Isaac Newton.

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|Mr Newtons Solutio of the Interest Probleme|

These

To Mr John Collins
at his house in Bloomsbury
next doore to ye three
Crownes in

London