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TrinTrinity CollCollege Feb 6 1669.Cambridge.

SrSir

Mr Barrow shewed mee some of yoryour papers in wchwhich I was much pleased at MonsrMonsieur Cassinis invention for finding yethe Apogœa & excentricitys of yethe Planets

For your Annuity Problem I have sent you the following solution to
consider of wtwhat use it ~~b~~ may bee. To know at what rate ($\mathrm{N}$ per cent) an Annuity of $\mathrm{B}$ is purchased for 31 yeares at yethe price $\mathrm{A}$. 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 yethe 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 $\mathrm{0,03111}$ 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 ${7}^{\stackrel{\u203e}{\mathrm{lib}}}$ yethe 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 when 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 wchwhich is all one if $\mathrm{A}\phantom{\rule{0.5em}{0ex}}\stackrel{\u203e}{\phantom{\u203e}\underset{\u203e}{\phantom{\mathrm{\u203e}}|}}\phantom{\rule{0.5em}{0ex}}6\mathrm{B}$, 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 Mr Barrow ytthat yoryour desire was only to have yethe 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 wchwhich is not to bee found in other Authors of better esteelme.
You make mention of another book of the same Author translated badly into latten by a German Gunner; wchwhich 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 ytthat understand Dutch.
In finding yethe Aggregate of the termes of a Musicall Progression there is one way by Logarithms very obvious (viz by bysubducting yethe logslogarithms of each denominator from that of the Numerator &c) wchwhich 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
YorYour troublesome Freind & Servant
Isaac Newton.