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

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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 $\frac{6Log\text{: of}\left(\frac{31\mathrm{B}}{\mathrm{A}}\right)}{100-50Log\text{of}\left(\frac{31\mathrm{B}}{\mathrm{A}}\right)}=Log\text{: of}\left(\frac{100+\mathrm{N}}{100}\right)$.

As for example if ${1200}^{\overline{\mathrm{li}}}$ bee payd at present for ${100}^{\overline{\mathrm{li}}}$ 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 $7^{\overline{\mathrm{li}}}$, $8^{\mathrm{s}}$, $7^{\mathrm{d}}$. Soe that the Annuity was bought at $7^{\overline{\mathrm{lib}}}$ y^e rate of $7^{\overline{\mathrm{lib}}}$. $8^{\mathrm{s}}$, $7^{\mathrm{d}}$ 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 ${18}^{\overline{\mathrm{lib}}}$ per cent, or w^ch is all one if {illeg} $\mathrm{A}\stackrel{‾}{\phantom{‾}\underset{‾}{\phantom{\mathrm{‾}}|}}6\mathrm{B}$, 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.

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|M^r Newtons Solutio of the Interest Probleme|

These

To M^r John Collins
at his house in Bloomsbury
next doore to y^e three
Crownes in

London