July 20th 1671.


I purposed to have given you a visit at ye late solemnity of or Chancellors creation; but I was prevented in yt Journey by ye suddain surprisall of a fit of sicknesse, wch not long after (God be thanked) I again recovered of. And since I am prevented from making a verball acknowledgment of yor undeserved favours, I must bee yet contented to do it in writing. In wch respect I find by yor last letter, yt I am still become more yor 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 a{illeg}|t| or next meeting, & that you would not for ye future put yor selfe to ye like trouble in sending any more books. I shall take it for a great favour if in yor letters you will onely inform mee of ye names of ye best of those bookes wch 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 ye discourse of infinite series, designing to illustrate it wth such problems as may (some of them perhaps) be more acceptable then ye 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 ye humour of completing them before ye impression of the introduction, because if I must helpe to fill up its title page, I had rather annex somthing wch I may call my owne, & wch may bee acceptable to Artists as well as ye other to Tyros.

There having some things past between us concerning musicall progressions, & as I remember you desiring mee to communicate somthing wch I had hinted to you about it, wch I then had not (nor have yet) adjusted to practise: I shall in its stead offer you somthing else wch I think more to ye purpose. Any musicall progression ab.ab+c. ab+2c.ab+3c.ab+4c &c being propounded whose last terme is ad: for ye following operation choose any convenient number \e/ (whither whole broken or surd) which intercedes these limits {illeg} 2mnb+d & mn; supposing b12c to bee m, & d+12c to bee n. And this proportion will give you the aggregate of the termes very neareby the truth.

As ye Logarithm e+12ce12c to ye Logarithm of nm, so is ae to ye {illeg}|d|esired summe.

Example. Suppose ye progression bee 1005.1006.1007.1008.1009.10010. That is a=100. b=5. c=1. d=10. m=4,5. n=10,5. 2mnb+d=6,3. mn=6,9, & e=6,6 ye number equally interceding those limits 6,3 & 6,9. And the operation will bee as follows.

e+12ce12c=7,16,1; its Log: is0,065929. & the Log: of that Logarithm is4,819076 0nm=10,54,5; its Log: is0,367976. &yeLog: ofytLogarithm is5,565819 0ae=1006,6; its Logarithm is __________________________________________1,180456 ___________ And hence the fourth proportionall its Logarithm is1,927199which
indicates 84,566 to bee ye desired aggregate. The same by adding ye severall termes together will bee found more justly to bee 84,5636 . But note that if there were more termes inserted into ye progression, (as suppose it was 1005.100512.1006.100612.1007 &c) the rule would still more approach to truth. And so it will in ye examples of usury 100106.100112.100118.100124 &c or 100108.100116.100124.100132 &c. Or in any < insertion from the left margin > other where the difference of the {illeg}|d|enominators beares a lesse proportion to the {illeg} denominator of the first terme. The ground of this rule I beleive you will easily apprehend by contemplating ye Hyperbola, what relation its area beares to such a musicall progressions. Farewell

Yor much obliged Servitor

I. Newton.

< text from f 11r resumes >

|Mr Newtons second Letter about a Musicall Progression|

To Mr John Collins at

Mr William Austins house {illeg} {illeg}|o|ver
against the Adam & Eve in
Petty France ,|in| Westminster.



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Professor Rob Iliffe
Director, AHRC Newton Papers Project

Scott Mandelbrote,
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