I purposed to have given you a visit at the late solemnity of our Chancellors creation; but I was prevented in that Journey by the suddain surprisall of a fit of sicknesse, which (God bee thanked) I have now recovered. But since I am prevented from making a verball acknowledgment of your undeservd favors, I must bee yet contented to do it in writing. In which respect I find by your last letter, that I still become more your debtor both for the care you take about my concernes & for Borrellius de motionibus. The last winter I reveiwed the introduction & made some few additions to it. & partly upon Dr Barro{ws} instigation, I began to new methodise the discourse of infinite series, designing to illustrate it with such problems as may (some of them perhaps) be more acceptable then the invention of working by such series. But being suddeinly diverted by some buisiness I have not yet had leisure to return to thos{e} thoughts & I feare I shall not before winter. But since you informe me there needs no hast I hope I may get into the humour of completing them before the impression of the introduction because if I must helpe to fill up its title page I had rather annex somthing which I may call my owne & which may bee acceptable to Artists, as well as the other to Tyro's.

There haveing some things past betweene us concerning musicall progressions, & as I remember you desiring me to communicate somthing which I had hinted to you about it, which I then had not (nor have yet) adjusted to practise: I shall in its stead offer you somthing else which I think more to the purpose.

Any musicall progression ab.ab+c.ab+2c.ab+3c.ab+4c&c being propounded whose last terme is ad. Suppose e is a meane proportion twixt b12c & d12c or any integrall or broken number that it is convenient by guesse which differs not considerably from it. suppose it intercede the limits bd & bc×dc very near it. & this proportion will give you the summe of all the termes very nearely.

As the Logarithm e+12ce12c to the logarithm of d+12cb12c, so is ae to the desired summe.

Examp suppose the progression bee 1005.1006.1007.1008.1009.10010. Then is a=100 b=5. c=1. d=10. & 612 intercedes {2bc×d+12cb+d} & bc2×d+c2 (that is {6&50 6310&4714}) which I therefore put for e. And work as follows.

045,002,2594,50 15,00 0(6,3 047,25 1,751,4134.(14712


e+12ce12c=76its log. is0,066917& The Log of that Logarithm,4,825731 d+12cb12c=1012412its log. is0,367977& The Log: of that Logarithm5,565820 ae=100612its logarithm is________________________________________1,187087 ‾‾‾‾‾‾‾‾‾‾‾‾‾ 1,927176 . the Logarith of the desired aggregate84|562 , substracte+12ce12c }add.d+12cb12c 100612 0the result, which is . 0|0

The same by adding the severall termes together will bee found more justly to bee 84|5636

But note that were there more termes inserted into the progression, (as suppose it was 1005.100512.1006.100612.1007.&c) the rule would still more approach to truth.

Secondly that the difference of the denominators ought to be

1x3.12xx12yy.yx::12zz12zz+4oz.o. oyyoxx2xxyy=yx×ozz4+2oz3 yyxx×z3=yx×2xxyy z3=2xxyyy+x 61.71 4,5.10,547,25 0.3010300 1.3064250 2.0423786 3,6498436 2,4637523 0.8212500

0,00000000,00007310 0,0659285.4,81907965 0,3679767.5,565820071,18045610 }11 6,74627617 8456600001,92719652 8456740001,92720302 0056600000 }11 18512583 17853298 00659285 6746275 0007198

To Mr Isaac
Newton fellow of
Trinity Colledge

© 2017 The Newton Project

Professor Rob Iliffe
Director, AHRC Newton Papers Project

Scott Mandelbrote,
Fellow & Perne librarian, Peterhouse, Cambridge

Faculty of History, George Street, Oxford, OX1 2RL - newtonproject@history.ox.ac.uk

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