I purposed to have given you a visit at ye late solemni{illeg}|t|y of or Chancellors creation; but that when I was prevented {illeg} in yt Journey by ye suddain surp{illeg}|ri|sall of a fit of sicknesse, wch (God bee thanked) I have now recoved|re|d. But since I am prevented {illeg}|fr|ō making a verball acknowledgment of yor favors undeservd favors, I must bee yet conte{illeg}|nt|ed to do it in writing. In wch respect I find by yor last letter, yt y{illeg} I must still become {illeg}|m|ore 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 methodis{illeg}|e| y{illeg}|e| discourse{illeg} of infinite series, designing to illustrate it wth such problems as may (some of ym perhaps) be more acceptable then the the invention of working by such series. But being suddeinly diverted by some buisiness out of Cambridg I have not yet had leisure to return to thos{e} thoughts & I feare I shall not renew them {illeg}|bef|ore winter. I am glad therefore \But since/ you are {illeg} \you informe me there needs no/ hast I hope I may get into ye humour of completing them before ye impression of ye introduction because I if I must helpe to fill up {illeg} /its\ title page I had rather annex somthing that \also wch I may call my owne & wch/ may bee acceptable to Artists{illeg}, as well as ye other to Tyro's.

There haveng {sic} som {sic} things past betweene us concerning musicall progressions, & as I remember you desiring me to communit|c|ate somthing wch I had hind|t|ed to you about it, wch I then had not (nor have yet) adjusted to practise: I shall in its stead of|fer| you somthing else wch I think more to ye purpose.

Any musicall progression AB.AB+C.AB+2C.AB+3 ab.ab+c.ab+2c.ab+3c.ab+4c&c being propounded whose last terme is {illeg} ad. Suppose e {illeg} \=/ meane proportion twixt b & d, or any integrall \or broken/ number betweene them interceding these limits {illeg} /a+d\. {illeg} e is a meane proportion twixt b12c & d12c or any integrall or or {sic} broken number greater then it but not greater then {illeg} |that it is convenient by guesse {illeg} wch differs not considerably from it.| |suppose it intercede the \limits bd & bc×dc/ very near it.| (supose not more differing then \the difference/ bee not greater then {+ bcd}) & this proportion will give you ye summe of all the termes very nearely.

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

Examp suppose {illeg}|y|e progression bee 1005.1006.1007.1008.1009.10010. Then is a=100 b=5. c=1. d=10. & {illeg} 612 intercedes {illeg} /{2bc×d+12cb+d}\ & bc2×d+c2 (yt is {illeg} {6&50 6310&4714}) wch I therefore put for e. And work as follows.
e+12c=11711its Log0,77815. e12c=6its Log0,84510. ‾‾‾‾‾‾‾‾‾‾‾‾ Diff.0,06705. 0 Log ofye Diff.1,17360. 0 3,82640. d+12c=1012,its Log1,02119 b12c=412,its Log0,65321 ‾‾‾‾‾‾‾‾‾‾‾‾ Diff0,36798 0 L ofye Diff. 4,56584 0 3,82640 Log ofaeor1006121,18709

| 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
/Whence\ the \ye desired/ aggregate desired is found to be 84|5621
And of {illeg} \The same/ by adding ye \severall/ seve termes together will bee found \more justly/ to bee 84|5636

But note yt were there more termes inserted into ye progression, (as suppose it was 1005.100512.1006.100612.1007.&c) the rule would still more approac{illeg}|h| to truth.

2dly that the difference of the termes \denominators/ ought to be two or

| 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

© 2022 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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