<43r>

Sr,

I rec'd yor former L're as well as yor later, and should have written to you sooner, but that I stay'd to think of something yt might satisfy yor Desire; But though I can not hitherto doe it to my owne liking, yet that I may not wrack yor patience too much I have here witt {sic} you what occurrs to mee, wch is only about facilitating ye Extraccon of ℞. The for Method might be applyed to determin all by every 1000th, as well as by every 100th ℞, but not with advantage, for it will require the Extraccon of ℞ to 14 or 15 places, besides a greater number of Addicons, Subduccons & Divisions in those greater numbers: And therefore I have rather sent you the following Notes about Extracting ℞.

1.) When you have extracted any ℞ by coon Arithmetick to 5 Decimal places, you may get the figures of the other 6 places by Dividing only the Residuum by double the Quotienttriple theqof the Quotientquadruple thecof the Quotient     for thesquareQu cubeQu square squareQu
Suppose B. the Quotient or ℞ extracted to 5 Decimal places, and C. the last Residuum, by the Division of wch you are to get the next figure of the Quotient, and D the Divisor (that is 2B or 3BB or 4B.c.=D & B+CD shall be the ℞ desired. That is, the same Division, by wch you would finde the 6th decimal figure, if prosecuted, will give you all to the 11th decimal figure.

2) You may seek the ℞ if you will, to 5 Decimal places by the logarithm's, But then you must finde the rest thus. Divide the propounded number oncetwicethrice by yt ℞ prosecuting the Division alwayes to 11 Decimal places, and to the Quotient add yesaidonce, & halfetwice, & a third partthrice, & a quarteroof the summshall be thesquareCubesquare squareIdesired.

For instance
let A be the numb, and B. its QCQQ ℞ extracted by Logarithms unto 5 decimal /places:\
and2)B+AB,AB0and3)2B+AB2, shall be the                and4)3B+AB3,QAB0CAB0root desiredQQAB0

Note yt you have according to my former Direccon but 76Q & 88C & 94QQ to extract, whereof 10 are exact ℞. But I think you will doe well to lett the Table of QQ alone, til you have done th' other two, and then, if you finde your time too short, print the Q. & C. ℞ without troubling yor selfe any further.

Sr, I am,       yor humble Srvant

Is: Newton

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Cambridge July 24th 1675

Copia vera

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<43v>

|For Mr Collins Copie Mr Newtons 2d L're conc: extraccon of ℞ |

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Professor Rob Iliffe
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