I received your former Letter as well as your later, and should have written to you sooner, but that I stay'd to think of something that might satisfy your Desire; But though I can not hitherto doe it to my owne liking, yet that I may not wrack your patience too much I have here writt you what occurrs to mee, which is only about facilitating the Extraction of ℞. The former 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 Extraction of ℞ to 14 or 15 places, besides a greater number of Additions, Subductions & Divisions in those greater numbers: And therefore I have rather sent you the following Notes about Extracting ℞.

1.) When you have extracted any ℞ by common 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 which 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 which 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 that ℞ 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 number, 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 that you have according to my former Direction 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 the other two, and then, if you finde your time too short, print the Q. & C. ℞ without troubling your selfe any further.

Sir, I am,       your humble Servant

Is: Newton

< insertion from the left margin >

Cambridge July 24th 1675

Copia vera

< text from f 43r resumes >

For Mr Collins Copie Mr Newtons 2d Letter concerning extraction of ℞

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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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