43.
SrSir,
I rec'dreceived yoryour former L'reLetter as well as yoryour later, and should have written to youu sooner,
but that I stay'd to think of something ytthat might satisfy yoryour Desire;
But though I can not hitherto doe it to my owne liking, yet that
I may not wrack yoryour patience too much I have here wittwritt youu
what occurrs to mee, wchwhich is only about facilitating yethe Extraccontion
of . The form̄mer 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 Extraccontion of to 14 or 15 places, besides a greater number of Addiconstions, Subducconstions & Divisions in those greater numbers: And therefore I have rather sent youu the following Notes about Extracting .
1.) When youu have extracted any by com̄mmon Arithmetick to 5 Decimal places, youu may get the figures of the other 6 places by Dividing only the Residuum by Suppose . the Quotient or extracted to 5 Decimal places, and . the last Residuum, by the Division of wchwhich youu are to get the next figure of the Quotient, and the Divisor (that is or or & shall be the desired. That is, the same Division, by wchwhich youu would finde the 6th decimal figure, if prosecuted, will give you all to the 11th decimal figure.
2) Youu may seek the if youu will, to 5 Decimal places by the logarithm's, But then youu must finde the rest thus. Divide the propounded number by ytthat prosecuting the Division alwayes to 11 Decimal places, and to the Quotient add
For instance let be the numbnumber, and . its extracted by Logarithms unto 5 decimal places:
Note ytthat youu have according to my former Direccontion but & & to extract, whereof 10 are exact . But I think youu will doe well to lett the Table of alone, til youu have done th'the other two, and then, if youu finde your time too short, print the & without troubling yoryour selfe any further.
SrSir, I am, yoryour humble SrvantServant
Is: Newton
Cambridge July 24th 1675
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