Copy letter from Newton to John Smith, dated 24 July 1675

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

\{\begin{array}{l} double the Quotient \\ triple the q of the Quotient \\ quadruple the c of the Quotient \end{array}\} for the ℞ \{\begin{cases} square \\ cube \\ square square \end{cases}

Suppose

B

. the Quotient or

℞

extracted to 5 Decimal places, and

C

. the last Residuum, by the Division of wchwhich youu are to get the next figure of the Quotient, and

D

the Divisor (that is

2 B

3 B B

{4 B}^{.c.} = D

B + \frac{C}{D}

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

\begin{matrix} once \\ twice \\ thrice \end{matrix}\}

by ytthat

℞

prosecuting the Division alwayes to 11 Decimal places, and to the Quotient add

y^{e} said ℞ \{\begin{array}{l} once, & halfe \\ twice, & a third part \\ thrice, & a quarter \end{array}\} \begin{matrix} of the summ \\ shall be the \end{matrix} \{\begin{array}{l} square \\ Cube \\ square square \end{array}\} ℞ desired.

For instance let

A

be the numbnumber, and

B

. its

\{\begin{cases} Q \\ C \\ QQ \end{cases}

℞

extracted by Logarithms unto 5 decimal places:

\begin{array}{l} 2) B + \frac{A}{B}, \\ and 3) 2 B + \frac{A}{B^{2}}, shall be the \\ 4) 3 B + \frac{A}{B^{3}}, \end{array}\} \begin{array}{l} Q \\ C root desired \\ QQ \end{array}

Note ytthat youu have according to my former Direccontion but

76 Q ℞

88 C ℞

94 QQ ℞

to extract, whereof 10 are exact

℞

. But I think youu will doe well to lett the Table of

QQ ℞

alone, til youu have done th'the other two, and then, if youu finde your time too short, print the

Q.

C.

℞

without troubling yoryour selfe any further. SrSir, I am, yoryour humble SrvantServant Is: Newton Cambridge July 24th 1675 Copia vera

For Mr Collins Copie Mr Newtons 2d L'reLetter conc:concerning extraccontion of

℞