# Letter from Newton [to John Smith?], dated 27 August 1675

Trinity Coll:

Aug: 27^{th}, 1675.

S^{r},

In the Theorems y^{t} I sent yo^{u} I perceive I com̄itted a mistake in transcribeing them from the papers where I had computed them. They should have been

$\begin{array}{l}\text{2)}\phantom{\rule{0.5em}{0ex}}\mathrm{B}+\frac{\mathrm{A}}{\mathrm{B}}=\sqrt{}\mathrm{A}\text{.}\\ \text{3)}\phantom{\rule{0.5em}{0ex}}2\mathrm{B}+\frac{\mathrm{A}}{\mathrm{B}\mathrm{B}}=\sqrt{}c:\phantom{\rule{0.5em}{0ex}}\mathrm{A}\text{.}\\ \text{4)}\phantom{\rule{0.5em}{0ex}}3\mathrm{B}+\frac{\mathrm{A}}{{\mathrm{B}}^{c:}}=\sqrt{}qq.\phantom{\rule{0.5em}{0ex}}\mathrm{A}\text{.}\end{array}$

In words at length: To finde the cube root of A to 11 decimal places: seek the Root by Logarithms to 5 decimal places, and suppose it B. Then square B, not by Logarithms, but by com̄on Arithmetick, y^{t} yo^{u} may have its exact square to 10 decimal places, and by this square Divide A to 11 decimal places, and to the Quotient add $2\mathrm{B}$: The third part of the Quotient shall be the root cubical of A: to 11 Decimal places. yo^{r} surest way will be to finde first the whole series of y^{e} Roots, B. by Logarithms, & try whether it be Regular by Differencing it: Then square those Roots by Nepeirs bones, and lastly Divide each Numb^{r} A. by the correspondent square, and add $2\mathrm{B}$ to each Quotient, and try the Resulting series againe by differencing it, whether it be Regular. If it be regular, I suppose yo^{u} know the differences will at last come to be equal: what is said of Cubes is easily applyable to Square=Squares: I would have given yo^{u} examples in numbers; but that I have lent my Bookes of Logarithms to a ꝑson, who is out of Towne.

yo^{r} humble Sirvant

Is: Newton

Copia ex^{r}

M^{r} Collins, I have left w^{th} the Maid yo^{r} book of Briggs Logarithms, and would request yo^{r} favour (if I might not be too troublesome) {illeg} to procure me the loane of Nepeirs Bones & the Booke of their use; I should in a few dayes returne them /J: Smith\

I finde all the Roots (found by the longest Radius of logarithms) false & uncertaine from the 8^{th} place of the Decimal onwards; though the logarithm it selfe & the work upon it be duly proved.