Letter from Newton [to John Smith?], dated 27 August 1675 Isaac Newton c. 341 words The Newton Project Falmer 2012 Newton Project, University of Sussex

1 f.

Published in H.W. Turnbull (ed), The Correspondence of Isaac Newton, vol. 1 (Cambridge: 1959), pp. 350-1

Copy letter from Newton to John Smith, dated 24 July 1675 [MS Add. 9597/2/18/43]
UKCambridgeCambridge University Library Macclesfield Collection MS Add. 9597/2/18/44
27 August 1675 England English Latin John Smith John Collins Unknown Cataloguer MathematicsCorrespondence Daniele Cassisa started tagged transcription Catalogue information compiled from CUL Janus Catalogue by Michael Hawkins Proofed by Robert Iliffe Code audit by Michael Hawkins
44.

Trinity Coll:College Aug: 27th, 1675.

SrSir,

In the Theorems ytthat I sent youu I perceive I committed a mistake in transcribeing them from the papers where I had computed them. They should have been $\begin{array}{l}\text{2)}\phantom{\rule{0.5em}{0ex}}B+\frac{A}{B}=\sqrt{}A\text{.}\\ \text{3)}\phantom{\rule{0.5em}{0ex}}2B+\frac{A}{BB}=\sqrt{}c:\phantom{\rule{0.5em}{0ex}}A\text{.}\\ \text{4)}\phantom{\rule{0.5em}{0ex}}3B+\frac{A}{{B}^{c:}}=\sqrt{}qq.\phantom{\rule{0.5em}{0ex}}A\text{.}\end{array}$

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 common Arithmetick, ytthat youu may have its exact square to 10 decimal places, and by this square Divide $A$ to 11 decimal places, and to the Quotient add $2B$: The third part of the Quotient shall be the root cubical of $A$: to 11 Decimal places. yoryour surest way will be to finde first the whole series of yethe Roots, $B.$ by Logarithms, & try whether it be Regular by Differencing it: Then square those Roots by Nepeirs bones, and lastly Divide each NumbrNumber $A.$ by the correspondent square, and add $2B$ to each Quotient, and try the Resulting series againe by differencing it, whether it be Regular. If it be regular, I suppose youu 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 youu examples in numbers; but that I have lent my Bookes of Logarithms to a person, who is out of Towne. yoryour humble SirvantServant Is: Newton I thank youu for yoryour intended present.
Copia exrextractaextrahatur Mr Collins, I have left wthwith the Maid yoryour book of Briggs Logarithms, and would request yoryour favour (if I might not be too troublesome) 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 8th place of the Decimal onwards; though the logarithm it selfe & the work upon it be duly proved. Mr Newtons rules about Pure Powers, for finding their rootes.