<2:1r>

Mr Newton Our former aquentance Imbold\e/neth me to troble you wth these few lines: wch is to Intreat your {fawor}{faivor} If yt an oppertunity will serve: To peruse over this Small Table of the Suns Azimuth for the Latitude of wch is for Every Hour of the day & night: \being/ from the Equinoctiall to 24 degrees of declination being Eather North or South for Every 4th degree of declination: the Azimuth Being from the East or west Eather North wards Or South wards{.} And Likwise this Table of the North Stars Azimuth which is shewed from the North for Every hour Eather above or belowe the pole wch is for Every 10th degree of Latitude from the Equinoctiall to ${60}^{degs}$ of Latitude: which I would intreate your favour to Examine & give me notice thereof whether the proportions hold good or not: vizd whether these Tables Shew the Sun or North Stars True Azimuth as I have heare shewed by these tables or not. As for this way of Azimuth it is by a way of my owne contriveing by such a way as I never see nor heard of before wch maketh me m{\o/}ore desirus to have your oppinion thereof: As for this way of callculation I do not at all conserne my selfe {illeg} \{illeg} wth/ allthe Altitude {illeg} {o}nley wth the Declination: so yt the Azimuth for Every Hour is found as soone o{illeg}|r| sooner or sooner then the Hourly Altitu wch is by such a duble way of workeing for ye Hour of l0 & 2 for North Declination & 10 & 2 for South Declination & the Hour of both 6es is wrought by the product of one Multiplication wch is to be Added to or Subtracted from a Nother Summ & then takeing $\frac{1}{2}$ thereof sheweth 6 Hours Azimuth: So yt the whole 24 hours Azimuth is wrought p the produce of five multiplications by being Added \or/ substracted fro\m/ An orther given summ wch is Allwayes a certane Summ for yt hour & Lat' to any decli\nat/{.} I have heare in some places not onely sett downe the degrees & minutes but have shewed by pricks {illeg} \to a/ quarter of a minute in some plac\es/{.} {illeg}|B|ut if I would have bene presise in this Calculation p this way of callculating of ye North Stars Azimuth I could have sett downe not onley the minute But likewise yt to one thousand part of a minute & yt by ye same Radius by wch this was Calcula\ted/{.} But this I Suppose is neare enough if yt ye proportions hold good wch I would desire you to Examine & give me notice thereof wch if it be within 14 dayes Af{illeg}|t|er the {dai{illeg}|t|} of this I may Receive it If yt you d{illeg}|i|rect your Letter to Mr William Kings Junior Silkeweaver In three horshoo alley in Old Streete in London I shall Receve it: Haveing no moore at prsent But my dutye to my {unc{illeg}|k|le} & my service to your selfe hopeing that you ar both well as praised be god I am I Remane

A your Command to serve you Arthur Storer

London Sept 4th: 1678

I sent you a former table by {unc{illeg}|k|le} Babington of ye North Stars \Hourly/ Altitude & Azimuth
But in yt {yof} table of the Suns Azimuth was Omitted wch I hope you have Receved before this{.}

<2:2v>

For his much Esteemed Frend
Mr Isaac Newton of
Trenity Colidg in
Cambridg:
{prs}

< insertion from higher up f 2r >

$\begin{array}{lllllll}{x}^{3}& +& 2a{x}^{2}& +& aax& -& abb\\ & -& b& -& 2ab\end{array}$    $\begin{array}{lllllll}{x}^{3}& -& a{x}^{2}& +& aabx& +& abc\\ & -& b& -& ac\\ & +& c& -& bc\end{array}$

$\begin{array}{llllll}-& aab& +& aac& +& 3abc\\ -& bba& +& bbc\\ -& cca& -& ccb\end{array}$    {illeg}$.abc.$    3 {illeg}   $\frac{-a-b}{c}+\frac{a-c}{b}+\frac{b-c}{a}$.   1    $\begin{array}{cc}\begin{array}{c}\left|\phantom{000}\begin{array}{ccccccc}& aa& +& 2ab& \phantom{000}& & ab\\ +& bb& -& 2ac& .& -& ac& .\\ +& cc& -& 2bc& & -& bc& \end{array}\right\\ \\ \\ \begin{array}{lllllll}& aabb& -& 2aabc& \phantom{000}& -& aabc\\ +& aacc& -& 2abbc& \phantom{0}.& -& abbc& \\ +& bbcc& -& 2abcc& \phantom{000}& +& abcc& \end{array}\\ \end{array}& \begin{array}{c}\begin{array}{ccc}\begin{array}{cc}& 4aa\\ -& 4ac& .\\ +& cc\\ & \end{array}& & \begin{array}{cc}& aa\phantom{4}\\ -& 2ac\end{array}\end{array}\end{array}\end{array}$

$\begin{array}{lllllllllllll}{x}^{5}& -& 4{x}^{4}& +& 6{x}^{3}& -& 4xx& +& x& +& a& +& b\\ & +& a& -& 4a& +& 6a& -& 4a& -& 4b\\ & & & +& b& +& 4b& +& 6b\end{array}$

$\begin{array}{lllllllllll}6& -& 24c& +& 3bb& +& 16aa& -& 24ab& +& 6bb\\ & -& 4& +& 1& & & -& 4\end{array}$

$\begin{array}{cc}\begin{array}{cccc}& 16aa& -& 28a\\ -& 28ab& +& 3b\\ +& 6bb& +& 6\end{array}& \begin{array}{cccc}& 36aa& -& 24a\\ & 48ab& +& 16b\\ +& 16bb& +& 16\end{array}\end{array}$

< text from f 2:2v resumes > < insertion from lower down f 2r >

$\begin{array}{ccccccc}{x}^{3}& +& 3xx& +& 3x& +& 1\end{array}$

$\begin{array}{cc}\begin{array}{r}\begin{array}{c}\begin{array}{lllllllllllllll}& & & & & & & & & & & & & +& 3\\ {x}^{4}& +& 3{x}^{3}& +& 3xx& +& x& +& a& \left(& {x}^{4}& +& 3{x}^{3}& +& 3a{x}^{2}& +& 1x& +& a\\ & +& a& +& 3a& +& 3a& & & & & +& a& +& bb& +& 3a& -& b\\ & & & & & & & & & & & -& b& -& ab& -& 3b& -& 3ab\\ & & & & & & & & & & & & & -& 3b& -& 3ab& +& 3bb\\ & & & & & & & & & & & & & & & +& abb& +& 3abb\\ & & & & & & & & & & & & & & & +& 3bb& -& a{b}^{3}\\ & & & & & & & & & & & & & & & -& {b}^{3}& -& 3{b}^{3}\\ & & & & & & & & & & & & & & & & & +& {b}^{4}\end{array}|\end{array}\\ \begin{array}{ccc}& & \\ x& +& b\end{array}\end{array}& x\end{array}$

$\begin{array}{lllllllll}1& .& 4& .& 6& .& 4& .& 1& .\\ 1& .& 5& .& 10& .& 10& .& 5& .& 1\\ 1& .& 6& .& 15& .& 20& .& 15& .& 6& .& 1\\ 1& .& 7& .& 21& .& 35& .& 35& .& 21& .& 7& .& 1\\ 1& .& 3& .& 3& .& 1\\ 1& & 2& & 1\\ 1& & 1\end{array}$

$\begin{array}{cc}\begin{array}{r}\begin{array}{c}\begin{array}{lllllllllllllllllll}{x}^{4}& +& p{x}^{3}& +& qxx& +& rx& +& s& \left(& {x}^{4}& +& p{x}^{3}& +& q{x}^{2}& +& rx& +& s\\ & & & & & & & & & & & +& a& +& ap& +& aq& +& ar\\ & & & & & & & & & & & & & +& aa& +& aap& +& aaq\\ & & & & & & & & & & & & & & & -& {a}^{3}& -& {a}^{3}p\\ & & & & & & & & & & & & & & & & & +& {a}^{4}\end{array}|\end{array}\\ \begin{array}{ccc}& & \\ x& -& a\end{array}\end{array}& x\end{array}$

$pp+2ap+aa$

$16+8a+aa.6+4a+aa$

$\begin{array}{ccccc}\begin{array}{rrrrrrrrrrrrrrr}1& .& 7& .& 21& .& 35& .& 35& .& 21& .& 7& .& 1\\ & & -5& & -30& & -75& & -100& & -75& & -30& .& -5\\ & & & & +9& & +45& & +90& & +90& & +45& & +5\\ & & & & & & -7& & -28& & -42& & -28& & -7\\ & & & & & & & & +3& & +9& & +9& & +3\\ & & & & & & & & & & +2& & +4& & +2\\ & & & & & & & & & & & & -2& & -2\\ & & & & & & & & & & & & & & -3\\ & & & & & & & & & & -13& & & & \\ & & & & & & -4& & -14& & -25& & -21& & -10\end{array}& \phantom{\text{_____}}& \begin{array}{rrrrrrrrrrrrrrr}1& & 7& & 21& & 35& & 35& & 21& & 7& & 1\\ .& & 2& & 12& & 30& & 40& & 30& & 12& & 2\\ & & & & & & -2& & -8& & -12& & -8& & -2\\ & & & & & & & & -6& & -18& & -18& & -6\\ 1& & +9& & +33& & +63& & +61& & +21& & -7& & -5\\ 1& .& 7& .& 21& .& 35& .& 35& .& 21& .& 7& .& 1\\ & & -10& & -60& & -150& & -200& & -150& .& -60& .& -1\\ & & & & +9& & +45& & +90& & +90& & +45& & +9\\ & & & & & & -7& & -28& & -42& & -28& & -7\\ & & & & & & & & +6& & & & & & \\ \phantom{0}& & & & & & & & & & & & & & \\ 1& & -3& & -30& & -52& & & & & & & & \\ & & & & -27& & & & & & & & & & \end{array}& \phantom{\text{_____}}& \begin{array}{l}\begin{array}{rrrrrrrrrrrrrrr}1& .& 7& .& 21& .& 35& .& 35& .& 21& .& 7& .& 1\\ & & -3& .& -18& .& -45& .& -60& .& -45& .& -18& .& -1\\ & & & & -\phantom{00}& & & & & & & & & & \end{array}\\ \phantom{0}\\ \begin{array}{rrrrrrrrrrr}1& .& -7& .& +10& .& +9& .& -10& .& -9\\ 1& .& 5& .& 10& .& 10& .& 5& .& 1\\ & & -7& .& -28& .& -42& .& -28& .& -7\\ & & & & 10& .& 30& .& 30& .& 10\\ & & & & & & +9& .& 18& .& 9\\ & & & & & & & & -14& & -14\\ 1& .& -2& .& -8& .& +7& & +1& & -10\end{array}\end{array}\end{array}$

${x}^{4}-6{x}^{3}+10xx-7x+1$

$\begin{array}{ccccc}\begin{array}{rrrrrrrrr}1& & 4& & 6& & 4& & 1\\ & & -6& & -18& & -18& & -6\\ & & & & +10& & +20& & +10\\ & & & & & & -7& & -7\\ 1& & -2& & +2& & -1& & -1\end{array}& \phantom{\text{____}}& \begin{array}{rrrrrrrrr}1& & 4& & 6& & 4& & 1\\ & & -2& & -6& & -6& & -2\\ & & & & 2& & 4& & 2\\ & & & & & & -1& & -2\\ 1& .& 2& .& 2& .& 1& .& -1\end{array}& \phantom{\text{____}}& \begin{array}{rrrrrrrrrrr}1& .& 5& .& 10& .& 10& .& 5& & 1\\ & & -2& & -8& & -12& & -8& & -2\\ & & & & -8& & -24& & -24& & -8\\ & & & & & & +7& & +14& & +7\\ & & & & & & & & +11& & +11\\ & & & & & & & & & & -10\\ 1& .& 3& .& -6& .& -17& & -2& & -1\end{array}\end{array}$

$\begin{array}{ccccccccccc}{x}^{5}& +& 3{x}^{4}& +& 3{x}^{3}& +& {x}^{2}& +& 3bx& +& b\\ & & & +& b& +& 3b& & \end{array}$

$\begin{array}{ccc}\begin{array}{l}\frac{b+3}{9}.\frac{3+9b}{9+6b+9bb}.\frac{9b+3bb}{1+bb+9bb}.\frac{1+3b}{9b}.\\ \frac{1}{3}+\frac{b}{9}.\frac{1}{3}+\frac{7}{9}b-\frac{1}{45}bb\end{array}& \phantom{000}& \begin{array}{lllllll}\genfrac{}{}{0}{}{1}{\phantom{9}},\frac{4}{9}& .& \frac{3,4}{4,4}& .& \frac{3,3}{4,4}& .& \frac{4,1}{9}\\ \genfrac{}{}{0}{}{1}{\phantom{6}},\frac{7}{16}& .& \frac{4,8}{7,7}& .& \frac{7,6}{8,8}& .& \frac{8,7}{7,7}& .& \frac{,7,1}{4,4}\\ \frac{1,11}{25}& .& \frac{5,15}{11,11}& .& \frac{11,15}{15,15}& .& \frac{15,11}{15,15}& .& \frac{15,5}{11}& .& \frac{11,1}{5,5}\end{array}\end{array}$

< text from f 2:2v resumes >