<2:1r>

Mr Newton Our former aquentance Imboldeneth me to troble you with these few lines: which is to Intreat your {fawor}{faivor} If that an oppertunity will serve: To peruse over this Small Table of the Suns Azimuth for the Latitude of which 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 which 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 which maketh me more desirus to have your oppinion thereof: As for this way of callculation I do not at all conserne my selfe with the Altitude {illeg} {o}nley with the Declination: so that the Azimuth for Every Hour is found as soone or sooner then the Hourly Altitude which is by such a duble way of workeing for the Hour of l0 & 2 for North Declination & 10 & 2 for South Declination & the Hour of both sixes is wrought by the product of one Multiplication which is to be Added to or Subtracted from a Nother Summ & then takeing $\frac{1}{2}$ thereof sheweth 6 Hours Azimuth: So that the whole 24 hours Azimuth is wrought per the produce of five multiplications by being Added or substracted from An orther given summ which is Allwayes a certane Summ for that hour & Latitude to any declination{.} I have heare in some places not onely sett downe the degrees & minutes but have shewed by pricks to a quarter of a minute in some places{.} But if I would have bene presise in this Calculation per this way of callculating of the North Stars Azimuth I could have sett downe not onley the minute But likewise that to one thousand part of a minute & that by the same Radius by which this was Calculated{.} But this I Suppose is neare enough if that the proportions hold good which I would desire you to Examine & give me notice thereof which if it be within 14 dayes After the {dait} of this I may Receive it If that you direct 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 present But my dutye to my {unckle} & 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 {unckle} Babington of the North Stars Hourly Altitude & Azimuth
But in that {thereof} table of the Suns Azimuth was Omitted which 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 >

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

$\begin{array}{llllll}-& aab& +& aac& +& 3abc\\ -& bba& +& bbc\\ -& cca& -& ccb\end{array}$    $.abc.$    3    $\frac{-a-b}{c}+\frac{a-c}{b}+\frac{b-c}{a}$.   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 >