Letter from Arthur Storer to Isaac Newton, dated 4 September 1678

<2:1r>

$\begin{array}{ccccc}\begin{array}{cc}& \begin{array}{c}\begin{array}{c}\phantom{0}\\ \phantom{0}\\ \phantom{0}\end{array}\\ \begin{array}{c}\text{Suns}\\ \\ \text{Hourly}\\ \\ \text{Azimuth}\\ \\ \text{Lat.}\\ \\ 52=00\\ \end{array}\\ \end{array}\end{array}& & \begin{array}{c}\begin{array}{c}\text{A Table of the Suns Azimuth for Every}\text{Hour of the Day Or Night Every}{\text{4}}^{\text{th}}\text{Degree of the Declination Eather North or South}\end{array}\\ \begin{array}{c}\begin{array}{ccccccccccccccccccccccccccccccc}\begin{array}{c}\phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}\end{array}& & & \begin{array}{c}\phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}\end{array}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \begin{array}{c}\phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}\end{array}& & & \begin{array}{c}\phantom{\text{\_\_\_\_\_\_\_\_\_\_\_\_\_\_}}\end{array}\\ & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & \\ \begin{array}{c}\text{North}\\ \text{Declin}\stackrel{\text{ t}}{\text{a}}\end{array}& & & \hfill \begin{array}{c}12\\ \\ 12\end{array}\hfill & & \hfill \begin{array}{c}1\\ \\ 11\end{array}\hfill & & \hfill \begin{array}{c}2\\ \\ 10\end{array}\hfill & & \hfill \begin{array}{c}3\\ \\ 9\end{array}\hfill & & \hfill \begin{array}{c}4\\ \\ 8\end{array}\hfill & & \hfill \begin{array}{c}5\\ \\ 7\end{array}\hfill & & \hfill \begin{array}{c}6\\ \\ 6\end{array}\hfill & & \hfill \begin{array}{c}7\\ \\ 5\end{array}\hfill & & \hfill \begin{array}{c}8\\ \\ 4\end{array}\hfill & & \hfill \begin{array}{c}9\\ \\ 3\end{array}\hfill & & \hfill \begin{array}{c}10\\ \\ 2\end{array}\hfill & & \hfill \begin{array}{c}11\\ \\ 1\end{array}\hfill & & \hfill \begin{array}{c}12\\ \\ 12\end{array}\hfill & & & \begin{array}{c}\text{South}\\ \text{Declination}\end{array}\\ & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & \\ & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & \\ \hfill \begin{array}{c}0\\ \\ 4\end{array}\hfill & & & \begin{array}{l}90=00\\ \\ 90=00\end{array}\hfill & & \begin{array}{l}71=13:\\ \\ 70=11\end{array}\hfill & & \begin{array}{l}53=46\\ \\ 51=58\end{array}\hfill & & \begin{array}{l}38=14\\ \\ 36=01\end{array}\hfill & & \begin{array}{l}24=28\\ \\ 22=03:\end{array}\hfill & & \begin{array}{l}11=55:\\ \\ 09=27·\end{array}\hfill & & \begin{array}{l}00=00\\ \\ \phantom{0}2=28\end{array}\hfill & & \begin{array}{l}11=55:\\ \\ 14=21\end{array}\hfill & & \begin{array}{l}24=28\\ \\ 26=47·\end{array}\hfill & & \begin{array}{l}38=14\\ \\ 40=20\end{array}\hfill & & \begin{array}{l}53=46\\ \\ 55=25·\end{array}\hfill & & \begin{array}{l}71=13:\\ \\ 72=10\end{array}\hfill & & \begin{array}{l}90=00\\ \\ 90=00\end{array}\hfill & & & \hfill \begin{array}{c}0\\ \\ 4\end{array}\hfill \\ \hfill \begin{array}{c}8\\ \\ 12\end{array}\hfill & & & \begin{array}{l}90=00\\ \\ 90=00\end{array}\hfill & & \begin{array}{l}69=01\\ \\ 67=41\end{array}\hfill & & \begin{array}{l}50=00\\ \\ 47=48:\end{array}\hfill & & \begin{array}{l}33=39\\ \\ 31=05\end{array}\hfill & & \begin{array}{l}19=33\\ \\ 16=54\end{array}\hfill & & \begin{array}{l}06=\stackrel{\phantom{·}}{56}\\ \\ 04=\underset{·}{20}\end{array}\hfill & & \begin{array}{l}\phantom{0}4=57\\ \\ \phantom{0}7=28\end{array}\hfill & & \begin{array}{l}16=\stackrel{\phantom{.·}}{44}·\\ \\ 19=\underset{.·}{08}\end{array}\hfill & & \begin{array}{l}29=01:\\ \\ 31=13\end{array}\hfill & & \begin{array}{l}42=19\\ \\ 44=13\end{array}\hfill & & \begin{array}{l}56=58\\ \\ 58=25\end{array}\hfill & & \begin{array}{l}73=\stackrel{\phantom{.·}}{01}\\ \\ 73=\underset{.·}{50}\end{array}\hfill & & \begin{array}{l}90=00\\ \\ 90=00\end{array}\hfill & & & \hfill \begin{array}{c}8\\ \\ 12\end{array}\hfill \\ \hfill \begin{array}{c}16\\ \\ 20\end{array}\hfill & & & \begin{array}{l}90=00\\ \\ 90=00\end{array}\hfill & & \begin{array}{l}66=07\\ \\ 64=16\end{array}\hfill & & \begin{array}{l}45=20\\ \\ 42=30\end{array}\hfill & & \begin{array}{l}28=18\\ \\ 25=14\end{array}\hfill & & \begin{array}{l}14=06\\ \\ 11=06\end{array}\hfill & & \begin{array}{l}1:\text{S}=38\\ \\ 1:\text{N}=12\end{array}\hfill & & \begin{array}{l}10=01\\ \\ 12=38\end{array}\hfill & & \begin{array}{l}21=30\\ \\ 23=54:\end{array}\hfill & & \begin{array}{l}33=22\\ \\ 35=\underset{·}{31}\end{array}\hfill & & \begin{array}{l}46=03:\\ \\ 47=51\end{array}\hfill & & \begin{array}{l}59=48\\ \\ 61=07\end{array}\hfill & & \begin{array}{l}74=34\\ \\ 75=17\end{array}\hfill & & \begin{array}{l}90=00\\ \\ 90=00\end{array}\hfill & & & \hfill \begin{array}{c}16\\ \\ 20\end{array}\hfill \\ \hfill 24\hfill & & & 90=00\hfill & & 62=01\hfill & & 39=14\hfill & & 21=49\hfill & & \phantom{0}7=53\hfill & & 4:\text{N}=09\hfill & & 15=20\hfill & & 26=20\hfill & & 37=\underset{·}{39}\hfill & & 49=\underset{·}{37}\hfill & & 62=24\hfill & & 75=58\hfill & & 90=00\hfill & & & \hfill 24\hfill \\ & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & \\ & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & \end{array}\end{array}\\ \\ \phantom{0}\\ \begin{array}{c}\text{A Table of the North Stars}\text{Hourly Azimuth}\text{From the North for Every Hour for}{\text{y}}^{\text{e}}\text{Latitude following}\\ \text{The Declination being}\stackrel{degs}{87}=\stackrel{min}{36}:\text{Being supposed for the yeare —}\underset{\_}{\text{1680}}\end{array}\\ \begin{array}{c}\begin{array}{ccccccccccccccccccccccccccccccc}& & & & & & & & & & & & & & & & & & & & & & & & & & & & & & \\ \begin{array}{c}\phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}\\ \text{Latitude}\end{array}& & & \begin{array}{c}\phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}\\ \text{North}\\ \text{Above}\\ \text{the pole}\end{array}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \begin{array}{c}\phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}\\ \text{North}\\ \text{under}\\ \text{the pole}\end{array}& & & \begin{array}{c}\phantom{\text{\_\_\_\_\_\_\_\_\_\_\_\_\_\_}}\\ \text{Latitude}\\ \text{of}{\text{y}}^{\text{e}}\text{place}\end{array}\\ 60& & & \hfill 00=00\hfill & & \phantom{0}1=20·\hfill & & \phantom{0}2=\underset{·}{34}\hfill & & \phantom{0}3=34:\hfill & & \phantom{0}4=18:\hfill & & \phantom{0}4=43:\hfill & & \phantom{0}4=\underset{·}{48}\hfill & & \phantom{0}4=33:\hfill & & \phantom{0}4=00:\hfill & & \phantom{0}3=14\hfill & & \phantom{0}2=15:\hfill & & \phantom{0}1=\underset{·}{10}\hfill & & 00=00& & & 60\\ 53& & & \hfill 00=00\hfill & & \phantom{0}1=05\hfill & & \phantom{0}2=06\hfill & & \phantom{0}2=56·\hfill & & \phantom{0}3=33\hfill & & \phantom{0}3=54\hfill & & \phantom{0}3=59\hfill & & \phantom{0}3=48\hfill & & \phantom{0}3=21\hfill & & \phantom{0}2=43\hfill & & \phantom{0}1=54\hfill & & \phantom{0}0=\underset{·}{59}\hfill & & 00=00& & & 53\\ 50& & & \hfill 00=00\hfill & & \phantom{0}1=01:\hfill & & \phantom{0}1=57:\hfill & & \phantom{0}2=44\hfill & & \phantom{0}3=19\hfill & & \phantom{0}3=39\hfill & & \phantom{0}3=\underset{·}{44}\hfill & & \phantom{0}3=33:\hfill & & \phantom{0}3=09:\hfill & & \phantom{0}2=33\hfill & & \phantom{0}1=\underset{·}{48}\hfill & & \phantom{0}0=55·\hfill & & 00=00& & & 50\\ 40& & & \hfill 00=00\hfill & & \phantom{0}0=\underset{·}{51}\hfill & & \phantom{0}1=37:\hfill & & \phantom{0}2=16\hfill & & \phantom{0}2=46\hfill & & \phantom{0}3=03\hfill & & \phantom{0}3=08:\hfill & & \phantom{0}3=00\hfill & & \phantom{0}2=40:\hfill & & \phantom{0}2=10\hfill & & \phantom{0}1=31:\hfill & & \phantom{0}0=47:\hfill & & 00=00& & & 40\\ 30& & & \hfill 00=00\hfill & & \phantom{0}0=44·\hfill & & \phantom{0}1=\underset{·}{25}\hfill & & \phantom{0}2=\underset{·}{00}\hfill & & \phantom{0}2=\underset{·}{26}\hfill & & \phantom{0}2=\underset{·}{42}\hfill & & \phantom{0}2=46:\hfill & & \phantom{0}2=\underset{·}{40}\hfill & & \phantom{0}2=22·\hfill & & \phantom{0}1=\underset{·}{56}\hfill & & \phantom{0}1=21:\hfill & & \phantom{0}0=42·\hfill & & 00=00& & & 30\\ 20& & & \hfill 00=00\hfill & & \phantom{0}0=40:\hfill & & \phantom{0}1=\underset{·}{18}\hfill & & \phantom{0}1=\underset{·}{50}\hfill & & \phantom{0}2=\underset{·}{14}\hfill & & \phantom{0}2=\underset{.·}{29}\hfill & & \phantom{0}2=33\hfill & & \phantom{0}2=27:\hfill & & \phantom{0}2=\underset{·}{12}\hfill & & \phantom{0}1=47\hfill & & \phantom{0}1=\underset{·}{16}\hfill & & \phantom{0}0=39·\hfill & & 00=00& & & 20\\ 10& & & \hfill 00=00\hfill & & \phantom{0}0=38\hfill & & \phantom{0}1=14\hfill & & \phantom{0}2=44\hfill & & \phantom{0}2=07\hfill & & \phantom{0}2=21\hfill & & \phantom{0}2=26\hfill & & \phantom{0}2=\underset{·}{21}\hfill & & \phantom{0}2=06\hfill & & \phantom{0}1=43\hfill & & \phantom{0}1=13\hfill & & \phantom{0}0=38\hfill & & 00=00& & & 10\\ 0& & & \hfill 00=00\hfill & & \phantom{0}0=37·\hfill & & \phantom{0}1=12\hfill & & \phantom{0}1=\underset{·}{42}\hfill & & \phantom{0}2=05·\hfill & & \phantom{0}2=19·\hfill & & \phantom{0}2=24\hfill & & \phantom{0}2=19·\hfill & & \phantom{0}2=05·\hfill & & \phantom{0}1=42\hfill & & \phantom{0}1=12\hfill & & \phantom{0}0=37:\hfill & & 00=00& & & 00\\ \begin{array}{c}\phantom{0}\\ \phantom{0}\\ \phantom{0}\\ \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}\end{array}& & & \begin{array}{c}\phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}\end{array}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}& & \begin{array}{c}\phantom{\text{\_\_\_\_\_\_\_\_\_\_\_}}\end{array}& & & \begin{array}{c}\phantom{\text{\_\_\_\_\_\_\_\_\_\_\_\_\_\_}}\end{array}\end{array}\end{array}\end{array}& & \begin{array}{c}\begin{array}{c}\begin{array}{c}\phantom{0}\\ \phantom{0}\\ \phantom{0}\end{array}\\ \begin{array}{c}\text{Suns}\\ \\ \text{Hourly}\\ \\ \text{Azimuth}\\ \\ \text{Latitude}\\ \\ \stackrel{degs}{52}=\stackrel{min}{00}\end{array}\\ \\ \begin{array}{c}\phantom{0}\\ \phantom{0}\\ \phantom{0}\\ \phantom{0}\\ \phantom{0}\\ \phantom{0}\\ \phantom{0}\\ \phantom{0}\\ \phantom{0}\\ \phantom{0}\\ \phantom{0}\\ \phantom{0}\\ \phantom{0}\\ \phantom{0}\\ \phantom{0}\end{array}\\ \begin{array}{c}\text{The}\\ \\ \text{Latitudes:}\\ \\ \text{The}\\ \\ \text{Declination}\\ \\ {87}^{-}=36\prime \\ \\ \text{North}\end{array}\end{array}\end{array}\end{array}$

M^r Newton Our former aquentance Imbold\e/neth me to troble you w^th these few lines: w^ch is to Intreat your {fawor}{faivor} If y^t an oppertunity will serve: To peruse over this Small Table of the Suns Azimuth for the Latitude of ${52}^{\mathrm{degrs}}{00}^{\min }$ w^ch is for Every Hour of the day & night: \being/ from the Equinoctiall to 24 degrees of declination being Eather North or South for Every 4^th 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 w^ch is for Every 10^th degree of Latitude from the Equinoctiall to ${60}^{\mathrm{degs}}$ 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 w^ch 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} w^th/ allthe Altitude {illeg} {o}nley w^th the Declination: so y^t the Azimuth for Every Hour is found as soone o{illeg}|r| sooner or sooner then the Hourly Altitu w^ch is by such a duble way of workeing for y^e Hour of l0 & 2 for North Declination & 10 & 2 for South Declination & the Hour of both 6^es is wrought by the product of one Multiplication w^ch is to be Added to or Subtracted from a Nother Summ & then takeing $\frac{1}{2}$ thereof sheweth 6 Hours Azimuth: So y^t the whole 24 hours Azimuth is wrought p the produce of five multiplications by being Added \or/ substracted fro\m/ An orther given summ w^ch is Allwayes a certane Summ for y^t 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 y^e North Stars Azimuth I could have sett downe not onley the minute But likewise y^t to one thousand part of a minute & y^t by y^e same Radius by w^ch this was Calcula\ted/{.} But this I Suppose is neare enough if y^t y^e proportions hold good w^ch I would desire you to Examine & give me notice thereof w^ch if it be within 14 dayes Af{illeg}|t|er the {dai{illeg}|t|} of this I may Receive it If y^t you d{illeg}|i|rect your Letter to M^r William Kings Junior Silkeweaver In three horshoo alley in Old Streete in London I shall Receve it: Haveing no moore at p^rsent 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 4^th: 1678

I sent you a former table by {unc{illeg}|k|le} Babington of y^e North Stars \Hourly/ Altitude & Azimuth
But in y^t {y^of} table of the Suns Azimuth was Omitted w^ch I hope you have Receved before this{.}

<2:2v>

For his much Esteemed Frend
M^r Isaac Newton of
Trenity Colidg in
Cambridg:
{p^rs}

< insertion from higher up f 2r >

$\begin{array}{lllllll}{\mathrm{x}}^3& +& 2\mathrm{a}{\mathrm{x}}^2& +& \mathrm{a}\mathrm{a}\mathrm{x}& -& \mathrm{a}\mathrm{b}\mathrm{b}\\ & -& \mathrm{b}& -& 2\mathrm{a}\mathrm{b}\end{array}$    $\begin{array}{lllllll}{\mathrm{x}}^3& -& \mathrm{a}{\mathrm{x}}^2& +& \mathrm{a}\mathrm{a}\mathrm{b}\mathrm{x}& +& \mathrm{a}\mathrm{b}\mathrm{c}\\ & -& \mathrm{b}& -& \mathrm{a}\mathrm{c}\\ & +& \mathrm{c}& -& \mathrm{b}\mathrm{c}\end{array}$

$\begin{array}{llllll}-& \mathrm{a}\mathrm{a}\mathrm{b}& +& \mathrm{a}\mathrm{a}\mathrm{c}& +& 3\mathrm{a}\mathrm{b}\mathrm{c}\\ -& \mathrm{b}\mathrm{b}\mathrm{a}& +& \mathrm{b}\mathrm{b}\mathrm{c}\\ -& \mathrm{c}\mathrm{c}\mathrm{a}& -& \mathrm{c}\mathrm{c}\mathrm{b}\end{array}$    {illeg}$.\mathrm{a}\mathrm{b}\mathrm{c}.$    3 {illeg}   $\frac{-\mathrm{a}-\mathrm{b}}{\mathrm{c}}+\frac{\mathrm{a}-\mathrm{c}}{\mathrm{b}}+\frac{\mathrm{b}-\mathrm{c}}{\mathrm{a}}$.   1    $\begin{array}{cc}\begin{array}{c}\left|\phantom{000}\begin{array}{ccccccc}& \mathrm{a}\mathrm{a}& +& 2\mathrm{a}\mathrm{b}& \phantom{000}& & \mathrm{a}\mathrm{b}\\ +& \mathrm{b}\mathrm{b}& -& 2\mathrm{a}\mathrm{c}& .& -& \mathrm{a}\mathrm{c}& .\\ +& \mathrm{c}\mathrm{c}& -& 2\mathrm{b}\mathrm{c}& & -& \mathrm{b}\mathrm{c}& \end{array}\right.\\ \\ \\ \begin{array}{lllllll}& \mathrm{a}\mathrm{a}\mathrm{b}\mathrm{b}& -& 2\mathrm{a}\mathrm{a}\mathrm{b}\mathrm{c}& \phantom{000}& -& \mathrm{a}\mathrm{a}\mathrm{b}\mathrm{c}\\ +& \mathrm{a}\mathrm{a}\mathrm{c}\mathrm{c}& -& 2\mathrm{a}\mathrm{b}\mathrm{b}\mathrm{c}& \phantom{0}.& -& \mathrm{a}\mathrm{b}\mathrm{b}\mathrm{c}& \\ +& \mathrm{b}\mathrm{b}\mathrm{c}\mathrm{c}& -& 2\mathrm{a}\mathrm{b}\mathrm{c}\mathrm{c}& \phantom{000}& +& \mathrm{a}\mathrm{b}\mathrm{c}\mathrm{c}& \end{array}\\ \end{array}& \begin{array}{c}\begin{array}{ccc}\begin{array}{cc}& 4\mathrm{a}\mathrm{a}\\ -& 4\mathrm{a}\mathrm{c}& .\\ +& \mathrm{c}\mathrm{c}\\ & \end{array}& & \begin{array}{cc}& \mathrm{a}\mathrm{a}\phantom{4}\\ -& 2\mathrm{a}\mathrm{c}\end{array}\end{array}\end{array}\end{array}$

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

$\begin{array}{lllllllllll}6& -& 24\mathrm{c}& +& 3\mathrm{b}\mathrm{b}& +& 16\mathrm{a}\mathrm{a}& -& 24\mathrm{a}\mathrm{b}& +& 6\mathrm{b}\mathrm{b}\\ & -& 4& +& 1& & & -& 4\end{array}$

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

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

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

$\begin{array}{cc}\begin{array}{r}\begin{array}{c}\left.\begin{array}{lllllllllllllll}& & & & & & & & & & & & & +& 3\\ {\mathrm{x}}^4& +& 3{\mathrm{x}}^3& +& 3\mathrm{x}\mathrm{x}& +& \mathrm{x}& +& \mathrm{a}& (& {\mathrm{x}}^4& +& 3{\mathrm{x}}^3& +& 3\mathrm{a}{\mathrm{x}}^2& +& 1\mathrm{x}& +& \mathrm{a}\\ & +& \mathrm{a}& +& 3\mathrm{a}& +& 3\mathrm{a}& & & & & +& \mathrm{a}& +& \mathrm{b}\mathrm{b}& +& 3\mathrm{a}& -& \mathrm{b}\\ & & & & & & & & & & & -& \mathrm{b}& -& \mathrm{a}\mathrm{b}& -& 3\mathrm{b}& -& 3\mathrm{a}\mathrm{b}\\ & & & & & & & & & & & & & -& 3\mathrm{b}& -& 3\mathrm{a}\mathrm{b}& +& 3\mathrm{b}\mathrm{b}\\ & & & & & & & & & & & & & & & +& \mathrm{a}\mathrm{b}\mathrm{b}& +& 3\mathrm{a}\mathrm{b}\mathrm{b}\\ & & & & & & & & & & & & & & & +& 3\mathrm{b}\mathrm{b}& -& \mathrm{a}{\mathrm{b}}^3\\ & & & & & & & & & & & & & & & -& {\mathrm{b}}^3& -& 3{\mathrm{b}}^3\\ & & & & & & & & & & & & & & & & & +& {\mathrm{b}}^4\end{array}\right|\end{array}\\ \begin{array}{ccc}& & \\ \mathrm{x}& +& \mathrm{b}\end{array}\end{array}& \mathrm{x}\end{array}$

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

$\begin{array}{cc}\begin{array}{r}\begin{array}{c}\left.\begin{array}{lllllllllllllllllll}{\mathrm{x}}^4& +& \mathrm{p}{\mathrm{x}}^3& +& \mathrm{q}\mathrm{x}\mathrm{x}& +& \mathrm{r}\mathrm{x}& +& \mathrm{s}& (& {\mathrm{x}}^4& +& \mathrm{p}{\mathrm{x}}^3& +& \mathrm{q}{\mathrm{x}}^2& +& \mathrm{r}\mathrm{x}& +& \mathrm{s}\\ & & & & & & & & & & & +& \mathrm{a}& +& \mathrm{a}\mathrm{p}& +& \mathrm{a}\mathrm{q}& +& \mathrm{a}\mathrm{r}\\ & & & & & & & & & & & & & +& \mathrm{a}\mathrm{a}& +& \mathrm{a}\mathrm{a}\mathrm{p}& +& \mathrm{a}\mathrm{a}\mathrm{q}\\ & & & & & & & & & & & & & & & -& {\mathrm{a}}^3& -& {\mathrm{a}}^3\mathrm{p}\\ & & & & & & & & & & & & & & & & & +& {\mathrm{a}}^4\end{array}\right|\end{array}\\ \begin{array}{ccc}& & \\ \mathrm{x}& -& \mathrm{a}\end{array}\end{array}& \mathrm{x}\end{array}$

$\mathrm{p}\mathrm{p}+2\mathrm{a}\mathrm{p}+\mathrm{a}\mathrm{a}$

$16+8\mathrm{a}+\mathrm{a}\mathrm{a}.6+4\mathrm{a}+\mathrm{a}\mathrm{a}$

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

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

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

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

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

< text from f 2:2v resumes >