Letter from Arthur Storer to Humphrey Babington, dated 19 September 1678

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S^ir

According to my desire I have Receved from M^r Newton that kindeness of' your promiss: which weare Conserning that Small Table of' mine w^ch was of' the North Star's Hourly Altitude & Azimuth The Contents of' w^ch Answer is in theese Lines: Cambridge Sept 11^th: 1678 M^r Storer I have Looked over your tables of' the North Star's hourly Altitudes & Azimuth: & doe not perceive but that they ar Suff'iciently {sic} Exact{.}

$\begin{array}{lllllll}\hfill \left.\text{Only in the Table of' Azimuths at}{\text{y}}^{\text{e}}\text{Lat'.}\stackrel{\mathrm{degs}}{0}=\stackrel{\min }{00}\text{at}{\text{y}}^{\text{e}}\text{Hours}\right\{& \begin{array}{ccc}& & \\ \phantom{:}& 8& :\\ & 4& :\\ & & \end{array}& \begin{array}{ccc}& & \\ \phantom{:}& 7& :\\ & 5& :\\ & & \end{array}& \begin{array}{ccc}& & \\ & \phantom{:}6& :\\ & \phantom{:}6& :\\ & & \end{array}& \begin{array}{ccc}& & \\ \phantom{:}& 5& :\\ & 7& :\\ & & \end{array}& \begin{array}{ccc}& & \\ \phantom{:}& 8& \phantom{:}\\ & 4& \phantom{:}\\ & & \end{array}& \\ \hfill \text{I put downe for these hours}& \stackrel{ d : min}{1=12}& 0=37& 00=00& 0=37& 1=12& \begin{array}{c}{\text{w}}^{\text{ch}}\text{is}{\text{y}}^{\text{e}}\text{Norths Star's Altitude in}{\text{y}}^{\text{t}}\text{Latitude}\\ \text{for}{\text{w}}^{\text{ch}}\text{hours}\text{ar put}\text{instead of}{\text{y}}^{\text{e}}\text{Azimuth}\hfill \end{array}\\ \hfill {\text{w}}^{\text{ch}}\text{weare}\stackrel{\_}{\text{p}}\text{my Table as}{\text{M}}^{\text{r}}\text{Newton sett down thus}& 2=\underset{·}{05}·& 2=19·& \phantom{0}2=24& 2=19·& 2=\underset{·}{05}·& \end{array}$

But if y^t I should presisely have sett downe the the minnute & fractionall part of' a minute as my Calculatio Gave it & by the very same Radius by w^ch that table of' the North Star's Azimuth was Calculated then it will shew it as followeth for those hours $—\stackrel{\mathrm{degs}}{2}—\stackrel{\min }{04}\phantom{..}\text{\&}..\frac{1614}{2216}$ parts of a minute w^ch is for y^e Hours of: $\left.\begin{array}{r}8\text{ \& }4\\ \text{\& – }4\text{ \& }8\end{array}\right\}$ & for y^eHours: $7\text{ \& }5\text{ \& }5\text{ \& }7$: $—2—19\phantom{.}\text{ \& }.\frac{170}{1770}:$ parts of \a min/ But for the Hours of Both Sixes: $\stackrel{\mathrm{degs:}}{2}=\stackrel{\min }{24}=\stackrel{\mathrm{″}}{00}$ w^ch is the Stars distance from the pole: But if I should have beene so presise w^th the degrees minutes & fractionall part of a Minute & so neare as was shewed by the calculation w^ch would have beene To the Thousand part of' a Minute or less: therfore I suppose it would have beene very Hard Tedious & Troblesome to have whroug\ht/ the same so neare by the Altitude: as I {illeg}|di|d there without it & by such a duble way of' working that no Less then 4 Hours Azim\{illeg}|u|th/ was shewed by the produce of' one simple multiplication casting away some five figures then Adding the Remainer To or Substrac\ting/ from a Nother constant given Summ: which is allwayes the same for y^e Lat' & Hour given But for y^e said tables he wrighteth that he doth not perceive but that they ar Sufficently Exact: Onley whether the Rule which I have computed the tables therby be Mathematically exact so y^t with {sic} without Error danger of' error it may be used to Compute the Altitudes & Azimuths of' Stars which ar very Remoate from the pole I \he/ cannot tell that he wrighteth is onley to be knowne from the demonstration of' the Rule grounded on Mathematicall principles for w^ch I think this same of' myne is & if he receve my Letter dated \about/ Sept y^e 4 or 5^th: he will see by it a table of' the Sunn or Stars Hourly Azimuth for Every 4^th degree of' the Sun{illeg}|s| \or Stars/ declination from y^e Equinoctiall to ${24}^{\mathrm{degrs}}$ Eather north or South: w^ch declination\s/ is Remoat from y^e pole w^ch I Calculated for y^e Latitud of' ${52}^{\mathrm{degs}}={00}^{\min }$ w^ch is shewed|th| very neare your Lat' at Cambridge Because he might the sooner Looke it over without troble any great troble for so it may chance y^t he may have them Allreddye callculated for y^t or your Lat' by him: But because I have beene at y^t panes To make these tables universall being halfe Calculated for All Latitudes vizd: from y^e Equinoctiall To y^e Pole I shall Heare Lay downe the Grounds of' theare proportions of' the Sun\s/ or any Stars Hourly Azimuth

$\begin{array}{cccc}\begin{array}{c}\begin{array}{l}\begin{array}{ccc}\begin{array}{cc}& \phantom{\text{Latitude}}\end{array}& \begin{array}{c}\\ \phantom{0}& \\ & \phantom{000}\text{A Table shewing the Equinoctiall hours Tangent}\phantom{000}\\ & \text{of Azimuth from 50: to 55 degrees of Latitude}\end{array}& \begin{array}{l}\end{array}\end{array}\\ \begin{array}{ccc}\begin{array}{cc}& \begin{array}{c}{\text{y}}^{\text{e}}\\ \text{Latitude}\\ \text{of}\end{array}\\ & \\ & \underset{\phantom{.}}{50}\\ & \underset{\phantom{.}}{51}\\ & \underset{\phantom{.·}}{52}\\ & \underset{\phantom{.}}{53}\\ & \underset{\phantom{.}}{54}\\ & \underset{\phantom{.}}{55}\end{array}& & \begin{array}{l}\\ \begin{array}{c}11\\ 1\end{array}& \begin{array}{c}10\\ 2\end{array}& \begin{array}{c}9\\ 3\end{array}& \begin{array}{c}8\\ 4\end{array}& \begin{array}{c}7\\ 5\end{array}& \begin{array}{c}6\\ 6\end{array}& \\ & & & & & & \\ \phantom{0}28587\phantom{0}& \phantom{0}13267\phantom{0}& \phantom{0}7660:& \phantom{0}4422\phantom{0}& \phantom{0}2052\phantom{0}& \phantom{0}\underset{\phantom{.}}{0000}\phantom{0}& \\ \phantom{0}29001\phantom{0}& \phantom{0}13459\phantom{0}& \phantom{0}7771:& \phantom{0}4486\phantom{0}& \phantom{0}2082\phantom{0}& \phantom{0}\underset{\phantom{.}}{0000}\phantom{0}& \\ \phantom{0}29408\phantom{0}& \phantom{0}13648\phantom{0}& \phantom{0}7880\phantom{0}& \phantom{0}4549\phantom{0}& \phantom{0}2112\phantom{0}& \phantom{0}\underset{\phantom{.·}}{0000}\phantom{0}& \\ \phantom{0}29805\phantom{0}& \phantom{0}13832\phantom{0}& \phantom{0}7986·& \phantom{0}4610\phantom{0}& \phantom{0}2140\phantom{0}& \phantom{0}\underset{\phantom{.}}{0000}\phantom{0}& \\ \phantom{0}30192\phantom{0}& \phantom{0}14012\phantom{0}& \phantom{0}8090\phantom{0}& \phantom{0}4670\phantom{0}& \phantom{0}2167\phantom{0}& \phantom{0}\underset{\phantom{.}}{0000}\phantom{0}& \\ \phantom{0}30568\phantom{0}& \phantom{0}14187\phantom{0}& \phantom{0}8191:& \phantom{0}4729\phantom{0}& \phantom{0}2194\phantom{0}& \phantom{0}\underset{\phantom{.}}{0000}\phantom{0}& \end{array}\end{array}\\ & \\ & \end{array}\end{array}& \begin{array}{c}\text{The Radius}\\ \text{Being}\\ \text{10000}\\ \end{array}\hfill & \phantom{0000}& \begin{array}{cc}& \begin{array}{c}\\ \begin{array}{c}\begin{array}{cc}& \text{A Table shewing the Roote of' hourly Azimuth}\\ & \text{Being for the Tangent of}{45}^{\mathrm{degs}}\text{of Declination:}\end{array}\end{array}\\ \begin{array}{ccc}\begin{array}{c}\begin{array}{c}{\text{y}}^{\text{e}}\\ \text{Lat'}\\ \phantom{00}\text{–}\phantom{00}\end{array}\\ \\ \underset{\phantom{.}}{50}\\ \underset{\phantom{.}}{51}\\ \underset{\phantom{·}}{52}\\ \underset{\phantom{.}}{53}\\ \underset{\phantom{.}}{54}\\ \underset{\phantom{.}}{55}\end{array}& & & \begin{array}{l}\\ \begin{array}{c}11\\ 1\end{array}& \begin{array}{c}10\\ 2\end{array}& \begin{array}{c}9\\ 3\end{array}& \begin{array}{c}8\\ 4\end{array}& \begin{array}{c}7\\ 5\end{array}& \begin{array}{c}6\\ 6\end{array}& & \\ & & & & & & & \\ \phantom{0}24836\phantom{0}& \phantom{0}12854\phantom{0}& \phantom{0}9089& \phantom{0}7421:\phantom{0}& \phantom{0}6655:\phantom{0}& \phantom{0}\underset{·}{6428}\phantom{0}& & \\ \phantom{0}24315\phantom{0}& \phantom{0}12585\phantom{0}& \phantom{0}8899& \phantom{0}7266:\phantom{0}& \phantom{0}6516\phantom{0}& \phantom{0}\underset{\phantom{.}}{6293}·\phantom{0}& & \\ \phantom{0}23787\phantom{0}& \phantom{0}12313\phantom{0}& \phantom{0}8706\phantom{0}& \phantom{0}7108:\phantom{0}& \phantom{0}6375:\phantom{0}& \phantom{0}\underset{.·}{6157}\phantom{0}& & \\ \phantom{0}23252\phantom{0}& \phantom{0}12036\phantom{0}& \phantom{0}8511& \phantom{0}6948:\phantom{0}& \phantom{0}6231\phantom{0}& \phantom{0}\underset{\phantom{.}}{6018}\phantom{0}& & \\ \phantom{0}22710:\phantom{0}& \phantom{0}11755\phantom{0}& \phantom{0}8313\phantom{0}& \phantom{0}\underset{·}{6787}\phantom{0}& \phantom{0}6085\phantom{0}& \phantom{0}\underset{·}{5878}\phantom{0}& & \\ \phantom{0}22162\phantom{0}& \phantom{0}11471\phantom{0}& \phantom{0}8112& \phantom{0}6623\phantom{0}& \phantom{0}5937\phantom{0}& \phantom{0}\underset{·}{5736}\phantom{0}& & \end{array}\end{array}\\ \\ \end{array}\end{array}\end{array}$

The use of' these Tables is thus for y^e Sun or any Stars Hourly Azimuth for these Latitudes the Declination being given multiply the sum of' the Root of y^e Hourly Azimuth, for y^e Hour & Lat' suppoased: by the Tangent of the Sun or Stars declination: Cutt of foure figuers To y^e Right hand from the product: The Remaner being added to or Substracted from the Equinoctiall hours Tangent of Azimuth: sheweth the Tangent of the hours Azimuth being counted from y^e East or West. Example Lat' 52: the Declination ${20}^{\mathrm{degs}}$ North I demand the Azimuth at :10 in y^e morning & :2: in the Affter noon: & at :6: in y^e mo^r & 6 past {noo\ne/} at 2. in the morning & 10 at night at theese six hours I demand the Sun or Stars Azimuth: Aganst y^e Lat' 52 under y^e hour $\begin{array}{c}10\\ 2\end{array}$ $\begin{array}{rrr}\text{I finde the sum of}{\text{y}}^{\text{e}}\text{Roote of}{\text{y}}^{\text{t}}\text{hour to be —}& 12313& \text{: then the declination being}{20}^{\mathrm{degs}}=00\mathrm{\prime }\text{the Tangent thereof is —}3640\text{:}\\ \text{by}{\text{w}}^{\text{ch}}\text{Tangent I multiply}{\text{y}}^{\text{e}}\text{Roote by thus —}& \underset{\_}{\phantom{0}3640}\\ & 492520\\ & 73878\phantom{00}\\ & \underset{\_}{\phantom{0}36939\phantom{000}}\\ \text{Rest.}& \mathrm{4481/9320}\end{array}$ $\begin{array}{rr}\text{the Equinoctall}{\text{tang}}^{\text{t}}\text{for}{\text{y}}^{\text{t}}\text{hour \& Lat' is —}& \underset{\_}{13648}\\ \text{the produce of}{\text{y}}^{\text{e}}\text{Roote of Azimuth is nearest —}& \underset{\_}{4482}\\ \text{by}{\text{Substra}}^{\text{ct}}\text{facit —}& \underset{\_}{9166}\\ \text{by Adition facit}\phantom{\text{ —}}& \underset{\_}{18130}\\ {\text{y}}^{\text{e}}\frac{1}{2}\text{of}{\text{y}}^{\text{e}}\text{produce facit}\phantom{\text{ —}}& \underset{\_}{2241}\end{array}$ /Three sums Heare I finded by Addition: Substr & $\frac{1}{2}$ y^e produce \$\frac{1}{2}$/ w^ch sums sheweth y^e = sheweth the Tang^t of 6 Hours Azimuth Thus{.}\

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First I finde by Substracting 4482: out of y^e \y^e Equ Tang^t/ 13648 Rest 9166: w^ch sum is the Tangent of ${42}^{\mathrm{degs}}={\underset{·}{31}}^{\mathrm{m}}$ for y^e Azim^th: of .10: mor & .2: past noon{.} Secondly I finde by Adding of — 4482 To — 13648: facit 18130: w^ch Sum is the Tangent of ${61}^{\mathrm{degs}}={07}^{\min }$: for y^e Azim^th: at 2 in y^e mo^r & at 10 at night{.} Lastly the halfe of — 4482 is — 2241: w^ch Sum is the Tangent of ${12}^{\mathrm{degs}}={38}^{\min }$: for y^e Azim^th: at 6 in y^e mo^r & 6: past noone{.} Thus by this Rule you finde 6 Hours Azimuth p the produce of one small multiplication: shewd by Additio Subst & $\frac{1}{2}$ the Sum{.} Againe I demand the Azimuth for theese 4 Hours of the day or night vizd for y^e Hour of 9 in y^e mo^r or 3 past noone, 9 at nigh or 3 in \mor/ The declinat ${20}^{\mathrm{degs}}$, Lat' $\stackrel{\mathrm{d}}{52}=\stackrel{\mathrm{´}}{00}$: $\begin{array}{rrrrr}\text{the Roote of these hours Azimuth is —}& 8706& & \text{the Equinoc}\stackrel{\_}{\text{t}}{\text{Tang}}^{\text{t}}\text{for}{\text{y}}^{\text{t}}\text{hour \& Lat' is —}& 7880\\ \text{being Multiplyed by the Tangent of}{20}^{\mathrm{degs}}\text{of declination}{\text{w}}^{\text{ch}}\text{is —}& \underset{\_}{\phantom{00}3640}& \text{— facit nearest}3169& \hfill \text{. . . . .}\hfill & \underset{\_}{3169}\\ & 348240& & \text{facit by Substr}\stackrel{\_}{\text{act}}& \underset{\_}{\phantom{0}4711}\\ & 52236\phantom{00}& & \text{facit by Addition}& 11049\\ & \underset{\_}{26118\phantom{000}}& \\ \text{Rest}& \mathrm{3168/9840}& \end{array}$

Heare I finde by substracting 3169 out of 7880: Rest 4711 : w^ch is the Tangent of $\stackrel{\mathrm{degs}}{25}=\stackrel{\min }{13}$: w^ch is y^e Azimuth for 9 in y^e mor or 3 past noon{.} But if I Add {illeg} 3169 to — 7880: facit 11049 w^ch sum is the Tangent of $\stackrel{\mathrm{degs}}{47}=\stackrel{\min }{51}$ w^ch is y^e Azim at 3 in y^e mor or 9 at night for North decli{.} But y^e Contrary hours for South declination: thus you may doe for any of y^t; said Latitudes to any Hour & declination{.}
To finde the Tangent of the Equinoctiall Hours Azimuth is Thus: Multiplye the Tangent of y^e Hours distance from :6: by the Sine of y^e Latitude Cut of 4 figures & y^e Remaner is the Tangent of y^e Azimuth: the Radius being 10000: To finde the Roote of Azimuth: or Tangent of ${45}^{\mathrm{degs}}$ of declination: Multiply y^e Cosine of y^e Latitude being doubled by the Secant of the hour Being counted from :6: Cutt of 4 figures to y^e Right hand from the product; then ha{illeg}|l|fe the Remaner shall be the desired Sum for the Roote of y^e Hours Azimuth: Now if you Add \or Substract/ this Root of y^e Hours Azimuth To the Tangent{illeg} of the saide Equinoctiall \Hours/ Azimuth it will shew the Tangent of y^e hours Azimuth for ${45}^{\mathrm{degs}}$ of Declination: Heare I have shewd|e|d the proportions by w^ch I Calculated the Sun' & The North Star's Hourly Azimuth w^ch I would have done when I sent M^r Newton my Last Table of y^e Sun's Hourly Azimuth w^ch I could not then doe by Reason of y^e shortness of my Time: Therefore I would Intreat your favour if y^t you pleace d|T|o do me that kindness once moore as to lett \desire/ him to Looke over theese proportions To see whether they hold Good yea or not for I have bestoed so much paines w^th them{.} In the Same Manner as I have shewed you by the Tables Heare \made/ universall: Tables for All Latitudes vizd from y^e Equinoctiall to y^e Pole{.} Therefore if M^r Newton could doe me y^t faivour as to Resolve me of w^ch in 6 or 8 dayes affter the dait of this He would much Oblidge me thereby: Haveing no moore at p^rsent But Returning you many thanc{illeg}|k|s for your Trouble & Great G kindeness at Boothby & Elcewhear with my True Love & Service To M^r Newton Returning him many thancks for his kindeness I Rest at yo^r Command your Nepheu to Seve you Arthur Storer

London Sept 19^th: 1678

Thanckes be to God All w^th us heare ar very well: \hopeing y^e like w^th you{.}/ I was w^th Coz W^m King this day who is very well & I hope will do well{.} Both hee & Shee mindes there buisness very Cloacely & I hope y^t he will have a very great helpe by her Especially in the Consernes of his trayde he imployeth a boy aprentice & a Jorneyman: All of w^ch I finde usally very hard at worke{.}

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For the Reverend D^oct Humphry
Babington at Trenitye Colidge
In Cambridge
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