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

<3:1r>

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 which was of the North Star's Hourly Altitude & Azimuth The Contents of which Answer is in theese Lines: Cambridge September 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 Sufficiently 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 that I should presisely have sett downe the minnute & fractionall part of a minute as my Calculation Gave it & by the very same Radius by which 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 which is for the Hours of: $\left.\begin{array}{r}8\text{ \& }4\\ \text{\& – }4\text{ \& }8\end{array}\right\}$ & for theHours: $7\text{ \& }5\text{ \& }5\text{ \& }7$: $—2—19\phantom{.}\text{ \& }.\frac{170}{1770}:$ parts of a minute But for the Hours of Both Sixes: $\stackrel{\mathrm{degs:}}{2}=\stackrel{\min }{24}=\stackrel{\mathrm{″}}{00}$ which is the Stars distance from the pole: But if I should have beene so presise with the degrees minutes & fractionall part of a Minute & so neare as was shewed by the calculation which 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 whrought the same so neare by the Altitude: as I did there without it & by such a double way of working that no Less then 4 Hours Azimuth was shewed by the produce of one simple multiplication casting away some five figures then Adding the Remainer To or Substracting from a Nother constant given Summ: which is allwayes the same for the Latitude & Hour given But for the 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 that without danger of error it may be used to Compute the Altitudes & Azimuths of Stars which ar very Remoate from the pole he cannot tell that he wrighteth is onley to be knowne from the demonstration of the Rule grounded on Mathematicall principles for which I think this same of myne is & if he receve my Letter dated about September the 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 Suns or Stars declination from the Equinoctiall to ${24}^{\mathrm{degrs}}$ Eather north or South: which declinations is Remoat from the pole which I Calculated for the Latitud of ${52}^{\mathrm{degs}}={00}^{\min }$ which is very neare your Latitude at Cambridge Because he might the sooner Looke it over without any great troble for so it may chance that he may have them Allreddye callculated for that or your Latitude by him: But because I have beene at that panes To make these tables universall being halfe Calculated for All Latitudes vizd: from the Equinoctiall To the Pole I shall Heare Lay downe the Grounds of theare proportions of the Suns 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 the Sun or any Stars Hourly Azimuth for these Latitudes the Declination being given multiply the sum of the Root of the Hourly Azimuth, for the Hour & Latitude suppoased: by the Tangent of the Sun or Stars declination: Cutt of foure figuers To the 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 the East or West. Example Latitude 52: the Declination ${20}^{\mathrm{degs}}$ North I demand the Azimuth at :10 in the morning & :2: in the Affter noon: & at :6: in the morning & 6 past {noone} at 2. in the morning & 10 at night at theese six hours I demand the Sun or Stars Azimuth: Aganst the Latitude 52 under the 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: Substraction & $\frac{1}{2}$ the produce $\frac{1}{2}$ which sums sheweth the = the Tangent of 6 Hours Azimuth Thus{.}

<3:1v>

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

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

London September 19^th: 1678

Thanckes be to God All with us heare ar very well: hopeing the like with you{.} I was with 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 that 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 which I finde usally very hard at worke{.}

<3:2v>

For the Reverend D^oct Humphry
Babington at Trenitye Colidge
In Cambridge
{4}