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    Re: Sumner's Line (Navigation question)
    From: George Huxtable
    Date: 2006 Feb 17, 00:44 -0000

    Frank's contribution has been most helpful. He wrote-
    | Here's some historical positional info for  Small's/Smalls Rock lighthouse:
    | I checked Bowditch 1826, 1842, 1849 and the  position listed in these
    | editions is
    | 51d 45'N 5d 36'W. That's very likely  the position known to Sumner.
    I agree. But it doesn't correspond very well with the Sumner's plotted position measured off his
    III, which I make to be 51deg 48' N, 5deg 38' W, just as Bill does. He appears to have plotted its
    position about 3 miles NNE of where Bowditch puts Smalls light prior to 1859. Not a big error, but
    outside what
    one would expect a seasoned mariner to plot, in a context where accuracy was important. He seems to
    have shifted the lighthouse from its true position (if we accept Bowditch) to a point that's exactly
    on his position line. In the circumstances Sumner was in, thick weather, not seeing the lighthouse
    until it was "close aboard", such a 3-mile gap could have been crucial, making all the difference
    between seeing the light, and sailing past it. If he had plotted the lighthouse in its true
    position, not exactly on his position line but 3 miles off it, would his tract have seemed less
    convincing? I think it would.
    | In the 1868 edition the position is about  three miles away:
    | 51d 43'N 5d 40'W.
    | In the 1883 edition, which is when the  US Navy took over and substantially
    | re-wrote the book, the position is the same  but more precise:
    | 51d 43' 14"N 5d 40' 9"W.
    | BTW, in all of these it's  listed as "Small's" (possessive).
    | In the 1918 edition of Bowditch, the  position for "Smalls Rocks: lighthouse"
    | (no longer possessive) is listed  as
    | 51d 43' 15"N 5d 40' 15"W.
    I agree  with that as the new position of the lighthouse after 1859.
    Raper (1864) gives it as 51deg 43.3N, 5deg 40.0 W
    Norie (1914) gives 51deg 43' 15" N, 5deg 40' 05" W.
    Modern Macmillan almanac 51 deg 43.25' N, 5 deg 40.15' W
    All agree well, though that may mean nothing more than that they have all been taken from a common
    The position given in the Times World Atlas gazetter, at 51deg 43' N, 5deg 30' W, seems to be
    aberrant and perhaps we should discard it.
    | According to various web sites, the first  Smalls Rocks light was built in
    | 1773 and stood until 1858. It consisted of an  octagonal cabin standing on nine
    | massive oak posts. A new lighthouse was built  from 1859-1861. It still stands
    | (apparently), and it is a circular stone tower  with a flared base. There's a
    | photo of it here:
    | http://jharadie.tripod.com/lights/lighthouse2/smalls_light.jpg with its former  paint scheme.
    Here's a fairly detailed history of the
    | lighthouses:
    | http://www.trinityhouse.co.uk/interactive/gallery/smalls.html
    I have followed up that useful website with a call to Trinity House, the UK lighthouse authority.
    They promise to look up the stated positiuon of the original Smalls light, and let me know within "a
    few days"
    | Are  there enough rocks out there to build the later lighthouse a few miles
    | away?
    Yes, I think so, though I have never sailed there. A friend is digging out his local charts for me,
    to be sure. One might have expected the lighthouse to be perched on the one rock that is always
    above water, Smalls Rock itself, for ease of construction. However, the dangerous reef extends from
    there toward the main shipping channel, so it would be more helpful to mariners to put the
    lighthouse further out along that reef, where the rocks cover at high tide (and leave Smalls Rock to
    mark itself, to some extent). I hope to know the answer soon.
    The newer light might well have been built on a different rock, which seems to have shifted it
    further away from the shipping channel, as far as we can tell. That seems to be a backward step. It
    was that I was thinking of when I referred to possible "migration" of the lighthouse.
    Sumner says nothing about the local tide conditions, but I estimate that at the time of his passage
    the tides were fairly neapish and had only recently started to go NorthWestward as he approached
    Smalls light. In which case I estimate the tide to be no more than half to one knot in a NW
    I've just been looking at "A History of Marine Navigation", by W E May (1973). I like its
    dedication, "to Commander A.V. Thamas, OBE, CStJ, JP, DL, Royal Navy, who has remained my firm
    friend although we shared an office for many years."
    But what interests me is that he has commented in some detail about Sumner, and has noticed
    discrepancies in Sumner's charting. I have  just loaded his relevant pages to the Nav-L blackboard.
    I don't agree with all he says, in the second paragraph of page 172, but he provides another view
    about Sumner.
    I wonder if anyone has tried checking over Sumner's time-sight calculation on page 15 of his
    pamphlet? He is using the formal given in my 1977 Bowditch in para 2106, "Longitude methods". The
    Goodwin method gives, effectively-
    versine t = [cos (lat -dec) - sin alt] / (cos lat cos dec)
    which gives HA as an angle which you can convert to a time. This is reasonably convenient for
    calculating with logs, as Sumner had to do, in the absence of a pocket calculator.
    When expressed as above lat and dec need to be given the right sign, + if North, - if South.
    A table of "log risings" is just another name for log versines, and can be found in some old sets of
    tables, and gives the result directly in hours.
    You have to be aware that Sumner's tables of natural sines would have given him numbers up to
    100,000, rather than up to 1, so sin 30 would be 50,000. When he takes logs of those differences,
    that gives him a result that's 5 greater than you might expect. The tables for log risings are
    adjusted to suit, being 5.00000 , not 0.00000, for an angle of 90 degrees (6 hours). Sounds
    complicated, but it's all aimed at making the arithmetic quick and simple (no negative numbers) for
    those that used the tables every day. I presume that this is what Sumner referred to as "Bowditch's
    3rd method".
    To work the problem by a calculator, rather than with logs, it's easier to use another expression
    from Bowditch para 2106,
    cos (HA) = (sin lat - sin lat sin dec) / cos lat cos dec,
    and then convert angle to hours by dividing by 15.
    Again, lat and dec should be North-positive.

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