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    Re: Certaine Errors in Navigation Corrected
    From: Gary LaPook
    Date: 2007 Sep 21, 15:50 -0700

    When computing great circle distances using Wright's method you can
    get better accuracy by using a modern plotting sheet such as the Radar
    Manoeuvering Board which has more even divisions and has a larger
    scale. It takes much less time to do the computation than it takes to
    discribe the process.
    
    gl
    
    On Sep 21, 1:29 pm, Gary LaPook  wrote:
    > In 1980 I discovered a book in my college's library which was a reprint
    > of Edward Wright's "Certaine Errors in Navigation" which had been
    > published in 1599. I still remember how pleasurable it was to read, it
    > was like sitting down with an old friend and discussing navigation. He
    > seemed like a thoroughly modern man. I highly recommend this book if you
    > can find a copy. There were other navigation books reprinted as part of
    > "The English Experience" series and they were also worth reading.
    >
    > Edward Wright was a mathematician who turned his interest to navigation
    > and was the first one to devise how to compute the table of meridional
    > parts necessary to create charts on "Mercator's Projection" and
    > published such a chart covering England to the Azores based on a voyage
    > he took with the Earle of Cumberland in 1589. I attached a copy of that
    > chart to a previous post.  For biographical information on Wright you
    > can go to:  http://books.google.com/books?id=uTsJAAAAIAAJ&pg=PA101&lpg=PA101&dq=c...
    >
    > In "Certine Errors" Wright laid out a method to compute the great circle
    > distance between two points on the earth using only a straight edge and
    > a pair of dividers! I have never seen this method described in any other
    > navigation text. Although it is now trivial to do this computation with
    > a calculator it think it might be interesting to look at Wright's
    > method. I am attaching 13 pages of his book that I photocopied back in
    > 1980 but I am missing several pages which would fall between the second
    > page of reproduced text (ending with "places.") and the next reproduced
    > page starting with "foote," (they didn't number their pages.) Don't be
    > intimidated by the type and spelling as you quickly get the hang of it
    > and can read it as quickly as modern text. There also appears to be a
    > couple of typographical errors that I will point out.
    >
    > I will also offer my own explanation.
    >
    > Wright starts out with special cases but I think it is easier if we
    > start out with the general case. Wright's description of the general
    > case starts on the last line of the third page of reproduced text ("If
    > the latitudes be not both equall...") and continues with a couple of
    > examples using London and Jerusalem ("Hierusalem") and London and Cusco
    > as the examples. The coordinates he uses for London are 51� 32' north
    > (which is as accurate as its modern latitude and a longitude of 22� (all
    > of his longitudes are east). ( I do not know where he got this value
    > since it does not comport with  his own chart (which would make it about
    > 16�) published in the same book, perhaps an earlier longitude using a
    > different prime meridian (perhaps the Cape Verde Islands) since he
    > couldn't be off by 6� between the Isle of Wright and London.) The
    > latitude of Jerusalem he gives as 32� north while its modern value is
    > 31� 16' north and he uses 11� south for Cusco while its correct value is
    > 13� 28' south.  The longitude he uses for Jerusalem is 68�, making it
    > 46� east of London while its true value is 35� 14' east, a longitude
    > error of 11�. He gives the longitude of Cusco as 295� making it 273�
    > east of London while its modern longitude is 72� west, the same as 288�
    > east of Greenwich showing an error of 15� between Wright's longitude and
    > the modern value.
    >
    > We will use Wright's values for this illustration, 51.5 for latitude of
    > the London, 22 for the  longitude of London; 32 and 68 for Jerusalem;
    > and 11 south and 295 for Cusco.
    >
    > Wright produces a diagram of the method however is is very complicated
    > since all the lines for his many examples are contained on one diagram.
    > We will draw only the lines we are interested in but will use Wright's
    > labels for the lines.
    >
    > Starting with the first example of figuring out the distance between
    > London and Jerusalem, we first draw a line across the base circle
    > diagram from the longitude of London, 22, "B" all the way across to the
    > opposite side at 202., "C." Wright would have us next do the same for
    > the longitude of Jerusalem but instead lets draw the line representing
    > the latitude of London. Add its latitude, 51.5, to 22 and place a dot on
    > the circle at 73.5, "E." Then taking a triangle or a plotter place its
    > straight edge on the line "B-C" and draw a line from "E" that is
    > perpendicular to "B-C," marking the intersection as "F." (Of interest is
    > that the length of "E-F" is the sine of 51.5 and the length from the
    > center of the circle (not labeled) to "F" is the cosine of 51.5.)
    >
    > Next we do the same for the position of Jerusalem, drawing a line from
    > 68, "M," across to 248, "J." Then we draw the perpendicular representing
    > Jerusalem's latitude, from 100 (68+32) "N" to intercept "M-J" at "O."
    > (The lengths of these lines represent the sine and cosine of 32.)
    >
    > Since London and Jerusalem are both in the northern hemisphere, we next
    > subtract the length of Jerusalem's latitude, "N-O," (the smaller value)
    > from the length of the latitude of London, "E-F," by taking a pair of
    > dividers and setting them to "N-O" and then placing one point on "E" and
    > the other point on the line "E-F" and marking this point "Q." (Note, if
    > you are following Wright's description at this point, there is a typo in
    > the text were it states "and P Q equall to N O" it should say "and E Q
    > equall to N O.") Next we determine the distance between the points "O"
    > and "F" (the intersections of the latitude lines with their respective
    > longitude lines) with our dividers and leaving one point on "F" we swing
    > the other point to the line "B-C" and mark it "P." Leaving that point on
    > "P" we next move the other point of the dividers to "Q," this is the end
    > of the chase. With the dividers set to the spacing between "P" and "Q"
    > we move the dividers to the scale on the circle and take out the number
    > of degrees between the divider points, this is the number of degrees of
    > the great circle joining the two places on earth. Multiply by 60 to
    > calculate the number of nautical miles.
    >
    > Does it work? Wright came up with "38 degrees and about 3/4 that is 775
    > leagues, which are 2325 miles." Using a modern digital calculator your
    > end up with 38.614 degrees, 2316.8 miles a difference of 8.2 miles or a
    > 0.3% difference. Not bad with just a straight edge and dividers.
    >
    > Wright uses London and Cusco to illustrate the case when the two places
    > are on opposite sides of the equator. The only difference is that
    > instead of subtracting the latitude of one from the other you add them.
    > You do this by extending the longer latitude line till it goes outside
    > of the circle and then marking on this extended line the distance of the
    > latitude of the other location. Using the diagram we already have and
    > pretending that Jerusalem's latitude is south instead of north, we
    > extend the line "E-F' some convenient distance outside the circle. Then
    > placing the divider points on "N-O" we take off that spacing and placing
    > one point on "E" we place the other point on the extended line outside
    > the circle and this becomes our new "Q," lets call it  "Q2." Then taking
    > the distance from "Q2" to our original "P" we take out the spacing and
    > place the dividers on the circle and find the number of degrees of the
    > great circle between London and a Jerusalem located in the southern
    > hemisphere. I come up with 92 degrees for a great circle distance of
    > 5520 miles. Using a computer I get 5565 miles, a difference 45 miles or
    > 0.8%.
    >
    > Wright uses Cusco to illustrate this case, drawing in lines "R-V" and
    > "S-T" in addition to the original lines for London. Taking the distance
    > between "S" and "F" and using it to establish "X, "(the equivalent of
    > "P" in the first example.) Then the line "E-F" is extended and point "Y"
    > is established by adding the length of  "S-T" to "F-E" which takes us
    > outside of the circle to "Y." The distance from "Y" to "X" (the
    > equivalent of "P to Q" in the first example) when placed against the
    > degree scale on the circle gave Wright "almost 97 degrees, of a great
    > circle that is 1940 leagues or 5820 miles." the calculator gives 96.74
    > degrees or 5804 mile, a difference of 16 miles or  0.3% difference.
    >
    > I don't know why it works and maybe somebody on the list can explain it
    > to me. I spent some time with the law of cosines and found a way to
    > reproduce this method using a calculator but you end up with a very long
    > and inelegant formula.
    >
    > BTW, it turns out that you never stop learning. I had always written the
    > law of cosines as c= square root of (a ^2 + b ^2 - 2ab cos C) but
    > working with it now I come up with c^2 = a^2 + b^2 -2ab cos C. I then
    > saw the pattern for the first time, the first part looks just like
    > Pythagoras' theorem. I then realized that the Pythagoras theorem is just
    > a special case of the law of cosines. When C=90� its cosine becomes zero
    > and the second part of the expression drops out leaving our old friend
    > Pythagoras!
    > --
    >
    > ___
    >
    >  English Experience.pdf
    > 22KDownload
    >
    >  Certaine.pdf
    > 392KDownload
    >
    >  Illustration.pdf
    > 151KDownload
    
    
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