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    Re: Plotting charts
    From: Frank Reed
    Date: 2009 Nov 13, 21:56 -0800

    Here's the instructions on plotting and using charts for Sumner Lines from the 
    1906 Bowditch (available on Google Books here: 
    http://books.google.com/books?id=4GcDAAAAYAAJ). This is just about the time 
    period when Sumner Lines were becoming standard navigation procedure in the 
    US Navy. Contrary to the popular legend, they did not catch on soon after 
    Sumner's original paper was published.
    381. The intersection of Sumner lines may be found either graphically or by computation.
    382. Graphic Methods.-Each line may be plotted upon the chart of the locality 
    in which the ship is being navigated and the intersection thus found. The 
    details of the plotting will be obvious, whether the line is defined by two 
    of its points, or by one point and its direction. This plan will commend 
    itself especially when the vessel is near shore, as the chart in use will 
    then probably be one of conveniently large scale, and it will be an advantage 
    to see where the position falls with reference to soundings and landmarks.
    383. When clear of the land it is often inconvenient to follow this plan; a 
    large scale chart may not be at hand, it may not be desired to deface the 
    chart with numerous lines, or the necessary space for chart work may not be 
    available. In such a case, the following method is recommended, as it 
    obviates the disadvantages of the other.
    To understand the principle of this method, suppose that the lines are defined 
    by the latitude and longitude of two points of each, and consider that they 
    are plotted on a chart which is constructed upon a sheet of elastic rubber. 
    It is evident that if, while holding it fast in the direction of the 
    meridians, we stretch this rubber along the lines of the parallels in a 
    uniform manner until the length of each minute of longitude is made to equal 
    a minute of latitude, the chart, while losing its accuracy as portraying 
    actual conditions on the earth's surface, still correctly represents the 
    positions of the various points in terms of the new coordinates which have 
    been created, namely, those in which a minute of latitude is equal to a 
    minute of longitude. Thus, if on the true chart a point is m minutes north 
    and n minutes east of another, on the stretched one it will still be m 
    minutes north and n minutes east, the only difference being that the minutes 
    of longitude will now be of a different length; and if on the original chart 
    the two Summer lines intersect at a point m minutes north and n minutes east 
    (on the original scale) of some definite point of one of the lines, the 
    intersection on the stretched chart will lie m minutes north and n minutes 
    (of the new scale) to the east of the same point.
    A stricter mathematical conception of the stretched chart and its properties 
    may perhaps be obtained by considering the chart of the locality to be 
    projected (with the eye at the zenith) upon a plane which passes through one 
    of the meridians and makes an angle with the plane of the horizon which is 
    equal to the latitude; each minute of longitude will then be increased by 
    multiplying it by the secant of the latitude, and thus becomes equal to a 
    minute of latitude.
    From a consideration of the properties of this hypothetical chart it may be 
    seen that the following rule may be laid down: If two or more Sumner lines be 
    plotted by their latitude and longitude upon any sheet of paper, using a 
    scale whereon latitude and longitude are equal regardless of the latitude of 
    the locality, the intersection of those lines, measured by coordinates on the 
    scale employed, correctly represents the intersection of the lines as it 
    would be measured upon a true chart.
    It follows from this that we may plot Sumner lines upon any piece of paper, 
    measuring the coordinates with an ordinary scale ruler, and assigning any 
    convenient length for the mile; the larger the scale the more accurate will 
    be the determination. Or, what is even more convenient, we may employ 
    "profile paper," whereon lines are ruled at right angles to each other and at 
    equal distances apart, in which case no scale ruler is needed.
    One caution must be observed in using this method; all longitudes employed on 
    the paper for any purpose must be those of the scale, namely, one minute of 
    longitude equals one minute of latitude. For instance, if the two Sumner 
    lines be taken at different times, in bringing the first up to the position 
    of the second by the intermediate run, that run must be laid down to scale; 
    that is, the easting or westing must appear as so many minutes of longitude, 
    not so many miles. To do this enter the traverse table with course and 
    distance run, and pick out latitude and departure; then, by means of the 
    middle latitude, convert departure into minutes of longitude and bring the 
    first line to the second by laying off so many minutes of latitude north or 
    south, and so many of longitude east or west.
    In the ease where the Sumner is defined by one position and its line of 
    direction, it is not correct to lay down the angle to the meridian on the 
    hypothetical chart, for all angles are distorted thereon. The best way is to 
    find another position on the line by assuming a second latitude ten or twenty 
    miles removed from that of the point given, entering the traverse table with 
    the angle that the line makes with the meridian as a course, and abreast the 
    latitude taking out the departure; convert departure into difference of 
    longitude, and plot the second point by its coordinates from the first."
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