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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:
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.

"FINDING THE INTERSECTION OF SUMNER LINES.

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."

-FER

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