In the art of pedagogy, much depends upon following the difficulties in the student’s mind; these remarks are offered with the hope that they may have some slight value of that sort.
It is an apt question whether plane sailing is not the most difficult for the student, especially the self-instructed one, to understand. At first it seems easy, but the mind is soon puzzled by the apparent utility of a method of dead reckoning avowedly based upon the violent assumption that the earth is flat. The plot only thickens with the realization that a complete answer to so many dead-reckoning problems includes the use of plane sailing (witness the solution for distance in a mercator sailing problem).
Most ordinary navigation texts deal with this point in a summary way, if at all. There must be many possible avenues of attack; one such would be to treat plane sailing as an adjunct to mercator sailing, rather than as a totally independent method. The student, once having grasped the principles of mercator chart construction, gives full faith and credit to mercator sailing, seeing that the solution by means of it is equivalent to plotting the problem on an imaginary mercator chart.
On the mercator projection, the location of both the meridians and the parallels can be described by means of measurements in meridional parts. Hence any line, such as a course line, drawn on the projection is measurable in meridional parts, as well as in minutes of latitude. The student is already familiar with m, the difference of latitude expressed, not in minutes of latitude, but in meridional parts.
Distance, measured in meridional parts, could be written dm. It is then apparent that dm/m = sec C. How many minutes of latitude, or sea miles, are represented by the measurement dm?
It is only necessary to multiply dmby the average quantum of latitude which, between the limiting latitudes of the mercator track in question, is represented by one meridional part. This quantum amounts to the fraction 1/m, or difference of latitude divided by meridional difference of latitude. A possible formula for distance, then, is:
d=1 dm/m
But dm/m = sec C, so the formula can be rewritten d = 1 sec C, as in plane sailing. This also means d/1 = sec C, which enables the student to draw the plane sailing triangle and to label two sides, entirely by deduction from mercator sailing.
The third side can be dealt with similarly. The easting or westing which constitutes the third side of the triangle, when measured in meridional parts, is simply the difference of longitude in minutes—this follows from the definition of a meridional part as the linear unit which, on the projection, represents one minute of longitude. How many miles are represented by the DLo., then, is the only question. The easting or westing is made along the track, hence the fraction \/m is still of interest:
p=1 DLo/m
But DLo./m = tan C, as the student already knows. So p = 1 tan C, and tan C = p/1. The third side of the plane sailing triangle is now understood as a measurement which amounts to the easting or westing in miles involved in covering a certain span of latitude on a certain course.
The student would now be in a position to reflect calmly that the plane sailing triangle, like the mercator sailing triangle, is a plane figure, but not necessarily part of a flat earth. Over a small area the measurement p nearly coincides with the length of the arc of the parallel passing through the destination, intercepted between the meridians of the destination and the place of departure. As the area in question grows larger, the measurement p, being a measurement on a plane surface, bears less and less resemblance to the measurement on the sphere. But measurements on the plane surface of the mercator projection do not strike the student as untrustworthy merely because they are made on a projection, and he will realize that plane sailing can be equally reliable if he understands its elements. In that connection, there is little doubt that the notion of departure is not an easy one for students to absorb.
The student wall have seen that easting or westing can be readily measured on the mercator projection, where the meridians are everywhere equidistant. The length of some arc of a parallel, as measured on the projection, represents a certain number of miles of departure over the earth’s surface. On earth, the situation is clearly different. Departure, not being motion toward some particular point (pole), must always be measured by indirection, although difference of latitude can always be measured directly as an arc of a great circle connecting the poles. Departure being distance made good in a direction 90° removed from that of the pole, must have the same value for n miles on course x as difference of latitude has for n miles on course x ± 90°. The points would be made to the student, then, that departure, though a bona fide element of travel over the earth’s surface, and not a fictitious measurement based on the assumption that the earth is flat, cannot be measured on the sphere without indirection, but can be measured on the plane surface of the mercator projection, and on the plane surface on which the triangle of plane sailing is drawn.
Be these things as they may, some slight expansion of the ordinary treatment of plane sailing would seem to recommend itself, so long as dead reckoning holds its place as part of a practical navigator’s education. Otherwise the instructor may well fumble for the proper reply when some member of his class inquires why he should accept the result of the “inaccurate” formula tan C = p/1 as the course made good at the end of some traverse. On the other hand, if the elements of plane sailing are fully taught the question would not be asked at all, or, if it were, it could be answered shortly: “Because you are using the correct value of the departure.”