About two years ago I read with much interest and profit Lieut. Commander Hepburn's article "Some Suggestions in Practical Navigation" which appeared in the Naval Institute Proceedings. His wide experience in the practice of navigation well qualifies him for the discussion of that subject and his article was very timely.
The bulk of the article, as readers will remember, dealt with the use of single position lines in order to determine the true position of the ship.
I had never laid down for my own guidance any particular rule, or set of rules, as to the method of making allowance for current in the crossing of position lines but was always guided by expediency, making such allowance in each case as the known conditions seemed to warrant.
Lieut. Commander Hepburn's article suggested a definite way of using lines, which might be relied upon in all cases to give dependable results.
From the diagrams and description accompanying the article it was not difficult to see that, where the current remained steady in set and drift, his method gave exact results. From the illustrations of the use of the method quoted it seemed, too, that it gave excellent results in cases of unsteady current.
I resolved to assume a number of different conditions, plot the hourly position of ship, and, applying Hepburn's method, see how accurate were the results.
The diagrams so plotted, numbered from I to IXa, are reproduced in the plates accompanying this article.
In each case the ship is assumed to steam on a steady course which is not the same in all the diagrams; the exact course is immaterial to the discussion, so is not stated. The ship's speed through the water is assumed to be ten knots and the D. R. position each hour is indicated by plain Arabic figures. In the diagrams one-quarter inch equals one mile.
The locality is taken as in 40° N. latitude, the date May 16, and the body observed the sun. The inclination of the position lines varies from hour to hour exactly as the sun's change in azimuth at the place and time given, beginning in some diagrams at 7 a. m., in others at 8 a. m. The absolute bearing of any line is not important and therefore not stated.
The long, thin, horizontal black line represents ship's D. R. track; the broken black line at each D. R. hourly position shows the set of the current during the hour, except in diagrams IX and IXa, in which the set of current is the same as in diagram VIII.
The small black circles indicate true position of ship at end of each hour. The assumed current conditions are stated under the diagram number.
The short dotted lines crossing ship's D. R. course are the advanced positions of the first position line (except in diagrams Via and IXa, to be explained later); the prime figures indicate intersection of these dotted lines with ship's D. R. course and mark equal divisions along the course from the beginning, or last "fix"; dotted circles mark the intersection of the advanced line with each succeeding line. It will be remembered that the method recommended was to advance the first position line each hour along the course a distance equal to the intercept on D. R. course between last fix and first position line. It is understood, of course, that all lines are obtained at hourly intervals, the first one being one hour after last "fix."
Let us begin with the case of steady current, diagram I. Here the true positions coincide with those obtained by advancing the first line by amounts equal to the intercept on D. R. course between fix O and where first position line cuts same course. Inspection of the diagram shows that this must be so, as we are dealing with a series of similar triangles. Therefore, when current is steady in set and drift, this method of crossing lines will always give us our true position.
Is the converse true? That is, if we obtain a series of positions lying in a straight line which, produced backwards, passes through the origin of the D. R. course, are such positions correct and is the current steady? Alas, no.
A little measurement will show that if, in diagram I, we have a current directly against us the first hour of two and one-quarter knots, the second hour one and one-quarter knots, third hour three-quarters knot, fourth hour one-half knot, fifth hour zero and sixth hour a current directly with us of one and one-eighth knots, our diagram will remain unchanged, except that the true hourly positions will all be on the D. R. course line.
Again, a little thought will show that we may have a current revolving in set and unsteady in drift which will give us the same position lines and give true positions all above the D. R. course line.
So we see that, though there is a presumption that a series of intersections that plot in a straight line which, produced backwards, passes through the origin, are true positions, yet we have no guaranty that such is the case.
Next we take the case, diagram II, where our current increases from one knot to four knots in as many hours. This is not an improbable case, as such conditions exist in the Florida Straits and elsewhere.
Here we get rather disquieting results. Our positions by intersection would indicate that we are being set to the left of our course, whereas the opposite is true and the plotted positions and true positions separate until, at the end of five hours, they are eight and one-quarter miles apart. Here the position line parallel to the D. R. course shows the great advantage of such a line, when we can get it. We always think of the set of current in terms of two rectangular coordinates with respect to our course. Are we being set ahead or back and to the right or left? A line perpendicular to the course alone answers the first question and a line parallel to the course the second.
Leaving for later consideration the additional dotted lines on diagram II, we pass on.
Diagram III shows a current of the same drift as diagram II but setting to the left instead of right of D. R. course. Our intersection of lines again gives false results, though this time they indicate truly the side to which we are being set. The final discrepancy in position after five hours is nine and one-quarter miles.
No comment is made for the present on the additional dotted lines of the diagram.
Diagram IV differs from diagram III only in that the current is decreasing instead of increasing. The results are quite plain and comment on diagram II applies to this case.
In diagram V we first take up a current which varies in drift from a minimum to maximum and back to minimum. The strength chosen is less than in preceding cases and represents conditions likely to be met with almost anywhere. Then results of plotting need no explanation, but they emphasize the fact brought out above—that we may get very erroneous indications.
Diagram VI differs from diagram V only in set of current. No further comment is considered necessary on it.
Diagram VIa reproduces all the conditions of diagram VI, except that the position lines are plotted at different angles relative to the D. R. course. Here our first position line cuts the D. R. course at an extremely acute angle, so we wait until we have three lines plotted, then move second line forward, moving it one-half the distance from 0 to 2', assuming current steady the first two hours.
Diagram VII is new only in that the set of current is at right angles to the D. R. course. We note that, in this case, our intersections give us very close approximations to the true positions.
In diagram VIII is first introduced variable set of current. Here the current sweeps around clock-wise, changing its set thirty degrees each hour. The drift is steady.
In this diagram our intersections give worse results than in any preceding case. For the seventh, eighth and ninth hours the lines intersect so far off the D. R. course that the intersection cannot be shown, but its position is indicated by arrows on position lines. Note that after nine hours the true position plots on the D. R. course though we should never suspect it from our intersections.
Diagram IX is drawn with current setting the same as in diagram VIII, but of variable strength. This is a condition such as is met with on Nantucket Shoals. Here the first position line passes through the D. R. position. The positions by intersection are not such as to give us confidence in sights when crossing the shoals.
Diagram IXa is a repetition of diagram IX, except for relative tearing of position lines to D. R. course. The same comment as to carrying forward the second line instead of the first applies here as in diagram VIa. The plotted positions agree pretty well with true positions until we get to the ninth hour when they differ by seven and three-eighths miles. (The plotted position is off the diagram but is indicated by arrow and distance, 47 mm.)
Another diagram was plotted in the rough, but not copied, wherein the plotted intersections were all on the D. R. course and close to D. R. positions, although the true positions ran all around. Such a case would give the navigator a firm but false conviction that the current was negligible.
After making these diagrams and poring over them many times and long, realizing the helplessness of the navigator, I felt with Kipling
"An' the end of it's sittin' and thinkin',
An' dreamin' Hell-fires to see."
Various tricks were tried with the lines and, on several diagrams, I could get a scheme which gave fine results for that particular diagram, but worthless results on the others. Unfortunately the navigator at sea does not know (with any certainty) which of the cases confronts him, therefore he must have a method which applies equally well to all cases, or he is no better off than before.
While not laying claim to having found such a method, I did hit on a scheme which, if applied to any of the diagrams, will give fairly good results.
Applying the rules which are given below to the diagrams, the greatest error obtained (diagram VIII) was two and one-half miles. Most of the cases give results not over a mile in error.
A serious shortcoming of the method is that it can be used only if a position line within five degrees of any one of four bearings can be obtained; that is, forty degrees of bearing out of the whole circle alone are available.
In order to use the method we must, after obtaining three or more position lines, obtain one which is within five degrees of being parallel to our D. R. course or perpendicular thereto.
The rules for the proposed method are but two in number and are alternative in character, that is, in some cases one rule is used, whereas in other cases the second rule applies.
The rules are given below, followed by an example of the application of each one. The examples are worked out graphically in the diagrams on the accompanying plates, so that it is hoped that what clearness is lacking in the statement of the rules will be compensated for by the simplicity of the examples and diagrams.
A few preliminary words and definitions are needed to avoid cumbersome circumlocutions in the statement of rules.
In all cases it is necessary first to have plotted three or more position lines, the last of which is parallel (or within five degrees thereof) to the D. R. course, or perpendicular (within the same limit) to that course. For the present we shall assume that the lines are all obtained at exact hourly intervals, the first one being one hour after the latest "fix."
The position line parallel to D. R. course or perpendicular to it I shall call the determining position line, as it is the one on which we determine our true position or fix. In any particular case we deal with either a parallel or a perpendicular determining line, but not with both, so no confusion can arise on this score.
All position lines plotted prior to the determining line will (probably) be oblique to the D. R. course, so I shall call them oblique lines, calling the one which immediately precedes the determining line the last oblique line. Others will be called first oblique line, second oblique line, etc., beginning with the first one obtained after latest "fix," will be spoken of as the origin.
The first oblique position line is, in all cases, to be advanced hourly by an amount equal to the intercept from the origin on the D. R. course. This follows Hepburn's rule. The intersections of the advanced first oblique line with its successors give us positions which we shall call assumed positions. Having no better information at hand we assume these intersections to be our true positions until we have further data.
With these preliminaries and with a reiteration of the statement that the procedure when the determining line is parallel to D. R. course is governed by one rule, and that the procedure in a case where the determining line is perpendicular to D. R. course is governed by the other rule, the rules themselves follow.
Rule I.—When the last of three or more position lines is parallel to D. R. course.
Having obtained our assumed position on the determining line, join that position to the origin by a straight line. On this straight line measure the distance intercepted between the last oblique line and the oblique line just preceding it. Call this distance "a." With "a" as radius and with center at intersection of straight line and last oblique line, strike an arc cutting determining line. Where this arc cuts determining line is our true position.
Example.—See diagram II. Here the line at fifth hour is our determining line. Line 5' intersects it in our assumed position. The straight line from origin to assumed position is shown dotted. The intercept on the dotted line between the third hour line and the fourth hour line is "a." The cross on fifth hour line is taken as true position.
In diagram II this comes within a mile of true position. In diagram III the error is one-quarter mile. The method is also shown in diagram IV, in first part of diagram VIII and in last part of diagram IXa.
Rule II.—When the last of three or more position lines is perpendicular to D. R. course.
Obtain assumed position on determining line. Measure the distance in a straight line from origin to this assumed position. Divide this distance by the total number of position lines plotted. Call the quotient "a." With "a" as radius and with center at origin, strike an arc cutting first oblique line. With this intersection as center and with same radius "a," strike an arc intersecting second oblique line. Continue this process until you strike an arc across determining line. This gives true position, provided the distance "a" is long enough to reach from last oblique line to determining line. It is usually too short to do this; in such cases, draw a straight line from intersection of arc with last oblique line perpendicular to determining line. Where this perpendicular strikes determining line is taken as true position.
Example.—See diagram Via. Here the entire method is shown except that the straight line joining origin and assumed position on determining line is not shown as it might make the diagram confusing.
The application of this rule is also shown in last part of diagram VIII and in first part of diagram IXa.
In all the diagrams the position lines are shown taken at equal and hourly intervals. It need hardly be said that equal intervals of other length may be used or unequal intervals may be used (if not too greatly unequal), care being taken to use proper proportions in plotting, carrying forward lines, striking arcs, etc.
It is to be regretted that a rule of more universal application cannot be proposed, but there is no doubt that, if sufficient interest is aroused in the effort, some one else will succeed where I have failed. Meanwhile it would be well if attention were given to the problem of obtaining instruments which will permit us to observe two celestial bodies at any hour, day or night, in clear weather.
Bearings and Distances
Among the other instructive points brought out by Lieut. Commander Hepburn is the neat rule about laying a course to pass a distant object at a given distance. This is what he refers to as the "one-fifth rule."
This is a very handy tool. It has a mate which I give, not that it is better, but simply because of a prejudice against anything which perpetuates the old "points" of the compass.
Remembering that a radian is fifty-seven and three-tenths degrees and that we make an error of less than five per cent by calling it sixty degrees, we have the rule that one-sixtieth of the distance from a ship to a distant object is the distance at which the ship will pass that object for each degree that it bears on the bow, provided we hold a steady course and there is no current set athwart the course. This rule may be used from zero degrees to thirty-five degrees with less than five per cent error.