There are several convenient thumb rules for piloting in places where only a single well-defined object is visible, of which the bow-and-beam bearing is probably the best-known. Another is doubling the angle on the bow; still another is the seven-tenths rule, given by Captain Gearing in Whole No. 123, U. S. NAVAL INSTITUTE PROCEEDINGS; and another is the 26½°-4.5° rule, given by Commander Muir in his excellent text-book.
While looking over Captain Gearing's note; in Whole No. 123, U. S. NAVAL INSTITUTE PROCEEDINGS, and Admiral Chester's discussion of it in a later number, it occurred to the writer recently that there is a general solution of the following problem:
Assuming a ship to be steering a steady course and to be unaffected by current or drift, find the relation between two bearings, from the bow, of the same fixed object ("relative bearings ") such that the run between bearings will equal the distance that the fixed object will be when it bears abeam.
In the figure these conditions are represented, AC being the ship's course, F the fixed object, and CF=AB by construction.
Evidently
CF (cot ?—I) = CF . cot ? (1)
Whence
cot ? = cot ?—I. (2)
Equation (2) expresses the relation sought. To put it in a convenient form for computation, let
cot ? = cosec2 x
tan ? = sin2x.
Then (2) becomes
cot ? = cot2 x
or (3)
tan ? = tan2 x
(Diagram found here in original article not replicated in this Word document.)
The following table has been computed from these formulæ. The angles in the first column are exact values; those in the second are the nearest quarter of a degree below the exact values in order to have a factor of safety. The marked pairs will probably be the most convenient ones to use, as they involve whole degrees only.
TABLE.
BEARINGS FROM AHEAD.
First Second First Second First Second
20 29¾ 30 53¾ 41 81¼
21 31¾ 31 56¼ 42 83½
‡22 34 ‡32 59 43 85¾
23 36¼ 33 61½ 44 88
24 38¾ 34 64¼ 45 90
‡25 41 35 66¾
26 43½ 36 69¼
26 44¾ 37 71¾
‡27 46 38 74¼
28 48½ 39 76¾
‡29 51 ‡40 79
This general solution includes the 26½°-45° rule, as it does the bow-and-beam rule. The use of the above table has the advantage, however, that the approximate determination of the distance off shore need not wait for the 45° bearing, as it must by the use of the 26½°-45° rule, or by the seven-tenths rule of Gearing. There are two whole-degree pairs by means of which such a determination can be made before the 45° bearing is reached. By the first pair (22°-34°) the observation is concluded when the second bearing is only 34°, while the first bearing itself is within a half-degree of the first bearing used in the seven-tenths rule, so that its accuracy is practically the same. The second pair (25°-41°) gives a check on the distance before it can be obtained at all by either of the two rules mentioned above. Of course the pairs having a fractional-degree value for the second bearing may also be used as further checks.
The use of the table is also more convenient than the method by plotting, though it does not fully take its place; and it is much more convenient than the use of tables 5A and 5B of Bowditch, though it does not fully take their place. Within the limits of its usefulness, however, the table appears to be an extension of present rules of thumb, and to be worth while.
A determination by this table is accurate to practically the same degree as by any other method in which the observations are made on the same bearings. It is obvious that the nearer to90° the bearing, the more accurate the observation will be, because the more sharply defined.
By noting the run between successive pairs of bearings, any component of current toward or away from the fixed object (at right angles to the course) will he shown. If there is a shortening of the distance run from earlier to later observations (or a shortening of time if the speed is constant), there is a component of set toward the fixed object, and vice versa. Results obtained from earlier pairs in which the bearings are small, while not as accurate as those obtained from later pairs, may yet be counted upon sufficiently to give warning. It is possible to get five whole-degree observations by the time that the fixed object bears 30° forward of the beam, as follows:
22°—34°
25°—41°
27°—46°
29°—51°
32°—59°
Of these the last three should be reasonably accurate.
In case of a current directly with the ship, the apparent distance that the ship will pass from the fixed object when it bears abeam, using the table, is less than the real distance; this is on the side of safety except in a channel where there may be danger on the opposite side if too far away from the fixed object.
If there be a current directly against the ship, the apparent distance that the ship will pass from the fixed object when it bears abeam, using the table, is greater than the real distance. This is the dangerous condition, and if there be reason to suspect such an adverse current, or such a component of current, a maximum allowance should be made for it, and the distance found by observation be correspondingly reduced. Thus, with a twelve-knot speed and a two-knot estimated current directly against the ship, the observed distance off the fixed object should be reduced one-sixth.
These remarks about the effect of current are so obvious that their mention may seem unnecessary, but the writer imagines that every officer of experience has seen occasions, as he certainly has, when it was only too evident that a consideration of current did not enter into the philosophy of officers using some one of the methods of observing a single fixed object.