Recently the writer on an occasion of duty found himself on a steamer bound from San Francisco to Coos Bay, Oregon. He made the acquaintance of the Captain, who extended to him many courtesies, and with whom he had many pleasant chats on topics both literary and professional. For the Captain is something of a poet; a poem of his on the Battle of Manila Bay had been read at the last banquet held by the First of May men in Washington; and he showed a wide and thorough reading of Military and Naval literature.
In passing Cape Mendocino he very kindly stopped his ship off Blunt's Reef Light-ship in order that there might be thrown into the boat that put off bundles of the latest newspapers and ephemeral literature that had been collected from the passengers and which the crew of the light-ship were sure to appreciate. That evening in his cosy cabin after some recitation of his poems and some discussion of historical naval battles, he spoke of the lightship and of the days when no light-ship was there. Then he produced a large tracing of the coast and of Blunt's reef, to show me his invention, something he had found out himself and which, he said, was the wonder of brother masters of shipping plying those waters. When he neared Cape Mendocino, he said, he always wanted to know how far his course, if he continued on it, would clear Blunt's reef. "Now," said he, "I doubled the angle." I understood what he meant, which was the application of the rule that if for any bearing on the bow of a fixed point that bearing be doubled the distance run is the distance of the point at the time of the second bearing.
"This rule is not new," I said. "No, of course not," he answered, "but what I do is this. When Mendocino is two points on the bow I take the time and patent log; and again when it is four point on the bow. That tells me how far off the Cape is. But what I want to know is how far off I shall be when it is a beam, and I want to know that before I get there, so as to change course if need be. So I have constructed this table, "showing me along table of distances for different runs between the two and four point bearings," -I have calculated these distances; I have run them out to many places, you see; this one to seven places just to see how far I could carry it." This was very interesting; the table was really carried out in one or two instances to that degree of refinement.
"But that is my invention, too, "I replied, meaning by that that it was something I had found out for myself, "only I don't resort to any table of distances; the table is unnecessary. All you have to do is to multiply the last distance run by seven-tenths and there you have it, how far from Mendocino you will pass if you maintain your course." The Captain looked incredulous. But by applying the seven-tenths rule to his tabulated distances they were found to be in practical agreement in every case. He was convinced, but to preserve his proper opinion of his own elaborate method exclaimed "That's all right, but who would trust a deck officer such as we now get, with multiplying anything by seven-tenths. They can't do it and I'll stick to my table." This naturally led to a discussion of the qualifications and acquirements of officers of merchant vessels, which is another story.
The fact remained that we felt ourselves co-discovers-of something of use and value. Most creditable to the master of this merchant coasting vessel is it that he had, unaided and untutored in mathematics, lighted on his handy problem in piloting. For myself, while I had seen the rule of "doubling the angle," in books, I had never seen in any text-book or book on Navigation, until I had run across Lecky's "Wrinkles," the prophetic feature of the problem, the part, too, which I think of greatest value. Lecky gives the multiplier as .71; it is really the one-half of the square-root of two [1.414], or .707. For practical application .7 will suffice. After I had lighted upon the rule I had frequent occasion to serve on examining boards for promotion of young officers and as occasions arose I have put this problem to them. "You are officer of the watch. The ship is steering North. When a point you intend to pass, off which lies a danger, bears N. N. E., you take the bearing and log readings, and again when it bears N. E., the run by log being 10 miles. If you continue on your course, how far off from the point will you pass?" I am obliged to state that in no instance have I lighted on an officer thus examined, all graduates of the Naval Academy, who could answer the question off hand. I have had fellow members of boards object to the question being put on the ground that it was a "trick question." Some, probably the majority, were able, finally,
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to work out the problem on paper. But that is a method hardly suitable for the bridge of a ship particularly if the night be stormy and you are encased in oil-skins. Some of these officers had been navigators of ships. Very few even knew what was meant by "doubling the angle on the bow" although I believe all knew the bow and beam “rule which is a particular case of the general rule. I have asked the question of graduates of this year, with the same result; none of them could answer off hand, correctly. Now the rule is a most simple, handy, and useful one. Fig. 1 shows its solution with the data I have usually presented with it.
It is seen that if bearings of the object be taken at 2 and 4 points on the bow (22 1/2° and 45°) seven-tenths of the run between bearings will be the distance the point will be when passed abeam.
The midshipmen who answered the question as stated, all said the abeam distance would be 10 miles, and referred me to their text-book, "Navigation and Compass Deviations, Muir, 1906," as their authority. Turning top. 214 of Muir, the prophetic part of the problem is given under the heading, Distance of Passing an Object Abeam. The rule stated is, "In case the first angle on the bow is 26 1/2° and the second angle is 45°, the distance run between the two bearings will be the distance of passing the object abeam, if course and distance are unaffected by current." Full of my own method, i.e., bearings taken at 22 1/2° and 45° which is an application of the general rule of "Doubling the angle on the bow," I hastily took the conclusion that the 26Y2° of Muir was a misprint for 22 1/2°, assisted there to by the fact that the rule immediately follows the paragraphs and figure devoted to "Doubling the Angle, "and that my midshipmen friends seemed also to associate the rule with doubling the angle. The error of my conclusion has been pointed out to me by friends. This rule in Muir is correct. To me it is entirely novel, and is quite an ingenious solution of the problem.
It is suggested to the author that in revision of his book a figure similar to Fig. 2 might be given illustrating and proving the rule. The basis of it lies in the angle 26Y2°, which is (approx.) the angle whose tangent is one-half. More nearly the angle is 26° 34', but 26 1/2° suffices for work at sea.
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We have two rules, then, for determining beforehand the distance a ship will pass when abeam of an object. Both are equally simple. Perhaps, as the master of the Breakwater preferred his own method with a table of distances carried out to millionths of a mile, I prefer the seven-tenths rule, because I had worked it out for myself and have always used it. It is granted that 2672° is not quite so sharp an angle as 22 1/2°, which is an advantage. But 26 1/2° is an odd sized angle as nautical practice goes, while 22 1/2° is two points and the second bearing is four points or twice two, and these angles are simply connected with compass courses when treated in points. I believe the seven-tenths rule will stick in the memory longer than a recollection of the angle 26 1/2°.
From a combination of these two rules there follows an interesting corollary; which is, that if bearings of an object at 22 1/2° and 26 1/2° on the bow be taken, then seven-thirds of the distance run in the interval will be the distance when abeam. Proof of this is shown in (b) of Fig. 2. Such a rule, better than guessing, might, on some rare occasion, be of value.
From the foregoing I believe it will not be amiss to re-state the above simple but valuable rules in piloting in order to inform those who do not know them and to impress on those who do know them their simplicity, handiness, and value and the great desirability of the officer of the watch applying them at all times while coasting. Of course corrections for current or tide in the distances run should always be noted.
Doubling the angle on the bow. —If for any bearing on the bow of a fixed point that bearing be doubled the distance run is the distance of the point at the time of the second bearing. The familiar problem of "bow and beam bearing' “ is but a particular case of the general rule.
To determine beforehand what distance your course will carry you from a given point. —Observe the object when it bears two points on the bow, and again when it bears four points. Seven-tenths of the corrected run will be the distance when the object is a beam; or, Observe the object when it bears 26 1/2° on the bow and again when it bears 45°. The corrected run will be the distance when the object is abeam.
One or the other of these rules, needing no calculations, no reference to tables, should be at the ready command of every officer navigating our ships.