These rules supposedly originated with Captain Clark of U.S.S. Oregon fame. At least the officer that passed them to me said he had received them from an officer who got them from Captain Clark. But it is almost certain that at most he did little more than collect and codify them. The rules themselves are so basic and so elementary that they doubtless have been in use in one form or another since Noah first started navigating. And yet it is surprising how few officers know them well enough to make of them the constant practical use they should. They reduce the seaman’s eye almost to the accuracy of the mooring board, and permit of astonishingly accurate simple maneuvers with no instruments or tables of any kind. The rules are:
The rule of the radian
The rule of the cosine
The use of the beam bearing
The Rule of the Radian
This rule is based on the fact that the radian is 57°. The rule uses 60° and says that the natural sine of any angle is the number of degrees in the angle divided by 60°. Thus the sine of 1° is 1/60, the sine of 5° is 1/12, the sine of 6° is 1/10, the sine of 20° is 1/3 and the sine of 30° is 1/2. Up to 30° the rule is almost exact.
The value of the rule lies in that the sine represents the transfer due to a change from the course you are on. Thus if you are on course north, and change course 6° you will be set 6/60 or 1/10 of the distance run to eastward (or westward) of north. If you run 2,000 yards, then you will be set sideways 1/10 of 2,000 or 200 yards. Its use is in applying this feature.
To take an instance. You are conning a ship on a short-range battle practice approach and find you are 200 yards outside your line. You are making 15 knots, or 500 yards a minute. If you change course 12° towards the target for 1 minute, you will have made a transfer of 1/5 of 500 yards or 100 yards. If you stay on the course 2 minutes you have crabbed the ship over the 200 yards. Practically if you use 2 minutes from the time you put the helm over to bring it back to the old course you will find that you have effected a transfer of 200 yards almost exactly. If it is only 50 yards you are off, using a 3° change of course you will transfer 1/20 of the distance run, or 25 yards a minute. A 2- minute change of course will bring you back on the line.
Or during the run you find that the base course lets you drift off 50 yards in 3 minutes. Now in 3 minutes you have covered 1,500 yards, so you have drifted off 1/30 of the distance run. One thirtieth of 60 is 2, so a 2-degree change of course will just compensate for the drift.
Or you swing into an unmarked swept channel. Your first cut after you are in it shows you are 200 yards from the center. A 6-degree change of course held for one mile, and you are right in the center of the channel.
Or you lay a course which is 5 degrees from the bearing of a light. Your range finder tells you the light is 20,000 yards away. A quick mental calculation tells you that 5/60 or 1/12 of 20,000 is 1,666 yards and that is the distance you will pass it abeam if you hold that course. If it isn’t enough, you can give an extra 5° to your course and pass the light 3,333 yards. Granting that it is generally preferable to lay off such problems on a chart, where you have a picture of the result, still often in piloting a quick mental decision is necessary and the law of the radian is of the greatest assistance.
And it is not only in navigation that the rule of the radian is valuable. As you become familiar with it a multitude of uses spring up. Gunnery frequently. Recently the question arose as to how low a plane had to be flying in order to be in the range of a gun with only 15° elevation. The rule of the radian gave the instantaneous answer, one quarter of the range. The more you use this rule the more versatile it becomes.
The Law of the Cosine
This is really more than the name implies in that the sine is used also. It requires the memorizing of the sine and cosine of 30°, 45°, and 60°. These are in tabular form:
If these are remembered the functions for intermediate angles can be obtained by interpolation with sufficient accuracy for all practical purposes.
The value of the law of cosines is chiefly in maneuvering where you want to gain distance to the flank without losing it along the path of advance. You are told, for instance, to form a scouting line, without losing distance ahead. The speed of advance is 20 knots. You don’t wish to use more than 25 knots, but you want to fan out as rapidly as possible. You have 5 knots excess speed, which is 1/5 or 20 per cent of the total speed you have available. This means that your cosine must be as close as possible to 0.80. If you fan out at 45° your cosine is 0.71. If you go out at 30° the cosine is .87. You do a rough interpolation and go out at 40° and you start the moment the signal is hauled down. Later you can work it out on the mooring board, and make any changes required. It won’t be more than two or three degrees (actually the change of course should be 37°).
By using this rule, I have seen a division commander snap the course signal for his division into the air so that it could come down with the squadron signal directing the basic maneuver. Perhaps it was a bit of showing off, but it is remarkable how quickly you can figure things like that in your head when you get in the habit of using the cosines—and the sines.
Another frequent use is determining how much speed you need to maintain the line of advance. You are in a column making 15 knots and you are ordered to go out on the beam 2,000 yards. You decide to go out with a 30° change of course. How much extra speed do you need not to lose bearing and how long will it take you? You know the cosine of 30 is 7/8, so that 15 knots must be 7/8 of your speed; you have to make 17 knots—(to be exact 17-1/7 knots). This will be 570 yards a minute. The sine of 30° is 1/2. So you are going to make 1/2 of 567 yards or 285 yards a minute transfer. You wish to make a total transfer of 2,000 yards, consequently you run for 2,000/283 or 7 (seven) minutes. We neglect small fractions, as this is rule of thumb, but again the results came out with surprising accuracy.
If you really start using these cosines and sines, you will be astonished to find how many uses you have for them, and how closely you can interpolate when you want more refined results. Thus while for rough calculations you can take the cosine of 30° as .90 instead of .87, if you want to use a cosine of .90 for .fairly accurate work you should use the angle 25°—whose sine by the law of the radian is 5/12. You don’t have to know what the natural cosine of 25° is. You do a mental interpolation. The cosine of zero degrees is one. The cosine of 30°is.87. By strict interpolation you get an angle 22$°. But the cosine changes more rapidly for the bigger angles so you use 25°. The actual natural cosine of 25° is .906. The above sounds like a long calculation, but with a little use and practice, the whole process is done in one mental operation.
The Use of the Beam Bearing
This is really one of the practical applications of the two foregoing rules, for a special purpose—that is obtaining a position in a formation.
You are on the beam of a column of ships and are ordered to take position in the formation. You are signaled its true course and speed. You put the beam bearing of the formation’s true course on your pelorus, and sight along it, to determine whether you are ahead or behind the abeam position of your place in the formation. Then by engines and helm, and using the law of cosines, crab in towards the formation at the same time walking your ship up or down until the line of bearing points at the ship next ahead of your position. Then as you get within 2,000 yards of the formation you gradually drop it aft so that it points to the middle of the gap you are to take. At the same time you gradually head the ship up as you get close. The result will be that you will make a seamanlike approach, and will slide into position without any radical last-minute change of course or speed, always upsetting when it comes to maintaining your position.
It is to be noted that all three rules are substitutes for the mooring board. They should not obviate the knowledge and use of the mooring board. But they should supplement it. They require a minimum of mental effort, they can be used almost instantly, and they give reasonably accurate results—more so than is implied by the phrase “Thumb Rules.” They serve as an excellent check on the mooring board, so much so that when the mooring board solution comes out radically different from the rule of thumb results, check your work carefully. Maybe you have done some poor calculating in working out the rule. But there is at least an even chance that you have made a mistake with your mooring board.