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    A Distance Off Method
    From: Dan Hogan
    Date: 1995 Jun 22, 11:35 PDT

    The following courtesy of
    	 The Navigation Foundation  (Tel. 301-622-8448)
    	 P.O. Box 1126, Rockville MD 20850
    	 The Navigator's Newsletter, Issue Twenty-three, Winter 1988-89
    Dan Hogan
    A Simplified Technique for Costal Piloting
    by Edward J. Nesbitt
    	This artricle offers a simple way to solve the frequent problem of
    determining how far offshore one is during coastwise sailing, or what is the
    distance to a lighthouse or buoy being passed.
    	Take two compass bearings in degrees on the point selected. The first
    should be when it's forward of the beam and the second when it's abaft the
    beam. Also record the times when each was taken and then determine the rate
    at which the degrees changed per minute by dividing the difference in
    bearings in degrees by the number of minutes between the readings.
    	You can find the answer "how far away you are," in nautical miles, by
    simply dividing that number into the speed of your boat (over the bottom) in
    	I've used this procedure for over five years but have never met anyone who
    knew of it before. I worked it out on an airplane flight after wondering how
    far away a particular mountain peak was and reasoning that there must be
    some way to compute it based on how fast the angle to it through the plane's
    window was changing. I'd thought of this before on a sailboat but didn't
    have the time to doodle about it.
    	There is a slight error (about 10 percent) due to this simplification which
    one could correct by reducing the the distance in the answer by 10 percent,
    and he'd be "bang on." But who knows a boats speed that accurately anyway?
    	This method has two advantages over the procedure usually employed: it does
    not depend upon the boat maintaining a steady course (which no small boat
    does) and you don't have to make your readings when the point in question is
    at a precise angle to your course. You can start and end when you think of
    it, although to keep the error to 10 percent, it's best to be within 10 to
    30 degrees of the abeam on both readings. But even if one reading was 40
    degrees from abeam and the other 10 degrees, the error (still high) would
    only be 18 percent and a conscientious navigator could compensate for it if
    he wanted.
    	A compass bearing is easily taken by a hand held compass, or by one on an
    RDF, or taken from the binnacle. Even if there is an error in the compass,
    it will be the same on both readings since they are only 30 to 60 degrees
    apart. The boat can deliberately change course by 10 to 20 degrees between
    readings with neglible effect on the accuracy of the answer.
    	To the scientifically curious who might be wondering how this simplified
    procedure works, it is due to the fortuitous coincidence that the rate of
    change of the trigonometric function of both the sign and the tangent, in
    units per degree, when the degree is small, is .0175 (2 pi divided by 360),
    and that by 24 degrees, which is in the expected range of angles before and
    aft of the beam used in this procedure, the tangent rate of change is only
    .0185 which is only about 10 percent greater than 1/60. The procedure uses
    boat speed in nautical miles per hour, and the degree change minute. The
    sixty minutes in each hour exactly cancels this out, at least with the 10
    percent that he answer is high. This procedure is therefore not a discovery
    but rather a simple observation. Just remember: divide the boat's speed by
    the degree change per minute.

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