Just recently I went out in the bay with someone who wanted to practice some of the basic coastal navigation skills that are the foundation for navigation yet now hardly used with the availability of cheap GPS receivers. Besides plotting a dead reckoning position, obtaining a fix by three bearings and undertaking a running fix, my student wanted to calculate a position using the dipping distance of a light together with its bearing.
Because of the curvature of the earth, the distance to the horizon is limited by ones height above sea level (Fig 1). For the same reason, lights or other objects ashore can only be seen if their height is sufficient for them to appear over the horizon. The effect of a distant light appearing first on the horizon and then getting higher as one approaches can be used to obtain a position. Once a light is sighted as it first appears on the horizon, a bearing is taken of the light using a hand bearing compass and an entry made in the ships log. The bearing provides a position line on the chart (Fig 2) while the height of the light and your height above sea level enable you to calculate distance from the light and hence obtain your position. Both Volume 2 of Bowditch (The American Practical Navigator) and Reeds Nautical Almanac provide a table to allow you to look up the dipping distance of a light using your height and the height of the light. An abbreviated table for this purpose is included here (Table 1). However you can also calculate the distance directly using a simple scientific calculator as shown:
Distance to horizon (nM) = 1.144 ?height(ft)
To compute the dipping distance of a light or other object you should add the distance from you to the horizon to that of the horizon to the object (Fig 3).
Example: You note that a lighthouse ahead of you has just appeared on the horizon. If your height above sea level is 5 feet and from the chart you see that the height of the light is 100 feet, what distance are you from the light?
Dist to horizon (for eye ) = 1.144 x .5 = 2.56nM
Dist to horizon (light) = 1.144 x ?100 = 11.44nM
Dipping distance = 2.56nM + 11.44nM = 14.00nM
The answer is 14 nautical miles which is the same as that shown in Table 1
While obtaining a position using dipping distances is restricted to night hours (when you can see the light), there is a similar technique which can be used during the day to find the distance to an object of known height using a sextant. If the angle between the horizon and the top of the object is measured accurately using a sextant then you may find the distance it is away by either using a table as shown here (Table 2) or by using one found in Bowditch Vol II or Reeds. Alternatively using simple trigonometry and a scientific calculator you can calculate the answer directly (Fig 4). The formula thus derived is:
Height (ft)
Distance (nM) = _____________________
6076 x Tan (angle in degrees)
Example: You sight a high range marker some distance away that has a height of 100 feet. With your sextant you bring the top of the range marker down to the horizon and record the angle as 28' (minutes). What distance are you from the range marker?
100 100
Distance (nM) =____________ = _________ = 2.02nM
6076 x Tan (28') 6076 x .00813
The answer is 2.02nM which corresponds to the value obtained using Table 2
When you are next out sailing try these techniques for yourself. Our recent practice was impressive and the positions obtained compared well with those given by a GPS which we had for comparison.
