I expect most DCA members know that the sextant can be a useful instrument for coastal navigation: it is not restricted to astro-navigation. They will be familiar with finding distance-off by vertical angle using either tables or the 565 rule.

Here is a method which takes the arithmetic out of the process. It requires a diagram which can be drawn carefully at home on winter evenings. With the aid of a straight edge, it gives direct reading of distance-off from the known actual height and vertical angle. It involves the use of a nomograph, which is drawn as follows:-

Draw two vertical, parallel lines spaced about four inches apart on a piece of paper.

Mark off the left hand line at one inch and one tenth inch intervals. Number the one inch marks 0, 100, 200, etc. from the bottom to the top (up to say 700). This is the actual height scale, and numbers represent feet. (The scale may also be marked off at 8.3 mm intervals, each representing ten metres, and numbered every 50 metres if so desired).

Mark off the right hand line at 1.2 inch and at 0.2 inch intervals. Number the 1.2 inch marks 0°, 1°, 2° etc. from the top to the bottom (up to, say 6°). This is the vertical angle scale with the 0.2 inch marks representing 10 minutes of arc.

Draw a diagonal line from 0 at the bottom of the left hand line to 0° at the top of the right hand line. This will become the distance scale.

Select suitable values for actual heights and calculate the corresponding vertical angles for distances-off. Suggested distances-off are:-

0.1, 0.2 ... 1.9, 2.0, 2.2, 2.4 ... 3.8, 4.0, 4.5, 5.0, 5.5, 6.0, 7.0, 8.0, 9.0, 10.0 miles.

To do this calculation use the rule:-

Distance-off (miles) = 0.565 x Actual height (feet) Vertical angle (minutes of arc)

re-arranged to give:-

Vertical angle = 0.565 x Actual height Distance-off

Lay a ruler across the scales cutting them at the actual height chosen and the vertical angle calculated for 0.1 mile off. Mark the point where the ruler crosses the diagonal scale and write ‘0.1’ by it. Repeat for all the other sets of figures.

To use the nomograph, select your object of known height, measure its vertical angle with the sextant, and lay a ruler across the nomograph cutting the vertical scales at the appropriate values. The distance-off will be shown at the point where the ruler crosses the diagonal scale.

Example of the use of the nomograph with an object 280’ high subtending a vertical angle of 2°38’ at the ship’s position. Distance-off is read from the diagonal scale as 1 nautical mile.

Does anyone use a sextant to measure horizontal angles and give a position fix from three marks? If so, they almost certainly use the ‘station point’ method or the usual rather tedious construction producing two intersecting position circles.

Here is an idea which simplifies the geometrical construction at the cost of some more ‘winter evening’ preparation and a little arithmetic.

That circle which passes through two of the marks and the ship’s position has a radius which is easily calculated:-

Radius = Distance between marks where θ = horizontal angle 2 x sinθ

1 Sit down and write out a table giving values of 2sinθ to four figure accuracy for values of θ between 0° and 90° at one degree intervals. It will not take long if you can borrow trig. tables and a calculator from the children! Given this table, the chart, and a measured horizontal angle between two marks, proceed as follows:-

Measure the distance between the two marks from the chart. 1 Look up the value of 2sinθ from your table. (If you want ½ degree accuracy, assume that the ½ degree values are halfway between successive whole degree values).

On the margin of the chart, multiply the distance between the marks by the figure from your table.

Set your pencil compasses to a distance equal to the result of the multiplication.

Draw two arcs cutting each other: one with the point of the compasses in each mark.

Set the point of the compasses where the arcs cut, and draw the position circle.

Repeat for the second pair of marks. The ship’s position is at the intersection of the two position circles.

Two points should be noted:-

a) If the horizontal angle is less than 90°, the centre of the position circle is between the ship’s position and the line joining the two marks.

b) If the horizontal angle is greater than 90°, the centre of the position circle is beyond the line joining the two marks. Also, subtract the horizontal angle from 180 before looking up 1 in your table. 2sinθ

Sounds complicated, but the plotting is easier than the conventional geometrical construction, especially if it is a bit bumpy out.