When I was a young man, it seems like a thousand years ago, I trained as a pilot in the Royal Air Force. I learned a few navigation tricks, a particularly useful one being the one-in-sixty rule.
The training plane we used to get our wings was the American-built T6 SNJ, or “Harvard,” as the British called them. We trundled round the countryside at 140 knots and a few thousand feet navigating mostly by eye. We received extensive training and practice in solving navigational problems in our heads.
A test on the way to wings qualification, which most students dreaded, was a 200-mile triangular cross country flight in which the pilot carried only a chart while an instructor in the back seat checked accuracy. One of the most important aids we learned was called the one-in-sixty rule. Typically, the student might find after 20 miles that a visual fix puts him three miles off course – what correction must he make and what is the wind doing?
A decade later, when I started sailing, I found that the same techniques were very applicable to coastal navigation.
What is the rule?
The rule is based on the premise that 1° is subtended at a distance of sixty units by one unit. The units can be miles, inches, light-years – anything so long as you use the same units. To be precise, the angle is 0.955°, but let’s not quibble. Expressed mathematically in navigational terms using GPS notation:
Angle = XTE x 60/range, where XTE is the cross track error.
Furthermore, up to about 20°, it is reasonably accurate to assume the angle is proportional to XTE, i.e., a 10-mile error at 60-mile range subtends 10°, 20 miles subtend 20° and so on.
For example, the hapless student pilot mentioned in the first paragraph finds himself three miles off course after flying 20 miles, corresponding to 9 miles in 60, or a course error of 9°. Let’s assume he flew his course accurately, the error must be due to the wind, which can then be calculated, although to do this completely the pilot has to know whether he is running early or late at that first fix.
The rule with GPS
Turning to a typical marine problem, look at Figure 1. A sailor is going from A to C, but at some point along the track he finds he is off course at point B. The GPS receiver will give the navigator several clues; his course will not correspond to the initial bearing and the unit will display a cross track error when the boat arrives at B. In general most sailboats do not hold a constant course, it varies all the time, and so the difference between bearing and course may not be very obvious; and the angular course error is not presented directly. The cross track error depends on the average error during the time to sail to B and thus, applying the rule is easy.
The distance to B is obtained by noting the reduction in the initial range. For example, if the range displayed on the GPS from A to C is 30 miles, let us say point B is 10 miles from A and the cross track error is 1 mile. The course error angle is XTE x 60/distance traveled or 1 x 60/10 = 6°. Assuming good helmsmanship, the error is caused by a current or leeway, the navigator must now correct for it. The GPS is now showing a revised bearing to point C, the change in bearing is simply XTE x 60/distance to go or 1 x 60/20 = 3°. If only the revised bearing is steered, the GPS will still show a difference between course and bearing and XTE will continue to increase. Both the angles calculated have to be applied to the new course; the navigator must change course by 9° (6°+3°) to arrive at C with no cross track error.
Suppose on an ocean passage the cross track error has increased to 4 miles due to difficulties in holding the course, which is off by, say, 10°. Then the wind changes and the GPS course now moves toward the destination bearing by 18°. The boat is now closing in on the rhumb line by 8°, a rate of 8 miles in 60, and will thus eliminate the error in about another 30 miles. A variation of this problem is the long tack/short tack situation when you can’t quite sail the rhumb line. Suppose close-hauled on the wind a long port tack is 12° shy of the bearing to the destination. You have 20 miles to go. How far must you sail on the short starboard tack to regain the original rhumb line? Easy – 12° corresponds to 12 in 60 or 4 in 20. Thus 4 miles at a right angle to the port tack will restore the original bearing.
The calibrated thumb
This use of the rule to find range has always amused my crew when I show them the trick. It is based on the fact that the average arm’s length is 30 inches and the size of the average thumbnail half an inch long. Thus the thumbnail viewed at the end of an outstretched arm subtends 1° at the eye. If you spot any object, such as a lighthouse, whose vertical height is marked on the chart, you can estimate the range by comparing the object’s subtended angle with your thumbnail. In this case: range = height x 60/angle.
For example, suppose you spy a lighthouse on shore, and it appears about half the height of your thumbnail, or half a degree. If the chart shows it is 100-feet high then the range is 100 x 60/0.5 = 12,000 ft., or about 2 miles. One of my crew was so intrigued by this trick that he calibrated his whole thumb in half-degree increments using a felt marker pen. He even turned the rule upside down by measuring the range of ships we encountered at sea by radar, and then he was able to estimate their length using his thumb horizontally.
Contributing editor Eric Forsyth has sailed hundreds of thousands of miles, including two circumnavigations.