Mental Maths for the OOW

The Rules tell you what to do. Mental maths tells you whether it is working. An officer of the watch makes most of their decisions on the bridge wing or at the radar, minutes or seconds from the event, and the chart table is too far away and too slow for any of it. What closes that gap is a small kit of approximations, quick to run and accurate enough for the con. This chapter gives you the kit.

The whole of it rests on three foundations: one rule for time, speed and distance, one family of rules for angles, and a handful of unit conversions to move between them. Every application that follows, from working out a contact’s CPA to timing a deceleration, is one of these three wearing working rig. Learn the foundations properly and the applications cost nothing to carry.

One caution before the methods. Everything here is con-grade, not fixing-grade. These approximations are for judging a developing situation, backing your seaman’s eye with a number, and catching the plan going wrong early. They do not replace the plot, the ARPA, or a fix, and BR45 makes the same point about its own rules of thumb. Where a figure matters legally or navigationally, get it properly.

Foundation one: the six-minute rule

A knot is one nautical mile per hour. Six minutes is a tenth of an hour. So in six minutes, a ship covers a tenth of her speed in miles: at 12 knots, 1.2 nautical miles; at 20 knots, 2.0. That single division by ten is the six-minute rule, and it is the backbone of every time and distance problem on the bridge.

For shorter distances, work in yards over three minutes. One knot is 2,025 yards per hour, close enough to 2,000, which makes one knot very nearly 100 yards every three minutes. So in three minutes a ship covers her speed times 100 in yards: at 14 knots, 1,400 yards; at 6 knots, 600. Per minute, a knot is about 34 yards, and if you prefer round numbers, almost exactly 100 feet.

PRACTICAL NOTE: The two clocks

Carry both versions and pick by scale. Miles and tens of minutes: six-minute rule. Cables and single minutes: three-minute rule, remembering a cable is 200 yards, so speed times 100 in yards is speed divided by two in cables. A ship at 16 knots covers 8 cables in three minutes.

Worked example. You have 7 cables to run to the wheel-over position at 14 knots. Seven cables is 1,400 yards. At 14 knots, you cover 1,400 yards in three minutes. Three minutes to wheel-over.

Foundation two: the radian rule and the sin rule

The radian rule is the angle-to-distance converter. From the definition of a radian it can be shown that at one nautical mile, 1° subtends 35.25 yards (32.32 metres), and with small angles at reasonably short ranges, 1° may be taken to subtend 35 yards at one mile without serious error (BR45 Vol 1, para 0221). BR45 Volume 6 rounds the same figure the other way for bridge use: approximately 33 yards, or 100 feet, or 30 metres per degree at one mile (BR45 Vol 6, para 0416b). Both are doctrine. Volume 1 gives the exact figure for pilotage; Volume 6 deliberately rounds down because, as it notes, multiples of 33 do not make for easy mental arithmetic under pressure, and working in feet only requires multiples of 100 (BR45 Vol 6, Table 4-3 note).

The rule scales linearly with range. Yards per degree is 33 times the range in miles: at half a mile, 16.5 yards per degree; at one mile, 33; at two miles, 66 (BR45 Vol 6, Table 4-3). Run it in either direction: degrees times yards-per-degree gives offset, offset divided by yards-per-degree gives degrees.

The sin rule is the same idea for angles too big to call small. For any angle up to about 60° off the bow, the sine of the angle is close to the angle in degrees divided by 60. So the lateral displacement of an object sitting an angle off your bow is:

displacement = range × (degrees off the bow ÷ 60)

At 30° the rule is exact, since sin 30° is precisely one half, and at 45° it reads about six per cent generous, which for con work errs the right way. An object one mile away and 30° off the ships head is 30 ÷ 60, one half, of a mile displaced: 5 cables.

PRACTICAL NOTE: One rule, three faces

The radian rule, the sin rule and the track-regain calculation in this chapter are a single mathematical fact used in three directions. Small angle, short range: 33 yards per degree per mile. Bigger angle: degrees over 60, times range. Wanting distance along track instead of across it: the same fraction inverted. If you remember only that a degree is a sixtieth, the rest reconstructs itself.

A sobering application of the same rule: radar bearings. A bearing error of 1° is equivalent to 35 yards at one nautical mile, or about one cable at six nautical miles (BR45 Vol 1, Ch 22). That is why radar ranges are trusted and radar bearings are treated with caution, and the radian rule is the reason in one line.

Foundation three: the conversions

These need to be automatic, because every method above assumes them: a nautical mile is 2,025 yards, called 2,000 for mental work; a cable is a tenth of a mile, 200 yards, about 185 metres; a knot is about 34 yards or almost exactly 100 feet per minute, and very nearly half a metre per second. And when two ships are on reciprocal courses their speeds add, while on the same course they subtract, so it is the relative speed you feed into the six-minute rule, not your own. A 6-knot speed advantage on another ship closes 600 yards every three minutes, regardless of what the water sees.

Application: how long to get there

The everyday one. Distance to run, speed known, time wanted. Choose the clock by scale: miles through the six-minute rule, yards and cables through the three-minute rule.

At 15 knots, how long to cover 5 miles? Fifteen knots is 1.5 miles per six minutes; 5 ÷ 1.5 is 3⅓ six-minute blocks, so 20 minutes. At 10 knots, how long to cover 3 cables? Ten knots is 1,000 yards per three minutes; 600 yards is three fifths of that, near enough 2 minutes. The arithmetic is deliberately dull. The skill is picking the clock that makes it dull.

Application: CPA by the sin rule

For a contact stopped in the water, or on a reciprocal course down your intended track, her CPA is the sin rule read directly:

CPA = range × (degrees off the bow ÷ 60)

A stopped fishing vessel at one mile, 30° off the bow, will pass 5 cables clear if both of you hold on. A buoy 15° off the bow at two miles passes 2 × 15 ÷ 60, half a mile: 5 cables again.

The restriction matters. The rule maps degrees off the bow to lateral offset from your track, which only describes the future if that offset is not itself changing. A stopped contact stays where she is; a reciprocal-course contact slides down a lane parallel to your own; but a crossing ship with her own way on is generating bearing movement of her own, and the number the rule gives you describes where she is, not where she will be. For a crossing contact, take a compass bearing and watch it, which is Rule 7’s method, and leave the sin rule out of it.

Application: regaining track

Displaced from track and steering back in, the sin rule runs backwards. Distance to run to regain track is the offset divided by the fraction, which is to say:

distance to run = offset × 60 ÷ degrees steered in

Displaced 500 yards to starboard of track, steering in 12°: 500 × 60 ÷ 12 is 2,500 yards to run before you are back on. And the six-minute rule prices the delay: at 12 knots, 2,500 yards at roughly 400 yards per minute is a little over 6 minutes. If that is too long, the same equation shows the cost of impatience the other way: doubling the cut to 24° halves the distance to 1,250 yards, at the price of a bigger alteration to swing on and off.

Application: the deceleration bearing

This is the one that looks like wizardry from the bridge wing and is nothing but the two foundations chained together. It comes from stationing, and BR45 Volume 6 builds it in two steps.

First, the deceleration distance. To find the distance over which the ship will decelerate, subtract the ordered speed from the stationing speed and multiply by the ship’s loss-of-speed figure in yards per knot; a Type 23 frigate decelerating at 33 yards per knot, reducing from 25 to 15 knots, takes 10 × 33, 330 yards (BR45 Vol 6, para 0416a). Your own ship’s figure comes from her Navigation Data Book, and gain and loss of speed data for warships is in BR45 Volume 7. The figure is the input to the method, not part of it.

Second, the radian rule turns that distance into a bearing. Taking station abeam, you want to finish decelerating as the guide comes onto the beam bearing, so you must start the reduction when she bears the deceleration distance early, converted to degrees at the stationing range (BR45 Vol 6, para 0416b and Table 4-3). At one mile that 330 yards is 330 ÷ 33, 10°; at half a mile, 330 ÷ 16.5, 20°.

Worked example at OOW manoeuvre scale. You are closing another ship from astern with a 6-knot speed advantage, coming up on her port quarter so that she lies to starboard of you, to stop abeam of her at 500 yards, wanting her to bear 090 when you get there, and your ship is credited with 10 yards per knot of deceleration. Deceleration distance: 6 × 10, 60 yards. At 500 yards, a quarter of a mile, a degree subtends about 8 yards. Sixty yards is 60 ÷ 8, 7.5°, rounded up for safety to 8°. So you order the reduction as she bears 082, and the last of the overtake washes off as she comes onto 090. Round the degrees up, not down: decelerating early costs a few seconds of station; decelerating late costs an embarrassing swim past her bridge windows.

Watch which way the bearing is drawing before you apply the degrees. With her to starboard the bearing increases towards the final bearing, so the deceleration bearing is the final bearing minus the degrees; with her to port it decreases towards it, and the same 8° would go the other way, an ordered reduction at 278 for a final 270. The degrees always come off on the approach side of the final bearing, which is to say before it in the direction the bearing is moving, and stating the side out loud when you set the problem up is what stops the sign error.

Application: rate of turn and radius

Rate of turn, speed and turning radius lock together by the same radian logic, and the mental form is:

rate of turn (degrees per minute) ≈ speed in knots ÷ radius in miles

The exact factor is 0.955, close enough to one to ignore. At 12 knots on a half-mile radius, about 23° per minute; wanting a one-mile radius at 18 knots, order about 18° per minute. Inverted, radius in miles is speed over ROT, which is the quick sanity check on whether the turn you have planned is one the ship can actually make in the water available.

Application: dipping range

At night the first hard information a light gives you is the moment it lifts over the horizon. The distance of the sea horizon in a standard atmosphere is approximately 2.08 times the square root of the height in metres, in nautical miles (BR45 Vol 1, Ch 8). A light dips or rises at the sum of two horizons, yours and its:

dipping range ≈ 2.08 × (√height of eye + √height of light), heights in metres

Height of eye 16 metres and a light charted at 25 metres: 2.08 × (4 + 5), about 18.7 miles. The square roots are why it pays to remember the perfect squares. BR45 adds the caution that a rising or dipping range can only ever be approximate, may be used as a position line only with particular caution, and never relied upon in isolation (BR45 Vol 1, Ch 22): height of tide and abnormal refraction both move it.

Application: making a timing gate

Required speed of advance is distance to run divided by time remaining, and the six-minute rule turns it mental: divide the minutes remaining by six to get blocks, then divide the miles by the blocks to get miles per block, which times ten is the speed. Eighteen miles to the gate and 80 minutes to do it: 80 minutes is 13⅓ blocks; 18 ÷ 13⅓ is 1.35; required SOA 13.5 knots. Run it the other way after every delay: the pilot boarding late by 12 minutes at 15 knots planned has cost 3 miles of advance, and the recovery speed over the remaining distance follows the same division.

Application: doubling the angle on the bow

An old single-object fix that costs nothing but attention. Note the angle on the bow of a fixed object and read the log; when the angle has exactly doubled, the distance run between the two observations equals your distance from the object at the second observation, because the triangle formed is isosceles. The tidiest case is 45° doubling to 90°: the run between them is your distance off abeam. It assumes a steady course and speed and no appreciable set, so treat the answer as an estimate with a position line’s worth of authority, not a fix’s.

Practising it

None of this stays sharp by being read. The arithmetic has to be quicker than the situation, which means drilling it until the methods run without conscious effort, and the Mental Maths Drill on the revision side generates randomised problems across every method in this chapter, marked with working shown. Ten minutes against the clock, a few times a week, is what turns a chapter you have read into a con you can trust.