Terrestrial & Coastal Navigation

A fix is only as good as the geometry behind it. Three position lines from bearings, ranges or transits, laid off from known objects and crossed on the chart, give you a fix; how much you trust that fix depends almost entirely on the angles at which the lines cross. Two lines meeting at a fine angle — say 20° apart — can shift the plotted position by a mile or more for a perfectly ordinary half-degree error in either bearing, because a small angular error swings a long way at a shallow crossing angle. The same half-degree error on lines crossing between 60° and 120° barely moves the plotted position at all, which is why a good navigator picks objects spread around the horizon rather than the three that happen to be easiest to identify.

Three lines from three objects rarely meet at a single point — they form a small triangle, the cocked hat, and the true position is taken as somewhere inside it, conventionally the centre, or the corner nearest danger if you are being cautious. A large cocked hat is an honest fix telling you it is uncertain. A vanishingly small one, especially from bearings taken and plotted in a hurry, should make you suspicious rather than pleased: it is a common habit, under pressure, to nudge the third bearing by a degree so the triangle "closes" neatly, which produces a fix that looks excellent and is actually fabricated.

Dead reckoning is the simplest possible statement of where the ship should be: take the last known position, apply the course steered and the distance run from the log, and plot the result. It uses nothing else — no current, no wind, no leeway. That is deliberate, because it encodes only what you actually commanded the ship to do, which makes it a useful sanity check: if a fix disagrees wildly with the DR, one of the two is wrong, and it pays to find out which before trusting either.

The estimated position goes a step further and tries to say where the ship probably is, by adding what you know about the water moving beneath her and the wind acting on her hull. That means layering on the tidal stream or current — its set and drift — for the time elapsed, and separately allowing for leeway, the sideways slip caused by wind on the ship's windage. An EP is a better estimate than a DR, but it remains an estimate built on estimates: the current you used may be a tidal-atlas figure for an average spring day when today is closer to neaps, and your leeway allowance is a judgement, not a measurement.

The examination slip that costs marks here is a labelling one: plotting a position that includes an allowance for current or leeway and then calling it a DR. The two are not interchangeable, and an examiner reading the symbol expects the working behind it to match — a DR symbol with a leeway correction folded in is wrong regardless of whether the arithmetic inside it happens to be correct.

Finding the course to steer is a geometry problem before it is an arithmetic one. Start by drawing the track you actually want to make good over the ground — the straight line from where you are to where you are going. From your starting point, lay off the current's set and drift as a vector for a convenient interval, usually one hour. From the end of that vector, swing an arc with radius equal to the ship's speed through the water for the same interval, and find where that arc cuts the intended track. The line from the end of the current vector to that cutting point is the course you must steer through the water; the length of track between your start point and the cutting point, divided by the time interval, is your speed made good over the ground.

That construction gives you a true course — a direction relative to the sea and the chart, nothing more. Two further corrections turn it into an instruction a helmsman can actually steer. First, leeway: wind pressing on the ship's side pushes her downwind of her heading, so if you simply steered the course from the diagram, the ship would end up down-wind of the intended track. The correction is applied into the wind — you steer that many degrees towards the direction the wind is coming from, not away from it, so the leeway drift cancels back onto the desired track.

Second, compass error: the course from the diagram, and the leeway correction applied to it, are both true directions, and you need a compass course to hand to the helmsman. This is the reverse of the familiar CADET rule for turning a compass bearing into a true one. Going from true to compass, you subtract an easterly error and add a westerly one — the opposite sense to compass-to-true, because you are undoing the same error rather than applying it.

The sailings are the trigonometry of a rhumb line, broken into pieces you can look up or calculate quickly. Departure is the east–west distance made good, expressed in nautical miles, and it relates to a change in longitude through the cosine of latitude — a mile of longitude is only a full nautical mile at the equator, and shrinks steadily towards the pole, so the same departure corresponds to a larger swing of longitude at high latitude than at low. Plane sailing takes a course and distance and resolves them into a change of latitude, distance times the cosine of the course, and a departure, distance times the sine of the course, treating a short stretch of the earth's surface as flat, which is accurate enough for the runs made between fixes on a coastal passage.

Two further distances belong in the same family because they share the same square-root-of-height relationship, not because they are sailings in the strict sense. Distance to the sea horizon depends on the observer's height of eye; the range at which a light is first seen at night depends on the light's charted height added to the observer's height of eye — the "dipping distance", the range at which a light rises above, or dips below, the horizon as you approach or leave it.

Speed made good — distance actually covered over the ground, divided by time — is not the same figure as the speed shown on the log, which measures speed through the water. Comparing the two after a run is one of the simplest ways to detect that a current is acting on the ship, and roughly how strongly, without waiting for the next fix.

Height of tide is the vertical distance between the sea surface right now and chart datum, and you need it to turn a charted depth into the actual depth of water under the keel at a given moment. Between the charted extremes of high and low water, the sea does not rise or fall at a constant rate — it moves fastest around the middle of the rise or fall and slows near each turn of the tide, tracing a shape close to a cosine curve. The standard method is to interpolate using that curve, whether the tidal curve printed for the port or the standard curve where no local one is available, entering it with the time before or after high water and the day's range, and reading off the height of tide.

The rule of twelfths is a mental-arithmetic shortcut for the same curve: it assumes the range divides into twelfths rising 1, 2, 3, 3, 2, 1 across the six hours between low and high water. It is close enough for a quick check at the chart table, but it assumes a textbook symmetrical semi-diurnal tide, which real ports do not always deliver — near springs and neaps, or anywhere the tide runs diurnal rather than semi-diurnal, the actual curve can diverge from twelfths by a significant margin. Use it to sanity-check a curve reading, never as the calculation you would defend afterwards.

Every depth and height on a chart is measured from a datum, and the datums in everyday use are not the same surface. Charted, or sounded, depths are given below chart datum, which is set low enough — close to the level of lowest astronomical tide — that the sea is very rarely below it, so a charted depth is close to the worst-case shallowest water you will find. Drying heights, shown for areas that uncover at low water, are given above chart datum, the opposite sense, because they describe how far a bank or rock stands out of the water rather than how far under it.

Vertical clearances — the height available under a bridge, cable or other overhead obstruction — are referenced to a different surface again: highest astronomical tide, the highest level the tide can be predicted to reach under average meteorological conditions. That is the deliberately pessimistic choice for air draught, because it gives the smallest clearance you should ever expect, whereas chart datum would give a falsely generous figure most of the time. Taking a clearance figure and reducing it by today's height of tide above chart datum, as you would for depth, silently overstates how much room is actually there under the bridge, because the reference surface is already close to the top of the tidal range.

Chart symbols and colouring encode these distinctions before you do any arithmetic at all: the tint marking water that dries or is very shallow at chart datum, soundings printed in ordinary type as depths below chart datum, and a bridge or cable's clearance printed as a single figure referenced to HAT unless the chart states otherwise. Reading the wrong datum for a given feature, rather than making an arithmetic slip, is how vessels find rocks and bridges that were marked all along.

The three worked examples below move through a course-to-steer problem with leeway and compass error, a tidal-clearance decision with a real go/no-go answer, and a running fix built on the bow-and-beam relationship — each one laid out the way you should show your working in the exam.