Module 03 · The line of position
Reading notes · Module 03 · Where you are
Four ways to know where you are not: a bearing, a transit, a sounding and a distance — and which of them to believe.
Open the interactive chapter →The same material with the chart, the drawings and the exercises.
What a position line is
One bearing, and what it buys you
You take one bearing of Brough Tower and it reads 010. Draw it back out of the tower and you get the line on the chart — 7.1 miles of it, running away to the south.
You are somewhere on that line. Not at its start, not at its end, not anywhere in particular along it — somewhere.
And here is the part worth having: you are nowhere else on this chart. Every square mile of water either side of that line has just been ruled out, and you have not touched an instrument since the compass.
Three boats, one bearing
Now watch what the line does not tell you. Three boats are out there right now, all on that same line, and all three have just taken the same bearing of the same tower.
The first boat is 2 miles off the tower. Hand bearing compass on Brough Tower, corrected to true: 010.
The second is a mile further out. Same tower, same compass, and the reading has not moved: 010. It has no reason to — the tower is still in the same direction.
And the third, a mile further still. 010 again. These three are 2 miles apart end to end and their compasses agree to the degree.
So which of the three are you? The bearing cannot say. That is what one line is worth: it gives you the line and leaves you to find your own place on it.
The bearing
A bearing is a direction, and it starts at you
You are here, and Skelvoe Tower is over there. Hand bearing compass up, the tower in the notch, and the card settles: 234. That is the whole instrument.
A bearing is a direction, and it runs from you to the object. Not the other way. The grey line is what your eye is doing over the top of the compass, and 234 is the angle it makes with north.
What the compass gives you is magnetic, and the chart is drawn to true. Correct it the way module 02 taught you — variation off the chart, deviation off the card — and you have a true bearing to work with. That part is not taught again here.
And two more things go in the book while you are at it: the time, to the minute, and the log reading. Every position line gets a time and a log against it, as part of taking it — not as a tidy-up afterwards. Why, at the end of this part.
And that is why you draw it backwards
Here is the trap, and it catches everybody once. You have 234 true. The one thing you cannot do is start the line where you are, because you do not know where you are — that is the entire reason you took the bearing.
The tower does know where it is. So the line starts at the tower and runs out along the reciprocal — 234 − 180 = 054 — back towards you.
There it is, running off this view and 9.0 miles down the chart. You are somewhere on it. The line is the same line either way round; what changes is that this end is nailed to something real.
A bearing, worked through
Two towers, and both of them are on the chart. You are lying off Brough Tower, and away to the south-east stands Wardhill Tower — 170 m of it, the highest thing on this coast.
Watch the whole thing done once. Every number below comes off the chart, and at the end the line you draw lands on a spot the chart already knows — which is how you will know you did it right.
You take the bearing. Hand bearing compass up, Wardhill Tower in the notch, and read it: 162 by compass. A hand bearing compass is held away from the boat's iron, so it has no deviation — what it gives you is magnetic and nothing else.
Correct it to true, the way module 02 taught you. The variation here is 6°W. Going up from the compass to the chart, west comes off — so 162 magnetic is 156 true. That is the direction you will draw.
And now turn it round, which is the step everybody forgets. You do not know where you are; the tower does. So the line is drawn from the tower, along the reciprocal — 156 − 180 = 336 — back towards you.
Lay it and draw it. There is the line, out of Wardhill Tower on 336. You are somewhere on it — one bearing never says more than that. And look where it goes: straight through Brough Tower, 6.21 M away, which is exactly where you said you were. The chart just checked your work.
The transit
Two things in line, and no instrument at all
Look back along the boat and there are two things in line: Skelvoe Tower in front and Halkirk Spire 1.45 M behind it, one sitting exactly over the other.
That is a position line, and you did not touch an instrument to get it. No compass, no protractor, no correcting. While the two stay in line you are on the line that runs through them — 039 one way, 219 the other — and the moment they open, you are not.
And it only works from one side. The line runs on past the spire as well, but that half is over the land the two of them stand on. Measured on this chart: out beyond the tower there are 2.5 miles of water you can be in; the other way, none at all. A transit you cannot float on is not a position line.
Why it is the best line there is
A bearing carries the compass inside it. Variation, deviation, the card swinging in a seaway, your own eye on the notch — every one of those goes into the line before you draw it. Topic 2 measured what that costs: one degree is 32 m at one mile and 97 m at three, and it grows the further off you are.
A transit has no compass in it at all. It is not that it is measured better — it is that nothing is measured. Two objects are either in line or they are not, and your eye settles that at a hundred metres and at three miles just the same.
So it does not open with distance. That is the whole of its advantage, and it is why every harbour that can put two marks up does it. When a transit is available, it beats anything you can measure.
The transit is not the pair: it is the line that leaves it
Stand back and the transit turns into what it really is: not two marks on a headland, but a line running out to sea from them, on 039.
The green stretch is the part you can be on. It starts 1.0 M out from the tower, because the first mile of the line is over the headland the tower stands on, and it runs 2.5 M before the water gives out. The line is longer than the useful part of it, and knowing which part is which is the navigator's job, not the chart's.
Anywhere along that green stretch, the two marks are in line and you are on the position line. Where along it, the transit does not say — the same as every other single line in this module.
The sounding
The line you did not draw, and it was there all along
Stand back off the chart and look at the colours. Skelmar is printed in four shades, and each one is a depth: green dries, dark blue is under 5 m, pale blue is 5 to 10, and Where one shade meets the next there is a contour: a line joining everywhere of the same depth, and the chart draws three of them.
A contour is a line, and it is a position line. The echo sounder gives you a number, that number puts you on one of those lines, and you did not use the compass and you did not need to see anything at all. This is the only line in the module you can get in fog, at night, with the visibility down to the pushpit.
But look at the shape of them. A bearing gives you a straight line and a transit gives you a straight line. A contour wanders — it goes round every headland and into every bay — and that is going to cost you something on the next screen.
7.4 metres, and where that leaves you
The sounder reads, you take the tide off it, and you are left with 7.4 m of charted depth. That is between the 5 m contour and the 10 m contour, so you are somewhere in the pale blue — and the chart has drawn that band for you already.
The rings mark them: there are 6 separate pieces of pale blue in this view. The 5 m contour wanders round every headland and back out again, and every time it does, it opens another place where 7.4 m is a perfectly good answer. The boat marked here is in one of them. It could be in any of the others.
That is what makes the sounding the weakest of the four. A bearing leaves you on one line; a sounding can leave you on a line that comes back and cuts you again three headlands away. It is still worth having — it just needs something else with it.
And the tide had to come off first. The sounder measures from the waterline and the chart measures from chart datum; the difference is the height of tide, and getting it is module 04. Until then the number is handed to you corrected.
The distance
Two ways to a distance, and neither gives you a line
There are two ways to get a distance without an instrument that plugs in. One is a vertical sextant angle: you know how high the object is, you measure the angle it subtends, and the two give the range. The other is rising and dipping: you note the moment a light first lifts over the horizon, and that moment is a distance.
And here is what makes them different from everything so far. A bearing gives you a line. A transit gives you a line. A sounding gives you a line that wanders. A distance gives you a circle — every point the same range from the object, all the way round.
The word «line» stops being the right word here, and it is worth saying out loud. A position circle behaves the same way as a position line in the one respect that matters: you are somewhere on it and nowhere else, and one of them is still not a fix.
A vertical angle, and the circle it draws
Skelvoe Tower stands 68 m above the sea, and the chart says so — the number in brackets under its name. Bring the sextant up, put the top of the tower on the waterline below it, and read the angle: 1.05°.
Height over angle gives range. 68 m subtending 1.05° puts you 2.0 M off — and that is the whole circle drawn here, not a point on it. The sextant does not know which way the tower bears; it only knows how far.
The boat is on the circle, and so is every other place on it. Put a bearing with it and the two cross; on its own it is one position line with a curve in it.
Rising and dipping, and which of the two runs out first
A light has two ranges and they are not the same thing. Its luminous range is how far the lamp throws — the number on the chart. Its geographic range is how far away it is still over your horizon, and that depends on its height and on yours. Standing in the cockpit your eye is about 2 m up, and this course works to that.
Hoyness is only 6 m up, and reaches 10 M. It goes under your horizon at 8.0 M — before the lamp gives out. The horizon decides, and the light is useful right up to the moment it dips.
Auskerry is 14 m up and would stay over the horizon out to 10.7 M — but the lamp stops at 8. The lamp decides, and the extra height buys nothing. Look at the chart: its 10.7 M circle is not drawn, because it does not fit on this sheet at all. That is the point made for you — its horizon is further away than the whole chart is wide, and the lamp ran out long before.
The smaller of the two is the one you get. That is the whole rule, and these two lights are the two ways round it goes.
Which one to trust
Four lines, and they are not worth the same
You now have four ways to get a position line, and a navigator has to know which to believe when they disagree. The order is not a list to learn: it falls out of what each one has inside it.
The transit carries no instrument at all. Two objects are in line or they are not; nothing is read, so nothing can be read wrong. It is first, and it is not close.
The vertical angle carries a sextant, which is read to a minute and is honest — but the sum it feeds turns a small angle error into a large distance error as soon as you are any way off. Good close in, poor far out.
The bearing carries the whole compass: variation, deviation, the card swinging, your eye on the notch. Every one of those is in the line before you draw it, and the further away the object, the more it costs.
And the sounding carries the tide — a number you had to get from somewhere else before the reading meant anything — and it gives you a line that wanders and can cut you in several places at once. Last, and still worth having when the visibility is gone.
The same three errors, at one mile and at three
Here are three of them measured against each other, drawn on the chart at the size they really are. All three are errors in the same thing: how far off the true line the position ends up.
Start at one mile. One degree of compass error puts the bearing 32 m off the true line. Half a degree on the sextant — with the tallest thing on this coast, Wardhill Tower, 170 m, which is the best case there is — puts the distance 0.11 M out, about 196 m. Already six times as much, and neither of them is a small error. The transit is the tick with no bar: no instrument, nothing to be wrong.
Now stand back to three miles, and look what happened. The bearing has grown to 97 m — three times, because the error is an angle and it opens. The sextant has grown to 1.20 M, which is 2219 m: not three times but eleven, because the angle it reads got small and half a degree is now most of it. And the transit is still a dot.
That is the rule you take away: every one of them gets worse with distance except one. So take the transit if there is one; take the sextant close in; take the bearing of the nearest decent object, not the most obvious distant one.
Closing
The four of them, on one chart
Here they are together, and you can see the shapes are not the same. The dashed line out of Brough Tower is a bearing. The green stretch off Skelvoe is a transit, straight too but nailed at both ends by two objects instead of read off a compass. The ring round Skelvoe Tower is a distance — a circle, not a line. And the marked patches are soundings: the same band, in four separate places, because a contour wanders.
Two straights, a circle and something that meanders — and every one of them does exactly the same job. It tells you where you are not.
That is the sentence this part opened with, and now you have four ways to earn it: a single position line does not put you anywhere. It rules out everything that is not on it, which on a chart this size is nearly everything — and that is worth having, and it is not a fix.
One line rules out. Two cross — and sometimes twice.
You can now get a position line four ways, you know which of them to reach for first, and you know what each one costs you at three miles instead of one. That is the whole of part 1, and none of it required anything that plugs in.
Look at the chart once more. The transit and the 2.0 M circle are both drawn, and they cross. Where two position lines cross, you are at the crossing — and that is a fix, which is a different animal from anything in this part.
Except look again: they cross twice. The transit runs straight through the middle of the circle, so it cuts it on both sides — and the two crossings are 4 miles apart. Two lines, a proper fix, and you still have to decide which end of the boat you are. Nothing you learned in this part settles that.
That is where part 2 starts. The angle two lines cut at — why lines meeting nearly head-on give a fix you should not trust, and why this one, cutting at nothing at all, gives two. And the cocked hat: what it means when you lay three lines and they make a small triangle instead of meeting at a point.
And one more thing, which is why you have been writing down the time and the log with every line you take. A position line is only true at the moment you took it. At five knots, five minutes is half a mile — so a line you laid a quarter of an hour ago is not where you are now, and neither are you.
That is also what saves you when there is only one object in sight. An old line can be carried forward by the course and distance you have run since, and crossed with a fresh one off the same object — two lines from one mark. It is called a running fix, it needs the time and the log you have been noting, and it is part 3. It is named here and taught there.
You have spent this part learning that one line is not enough. Part 2 is about how many more you need, how they should cut, and how much to believe the answer.
Open the interactive chapter →The same material with the chart, the drawings and the exercises.