Directions and angles

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Points of the Compass

	Most people know the four main points of the compass, North, South, East and West (see left). These are known as the cardinal points of the compass. If you can't remember which way round 'West' and 'East' are, they read 'we'. However, there are 32 points of the compass. The best way to see this is to build it up a bit at the time. First draw lines half-way between the cardinal points (see right).
	Now label them (see left). Half way between North and East is North East, half way between North and West is North West, half way between South and East is South East, half way between South and West is South West. The North or South always comes first, and the East or West comes second. These eight points (N, NE, E, SE, S, SW, W, NW) are known as the principal points of the compass. Now draw more lines between the existing lines (see right).
	Half way between East and North East is East North East. The single point (North, South, East, West) comes first, and the double point (North East, North West, South East, South West) comes second. This does make an unambiguous term, although you may be getting confused at this point! You may prefer to think of it as East of North East. This makes sixteen points so far. Draw more lines between the existing lines (see right).
	To label these last points, choose an unnamed point. It will be next to one of the eight principal points (with a single name or a double name). Write that down followed by 'by'. This new point is either North, South, East or West of this point (going either clockwise or anti-clockwise). So write that down. So the point clockwise of North is North by East. The point anti-clockwise of South West is South West by South. The complete 32 points starting from North and going clockwise are: N, N by E, NNE, NE by N, NE, NE by E, ENE, E by N, E, E by S, ESE, SE by E, SE, SE by S, SSE, S by E, S, S by W, SSW, SW by S, SW, SW by W, WSW, W by S, W, W by N, WNW, NW by W, NW, NW by N, NNW, N by W, N This is all rather complicated, and there isn't room on the left to mark them! But if you rest your mouse pointer on one of the blue dots on the left, it will tell you what that point is.

Ordinal points of the compass

The cardinal points of the compass are North, South, East and West. I have been asked about ordinal points of the compass. I don't think this is a British phrase - rather an American one. But there seem to be several possible meanings:

• The 'cross points' from the cardinal points - NE, NW, SE, SW
• Any of the principle points of the compass - N, NE, E, SE, S, SW, W, NW
• Any point on a compass rose which is marked with a small line rather than a big line - so not north, south, east or west, but anything else (an example on the web gives a compass rose marked in 30 degrees points, and calls these ordinal points!)
• A point on the compass (such as NE) rather than a degree reading

I would be interested to hear from anyone who could explain this. It all sounds very odd to British ears, since cardinal numbers are the 'normal' numbers - one, two, three, etc. - while ordinal numbers are the numbers in order - first, second, third, etc. So I don't see how you can divide up a compass rose in this way. It does make sense to call north, south, east and west the cardinal points, since those are the main points of the compass, and 'cardinal' means 'of fundamental importance'.

Examples of compasses

These are conventional compasses. The compass on the left has a fixed dial. Normally you would turn it until the needle is pointing to North on the dial, but you can see that I haven't bothered. Once the needle is aligned, you can read off the directions.

The right-hand compass has a pale blue floating dial. There is a bubble of air. You can see that the red arrow points North, and that West and East are automatically in the right place. This makes it much quicker to use. It is surrounded by a black dial with degrees, if you want to use this method instead. But these are fixed, so you would need to rotate the compass to use them. I haven't bothered to do this, either.

If you are walking with a map, and wanted to use a compass to get your bearings, you first get the compass to point to North. Then you can tell where various parts of the landscape are, for example "That mountain is South West of me." If you want to go in a particular direction, you find a landmark in that direction, put the compass away, and head for that landmark. When you reach it, you use the compass to find another landmark in the correct direction, and continue.

If you look at the pictures below, you may notice that West and East seem to be reversed. This is not a mistake!

	Imagine if you were in a boat on the sea. You can't use landmarks to fix your direction as there aren't any in the open ocean. You need to fix the compass to a surface so you could look at it without using your hands, which you need for steering. Also, the boat may be constantly driven off course, by wind or current, so you also need to look at it all the time to steer by. However, if you have a fixed indicator dial, once you have fixed the compass, the indicator dial would always point the same way, since that is fixed to the compass. Only the needle moves. So North on the dial is always the way the ship is pointing, regardless of where real North is. You use the needle to give the direction that you are steering. Now, if you are heading North, the needle will point North, and that is fine.
	But if you wanted to go East and you used a conventional compass, when the needle points to the East, the North on the compass (which is the direction that the boat is facing) is actually facing West! If you don't understand this, then look at the compass on the right. The red needle points North, and the figures in blue show the other cardinal points of the compass relevant to this. These are the 'real' directions. The letters in black show the compass points marked on the compass, and here 'North' is the direction that the boat is steering in. You steer the boat so the red needle points to the black 'East' (remember that you can't see the blue letters). This makes the boat steer towards to blue West, which is 'real' West'. Not good!
	So either you have to remember to reverse West and East all the time (which is silly) or the compass itself must reverse them for you (see left). This looks very strange when steering North, but it doesn't matter, as you are not concerned with East and West then, only North, which is OK. When you want to steer East, then you find that steering the boat so the needle points to the black mark for East does really take you in the direction East (see right).

The compass above looks like a surveying compass. Here, you aren't fixing the compass to a vehicle, but you are sighting on an object using the sight-lines (which are fixed to the body of the compass, and so to the direction of North), and using the needle to read off the direction, so you get similar reversal problems.

This problem is resolved nowadays by having a floating indicator dial. In these, the whole dial, with North, South, East and West on it, is magnetised, and floats in a liquid. This means that the whole dial floats round and points to North, and so the points are in the correct place without any turning of the compass. So you can determine your direction without touching the compass, and without the dial being reversed. You can easily buy these as car compasses. However, this requires a compass with a sealed container with the liquid in. Perhaps this was harder to manufacture before plastics. It may also be difficult to sight using such a compass.

Here are some photos of a compass with gradians. There were 400 gradians to a circle rather than 360 (see below). I suspect that this is a French compass as the letters are N (nord), S (sud), E (est) and O (ouest).

Other ways to give directions

Points of the compass are not in common use anymore. Now directions are given by degrees (see below). North is zero, and the degrees go clockwise. There are 360° in a circle. So East is 90°, and West is 270°. This has the advantage that all bearings are equally simple, but it doesn't sound so romantic. See conversion.

A correspondent sent this to me:
"It is interesting to note that to this day, whilst circles are divided into 360 degrees and compass courses are given in degrees (example; to steer East the course is given to steer 090) the older system of compass points is still very much used as a quick system of reporting bearings of objects relative to the 'ships head'.  thus if a lookout say, wanted to report a white light roughly 45 degrees on the Starboard bow he would call out "white light four points to starboard!"  As there are 32 points in a full circle this gives 8 in a quarter (in this case the Starboard beam) so four points would be roughly halfway between the bow and the beam, looking forward on that side. Very quick, and rough but the system does work - it's been in use for many years - since the days of sail, so unlike the modern idea of change for change's sake it is left well alone!   Why change a system that works?"
This is using points of the compass as a measure of angle rather than a direction. A point is 11.25°, so his 'four points' is 45°.

People use different ways to describe a direction. The compass points will only work if you know where North is. For example, on the South coast of England, they mark on the road N,S,E,W because everyone knows where the sea is, and roads tend to run along the coast, or head towards or away from the sea. A correspondent who wrote to me about paper used North to mean the top of a piece of paper, presumably because maps are supposed to have North at the top (although I've known some that don't - very disconcerting!)

Left and Right - However, when you are giving directions to a lost car-driver, then youdon't talk about "Turn North." You assume, rightly, that a lost driver is unlikely to know where North is. Indeed, you may not know yourself (but at least you aren't lost). You say "straight ahead" or "turn left". Some people find knowing their left from their right easy while other people have problems. But everyone knows if they are right-handed or left-handed. If you are wondering where 'left' is, then pretend to write something in the air. You will automatically use your writing hand, and you will know which that is.
It is notorious that the British (and a few other countries) drive on the left of the road while the rest drive on the right. The British have a reason for this. We drive on the left, because horses travelled on the left, and the reason for this is that you mount a horse on the left, and you don't want to mount a horse in the middle of the road! And you mount a horse from the left, because if you are wearing a sword, and if you are right-handed, the sword hangs from the left hip, and if you try to mount a horse from the right, it's liable to get between your legs, which is unpleasant when you sit on the saddle. Isn't that all logical? I wonder why the rest drive on the right. An Australian correspondent says: "I wanted to mention (because we drive on the left also) another historical reason for driving on the left which was to keep the right hand free to either shake hands or wave with passers by - or to engage in sword or gunplay with a passing enemy. Apparently this was a defining factor in Napoleon's conversion of captured countries to right hand driving (because he was left-handed - I am as well so I understand his reasoning!). And this is apparently another historical reason for the English and therefore Australian retention of left hand driving - because we weren't converted by the French!"
Now Napoleon being left-handed makes sense (it was all obviously the French's fault, as usual). It doesn't explain America, unless they were being deliberately anti-British but drew the line at metric measurements. I'm not so sure about fighting. When knights on horseback jousted (see right), they didn't pass right side to right side as you might expect, even though that side was where their spears were. They passed left side to left side, as that was where their shields were, and the shields were supposed to protect them. However, pistols would make more sense. If you were passing an enemy on the right and were right-handed, you risked putting a bullet through your horses head. But cavalry never seemed to use guns.
Sailors use starboard to mean the right side of the boat, and port to mean left. A mnemonic to help you remember: "There's some port left." This refers to port wine, which is red, and red lights show port, with green lights showing starboard. You may also find it helps if you know that 'starboard' comes from 'steer-board', the oar hung over the side of a Viking ship to steer it. It would be on the right-hand side as Vikings would be mostly right-handed. Apparently, most aeroplanes also use port and starboard, except planes in the Navy. If you have planes on a aircraft carrier, then they may be facing a different way to the direction of the ship. Port and starboard are always the left and right of the ship, so the planes are described as having right and left sides instead, to avoid confusion.

Clockface - 'Left' and 'right' are OK for simple turns, but you may wish to describe where something is, and want to describe an angle other than 90°. A simple way to do this is to give directions is to use a clockface, such as 'two o'clock', or 'half past ten'. This imagines that twelve o'clock is straight ahead. So two o'clock is 60° to the right. Once explained, anyone can use this system, while degrees tend to fluster people who think they are bad at Maths! But it's not quite as simple as that. The Oxford English dictionary said that in a plane, twelve o'clock could mean directly above you, so "Bandits at six o'clock" meant enemy aircraft below you. Planes work in three dimensions! There is a different meaning, used by artillery in World War II. An observation post guided guns firing or described where a shot had landed. It still uses the clockface, but twelve o'clock is north, rather than the direction that you're facing. Obviously the observation post and the gunners might be facing in different directions, but north stays as north.

To summarise this section, there are two ways of describing direction. You can give a fixed direction, such as North, or you can describe direction relative to which way you are facing, such as Left. We use one or the other naturally, without realising what we are doing, even it is not necessarily logical. For example, you are probably using a mouse or similar device now, without realising how silly it is. I used to introduce people to the internet, and that involved sometimes teaching them how to use the mouse. So I started by asking them to put their hand on the mouse and move it left and right, and watch how the pointer on the screen moved left and right. Then I asked them to move the pointer up the screen. One elderly woman lifted the mouse in the air. I was delighted! She was the only person to try to use the mouse logically. After all, why should moving a mouse forwards make the pointer move upwards? But you don't think about it, do you?

Degree

To describe direction, you need to describe an angle, even if you use a word such as East or Right. Degrees are the common way to describe angles. The abbreviation for 'degree' is °. There are 360° in a circle (or 'full turn'). This is a useful figure, since it can be divided in many ways, so if you want to divide a round cake into 3, it's 120 degrees, 4 pieces are 90 degrees, 5 pieces are 72 degrees, 6 pieces are 60 degrees, and so on. One degree is a very small amount, but if you want something smaller, then there are 60 minutes to a degree, and 60 seconds to a minute. So there are 1,296,000 seconds to a circle! The fascination with the number '60' came from the Babylonians, who had a base 60 number system.

Sailors now steer using compass courses given in degrees rather than points of the compass.

My father, who was a gunner in World War II, said that to estimate small numbers of degrees of direction, you stretch out a straight arm, with clenched fist, knuckles upwards. The distance between the knuckles of the first and second finger was about three degrees, while between the knuckles of the second and third, or third and little finger was about two degrees. My husband who studied astronomy, thinks that a little finger nail is about one degree if you hold your arm out-stretched.

If you want some practice with measuring angles in degrees, try visiting my website about angles.

Radian

In Mathematics, we meet another way of measuring angles - radians. There are 2 π radians to a circle, where π is the ratio of the diameter of a circle to its circumference (round its edge). π (pronounced 'pi') is a very strange number, approximately 3.1416, but however many places of decimals you write, you never reach the accurate figure of π, and the number sequences never start to repeat itself. This is called an irrational number, and π is, perhaps, the most famous example. I am tempted to call mathematicians pretty irrational themselves to use a unit where you not only don't get a whole number of them to a complete circle, you don't even get a rational number (or fraction)! They will defend themselves by stating that if you cut a slice of cake with angle of one radian, then its rounded edge is the same length as its straight sides (or the radius of the original circle). There are also formulae which are easier to deal with if you use radians rather than degrees. The abbreviation of 'radian' is 'rad'.

Mil

Another angle unit is the mil. This is a military unit for defining angles. The name derives from milliradian, and they are used because an angle of X mils is X metres wide at a distance of one kilometre. This means that if you drop a shell 200 metres to one side of the target (according to your map) and it's 4 km away, 200 divided by 4 is 50, so you swing your aim by 50 mils. Note that current OS maps have the magnetic and grid deviations shown in mils as well as degrees. There are 6283.1853 mils in a circle, but they seem to be rounded off to various values in use, such as 6283 and 6280. However, the US military made things 'simpler' by standardizing on 6400 mils in a circle. To make things even more interesting, the Russians, and perhaps others in Europe, use 6000 mils in a circle! A correspondent explains this:
"Mils were first used for military purposes by the German Wehrmacht during World War II. After the war, other armies started using mils instead of degrees. NATO armies adopted the German standard of rounding 6283 up to 6400. Warsaw Pact armies also adopted the system but rounded 6283 down to 6000. The relationship between width at the target, range from observer to target, and angle is described by the mil relation: a target one mil wide in your binoculars and one metre in width is one kilometre away. This relation, which is good both for range estimation and for giving corrections to artillery, is also known as the W = RM rule. Nevertheless, the W = RM rule is not exact - it is just a useful approximation when using angles up to about 400 mils."

Gradian

There is a metric measure of angle called the gradian. There are 400 to a circle, so 100 to a right angle. On the right is a gradians protractor. It is in the Museum of the History of Science in Oxford. The abbreviation for 'gradian' is 'grad'. I thought that grads was a historical curiosity but a correspondent told me:
"Grads are still in use, albeit in a restricted range of uses. They are still used a bit in parts of Europe and were common in photogrammetry not all that long ago (aerial photo measurement work)."
In fact, in scientific calculators, you are often given the choice of degrees, radians or grads, much to the bewilderment of people who have never heard of grads.
Another correspondent from Stockholm says
"A 400 degree compass is used by people that have a hobby of running through woods from point to point. In Sweden we call it Orienteering."
In Britain we call it Orienteering as well, but I think we use a conventional compass with 360 degrees.

Gradient

Another use of angles is measuring the gradient (or grade) of a road. (This has nothing to do with gradians, by the way). However, gradients are not measured using any of the above units. The grade of a road is measured as a percentage. For instance, if the road rises 25m for every horizontal distance of 100m, the road would be a 25% grade (you have travelled 25% vertically of the distance that you travelled horizontally). This 25% grade road would have an angle of about 14° from the horizontal. A 100% grade road would have an angle of 45°.

In Britain, we used to describe a road as, for example, one in ten. This was generally described as ten units along, horizontally, and one unit up, vertically. That would be a tangent measurement. However, it was usually done as ten units along the surface of the road (which was sloping, of course), and one unit up. That would be a sine measurement, and would be easier to do, since you could measure the 'along' distance actually on the road itself. For practical slopes, sine and tangent would be very similar, so it wouldn't matter which you used! The Highway Code give the sign on the right for a steep hill, although it also says that gradients may also be shown as a ratio, i.e. 20% = 1:5 so obviously Britain, as usual, is using both methods.

© Jo Edkins 2009 - Return to units index