So for some inexplicable reason you may not know exactly where you are on the map, and might consider it useful to find out. Hey, even Daniel Boone got a bit confused at times. To paraphrase from various versions - “I can't say as ever I was lost, but I was bewildered once for three days”. I too have been temporally bewildered on a handful of occasions - I'm pretty sure every one has - but in my defense I was generally dropped in the middle of nowhere from the back of a Unimog or Land Rover in places where a map could have been suitably replaced by a sheet of sandpaper (desert) or a cos lettuce (jungle). I digress.
Finding exactly where you are requires taking a resection (or triangulation). A resection consists simply of identifying 3 or more features such as spot heights or hills, radio masts, trig points and so on, taking compass bearings from your location to these points, plotting the back-bearing, and identifying the triangle within which you are located. Usually this will be within 100m or so (even better with good points). To do this you need a reasonable map, some broad sense of where you are (ie. are you actually on that map, and if so do you know what grid square(s) you are probably within!), and it helps to have a compass. No, I'm not being totally facetious there, you can do it roughly by eye also. This is a skill that should be continuously practiced.
Move to somewhere with a good vantage. It's easier to start from high ground yourself - you need to be able to see the points you are taking bearings to.
Orient the map North-South. It's always harder reading it upside down...
Locate high points as initial references in preference to low features. Hills, crags, spot-heights and so on are more distinct than creek junctions. Hills rarely change position and are usually obvious, but creek beds can change and junctions can be flat and relatively indistinct. Man-made features such as reservoirs are obvious exception.
Select points around the compass circle. Try to take bearings on points distributed around the circle. Optimally, points at 120°/2100 mils would be ideal to space them around the compass, but in reality right angles are more intuitive to deal with. Operationally, 20-30° or more will be ok. This spread is necessary to reduce the relative effects of small errors in taking the bearing, and provides a nice little triangle of uncertainty within which you can be pretty sure you are located.
Use points further away first to get yourself in the general vicinity. This is where your practice in estimating distances from the map comes in. You should at least know whether a feature is 1,2,3 up to 10 km away from you. The initial resection with points further away is particularly necessary when you are really not sure where you are. There will be greater error in the initial position because of measurement error in your bearings, but you should get your position to within a few 100m or better, depending on the distinctness of the reference point and your skill at taking bearings. Remember 1 mil subtends 1m at 1km, so if you are 200 mils off a point 2 km away, you will be 400 m off the point you are taking a back bearing from.
Take the bearing. Record, and calculate the back-bearing. The more precise you are at measuring the bearing, the better. This is also where you can see the benefit of readily distinguishable features. A bearing on a knoll will be subject to the uncertainty of 'where exactly is the highest point?', assuming of course a spot-height actually exists on the map.
Once you've worked out the ball park of where you are then use closer points to pinpoint your position. You should get your position to at least 100m (6-figure grid reference) or 10m (8-figure grid reference). As an aside, during war if bodies needed to be buried (eg. in Vietnam) it was usual practice to use 8-figure positions.
Transferring back bearings to the map
Now we get out the trusty protractor/romer, or use the compass baseplate to scribe the back bearings onto the map. Personally I prefer a protractor for the precision, but the baseplate is totally acceptable. From each of the postulated points, trace the back-bearing. If you have correctly identified the points, and are super-precise with your bearings, then the three lines should intersect at your exact position. The smaller this triangle of error the better.
Reality check - terrain association
As with pretty much everything in navigation, always question and check your assessment of where you are. So you have a nice tight triangle on your map - does this coincide with what you can see around you, and are all the landscape features you should expect to be there from the map actually there, where you would expect them to be? This is not being excessively paranoid. There is a tendency amongst folks to bluster the map into our version of where we think we are. Landscapes can have characteristic configurations as a function of its geology - it is quite possible to have 3 similar-looking points at similar angles from each other along a river valley, for example. To any statisticians out there, this can be likened to an iterative local optima. I digress.
Part of this uncertainty can be reduced by following my first suggestion - pick big, very obvious features further away first. There are going to be fewer of these combinations than smaller groups of features. Next, challenge your perception of looking at the map and predicting what features should be around you, on which bearing, and looking to see if in fact they are there. Conversely, look around and make sure any obvious feature is indeed present on the map.
Lets look at an example. I've been scrambling around on rock faces - not for any particular reason other than to get away from it all. I started from the refugi at Estanys de la Pera, travelling approximately southwest for a few km. I've now come up with the brilliant idea of linking up to the GRP to get to Naturlandia which is somewhere over to the west. This will be a lot easier if I figure out where the hell I am.
I know within a few grid squares where I should be. I see one big peak not far to the east-northeast, and another west-northwest. I'm pretty sure that one is the lookout at Port Negre. I'll start with those two. I shoot bearings of 330° and 60° to each of these, and calculate the back-bearings. I'm in Andorra, so the Magnetic to Grid difference is negligible, otherwise I would convert from Magnetic to Grid. Turning in the opposite direction to spread my points around the circle I look for another good feature. The only good candidate is a knoll on the left side of a saddle. It's a bit indistinct so I'm going to be cautious with it but I take a bearing of 218°. Looking at the map, I'm pretty happy that Port Negre and torre dels Soldats should be the obvious features given the grid squares I know I'm in. A cursory examination of the terrain supports this too - I'm overlooking a river valley with scree slopes running downhill East-West. I plot the three back bearings, which form quite a nice triangle of error. This is a good sign - if I was working with the wrong features, then there would be no good intersection, unless I was pretty unlucky.
Now for the reality check. I'm on a spur, which slopes sharply on the southern side but is even running NNW. I'm looking over a scree slope which terminates into a river bed which runs NW. This fits with the map.
Now, if I'm right, there should be a weak saddle south of torre dels Soldats (for those of you who speak Catalan - you should get why I love that name) ending at spot height 2612 which is about 800m south of the torre. From my position, this should lie at around 120° from me, and about 3 handspans to my right from the torre. Bingo, there it is! The valley to the northwest should terminate 20° west of port Negre, as it does. I'm happy with the resection!
Lets say you are happily travelling along a well marked - not too convoluted and relatively linear feature like a road or river. We can be a bit lazy in our resection and just get bearings to one or two points, plot them on the map and use the location of the linear feature as our third reference. My apologies for the singularly uninformative heading name, this is a US army term. Not really a method in itself, just another little trick.
Sometimes you may have only one recognisable feature on a map. It is still possible to get a position, but you have to move! In Brotherton's terminology this is a running bearing. It's sort of a hybrid method the military use to fix the position by taking bearings from different known points, kind of a reverse resection. A running bearing works this way.
1. From your position, take a bearing to a prominent feature - in this example, pica Roja. Record the back bearing. Make sure you work in Grid! So in this case, I'm estimating I'm starting on a line 40° from pica Roja. I just don't know where yet.
2. Pace on a bearing (it's preferable to pick a bearing that is kind of in the direction you want to go) until the bearing on your prominent feature has changed by at least 30°. In this example, I followed 260° - so I know my course parallels the black dashed arrow on the illustration - for 1200m. Take a bearing to the feature (pica Roja), and plot the back bearing.
3. Use your ruler and slide it between the V you've marked from the bearings on the map, parallel with the bearing you paced, until the V lines are separated by the distance paced. In this case, 1200m represented by the blue line. The intersection of the second back bearing and the ruler is your estimated position ± some function of your bearing and (more importantly) your distance estimation error. A quick check of the topography around you, and any other resection points will help confirm your position.
There is no particular geometrical magic to this method. We have a fixed triangle vertex (our prominent point), the two lines of our V from that point, and we just need to slot a fixed length baseline (or paces) to the triangle, at the angle of our paced bearing. Easy.
We can employ some high school geometry if we wanted to and be really clever to impress our friends. If our prominent object is between 9-46 degrees from our paced course, we can pace until the deviation of the object is twice the initial bearing - for example if it was 25° from our course initially, we pace until it is 50° from the course. The distance you paced is the same as the distance to the object. What magic is this, you ask? This is one of the properties of an isosceles triangle based on the inside angle of the sides. There are a few other tricks you can employ with the good old triangle rule: SOHCATOA - Sin=Opposite/Hypothesis, Cosine=Adjacent/Tangent, and Tangent=Opposite/Adjacent. But, given it's easier to just measure off the map, only do these if, like me, you have no life.
Oops... It didn't work! I still don't know where I am
There are a few indicators that there was a problem with your resection. First - the ground around you obviously is not the same as it should be according to the position you fixed. Second, the triangle of error may seem a bit big. Third, you may not even have a triangle at all, just a mess of lines. First things first, recheck your bearings and back-calculations. Second, evaluate each point you took a bearing to. Weight the ones you are really confident in, and re-examine the ones you are less certain on. For example, you might have taken a bearing on a spot height that is not actually visible due to intervening ground. This is not uncommon in very high country.
Do a careful study of the map. Are there any other similar groups of features that could yield a similar result? If there are some competing locations, then you will need to explore and evaluate further. From experience, errors are more likely due to incorrectly identifying the feature you took a bearing to. You may need to do a cardinal point search to check there are no features you missed. Shifting position 100m or so can give you a different perspective - Oh look! That's where the spot height was!