Some hints on old-fashioned navigation
Horizontal sextant angles
Some explanations on the web are either very verbose or very complicated, if not wrong. In essence, to fix your position, you need three landmarks (we’ll call them A,B and C) – With sextant (horizontally), measure the angle between two of them. Subtract this angle from 90 degrees to get either a positive or a negative number. On your chart, draw baseline between the two landmarks (A and B) and mark off that angle (at each landmark) – positive numbers towards you or negative numbers away from you. Mark where they intersect. With compasses, draw an arc centred on that intersection and passing through both landmarks. Do the same with the other pair of landmarks. Where the arcs intersect is where you are. To confirm you did it right, do the plots again using A and C and the three arcs should intersect at the same place (which is where you are).
Angle measured between landmarks A and B is 40 degrees, so angle to mark off from each end of the baseline AB is 50 degrees (90-40). This number being positive, draw lines with angles from baseline 50 degrees towards you. Compasses centred on this intersection draw arc through A and B.
Angle between B and C is 80 degrees, so angle from baseline BC = 10 degrees, again towards you. From intersection (as the centre) draw arc through B and C. Where arcs intersect is your position.
Confirmation (not shown): Angle between A and C is 120 degrees, so angles from baseline AC = -30, this time draw these lines at angles 30 degrees away from you (because of the negative number). Draw the third (confirmatory) arc.
Here is a verbose but correct explanation on YouTube
It is possible to perform the same process using a hand bearing compass if you are able to get precise relative bearings but uncertain of the magnetic variation or deviation hence doubtful result from plotting standard cross-bearings.
This is the method for estimating position through knowledge of speed through the water (or distance measured by log) combined with tidal flow. It is most valuable when landmarks are not available to plot positions. When a single landmark is visible, dead reckoning can be combined with bearings of the single landmark to produce a “running fix”. The origin of the term “dead” is obscure but is not thought to be an abbreviation of “Deduced”.
For basic dead reckoning, plot your last known position (A) on the chart and note time. Maintain a record of speed and direction (or distance through water by log). After a known interval, (say one hour) estimate distance and direction traveled through the water and mark this (B) on the chart. From that position also estimate distance and direction carried by the tide and plot that from B. The combination (as a vector diagram) gives the estimated position at the end of that time period. This then forms the starting point for the next interval.
If you only have one landmark visible and have a reasonable idea of your speed and direction (by dead reckoning) you can estimate your position by two successive compass bearings on the single landmark. First take a compass bearing of the object and plot this on the chart. Then after an appropriate interval, estimate your distance and direction traveled. From any point on your first bearing line plot this distance and direction, and draw a second line through the resulting point, parallel to the first bearing line. At the same time, take a second bearing of the object. Plot this second bearing and where it intersects the transposed first one is where you are.
Some common ‘rules of thumb’
I believe these work only in the absence of significant cross tide, and provided the estimate of course made good allows for any following or contrary tidal stream.
“Double the angle on the bow”
This can be done with compass or a pelorus. The rule is that if you estimate the angle between an object (e.g. lighthouse) and the bow of the vessel (say it’s 25 degrees) and you estimate your course made good between then and when that angle has doubled (to 50 degrees in this example) then the distance you have travelled equals your distance from the object.
This and other rules of thumb are described in this link.