Since we don't know where we are, we need to assume a position. Since Bernard was the helmsman, we chose a nearby position on land. My nautical chart has 1° grid spacing with latitude/longitude labels on the edges, so I know the assumed position is 51° N 001° W. You need to know the latitude and longitude of the assumed position and its precise location on the chart for the method to work.
A nautical almanac will tell you the altitude of a navigation star from the assumed position at the time of your observation. Those heights above the horizon are called Hc. The almanac will also tell you the bearing from the assumed position to the star. If your boat was at the assumed position, your observed altitudes Ho would match the calculated altitudes Hc. Comparison of Ho and Hc determines which location is closer to the given star, your observing location O or the assumed position AP. Think of it this way...if the star stood still and you sailed toward it, it would appear to rise higher in the sky. So whatever position is closer to the star yields the higher altitude for the star.
For each star shot, subtract the Hc from the Ho and note the difference and its sign. Multiply the arc minutes by 2 km to turn them into distances. Use the sign to determine whether you are closer or further than AP from the star. Here is what we can say right now based on our measurements...
Our unknown position is nearly the same distance away from Alioth as the known assumed position.
Our unknown position is much closer to Gienah than the known assumed position.
Our unknown position is much further from Elnath than the known assumed position.
Let's put that information on the chart using the in-game mapping tools.
I have a typo in the picture above. The second blue line should read Hc > Ho...I wil fix it later...