GPS is a wonderful creation that has raised our expectations in positions to unrealistic but attainable levels. It is not uncommon to see a GPS touting an accuracy of a few feet! You might want to achieve these same results with a sextant, only to have your dreams dashed in the waves.
The goal of accuracy for a sextant is 25 nautical miles. Yes, not feet, but miles. While a GPS is considered good if it can place you within 10 feet of your true position, a person calculating their position with a sextant on a boat at sea is considered good if they can calculate their position to be within 151,903 feet!
This might sound ridiculous, but the reasoning behind it is rather sound. Celestial Navigation is not to plot your position and get you to slip between a reef and a wreck on a chart, that is the realm of visual navigation. Celestial Navigation is to get you to your next landfall. Land can be spotted from very far away, and if you can get close enough to see it, you can then sail towards it, letting visual navigation take over from there.
When calculating your position, you can use very VERY precise measuring techniques which will let you calculate your position down to under 400 feet. The problem is this sort of refined measuring is done on land, which is not moving, and at rest, which is not sailing. The measurements can then be repeated daily to finally work down the exact numbers and find your precise position.
On a yacht, you are sailing forward, so your measurements are off to begin with. Next, you are on a pitching deck, being rocked side to side as you ride up and down waves. You have to hold a sextant perfectly stable while being tossed around as your altitude changes constantly!
If you take a course on celestial navigation or read about it in a book, it will typically have the calculations carried all the way down to the second (of time and location). 1 second of time is 1/60th of a minute. 1 second of latitude & longitude is 1/60th of a nautical mile in latitude, and 1/60th of a nautical mile at the equator to nothing at the North Pole. If you try to carry your calculations down to the second, you will find that your not going to gain much in accuracy for the amount of additional work you need to put in. It is best to take all of your recordings and calculations down the the minute (of time and position).
Think about it, when you take your noon site and record the time. If you record to the second, you will then have a very accurate reading. If you take it to the minute, you could be off by as much as 59 seconds! But what does 59 seconds come out to be in the grand scheme of things? The boat is moving, you are pitching around, and if all you need to do is get a general idea of where you are, why do all the extra math?!
If you take your noon site at 16:00 vs 16:01, the difference in your longitude will be minimal
16 : 00
-12 : 00
4 : 00
16 : 01
-12 : 00
4 : 01
60 + (0.25x60)
What this means is that your longitude is going to be calculated as one of these two positions. If you were to measure the time of the noon site down to the second, your answer would fall somewhere in between 60*0'W and 60*15'W. Yes, that would be a much more accurate calculation, but you can assume that your true position is somewhere between 0' and 15', which means that you are not that far off!
Imagine that the true time of the noon site was 16:00:30, this would mean that the true position would be around 60*07'30". Your calculations would only have been around 7.5 nautical miles off. Not bad for skipping an entire section of calculations!
You might be wondering how you can verify what your true position is out at sea and how you would then calculate your error, and the answer just makes it more logical to stop the calculations at the minute. You check your calculated position with a GPS reading of your present coordinate at that time. The GPS, which is accurate down to a few feet, will tell you where you actually are; so why go through all the extra headache and math to figure out your position to a few miles closer when at the end of it all, you will verify it with a GPS?
In practice, I have found that I am usually within 4nm of our true position. This is far lower than the acceptable 25nm radius of error, and this lets me know that should all the GPS's in our yacht fail, I can trust my calculated coordinates to get us to land. At first, you might find that you are way off, but with practice, you will zero in on a very accurate noon site while only taking your measurements and calculations down to the minute.