Mike,
The purpose of an assumed position (AP) has nothing to do with the underlying math.The purpose is to make the tables compact enough to carry aboard. The AP is simply a contrived "trial position" somewhere close to where we expect to find ourselves which ALSO is chosen to match the limited number of solutions for Hc and Z listed in the tables. The number of tabulated solutions is diminished by only listing those solutions for starting points (AP's) that have a whole degree of latitude and a whole degree for the Local Hour Angle (LHA). This eliminates thousands of solutions that one would find with the partial degrees of these values between those listed.
For direct calculation by computer app or scientific calculator, and some tables that do not require an AP (H.O. 208 Ageton's and its derivatives) we use our dead reckoning (DR) position as a starting point but it amounts to the same thing: it is just a trial starting point that we think is somewhere close to where we will eventally find our fix.
Thus when using a calculator to do your sight reduction or an app such as Sight Calc you don't need to derive an AP, although you certainly can.
When using most tables you need to "Assume" as in "Deliberately take up" a contrived position to suit the tables which is also close to where you think you are.
The latitude part is easy: You simply use the whole degree of latitude nearest to your DR. If your DR is exactly half way between two whole degrees of latitude it doesn't matter if you pick the whole degree larger or smaller -- either will work equally well.
The AP longitude is not dependent upon the AP latitude or the change from your DR latitude to the AP latitude whatsoever. Instead it is only dependent on the body's LHA which is how far west that the body's geographic position is from your DR measured in degrees of longitude. It is highly unlikely that this LHA will compute to be a whole degree, so to use most tables we need to contrive an AP longitude to use where that will be be the case.
If you are in Western longitudes, to find the LHA from the desired AP you will be subtracting the AP longitude from the Greenwich Hour Angle (GHA) of the body calculated for the time of the sight using almanac data. Since you need this to come out to a whole degree (no remaining minutes and tenth's) you make up your new AP longitude by using the whole degrees of your DR longitude but then append to that the minutes and tenths of the GHA. Now when you subtract the AP longitude from the GHA to get the AP's LHA it will be a whole degree with no minutes and tenth's -- which is necessary for the tables.
Things are different in Eastern longitude because you will now ADD your AP longitude to the GHA of the body. Therefore you start with a whole degree of longitude near to your DR and append to that the necessary minutes and tenth's so that the addition comes out to be a whole degree for LHA from the AP. These minutes and tenth's are found by subtracting the minutes and tenth's of the GHA of the body from 60.0' the result being what you need to tack onto the whole degree of Eastern longitude from your DR to get an AP that has a whole degree LHA.
There has been a pretty extensive discussion in the General Forum lately on this including examples as well as covering some further nuances. If you start with "LHA iin Eastern longitudes ?" you should get a lot of your questions answered.
If not, then ask away. We are here to help one another.
PeterB
The purpose of an assumed position (AP) has nothing to do with the underlying math.The purpose is to make the tables compact enough to carry aboard. The AP is simply a contrived "trial position" somewhere close to where we expect to find ourselves which ALSO is chosen to match the limited number of solutions for Hc and Z listed in the tables. The number of tabulated solutions is diminished by only listing those solutions for starting points (AP's) that have a whole degree of latitude and a whole degree for the Local Hour Angle (LHA). This eliminates thousands of solutions that one would find with the partial degrees of these values between those listed.
For direct calculation by computer app or scientific calculator, and some tables that do not require an AP (H.O. 208 Ageton's and its derivatives) we use our dead reckoning (DR) position as a starting point but it amounts to the same thing: it is just a trial starting point that we think is somewhere close to where we will eventally find our fix.
Thus when using a calculator to do your sight reduction or an app such as Sight Calc you don't need to derive an AP, although you certainly can.
When using most tables you need to "Assume" as in "Deliberately take up" a contrived position to suit the tables which is also close to where you think you are.
The latitude part is easy: You simply use the whole degree of latitude nearest to your DR. If your DR is exactly half way between two whole degrees of latitude it doesn't matter if you pick the whole degree larger or smaller -- either will work equally well.
The AP longitude is not dependent upon the AP latitude or the change from your DR latitude to the AP latitude whatsoever. Instead it is only dependent on the body's LHA which is how far west that the body's geographic position is from your DR measured in degrees of longitude. It is highly unlikely that this LHA will compute to be a whole degree, so to use most tables we need to contrive an AP longitude to use where that will be be the case.
If you are in Western longitudes, to find the LHA from the desired AP you will be subtracting the AP longitude from the Greenwich Hour Angle (GHA) of the body calculated for the time of the sight using almanac data. Since you need this to come out to a whole degree (no remaining minutes and tenth's) you make up your new AP longitude by using the whole degrees of your DR longitude but then append to that the minutes and tenths of the GHA. Now when you subtract the AP longitude from the GHA to get the AP's LHA it will be a whole degree with no minutes and tenth's -- which is necessary for the tables.
Things are different in Eastern longitude because you will now ADD your AP longitude to the GHA of the body. Therefore you start with a whole degree of longitude near to your DR and append to that the necessary minutes and tenth's so that the addition comes out to be a whole degree for LHA from the AP. These minutes and tenth's are found by subtracting the minutes and tenth's of the GHA of the body from 60.0' the result being what you need to tack onto the whole degree of Eastern longitude from your DR to get an AP that has a whole degree LHA.
There has been a pretty extensive discussion in the General Forum lately on this including examples as well as covering some further nuances. If you start with "LHA iin Eastern longitudes ?" you should get a lot of your questions answered.
If not, then ask away. We are here to help one another.
PeterB