How can finding longitude require you to already know your longitude?

That is exactly my question too. The whole thing requires the navigator to first know their longitude. How would that be established?

I am a beginner, but this is how I understand it. I hope someone will review and correct me where I may be wrong. Longitude is determined from the difference between the time at the Local Apparent Noon and the time at Greenwich (GMT/UTC). This is why it was important to develop a highly accurate timepiece. Starting from no information, but having two clocks, one set to GMT and one set to Local time, take sightings up to 30 minutes before Local Noon and continue to about 30 minutes after. We could be in the East or West of the time zone but since a time zone is one hour then we can't be more than 30 minutes away from the meridian, I assume. These sightings will give a table similar to the table in the course material with the columns "Hours Minutes Seconds Altitude" From these sightings we find that the highest altitude of the sun is 77 deg 04.4 minutes. The time at which the sun is highest during the day is the LAN. In the course material table this occurs at 12:20:04 Next we compare the LAN time with GMT. In the General Data in the course materials it is given as 16:20:04. This would be obtained in real life by looking at our second timepiece set to GMT. So now we know that the difference is 16:20:04 - 12:20:04 ---------------- 4:00:00 Since 1 hour is 15 deg then our longitude is 4 x 15 = 60 deg West of Greenwich. Clearly this is not correct. Higher up in the material the text states that the meridian pass at Greenwich is 11:59:00. Using this time we should get 16:20:04 - 11:59:00 --------------- 4:21:04 This computes to (4 x 15) + ((21/60)*15) = 60 + 5.25 = 65.25 deg west of Greenwich, which is our longitude, and that is the Longitude assumed in the example from the start. This way you need your 2 timepieces and the Almanac, nothing else to determine longitude.

Think back to coastal navigation and how we are taught to keep a plot of our course. We start with a fix. From the start fix we create a DR (dead reckoning) plot. Periodically we take fixes and then compare the fix to the DR position of the same time. The same concept in Celestial Navigation. We keep a DR plot and we will have a DR plot for, in this case, noon, June 8, 2004. This provides the "estimated" lat & long. From the DR we have an estimation of four long and therefore we should know which time zone we are in relative to Greenwich. Unfortunately the course materials does not do a great job of explaining or demonstrating this.