You can’t use terrestrial objects to give a scale to astronomical objects in a picture, because it depends on how close the camera is to the foreground object and what the zoom is. You have to either compare it to other astronomical objects, or to a phenomenon like a rainbow that always has the same angular size.
Back to the maths. Don’t you also need to record the exact time of the observations? I’m not sure that would have been real easy on Gilligan’s island.
You do need times, but you can make your observation at any time that you can know. Observations made three nights in a row, at the moment when Sirius rises each night, for instance, would work fine. Or more realistically, every night for a month at that time.
You would still need ephemeris tables from a book, but this being the Professor, I think we can assume that he’s memorized all of them.
