NavList:

Alex,

Thanks for getting the ball rolling on this.

You suggested:
"The main part, of course is how to measure any angles with a rope alone."

Ah, but I think you have allowed that assumption to lead.

Look carefully at the tools available to you. One of them has a precision-measured scale built right in. Find one of the explanatory pages in your almanac, and rip it out! Each line of text is perfectly spaced from the lines above and below it. With a little manual measuring (presuming you know the span of your own hand with outstretched fingers to the nearest cm, or quarter inch) and maybe by some basic knowledge of typography and font sizes, you can turn those lines of text into a beautiful scale. And if you know the actual page sizes (depends on the era, but possibly a standard size), then you can measure off any other length. After that, you're all set to measure angles: 3438 × size / distance yields minutes of arc, where 'size' is the measured linear distance across the line of sight on the scale and 'dist' is the measured linear distance from your eyes 'out' to the scale.

You added:
"I propose this: find two nearby stars, compute their distance from almanac"

Yes, this is a great way to test and calibrate the angle measuring process. It's worth noting that the castaway can make life much easier by using a pair of stars separated mostly in Declination (with nearly identical RA / SHA). The "Pointer Stars", Dubhe and Merak, in the Big Dipper will work nicely. Then the computation of the distance between them is just Dec(Dubhe)-Dec(Merak) of 5°22' ...near enough for this purpose!

You mentioned that Kelvin "included a  'light to read the almanac'. But he did not include a pencil to do calculations!! (Paper is not so necessary since one can use the blanc pages and/or margins of the almanac. A flat ground covered with fine sand can in principle replace a pencil."

Ha. Yes. :) Or perhaps Kelvin intended that the castaway should find find some dry tree branch, learn to make fire (using Classical Greek methods, of course), and burn that branch to make charcoal to work the calculations. ;) Yeah, I think the smart castaway here would snap a twig off that branch, skip the fire, and use the twig to draw and calculate in the sand. 

You wrote:
"I estimate that the proposed method with extreme care and dexterity can give maximum 1 degree accuracy in measuring the angles."

To this I will say: No! You're off by a factor of five or even ten! Get outside and try this! :) You can measure angles to one or two tenths of a degree up to a maximum angular elongation of maybe 10°-15°. And notice that this range limit is consistent with Kelvin's original statement that the bright star is "near" the Moon. He doesn't specify, but it's certainly a fair, reasonable reading to say that "near" means less than 10°, right? For that matter, if by "near" the Moon, Kelvin meant less than, let's say, 2.5°, then the Moon itself would serve as a good angular scale allowing accuracy much better than one whole degree.

And what of parallax correction? That great task of clearing lunars?! This can be handled very simply by drawing a picture at the level of accuracy expected from the observations.

Next you caught Kelvin in an error! You wrote:
"So your ability to tell Altantic from Pacific ocean depends on where exactly you are."

Yes, indeed. The puzzle is not well-specified. There are points with exactly the same longitude, specifically 80°W, that are in both oceans --at 30°N, you're in the Atlantic; at 5°N, you're in the Pacific. The Atlantic and Pacific are not completely exclusive by longiude. Kelvin, if grading such a complaint in an answer, might counter by suggesting that you should use your primitive angle-measuring tools to get latitude, too. 

You continued:
"Determining day of the month is not difficult if you have almanac with Lunar distances."

You don't need tabulated distances if the almanac includes a basic star chart, or if you are able to draw your own star chart in the sand... Recovering the Greenwich date to the nearest quarter of a day or so is unambiguous by the Moon's position. This is the crudest form of "lunar distances" work and requires no precomputed tables. And incidentally, John Letcher made this point in his 1970s book, with the regrettably long and misleading title "Self-Contained Celestial Navigation with H.O. 208". 

Sometimes... you can also, absolutely, determine the day by the phase of the Moon... but the success of this method depends on the phase of the Moon. Near Full Moon, you might go four days in a row with no observable difference in phase. So then it doesn't work. But near First and Last Quarter (Moon near 50% illuminated), the difference from one day to the next is clear and obvious. Indeed the day-by-day change in the phase of the Moon near Half Moon is one of the first things we look at in each of my "Lunars" workshops. I would say that we can determine the day by the phase of the Moon for about ten days, total, out of the 28-day lunar month.

We also need to figure out one other major and important step in this process of determing longitude by using the Moon. You need local time. This is where I would use that 'rope' in the kit. Find a small tree with a not overly-leafed branch. Tie the rope to the branch so the low end is left hanging a few feet off the ground. Lay down on the ground under it, and by sighting along the rope estimate the RA of the zenith. This is identical to Local Sidereal Time. Then what...? :)

Frank Reed