NavList:

Lars, you wrote:
"Frank wrote that table II is limited to something like 78.5 degrees for positive results. As the table runs right up to (almost) 90 degs this is obviously not true. The actual limitation of the formula is 101.5 degs"

Thanks, Lars! A sign error on my part. I saw 0.2 and thought -0.2. The 0.2, as it stands, serves no purpose.

The pointless limitation on his Table I remains (maybe we should put a stamp on them: "Invalid for Kochab sights; not for Arctic use!"). I'm kidding, of course.

Just to test it all out and re-assure myself that it would work, I set up a little spreadsheet. Instead of Martelli's rather bizarre choices, tabulate these quantities (to 4 or 5 decimal places, as desired):
Table A: -log(cos(x)) ;x from 0 to 89 59'.
Table B: 1+cos(x) ;x from 0 to 90 degrees.
Table C: 1-sin(x) ;x from 0 to 90 degrees.
Table D: 10+log(x) ;x from 0.0001 to 1.0000.
Table E: 10+log(1-cos(x)) ;x from, perhaps, 15 to 165 degrees and also listed by hours from 1 to 11 hours.

Apologies in advance if I've written anything down incorrectly. This is far simpler than the Martelli version but solves the exact same problem starting from the same equation. The rules would be the same as Martelli, except the "subtract two" in step 5:
1) Enter Table A twice with Latitude and then Declination. Record those logs.
2) Calculate Diff = Lat~Dec (Lat-Dec if same name, Lat+Dec otherwise).
3) Enter Table B with Diff. Record that number.
4) Enter Table C with the Altitude. Record that number.
5) Add the two numbers from steps 3 and 4. Subtract two from that sum.
6) Enter table D with the result of step 5. Record that log.
7) Add the logs from steps 1, 2, and 5.
8) Seek the sum from step 7 in table E. Read out the Apparent Time (AM or PM) or the hour angle.

That's it. No limitation in Lat or Dec, easily incorporates hour angle along with times, no mysteries at all, and easy to re-generate using simple software today. Of course, the standard historical method is just as easy, barely more work, and requires no specialized tables. And by the way, I've followed Martelli listing the above quantities as separate tables, but tables A, B, C, and E can be combined as was commonly done with trig tables. And again, table D is just a table of common logarithms (adding 10 was standard practice). Note that even the very small extra step of subtracting two could be eliminated by tabulating 10+log(x-2) in table D.

Martelli's Tables may have been mysterious decades ago, but they certainly weren't genius. In fact, quite the opposite. They were crudely designed and idiosyncratic, even bizarre. The best spin I can put on it is that their peculiar construction was "marketing genius". Unfortunately, it's more likely that they reflect limited mathematical competence. It appears to me that Martelli "hacked" them together, perhaps borrowing little tricks from obsolete lunars clearing methods.

Martelli's Tables are an intriguing novelty, of course. Fun to play with. An "artifact", as Hewitt put it. But let's not fall for the same cult-like belief that had some navigators, decades ago, clutching to them as the last vestige of a by-gone era. I have to say, I feel sorry for readers of NavList messages who are relatively new to celestial navigation and may have gotten the impression that these tables are important or somehow valuable today. They're not, except as "quaint novelties".

-FER

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