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    Re: Modified Martelli Table V
    From: Frank Reed
    Date: 2013 Jan 3, 11:38 -0800

    Hewitt,

    Just to reiterate, FUNDAMENTALLY all of the Martelli tables are in Bowditch and other navigation manuals. His Table I is a table of log cosines (plus a constant). His Table II is a table of natural cosines (plus a constant). His Table III is a table of natural sines (negated, plus a constant). His table IV is a table of common logarithms (negated, plus a constant ...or equivalently, a table of "proportional logarithms" plus a constant). His Table V is a table of log haversines (negated, plus a constant ...or many other equivalent descriptions). These are all simple tables which could have been combined into a slim volume by anyone. But who would buy such a volume if they're already in Bowditch or Norie? Ahh, but mix in a little Italian magic and peddle it to gullible British mariners, and now you've got a product that sells! A big advantage of short, highly-specialized methods like Martelli's is that the tables can be truncated to save pages and to make lookups easier. For example, the range for his tables I, II, and III, is 0 to 90 degrees. That's more convenient than making tables that cover 0 to 360 degrees (it doesn't save pages --it saves the navigator from having to learn multiple entry tables). But then again, that means that they are much less useful for any other problem.

    Incidentally, I think I've got a "good" reason now for his odd choice of sexagesimal (units, minutes, seconds) output from his log tables. It's easier to look up logs in sexagesimal tables because you can head each page with a certain number of "units" (call them hours or degrees, it makes no difference) and then head the columns with a certain number of "minutes" and then each row is labeled with "seconds". And as it happens a small book can easily display 60 rows per page. This saves space in the table of common logarithms (Table IV) though at the big expense of making those logarithms useless for any other problem. And since logarithms were so useful to a navigator, any navigator carrying Martelli's tables with their obfuscated logarithms would necessarily also have to carry a redundant normal table of logarithms (probably in a larger volume, like Bowditch or Norie).

    -FER


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