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    Re: Constructing A Logarithm Table
    From: Frank Reed
    Date: 2009 Jan 9, 21:17 -0800

    Hewitt, you wrote:
    "Doing this made me wonder how a log table is made."
    With a spreadsheet?? :-)
    And you asked:
    "Do you have to slog by trial through 90 degrees 1' of arc at a time or is there another way?"
    First, I do believe you have just invented a better expression for those "S 
    Table" logarithms than Pepperday's "help numbers". They should be called 
    Historically, most of these tables were made by using other people's work. In 
    other words, you would find some algorithm that would generate the logarithms 
    you need in the fewest possible steps from other tables which had been 
    previously published. That may sound simple, but there's a real art to this. 
    To give a trivial example, suppose I want a table of logsecants. Well, since 
    the the secant is 1/cosine, if I can get a table of logcosines, then a 
    logsecant is just -logcosine or, the way they did things 200 years ago, it 
    would be 10-logcosine. That's not too much work. Also, in sections where the 
    rate of change is not too fast, you can do a lot of the work by differencing. 
    Also, differencing (comparing the difference between neighboring entries in 
    the table) is the best way to find errors.
    The obvious disadvantage from trusting other people's tables is that you will 
    end up propagating their errors, if any. If you do "slog" through each 
    calculation by long-hand, you may end up discovering thousands of errors in 
    the published tables, usually in the insignificant final digit of a 
    logarithm. You can then use those discoveries when marketing your work by 
    claiming that previous works include "thousands of errors". Conveniently, you 
    can easily point to all of those errors since you've worked them out. 
    Meanwhile, anyone on the other side of the table would have to completely 
    re-calculate *your* tables to find the thousands of small errors, which 
    almost certainly have crept in, in your new tables. 
    And that's just how it went down back in 1799. Bowditch re-calculated many 
    standard tables in Moore's "New Practical Navigator" and found "thousands of 
    errors" (mostly totally insignificant) and his publisher Blunt used those 
    discoveries to market the "New American Practical Navigator".
    To be fair, there were some other significant advantages of Bowditch's version 
    compared to Moore's. It was better written, in my opinion, though still very 
    much in an 18th century style. It included a modest improvement in lunar 
    calculations. And above all, it was "localized" for the American coasts and 
    shipping markets. Some would add that Bowditch also fixed the "leap year bug" 
    in Moore's book, but I think that's unfair since later editions of Moore had 
    already fixed that years before it became relevant (the bug was that earlier 
    editions had marked 1800 as a leap year in revolving tables of the Sun's 
    declination, but century years, like 1800, are not leap years, unless 
    divisible by 400).
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