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    Re: Constructing A Logarithm Table
    From: Hewitt Schlereth
    Date: 2009 Jan 10, 16:36 -0400

    Hi Frank -
    
    'Slog' and 'Co-slog'. I dig it. (In high school we use to call them
    'slime' and 'co-slime').
    
    Anyway, I've never had occasion to grapple with a spread-sheet.
    (Hmm... No. I won't go there. :-) ). I imagine there is an algorithm
    for spread-sheets that would produce a table of logarithms in a blink,
    but the whole exercise must by now be academic. For ten dollars you
    can have a scientific calculator that'll deal with S and CoS directly.
    
    Come to think of it, there's a nice little book that has the S/CoS
    formulas for direct solution of the undivided triangle for Hc and Z -
    backup is the NAO reduction table with a more lucid explanation.  Also
    has a long-term almanac of Aries and Sun; pub. by Starpath School of
    Navigation.
    
    Speaking of lucid, thank you for your sketch of the historical context of logs.
    
    Hewitt
    
    On 1/10/09, frankreed{at}historicalatlas.com  wrote:
    >
    >  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 
    "slogs".
    >
    >  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).
    >
    >  -FER
    >
    >
    >
    >  >
    >
    
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