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    Re: Constructing A Logarithm Table
    From: George Huxtable
    Date: 2009 Jan 14, 20:50 -0000

    Hewitt Schlereth, in [6968], asked how log tables were constructed, which I 
    will deal with later. First, he referred to Mike Pepperday's "S" table, 
    suggesting that its used logs to an unfamiliar base of 11.5.
    
    A log to such an awkward base can easily be converted, by multiiplying  by 
    an appropriate constant, to the  base-10 logs that became familiar to many 
    of us in our schooldays, to provide a much simpler way of handling the 
    matter. Hewitt gave us no numerical examples in his first posting, and only 
    one point in his second, in which 30� was tabulated as 28.385.
    
    Peter Fogg quoted an anonymous source as stating that Pepperday's S table 
    used the relation
    100*log (1/sin s)/log 11.4953
    with all logs to base 10, and that's a useful simplification of Hewitt's 
    analysis. Without more numbers being provided for different angles, I'm not 
    able to confirm that a log (1/sin) law is what Pepperday's table is based 
    on, but it certainly fits the figure provided at 30�. If so, it could be 
    further simplified a bit to-
    94.2929*log(1/sins)   , where the log is the familiar base-10 variety, 
    readily available from log tables and calculators.
    
    I've no idea where the constant 94.2929 comes in, but presumably it has some 
    convenience for Pepperday in working the tables.
    
    =======================
    
    Now for how log tables were constructed.
    
    Originally, when the notion came to Napier as early as1614, they were 
    "natural" logs, not to base-10 but to base-e, where e is an irrational 
    number equal to 2.781828... and these logs still have a certain utility in 
    maths and physics. Briggs in 1624 saw the simplicity that would result from 
    using 10 as a base instead.
    
    For anyone that's seriously interested in how the business of logs 
    developed, Google Books have kindly scanned Henry Sherwin's Mathematical 
    Tables, the 1717 edition of which you can download free. Just in case it 
    takes anyone's fancy, I will summarise its contents.
    
    He offers a thorough introduction describing the developments of the 
    previous century, which may be a bit hard-going because it uses notation and 
    conventions that differ a bit from modern practice. He quotes from his 
    predecessors about the series expansions involved and some of the 
    short-cuts. Producing those tables must have involved colossal labour.
    
    Next he gives a table of what are, effectively, 7-figure logs, to the base 
    10. Modern log tables would give these for numbers in the range from 1 to 
    10, to which nowadays we would give the values of 0.0000000 (for log 1) 
    through 0.3010300 for log 2, to 1.0000000 (for log 10). Sherwin, instead of 
    1 to 10, gives that range as 10000 to 99999, and the logs therof as 000.0000 
    (for log 10000) through 301.0300, to 999.9956 (for log 99999). So, what he 
    is is actually tabulating is 1000* log (N/10000). Once that difference has 
    been accepted, his log tables are usable today.
    
    But that's not all. As navigators, we need logs of trig functions, and 
    Sherwin provides a table of sine and log sine, tan and log tan, sec and log 
    sec, for all angles from 0 to 90�, in increments of 1 arc-minute. Of course, 
    by using the complementary angle (i.e. subtracting from 90), it's easy to 
    obtain cos, cot, and cosec also.
    
    The trig functions, such as sine, are multiplied by 10000 compared with 
    modern tables, so for example sin 45� is given as 7071.068, whereas we 
    normally take it to be 0.7071068,. Logs of trig functions always have 10. 
    added so that log 45� becomes 9.8494850, whereas we would write it as 
    (algebraically) -0.1505150, or for easy calculation ("bar"1). 84984850, or 
    in mariner's notation 9.8494850, just as Sherwin does.
    
    Sherwin goes on to tabulate natural and log versines, for every minute to 
    45�, again to 7 figures, and gives a comprehensive traverse table, for plane 
    sailing calculations.
    
    Next is an explanation of how logs are used in the solution of spherical 
    triangle problems.
    
    And finally, there's an extraordinary final financial section, on how logs 
    come into the calculation of compound interest and annuities, written by - 
    guess who?
    Edmund Halley, Professor of Geometry at Oxford, later the predictor of 
    Halley's Comet and Astronomer Royal, mariner who first mapped Earth's 
    magnetic field from at sea, and discoverer of the first succesful method of 
    determining longitudes from observations of the Moon.
    
    That's what you got for your money if you bought Sherwin's Mathematical 
    Tables.
    
    George.
    
    contact George Huxtable, at  george{at}hux.me.uk
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. 
    
    
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