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    Re: Constructing A Logarithm Table
    From: Hewitt Schlereth
    Date: 2009 Jan 14, 18:05 -0400

    George -
    Thanks very much for your explication of logarithm tables.
    I didn't mean to be cryptic about how I unraveled Pepperday's "S"
    values - to my satisfaction at any rate. The fact that his method
    involves adding and subtracting what he calls "help numbers" to solve
    five formulas for the divided nav triangle indicated they were
    His formulas are the same as Ageton's but inverted to sines and
    cosines instead of Ageton's cosecants and secants.
    I recalled from high school that a log is the exponent to which a base
    number must be raised and decided to see if I could discover the base
    and the exponent which would give the sine of 30�, 0.5.  I used the
    X^Y button on my HP35s. From Pepperday's intro I got the idea he had
    multiplied his logs by something to get the decimal point in a
    "helpful" place so I figured the log he gave  for 30� (28.385)
    probably started life as .28385.
    So, I plugged away trying various base numbers for X and repetitively
    using .28385 and -.28385 for Y. 10 didn't work and 2 didn't work and e
    didn't work and 8 d8dn't work. 12 gave me too much and 11 too little;
    so I worked up from 11 in 0.1 steps till I hit 11.5.
    11.5 to the -.28385 power is 0.49994 on the HP35s, which was close
    enough to 0.5 for me. To relocate the decimal point  and get rid of
    the minus sign I figured he'd multiplied by -100.
    I then did a check for 25� (sin .42262) with P's log of -.35271 and got .42255.
    Checking 45�, 6', 15' and 30' produced similarly "close-enough-for-me" results.
    Doing all this sure took me down memory lane. I remembered logs okay
    in high school but didn't recall any exercises in constructing a
    table. And that's what prompted me to query Frank, even though I guess
    the need for such tables is pretty much academic these days. Unless
    you want to do a riff on Ageton, that is. :-)
    On 1/14/09, George Huxtable  wrote:
    >  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@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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