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## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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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@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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