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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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