# NavList:

## 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: Frank Reed
Date: 2009 Jan 9, 21:17 -0800

```Hewitt, you wrote:

"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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