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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: Index Card for Trig/Log Table
From: Greg Rudzinski
Date: 2009 Sep 10, 04:30 -0700

```George,

I am using the trig log table like a slide rule and keeping track of
the decimal point by rounding the cosines to a single digit and
multiplying out an approximate product.

Greg

On Sep 10, 3:15�am, "George Huxtable"  wrote:
> Andres Ruiz commented, about Greg Rudzinski's index-card method
>
> For Lat = 34�10'=34.1667�
> And table vlue is:
> 100000*LOG(10*COS(RAD(Lat))) = 91771.9422 = 91772
> Where LOG is the logarithm to base 10
>
> ======================
>
> from George-
>
> There's nothing wrong here, in that a log of a number less than one, such
> as -0.0823, is in navigator's convention turned into a positive value by
> adding 10.0 to it, (equivalent to multiplying the original cosine by
> 10,000,000,000. ), so it becomes 9.9177. Greg omits the integer preceding
> the decimal point, just concerning himself with the "mantissa" part, which
> becomes .9177, or actually, as Greg quotes an additional digit, .91772.
>
> ================================
>
> But difficulties can arise with the simplifications and short-cuts that Greg
> has adopted, in certain circumstances.
>
> For example, in [9685], Greg wrote- "The result is accurate to four places
> even when the the SIN product is calculated by slide rule.".
>
> Only sometimes (most times, perhaps] will that be the case. Let's assume
> that the multiplication can be done to one part in 1000 by slide-rule (is
> that fair?).
>
> The formula used is -
>
> sin h = cos lat cos dec cos (LHA) + sin lat sin dec.
>
> In the case Greg showed, that second term, product-of-sines, is much smaller
> than the product-of-cosines first term, so finding it to 1 part in 1000 �is
> quite adequate for getting the overall sum to 1 part in 10,000. In many
> cases where the Sun is observed from low latitudes that will be true, as sin
> dec never exceeds 0.4. But if the first term happens to be small, such as it
> will if LHA gets near to 90�, (with the Sun near 6am or 6pm, for longitude)
> , or with a star of high declination, then the second term will dominate,
> and then a slide-rule calculation will be insufficiently precise.
>
> ========================
>
> Also, Greg seems to have ignored the whole-number prefix that precedes the
> decimal point in his logs, a procedure which has worked for the example he
> chose. For each of the log-cosines, there should be a 9 before the decimal
> point, and when those logs are summed, they should come to 29.88205, not
> 2.88205 as he indicated. By discarding two 10's, as the navigators-log
> convention allows, we get it to 9.88205, and by looking up that number in
> inverse logs, the result will be 0.7622, as shown. Greg has discarded his 2.
> and in its place has somehow substituted a 9. before looking up that
> antilog., which has happened to give the right answer.
>
> But if his sum-of-cosines had happened to be somewhat less, so the result
> summed to 28.xxxxx rather than 29.xxxxx, then he would have had to search in
> quite a different part of his antilog tables, the part commencing with
> 8.xxxxx rather than with 9.xxxxx. So if he fails to take account of those
> "characteristic" numbers, before the decimal point, he will sometimes get
> into serious trouble.
>
> I invite Greg to try a calculation using a LHA of 85�, for which the cosine
> is .08716, and the log cosine is either -1.05970 or, in navigator's parlance
> (adding 10.0) 8.94030. Unless he takes account of that prefix being 8.
> rather than 9., his answers will be very wrong. Those prefix integers need
> to be accounted for, not just ignored, as Greg appears to have done.
>
> If I've misunderstood his procedure, perhaps he will put me right.
>
> George.
>
> contact George Huxtable, at �geo...@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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