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

    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.
    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�
    > LOG(COS(RAD(Lat))) = -0.0823
    > 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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