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    Re: Index Card for Trig/Log Table
    From: Gary LaPook
    Date: 2009 Sep 11, 00:20 -0700

    And we found that using the Bygrave formulas with a ten inch slide rule 
    provides greater accuracy than when using the sine-cosine formulas. See:
    George Huxtable wrote:
    > A few comments about Greg's posting [9697].
    > He wrote- "I choose to drop the characteristics on the index card then keep 
    > track of the decimal by approximation."
    > Fair enough. That's a perfectly valid way to do the job, though it adds 
    > quite a lot of complication; which he didn't mention when introducing us to 
    > his procedure. That explains why he had to look up, and note down, the 
    > cosines of the three angles, as well as the log-cosines, which would 
    > otherwise have been sufficient. Then he can multiply them together, to get a 
    > rough idea of the product. Simply estimating the product in-his-head should 
    > suffice. That then allows him to put the decimal point in the right place 
    > when the antilog gives the precise product-of-cosines.
    > But isn't that a bit long-winded and unnecessary, when it can be bypassed, 
    > simply by working the log calculations to include the integer before the 
    > decimal place?
    > That calls for writing down fully the log-cosines of the three angles, as 
    > 9.91772, 9.99804, and 8.94030, as he has done below, summing them to give 
    > 28.85606, as he has done below. Then discarding two tens from the result, to 
    > arrive at 8.85606 (not 2.85060, as he has shown). Then the next step is to 
    > use inverse logs, of 8.85606, to arrive at 0.07179, automatically. It's just 
    > what such logs were invented for.
    > For LHA 85� :
    >                                    appx(.07)  appx(.05)
    >              Log Cos      Log       Cos       Sin (by slide rule)
    > Lat 34� 10'  9.91772      91772     .8274     .5616
    > Dec  5� 26'  9.99804      99804     .9955     .0947
    > MA  85� 00'  8.94030      94052     .0872     +.0532
    >             28.85606    2.85628  ............. .0718
    >  Inv Log      .07179     .07183                .1250  Inv Sin
    >                                              Hc 7� 11'
    > ===============================
    > Then the next step is to add to that the product of sines of lat and dec. In 
    > this case, it can be seen that the two terms in this sum are of similar 
    > magnitude, so a 1 in 1000 error (if that's accepted as reasonable, from a 
    > slide rule) ends up as 1 in 2500 error in the sum, which would create an 
    > overall error of well over a minute in the result, from that cause alone. 
    > It's why slide rules are not really acceptable for such calculations. 
    > Instead of multiplying those sines by a side rule, log sines could be 
    > extracted instead, added, then the result found by an inverse log (antilog) 
    > table. Then that has to be summed with the product-of-cosines, found 
    > previously, and then the inverse sine obtained as before.
    > Those log sin calculations could be completed in the time saved by not 
    > having to work, and multiply, those cosines. And the result would not have 
    > lost any of its precision. What I'm trying to point out is that in this case 
    > the old ways were the best, and Greg hasn't saved time, but has lost 
    > precision, by doing it differently.
    > Of course, better ways of working that calculation, by logs, were developed 
    > to avoid that awkward addition.
    > 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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