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George,

I did your suggested example with an extra set of log calculations
just for comparison purposes. The straight sin cos entries only take
about a minute to look up and offers a quick way to double check with
an electronic calculator or slide rule if so desired. The
approximations are done in the head and need not be written down. The
index card method as originally presented came out on top as my
personal method of choice for relatively quick error free results.
Others might want to try a few different variations to see what works
best for them.

Greg

On Sep 10, 12:31�pm, "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 �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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