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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:

http://www.fer3.com/arc/m2.aspx?i=109019&y=200907

gl


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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