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    Re: slide rule sight reduction accuracy
    From: Gary LaPook
    Date: 2009 Jun 19, 13:32 -0700

    You are right about how long a slide rule gets if it wants to cover all 
    values on the cotangent scale. To cover the range of 1' to 89º 59' would 
    cover 70 cycles, like gluing 70 slide rules end to end. For each cycle 
    of a slide rule the log (cotan) changes .1. The log (cotan) of 1' is 3.5 
    and the log (cotan) 89º 59' is -3.5, a range of 7.0 covering 70 cycles. 
    Each cycle on the Bygrave covers one spiral.
    Bygrave restricted his slide rule to the range of 20' to 89º 45' so he 
    could restrict its size to only 44 cycles, and my version is only 37 
    cycles, covering 55' to 89º 15' , both version give up a very slight 
    amount of range but there are procedures to deal with these rare cases.
    The reason I mentioned the special case where Az exceeds 85º is that to 
    compare the accuracy of the Bygrave with other methods you should either 
    ignore these cases, due to atypical lowered accuracy, or follow the 
    special procedure for these cases, as a navigator would, of exchanging 
    lat and dec and redoing the computation.
    I am quite interested in seeing what your simulation might show both for 
    using the Bygrave procedure on a normal 10 inch slide rule and also with 
    my flat version of the Bygrave.
    Brad Morris wrote:
    > Hi Gary
    > In my simulations (recently posted), you will find that I obtain accurate 
    answers all the way out to 90 degrees.
    > The reason is simple, since my simulation has the table of log values for 
    each minute of arc to 90 degrees (and they are all offset by 1e-11 degrees), 
    then the table is fully populated.  The trig functions of (cosine, cotangent) 
    or (secant, tangent) never divide by zero when offset in that manner.  The 
    offset of 1e-11 in degrees does not affect the output, particularly when we 
    only care about the nearest minute in result.
    > Further, Bygrave couldn't do this because the distances become huge on the 
    slide rule.  You will find the same problem with the Flat Bygrave.  While 
    possible to create a scale based on these distances, it isn't practical to 
    make such a scale.  Bygrave mentioned such problem in his patent.  I have 
    been playing with his proposed solution, no progress just yet.
    > My simulation can represent the scales, because LOG(ABS(Trig Function)) is 
    not represented by a distance, rather it is represented by a numerical value.
    > As such, I would think other simulations, provided they are based on 
    numerical values and not deriving the value from slide rule distance scales, 
    would not suffer from this issue.
    > Best Regards
    > Brad
    > -----Original Message-----
    > From: NavList@fer3.com [mailto:NavList@fer3.com] On Behalf Of Gary LaPook
    > Sent: Friday, June 19, 2009 7:06 AM
    > To: NavList@fer3.com
    > Subject: [NavList 8720] Re: slide rule sight reduction accuracy
    > Paul Hirose wrote:
    >> Regarding the Bygrave simulation, it can be done if I know the rules.
    >> For instance, azimuth is obtained by taking an arctangent, but that
    >> function yields a result between -90 and +90. There must be some rule to
    >> convert that to the range 0 to 360.
    > But to do a simulation you do not need to get to the step of converting
    > Az to Zn since the Hc is computed from Az not from Zn.
    > In a simulation, watch out for the special case where Az is greater than
    > 85º since Hc becomes very sensitive to Az in that range and even
    > rounding up or down half a minute can make a significant change in Hc.
    > gl
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