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    Re: Towards a basis for Bruce Stark's Tables
    From: John McKeel
    Date: 2003 Jan 3, 09:22 -0700
    RE: Towards a basis for Bruce Stark's Tables

    Good Morning Fred,

    Just a quick point of clarification. Were you using an artificial horizon with the Mark 3? I am using a Mark 2 twice a day to practice with in the parking lot outside my office along with the Davis artificial horizon and some used motor oil. My results are regularly within 2 miles and often much closer than that. I have been able to do lunar sights and on the rare morning, Venus, with the same results.

    It wasn't that way when I began though. The Davis "telescope" is a bit "persnickity" and the IC can change radically from sight to sight. I also found Bruce Bauer's suggestions (The Sextant Handbook) for checking the SD very helpful.

    I am looking forward to the day when I can buy a "real" sextant. Having said all that, I'll go back to "lurking" and learning. This new hobby and the posts on this list have sent me back to the community college to take the math classes I skipped in favor of Greek and Latin!

    Cheers,
    John McKeel
    Phoenix

    -----Original Message-----
    From: Fred Hebard [mailto:Fred{at}ACF.ORG]
    Sent: Friday, January 03, 2003 12:35 AM
    To: NAVIGATION-L{at}LISTSERV.WEBKAHUNA.COM
    Subject: Towards a basis for Bruce Stark's Tables


    OK, some history of this major obsession of mine,

    I used to sail and do coastal piloting, and always wanted to do celestial, but have lived inland for 30 years.  Then I noticed the price of a Davis Mark 3 on the Celestaire site and bought it.  Then of course spent hundreds of dollars on sight reduction books when I couldn't get my position down to better than 20 miles, trying to figure out what was going on.  Thought maybe elevation above sea level was involved, but in my case that's a half mile difference in a 6000 mile radius of the earth; elevation affects lunar parallax and refraction a bit, but not much more.  Finally got my position to about 2 miles once.  Decided I needed a better sextant.  Got a 1948 Husun Mate on Ebay for a little more than $200 because the batteries had burst the handle; but the enamel was not noticeably chipped in any of the photos and all the parts were there.  Found I would have to remove the index arm to remove the handle for repair.  Delayed doing that.  Ran a lunar and  got to within 12 seconds of GMT!  Maybe that Husun was OK, but handle needed fixing.  So removed index and fixed handle, but feared destruction of the calibration due to a small blunder while removing index.  To check calibration, I would like some accurate interstellar distances, thus need to apply Borda's method to correct for refraction.  But how do all those tables in Bruce Stark's book work?  A far more abstruse question.

    Table K, one of the big ones, is log(haversine()).  Found a reference to Gauss' formulas at http://mathworld.wolfram.com/SphericalTrigonometry.html .  They are similar to Napier's formulas.  So one of those must be the formula for the Gaussian table.  It clearly is not the normal approximation to the binomial.

    I have gotten about as far as this as I have time, and sure would like to see Bruce Stark's method laid out in detail.  It appears to be a fairly straight-forward application of log haversines to spherical triangle trigonometry.  He appears to be using Borda's method for clearing the distance, and the standard method for computing intercelestial distances (with no correction for refraction).

    Here is as far as I've gotten with his method for computing intercelestial distances.  Let del(GHA) be the absolute value of the difference in GHA between the two bodies, del(dec) be the corresponding value for declinations.  ~ is the operator for finding the absolute value of a difference.  Mdec is the declination of one body and Sdec the declination of the second;  logs are to the base 10; hav is haversine; archav its inverse; and Gauss are values from his Gaussian table:

    =archav{-Gauss[(log(hav(del(GHA))) + log(Mdec) + log(Sdec)) ~log(hav(del(dec)))]
    + [log(have(del(GHA))) + log(Mdec) + log(Sdec)) ___or___
    + log(hav(del(dec)))]}.


    The ___or____ function chooses the lesser of the two values.

    So what is the Gaussian, and how do you hook all these guys together in standard mathematical notation?  Hopefully, this will help and inspire somebody, maybe Bruce, to lay this out!

    Thanks,

    Fred



    --
    --------------------------------------------------------------------------
    Frederick V. Hebard, PhD                      Email: mailto:Fred{at}acf.org
    Staff Pathologist, Meadowview Research Farms  Web: http://www.acf.org
    American Chestnut Foundation                  Phone: (276) 944-4631
    14005 Glenbrook Ave.                          Fax: (276) 944-0934
    Meadowview, VA 24361
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