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    Re: I couldn't resist!
    From: John Huth
    Date: 2012 Oct 19, 14:27 -0400
    I think as long as you are only measuring a few years in advance, you can probably get declinations to about 20 arc minutes.

    As I mentioned earlier, I used the classic Euclidean bisection of angles to make a quadrant.   I think about 20 arc-minutes was my precision.   It took a bit of effort, but was doable with material available.

    If you then look at declination, you can start with a sinusoidal approximation, with the zero crossing on, say, March 21st.   The deviations from a sinusoidal approximation are around 1.7 degrees, give or take, depending on year.   Then, the Julian calendar is decent enough that you can factor in leap years, so you can project some number of years into the future.    

    The deviations from the sinusoid were done by Ptolemy with the addition of an epicycle to the 'orbit' of the sun around the earth.   He also showed that the epicycle could be reproduced by having the earth's position in a circular orbit of the sun displaced.   Both techniques gave a decent table of declinations.

    As I mentioned 20 arc-minutes seems to me to be about right for the tools available around that period.  This is modulo  issues like Mars, which obviously was waay out of kilter some of the time (6 degrees, but that was due to the difference in geocentric circles and heliocentric ellipses).   

    I seriously doubt that a shot could be taken to better than a degree with a mariner's astrolabe.   They were just too small and crude.   So, 20 arc-minutes is just fine for an ephemeris.

    On Fri, Oct 19, 2012 at 2:04 PM, Lu Abel <luabel{at}ymail.com> wrote:
    Interesting points, thanks!

    I had only heard that the Portuguese "had constructed" a Sun declination table "just in time as they went south of the Equator" in their exploration of the West African coast, but I never thought of the technical issues you raise (which in turn make one wonder how accurate those tables were).  I guess it's just too easy to relax in these days of "having it" and not realize how difficult it was to go from "not having it" to "having it."

    Kinda like trying to explain why a difficult-to-use, often impossible-to-get-a-shot method of navigation was so, so important for the past 400 years save the last dozen or so where we have an infinitely better thing called GPS.  Or explaining calculations using logarithms or mechanical calculators....

    From: Brad Morris <bradley.r.morris---.com>
    To: NavList@fer3.com
    Sent: Friday, October 19, 2012 10:47 AM

    Subject: [NavList] Re: I couldn't resist!

    Hi Lu
    Let us begin with the ephemeris.  In order to construct a declination table one needs the following.
    1) An measuring device which is aligned to your meridian. 
    2) The measuring device must have an accurate division of the circle.
    3) Sufficient numbers of measurements such that any cyclic nature can be statistically derived
    4) A system of a calendar which accurately predicts the point within the cycle from which we extract the predicted value from the declination table.
    There may be other contributors, but these are the fundamentals which occur to me.
    Errors in meridian alignment will contribute to a systemic error in observation.  The altitude measured will always be wrong.  Errors in the scale, as produced by the same tools that created the astrolabe, will result in a reproducible result, albeit consistently long. 
    Sufficient numbers of measurements to determine the cycle and to derive a statistically valid result (in the absence of a calendar) will require at least 2 cycles, but more probably 6.  That's roughly 24 years of measurements, assuming you never miss a day due to overcast.  Mathematical interpolation to fill in missing points would have been challenging in the day.
    Finally, the calendar.  The calendar was not understood in Columbus' day, awaiting the 1500's re-alignment of the calendar with the religious holidays.  So which date to extract data from the table?
    In my view, this is the first contributor to in-accuracies.
    Best Regards
    On Oct 19, 2012 1:25 PM, "Lu Abel" <luabel{at}ymail.com> wrote:
    It would be interesting to know why Dias's measurement was off by such a large amount. 

    As I understand it, one of the achievements of the Portuguese navigators (who were the best in the world in the 15th century) was development of a declination table for the sun allowing sun shots to be used for calculating latitude (especially important south of the Equator where there's no Polaris to shoot).

    So to what do we attribute Dias's error?   Was the declination table off?  Was his "noon" shot off?  Were his measurements mis-recorded?   Or what?   I doubt that it was the crudeness of his instrument (astrolabe?  staff?) because even the crudest instrument wouldn't be off by 10 degrees.   Heck, I could measure up from the horizon by fist-widths and be off by less than 10 degrees!

    From: Apache Runner <apacherunner---.com>
    To: NavList@fer3.com
    Sent: Friday, October 19, 2012 9:26 AM
    Subject: [NavList] Re: I couldn't resist!

    All Polaris or eclipse, according to Morison.    I'm attaching a few pages I put together summarizing his measurements. Again, this is all from Morison's biography, I just pulled it all together.   

    What's strange about Polaris is that he has to do a 4 degree correction to get latitude depending on where it was in the sky (e.g. relative position to Cassiopeia and the Big Dipper).   

    As far as I can tell he only used the Regiomontanus information for eclipses and did the other sightings on Polaris because he memorized this 4 degree trick with Polaris.  

    Bartholomew Dias, on the other hand, didn't have the luxury of this Polaris trick, and I think he used a sun-shot for the latitude of the Cape of Good Hope (and was off by nearly 10 degrees, too).

    On Fri, Oct 19, 2012 at 11:53 AM, Brad Morris <bradley.r.morris---.com> wrote:
    No noon sunshots?  Okay, there's the end of my wild a**ed guess (wag). 
    I had assumed he did, based upon my memory of the following items.  (1) The ephemeris compiled by Regiomontanus included tables of solar declination (2) That the ephemeris was of the period (3) That Columbus had instruments capable of the measurement of the sun's altitude at meridian crossing.
    Odd to think that he did not put these items together to sail a latitude line. 
    On Oct 19, 2012 10:33 AM, "Apache Runner" <apacherunner---.com> wrote:
    I don't have all of Samuel Eliot Morison's bio of Columbus memorized, but I can try to summarize from memory.

    Western celestial nav was still in it's birthing stages when Columbus made his voyages.  I forget the precise year, but Don Joao II sent an astrologer on a Portuguese vessel down the coast of Africa in 1480.  He took a sighting of the sun and was off in latitude by about 7 degrees or so.

    Bartholomew Dias took a shot at the Cape of Good Hope and sighted 42 degrees when it should've been 33 or so.

    Columbus knew some of these shots, and took an ephemeris with him, quadrant and astrolabe.   On his first voyage, he took a shot of Polaris, which required a 4 degree correction back then.   In Cuba he got a latitude of 42 degrees (i.e. Boston's latitude).   Morison speculates that he misidentified Polaris.   Later in Haiti, he got a better shot, but was still pretty far off.   On his return, he tried a shot on approaching the Azores, but the wind affected his instruments too much. 

    On later voyages he did somewhat better.   On his 4th, he got a pretty good value of latitude for Saint Anne's Bay in Jamaica.   He also did a longitude measurement by using Regiomontanus' tables for an eclipse, but was off by a huge amount.   

    On his 3rd voyage, he took three shots of Polaris while in the doldrums, but one of them was way off on one of them and concluded that there was a large bump in the ocean.

    So, very much experimental - he wasn't really using the stars to navigate, but he was playing around with it.

    Most of the navigation was dead reckoning using magnetic compass and log line.   Only later in the 1500's, however, did people seriously try to be more systematic in correcting for magnetic variation.  

    On Thu, Oct 18, 2012 at 8:24 PM, Brad Morris <bradley.r.morris---.com> wrote:
    (Okay,no list comments on the fact that Columbus didn't practice celestial,I only learned that later)
    Didn't he cross the ocean on a latitude line?  Most likely by using the noon altitude of the sun?  That's celestial!
    On Oct 18, 2012 7:20 PM, "Lu Abel" <luabel{at}ymail.com> wrote:

    I totally agree on taking a run of sights.  I know of no text (either standard ones such as Dutton's or celestial navigation specific texts) that doesn't suggest this.  Some books even suggest plotting them on graph paper; where a consistent rise or fall in the curve and "outliers" are easily spotted.  In fact one can push beyond that and do a least-squares fit of a line to the data (impracticable in real life, but gives the mathematicians joy). 

    I also agree with "the more, the merrier" implied in your quote from Dutton's.  I think that a "starburst" of LOPs will quickly show any LOPs that are outliers and should be eliminated.

    But nowhere in Duttons can I find a suggestion that one not trust two LOPs or that one's position must be in the center of the cocked-hat formed by the intersection of three LOPs.     And there was heated discussion on this list about the latter, starting with the simple fact that if one assumes there's a 50/50 chance of one's true LOP being on one side or the other of the plotted LOP then there's only a 1 in 8 chance of the fix being inside the cocked hat.

    Last but not least, Debra said she's a beginner in celestial navigation.  I've taught a lot of beginners and in my experience it's a whole lot easier to help them see the magic of celestial when they can start with sights taken during daylight (ie, sun or moon).  I remember when I reduced my first sight 30 years ago (a sun shot) and I felt like Christopher Columbus when I reduced to an LOP only a mile or so from my KP!   (Okay, no list comments on the fact that Columbus didn't practice celestial, I only learned that later)

    Celestial has a steep learning curve -- handling the sextant properly, reading it properly (especially the vernier), measuring IE, bringing a body down correctly, rocking the sextant, etc, etc, not to mention timekeeping, and sight logging.  (In the last celestial class I taught we did Sun shots three hours apart and the second set of sights was for almost every student better than the first set for consistency and closeness to our KP).   So why make it more difficult by insisting that even a beginner has to take shots at twilight?

    My 0.0153 Euro's worth.


    PS - my Dutton's, which is the 1976 edition, seems to have lost a chapter -- your quotes are in my paragraph 2509.   Any reader have an idea what was dropped between 1972 and 1976?

    From: Brad Morris <bradley.r.morris---.com>
    To: NavList@fer3.com
    Sent: Thursday, October 18, 2012 3:07 PM
    Subject: [NavList] Re: Line of Position Fix

    Hi Lu
    Dutton's Navigation & Piloting
    Twelfth Edition
    Naval Institute Press
    PARAGRAPH 2609
    A fix is obtained when two or more lines of position are crossed,adjusted to a common time. ... At twilight, in clear weather, he will observe a minimum of five stars, well distributed in azimuth.  He will make 3 observations of each body. ... This gives a check on the consistency of observations and therefor their probable reliability.
    End quote
    The purpose of taking 15 observations of 5 bodies is to provide a more reliable fix, consistent for each body (3 observations), and consistent among bodies (5 stars). 
    If it was good enough for US Navigational Officers at the height of celestial navigation, then that's good enough for me.
    In deference to your considered opinion, you may follow the practice you see fit.
    Brad Morris
    On Oct 18, 2012 5:42 PM, "Lu Abel" <luabel{at}ymail.com> wrote:
    I agree that even on land LOPs are likely to be off by a mile or even more and so fixes have that uncertainty; it's just the nature of celestial navigation.

    But I have to question the claim that getting three LOPs will give a better fix.  As someone pointed out in a discussion on this list a few years ago, there's only a one in eight chance of a fix even being inside the cocked hat and there's no certitude at all that the fix will be in the center of the hat.

    From: Brad Morris <bradley.r.morris---.com>
    To: NavList@fer3.com
    Sent: Thursday, October 18, 2012 12:45 PM
    Subject: [NavList] Re: Line of Position Fix

    Hi Debra
    The best two times of the day are the civil twilights.  Those are just before dawn and just after sunset.  The best bodies to use are the 57 navigational stars.
    Just as a note, you will want 3 bodies in your fix.  It is unlikely that 2 LOPs will cross at your position, in general, and 3 will produced a 'cocked hat'.
    Best Regards
    Brad Morris
    On Oct 18, 2012 3:23 PM, "Debra Hillman" <wombatroo---.com> wrote:
    Could somebody help me with my problem . I am learning on my own at present from books on celestial navigation.I need to get a fix with two lines of position but not sure which celestial body other than the sun and the best times of the day to achieve the fix.

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