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    Re: Ephemerides at Columbus' Time
    From: Brad Morris
    Date: 2012 Oct 21, 13:44 -0700

    Hi John

    I get the sense that your friend Owen has actual copies?  Some these would be incunabula, or books published before 1500.  In a single word: wowiewowwowwow!  Okay, maybe that's just one too many wows to be considered one word.

    If so, I am extremely jealous!  In my wildest dreams I would hope for such a book.  Owen lets you touch them?  Good golly miss molly!

    One of my ebay search terms is ANY incunabula.  Some sellers advertise as post-incunabula.  Those are published after 1500.  Even marginal incunabula of random topic are $$$$$. 

    Best Regards

    On Oct 21, 2012 4:21 PM, "Apache Runner" <apacherunner@gmail.com> wrote:
    Hi, Brad - 

    You're probably right that the shift from date to longitude is the trickiest.   On that point, I tried to construct my own ephemeris based on the Ptolemaic scheme.   If you screw up that conversion, I found that at certain times of the year, you can be off in declination by as much as 1.7 degrees. In this case, I was doing a simple mapping of (days since the vernal equinox)/365.24 onto the ecliptic.   

    On the other hand, with some care, I maintain you can probably get down do about 20 arc-minutes if you do that conversion more carefully - just based on experience.

    Owen Gingerich offered to have me over to his office and spend an afternoon with his ephemerides and look at the accuracy one achieves in solar declination going through the whole process.   This won't be until something around Thanksgiving, so I won't have anything to report back until then.       

    We'll see if my 20 arc-minutes prediction holds.   That's a guess on my part, but I'll find out.

    If someone wants to try this with available reproductions, I'd be interested to hear the answer.   I won't have time to do this for about a month. 


    On Sun, Oct 21, 2012 at 2:43 PM, Brad Morris <bradley.r.morris@gmail.com> wrote:

    Hi John

    You wrote:

    Then you go to some other section of the book and look up declination versus longitude to get a declination.

    From Cambridge, we have:

    Regiomontanus produced the "Tabulae directionum,produced with the assistance of Martin Bylica,were primarily for astrological use and seem to have been correspondingly more popular: they were first printed in 1490,and went through eleven editions up to 1626. In addition to tables for calculating horoscopes,the collection included a table of the declination of the sun for every degree of longitude in the ecliptic ..."

    Since the pioneers would know their date (I concede on this point), they would merely have to transform from date to ecliptic longitude.  From http://farside.ph.utexas.edu/syntaxis/Almagest/node34.html we have equations 104 through 107 which perform that exact transformation.

    I count several multiplication and some trigonometric functions.  These would have been extremely challenging for the era.  No calculators or computers.  Gunter's rule was 100 years into the future.

    I'll grant a perfect declination table.  The error is most likely in the reduction from date to ecliptic longitude.

    Best Regards

    On Oct 21, 2012 1:24 PM, "Apache Runner" <apacherunner@gmail.com> wrote:
    I'm meeting with Owen Gingerich to go over the structure of his copies of Regiomontanus's ephemerides.   Unfortunately it won't be for a  month or so.  I'll certainly report back. 

    One way I could see structuring an ephemeris would be something like the following:  report longitude versus date/time.   Then you go to some other section of the book and look up declination versus longitude to get a declination.     

    In the approximation that planets are confined to the ecliptic, then this works reasonably well, particularly for the purposes of astrology.   As far as Bartholomew Diaz and using a solar declination, with errors of 10 degrees, small differences of a degree or so aren't going to be of great import. 

    On Sun, Oct 21, 2012 at 11:29 AM, Apache Runner <apacherunner@gmail.com> wrote:
    Just to add - ecliptic coordinates are helpful for astrologers, as the audience typically wants to know where planets are located, and since the planets mostly are confined to the plane of the ecliptic many ephemerides of the period used an ecliptic system.  

    Also - I only recently learned this from none other than....this group!   I was trying to decipher some odd entries in an online ephemeris.

    On Sun, Oct 21, 2012 at 10:30 AM, Apache Runner <apacherunner@gmail.com> wrote:
    I have to go back to Owen and get him to open them up again for me.  I know that there were a number of different kinds extant from Regiomontanus, so I'm going on memory what Owen showed me.   There was definitely planetary and solar data on one side of a page and astrological data on the other.  

    A quick look at the literature online, I found "A Survey of European Astronomical Tables in the Late Middle Ages."

    They compare tables of solar declination in three sources, using the argument of the Sun's longitude for declination.

    The three tables have a maximum declination of 23 30, 23 33 and 23 30.

    Regiomontanus has a maximum declination of 23 28, according this this source.   

    In another book, there's a discussion as to whether an astrologer named Zacuto copied one set of Regiomontanus' tables for Don Joao II.

    Certainly there were multiple tables made by Regiomontanus, and I'm fairly sure that some of them had solar declination in them.

    In comparing the different sets of declination data in the above book, it looks like they agree to a few arc-minutes around the equinoxes and the solstices, but have more divergence midway between the two. 

    Eg.   with an argument of 30 degrees, they have 11 31 11 30 and 12 17 for the declination. 

    On Sun, Oct 21, 2012 at 3:39 AM, <666@poorherbert.org> wrote:
    Dear Apache Runner,
    I am curious to know what tables you have looked at. The ephemerides of Regiomontanus, like any other that I have seen from that period, do not contain RA or Dec of anything. The longitude of the sun, moon and planets is all you get, plus the longitude of the ascending node of the moon. To obtain the declination of the sun, further computation and/or auxiliary tables are required. Even Origanus, 100 years later, provides only the latitude of the moon in addition to the above. Not exactly seaman-friendly! 
    Herbert Prinz
    -------- Original Message --------
    Subject: [NavList] Re: I couldn't resist!
    From: Apache Runner <apacherunner@gmail.com>
    Date: Fri, October 19, 2012 10:37 am
    To: NavList@fer3.com

    I'm quite sure the ephemerides of that period were quite accurate at that level.   I have a colleague who has a collection of ephemerides from Regiomontanus, Copernicus, and Kepler.   They're fascinating to look at.

    There are two pages, side by side.   One contains information like declination and RA, and the other page has astrological information.   Evidently this was quite common.

    The construction of solar declination was probably accurate to about 20 arc-minutes or better.   I can get this from my colleague, but I tried constructing a table using Ptolemy's method, and somewhat crudely and got to this level, so I'm quite sure that Dias had an ephemeris that was accurate to about this level.

    I suspect more one of two plausible problems for Dias.   The Portuguese in that period had a modified astrolabe.   Rather than being filled up with a star finder and a projection of altitutde/azimuth coordinates, it had four cut-outs on it to limit wind buffeting.   Still, they got buffeted by the wind.   

    I don't know offhand whether Dias went ashore to take the sighting, but it's possible in my thinking that he might have taken it on ship-board, and the instrument got pushed around in the course of taking the shot, and he didn't notice this.   It was at a time when the techniques were still in their infancy.   

    On Fri, Oct 19, 2012 at 1:24 PM, Lu Abel <luabel@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@gmail.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@gmail.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@gmail.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@gmail.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@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@gmail.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@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@gmail.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@hotmail.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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