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    Re: Ephemerides at Columbus' Time Earlier: I couldn't resist!
    From: Brad Morris
    Date: 2012 Oct 29, 15:37 -0700

    Hi Herbert

    To perform the procedure you advocate, we will require a single table look up function, also known as linear interpolation. 

    I count 3 subtractions, 1 division, 1 multiplication and 1 addition to perform this task.  While the subtractions and addition may be well within the grasp of the ancient mariner, I do consider the multiplication to be hard and the long division well nigh impossible for them.

    It was for this reason that logarithms were created, that is, to eliminate those pesky multiplications and divisions. 

    So yes, in my opinion, the two table solution advocated is not obviously useful.  The proof of this is that the direct table was available no later than 1509, as Wolfgang points out, a mere 17 years after Columbus.

    I am still patiently awaiting confirmation that the first of the two tables (date to sun's longitude) was available.  I do hope we can have confirmation either way.

    Best Regards
    Brad

    On Oct 29, 2012 5:51 PM, <666{at}poorherbert.org> wrote:
    It does not strike me as excessive to look up the noon longitude of the sun in an ephemeris and to use this value as argument for one further lookup in a declination table. (Dependent on step width of the argument in the declination table, interpolation may be required.)
     
    In most modern celnav procedures you similiarily use the NA together with sight reduction tables (such as HO 229). Are these all useless procedures in your opinion?
     
    Herbert
     
    P.S.
    As I am writing this in 40kts of wind gusting to 55, using a depleting UPS and from a fourth-world-country that still uses mostly overhead power lines, I might not be able to follow this thread in the coming days.
     
     
    -------- Original Message --------
    Subject: [NavList] Re: Ephemerides at Columbus' Time Earlier: I
    couldn't resist!
    From: Brad Morris <bradley.r.morris---.com>
    Date: Mon, October 29, 2012 10:58 am
    To: NavList@fer3.com

    Hi Herbert
    If there are a significant number of operations and mathematical transactions, the chance for error is high.  That's what I mean by usable.  A table is usable when the person using it is capable of deriving a meaningful result.
    Brad
    On Oct 29, 2012 1:41 PM, <666{at}poorherbert.org> wrote:
    What do you mean by "usable"? What's wrong with a table of the sun's declination versus its longitude? Such a table you can find in the Almagest.
     
    Herbert
     
    -------- Original Message --------
    Subject: [NavList] Re: Ephemerides at Columbus' Time Earlier: I
    couldn't resist!
    From: Brad Morris <bradley.r.morris---.com>
    Date: Mon, October 29, 2012 10:31 am
    To: NavList@fer3.com

    Hi Wolfgang
    In order to be usable, the declination table should be the sun's declination vs the day.  When did such tables first appear?
    Regards
    Brad
    On Oct 29, 2012 1:02 PM, "Wolfgang Köberer" <koeberer{at}navigationsgeschichte.de> wrote:
    Just a short remark on Bartolomeu Dias: There is no indication that Dias was “off by nearly 10 degrees” other than two marginal notes written by Columbus or his brother Bartholomew in Columbus’ copies of Pierre d’Ailly’s “Imago Mundi” and Pius II’s “Historia rerum ubique gestarum” in which he alleged this. No Portuguese source mentions such a fact.
    Leaving aside the question who really wrote this marginal note and when, there are several reasons to doubt the story: The Portuguese who had opened up a route to the East which was clearly in the realm that was granted to Portugal would certainly not tell a foreigner useful facts about that route. So there is a possibility that he was just misinformed. Another possibility would be that Columbus was spreading misinformation himself for obvious reasons.
    One thing is not possible, though: that Dias was really 10 degrees off. He was accompanied by first rate navigators, among them Pero de Alenquer, who would certainly not commit such an error. Apart from that the latitudes that are found in the first manuals of navigation a couple of years later are rarely off more than 30 minutes. And they were taken by the same instruments – astrolabes and quadrants – because the cross staff was not in use by navigators then, they are first mentioned on board in the 1520s. These instruments give quite good results as has been shown by experiments undertaken by Nicolas de Hilster (see his homepage) and Jose Malhao Pereira (Experiências Com Instrumentos e Métodos Antigos de Navegação Lisboa 2000), so an instrumental error of that magnitude is clearly out of the question. The same goes for the declination tables that are off only a few minutes according to modern calculations.
    Wolfgang
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