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    Re: Azimuth and Declination formulae
    From: Wolfgang K�berer
    Date: 2005 Jul 25, 13:27 +0200

    Some comments on Lu Abel`s remarks on the history of tables.
    Ad 2.: Portuguese historians of navigation agree that there was no such
    thing as " Prince Henry's institute". Tables of the declination of the sun
    were produced long before the mid-1400's; they gave the declination of the
    sun in relation its position on the ecliptic. The innovation made in
    Portugal at the end of the 15th century was to introduce tables of the sun`s
    declination with respect to the day of the year - first in a single table
    serving for all years (and introducing corresponding errors), later for a
    four year cycle. The four-year tables were available at the time of the
    Magellan circumnavigation (and probably developed for that).
    Manuals of navigation in all European languages - starting with Pedro de
    Medina`s "Arte de navegar" (1545) have contained such tables since the
    middle of the 16th century; they also had tables of star positions, though
    it is hard to imagine that they were of any use.
    Although almanacs for seamen - like Tapp`s "Seaman`s Calender" were printed
    around the turn of the 17th century, almanacs as we know them were only
    printed at the end of the 18th century, starting with the "Nautical
    Ad 3.: Tables for facilitating computations in sight reduction have existed
    since the "Tables requisite..." that accompanied the "Nautical almanac...".
    The real breakthrough for "short inspection tables" to look up precomputed
    values of hc and az came with the advent of long-range bombing in WWII where
    airplane navigators did not have much time to compute positions from star
    observations. The British development of these tables is described in the
    following article: Anderson, E.W. and D.H. Sadler; The Genesis of the
    E.A.N.T.s., in: Journal of the Institute of Navigation, Vol.  6 (1953), 333
    - 357.
    W. K?berer
    -----Urspr?ngliche Nachricht-----
    Von: Navigation Mailing List [mailto:NAVIGATION-L@LISTSERV.WEBKAHUNA.COM] Im
    Auftrag von Lu Abel
    Gesendet: Mittwoch, 20. Juli 2005 16:35
    Betreff: Re: Azimuth and Declination formulae
    Peter Fogg wrote:
    >>From Lu Abel
    >>...The whole reason for versines and haversines was to allow sight
    >>reductions to be done using logarithms (and therefore the requisite
    >>multiplications become additions); but logs are not defined for negative
    >>numbers, hence the need to shift everything to have a positive value.
    > So in order to use haversines log tables are needed?
    The opposite:  In order to use logs (optional, but it makes multiplying
    4 or 5 digit numbers a hell of a lot easier!), you need to use haversines.
    I'm not the expert on the history of navigation like some of our other
    list members, but in historical order:
    1.  Sight reduction formulae (actually, spherical triangle formulae) --
    developed by Euclid and pals 2500 years ago.
    2.  Nautical Almanacs.  Prince Henry's institute in Portugal produced
    the first tables for the sun's declination in the mid-1400's; it
    wouldn't surprise me if Arab astronomers had produced much more data
    much earlier.  With the explosion of interest in astronomy over the next
    couple of centuries, it also wouldn't surprise me if almanacs much like
    today's existed by the end of the 17th century.
    3.  (Long pause -- like 3 century's worth)  Sight reduction tables.
    Bottom line:  for many of the great explorations of the 17th and 18th
    (and perhaps even 19th) centuries (and for simple ship-borne commerce,
    too!), navigators could take sights easily.  But sight reduction was
    difficult.  Without reduction tables (like our HO229 and its
    predecessors), navigators had to do the equivalent of what many of us do
    today -- run through the sight reduction formulae -- but without the
    benefit of a calculator.
    Multiplying multi-digit numbers ain't fun and, more important, is error
    prone.  Being able to ADD the logs of those two numbers is a lot quicker
    and simpler.  Hence haversines.
    Lu Abel

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