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    Re: Classification of the methods for clearing the Lunar Distances
    From: Jan Kalivoda
    Date: 2003 Apr 11, 22:36 +0200

    ----- Original Message -----
    From: "Bruce Stark" 
    Sent: Thursday, April 10, 2003 7:17 PM
    Subject: Re: Classification of the methods for clearing the Lunar Distances
    > Never you mind what that writer in your 1962 Bowditch said about haversines.
    > Haversines were used by navigators before that fellow's grandfather was born.
    > It's just that the weren't called "haversines." They were called things like
    > "logarithms of the horary angle," "log sine square . . .," etc. They saved
    > two steps in working a time sight. Not every set of navigation tables had
    > them.
    Yes, it is true. The late date of introducing haversines into the nautical 
    astronomy in the 1962 Bowditch is valid in some extent only in the relation 
    to Marcq St. Hilaire method, the "new navigation" of the end of the 19th 
    century. It broke through only from 1890 or so, at first in navies. And at 
    the beginning various formulas circulated for reducing its "intercepts". 
    Thereafter the standard cosine-haversine formula won out from cca 1905 and 
    from 1910 the "short" methods started to be used.
    In 1905 Percy L.H.Davis published his "Requisite Tables", where the 
    log-haversines and nat-haversines were printed in two adjacent colums of one 
    table for the (nearly) first time, so that one could get across from logs to 
    natural values and vice versa without finding out the auxiliary angle 
    ("theta" in the cosine-haversine formula) itself. This helped the haversine 
    formula to the popularity very much. Rose added the third column with the log 
    cosine only after 1950.
    For computing the local hour angles of celestial body from old "time sights" 
    (to obtain longitude by comparing H.A. of a body with the chronometer or 
    "lunar" GMT) the versines or haversines were used at sea from 1767 
    (Maskelyne) at least. Their another name (in addition to the pseudonyms cited 
    by Bruce) was often "log risings". But combined tables of their natural and 
    logarithmic values were not published until 1905 - with the great exception 
    of Douwes' tables from 1775, printed in Amsterdam. These tables offered the 
    natural and logarithmic values of versine in two adjacent columns in the 
    exactly same arrangement, as proposed by Davis in 1905! Unfortunatelly, the 
    language barrier blocked this superb concept from being introduced into the 
    use of Anglo-Saxon and French navigators for 140 years.
    Jan Kalivoda

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