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I thank both to Herbert Prinz and George Huxtable for their discussion. The 
remark of Herbert that LOP navigation was to wait for the time when 
chronometers became common and then appeared very quickly with Sumner, seems 
very clever. The GMT obtained from lunar distances was so inaccurate and  so 
rarely obtained that the idea of LOP hadn't soil to originate before 
chronometers.

All formulas for computing the hour angle from declination, latitude and 
altitude, given by Bowditch, Sumner, Cotter, Herbert and George, belong to 
the "pre-haversine age". After haversines tables had appeared in nautical 
tables collections (Bowditch didn't contain them up to 1851), the pertinent 
formulas, constructed already at the beginning of 19th century, became much 
more popular from cca 1850:


On British merchant ships the Norie's formula was very common:

hav (H.A.) = sec (lat)  sec (dec) cos (lat + codec + alt) cos [(alt + 90) - 1/2(lat + codec + alt)]


In the British Navy the much handier Inman's formula was used for many decades:

hav (H.A.) = sec (lat) sec (dec) SQRT {hav [coalt + (dec ~ lat)] hav [(coalt - (dec ~ lat)]}

(see Cotter, p.252; for this formula, the tables of the log half haversines 
had been inserted into the Inman's tables and remained there to 1918, at 
least)


And at the end of days of Sumner LOP's, at the beginnning of the 20th century 
(before Saint-Hilaire broke the bank), this was the last formula for the 
"time sights":

hav (H.A.) = sec (dec) sec (lat) [hav (coalt) - hav (dec ~ lat)]

This short formula became popular only after Percy L.H. Davis had published 
the haversine table with log values and natural values in two adjacent 
columns in 1905 for the first time (after the Dutch in cca 1775, unnoticed).


Jan Kalivoda

   

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