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I would suggest Letcher's Feb 1964 article in Yachting is the earliest
"modern" rediscovery of the longitude by lunar altitude methods.

As Frank points out, such methods were well known more than a hundred
of years ago.  The Ashe discussion is from: Monthly Notices of the
Royal Asronomical Society, Vol XII, nov 1851-june 1852, page 179.
(Available from Google books.)  Many of the astronomy and navigation
texts from the 1800's (and there are quite a large number on google
books) have references to longitude by lunar altitudes.  Since this
was before LOP, most discuss only longitude assuming latitude by one
of a number of other methods.  See for example, New Methods of finding
the Longitude at sea or on shore, by Yarrow and Lynn, 1826, also on
google books.

Below is an example using one of these methods with the data from
Bennett's recent "Longitude from Lunar Altitudes Simplified".

Outline:

Find local hour angle of star given altitude, latitude, and
declination of star.  Find local sidereal time given LHA (sometimes t
when measured +/-) and RA for star.

Extend LST to time and place of lunar observation.

Find LHA of moon given altitude, latitude, and declination of  moon.
Interpolate into RA of moon as a function of time to find time of RA
that gives correct LST when combined with LHA.



HAMEL@ 6 54 23

h       L       d

17.440000   41.258333   23.491667

fm sextant  known lat   alm dec



    cos t=  0.053432    calc fm h L d

    t(deg)= 86.937139   calc fm cos t

    t(hr)=  5.795809    calc fm t(deg)

    RA(hr)= 2.125222    fm almanac@6 54 23

    lst(hr)=7.921031    55.261890   15.713400

                min     sec





time dif st 2.246049





MOON@ 9 7 55

h       L       d

37.608333   41.401667   28.538333

fm sextant  known lat   alm dec



    cos t=  0.446633    calc fm h L d

    t=  63.472146   calc fm cos t

    t(hr)=  4.231476    13.888584   53.315014



lst(hr)=10.167081   10.024851   1.491079    required

ra(hr)= 5.935604    56.136268   8.176065    required



RA@9 0 0    89.008333   5.933889    56.033333   2.000000

RA@10 0 0   89.598333   5.973222    58.393333   23.600000



new time=   9.043616    2.616977    37.018614

delta time= -0.088328   -5.299690   -17.981386



Final answer, 5 min 18 sec vs published 5 min 47sec.  As close as
might be expected for the accuracies involved.





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