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Ed - I'm currently trying to understand how to do this from an Occultation which is similar (there's one on Sept4/5 with Aldebaran over the UK). If you get a copy of the 1861 Bowdith and look at page 409 you will see an explanation. There are various other sections worth reading as well. I find it quite hard to follow but hope to get there eventually. If anyone has a more straightforward explanation I'd appreciate it.

Extract Below

Regards

Bob Crawley

PROBLEM VI.
Given the latitude of the place, and the apparent times of the beginning and end of a solar
eclipse, countedfrom noon to noon, according to the method of astronomers, tofnd the
longitude of the place of observation.
In the rule for solving this problem, references will be made to figure 12, Plate XIII, in
which DSE represents a small arc of the ecliptic; S, the place of the centre of the sun
supposed at rest ; F, L, the apparent places of the centre of the moon at the beginning and
end of the eclipse respectively 5 FD, SC, and AEL, are perpendicular to UE ; FA parallel
to DE, and SB perpendicular to FL. Then it is evident tiiat FD, LE, represent the apparent
latitudes of the moon, which fall below DE if south, above if north ; and SF, SL, represent
the sums of the corrected semi-diameters of the sun and moon, at the beginning and end of
tlie eclipse respectively.
RULE.*
To tlie apparent times of the beginning