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Regarding Halley's calculational method for determining the Moon's true
position... In fact, he didn't need one as long as he had a diligent
astronomer back home willing to measure the Moon's position as frequently as
possible. For mapping and surveying, where the position determined by
observations can wait to be calculated when we get home (as opposed to live
navigation where we want the position on the spot), we don't need any lunar
theory at all. Imagine an alternate history where the reflecting sextant or
a similar instrument was invented 2500 years ago. If some king or emperor
with money to burn had decreed it necessary, astronomers could have been
sent to the far corners of the known world. They could have determined
longitudes as accurately as any late 18th century lunarian. A team of
astronomers in the imperial capital would measure lunar distances every hour
of every night, and travelling observers would measure the same distances
whenever they could in distant lands. Then, upon returning home, with some
clearing calculations (which would have been possible given the state of
mathematics though definitely challenging) those observations made in
distant lands could have been compared with the reference observations made
at home and as easily as that the world would have been mapped --to an
accuracy as good as world maps in 1800 (in the real world). Of course they
would need to understand lunar parallax (Hipparchus probably did) and they
would have needed a basic refraction table (which would have been discovered
almost immediately after the invention of the double-reflecting sextant). Of
course, an accurate map like that is nearly useless economically unless you
have a 'live' navigation method to go along with it.

 -FER



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