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Brad, trying to find the longitude of the Moon, had written-
" I just couldn't understand why the latitude was so small, and assumed it 
was due to the heliocentric nature. "

and Frank added-

"It would probably be useful to you to get a copy of the Nautical Almanac 
from that period. Then you'll be able to see exactly which data were being 
published. There are a bunch of old Nautical Almanacs on Google Books. I've 
got some links to some examples here: 
http://www.fer3.com/Mystic2008/navbooks1.html
and there are many more available at Google Books if you search around a 
bit."

=====================

Old almanacs are certainly instructive, and interesting in themselves, now 
that they are becoming readily available.

Moon longitudes and latitudes were given in  the early nautical almanacs at 
just two moments each day,  Noon and Midnight. Initially this was at 
Greenwich apparent time; from 1834, Greenwich mean time.

There are one or two problems anyone needs to be warned about when delving 
into old almanacs for ecliptic longitudes. Although Right Ascension was 
measured, in degrees up to 360º (or sometimes, in hours up to 24h), 
Eastwards around the sky starting at Aries, celestial (ecliptic) longitude 
was not measured in quite the same way. Longitude was also measured 
Eastwards around the sky from Aries, but around a different arc tilted at 23 
and-a-bit degrees, and to measure it in just the same units, 0º to 360º, 
would have been simple and logical. Instead, however, in early almanacs it 
was always measured in "Signs", each 30º wide, so there were 12 such signs 
around the circle. The first such sign was a block labelled with the 
sign-of-the-Zodiac name of Aries, 0º to 30º in longitude, the starting point 
of the whole thing, 0º in Aries, being literally "the first point of Aries", 
which today, we shorten to just the word "Aries". Then, after Aries, the 
next Zodiac sign is Taurus, 30º to 60º, and so on. The list is -
S      range
0    0º to 30º Aries
1  30º to 60º Taurus
2  60º to 90º Gemini
3  90º to 120º Cancer
4  120º to150º Leo
5  150º to 180º Virgo
6  180º to 210º Libra
7  210º to 240º Scorpio
8  240º to 270º Sagittarius
9  270º to 300º Capricornus
10 300º to 330º Aquarius
11 330º to 360º Pisces
and then restart at Aries again.

Unfortunately, emailese can't show those fancy symbols that go with the 
names, so I will add them as an attachment, taken from the first Almanac , 
for 1767. They can be useful, because some early almanacs didn't provide a 
useful "Sign-nunber" S, as Maskelyne did in his nautical almanac, nor a 
sign-name, but just the symbol itself. In which case, you need to know the 
symbol to put a given longitude into a sensible range of degrees. The 
symbols varied a bit, over time and in different countries.

The Sun, which has an ecliptic latitude close to zero, travelling round the 
line of the Ecliptic itself, stays within a sign for about a month. The 
Moon, with longitude increasing by about 12º per day, is in and out of a 
sign in about two and a half days.

I don't know at what date such "signs" units were dropped in favour of 
straight 0º to 390º notation, but they had certainly gone by 1864, probably 
much earlier.

Another problem is this. Because predicting a Moon longitude and latitude, 
by hand-calculation, involved such immense labour, Maskelyne asked for them 
to be calculated as seldom as possible, and settled on an interval of 12 
hours. But his lunar distances needed to be provided at 3-hour intervals, so 
Moon longitudes and latitudes were also needed, for that purpose, at 3-hour 
intervals. So interpolations were needed, between those Noon and midnight 
values, but the Moon's motion is such that simple linear interpolation would 
have been hopelessly inaccurate. What Maskelyne had to do was, effectively, 
to fit a curved line to the nearest four predicted values, and then to 
interpolate using that. So we, like Maskelyne, can't interpolate between the 
noon and midnight predictions without using similar tricks.

How accurate were those predictions anyway? They were provided to the 
nearest arc-second, but that's simply because that was the next unit of 
angle smaller than an arc-minute. The claim was that they were good to a 
half-minute, but I doubt if they came reliably within that error-limit for 
many years.

As a random example, for Greenwich apparent noon 12 March 1767, the Nautical 
Almanac gives Moon long 4S 17º 58' 28", so we have to multiply the 4 "signs" 
by 30 and add 17 to get a longitude of 137º 58' 28", taking the 28" with a 
grain of salt.

All in all, you are far better nowadays to get predicted positions, Moon, 
Sun, or Planets, for those early dates, from a modern prediction program. 
It's odd but true that we know now where those bodies were, then, far better 
than they did at the time. However, I don't know where you will find modern 
predictions showing ecliptic latitudes and longitudes directly. If somebody 
does, please say. My own Skymap display program will display a grid of 
ecliptic lat and long on top of a picture of the sky, but won't show the 
calculated values. What I think you have to do is to obtain precise modern 
predictions for RA and dec, then convert to ecliptic long and lat using the 
Meeus expressions I copied into an earlier posting. But if you are trying to 
work to great precision be sure to use the contemporay value for the Earth's 
orbital tilt.

George.

contact George Huxtable, at  george@hux.me.uk
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.


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