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

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

If you worried that "straight 0� to 390� notation" was yet another odd way 
to measure angles, think again. In was just a slip-of-the-keyboard for 360. 
Sorry about that.

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.
----- Original Message ----- 
From: "George Huxtable" 
To: 
Sent: Saturday, February 07, 2009 11:51 AM
Subject: [NavList 7292] Re: Fix by Occultations


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 360� (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, 0� to 360�,
would have been simple and logical. Instead, however, in early almanacs it
was always measured in "Signs", each 30� 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, 0� to 30� in longitude, the starting point
of the whole thing, 0� 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, 30� to 60�, and so on. The list is -
S      range
0    0� to 30� Aries
1  30� to 60� Taurus
2  60� to 90� Gemini
3  90� to 120� Cancer
4  120� to150� Leo
5  150� to 180� Virgo
6  180� to 210� Libra
7  210� to 240� Scorpio
8  240� to 270� Sagittarius
9  270� to 300� Capricornus
10 300� to 330� Aquarius
11 330� to 360� 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 12� 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 0� to 390� 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 17� 58' 28", so we have to multiply the 4 "signs"
by 30 and add 17 to get a longitude of 137� 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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