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    Re: Accuracy of 1767 almanac
    From: Frank Reed
    Date: 2013 Aug 8, 16:14 -0700

    Peter, you wrote:
    "I guess disparities this small are probably not worth worrying about
    for the purpose of looking at historical almanac error, which by all
    accounts was still ~10 arcsec at the end of the lunars era."

    I did a bunch of test years of lunar distances a while back, comparing the published distances with nearly exact modern values. Today I happened upon my spreadsheet with the numbers. I tested tabulated lunar distances for the years 1770, 1790, 1795, 1800, 1811, 1813, and 1845. I chose those years because I was trying to test out a couple of historical details regarding improvements in the calculated distances. From 1770 through 1800, the scatter in distances is relatively stable. They show a typical mean error of about one-quarter of a minute of arc and a standard deviation of about one-third of a minute of arc. This is consistent with a relatively uniform distribution of errors over the interval from -0.30' to +0.75'. That mean error explains a consistent bias eastward of about 0.125° in longitudes by lunars obtained in the late 18th century. That, by the way, apparently accounts for nearly all of the "easting" bias in Cook's longitudes described in that neat little video summary of the analysis of Cook's lunars which we discussed a few months back, assuming I've got my signs right (otherwise the bias is twice as large!).

    In the early 19th century, the lunar tables were improved significantly. I don't know the exact date that the prize-winning lunar motion theories of Bürg and Burckhardt filtered into the published almanacs (which were prepared some years ahead), but the first implementation was around 1810. By 1811, the mean error in the tabulated lunar distances is under 0.1 minutes of arc and the s.d. is about 0.2'. By 1820, those values were roughly halved again. At this date, the tables are essentially perfect for navigational use. Unfortunately for those hard-working calculators, this was just about the time that lunars were disappearing from use aboard major British vessels, though they lasted another thirty years aboard American vessels. Presumably due to this "lack of interest", the lunar tables in the British Nautical Almanac slowly succumbed to drift towards the middle of the century. By 1845 the mean error had climbed back to 0.2' though the standard deviation was still less than 0.1'. This drift was apparently the source of the error which was seen when the British Nautical Almanac was compared against the newly-published American Ephemeris & Nautical Almanac starting around 1855. It was easy to fix by dusting off the original tables (from which the lunar distances were calculated) and updating them based on contemporary observations of the Moon. Of course by then all of this was really of no further interest to navigation.

    -FER
    PS: My "mean error" and "standard deviation" numbers listed above are based on a relatively small number of cases in each calendar year, so I trust them to less than two significant figures (but better than 1 sig fig). My samples of lunars were based on my statistics of actual usage so I looked at lunar distances strongly biased in favor of Sun-Moon distances but with some star-Moon distances thrown in, too, in a ratio of about four-to-one.

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