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    Re: Nautical Almanacs
    From: Frank Reed
    Date: 2010 Nov 25, 17:34 -0800

    George, H, of the nautical almanacs:
    "and the American version commences life by wholesale copying from the British?"

    Well, in a way... There WERE copies of the British "Nautical Almanac and Astronomical Ephemeris" published in the US in the early 19th century. Blunt and various other publishers had them printed. But these were unlicensed re-printings. No one suggested that they were anything beyond that (except to the extent that they were sometimes published after the official "errata" came out so it was possible to "clean up" printing errors found in the original). In fact, it was important commercially to say that the data were from the British Nautical Almanac. But that changed in 1852. The "American Ephemeris and Nautical Almanac" which started publication in the 1850s was a new work, independently calculated and laid out in a very different fashion.

    You wrote:
    "It seems likely to me that the overall error will be dominated
    completely by item 1, compared to which the others are all negligible."

    Sure. But I have exact values for the lunar distances in the period, so there's no difference in the work either way. It's a simple comparison. Really the only difficulty is reading some of the digits in the earliest editions since the printing was not that great and the scanning process has only made it worse.

    And you wrote:
    "I understand that the Moon predictions in the American almanac became
    more accurate on account of their earlier adoption of Brown's lunar theory.
    It would be interesting to see how that worked out in practice."

    Actually, according to Peirce who was the principal calculator of the American almanac (I can't find the exact reference right now), the higher accuracy of the American almanac when first published, which caused some embarrassment for the British calculators, had a simpler cause. They were using the same theoretical model of the Moon's motion as the British almanac, but the British ephemeris calculators had not updated their lunar tables in decades (the constants and coefficients were supposed to be updated regularly by comparing against observations), so they had slowly drifted away from the high quality that they had early in the century. I haven't independently confirmed that yet to my satisfaction, but here are some numbers to get you started:
    year s.d.diff
    1770: 0.3'
    1790: 0.3
    1795: 0.3
    1800: 0.3
    1811: 0.15
    1820: 0.1
    1845: 0.2

    As you can see, the standard deviation of the difference between the published lunar distances and the actual lunar distances (known from today's very accurate ephemerides) is stable until the beginning of the 19th century. The new lunar tables of Berg etc. are employed starting in 1807 (if I remember correctly) which means that they were first seen in the Nautical Almanac for 1811 since Maskelyne and his calculators were four to five years ahead of the game in this period. Incidentally, these changes are all outlined in Maskelyne's introductions to the Nautical Almanacs.

    The very high quality of the lunar tables around 1820 seems to fit nicely with David Thomson's claims published in 1825 about the expected accuracy of longitude by lunars (more below). Regarding these standard deviations, they're based on data sets of only twenty individual cases from each year which is why I am only quoting them to the nearest tenth of a minute of arc (except 1811 where I decided not to round). The cases selected were biased based on the common practice of historical lunars. I selected two-thirds Sun-Moon lunar distances, one-third Sun-star distances.

    By the way, for anyone following along who isn't familiar with this issue, the standard deviation error in the lunar distance tables would not simply be added onto the standard deviation of the observational error to get the net error. Rather, standard deviations add as the square root of the sum of the squares. So if you're looking at lunars in 1790 and you believe that an observer can get observations with a standard deviation of 0.3 minutes of arc, then combined with the 0.3' s.d. error in the tables, the net system error would be sqrt(0.3^2+0.3^2) or 0.42'. If in addition, you expect some calculational error from a particular method of clearing lunars that would give its own s.d. error of 0.2' which would not be surprising in 1790, then the net system error would have a standard deviation of sqrt(0.3^2+0.3^2+0.2^2) or 0.47'. By 1820, with the s.d. error in the lunar tables reduced to 0.1' and the calculational error at 0.1' or below, and when averaging sets of four or more the observational error could also be reduced to 0.1' or 0.2', the net error would have a standard deviation of only 0.17' to 0.24' corresponding to a standard deviation error in longitude of about 5 to 7 minutes of arc. Since this is the 1 s.d. error, that means that the longitude would be within those limits just about two-thirds of the time. That's nicely consistent, as I noted above, with David Thomson's statement in 1825 that "the longitude deduced from a set will seldom differ 10 miles from the truth." Naturally, the word "seldom" is open to interpretation.

    -FER


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