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    "Attainment of Precision" article (1964)
    From: Frank Reed
    Date: 2009 Jul 3, 20:30 -0700

    In message 8851, Douglas posted an article by Robert B. Gordon entitled 
    "Attainment of Precision in Celestial Navigation" published in 1964.
    Here's the link to it: http://www.fer3.com/arc/imgx/ACPN_s.pdf
    First, as an administrative note, I should mention that this article is 
    apparently still under copyright, so if the copyright holders request it, I 
    will have to remove it. Until then, grab a copy while it's available.
    The author of this article is Robert B. Gordon who went on to become a 
    professor at Yale specializing in "archaeometallurgy" (he is now retired). 
    Back in the early 1960s, Gordon was a young (early 20s?), recent engineering 
    school graduate with a passion for boating and an interest in celestial 
    navigation. He combined these interests to generate this article. 
    Incidentally, for those of you who've been to the "Treworgy Planetarium" at 
    Mystic Seaport, the first director there (when it was known as the "Seaport 
    Planetarium") was F.W. "Fritz" Keator who was a professor in engineering at 
    Yale and a prominent celestial navigation educator in the 1940s/50s. I have a 
    hunch he was personally responsible for convincing Gordon to write up his 
    celestial navigation experiments and try to get them published. I never met 
    either of them, though I've heard a great many stories about Professor Keator 
    from Professor Treworgy (the latter professor-ship is honorary --awarded by 
    the famous "Frank Reed School of Navigation").
    Gordon's article is about the accuracy of his experiments in standard line of 
    position sights. It's not about star-to-star sights, and he makes no attempt 
    to test the accuracy of the sextant independently of the accuracy of the 
    It's fun to note that Gordon's experiments were done in the same areas where 
    I've done many of my navigational experiments: Long Island Sound and Block 
    Island Sound. These are friendly waters where one is never out of sight of 
    land, and celestial navigation is therefore largely superfluous (even in 
    1964). Back then, the big advantage of working in such areas was that the 
    observer's position could be determined with some accuracy by taking bearings 
    of the numerous lights and other navigational aids available. Gordon mentions 
    that many of his observations were taken from a small boat. It turns out he 
    means a 16 foot boat. Yeah, that's a fairly small boat for celestial 
    navigation. Impressive!
    Gordon describes his setup for measuring dip (for land-based observations) 
    which is certainly interesting. He goes on to describe his computational 
    methodology which seems fine, and it's a reminder of how much more difficult 
    experiments like this were over forty years ago. He had to do all his 
    computation by hand or at best aided by a "desk calculator". He makes an 
    interesting case for dropping "outliers" in data which is an issue that comes 
    up occasionally on NavList. He recommends Chauvenet's condition.
    Gordon describes how he set about determining the index correction for his 
    sextant. It's interesting that he reports that the index error varied over a 
    range of 2' in five months. Some NavList members have commented that they 
    consider significant variability of IC (at least in a metal sextant) as 
    evidence that the instrument is no longer in good condition. I think this is 
    an open issue personally and would like to hear more (modern) data on it.
    Gordon attempted to get IC for his sextant using star sights. His results are 
    amazingly poor --I can't imagine any sextant user today not being amazed by a 
    range of FIVE minutes of arc. If my notes are right, he only attempted this 
    with his 2.5x scope which he later described as optically low in quality. 
    This might be part of the problem, but it's not consistent with the low 
    standard deviation of his altitude observations with the same scope reported 
    later. He talks about "imperfect" star images, and this surely suggests some 
    optical flaw, either in his telescope or in his eyes (again, dark-adaptation 
    can make star images un-necessarily "blurry"). Based on these poor results 
    estimating IC, Gordon adds a final comment saying that he doesn't think 
    star-to-star sights would work for testing a sextant's arc. This is an 
    extrapolation. He has no data on this point.
    Gordon next tries horizon sights for index error, and he discovers that he can 
    get excellent results with no telescope. This is a case of "vernier acuity" 
    of "hyper-acuity" though Gordon does not apparently know this. Incidentally, 
    this has come up before on NavList. I hate to be always quoting myself 
    (humble me --ha!), but here's how I described it way back in February, 2005: 
    "This hyperacuity only happens in these very specialized tasks, like a kink 
    in a long straight line. And the interesting thing for us is that some of 
    these tasks should be very relevant for sextant use. A step in the horizon 
    should be discernible at much finer resolution than the diffraction limit 
    (but it ought to be a sharp step so a telescope may actually make things 
    worse by providing a smooth transition across the field of view --thinking 
    aloud here)." And in fact, that's just what you find. I've experimented with 
    this on numerous occasions. If you look at the horizon through a sextant with 
    its telescope removed (wear your eyeglasses), you can get an excellent value 
    for IC --as good as you can get with a 7x telescope using other tests.
    Gordon dismisses using the common technique of aligning the Sun with itself 
    (aligned limb-to-limb). He apparently believes that this is too difficult 
    aboard a vessel. This contradicts a couple of centuries of experience among 
    celestial navigators, but for his small boat conditions, it's believable.
    Gordon finally gets to his data and his results. He finds a standard deviation 
    of about 0.4 minutes of arc for daylight LOP sights with a 2.5x scope (and 
    possibly with the 3x scope). For star observations, reported later in the 
    article, his standard deviation is about 0.6 minutes of arc. These results 
    correspond quite well with the normal range expected for standard LOP 
    navigation, though, in my experience, many navigators have reported results 
    closer to 1.0 minutes of arc. That is, they will say that they trust their 
    LOPs within +/- one nautical mile about two-thirds of the time. Gordon 
    doesn't mention what times of the year he did his sailing. This is relevant 
    because anomalous dip is quite common in Long Island Sound except in the 
    summer. Except in summer, I would bet on an s.d. of 1.0' with occasionally 
    significantly worse results due to anomalous dip. He does discuss this later 
    when he talks about systematic errors, but I can't tell whether he has chosen 
    his observations for his analysis categories based on this or not.
    Next we reach a section of Gordon's article which is just a mess. This is 
    where he discusses the resolution of sextant telescopes. He states that the 
    resolution of the telescope has to be tested with the scope "as mounted on 
    the sextant". This just isn't true. The mirrors have nothing to do with 
    resolution, unless they're very poor. He says he tested the resolution of his 
    telescopes by measuring the angular diameter of the star Altair. This is a 
    terrible idea, and it has little, if anything, to do with the resolution of 
    his telescopes. He gets ridiculous numbers for scope resolution (and he 
    should recognize that they're completely inconsistent with his LOP data). He 
    also compares this with something he calls the "theoretical resolving power". 
    What he's refering to is the diffraction-limited resolution of the 
    telescope's obective which has very little relevance here. At most this 
    theoretical resolution tells us the maximum magnification that we should use 
    in a given telescope. This applies only to telescopes which have different 
    oculars which can be swapped to provide different magnifications. Such things 
    were common in the 19th century on sextants, but they're rare today. A 
    sextant telescope is almost never supplied with an ocular magnifying beyond 
    the diffraction limit, and usually a sextant telescope has only one ocular 
    and thus one resolution, which is generally well above the diffraction limit. 
    Gordon mentions that one of his telescopes is optically poor and apparently 
    suffers from chromatic aberration. This is an important point. Though a 
    sextant telescope may be designed and sold as (e.g.) 3x or 7x in 
    magnification, if the optics are bad the corresponding resolution will not be 
    achieved. By the way, since Gordon has provided a worthless method for 
    assessing telescope resolution, I should mention one simple alternative: an 
    ordinary eyechart. Print one out (you can find them online; the scale as 
    printed does of course matter). So-called 20/20 vision corresponds to a 
    resolution of one minute of arc. You should be able to read that line on an 
    eyechart with normal corrected vision at 20 feet (+/- how much?). Now grab 
    your sextant telescope. If it's a 3x scope, you should find that you can read 
    that same line from 60 feet, assuming that there are no problems with the 
    optics. Your eye sees the same image three times enlarged by the optics, but 
    three times reduced from being that much further away --so the net is the 
    Gordon goes on to discuss sights in deep twilight and the middle of the night, 
    and he has some interesting ideas for a sextant optimized for such sights. 
    These are clever thoughts, interesting to consider, but of course no longer 
    of much relevance today.
    Next we get to Gordon's discussion of anomalous dip and anomalous refraction. 
    He claims that he finds anomalous refraction (not just dip) with observations 
    of objects above 30 degrees altitude. This conflicts directly with very long 
    series of astronomical as well as celestial navigation experiments, and it 
    conflicts with refraction theory, too. I can only conclude that Gordon 
    somehow failed to control properly for the very systematic errors that he is 
    attempting to analyze. The most likely explanation is that his sextant 
    suffered from insufficiently measured arc error. This is just the thing that 
    star-to-star sights would have helped him to identify --if he had understood 
    them. He "hypothesizes" that this he is seeing unusual refraction due to 
    tilted or curved surfaces of constant density in the atmosphere. For an 
    engineering student, it's not a bad hypothesis, but if he had consulted with 
    a meteorologist, an astronomer, or someone versed in atmospheric science, he 
    would have known better. The hypothesis is untenable.
    Gordon concludes with some general observations about the accuracy of sights 
    (which he says can be "within 0.2' under good daylight conditions") and he 
    adds some thoughts on increasing the accuracy of deep twilight and night star 
    All in all, it's an interesting little paper and also a delightful story --the 
    image of this young guy sailing single-handed around Long Island Sound and 
    recording so much data is just great. I would add that even in its positive 
    points, this article contains nothing we haven't discussed before, often in 
    greater detail here on NavList. Its errors, while few in number, are 
    remarkably bad. 
    Navigation List archive: www.fer3.com/arc
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