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    From: Douglas Denny
    Date: 2010 Nov 25, 04:35 -0800

    This is a second try at posting my comments for discussion. I sent it some days ago, when it was not published by Frank Reed. I cannot do so myself with the code I used to use, as it has been withdrawn by Frank Reed. To disallow my reply to his comments on the subject after his mentioning my name personally and making comments, is no less than self-interested censorship in very poor form.

    It would appear Frank Reed is frightened of criticism, and cannot stand counter-argument to his cherished beliefs. It seems he refuses to publish this refutation of mine, of his claimed 20 second of arc as a consistently achievable measurement with a hand-held sextant for Lunar Distances.

    Is he to have his opinion only on this website prevail and not let others have their opinions aired? Is this fair? Is this the American way? It is certainly not British.

    I have sent it by e-mail to two others on this 'Navlist' so they know the content already. I think others might find the content interesting. So: I challenge Frank Reed to publish this offering of mine now. Here it is again:

    It will be my last posting here I think.


    Frank Reed wrote:-

    "...Denny wrote:-""It has still not disinclined me from my final conclusion however, that one minute of arc (i.e. half a moa absolute from the 'truth') is about the best one can manage under normal circumstances.""

    This, of course, is EXACTLY what Feynman was talking about. This isn't his "conclusion" based on his data --it's the pre-conceived notion that he took into this process.
    And Douglas wrote:- ""And yes, I have discovered there was another bug in what I was doing, but it was not not the programming of the calculator. That is correct.""

    And there are no doubt more. This is one of the problems that crops up now and then when modern celestial enthusiasts attempt lunars. They frequently attempt to "roll their own" solutions when their are excellent, easy methods available from the historical period, as in Bowditch and Norie, and also excellent modern solutions like Bruce Stark's tables or like my software tools (which have been well tested by many users so I feel comfortable calling them "excellent" despite the way it sounds). And having "rolled their own" creative solutions, they get results which are mediocre and then blame the method itself...."

    This is highly disingenouous, and ignores the facts of what I have actually written.

    In order to denigrate my assertions Frank uses accusations that:
    1) I am not being objective (inciting Fenman etc) and
    2) The admitted (small) errors in my calculation method are the cause of my dissent to _his_ stated assertions and which I dispute, that it is possible to obtain consistently 20 second accuracy in measurements for Lunars with a hand-held sextant.

    Let me deal with these smears:

    My conclusions are based on my own measurements and observations and _not_ on the calculations.
    Moreover, those measurements I made with a fairly good sextant and with a fairly good x8 telescope - the East German Zeiss Frieberger sextant - which I think is as good as the West German Plath which of which the Zeiss is a copy. (Though I would readily bow to Bill Morris' judgement on this if he cares to give his opinion on a comparison).

    My criticisms are not in the calculations side of it - that is solved with modern calculators; and the Lunar Distance Tables are as near exact as you want (which was a serious criticism of them in the nineteenth century); and my calculations though not perfect with two small errors found, the inaccuracies were mainly in the altitudes which, as Frank rightly points out, are not so sensitive an issue in the lunar calculation anyway.

    No, ... the conclusion I have come to that within one minute of measurement accuracy (plus or minus half a minute of arc) absolute is about as good as you can expect is based on the _spread_ of my results in my experimentations, not the absolute value which would be affected by the slight errors in calculation of slightly incorrect altitudes and which are negligable anyway in this discussion.
    I am only obtaining a results spread of (plus or minus) within six minutes of time, which equates to a measurement accuracy of around one to three minutes of arc.

    Whereas the calculations were slightly incorrect for the reasons given, and therefore lay me open to his accusation, I repeat, my conclusions are based on my experience of the _measureing process_ and hence the inherent difficulty and variability which is the inevitable consequence of trying to place a star (or in most of my measurements, Jupiter) onto the image of the Moon's limb with a hand-held sextant.
    To expand on this:-

    Even with a good sextant, (and one which you could reasonably say was 'perfect' with no errors of index error etc); and even when steadied on a tripod as I have done, it is completely clear to me that it is a very difficult task indeed to get the correct placement of star/Moon exactly right with it just touching the rim.
    And THIS is where I believe it is just _not_ possible to get down to 20 second accuracy consistently, when Frank says it is. I know he says with the certitude of telescope optics that you can see down to seconds of arc etc, but I still I disagree entirely. In fact I just don't believe it. Putting that slightly blurry (which even a good telescope has) of light onto the moon's rim is not so exact a process as he says is possible. I have tried it. That is what the experiment I carried out was all about. You can put Jupiter definitely on one side just touching as you think; then on the other side just touching as you think ...and you wouldn't know the difference within a minute and half of movement of the micrometer drum.
    I would like to see him in person perform this with a number of Lunar sights to prove it his 20" accuracy claim.

    Any star in any telescope is not a point source of light it is a blur-circle of a certain diameter plus distortions/aberrations; and Jupiter is a tiny disc plus aberrations. Placing this aberrated disc onto the rim of the Moon (also changed in true position with irradiation to an unknown amount depending on the illuminance of the Moon's image and the operator themselves) makes for a variation of about one minute of arc I would say. You would have to be very skilled indeed to be sure (for certain sure that is!) that it is not the outside of the aberrated disc you are seeing on the Moon's rim and not the inside of the aberrated disc - when in fact it should be the _middle_ of the aberrated disc right on the rim's edge. A one minute of arc inaccuracy is the variation here I find - or even more with a wobbly image of Moon/star as one actually has with a hand-held sextant.

    Your calculations can be as correct as you like, (to within sub-second arc accuracy if you want!) but you are not going to be able to overcome this inherent measuring inaccuracy without land-based, sophisticated instruments. We are talking hand-held sextants if you please! .. and at sea too!
    I think Frank's obsession with 'Lunars' has obscured _his_ objectivity to the practicalities of the whole business, so to accuse me of lack of same and cite Feynman as a riposte is a bit rich!

    Let us see what previous experts in navigation have to say about Lunars:

    It is instructive to consider what two of the great works navigational expertise have to say about it.
    i.e. Liet. Henry Raper RN. in his 'Practice of Navigation' (my copy is 1849) when Lunars were in full swing; and Captain Lecky's 'Wrinkles in Practical Navigation' (my copy in 1899) when Lunars were very much on their way out of use.

    Raper first, who you would expect to be completely objective about them as they were in full use at that time:

    Raper: " The true distance is affected by errors of observation, and by errors of computation. An error in the distance, of whatever kind, produces on the average, about 30 times its amount in the longitude; thus 1" of error of distance produce about 300" or 5' of longitude.
    The observed distance is liable to the ordinary errors of angular distance, the chief of which are, perhaps, most usually that due to defect of parallelism of the telescope, and that arising from making the contact above or below the centre of field . Irradiation is also included in the errors of observation."

    You will note I have already commented on irradiation on this forum - only for it to be poo pooed by Frank previously.
    Irradiation was such an important issue for the space sextant measurements for the Gemini programme they went to enormous trouble to assess it and try to eliminate it. There are a number of papers by NASA (available on the internet, and links given by me here) on irradiation at the Ames laboratory, and done over a number of years which I have mentioned on this forum - only to have them ignored by Frank Where is the objectivity there Frank? Irradiation is a very definite error-inducing factor for moon/star observations and can be some minutes of arc - swamping other issues completely.

    I have also mentioned on this forum regarding my experiments, that I too have noticed over one minute at least and possibly more than two minutes difference in observing from the centre of field of the telescope towards the edges, which is very easily done as the arm tires and drops. This is a standard prismatic effect of all lens systems for off-centre rays - which, I note, Raper describes quite nicely above, (and incidentally it is the first time I have seen Raper's comment having just looked it up).

    Raper again: "....The effects of errors of a few minutes in the altitude are insensible. hence and ill-defined horizon ins no great detriment to a good observation; and hence, also, in computing the altitudes, precision is not essential. this last remark is worthy of attention, since the calculation of altitudes is a heavy addition to the work of a lunar....".

    Raper also says: "...When the altitudes are not observed they must be calculated. M.T. is supposed to be given...".

    These two paragraphs above show conclusively the whole business of a _calculated_ altitude depends on a known assumed local Mean Time within reasonable limits of absolute, to provide for altitudes "within a few minutes" for the lunar part of the calculation.

    It was this which initially puzzled me about calculated altitudes - because you cannot calculate altitudes for a lunar without knowing fairly closely what your local mean Time is. In other words, it is a reliance on, and justifiable use of an approximation, by utilising the _practical_ point about the insensitivity of the altitudes in the lunar calculation.

    Raper suggests three or five observations, and even suggests corrections for barometric pressure and temperature and has interesting suggestions for the observer:

    "Raper: "..Since the most rapid change of distance is about 1deg-48' in three hours, the effect of 1' error of distance is 25' of longitude: or the effect of 15" of error of distance is 6' of longitude - in the most favourable case..."

    Raper: "...When at sea, the ship has much motion, the observer fixes himself firmly in a corner, or lies on his back on the deck, in order to remove as much as possible, the sense of bodily effort and inconvenience which disturbs the eye and attention..."

    At this time the tables were a problem:

    Raper: "..After the result has been obtained with the utmost care, there still remains the error of the lunar tables, which appears to be about 0."5of RA or 4' of longitude.

    So in the mid-1800s evidently it was expected to have an error of at least 4' of longitude no matter how accurate you were with your observations / calculations.


    And now to Captain Lecky: fifty years onwards from Raper's comments in the mid-nineteenth century to the turn of the century into the twentieth.
    He is, to say the least, utterly dismissive of 'Lunars'.

    Lecky: "... This brings us to 'Lunars', the only astronomical method of finding Greenwich mean Time, with any degree of accuracy, which can be used for this purpose on board ship.
    Though still clung to by some of the few of the ancient ones, who, in their snug retirement, write to the papers and magazines under various assumed names; 'Lunars' are rapidly dying out along with their advocates, and the rising generation mostly look upon them in an unsympathetic spirit as "fancy navigation". Excellent chronometers can be purchased brand new for £26 to £30; when second-hand, and equally good, for much less: in fact they have become a drug on the market. On long voyages the better class vessels seldom carry fewer than three........
    ........ Thanks to the mathematicians, who from the time of Newton have given to what is called the 'Lunar Theory' in Physical Astronomy the perfection which it now possesses, we can tell for years in advance, where the Moon will be relatively to the stars, at any moment of Greenwich Time, more accurately than it can be observed at sea, and almost as accurately as it can be observed at a fixed observatory on shore. Hence the error of the clock is known more accurately than we can read its indications at sea, and the accuracy with which we can find the Greenwicvh Time by it is practically limited by the accuracy with which we can observe the Moon's place relative to Sun, planet or star. This unhappily, is.. ** very rough in comparison with what is wanted for navigation.** (my emphasis- Douglas).

    The Moon performs her orbital revolution in 27.321 days, and, therefore moves at an average rate of 0.55deg per hour, or 0.55 of a minute of angle per minute of time. Hence to get the Greenwich Time to one minute of time, or longitude within 15', it is necessary to observe the Moon's position accurately to half a minute of angle. **This can be done, but it is about the most that can be done in the way of accuracy at sea.** (my emphasis - Douglas).
    ...... In the case of Lunars it is done by measuring, with the sextant, the angular distance of the moon from the star, as nearly as may be, in the great circle of the moon's orbital motion. Thus, supposing the ship to be navigating in tropical saes, where a minute of longitude is equal to a mile of distance, a careful navigator, with a good sextant, whose errors he has carefully determined, can, by one observation of the lunar distance, find the ship's place within 30 miles of East and West distance.

    If he has extraordinary skill, and has bestowed extraordinary care on the determination of the errors of his instrument, he may, by repeated observations, attain an accuracy equivalent to the determination of a single lunar distance within a quarter of a minute of angle, and so may find his ship's place within 7 miles of east and west distance; but, _practically,_ we cannot expect that a ship's place will be found within less than 20 miles, by the method of lunars, in tropical seas, or within 10 miles in latitude 60deg; and to be able to do even so much as this is an accomplishment which not even a good modern navigator, now that the habit of taking lunars is so much lost by the use of chronometers , can be expected to possess.
    To be able, therefore, to place any reliance on 'Lunars' requires a really first class observer, and constant practice, and **even then the results are at best but approximate**." (my emphasis - Douglas).

    Lecky also delights in quoting Raper:-

    Lecky:- "... Raper also says: ""Great practice is necessary for measuring the distance successfully; and the application of so many small corrections as are necessary where accuracy is required is, even with extraordinary care and some skill, scarcely compatible with extreme precision.""

    Lecky goes on to mention the kind of accuracies actually obtained at sea:-

    Lecky:- "..The Rev G. Fisher, in the appendix to Captain Parry's second voyage, states that the mean of 2500 Lunars observed in December differed 14' from the mean of 2500 observed in March following; and that the mean of the observations made in the same summer differed 10' from these last, or 24' from the first.
    Captain King, in his survey of Australia, notices a discrepancy of a similar kind to the amount of 12' at the Golbourne Islands."

    Lecky also mentions a photographic method, which is being mentioned here on this forum as if a new invention; but...not so; here is Lecky again in 1899:-

    Lecky:- " If however, Lunars are of little service to the average navigator, the recent discovery of Dr. Schlichter has enhanced their value for the determination of secondary meridians, and the checking, by an independent method, of those already determined by telegraphic time comparisons.
    Dr. Schlichter's method invokes the aid of photography - so much used of late in astronomical research. By obtaining a parallel series of photographs of the moon and a fixed star or planet on one plate, and afterwards measuring their distances on the plate with a micrometer, he claims to be able to fix the longitude to within 6" or less than half a second of time. The method combines accuracy with simplicity, and when developed will no doubt be extensively used in the future by surveyors and explorers."

    Lecky finishes with:-

    "Here we finish with Lunars. Requiescant in pace. They have had their day".

    I'm with Lecky and Raper regarding the accuracy issue of Lunars - and note: that is with my own observations objectively assessed.

    I comprehensively refute Frank's assertions that 20" accuracy is routinely possible in Lunar distance measurements.

    Douglas Denny.
    Chichester. England.

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