NavList:

D Walden wrote, about the method suggested for determining azimuth by backwards use of the specific sight reduction tables:

Pretty slick, eh?  

Certainly is ingenious.

The sight reduction tables are based on the cosine formula, in which an angle or a side opposite can be made the subject of an equation, ie; altitude/local hour angle, declination/azimuth angle. The tables are used to derive an angle from a triangle in which the other sides/angles are known or determinable, so can be used backwards as recommended to find azimuth.

The practical result is that a navigator who is relying on this ‘one stop shop’ has yet another method available for the determination of azimuth. Two others are already offered: quick and simple to use look-up tables that 98.8% of the time give the azimuth accurate to within 2 degrees (more about this below) and the Weir diagram method.

The Weir diagrams use a graphical method. Presumably it is accurate (no nit-picker has yet emerged to propose the contrary) but the nature of this beast means that precision is literally in the hands of the user. At best it is precise, I guess, to about the nearest degree of azimuth.

And so it goes with this book: "The Complete On-board Celestial Navigator" offered as a back-up method of celestial navigation for yachts to complement electronic dependency. It uses data expressed and calculated to the nearest minute of arc, or nautical mile. Its strength lies in bringing together everything a celestial navigator needs within the one handy book (apart from the sextant and timekeeper). It proposes that an azimuth correct to within a degree or two is sufficient for the purposes of the yachtsperson for whom it is designed. In this age of declining interest in celestial navigation the book seems to have found a niche market and has done well, and is now into its second edition.

 As a rule of thumb, celestial navigation practiced from the deck of a small yacht under average offshore conditions gives a result that, if it is within 10 nautical miles of the actual position, can be counted as a good result. If within 5 miles then an excellent result. Once this is accepted then methods and tools designed for other uses, eg; ships with their more elevated viewing platform and less swell-induced motion, may not be the most appropriate. They may be more precise; typically using angles expressed and calculated to the nearest tenth of a minute. Certainly as a consequence they are more bulky, typically many different volumes are needed to complete the navigator’s necessary books and tables, but the extra precision tends to be wasted effort. What is the point of determining the position – you are almost certainly not at – to an irrelevant precision? The Bennett yacht book produces a result appropriate for its purpose. It is accurate but limited in precision.

Does this lack of precision mean the methods are less accurate? No; precision and accuracy are different concepts and neither has necessarily a monopoly on virtue (as in a “good” method or result). As an example, let’s look at a sundial. If correctly set up and the necessary corrections made (for the equation of time and the difference in longitude between the sundial and that of the time zone used) then it is entirely accurate. But not particularly precise; the sun’s fuzzy shadow limits that. It is accurate but limited in precision. On the other hand a mechanical contrivance typically indicates the time to the nearest second, while never being wholly accurate. It is precise but limited in accuracy.

Horses for courses. The author, George Bennett, co-wrote another book when he was the Professor of the School of Surveying at the University of New South Wales, about field astronomy for surveying. Essentially position location using a theodolite and familiar celestial methods for surveying purposes, it was once useful, eg; to mark out the boundaries of vast pastoral holdings in the Australian outback. The parts of that book I found fascinating were the different methods usable to wring the most accuracy possible from the raw material of multiple sights. Some methods were statistical, at least one was graphical and involved invoking circles and triangles from the initial four or more position lines (three, alas, is not the best number) the result looking like some magician’s complex magic spell. For those of dim memory, some of this has already been detailed on this List in the past. How close could they get? To the nearest second of arc, it seems. That’s about 30 metres on the earth’s surface. Not bad. Horses for courses.

The Bennett yacht book takes a different approach. One person only has found fault with this – the infamous Huxtable example – and the nature of that complaint was that if using the azimuth look-up tables only, at an unlikely confluence of carefully contrived half degrees each of declination, local hour angle and altitude; significant error could result. It presumes that the navigator would be wholly dependent on these tables – ignoring Weir over the page – and would not already have a good idea of azimuth anyway, eg; via a corrected compass bearing. Anyone who emerges from their cosy navigational armchair and ventures out upon the wide blue yonder tends to find that determination of azimuth for plotting purposes is perhaps the least demanding aspect of celestial navigation while offshore in a small boat. 

To put this specific charge into perspective, a statistical analysis of the likelihood of significant error using the azimuth tables has been made and is available at:

http://gbennett.customer.netspace.net.au/azimuth/azimuth.htm

Taking and analysing the whole gamut of possible results using combinations of declinations, local hour angles and altitudes 121.7 million times, the resulting indicated azimuths are found to be correct to within 2 degrees 98.8% of the time (and correct to within 1 degree 93.6%).

What does this mean? Well, if an infinite number of sailors were bobbing about the oceans, all relying solely on these tables morning, noon and night, while spending one third of their time at sea (which is roughly correct, live-aboard voyagers spend most of their time in port) then; if I’ve done my sums real good, it would on average take the best part of several thousand years for any one of them to come across an error of 15 degrees or more. It is unlikely to an absurd extent. It is a furphy. It is a paper tiger. It is a chimera. It is a nonsense. You should be so lucky. You’d have a better chance of winning the lottery. It ain’t gunna happen.

Another example: Is it possible that life on earth is going to be taken out by an asteroid? Yep. Is it likely to happen? Well, it is probably inevitable if we wait long enough. Some people believe this has led to extinction events in the past, as with the disappearance of the dinosaurs 90 odd million years ago. Should we worry about it?  Nah. That would be a waste of good worry. It is unlikely to an absurd extent. It is a furphy. It is a paper tiger. It is a chimera. It is a nonsense. Etcetera.

To propose the possibility of something happening as being a serious cause for concern without considering the probability of that event occurring is naïve at best and dishonest at worst. Or is it simply a case of sour grapes? Or is the obsessive search for errant nits, artificially concocted when needs be, an under-appreciated art form all of its own?

The probability of significant error being encountered while using these tables is remote beyond the need for concern by practical navigators. Who presumably have their wits, their compass, and the Weir tables to comfort them. Plus, now, the possibility of using the sight reduction tables backwards to find the azimuth both accurately and with more precision than needed, thanks to D Walden.

They also have an explicit warning in the second edition. It reads:

“Warning: If you try to drink your coffee while it is still too hot then you just might burn yourself. Y’all take care now, and have a nice day.”

No, that’s just my little joke. It really says:

“In extreme cases, the table should be interpolated when observations have been made in the vicinity of the prime vertical and/or when LHA, declination, and latitude require substantial rounding off before using the table. When in doubt, use the Weir diagrams.”

Having used these tables reasonably extensively – certainly many more than a hundred times –  I have never come across a situation where the tables’ indicated azimuth was significantly incorrect. However should my asteroid arrive and the calculated result give concern, ie; disagree seriously with my corrected compass bearing or any other check method, I could probably find my way to the next page. If I didn’t prefer to use Weir to begin with. If I hadn’t heeded the warning to beware, eg; the extremely rare coincidence of half values for all variables. In any case, it would only, at worst, be one position line skewed; it wouldn’t affect the other two – unless we want to consider how likely it would be to have all three variables leading to the other azimuths each of all half degrees as well –  so may not even result in a disastrously wrong result, given the practical limitations on accuracy of cel nav from the deck of a small craft. Horses for courses. As to great passion – ah, we live in hope, but if this engenders any emotion at all its more akin to bemusement.

More information about the Bennett yacht book can be found at the abbreviated version of the link above:

http://gbennett.customer.netspace.net.au/

Congratulations to D Walden for bringing the backwards method forward. More useful grist for the mill.

From: Navigation Mailing List [mailto:NAVIGATION-L@LISTSERV.WEBKAHUNA.COM] On Behalf Of d walden
Sent: Wednesday, 23 November 2005 10:04 AM
To: NAVIGATION-L@LISTSERV.WEBKAHUNA.COM
Subject: Bennett's '...Celestial Navigator' --An improved Zn calculation

With some trepidation, I raise again the question of using '...Celestial Navigator' to obtain Azimuth.  Using the infamous Huxtable example:
Dec=55-30
LHA=54-30
Alt=61-30
George H. didn't give the corresponding Lat, but it can be found to be:
Lat=60-18

First going through the altitude calculation using the Bennett work form, on page 168, to generate altitude from given Dec, LHA, and Lat.

line 13 Local Hour Angle          54-30 -> 8841
line 14 DR Latitude             N 60-18 -> 3974
line 15 Declination             N 55-30 -> 3217
                                          ______
line 16 (theta=28-04)                 SUM 16032 -> RES  11760
line 17 Latitude ~ Declination
        (ABS(Lat-Dec))             4-48  ------------>    351
                                                       ______
line 18 Computed Altitude         61-30.5 <------  ALT  12111

Now for the new method to calculate Zn.  In a sentence, use Bennett's table 'backwards' substituting Alt for Dec, and Dec for Alt.  The final Z will be the LHA value.  Continuing with the infamous example from above:

remember, substitute Dec for Alt
line 18 Computed Altitude         55-30  --------> ALT  17587
now, Lat~Dec becomes Lat~Alt                           ______
line 17 Latitude ~ Declination     1-12  -------->         22
now, calculate what RES must be for sum to equal ALT, (17587-22)
line 16  (theta=34-29)                SUM 13763 <- RES  17565
                                          ______
now, we have the sum of three, we know two, so we can solve for the third.
remember, substitute Alt for Dec
line 15 Declination              61-30.5 -> 4187
note, line 14 is the same as above
line 14 DR Lat                 N 60-18   -> 3974
for SUM to be correct, line 13 must be 13763-4187-3974
note use top of column LHA value as Z
line 13 Local Hour Angle           75-07 <- 5602
now, we apply our one rule, if LHA(the real LHA)<180, Zn=360-Z, else Zn=Z
So, Zn=360-(75-07)=284-53  Exactly the ATAN2 formula result!

Note, there is! a typo, which I don't recall seeing mentioned before, in Bennett's response to Huxtable: "If, however, the Tables are interpolated (X=460) the azimuth is found to be 255 or 285 (not 075 or 105) which compares favourably with the results from direct calculation of 255.3 and 254.8."  The last number should be 284.8, as above.

Pretty slick, eh?  
(Some adjustments of signs for special cases are left to the reader as an exercise.)