NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Peter Fogg
Date: 2005 Dec 3, 10:01 +1100
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
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:
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.)