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A long time after many list members, I have obtained the copy of "Tables for 
Clearing the Lunar Distance and Finding G.M.T. by Sextant Observation" 
(edited in 1995 and then in 1997 for the second time) from Bruce D. Stark, 
the valued member of this group. Sorry for this delay, but here in Central 
Europe, maritime only in Shakespeare's poetic licence,  there are 
difficulties in getting nautical titles at all. If you know all the following 
already, I am sorry for bothering you.


But after I had studied Bruce's Tables and his explanatory texts sent to this 
group for some days, I was astonished by their ingenuity. They don't repeat 
old solutions mechanically, but are significantly better than renowned works 
of the past, although they don't misuse the modern technical possibilities 
and go the fully traditional way of tabular and paper solution. It had to be 
an intellectual adventure to compose them and it is a delight to study them.

Let me consider them in the historical perspective. I won't repeat information 
already published in the group (you can now read it at 
http://web.dkm.cz/kalivoda/LunDistClass.htm). I would only remind here that 
two classes of methods for clearing Lunar Distances (LD') exist:

The "approximate" methods grew ripe relatively quickly and 50 years after the 
first volume of the Nautical Almanac had been published, they had reached the 
state of perfection with David Thomson in 1824. After that date no 
significant development in this field took place. These methods were very 
popular at sea during the whole 19th century for their speed, simplicity and 
for the important fact that they required the use of 4-digit logs only. And 
in spite of it they permitted the (nearly) same accuracy as their 
counterparts - see an exception immediately below.

(At http://web.dkm.cz/kalivoda/Thomson.pdf you can read detailed description 
and commentary on Thomson's "Lunar and Horary Tables for new and concise 
Methods of performing the Calculations necessary for ascertaining the 
Longitude by Lunar Observations or Chronometers...", London 1824 and 
subsequent sixty seven editions up to 1880).

Approximate methods had two great drawbacks. Firstly, the most popular and 
most widely used ones didn't allow the user to take the effect of 
non-standard refraction upon the measured distance into account, or they 
allowed it only by very bothersome procedures that would have deprived them 
of all their advantages, if used. This gap could only exceptionally create an 
error greater than 30" in the cleared distance, which was not a tragedy. But 
nevertheless, with these methods and in tropical (or Arctic) latitudes, the 
navigator had always to doubt a bit of the reliability of his LD, if he used 
the Moon or the other distance body in a lower altitude than some 20 deg.

Secondly, the auxiliary tables necessary for use of these methods were very 
scarce in giving details of their structure and genesis. The sailor had to 
use them or reject them, but he could not make his own opinion about them. 
Some of these tables were checked by mathematicians, but only many years 
after their publication. Some were found very accurate (Thomson), some rather 
inaccurate (Elford), but without any impact on the sea practice.

It is no wonder that teachers of navigation hid the most popular "approximate" 
methods from their learners and that sailors with less fatalistic point of 
view sought another solutions. Such solutions were offered by the second 
class of methods for clearing LD's, by the "rigorous" methods. These methods 
were absolutely lucid for men that wanted to understand them. They gave the 
full control of the calculation, allowed every sort of corrections, the 
correction of refraction necessary for real atmospheric conditions, needless 
to say,  included. But their drawback was their relative complexity and above 
all the necessity to use the 6-digit logs in computing and to switch from log 
values to natural values of trig functions alternately while solving them.

Old astronomers and arithmeticians used to say that each further digit of logs 
used in calculation increased its lenght and tediousness by a half at least. 
If so, the difference between the work with 4-digit or 6-digit log tables was 
palpable. In our days, when we have the accuracy of a calculation up to 10 
digits and more at our disposal within the reach of one button of a hand 
calculator, we cannot imagine what a burden everyday logarithmic calculations 
created for ordinary navigators of 19th century.

Therefore, new rigorous methods for clearing LD's arose again and again during 
the 19th century and none of them was fully successful. They were pressed 
upon students of navigational courses, but in the sea practice probably only 
few fans and some snooty navy officers used them. Their main drawbacks 
mentioned above remained.


Up to Bruce Stark in 1995/1997.


Above all, Bruce derived and uses the very apt formula for reducing LD. Here it goes:

hav D = hav (M~S) + (cos M cos S sec m sec s) ? SQRT{hav [d-(m~s) hav [d+(m~s)]}

M,S,D - true geocentric altitudes of the Moon and the Sun/star and distance of them
m,s,d - apparent, i.e. observed values

(Maybe it would be useful to consult the excellent article of George Huxtable 
on logarithmic computations sent to this group 
(http://www.i-DEADLINK-com/lists/navigation/0306/0008.html), while reading 
the following text.)

The formula seems horrible, as all "rigorous" formulae do, but with Bruce's 
comfortable tables and work sheets only a sharp pencil is needed for quickly 
resolving it. And its extraordinary and never before achieved advantage is 
evident: the term (cos M cos S sec m sec s) excepted (which is taken from 
tables by inspection), only one trig function - haversine is needed for 
computing!

And more: the haversine is extraordinary suitable at this place, as 5-digit 
log tables of it suffice to obtain the accurate result within the range of 
some arc-seconds. As you know, the haversine of an angle is the squared sine 
of the half angle. The squaring beneficially enlarges the differences of log 
mantissas between subsequent function values in useful intervals and the 
halving moves the used angle arguments farther from the right angle, where 
the sine would be very unreliable. Thank to both these features, the use of 
only 5-digit log haversine tables can be accepted. It would be impossible 
with the sine or cosine, so frequently used in old rigorous formulae. See 
bellow the third reason permitting the use of only 5-digit values.



The second Bruce's deed is the manner how he had solved the problem with the 
addition in his formula. Such addition makes the straightforward logarithmic 
solution of the equation impossible (see George Huxtable's text mentioned 
above). Additions, mostly inevitable in rigorous formulae for clearing LD 
even after torturing them by the most sophisticated trigonometric 
transformations, used to be overcome by jumping between log and natural 
values of trig functions. Of course, each such jump enlarged the time and 
effort demanded by the method and increased the maximal possible error of the 
result.

Bruce Stark goes another way. He uses the Gaussian logarithms that make 
possible to remain in world of logarithms all the time of calculation and 
transform an addition of natural numbers to the addition and subtraction of 
their common and special logarithmic values by use of a special table. It is 
much easier than to convert logs to their natural values, to add them and 
again to convert them to logs. And moreover, Gaussian logs yield greater 
accuracy of result than the traditional computing method and help 5-digit log 
values to be sufficiently accurate for this method.

The use of "Gaussians" by Bruce is original in the field of navigation. I 
don't know another example of using them by seamen or aviators - with the 
exception of Soviet navigators, which had Gaussians in their standard table 
sets up to cca 1960. The Gaussians were probably regarded by Stalin's 
comissars as opponents of Anglo-Saxon cosmopolitan and aggressive haversine 
that wasn't allowed to the Soviet navigational practice. But in Bruce's 
hands, Gaussians coact peacefully with haversines in rationalizing the LD 
procedure to the level unknown so far.



The third asset of Bruce is his method of obtaining reference lunar distances 
that are to be compared with the cleared distance for obtaining G.M.T. One 
would say that after these distances had disappeared from nautical almanacs 
in 1907-1924, the death of lunars was imminent. Who was bold enough to tell 
sailors to compute reference distances by hand?

But Bruce Stark changed this handicap to the contrary. He proposed the formula 
for obtaining the reference distances to be compared that is absolutely 
conformal with the notoriously known haversine formula for finding the 
altitude in Marc St.Hilaire's method. Therefore, with the prepared work sheet 
the time and effort for computing them is pressed  to an absolute minimum 
possible. And because with modern almanacs at sailor's disposal one can 
compute such reference distances for each hour without any interpolation of 
GHA and declination, the interpolation of G.M.T. from them is much more 
accurate that in the times when 3-hours almanac intervals were common for 
tabulated distances. For user of Bruce's Tables this makes possible to 
evaluate even very short distances that would have unusable second 
differences in three hours intervals. And as Bruce Stark emphasizes, such 
short distances are the easiest ones to be observed from small sailing ships 
of archeonavigators riding their!
 hobby of the celestial navigation.



Other advantages of Bruce Stark's tables I can mention only briefly, so that I 
could end this article soon enough. They are e.g.:


- Shifting from arc-seconds to hundredths of arc-minutes, which agrees with the custom of modern seamen

- Very handy "inside-out" tables reducing the demand for place

- Combining the corrections of altitudes for dip and semidiameters in one table

- If the user doesn't care about the principles, he needn't even understand the idea of logarithm


After Bruce Stark disclosed the principle of his work for Nav-L during the 
last two months, every navigator (fondling the GPS in his pocket) can revert 
to the sea history in his practice very easy. And he can be sure that with 
these Tables, the history of Lunar Distances is consummated now and the long 
line of rigorous methods for clearing them ends successfully - only in our 
days.



Jan Kalivoda

   

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