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As you all know very well, the key step in the finding the GMT by lunar 
distances is to compare the distance measured by the sextant or the repeating 
circle with the values tabulated in almanacs (after 1767, when the first 
volume of the Nautical Almanac was published by Nevil Maskelyne; in ten 
previous years another interesting method was used by a handful of informed 
navigators - rather navigating astronomers).

But the measured distance is "dirtied" by the effects of refraction and 
parallax on the altitudes of both bodies (although the parallax of the other 
body was often neglected, even in the case of Sun or Venus; stars have 
absolutely negligible daily parallax, of course). Therefore this measured 
lunar distance must be "cleared", i.e. reduced to the theoretical value that 
would be observed from the Earth's centre in vacuum and only then it can be 
compared with tabulated values of the Almanac so as to obtain the GMT.

This "clearing" is difficult part of "lunars" and about hundred procedures 
were devised for this purpose, beginning from 1750/1759 when the Frenchman 
Lacaille (La Caille, known by creating several names for faint southern 
constellations, too) proposed the first one applicable on the basis of 
studies of his countryman Jean Morin, who had analyzed the problem in 1633.

Maybe it would be of some profit to classify these methods according to their 
principles. I will try it as a modest additamentum to the valuable book of 
Charles Cotter "A history of nautical astronomy", London 1968, which pays 
little attention to older and the most important and renowned methods from 
the times before 1850, when the "lunars" were at their best.


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We can distinguish four classes of these methods, which are remotely similar 
to the classes of the methods for reducing sights by "Marcq St Hilaire 
(intercept) method", the only method for using celestial lines of position 
surviving in today's navigation. These are in the order of their increasing 
length, difficulty and logical clearness and beauty (in my eyes):


- software solutions; quite common now and not unknown in the first half of 19th century!

- inspection tables (compare HO 214, 218, 240, 229 and ancient Ball's tables, firstly edited in 1907)

- "short" methods (compare Ageton's method in HO 211, Dreisenstock's method in 
HO 208, Smart, Ogura, Aquino etc.; in these methods short tables with 
auxiliary values are provided that are combined to obtain the end result; 
these tables were much less bulky and expensive than the inspection tables, 
but their use was more difficult and time-consuming)

- rigorous solutions (compare cosine-haversine formula)



1. Software solutions

Yes, the third mechanical computer of human history (preceded by Descartes' 
and Leibniz' machines) was created for computing the corrections of lunar 
distances. Its designer was Charles Babbage (1792-1871), who projected this 
programmable mechanical device together with Byron's daughter Ada after 1822. 
The machine was programmed by predecessors of punched cards. Its prototype 
survived to our days, but did never function.



2. Inspection tables for clearing lunar distances

The plural is not appropriate - only one such work appeared. It was "Tables 
for correcting the apparent distance of the Moon and a Star from the Effects 
of Refraction and Parallax", Cambridge 1772, in folio. It is commonly cited 
as "Cambridge Tables", or sometimes as "Shepherd's Tables" (A.Shepherd was 
the author of the preface, but took no part in computing the tables). They 
were computed and edited in the first spell of enthusiasm for lunars, after 
Tobias Mayer's lunar tables were edited in 1770 and used even before in the 
manuscript form by Maskelyne for editing the first volumes of Nautical 
Almanac.

Cambridge Tables were an incredible deed. After 4 pages of foreword and 7 
pages of instructions 1104 (thousand hundred four) pages follow with up to 
370 corrections on each page, together cca 300000 values. Corrections were 
computed and arranged for each degree of lunar distance from 10 to 120 
degrees. Each degree of distance occupied 3-14 pages. For each degree of 
distance all possible combinations of Sun's and Moon's altitudes (stepped by 
one degree) were evaluated and the corrections of apparent lunar distances 
(further L.D.'s) for Moon's horizontal parallax of 53 arc-minutes and the 
mean refraction were given. Other two table columns gave the corrections for 
the actual Moon's horizontal parallax and the actual air temperature and 
pressure. Of course, triple interpolation was needed, but second differences 
were negligible, rarely exceeding 3 arc-seconds. Small table for correcting 
for horizontal parallax of the Sun (9 arc-seconds) was given. Planets were 
not yet used for !
L.D.'s in that time.

The head of the working group of calculators was probably Israel Lyons, who 
prepared the clever method of computations (one of "short" methods, mentioned 
below), too. After editing this giant work, he took part in Phipps' polar 
expedition in 1773, but died at home in 1775 in the age of 36 years.

Of course, these folio tables were too bulky, cumbersome and costly to gain 
any popularity at sea. Very small number of their copies survived to our days 
in great libraries.



3. "Short" or "approximate" methods

Imagine the triangle in the sky with the vertices Z - zenith, S - true 
Sun/star and M - true Moon. And another triangle with wertices Z - zenith, s 
- apparent=observed Sun/star and m - apparent=observed Moon. The two 
triangles have the common vertex (and angle) Z and their two sides (zenith 
distances of the four bodies mentioned!) crossing at Z and perpendicular to 
the horizon coincide for the most part of them: s lies above S, as the daily 
parallax (which always lowers the apparent=observed body below 
true=supposed-to-be body for an observer on Earth's surface) of the Sun or 
planet (not mentioning the stars) is always much smaller then the effect of 
refraction (which always raises the apparent body above true body). On the 
contrary, m lies below M, as her great daily parallax is always greater then 
the effect of refraction. As a result, the third sides (apparent and true 
lunar distance!) of both triangles, ms (apparent=observed L.D.) and MS ( 
true=cleared L.D.) cross each o!
ther at the common point X. But the sections mM and sS are very short (half 
degree at most, but mostly shorter), which is essential for further 
procedures.

Therefore if we drop perpendiculars from the points M and S to the side ms 
(apparent L.D.) and vice versa, we can trigonometrically deduce approximate 
equation permitting to reduce ("clear") the apparent L.D. to the true L.D. 
(Here you can see a very remote similarity with Ageton's and other methods 
for resolving the nautical triangle; but these are not approximate in any 
degree, only their use of perpendiculars to triangle sides is somewhat 
similar.)

(M,S,m,s are meant as centres of bodies - the limbs are measured, of course, 
but applying the corrections for the semidiameters of bodies, one obtains the 
values for centres. I neglect all three efects of ellipsoidal earth's shape 
on clearing L.D., too; they can make a maximal error of 13 arc-seconds in the 
true distance cleared, when neglected.)

The final approximate formula can be confirmed directly by the calculus 
(Taylor's polynoms), too, but the spherical trigonometry alone can find the 
long line of always diminishing trigonometrical terms of corrections allowing 
for effects of parallax, refraction and their combinations on a measured 
lunar distance. 10 (ten) terms were sometimes used for calculation! This 
formula is called "approximate", as it is not derived strictly, but only in 
gradually approaching steps and terms; but when sufficient number of terms is 
included, its accuracy leaves nothing open.

The first methods of this kind were the methode of Lacaille (1759) and Lyons 
(1766); both were mentioned above. Another was Witchell's method from 1772 
(the "fourth method" of Bowditch). But their formulas were too complicated 
for seaman's everyday use, therefore Dunthorne's and Borda's rigorous methods 
(see below in the fourth chapter) were more popular then.

But from the beginning of the 19th century seamen were not left alone with 
these approximate methods. Many proposals of simpler procedures appeared:

D,d - true and apparent=observed lunar distances
M,m = true and apparent=observed ALTITUDES of the Moon (NOT its centres as above!)
S,s = true and apparent=observed ALTITUDES of the Sun/star (NOT its centres as above!)
HP = horizontal parallax of the Moon

The formula for the sea practise, as introduced from 1810:

D = d - HP sin s cosec d + HP sin m cot d + MYSTERY

The navigator only computed the two first corrections by logarithms of 
trigonometrical functions to 4 figures and by proportional logarithms 
originally tabulated by Maskelyne for interpolating the tabulated L.D.'s in 
the Nautical Almanac; that were two greatest terms of Moon's parallax in the 
"approximate" equation, mentioned above.

And the MYSTERY was the "third correction", tabulated according to the values 
of Moon's and Sun's/star's altitudes observed and of lunar distance observed.

The main difference between various methods of this numerous class was, how 
many secondary terms (from these remaining eight terms in the "approximate" 
equation) were taken into account; the authors seldom stated these details 
and published their tables as they were - sailor, take it or leave it!

The second difference between various tables was their step, of course, and 
consequently the amount of the interpolation needed. Several were even 
arranged as nomograms, in a graphical form.

The first table of this kind (after two unpublished or unnoticed predecessors) 
was the publication of merchant master Elford from Charleston, which appeared 
firstly in 1810 and was several times reedited and many times stolen by other 
"authors" up to the end of 19th century. Elford's table of the "third 
correction" included only two greatest terms of refraction, leaving other six 
smaller refraction and parallax terms aside.

The same value is given in the "Set of linear lables for correcting the 
apparent Distance of the Moon from the Sun or a fixed Star for the effect of 
Refraction", edited by well-known J.W.Norie in 1815 in London. That work 
contained 24 nomograms, from which the "third correction" could be taken 
without any interpolation with the precision of 2 arc-seconds. This set was 
popular, but never edited again, as original engravings of nomograms were 
difficult to obtain. So was Norie protected from thiefs that irritated Elford 
so much and so often. But sailors had to leave this tool.

But the most prominent author of the tables in this class was David Thomson, 
who published the workhorse of British navigators in the first half of the 
19the century: "Lunar and Horary Tables for new and concise Methods of 
performing the Calculations necessary for ascertaining the Longitude by Lunar 
Observations or Chronometers..." (London 1820). In 1851 the 42th edition 
appeared! And again was his main table accepted (i.e. stolen) into many other 
nautical tables collections.

It was an ace of nautical tools in that time. Firstly, it gave on 51 pages (so 
that no interpolation was necessary) the value of the mysterious "third 
correction", allowing (as opposed to Elford and Norie and others) for further 
smaller terms of the complete approximate formula. It brought the improvement 
of 40-60 arc-seconds to the precision of corrections in some (not very 
frequent) unfavourable situations. A small table was given for reducing the 
parallax effect of the other body used.

Secondly, the Thomson's table set included auxiliary tables for computing the 
first two Moon-parallax corrections of the simplified formula mentioned 
earlier that the seaman had to resolve directly. Taken together, Thomson's 
tables permitted the shortest method for clearing lunar distance ever 
contrived - it was shorter than reducing the Sumner line by cosine-haversine 
method.

And many other useful tables were included, e.g. for resolving "time sights" 
(i.e. measuring altitudes of celestial bodies for computing their local hour 
angle to be compared with the chronometer time or "lunar" time for "finding" 
the longitude) by cosine-haversine method, tables for finding azimuths of 
celestial bodies and so on.

David Thomson went the long route from the ordinary soldier and seaman to the 
merchant master. He died in 1834 in Mauritius as a storekeeper, unknown and 
enigmatic personality. He never specified the method of computing his main 
table of the "third correction". It was guessed that he had to compute 30000 
lunar distances directly and to interpolate another 50000 values so as to 
construct this table. His results were proved to be independent of "Cambridge 
Tables" and are better than theirs in the average. But his caginess about his 
computing method prevented his table from entering into the navigation 
courses and navigation practise aboard navy ships, which were not insured.

The Thomson's method and tables (after being simplified) were taken over by 
Bowditch as his "second method" for clearing the L.D.'s., as Bowditch states 
expressly (he spells him "Thompson", but in my other sources the name always 
sounds "Thomson") The "first method" and "third method" of Bowditch, which 
were devised by himself, and his "fourth method", improved from Witchell's 
procedure (see above), were "short/approximate" methods, too, but they were 
rather obsolescent after 1810, as their lenght and greater number of 
necessary arithmetical operations in comparison with Thomson's "second 
method" prove in Bowditch's examples. (The "first method" stood in the 
appendix in the first Bowditch's editions and only later he shifted it into 
the main text to the head before the Thomson's method - the sign of author's 
growing self-confidence.)

Of course, in the second half of the 19th century some other 
"short/approximate" methods appeared that didn't resemble the Elford/Thomson 
solution. Some are mentioned in Cotter's book. Another was the method of the 
French astronomer Chauvenet that replaced all other older methods in 
"American Practical Navigator" in the year 1888 (the pertinent pages were 
scanned and published on the web by Dan Allen for this group). This method, 
in contrast to the all mentioned above, was capable to take into account ALL 
effects of ellipsoidal Earth's shape and temperature/barometric corrections 
of mean refraction values. In competition with widely used chronometers and 
owing to very precise lunar positions in almanacs from 1880 (Newcombe's 
superb equations of planetary and lunar motions began then to be used for 
ephemerides), the editors supposed in this year that "lunars" should be given 
a more precise, although more laborious method in the "American Practical 
Navigator" to survive, at le!
ast for checking the chronometers.

Maybe the method of Bruce Stark is the last method invented in this class, but 
I don't know anything about it.



4. Rigorous methods for clearing the lunar distances

The most logical class comes the last. Take the triangle zenith - true 
Sun/star - true Moon and the second triangle zenith - apparent Sun/star - 
apparent Moon once more. They have the common vertex and angle at zenith. 
This permits to compare the basic trigonometric equations for both spherical 
triangles and deduce various straightforward trigonometric formulas for 
finding the true lunar distance, when apparent=observed lunar distance and 
apparent=observed and true altitudes of both bodies used are known (we can 
obtain the true altitudes from apparent=observed altitudes very quickly by 
allowing for refractions and parallaxes).

So again:

D,d - true and apparent lunar distances
M,m = true and apparent altitudes of the Moon
S,s = true and apparent altitudes of the Sun/star
A = auxiliary value

Two most popular methods of this class were Dunthorne's and Borda's method. I 
won't write out their deduction, only the final forms:


Dunthorne (1766): cos D = cos(M-S) + cos M cos S sec m sec s [cos d - cos(m-s)]

Mackay improved this form by using versines instead of cosines in 1819, 
removing the small incovenience of changing the sign of cosine at 90 degrees 
by this substitution. Young's formula from 1856 is very similar to the 
original Dunthorne's form.
The Dunthorne's method was very popular in German speaking countries and in 
Scandinavia up to the beginning of 20th century, at least in navigation 
courses.


Borda (1778):
 cos A squared = cos M cos S sec m sec s cos[(m+s+d)/2] cos[(m+s-d)/2]
 sin D/2 squared = sin[A + (M+S)/2] sin[A - (M+S)/2]

I cannot understand, why this cumbersome method gained such popularity. But it 
was widely used in France and other Romance speaking countries and many 
successors devised similar formulas: Delambre, Krafft (a bulky volume of 
auxiliary tables in 600 pages were collected for that method by Mendoza del 
Rios in 1801) and others.


In all these equations the term (cos M cos S sec m sec s) returns again and 
again. It was called "logarithmic difference" and tabulated in an inspection 
table according to the apparent altitudes of the Moon and of the other body. 
An error of some 3-5 arc-seconds arose from its use, but this was considered 
tolerable before 1850.

The great disadvantage of all rigorous methods was that they requested the use 
of logarithms to 6 figures (and some theoreticians frowned at it, vainly 
requesting the use of the logarithms to no fewer than 7 figures), whereas the 
approximate methods were quite satisfied with logarithms to 4 figures with 
the same accuracy. The difference in difficulty of computations is manifest.

On the other side, all rigorous methods were capable of all three corrections 
for ellipsoidal Earth's shape and of corrections for the actual thermometer 
and barometer values (effects on the mean refraction), whereas these 
corrections are difficult or impossible to use in the most approximate 
methods (except from tedious Chauvenet's method, see above). And each step of 
calculation was under the full control of navigator in rigorous methods, 
where one can be sure that if logarithmic tables are correct (which could be 
guaranteed almost surely even in the 18th century), the result depends only 
on navigator's sextant, hand and mind. Approximate methods with their 
mysterious tables required a bit fatalistic seaman (which was surely the 
frequent case).



Thank you for your corrections and supplements.


Jan Kalivoda

   

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