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Approximate methods for clearing the Lunar Distances - some details
From: Jan Kalivoda
Date: 2003 Apr 13, 14:07 +0200

```In my previous article on classification of methods of clearing the lunar
distances, I didn't want to burden its text by greater details. But some of
you replied that they would like to read more about the approximate methods
for clearing the lunar distances ("lunars", LD's) and about the deduction of
their basic formula. Therefore I can perhaps try to give more details now -
only for the interested guys.

Meanwhile Arthur Pearson prepared a nice picture for this purpose on his
him very much, it was beyond my graphic abilities. Use this picture as the
reference, as I will do onward:

http://members.verizon.net/~vze3nfrm/Nav_L_Graphs/ApproxLD.JPG

(In this picture and in the following text you can always take a star or a planet for the Sun ! )

Consider the spherical triangle: Zenith - true Moon - true Sun; and the second
one: Zenith - apparent Moon - apparent Sun. They have the common angle Z'.
Therefore we can deduce various strict fomulas for the true LD between the
true Sun and the true Moon, using the apparent LD and four apparent and true
zenith distances in these two triangles. This is the way of rigorous methods.

But as the distances apparent Sun - true Sun and apparent Moon - true Moon are
very small in comparison to other elements of both triangles, we can also
search only for small corrections that can be algebraically added to the
apparent=observed LD to obtain the true LD. This is the way of approximate
methods.

These corrections can be found by two procedures: by the calculus or by using
the spherical trigonometry of perpendiculars dropped between the vertices and
sides of two triangles mentioned (see the lower part of picture). The end
formula CAN be the same in both cases (as always in the nautical astronomy,
several equivalent solutions are possible, according to the aim of their
inventors).

The trigonometrical deduction would require some hundred lines; so I confine
myself to the calculus procedure.

Symbols:

Z - zenith
Z' - angle at zenith (difference of azimuths of both bodies)
M, S - true Moon and Sun
m, s - apparent Moon and Sun
m', s'  - angles at m, s
D - true lunar distance (distance MS)
d - apparent (observed) lunar distance (distance ms)
A, a - apparent altitudes of the Moon and the Sun over the horizon
x - the distance Mm, i.e. the difference of the Moon' parallax in A and the refraction in A
y - the distance Ss, i.e. the difference of the Sun's parallax in a and the refraction in a
P, p - HORIZONTAL parallax of the Moon and Sun
(parallaxes in altitudes A, a can be obtained by formulas: P cos A , p cos a)
R, r - refractions in altitudes A, a

The starting point is the Taylor's polynom for the small changes x,y of two
sides Zm, Zs of the spherical triangle Zms, forming the constant angle Z':

D = d + y cos s' - x cos m' + � (y� sin� s' + x� sin� m') cotg d + x y sin s'
sin m' cosec d  .....  (an infinite number of smaller terms follow that can
be and were fully neglected)

Of course, this form is unusable at sea. Now three substitutions should be made:

y = r nearly, as the maximal parallax of the Sun is 9 arc-seconds, of Venus 33
arc-seconds and and of Mars 23 arc-seconds; therefore the parallax of the
second body can be reduced by a special term at the end of procedure and
neglecting it at the beginning of arrangements will not induce a significant
error in the further treatment of the formula; Jupiter, Saturn and stars have
the daily parallax zero for nautical purposes

x = P cos A - R   ; difference of Moon's parallax in its observed altitude and
of Moon's refraction in its observed altitude!

cos m' = (sin a - sin A cos d) / (cos A sin d)   ; this is the clever
substitution of Israel Lyons (1766) into the term (x cos m'), resulting from
the spherical cosine theorem for the triangle Zms; this substitution is
essential for the use of the whole formula in this branch of approximate
methods

So after these substitutions and some following bothersome, but elementary
trigonometric arrangements (let me jump them over now, I can send them to you
individually, if you like), we obtain the basic formula for the most
important approximate procedures and their tables for clearing the lunar
distances from 1810 to the death of lunars:

D      =     d     -

(1)            -         P sin a cosec d    +

(2)           +         P sin A cotg d      +

(3)           +         r cos s'     +       R cos m'       +

(4)           +         � P� cos� A sin� m' cotg d     -

(5)            -         P R cos A sin� m' cotg d        +        P r cos A sin s' sin m' cosec d      +

(6)           +         � (r� sin� s' + R� sin� m') cotg d      -      R r  sin s' sin m' cosec d      -

(7)           -          p sin A cosec d     +    p sin a cotg d

Remarks:

1,2 - two greatest terms for the Moon's parallax; these two corrections were
computed by the sailor himself by logarithms to 4 places and proportional
logarithms; thanks to Lyons' substitution, sailor hadn't to be tortured by
computing the angle at the apparent Moon and worked only with the horizontal
parallax of the Moon, with both observed altitudes and with the observed
distance; Thomson has provided four auxiliary tables to speed up these two

3 - two greatest terms for both refractions; here the "THIRD CORRECTION"
begins; ALL THE FOLLOWING (except line 7) was tabulated in ONE table
according to observed altitudes of both bodies and their observed distance;
Elford as the inventor of such sort of tables (1810), then Norie (1815) and
many others tabulated only the value of this line (3), leaving all other
terms aside

4 - another term for the Moon' paralax; the value of this line was tabulated
for the first time by Maskelyne in the table No.13 of his "Tables Requisite"
= table No.35 of Norie's "Epitome of practical Navigation" = table No.20 of
Bowditch' tables (??? I am not quite sure of this last assertion); these
tables were used in older approximate methods (Lyons, Witchell, Maskelyne,
Bowditch) preceding the Elford's deed in 1810; they could be used together
with Elford's (and similar) tables and Norie's nomograms, too

5 - two terms for combined effects of the moon's parallax and both refractions

6 - two smaller terms for both refractions

7 - two greatest terms for the parallax of the second body, if any; it was
given by a special table by Thomson; all smaller terms of this parallax
remained fully neglected as quite unimportant

It was my beloved Thomson, who brought this procedure to the state of
perfection. He never stated his method for computing his pithy table of the
"third correction" (firstly 1824, the 67th edition in 1880, according to the
kind information of Bruce Stark), which didn't require any interpolation, as
its steps were very small. (Compare the table No. 48 in the Bowditch' old
editions (to 1851 at least), which is the reprint of Thomson's main table,
although it is called "Third correction in Lyon's improved method" in the
"second method" for clearing LD's.)

From Thomson's results it had been ascertained that he included ALL terms of
the approximate formula given above. He couldn't compute them step by step,
as it would be an absolutly hopeless effort for one man, even for many. From
some his remarks (and above all from remarks of G.Coleman, the editor of
Norie's Epitome after Norie's death, who was purportedly in contact with
Thomson in his youth) it is probable, that Thomson computed only the first
two terms from the lines (1) and (2) above and then the whole difference
between true and apparent distance by some rigorous method. And by
subtracting those two terms from the whole difference found, he obtained the
value of remaining terms from the lines (3) - (6). The effect of the parallax
of the second body from the line (7) he tabulated in a special table, as said
above.

The improvement of accuracy of Thomson' table in comparison with Elford and
Norie is the most evident, when the lunar distance itself and both altitudes
are small (and if altitudes are nearly the same - this unfavourable case is
seldom in lower latitudes). So for both altitudes and lunar distance of 20
degrees, Elford gave the "third correction" of 22 arc-seconds, Norie the
value of 19 arc-seconds, but Thomson (and Bowditch) 88 arc-seconds! And this
greater value is the true one. The importance of smaller terms in the
approximate formula (lines 4 -6) is manifest by that.

The general accuracy of Thomson's main table of the "third difference" was
very high. It had been found out by later trial calculations that the error
is under 2 arc-seconds in the most cases, and only seldom it attains 5-6
arc-seconds. Of course, isolated errors of much greater degree can hide in
the tables (which were never recomputed as the whole), as in all tables for
"approximative" methods. Slocum mentioned one such case in the record of his
circumnavigation - near the island Nukahiva in Marquesas, he ascertained "an
error in the important logarithm of the tables" during clearing the lunar
distance. I suppose that he had some table for approximative methods in hands
- he could not verify the values of the decadic logarithms themselves aboard,
I guess. Or does understand anybody this interesting place of Slocum's book
in a different manner?

But the accuracy of Thomson's tables, as mentioned, must be understood only
for their default conditions. All tables of the "third difference" for
clearing LD's by "approximate" methods were liable to the important drawback:
they were to be computed for the MEAN EQUATORIAL horizontal parallax of the
Moon (57,5 arc-minutes) and for the MEAN refraction (for 30 inches of
barometric pressure, 55 Fahrenheit degrees of temperature). And because their
values were the lump of effects of both the parallax and the refraction,
allowing for the actual Moon's parallax, for the actual atmospheric
conditions and for the effects of ellipsoidal Earth's shape was very
difficult. I won't enter into these details here. But the maximal error
created by using only the mean Moon's parallax (57,5 arc-minutes) was 10
arc-seconds at short LD , 5 arc-second at LD of 40 degrees and further
decreasing. The combined maximal amount of all three effects of ellipsoidal
Earth's shape was 13 arc-seconds. The !
effect of not standard atmospheric conditions could be considerable, of
course, above all in the cases, when the LD ran vertically to the horizon -
the error of 60 arc-seconds is then exceptionally possible.

But one had always to expect a MINIMAL error of 30 arc-seconds in measuring
lunar distance aboard, even after averaging a set of observations. And the
tabulated Moon's positions in almanacs (and therefore the tabulated lunar
distances as well) could be in error of 1 arc-minute before 1820 and 20
arc-seconds before 1880. So an exaggerated accuracy of procedures and
auxiliary tables for clearing lunar distances was considered meaningless by
sailors (not by theoreticians and teachers of navigation). In the second half
of 19th century the more precise forms and tables of approximate methods
appeared (Chauvenet after  1850, Bolte in 1894 and certainly other ones). But
they became so complicated in use that they provided no advantage in
comparison to rigorous methods. So the generic line "(Lyons) - Elford -
Thomson - epigons of both" prevailed up to the end of lunars.

Of course, Thomson's empirical procedure and table constructed by it could not
be taken into the official navigation handbooks and courses. Thomson's
procedure and tables remained an illegitimate darling of sailors for 80
years, but they never obtained the status of their reputable bride.

Jan Kalivoda

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