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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 delightful website devoted to "lunars" (http://www.ld-DEADLINK-com/). I thank 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 calculations made by seaman 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 table of contents; but Thompson=Thomson is cited in the main text at 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