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## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Lunar distances - shot clearance methods**

**From:**George Huxtable

**Date:**2004 Sep 12, 12:19 +0100

Henry Halboth revisits Arnold's lunar distance method, to tie up some loose ends. At the foot of this posting, I will restate my earlier transcription of Arnold's text into emailese, as quoted by Henry, but amended to correct a few things that went wrong with his original text. I commented on Arnold's simplifications of Mendoza, on 5th Sept., as follows- "This seems a worthwhile simplification on Mendoza's method.. However, it comes at a price. Arnold has had to supply separate tables for correcting Moon, Sun, and stars, though he can avoid the need to use separate refraction and parallax correction tables. And Arnold has lost some flexibility in this simplification. When Mendoza calculated refraction, it was possible, if he thought fit, to apply corrections for a non-standard atmosphere. If a lunar was taken to Venus or Mars, which require non-standard parallax corrections, that parallax, if known, could readily be applied. (Does anyone know when lunar distances to planets started to appear in the Almanac?) As I see it, however, Arnold's method, including standard refraction and parallax in his tables I, II, and III, was inflexible in that it would have been unable to adapt to such requirements." And Henry has responded- >As regards Arnold's tables being somewhat restrictive by reason >of constraints in altitude correction, please note the use criterion >to be the bodies apparent altitude or the correction thereto; how either >is calculated is not mandated and may be at the user's option - >whatever refinement in the way of temperature and latitude >corrections for refraction and parallax may certainly be applied >at discretion. ================== Reply from George- Well, I don't wish to make a "big thing" of this resulting lack of flexibility, because I don't regard it as an important matter in practical terms. But I question Henry's argument here. The difficulty is that the altitude correction of the Moon (and the same applies to the Sun or star) enters into Arnold's calculation, not once but twice. Taken from the transcription below is the sentence- "Add to the apparent distance the first correction and the correction of the sun or star's altitude, and subtract the sum of the second correction, and the correction of moon's altitude will be the corrected distance." I think that may perhaps be slightly garbled, but its meaning is clear enough. Perhaps it should really have ended something like "...and subtract the sum of the second correction and the correction of moon's altitude, and the result will be the corrected distance." No matter. As far as that sentence is concerned, Henry is quite correct. That "correction of Moon's altitude" can be worked out by the navigator as he chooses, and he may include, or else disregard, temperature / pressure corrections to refraction (same for Sun or star). BUT that altitude correction (which combines refraction and parallax) enters also into Table I (and table II or III, as relevant). Arnold states- "Enter Table I with the Moon's apparent altitude and horizontal parallax, and take out the corresponding logarithm, which place in the first column." and Henry himself describes the construction of that table as- "Tabular log = log sin (30 deg) + log cos Moon's apparent altitude + [prop-log] of Moon's altitude corrections." So built into table I (for the Moon, and similarly into tables II or III for star or Sun) is its own "Moon's altitude correction" (for refraction and parallax), which the table can deduce from its knowledge of the altitude, and which the user has no means of tinkering with. That's what prevents him from applying corrections to refraction for extreme climatic conditions, or allowing for the parallax of Venus or Mars. I wonder if Henry is convinced. As I said, it's not a big deal. ================= Henry continues- One cavalier treatment, not previously mentioned, is that >Arnold advocates a standardized observed altitude correction to obtain >the apparent altitude; under a Rule III, he advocates, across the board >... > >"To the moon's observed altitude, add 12', if the lower limb be taken, >but if the upper limb be taken, subtract 20'. to the observed altitude of > >the sun's lower limb add 12', and from the star's observed altitude >subtract 4', and you will have their apparent altitudes." Of course, >we know this to be technically incorrect - perhaps it is simply a >reflection of the often expressed opinion that an error of a few >minutes of arch in altitude does not materially affect the result in >clearing the distance. Comment- This seems a crude approach, but it's a perfectly valid approximation for the purpose of correcting lunars, because the altitude corrections only vary slowly with altitude itself, and because it's those corrections, rather than the altitudes, that are so vitally important for clearing the lunar distance. ============== I should point out that anyone who, like me, tries to get to the bottom of Mendoza's approximate method, and Arnold's method, will find a description in Cotter's "A history of nautical astronomy" , pages 227 to 231, of Merrifield's method. This is very similar to Mendoza and Arnold, and Cotter's analysis showed the way that the trig was done. Without that, I couldn't have tackled Mendoza's or Arnold's recipes. Taking Cotter's description at face value, the main difference seems to be this- Merrifield simply disregards the final correction term, provided by Arnold's Table VII. George. =================================================== "A short Method of Correcting the Apparent Distance of the Moon from the Sun or Star. Invented by the Author. Rule Add together the apparent distance and apparent altitudes, and take half their sum; The difference between the half sum and the sun or star's altitude, call the first remainder. The difference between the half sum and the moon's apparent altitude, call the second remainder. Set down- The sine of the apparent distance in two columns The secant of the half sum also in both columns The cosecant of the first remainder in the first column And the cosecant of the second remainder in the second column. Enter Table I with the Moon's apparent altitude and horizontal parallax, and take out the corresponding logarithm, which place in the first column. Enter Table II, if a star is used, or table III, if the sun is used, and take out the corresponding logarithm, which place in the second column. The sum of these four logarithms, rejecting the 10's in the indexes, in the first column. will give a proportional logarithm of the first correction. And the sum of the four logarithms in the second column, rejecting the 10's in the indexes, will be the proportional logarithm of the second correction. Add to the apparent distance the first correction and the correction of the sun or star's altitude, and subtract the sum of the second correction, and the correction of moon's altitude will be the corrected distance. Then enter Table VII, with the corrected distance at the top, with the difference of the first correction, and the correction of the moon's altitude in the left side column, and also in said table with the correction of the moon's altitude in the left side column, and take out two corresponding numbers. The difference between the two numbers is to be added to the corrected distance when less than 90 degrees, or subtracted if above 90 degrees." Henry Halboth adds the following notes about the tables- "Relevant included tables". Table I = A table of logarithms against top entries of Moon's Apparent Altitude and with side entries of Moon's Horizontal Parallax. This table is constructed/calculated as follows... Tabular log = log sin (30 deg) + log cos Moon's apparent altitude + [prop-log] of Moon's altitude corrections. Table II = A table of logarithms agains Star's Apparent Altitude This table is constructed/calculated as follows... Tabular log = log sine (30 deg) + log cos Star's apparent altitude + [prop-log] of Star's altitude correction. Table III = A table of logarithms against Sun's Apparent Altitude. This table is constructed/calculated as follows... Tabular log = log sine (30 deg) + log cos Sun's apparent altitude + [prop-log] of Sun's altitude correction. Table VII = A table of corrections against corrected distances across top and the difference between the first correction and moon's altitude correction or the moon's altitude correction alone as side entries. This table provides a third correction to the Distance for parallax and refraction.- it is essentially Norie's Table XXXV, presented in a slightly different manner. There are other tables included which are essentially as contained in any navigational epitome, i.e., altitude corrections for the Moon, Sun, and Stars, etc., which are unnecessary of comment at this time." ================================================================ contact George Huxtable by email at george@huxtable.u-net.com, by phone at 01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. ================================================================