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

**Classification of the methods for clearing the Lunar Distances - corrected version**

**From:**Jan Kalivoda

**Date:**2003 Apr 13, 14:05 +0200

Thanks to numerous comments and corrections from several list members, I can send a corrected version of my article with same subject from 7th April into the list. I supposed that it would be appropriate. Jan Kalivoda ===================== CLASSIFICATION OF THE METHODS FOR CLEARING THE LUNAR DISTANCES 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, e.g. Circinus, Fornax, Horologium etc.) 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. ================ 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 the 19th century! - inspection tables (compare HO 214, 218, 240, 229 and ancient Ball's tables, firstly edited in 1907 - compare HO 208) - "short" methods (compare Ageton's method in HO 211, Dreisenstock's method in HO 208, Aquino (HO 200), Smart, Ogura, 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 the 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 (onwards LD'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 LD! '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 have 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 its 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 LD) and MS ( true=cleared LD) cross each other! at the common point X. But the sections mM and sS are very short (one 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 LD), we can trigonometrically deduce approximate equation permitting to reduce ("clear") the apparent LD to the true LD (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 LD, 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 the approximate equation. 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 practice, as introduced from 1810: D = d - HP sin s cosec d + HP sin m cotg d + MYSTERY The navigator computed only the two first corrections by logarithms of trigonometrical functions to 4 figures and by proportional logarithms originally tabulated by Maskelyne for interpolating the tabulated LD'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 the 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 in 1810 for the first time and which 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 1824). In 1851 the 42th edition appeared, in 1880 the 67th edition! (Thanks to Bruce Stark for this information.) 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 90 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 practice 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 LD'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 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 American astronomer William 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 (Newcomb'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 survi! ve, at least for checking the chronometers. 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 strict 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 1793 (he edited tables for this purpose in 1809), 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] Another slightly simpler forms of these equations are published. In spite of it, 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 requiring 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 certainly the frequent case). The last rigorous method invented is the very ingenious procedure of Bruce Star from the last years, which overcomes the disadvantages of rigorous methods and retains their advantages. Unfortunately, I don't know much about it, hence I cannot give any details. Thank you for your further corrections and supplements. Jan Kalivoda