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A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Re: Lunars Tables
From: George Huxtable
Date: 2007 Dec 5, 19:57 -0000
From: George Huxtable
Date: 2007 Dec 5, 19:57 -0000
Andres Ruiz wrote: "Both with the best method for clearing a lunar distance, for me: the Young�s formula." Aqnd Frank Reed commented- Ya know, unless you get into historical things, like working out the calculation by logarithms or maybe using tables of so-called "logarithmic differences," there's really no meaningful difference between "Young's formula" and the common cosine formula. They differ only by a trig identity. Once you put these calculations on a computer, they are identical (except for a very small difference in computation time, on the order of one microsecond for each cleared lunar). The attention given to Young's formula in Cotter's "History of Navigation and Nautical Astronomy," and repeated by some who have read that book, is completely out of proportion to its significance. Young's formula is a footnote in the history of lunars. This mistaken attention appears to be due to the fact that Cotter focused on publications from the period when lunars were obsolescent --post-1850. ====================== Comment from George- I think the point has been missed, in Cotter's book and by Frank also, about what Young was trying to do. He was an academic, a prof. of mathematics at a college in a UK seaport, rather than a practising navigator, and he was interested in the teaching of navigation, and trying to get students to understand what they were doing. All the other texts relied on logarithms, which was, in those days, just about the only way of doing precise arithmetic calculations, if you wanted to do anything more complex than simple addition and subtraction.. However, using logs presents two enormous problems. 1. Having gone "into logs", it's then easy to do multiplications, divisions, raising to powers, and all that. However, you can't add and subtract intermediate quantities, that come into an equation, without coming "out of logs", into the real world, doing the addition / subtraction, and then taking the log of that intermediate result (not without the clever trick that Bruce Stark uses in his lunar tables, a special table of addition / subtraction, or Gaussian, logs). 2. Logs are only meaningful for positive quantities. So lunarians went to enormous trouble to fiddle their trig to allow logs to be used. Instead of sines and cosines, which are negative for certain angles, haversines were invented, always positive. And the trig formulae were distorted and twisted, so that any additions and subtractions, if they were unavoidable, were at the start or the end of the calculation, separate from the part where logs were used. And this is why there were so many different "solutions" to the lunar problem; theorists vied with each other to minimise the number of arithmetical steps that were called for to get an answer. Young provided a solution to lunars using logs. But he advocated an alternative way of doing the job, that didn't call for logs at all, instead requiring the user to do one precise "long multiplication", to 6 figures, and then one equally-precise "long division". Without access to calculators, students were expected to be much handier at such figure-work than they are today. When Andres and Frank state that they are using "Young's method", I suspect that both are referring to what Cotter described as "Young's method" in his book, "History of Nautical Astronomy", for obtaining the corrected lunar distance, D, from the observed lunar distance, d. This calculation always required, as an auxiliary measurement, the observed moon altitude m, to which parallax and refraction adjustments were applied to give a corrected moon altitude M; and also, the observed sun or star altitude s, similarly adjusted to provide a corrected altitude S. Let's start from the basics- If you construct two slightly-different navigational triangles, joining the observer's zenith to the Sun amd the Moon; in one case to the observed positions m and s, and in the other case to their true positions M and S, and recognise that the azimuths of Moon and Sun are unchanged by the altitude corrections, then we can say, in one case - cos (azimuth difference) = {cos D - sin M sin S } / (cos M cos S) and in the other- cos (azimuth difference) = {cos d - sin m sin s } / (cos m cos s) so therefore- {cos D - sin M sin S } / (cos M cos S) = {cos d - sin m sin s } / (cos m cos s) which we can write as cos D = {cos d - sin m sin s } {cos M cos S } / {cos m cos s } + sin M sin S That's about as basic as you can get, as an expression for the corrected lunar distance D. With a computer, or a pocket calculator, it presents no problems at all. Young tinkered with it a bit, to turn it into- cos D = {cos d + cos (m + s)} {cos M cos S} / {cos m cos s } - cos (M + S) which is what Cotter quotes, as "Young's method", and presumably what Andres and Frank are referring to also. This has just one advantage. It calls only for a table of cosines: no sines or other functions are needed. That enabled Young to incorporate a nice big table of cosines in early editions of his book "Navigation and Nautical Astronomy". These were labelled "natural cosines" to differentiate them from the more-common log. cosines, and were presumably to 6 decimal places (unfortunately, that table has been excised from my much-later 1903 edition). Of course that was a simplification only when large lookup tables were called for, and makes no difference when things are done by calculator or computer. Cotter commented " The computation of cos D, using Young's method, is shortened considerably if a table of logarithmic differences is used to evaluate {cos M cos S} / {cos m cos s }". That shows Cotter's misunderstanding of what Young was trying to achieve. A table of log differences would help only if the whole calculation was being done by logs. Young showed, for completeness, how that could be done, but his point was quite different. Instead he recommended that it should be done without using logs at all. To that end, he rewrote the expression for cos D as- {cos d +cos (m + s)} {cos (M + S) + cos (M - S)} / {cos (m + s) + cos (m-s)} - cos (M + S) At first sight that looks more complicated, but you can see that each of the three terms in curly brackets is just the sum of two cosines, taken from the cosine table. First he needed the cosines of 5 angles, d, M + S, M - S, m + s, m - s. Note, by the way, that because cosine of a negative angle is exactly the same as for a positive angle, with the terms involving subtracted angles, you can make that subtraction in the easiest way, taking the lesser from the greater. Sometimes the sum of two angles will exceed 90 degrees, resulting in a negative cosine, which sign must be observed. Anyway, those three terms in curly brackets could be readily arrived at, with just the cosine table, and no need for logs. Next comes the difficult bit, when you try to do the job without logs, which asks you to multiply the first two terms in curly-brackets together, maintaining a six digit accuracy, then dividing the result by the third term in curly brackets, still maintaining that accuracy. That is, a rather long long-multiplication, followed by a rather long long-division, done without arithmetic errors. Beyond most of us now, I expect, but in the 1860s navigators were expected to be fluent and precise with such "figuring". Finally, that last term cos (M + S) had to be added, and the result gleaned from the cosine table, using it in reverse, and taking the complement from 180 degrees if the result was negative, as it would be for a large lunar distance. That difficult arithmetic the price you had to pay for working without logs, and Young argued that it was a price worth paying, in terms of overall simplification. Whether he was right or wrong about that, Cotter misunderstood what he was getting at. It's all, of course, completely irrelevant today, with our electronic tools, just as Frank says. George. contact George Huxtable at george@huxtable.u-net.com or at +44 1865 820222 (from UK, 01865 820222) or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to NavList@fer3.com To , send email to NavList-@fer3.com -~----------~----~----~----~------~----~------~--~---