# NavList:

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

**Re: Easy Lunars**

**From:**George Huxtable

**Date:**2004 Apr 29, 21:11 +0100

Riffling through the dross in my in-box, among the offers for Viagra and enlarging my penis (how CAN they tell? I ask), I found this nugget on "Easy lunars" from Frank Reed. I liked his clear explanations, and his simple approach to estimating a land-bound rough-altitude of the Sun. In a flat and gridded city such as Chicago, with many shoebox buildings, there will be no shortage of parallel horizontal lines to converge at a vanishig-point on the horizon. Perfectly good enough in accuracy for estimating Sun altitude, to allow for the small correction for Sun refraction. Clearly, Frank's proposed method expects the user to have some sort of calculator, with trig functions. In that case, I think further simplifications might be made. Follow his explanation until you get to- >And now to clear the distance... At this point, we have the observed lunar distance d, which should already have been partly-corrected for index error and for semidiameter(s), but not yet for the effects of parallax and refraction. We have the Moon's altitude m, which should already have been partly-corrected for index error, dip, and semidiameter, but not yet for the effects of parallax and refraction. And for the other-body, which might be the Sun, or a star, or a planet, we have its altitude s, again partly-corrected for index error, dip, and (if relevant) semidiameter, but not yet for the effects of parallax (if relevant) and refraction. It's easy to correct m and s for parallax and refraction, by adding an amount for parallax (can be up to 1 degree for the Moon) and subtracting an amount for refraction. These corrections correspond to Frank's quantities dh_Moon and dh_Sun, and he has explained how to obtain them. This gives us two corrected quantities, M for the Moon, which because of its great parallax will always exceed m, and S for the other-body, which will always turn out to be less than s. Now, from all this, we need the corrected, or "cleared" lunar distance D, which allows for the combined effect, on the lunar distance, of those corrections to the altitudes of the Moon and the other-body. And this comes straight out from one equation, as follows- D = arcCos ((( cosd - sinm sins) cosM cosS / (cosm coss)) + (sinM sinS)) And that's it! This is the 'cleared' observed lunar distance, to be interpolated between the predicted lunar distances which were calculated for successive hours, just as Frank explains. Of course, to work this on a calculator, all the degrees-and-minutes quantities have first to be converted to decimal degrees. That expression for D (or its equivalent) can be found in Cotter's "History of nautical astronomy" as a stepping stone in deriving other methods; Borda's, Young's, and Krafft's. I hear you say- "Well, if it's that simple, why didn't they use that formula in the 18th/19th centuries?". I will explain. Really, it's because they didn't have pocket calculators. Just think about the percentage accuracy that a navigator required for calculating a lunar. He needed to measure to the ultimate accuracy of which his eye, and his sextant, were capable. To better than an arc-minute, perhaps to 0.2 arc-minutes after some averaging, in an angle that can extend to well over 100deg of arc. So that's to 1 part in about 30,000. Even today, there are not many quantities we measure to that accuracy, outside a precision workshop. And to avoid any dilution of that accuracy, he wanted his calculations to be done to an even better standard, to 1 part in 100,000. That's why 5-figure, and even 6-figure log trig tables were called for by navigators. In those days using logs was a matter of necessity: there was no way a mariner could reliably process such numbers by long-multiplication and long-division. But there were some terrible snags about using logs, and log-trig tables. First, the log of a negative number, or of zero, has no meaning, and trig formulae had to be bent and twisted to avoid such numbers arising (which is why haversines arose). Second, although logs were fine for expressions where trig quantities had to be multiplied and divided, if it came to additions or subtractions part way through, it was necessary to "come out of logs" by using a lookup table, do the additions/subtractions, then take logs again. The extra time and trouble, and the extra likelihood of error, that this entailed, meant that it was avoided at all costs by further twisting of the trig formulae used. Third, as more accuracy was required of log trig tables, they became bulky books, with hundreds of pages to thumb through in each lookup procedure. In general, pocket trig-calculators avoid all those problems, and allow us to go back to the simpler, basic, underlying trig formulae. Most such calculators work trig quantities to quite sufficient accuracy. My own ageing Casio works to 1 in 10 billion. If James Cook had been offered such a device, he would have welcomed it: though no doubt he would have wanted to make yet another circumnavigation, to test it out against his log trig tables, before really trusting it. Frank goes on to say- >The method for clearing a lunar that I've described above is very similar to >the old methods of Witchell and Thompson and others which were popular in the >19th century (compare Witchell's method in the online copy of Norie --URL >below). Those methods depended on logarithms for their calculation of A >and B and >they used various trigonometric identities to put those calculations in a form >suitable for logarithms, so you'll see tangents and cotangents,e.g., instead >of sines and cosines, but that's just a re-arrangement of terms. There is no >fundamental difference. I don't know anything about Witchell's method, but I do have a copy of "Lunar and Horary tables", by David Thomson (not Thompson). This must have been remarkably popular in its day, as my 1857 copy is already in its 52nd edition! This method adds three correction terms to the observed distance to clear it, just as Frank's method does. But his method appears to differ more from Frank's than he implies. From my limited understanding (it's all rather obscure) it appears that the first two of Thomson's terms relate only to the corrections due to Moon parallax, and that the refractions and any Sun or planet parallax are lumped together into the third correction term (which Jan Kalivoda would describe as the "magic" term). Thomson's procedure, and Frank's too, offer a clever way of avoiding the requirement of extreme precision in the calculation of lunars. They are both approximation techniques which take the observed lunar distance and simply add some small corrections to it. As those corrections never amount to more than 1 degree at most, evaluating those corrections to 1 part in 1,000 is more than sufficient to get a good enough answer. As a result, Thomson's tables became quite thin ones, and easy to handle. In those days, it was a very practical way to correct lunars. But I suggest the relative convenience of that method, and its like, have now been eclipsed by the pocket calculator, because it can offer all the necessary precision at the touch of a button, can handle positive and negative trig quantities, and is unworried about mixing in sum terms with product terms. For those that have such a trig-function (and inverse) calculator, especially one that can be programmed-up with the expression given above, the "clearing" of a lunar loses all its terrors. Thanks to Frank Reed for a really stimulating posting. George. ================================================================ 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. ================================================================