NavList:

Ed, you wrote:
"results compare very well with the Bowditch-Thompson method he also presented in his workshop (and I also programmed except for the 3rd correction which I must use paper tables for)."

Depending on the limits of your calculator, you can reproduce those tabulated values. 

You may recall Baron von Zach, who wrote about his surprise discovering that American navigators shot and worked lunars in 1817 aboard George Crowninshield's yacht Cleopatra's Barge, and his even greater surprise that the ship's chef (whom he first met holding a big knife in one hand and a bloody chicken in the other) could also work lunars and knew multiple methods for doing the computations. I spoke about him in one of my presentations in Mystic in November. Years later, when he normally styled himself the Baron de Zach, he pubished an article about Thompson's then-popular tables in which he did Thompson the enormous favor of proposing a computationally expensive method for producing those paper tables of the third correction. The effort required to calculate them by the baron's suggested method was prohibitive and quite possibly kept other authors of tables from copying his method (except Bowditch, who simply copied the tables in 1837). Today, of course, computation is dirt cheap, so it's quite possible to regenerate Thompson's "third correction" using the arduous technique proposed by the baron, if we choose.

Baron de Zach was an enthusiastic albeit amateur mathematician, and it appears that he did not understand the concept of the "series solution" to the problem of clearing lunars. In a series solution, we look at the large corrections which are linear (directly proportional) to the two bodies altitude corrections first. Then we look at terms quadratic in the corrections, for example proportional to the Moon's parallax squared. Since we're squaring small numbers, the quadratic correction is often negligible. And next we should (but never do!) calculate corrections proportional to the cube of the altitude corrections, the fourth power, and so on, ad infinitum. Anything that ends ad infinitum can sound bigger than it is. In the real world, including this case of clearing lunars, we can normally truncate a series of terms at a very low order. The mathematicians of nautical astronomy had learned at an early date that they needed one piece of the quadratic correction --the term proportional to the Moon's altitude correction squared. A slight refinement could be made by including the second piece of the quadratic correction which is proportional to the product of the Moon's altitude correction and the other body's altitude correction, and that's all that's required, even in rather extreme circumstances. We need go no farther. In short, to produce his more accurate third correction table, Thompson only needed to add on one small correction to tables which had preceded his. But Baron de Zach never undertood this, and many who followed him also never understood it. Instead the baron suggested that Thompson calculated each case, every possible example of a lunar, using the computationally-intensive direct triangle approach and then subtracted off the Moon's linear parallax term.

We can write a symbolic representation of the direct triangle solution as
  LD' = ƒ(h₁, h₂, LD) = acos(...a rather long expression...).
where LD is the original, observed lunar distance ange, h₁ and h₂ are the altitudes of the Moon and other body, and LD' represents the quantity that we are looking for --the "cleared" or "corrected" lunar distance angle. It's not hard to write down that "rather long expression". It depends on the constancy of the difference in azimuth between the two bodies. There's a lot of detail hidden in that ƒ( ) or "function of" notation but the principle is simple. This is the long, computationally-intensive direct solution.

Or we can travel the series path:
  LD' = LD + HP·cos(h₁)·cos(α) + terms linear in refraction + quadratic terms + higher-order terms.

In a series, we add on a laundry list of corrections directly to the initial base: the observed lunar distance. Note that the angle α (alpha) is the angle at the "corner" where the Moon is found in the spherical triangle consisting of the Moon, Zenith, and the other body, and cos(α) is what I call the "corner cosine" which is the geometric expression for the fraction of the Moon's altitude correction that "acts along" the lunar arc. That term, proportional to the Moon's HP, is the only trig work we really have to do. All the rest can be dumped into a lookup table. This was the key trick that evolved into Elford's tables and Turner's tables and others and eventually led to Thompson's commercially quite-successful tables.

Rewriting the expression above to emphasize that split, we have:
  LD' = LD + HP·cos(h₁)·cos(α) + TABLE,
and we can calculate the value that I'm calling TABLE either as 
  TABLE = two terms linear in refraction + two quadratic terms (+ higher order terms which we drop).
This is almost certainly how Thompson actually compiled his table.

Or we can follow the baronial suggestion. We can get LD' for each and every case by the long, direct method and then use it for an alternate expression for this tabulated set of values:
  TABLE = ƒ(h₁, h₂, LD) - LD - HP·cos(h₁)·cos(α).
That's a lot of work, but it has the advantage of "seeming" to be computationally exact. Many nineteenth century commentators were seduced by the concept that this methodology integrated the quick ease of the series methods with the mathematical "rigor" of the direct triangle solution. This seduction continued right into the twenty-first century, and before I joined "Navigation-L" (the mailing "list" which gave us the "List" in "NavList") there were quite a few messages along these lines. It was all wrong...

For modern computation, the distinction is immaterial. Naturally, there's something to be learned from each approach, and it's quite satisfying to see that the numbers line up.

This started out as a short note... Oy.

Frank Reed