NavList:

Jacques, you wrote:
"It seems to me that there are TWO Lyons methods for clearing the distance."

I can make an educated guess. I don't know what sources you're using where you're finding these two methods (could you tell us?), but I suspect that this is just semantics. There is a computational recipe that was the "original" Lyons method, but since it was among the earliest of the series solutions, any later series method can be categorized as a variant on Lyons method. In the world of taxonomy and classification, there are always two camps: splitters and lumpers. The splitters take every small difference and slap a label on it. The lumpers group things together and focus on common properties and collect together all "cousins" under one name. With respect to the method of Lyons, I have seen historical sources that will label ANY series method as a Lyons method. Most, however, limit this to one specific computational method, which actually is a distinct, "cleaned up" version of the one originally published under the name of Israel Lyons.

You described the two possible methods as follows:
"First method : calculation of two corrections (as Whitchell) for sun (or star) and moon ;
Second method : calculation of one correction for moon parallax, and correction of refraction for linear tables, but the formula for parallax is différent of formulas of first method."

The "first" one sounds like the standard Lyons method. The difference from Witchell and Bowditch (also known as the method of Mendoza Rios, thanks to Norie) is the manner in which the geometric factors or "corner cosines" are calculated. Most of the common series methods group the calculation as
d' = d + δh1*A + δh2*B + (δh1)^2*C + ...
where d' is the corrected distance, d is the observed (center-to-center) distance, δh1 is the Moon's ordinary altitude correction and δh2 is the other body's ordinary altitude correction. A, B, and C are geometric factors (the first two are the "corner cosines") that have to be calculated, and that's what distinguishes Witchell from Lyons and the others. The standard Lyons method calculates A and B basically using the common cosine formula for the spherical triangle, while Witchell uses right spherical triangles, and Mendoza Rios/Bowditch use the so-called "haversine" formula. It's the same goal. They calculate factors which multiply the altitude corrections. The results are the linear terms in the series expansion. Generally the quadratic correction (with its parameter C) is small enough that it can be tabulated. Witchell and Lyons and the rest can all use that same table. Other later methods added in the next quadratic corrections, but this was really un-necessary. It was simple to "slipstream" those changes into the other methods: replace the old table for the quadratic correction with a new table. Superficially, the method is unchanged.

The "second" method you describe sounds to me more like Norie's "linear tables" from c.1820. The expression "linear tables" can be confusing. It does not refer to the linear terms in the series expansion (in modern terminology). Instead, in that era the English words "linear tables" referred to graphic diagrams (they were "tables" built from drawn "lines", hence "linear tables"). Such diagrams were expensive to produce, but to some extent this was commercially useful to Norie since it made his "linear tables" more difficult to steal. Norie's linear tables, like a number of other calculational recipes from this period, split up the altitude corrections into separate parallax and refraction portions. This has a small advantage since the combined refraction, for standard conditions, can be tabulated separately. If you come across John S. Letcher's excellent 1978 work, there is a nice chapter on lunars, and he advocates a method which he concocted himself that similarly splits off the refraction components.

A further refinement of this process combines the quadratic correction with the tabulated refraction correction. That leaves only one long-hand calculation requiring logarithms: the calculation of the Moon's linear term but with parallax only in the correction δh1. That is, the lunarian navigator had to work up δh1*A, and since there's a cos(h1) in the Moon's parallax correction and also in the denominator of A, that cancels out and the work is fairly short. The rest, which included refraction and the quadratic corrections, was just a "look-up" in a rather large, specialized table. The most successful in this category was Thomson's method which was even borrowed wholesale into Bowditch's Navigator starting in 1837.

-FER

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