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From: Frank Reed
Date: 2016 Oct 3, 14:01 -0700
Francis, you wrote:
"I have been experimenting with the Thompson lunar clearance method as described by you:
P1 = HP · sin(Hm) / tan(LD)
P2 = HP · sin(Hs) / sin(LD)
and
P = P1 - P2.
This is the effect of the Moon's parallax along the lunar arc."
One can derive those equations directly from the standard series expansion for clearing a lunar. That's how I did it, and I should note here that this is the linear contribution of the Moon's parallax. Also bear in mind that this is a general set of equations for the first-order calculation of the cleared distance. There's nothing specifically "Thompson" about it, and it's useful in many contexts. Deciphering Thompson's own methodology depends on understanding that he made his tables specialized to the task, rather than publishing generic tables of logarithms of trigonometric functions. He says to get the logarithm "of the Moon's apparent altitude" and add it to the "logarithm T of the apparent distance". But if you look at the tables where you're supposed to find these logarithms, they are actually logarithms of trig functions (and possibly with offset arguments). The logarithm of the Moon's altitude is actually the log sine of the Moon's altitude. And the log "T" is actually the log tangent of the distance. There are little tricks in the tables that may throw you off, but if you stick to the Bowditch version, those are mostly removed. Thompson created his tables in this streamlined form both for ease of use and also, I speculate, to render them somewhat difficult to reverse-engineer.
Frank Reed
ReedNavigation.com
Conanicut Island USA
