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Thank you for providing the detailed photos of this device and its 
documentation. It's a neat little calculating machine and complements nicely 
the recent discussions about using slide rules to work celestial sights. 
While standard altitude calculations for modern celestial navigation really 
test the accuracy limits of most slide rules, the calculations for lunars are 
a different story. Clearing lunars, if done according to a certain plan, 
requires only three significant digits of accuracy, and that's within the 
capabilities of common slide rules and this planisphere device.

The "notice" or documentation for the planisphere outlines in good detail how 
the calculation works. He's proceeding by a fairly standard "series 
expansion" solution to the problem of clearing lunar distances. Several pages 
are devoted to deriving the basic equation, his equation "(1)". He seems to 
be unaware that he is re-inventing the wheel (in this case, literally) or at 
least he's not worried about the fact that he's re-inventing it. This is a 
common phenomenon in articles about lunars from the latter half of the 
nineteenth century. His equation "(1)" with minor variations in grouping the 
terms had been well-known to nautical astronomers and mathematicians who 
studied lunars for the better part of a century. It consists of two linear 
terms and between one and three quadratic terms (depending on where the 
series was truncated). You can find the same equation derived in the rather 
long paper by Mendoza y Rios published in the Transactions of the Royal 
Society back in 1797. Here's the relevant page from that article: 
http://www.historicalatlas.com/lunars/myr/pagevu1.asp?num=98.

Incidentally, all of the methods for clearing lunars which were published in 
Bowditch through 1880 worked the problem using almost the same math. The 
differences among these methods were primarily "accounting" and certain 
clever tricks for avoiding subtraction when possible. One could group 
together the various pieces of the quadratic calculation into different 
tables, and that's mostly what's going on in the different methods. One of 
the last methods for clearing lunars published in practice was included in an 
appendix in the British Nautical Almanac in the early 20th century (the one 
used by mariners, the "Nautical Almanac, abridged for the use of seamen"), 
and that, too, was a series solution which used a clever trick advocated by 
Airy for eliminating the quadratic corrections (by doing the linear 
correction twice and then averaging).

Although I haven't done any actual comparisons of computation time, I doubt 
that this planisphere would have saved any time compared with the nicely 
organized tables like Bowditch's First Method or Thompson's tables (also 
published in Bowditch as the Second Method after 1837). Its creator seems to 
have fallen for the same mis-conception that plagued a large number of 
mathematicians in the late 19th century. They frequently seemed to believe 
that lunars were not popular anymore, not because they were no longer of 
practical value at sea, but rather because the math was too complicated. 
Again and again in publications like the MNRAS (Monthly Notices of the Royal 
Astronomical Society) you'll find mathematicians re-inventing various tricks 
for working the series expansion solutions to clearing lunars and stating 
that their "new" solution will finally make clearing lunars "quick and easy 
for the common man". It's a noble goal, of course, but since many of them 
were working in relative isolation, they simply didn't realize that they were 
solving a problem which was not only already solved but which had also fallen 
into obsolescence.

By the way, it should be relatively easy to build a working model of this 
planishere and it would certainly be an entertaining way of clearing lunars 
even if it was un-necessary historically. Any volunteers?

-FER




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