# NavList:

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

**Re: Planisphere pour les Distances Lunaires**

**From:**Frank Reed

**Date:**2009 Jun 2, 19:42 -0700

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 --~--~---------~--~----~------------~-------~--~----~ Navigation List archive: www.fer3.com/arc To post, email NavList@fer3.com To unsubscribe, email NavList-unsubscribe@fer3.com -~----------~----~----~----~------~----~------~--~---