NavList:

Hanno, you wrote:
"re: vertical circle forcing Despite some lengthy effort to find a description I was not successful. Maybe I used not the correct term(s)? Taking advantage of your invitation to ask: Please be so kind to post some pointers where to find a description or, if not too long, to post a description again."

Just to clarify, my invitation to "ask" referred to a subsidiary point about altitude accuracy. We can talk about 'vertical circle forcing' at some point in the near future, but I think this would be a major distraction at the moment. I brought it up in the original post in this topic, both because it's something distinct from the other two main categories of lunars solutions, and also because it has a problem which also applies to your "hav-Doniol" tables --it's not terribly useful unless we can somehow transport it "by time machine" to navigators living fifty years ago (50 years for the hav-Doniol methodology, 200 years for my 'vertical circle forcing' methodology of lunars). I do have some ideas for you (and Greg) on how to make an impact with hav-Doniol even in the 21st century, and I'll post on that later, so it's not quite as bad as having a new method for working lunars by hand, but there's still a close analogy.

Hanno, in one message you referred to lunars calculations "a la Pearson" and then later you mentioned his full name, Arthur Pearson. Maybe I can clarify this for you... Arthur Pearson was merely a modern navigation enthusiast with a particular interest in lunars. In short, he was a typical NavList member. Back from about 2002 to 2004, Arthur Pearson was an active member of this community (that's back when this still was an "email list" and was known as "Navigation-L"). Here's a complete index of his messages in the NavList archives. He compiled some summary articles on the community's consensus opinions on lunars at that time, much of which was good, some of it misguided. He also created a very nice spreadsheet which allowed a user to enter the Moon's almanac data and the backyard observer's lunar observation details, and it would clear the lunar distance using the standard "direct triangle" computation which I described in the opening post of this subject thread here. This method of clearing lunars has been understood as the baseline approach since the 18th century. This was not "his method" of clearing lunars, and there's nothing about this that is "a la Pearson". Understand? And his spreadsheet is now thoroughly obsolete since it required the user to enter almanac data. It has been replaced by many other tools, and as early as 2004, it was completely superseded by my lunars clearing software 'web app' which you can access here (under 'Clear a Lunar'). Incidentally, Arthur Pearson became unreachable in 2004, and I have to assume that he passed away or became incapacitated at about that time. His web site soon fell into disrepair. A web site domain name which he had acquired for lunars eventually expired, and it was scooped up by a domain-merchant (now on sale for $2495... it's only worth about $20 --if that).

I mentioned in my earlier post that there are numerous variants on the basic methods that are all derived by trig identities. That is, they change the steps of the calculation, change the recipe, but are otherwise mathematically identical (in the same sense that sin²(x/2) is mathematically identical to (1-cos(x))/2 which is also defined to be hav(x)). In the late 18th century there was a great abundance of variations on the two main types of clearing methods. I think you, and others, may enjoy reading an article from that period that reviewed and explored these various mathematical variations. It was read before the Royal Society in December 1796 and published in the Transactions of the Royal Society in 1797. The paper was written by one of those international scholars who personified the Enlightenment period in the late 18th century. He was a Spaniard, Don Jose de Mendoza y Rios, who lived in England, and normally wrote in French. I'm attaching his article from the Transactions on problems in nautical astronomy (see below). It's in French. Hanno, I think you read French, n'est-ce pas? For anyone else, this is technical French, so with a dictionary or an online translation tool at your side, it's not difficult to read.

The paper by Mendoza Rios (the shorthand version of his name often used in contemporary English sources) is divided into two major parts. Starting on page 44 he discusses the mathematics of the problem of latitude by double altitudes, the so-called Douwes problem, which fascinated mathematicians in this period. On page 74, he discusses the "time sight", and on page 75, you'll find the standard equation for the time sight (he attributes this to Borda) which was used from the late 18th century right through the middle of the 20th century. It's interesting to note that this standard time sight formula is an implicit haversine formula but it was not known by that name at the time. There was really no need for a separately named function in this context. At the beginning of the article, Mendoza Rios introduces notation for a variety of additional trig functions including the versine (written out Latinate as "sinus vers" and abbreviated as "sin. v." in most of the paper) as well as others like the "su cosinus vers" which is actually just 'one plus the sine' of an angle, but the "haversine" name was not used in this era (and in fact the late 19th century "haversine" name breaks the international Latinate naming rule since the 'ha' is obviously English for 'half').

Part 2 of Mendoza Rios's paper, which begins on page 77, is devoted to the methods for correcting the lunar distance. Naturally he dives right in with the standard "Direct Triangle" solution and then proceeds to apply various trigonometric identities to produce over a dozen equivalent expressions, many involving versines. Buried in there, no doubt, is a progenitor of the more recent Doniol haversine solution of the spherical triangle. I should emphasize that all of these variations are mathematically identical to the basic "cosine formula" solution and they depend on the fact that the difference in azimuth is constant when transitioning from the observed altitudes and lunar arc to the corrected altitudes and the corrected lunar arc. Incidentally, Henry Halboth, who was working lunars even in the 1940s, mentioned in a recent NavList message that he enjoyed clearing lunars using a "versine method".

By the way, many people in recent years have been introduced to the mathematical process of clearing lunars via John Karl's book. That book employs some unusual (and un-necessary) variations in this process and in the terminology. Rather than calling the difference in azimuth simply that, or maybe dZ, he chooses to refer to this as RBA or "relative bearing angle". He then proceeds with the standard "direct triangle" solution, exactly what you find on page 79 of the Mendoza Rios paper published way back in 1797 and exactly what was implemented in Arthur Pearson's spreadsheet back in 2003, when you peel away the idiosyncracies.

Beginning on page 96, Mendoza Rios describes the "Series Solution" to clearing lunars, which he calls "Approximate Methods" though he makes very clear that these methods are not approximations in any meaningful navigational sense and fully satisfy the accuracy requirements of the lunarian navigator. At the top of page 103, Mendoza Rios write the general solution to quadratic order in terms of the factors which I call "corner cosines". In his paper, he writes cos(L) for the cosine of the angle at the Moon (L for 'lune') and cos(S) for the cosine of the angle at the Sun. He goes on the demonstrate a method for calculating cos(L) and cos(S) in the principal terms that "requires no distinction of cases" despite being somewhat longer in the working (see PS). This is the same transformation found in Borda's "time sight" formula which I mentioned above.

After discussing all these various methods and their variants and demonstrating how one can analyze any --quelconque-- method by fitting it into the zoology of methodologies in this paper, Mendoza Rios closes the discussion with a tease. He explains that he is a fan of a certain versine procedure for clearing lunars, but that his set of tables for this purpose is not yet ready ...soon to be published. In fact this paper itself, presenting all of this detailed analysis, may well have been a sort of advertisement of Don Jose's mathematical skills designed to encourage the funding that he needed to complete his large volume of tables for clearing lunars. Mendoza Rios was under the tutelage of Joseph Banks at this time, and Banks may well have suggested an article to enhance his standing (to be clear, this is just a speculation on my part). Those tables were, in fact, funded eventually, and the tables of Mendoza Rios were relatively popular as an efficient and fast method of clearing lunars for more than a decade. However, as I noted in my original post on this topic, the much shorter and more efficient series methods eventually won out for the last few decades of the use of lunars at sea.

Frank Reed
ReedNavigation.com
Conanicut Island USA

PS: The transformation of the calculation of the corner cosines described by Mendoza Rios which "requires no distinction of cases" is the same small improvement in working lunars which was published by Nathaniel Bowditch c.1799 and which was wildly aggrandized by later accounts, still found in the introduction to the American Practical Navigator in the late 20th century. It wasn't perceived quite the same way at the time. In fact, Bowditch was accused of "mutilating" this Mendoza Rios technique by Norie, the author of a major competing navigation manual. The coincidental timing here, that Bowditch introduced this mathematical transformation as his own just a few years after it was published by Mendoza Rios led to some controversy at the time, and it's not beyond the realm of possibility that Bowditch learned about the trick from the Mendoza Rios paper and left that part out as he was being showered with accolades as one of the young nation's (the USA's) finest mathematicians. It's also quite possible that Bowditch happened upon the trick independently. This isn't rocket science...

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