# NavList:

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

**Re: Easy Lunars**

**From:**Frank Reed CT

**Date:**2004 May 3, 18:16 EDT

George H wrote:

"Clearly, Frank's proposed method expects the user to have some sort of calculator, with trig functions. In that case, I think further simplifications might be made. "

I'll start off by saying that that wasn't really the point of 'Easy Lunars'. There is only an insignificant difference in 'calculational efficiency' between the two ways of clearing a lunar and no difference in accuracy. The post I made was designed to make one specific type of calculation with deep historical roots "comprehensible", and I'm fairly confident that it did that.

George wrote:

"D = arcCos ((( cosd - sinm sins) cosM cosS / (cosm coss)) + (sinM sinS))

And that's it!"

Yep, the standard "triangle method". There are two general categories that cover most of the mathematical techniques for clearing lunars. Historically they were known as "rigorous" and "approximative" but these names are distinctly misleading. I prefer to call them "triangle solutions" and "series solutions". The series solutions are every bit as good as the triangle solutions within the normal limits of the lunars problem generally. They solve the same problem to the same level of accuracy. I do agree though, that if you aren't worried at all about history or pedagogy, then the triangle solution [which you posted] is the way to go. When I do this sort of calculation on a computer, I do the triangle solution unless the purpose is to demonstrate something about the nature of a series solution. But I think there is significant historical and pedagogic value in the series solutions.

And George wrote:

"I hear you say- 'Well, if it's that simple, why didn't they use that formula in the 18th/19th centuries?'. I will explain. Really, it's because they didn't have pocket calculators."

But they did use the triangle solutions AND the series solutions in the 18th/19th centuries. The real question is why did some navigators PREFER the series methods over the triangle methods?? Half of the answer to this question is, just as you say, they had no "pocket calculators". That is, computation was expensive and time-consuming and the triangle methods were more of a nuisance on this score. But there is another reason which still applies today: pedagogy. The linear terms of the series methods are very easy to interpret. The idea that "A" and "B" (in my terminology) are the factors that determine how much of the altitude correction acts along the arc of the measured lunar is relatively easy to explain and remember. It's easy to interpret these as the "percentage" of the altitude correction that gets applied to the lunar. It's even possible to estimate the values of A and B by eyeball when you're taking the sight and use those estimates as a check on your work. Checks for "reasonableness" of the numbers in lunars calculations are hard to come by and undoubtedly one of the reasons that some navigators have found them frustrating.

George also wrote:

"I don't know anything about Witchell's method"

I bet you do, but you don't recognize it by name. You've got an old copy of Norie, don't you? If not, hit that link on the Mystic Seaport library web site. It's "Method III" in Norie, if I remember correctly. It's also in Moore and Bowditch. Norie, unlike Bowditch, gives credit to the originators of the methods he includes in his text. Witchell's method is what I think of as the "cot-tan-cot" method because you need a series of log cotangent, log tangent, and then another log cotangent in the main part of the calculation. You can always spot it by that cot-tan-cot signature. These are simply old-fashioned logarithm-based steps to calculate the A and B factors (the "corner cosines"). I've been working through a logbook from 1809/10 where all the lunars are worked out in detail by Witchell's method, but it's interesting that he never applies the quadratic correction (the "Q" from my previous post). Maybe he never learned it... possibly some accounts of the method failed to include it... maybe the captain wasn't concerned by errors of a minute of arc or so in his cleared lunar.

And (of Thompson's method) George wrote:

"This method adds three correction terms to the observed distance to clear it, just as Frank's method does. But his method appears to differ more from Frank's than he implies. From my limited understanding (it's all rather obscure) it appears that the first two of Thomson's terms relate only to the corrections due to Moon parallax" etc.

It's just book-keeping. The differences among the various series methods usually amount to deciding how much of the altitude correction to "fold into" the calculation of the A and B factors. You remember Letcher's method? He does something similar. He drops the refraction calculation into the calculation of A and B. This is supposed to save the effort of looking up the actual altitude correction in the Nautical Almanac. But it's not that much effort, and it's something that every navigator knows how to do (so why not take advantage of that familiarity?). As Letcher notes, this has the disadvantage of excluding any possibility of temperature and pressure corrections and it also means that the parallax of the Sun and nearby planets cannot be included. Other 19th century methods, like Thompson's (if I understand it correctly), shifted around the refraction and parallax terms in similar ways.

And:

"But I suggest the relative convenience of that method, and its like, have now been eclipsed by the pocket calculator"

Sure. Very true. And calculators have been eclipsed by computers (and even cell phones that can run Java). And for that matter LUNARS have been eclipsed by chronometers for some fifteen decades now!

Anybody thinking about doing lunars has to ask a series of questions:

* Why am I considering doing lunars? For a practical backup in the unlikely event that all chronometers and timepieces become unreliable (but almanac data is still somehow available)? For amusement and challenge and a test of my skill? For historical "re-enactment" and education? For bragging rights?

* What level of accuracy do I hope to achieve? Accuracy comparable to some historical time period? The absolute maximum accuracy possible? Whatever is reasonable given the limitations of my sextant and my skill?

* How will I handle the computations? Using completely historical paper methods? If so, WHICH of the dozens of historical methods? Using the best modern paper method (very likely Bruce Stark's tables)? Using a calculator but doing some aspects of the calculation on paper to get a "feel" for the historical methods without the tedium? By computer with no paper work at all? Via the Internet?? The last two would apply to people who enjoy the sextant process but don't like the math. It should also apply to anyone who wants to get as much practice as possible doing actual sights.

* Will I teach lunars to other navigators? How many variations are there on the answers to these questions among my potential students??

For anyone messing around with lunars, the answers to the above questions will shift and change many times. Lunars are NOT practical sights. They're historical curiosities or a backup method that's almost certainly never going to be used. They're fun, challenging, but nowhere near as hard as their reputation suggests. Since they are not practical, there's no real need to seek a perfect method. Interestingly enough, star-star sights, which are very similar to lunars, have some continuing practical value since they can be used to assess arc errors. Like lunars, you can clear star-star sights either "by series" or "by triangle".

And just to repeat for anyone who made it to the bottom of this post:

Lunars are Easy! Now get out there and shoot some...

Frank E. Reed

[ ] Mystic, Connecticut

[X] Chicago, Illinois

"Clearly, Frank's proposed method expects the user to have some sort of calculator, with trig functions. In that case, I think further simplifications might be made. "

I'll start off by saying that that wasn't really the point of 'Easy Lunars'. There is only an insignificant difference in 'calculational efficiency' between the two ways of clearing a lunar and no difference in accuracy. The post I made was designed to make one specific type of calculation with deep historical roots "comprehensible", and I'm fairly confident that it did that.

George wrote:

"D = arcCos ((( cosd - sinm sins) cosM cosS / (cosm coss)) + (sinM sinS))

And that's it!"

Yep, the standard "triangle method". There are two general categories that cover most of the mathematical techniques for clearing lunars. Historically they were known as "rigorous" and "approximative" but these names are distinctly misleading. I prefer to call them "triangle solutions" and "series solutions". The series solutions are every bit as good as the triangle solutions within the normal limits of the lunars problem generally. They solve the same problem to the same level of accuracy. I do agree though, that if you aren't worried at all about history or pedagogy, then the triangle solution [which you posted] is the way to go. When I do this sort of calculation on a computer, I do the triangle solution unless the purpose is to demonstrate something about the nature of a series solution. But I think there is significant historical and pedagogic value in the series solutions.

And George wrote:

"I hear you say- 'Well, if it's that simple, why didn't they use that formula in the 18th/19th centuries?'. I will explain. Really, it's because they didn't have pocket calculators."

But they did use the triangle solutions AND the series solutions in the 18th/19th centuries. The real question is why did some navigators PREFER the series methods over the triangle methods?? Half of the answer to this question is, just as you say, they had no "pocket calculators". That is, computation was expensive and time-consuming and the triangle methods were more of a nuisance on this score. But there is another reason which still applies today: pedagogy. The linear terms of the series methods are very easy to interpret. The idea that "A" and "B" (in my terminology) are the factors that determine how much of the altitude correction acts along the arc of the measured lunar is relatively easy to explain and remember. It's easy to interpret these as the "percentage" of the altitude correction that gets applied to the lunar. It's even possible to estimate the values of A and B by eyeball when you're taking the sight and use those estimates as a check on your work. Checks for "reasonableness" of the numbers in lunars calculations are hard to come by and undoubtedly one of the reasons that some navigators have found them frustrating.

George also wrote:

"I don't know anything about Witchell's method"

I bet you do, but you don't recognize it by name. You've got an old copy of Norie, don't you? If not, hit that link on the Mystic Seaport library web site. It's "Method III" in Norie, if I remember correctly. It's also in Moore and Bowditch. Norie, unlike Bowditch, gives credit to the originators of the methods he includes in his text. Witchell's method is what I think of as the "cot-tan-cot" method because you need a series of log cotangent, log tangent, and then another log cotangent in the main part of the calculation. You can always spot it by that cot-tan-cot signature. These are simply old-fashioned logarithm-based steps to calculate the A and B factors (the "corner cosines"). I've been working through a logbook from 1809/10 where all the lunars are worked out in detail by Witchell's method, but it's interesting that he never applies the quadratic correction (the "Q" from my previous post). Maybe he never learned it... possibly some accounts of the method failed to include it... maybe the captain wasn't concerned by errors of a minute of arc or so in his cleared lunar.

And (of Thompson's method) George wrote:

"This method adds three correction terms to the observed distance to clear it, just as Frank's method does. But his method appears to differ more from Frank's than he implies. From my limited understanding (it's all rather obscure) it appears that the first two of Thomson's terms relate only to the corrections due to Moon parallax" etc.

It's just book-keeping. The differences among the various series methods usually amount to deciding how much of the altitude correction to "fold into" the calculation of the A and B factors. You remember Letcher's method? He does something similar. He drops the refraction calculation into the calculation of A and B. This is supposed to save the effort of looking up the actual altitude correction in the Nautical Almanac. But it's not that much effort, and it's something that every navigator knows how to do (so why not take advantage of that familiarity?). As Letcher notes, this has the disadvantage of excluding any possibility of temperature and pressure corrections and it also means that the parallax of the Sun and nearby planets cannot be included. Other 19th century methods, like Thompson's (if I understand it correctly), shifted around the refraction and parallax terms in similar ways.

And:

"But I suggest the relative convenience of that method, and its like, have now been eclipsed by the pocket calculator"

Sure. Very true. And calculators have been eclipsed by computers (and even cell phones that can run Java). And for that matter LUNARS have been eclipsed by chronometers for some fifteen decades now!

Anybody thinking about doing lunars has to ask a series of questions:

* Why am I considering doing lunars? For a practical backup in the unlikely event that all chronometers and timepieces become unreliable (but almanac data is still somehow available)? For amusement and challenge and a test of my skill? For historical "re-enactment" and education? For bragging rights?

* What level of accuracy do I hope to achieve? Accuracy comparable to some historical time period? The absolute maximum accuracy possible? Whatever is reasonable given the limitations of my sextant and my skill?

* How will I handle the computations? Using completely historical paper methods? If so, WHICH of the dozens of historical methods? Using the best modern paper method (very likely Bruce Stark's tables)? Using a calculator but doing some aspects of the calculation on paper to get a "feel" for the historical methods without the tedium? By computer with no paper work at all? Via the Internet?? The last two would apply to people who enjoy the sextant process but don't like the math. It should also apply to anyone who wants to get as much practice as possible doing actual sights.

* Will I teach lunars to other navigators? How many variations are there on the answers to these questions among my potential students??

For anyone messing around with lunars, the answers to the above questions will shift and change many times. Lunars are NOT practical sights. They're historical curiosities or a backup method that's almost certainly never going to be used. They're fun, challenging, but nowhere near as hard as their reputation suggests. Since they are not practical, there's no real need to seek a perfect method. Interestingly enough, star-star sights, which are very similar to lunars, have some continuing practical value since they can be used to assess arc errors. Like lunars, you can clear star-star sights either "by series" or "by triangle".

And just to repeat for anyone who made it to the bottom of this post:

Lunars are Easy! Now get out there and shoot some...

Frank E. Reed

[ ] Mystic, Connecticut

[X] Chicago, Illinois