# NavList:

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

**Lunars: series vs. triangle methods**

**From:**Frank Reed CT

**Date:**2004 Sep 26, 20:53 EDT

Fred H wrote:

"The approximate lunar methods, of which Frank Reed's is one, were

developed for ease of calculation using tables."

That is only partially true. Like so many on this list and in the history of lunars, you have been severely misled by the pedantic, archaic terminology "rigorous" and "approximate" to describe methods of clearing lunars. The so-called "approximate" methods are simply a series expansion of the standard triangle solution, mistakenly called rigorous. Let me give you an analogy. Suppose I tell you that the definition of the function sin(x) is the ratio of the opposite side of a right triangle to its hypotenuse. Suppose a mathematician then says that the definition of sin(x) is

sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ...

Which is right? Is the infinite series a mere "approximate" method while the geometric definition is rigorous? In the late 18th century, mathematicians did not yet understand the exact equivalence of these approaches. They are the SAME THING. Likewise, the triangle solutions to the problem of clearing a lunar and the series solutions are the SAME THING. And BOTH methods lead only to approximate solutions of the lunars clearing problem since they do not include all possible effects which can distort a measured lunar distance. It is simply not the case that series solutions are worse than solutions by direct triangle.

In the case of the series solutions, there is an additional approximation that can in some cases cause confusion. Just as the expansion sin(x) = x is valid to some degree of accuracy over some range of the variable x, while sin(x) = x - x^3/3! is valid over a much wider range of the variable x, so the series solutions of the lunars problems have limits based on the degree of extension of the series.

And:

"Using a calculator rather than tables, there is a sharp diminution in the difference in

difficulty between the approximate and exact methods."

That's true as far as keypresses are concerned. So why use a series method at all? Two reasons:

1) They were used historically. It gets you much closer to the methods recommended by Bowditch and others and actually used at sea. Among the historical methods, those of Maskelyne, Lyons, Witchell, Mendoza Rios, Bowditch, and Thomson were all minor variants on the standard series solution. It is true that the French, for example, prefered direct triangle solutions, but they used lunars much less actively. There is nothing comparable to the 19th century "Bowditch" for the French because they didn't bother with lunars much.

2) There are several pedagogic issues that are greatly clarified by understanding lunars from a series expansion point of view. I'll add more on this in the next paragraph.

And Fred added:

"An advantage of the exact methods is that one _might_ become more aware of the

underlying math and geometry, which are obscured in the series expansions underlying the approximate methods. "

That is completely untrue, Fred. There are numerous examples in applied mathematics where a series expansion provides far GREATER physical insight than a (so-called) rigorous solution. Sticking to the immediate subject, consider the series expansion for lunars. The corrected distance, d', is related to the observed distance, d, by a simple equation

d' = d + dh1*A + dh2*B + Q +...

where dh1 is the Moon's altitude correction, dh2 is the Sun or star's altitude correction, the quantities A and B are the "corner cosines" that tell us what fraction of the altitude corrections act along the measured lunar distance arc, and Q is the small quadratic correction. I've included the "..." (ellipsis) to remind you that this is technically an infinite series though in any actual practice we never need to go beyond quadratic order. This series expansion can be derived very easily using straight-forward calculus from the standard triangle solution (naturally it can --they are the same thing fundamentally). If you want to try it yourself, it is a two-dimensional Taylor expansion in the variables h1 and h2 (the altitudes of the Moon and Sun). The quantities A and B have a very simple mathematical interpretation. I call them the "corner cosines". They are the cosines of the angles in the spherical triangle Sun-Moon-Zenith at each of the bodies. You never need to evaluate these angles directly; only their "cosines" so we can just call those numbers A and B.

So what can we learn from this series approach? Well, for starters, we can make good estimates of the corrected distance without any calculation. This is a BIG plus for error detection. For example, a few days ago you shot some lunars of the Moon. When you were looking at it, the Moon's horns were nearly vertical, right? Now you know that the line perpendicular to the horns points directly to the Sun. So if the line through the horns is pointing straight up (towards the zenith) and the line to the Sun is perpendicular to that, then the corner angle is just about 90 degrees. Follow that? If not, I'll put together some diagrams and make a little web page of it. Assuming you're following along... If that corner angle is 90 degrees, then A is going to be nearly zero. That means that the Moon's altitude correction acts perpendicular to the arc of the lunar distance and will add next to nothing to the corrected distance. Another way to think about this: the plane of your sextant would have been almost exactly perpendicular to a vertical line through the Moon. There is more... But if you dismiss series solutions as merely "approximate" while triangle solutions are that Holy Grail of "rigorous" math, you'll never catch on.

There are insights to be gained into the history as well as the meaning of lunars by looking at series solutions. Oh, and by the way, in no way do I claim that my "Easy Lunars" article was about MY method of clearing lunars. It's not "Frank Reed's method" of clearing lunars. It's the "standard series expansion for clearing lunars". The explanatory text and description are my words, and naturally I expect that authorship to be respected, but the mathematical concepts there --the expressions for the series expansion-- have been common knowledge (as common as anything regarding lunars can get!) for well over 200 years. If you want to see it all in historical context, read the 80-page article by Mendoza Rios from 1796 that I posted recently on my web site:

http://www.HistoricalAtlas.com/lunars/myr/

It is unfortunate that so many of the people interested in lunars on this list have turned to Cotter's "History of Nautical Astronomy" for their information. While much of the rest of the book is good, albeit not great, the chapter on lunars is really very poor. Cotter mentioned the article by Jose de Mendoza y Rios which I've refered to above, erroneously calling him "Mendoza del Rios" but it is very clear from the comments and the rest of the chapter that he never read the article. As far as I can tell, he read only a small sampling of the literature on lunars, and mostly that written in the late 19th century when lunars had become the domain of antiquarians and pontificating land-based mathematicians. Cotter's analysis of "Merrifield's method", a very late re-hash of a series method, is a muddled mess that will teach you next to nothing about the concept underlying series methods for lunars. The whole chapter is un-historical junk. It is not a history of lunars; it is a history of Cotter's very limited familiarity with lunars.

Frank R

[ ] Mystic, Connecticut

[X] Chicago, Illinois

"The approximate lunar methods, of which Frank Reed's is one, were

developed for ease of calculation using tables."

That is only partially true. Like so many on this list and in the history of lunars, you have been severely misled by the pedantic, archaic terminology "rigorous" and "approximate" to describe methods of clearing lunars. The so-called "approximate" methods are simply a series expansion of the standard triangle solution, mistakenly called rigorous. Let me give you an analogy. Suppose I tell you that the definition of the function sin(x) is the ratio of the opposite side of a right triangle to its hypotenuse. Suppose a mathematician then says that the definition of sin(x) is

sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ...

Which is right? Is the infinite series a mere "approximate" method while the geometric definition is rigorous? In the late 18th century, mathematicians did not yet understand the exact equivalence of these approaches. They are the SAME THING. Likewise, the triangle solutions to the problem of clearing a lunar and the series solutions are the SAME THING. And BOTH methods lead only to approximate solutions of the lunars clearing problem since they do not include all possible effects which can distort a measured lunar distance. It is simply not the case that series solutions are worse than solutions by direct triangle.

In the case of the series solutions, there is an additional approximation that can in some cases cause confusion. Just as the expansion sin(x) = x is valid to some degree of accuracy over some range of the variable x, while sin(x) = x - x^3/3! is valid over a much wider range of the variable x, so the series solutions of the lunars problems have limits based on the degree of extension of the series.

And:

"Using a calculator rather than tables, there is a sharp diminution in the difference in

difficulty between the approximate and exact methods."

That's true as far as keypresses are concerned. So why use a series method at all? Two reasons:

1) They were used historically. It gets you much closer to the methods recommended by Bowditch and others and actually used at sea. Among the historical methods, those of Maskelyne, Lyons, Witchell, Mendoza Rios, Bowditch, and Thomson were all minor variants on the standard series solution. It is true that the French, for example, prefered direct triangle solutions, but they used lunars much less actively. There is nothing comparable to the 19th century "Bowditch" for the French because they didn't bother with lunars much.

2) There are several pedagogic issues that are greatly clarified by understanding lunars from a series expansion point of view. I'll add more on this in the next paragraph.

And Fred added:

"An advantage of the exact methods is that one _might_ become more aware of the

underlying math and geometry, which are obscured in the series expansions underlying the approximate methods. "

That is completely untrue, Fred. There are numerous examples in applied mathematics where a series expansion provides far GREATER physical insight than a (so-called) rigorous solution. Sticking to the immediate subject, consider the series expansion for lunars. The corrected distance, d', is related to the observed distance, d, by a simple equation

d' = d + dh1*A + dh2*B + Q +...

where dh1 is the Moon's altitude correction, dh2 is the Sun or star's altitude correction, the quantities A and B are the "corner cosines" that tell us what fraction of the altitude corrections act along the measured lunar distance arc, and Q is the small quadratic correction. I've included the "..." (ellipsis) to remind you that this is technically an infinite series though in any actual practice we never need to go beyond quadratic order. This series expansion can be derived very easily using straight-forward calculus from the standard triangle solution (naturally it can --they are the same thing fundamentally). If you want to try it yourself, it is a two-dimensional Taylor expansion in the variables h1 and h2 (the altitudes of the Moon and Sun). The quantities A and B have a very simple mathematical interpretation. I call them the "corner cosines". They are the cosines of the angles in the spherical triangle Sun-Moon-Zenith at each of the bodies. You never need to evaluate these angles directly; only their "cosines" so we can just call those numbers A and B.

So what can we learn from this series approach? Well, for starters, we can make good estimates of the corrected distance without any calculation. This is a BIG plus for error detection. For example, a few days ago you shot some lunars of the Moon. When you were looking at it, the Moon's horns were nearly vertical, right? Now you know that the line perpendicular to the horns points directly to the Sun. So if the line through the horns is pointing straight up (towards the zenith) and the line to the Sun is perpendicular to that, then the corner angle is just about 90 degrees. Follow that? If not, I'll put together some diagrams and make a little web page of it. Assuming you're following along... If that corner angle is 90 degrees, then A is going to be nearly zero. That means that the Moon's altitude correction acts perpendicular to the arc of the lunar distance and will add next to nothing to the corrected distance. Another way to think about this: the plane of your sextant would have been almost exactly perpendicular to a vertical line through the Moon. There is more... But if you dismiss series solutions as merely "approximate" while triangle solutions are that Holy Grail of "rigorous" math, you'll never catch on.

There are insights to be gained into the history as well as the meaning of lunars by looking at series solutions. Oh, and by the way, in no way do I claim that my "Easy Lunars" article was about MY method of clearing lunars. It's not "Frank Reed's method" of clearing lunars. It's the "standard series expansion for clearing lunars". The explanatory text and description are my words, and naturally I expect that authorship to be respected, but the mathematical concepts there --the expressions for the series expansion-- have been common knowledge (as common as anything regarding lunars can get!) for well over 200 years. If you want to see it all in historical context, read the 80-page article by Mendoza Rios from 1796 that I posted recently on my web site:

http://www.HistoricalAtlas.com/lunars/myr/

It is unfortunate that so many of the people interested in lunars on this list have turned to Cotter's "History of Nautical Astronomy" for their information. While much of the rest of the book is good, albeit not great, the chapter on lunars is really very poor. Cotter mentioned the article by Jose de Mendoza y Rios which I've refered to above, erroneously calling him "Mendoza del Rios" but it is very clear from the comments and the rest of the chapter that he never read the article. As far as I can tell, he read only a small sampling of the literature on lunars, and mostly that written in the late 19th century when lunars had become the domain of antiquarians and pontificating land-based mathematicians. Cotter's analysis of "Merrifield's method", a very late re-hash of a series method, is a muddled mess that will teach you next to nothing about the concept underlying series methods for lunars. The whole chapter is un-historical junk. It is not a history of lunars; it is a history of Cotter's very limited familiarity with lunars.

Frank R

[ ] Mystic, Connecticut

[X] Chicago, Illinois